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RAILROAD VEHICLE
DYNAMICS
A Computational Approach
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RAILROAD VEHICLE
DYNAMICS
A Computational Approach
Ahmed A. Shabana
Khaled E. Zaazaa
Hiroyuki Sugiyama
Boca Raton London New York
CRC Press is an imprint of the
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Library of Congress Cataloging-in-Publication Data
Shabana, Ahmed A., 1951Railroad vehicle dynamics : a computational approach / Ahmed A. Shabana,
Khaled E. Zaazaa, Hiroyuki Sugiyama.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4200-4581-9 (alk. paper)
1. Railroad cars--Dynamics--Mathematics. 2. Railroad cars--Mathematical
models. I. Zaazaa, Khaled E. II. Sugiyama, Hiroyuki, 1974- III. Title.
TF550.S48 2007
625.201’51--dc22
Visit the Taylor & Francis Web site at
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2007060396
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Table of Contents
Preface.......................................................................................................................xi
Acknowledgments....................................................................................................xv
Chapter 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Railroad Vehicles and Multibody System Dynamics......................................2
1.1.1 Generality .............................................................................................2
1.1.2 Nonlinearity..........................................................................................4
1.1.3 Implementation of Railroad Vehicle Elements....................................6
Constrained Dynamics .....................................................................................9
Geometry Problem .........................................................................................11
1.3.1 Differential Geometry ........................................................................12
1.3.2 Rail and Wheel Geometry .................................................................14
Contact Theories ............................................................................................17
1.4.1 Creep Forces ......................................................................................17
1.4.2 Wheel/Rail Creep Theories................................................................18
General Multibody Railroad Vehicle Formulations....................................... 18
1.5.1 Constraint Contact Formulation.........................................................19
1.5.2 Elastic Contact Formulation ..............................................................20
Specialized Railroad Vehicle Formulations...................................................20
Linearized Railroad Vehicle Models .............................................................23
Motion Stability .............................................................................................24
Motion Scenarios ...........................................................................................27
1.9.1 Hunting...............................................................................................28
1.9.2 Steady Curving...................................................................................28
1.9.3 Spiral Negotiation ..............................................................................30
1.9.4 Twist and Roll ....................................................................................30
1.9.5 Pitch and Bounce ...............................................................................31
1.9.6 Yaw and Sway....................................................................................31
1.9.7 Dynamic Curving...............................................................................31
1.9.8 Response to Discontinuities...............................................................32
Chapter 2
2.1
2.2
Introduction ........................................................................................1
Dynamic Formulations ....................................................................35
General Displacement ....................................................................................36
Rotation Matrix ..............................................................................................37
2.2.1 Direction Cosines...............................................................................38
2.2.2 Simple Rotations ................................................................................41
2.2.3 Euler Angles.......................................................................................41
2.2.4 Euler Parameters ................................................................................45
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2.3
Velocities and Accelerations ..........................................................................49
2.3.1 Velocity Vector ...................................................................................49
2.3.2 Acceleration Vector ............................................................................50
2.3.3 Generalized Orientation Coordinates.................................................51
2.3.4 Singular Configuration.......................................................................53
2.4 Newton-Euler Equations ................................................................................58
2.5 Joint Constraints.............................................................................................62
2.5.1 Spherical Joint....................................................................................62
2.5.2 Revolute Joint.....................................................................................63
2.5.3 Cylindrical Joint.................................................................................64
2.5.4 Prismatic Joint....................................................................................65
2.6 Augmented Formulation ................................................................................66
2.7 Trajectory Coordinates...................................................................................70
2.7.1 Velocity and Acceleration ..................................................................72
2.7.2 Equations of Motion ..........................................................................74
2.8 Embedding Technique....................................................................................76
2.8.1 Coordinate Partitioning and Velocity Transformation.......................77
2.8.2 Elimination of the Constraint Forces.................................................78
2.8.3 Reduced-Order Model........................................................................78
2.9 Interpretation of the Methods ........................................................................80
2.9.1 Kinematic and Dynamic Equations ...................................................80
2.9.2 Augmented Formulation ....................................................................83
2.9.3 Embedding Technique........................................................................84
2.9.4 D’Alembert’s Principle ......................................................................85
2.10 Virtual Work...................................................................................................86
Chapter 3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Rail and Wheel Geometry...............................................................89
Theory of Curves ...........................................................................................90
3.1.1 Arc Length and Tangent Line............................................................90
3.1.2 Curvature and Torsion........................................................................91
Geometry of Surfaces ....................................................................................92
3.2.1 Tangent Plane and Normal Vector .....................................................94
3.2.2 First Fundamental Form.....................................................................95
3.2.3 Second Fundamental Form ................................................................96
3.2.4 Normal Curvature...............................................................................99
3.2.5 Principal Curvatures and Principal Directions ................................100
Rail Geometry ..............................................................................................103
Definitions and Terminology .......................................................................106
Geometric Description of the Track ............................................................108
Computer Implementation ...........................................................................111
3.6.1 Track Segment Types.......................................................................112
3.6.2 Linear Representation of the Segments...........................................112
3.6.3 Derivatives of the Angles.................................................................114
Track Preprocessor.......................................................................................116
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3.8
3.7.1
3.7.2
3.7.3
3.7.4
Wheel
Chapter 4
4.1
4.2
4.3
4.4
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Contact and Creep-Force Models ................................................127
Hertz Theory ................................................................................................128
4.1.1 Geometry and Kinematics ...............................................................128
4.1.2 Contact Pressure...............................................................................133
4.1.3 Computer Implementation ...............................................................138
Creep Phenomenon ......................................................................................140
Wheel/Rail Contact Approaches..................................................................145
4.3.1 Exact Theory of Rolling Contact.....................................................146
4.3.2 Simplified Theory of Rolling Contact .............................................147
4.3.3 Dynamic and Quasi-Static Theory ..................................................147
4.3.4 Three- and Two-Dimensional Theory..............................................147
Creep-Force Theories...................................................................................147
4.4.1 Carter’s Theory.................................................................................147
4.4.2 Johnson and Vermeulen’s Theory ....................................................149
4.4.3 Kalker’s Linear Theory ....................................................................150
4.4.4 Heuristic Nonlinear Creep-Force Model .........................................153
4.4.5 Polach Nonlinear Creep-Force Model .............................................154
4.4.6 Simplified Theory.............................................................................156
4.4.7 Kalker’s USETAB ............................................................................159
Chapter 5
5.1
Track Preprocessor Input .................................................................117
Numerical Integration ......................................................................118
Track Preprocessor Output ..............................................................120
Use of the Preprocessor Output during Dynamic Simulation ........121
Geometry ..........................................................................................123
Multibody Contact Formulations.................................................161
Parameterization of Wheel and Rail Surfaces.............................................162
5.1.1 Track Geometry ...............................................................................163
5.1.2 Wheel Geometry ..............................................................................165
Constraint Contact Formulations .................................................................165
5.2.1 Contact Constraints ..........................................................................166
5.2.2 Constrained Dynamic Equations .....................................................167
Augmented Constraint Contact Formulation (ACCF) ................................168
Embedded Constraint Contact Formulation (ECCF) ..................................171
5.4.1 Position Analysis..............................................................................172
5.4.2 Equations of Motion ........................................................................173
Elastic Contact Formulation-Algebraic Equations (ECF-A).......................174
Elastic Contact Formulation-Nodal Search (ECF-N)..................................177
Comparison of Different Contact Formulations..........................................178
Planar Contact ..............................................................................................179
5.8.1 Intermediate Wheel Coordinate System ..........................................181
5.8.2 Distance Traveled.............................................................................182
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5.8.3
5.8.4
Profile Parameters ............................................................................184
Coupling between the Surface Parameters ......................................185
Chapter 6
Implementation and Special Elements ........................................187
6.1
General Multibody System Algorithms.......................................................188
6.1.1 Constrained Dynamics .....................................................................188
6.1.2 Penalty and Constraint Stabilization Methods ................................189
6.1.3 Generalized Coordinates Partitioning ..............................................191
6.1.4 Identification of the Independent Coordinates ................................194
6.2 Numerical Algorithms — Constraint Formulations ....................................194
6.2.1 Augmented Constraint Contact Formulation (ACCF) ....................195
6.2.2 Embedded Constraint Contact Formulation (ECCF) ......................201
6.3 Numerical Algorithms — Elastic Formulations..........................................205
6.3.1 Elastic Contact Formulation Using Algebraic Equations (ECF-A)....206
6.3.2 Elastic Contact Formulation Using Nodal Search (ECF-N) ..............208
6.4 Calculation of the Creep Forces ..................................................................210
6.5 Higher Derivatives and Smoothness Technique ..........................................211
6.6 Track Preprocessor.......................................................................................214
6.6.1 Change in the Length Due to Curvature .........................................216
6.6.2 Use of the Preprocessor Output during Dynamic Simulation ........218
6.7 Deviations and Measured Data....................................................................219
6.7.1 Track Deviations ..............................................................................220
6.7.2 Measured Track Data .......................................................................222
6.7.3 Track Quality and Classes ...............................................................223
6.8 Special Elements ..........................................................................................225
6.8.1 Translational Spring-Damper-Actuator Element .............................227
6.8.2 Rotational Spring-Damper-Actuator Element .................................230
6.8.3 Series Spring-Damper Element .......................................................231
6.8.4 Bushing Element ..............................................................................232
6.9 Maglev Forces ..............................................................................................236
6.9.1 Electrodynamic Suspension (EDS)..................................................236
6.9.2 Electromagnetic Suspension (EMS) ................................................237
6.9.3 Modeling of Electromagnetic Suspensions .....................................237
6.9.4 Multibody System Electromechanical Equations............................240
6.10 Static Analysis..............................................................................................242
6.10.1 Augmented Constraint Contact Formulation...................................242
6.10.2 Embedded Constraint Contact Formulation ....................................244
6.10.3 Line Search Method.........................................................................245
6.10.4 Continuation Method .......................................................................246
6.11 Numerical Comparative Study.....................................................................247
6.11.1 Simple Suspended Wheelset ............................................................247
6.11.2 Complete Vehicle Model..................................................................248
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Chapter 7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
General Displacement ..................................................................................256
7.1.1 Trajectory Coordinate System .........................................................256
7.1.2 Body Coordinate System .................................................................258
7.1.3 Generalized Trajectory Coordinates ................................................259
Velocity and Acceleration ............................................................................260
7.2.1 Velocity of the Center of Mass........................................................260
7.2.2 Acceleration of the Center of Mass.................................................261
7.2.3 Angular Velocity and Acceleration ..................................................262
Equations of Motion ....................................................................................264
Trajectory Coordinate Constraints...............................................................265
7.4.1 Numerical Example..........................................................................266
7.4.2 Use of the Cartesian Coordinates ....................................................269
Single-Degree-of-Freedom Model ...............................................................272
Two-Degree-of-Freedom Model ..................................................................277
Linear Hunting Stability Analysis ...............................................................280
7.7.1 Model 1 ............................................................................................287
7.7.2 Model 2 ............................................................................................288
Chapter 8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Specialized Railroad Vehicle Formulations .................................255
Creepage Linearization .................................................................291
Background ..................................................................................................291
Transformation and Angular Velocity..........................................................295
8.2.1 Matrix Identities...............................................................................295
8.2.2 Definition of the Angular Velocity ..................................................296
Euler Angles.................................................................................................298
Linearization Assumptions...........................................................................300
Longitudinal and Lateral Creepages............................................................301
Spin Creepage ..............................................................................................305
Newton-Euler Equations ..............................................................................306
Concluding Remarks....................................................................................309
Appendix A
Contact Equations.......................................................................313
Appendix B
Elliptical Integrals.......................................................................319
References .............................................................................................................321
Index......................................................................................................................333
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Preface
The methods of computational mechanics have been used extensively in modeling
many physical systems, including machines, vehicles, mechanisms and robotics,
space structures, and biomechanical and biological systems, among many others.
Multibody system techniques, in particular, have been used successfully in the study
of various and fundamentally different applications. This success can be attributed
to the generality and flexibility of these techniques, which facilitate tailoring the
general formulations to a specific application. The aim of this book is to present a
computational multibody system approach that can be used to develop complex
models of railroad vehicle systems that include significant details. One of the important features that distinguish railroad vehicle systems from other multibody system
applications is the vehicle/rail (guide) interaction. Special force or kinematic constraint elements must be included in the multibody system algorithm if the vehicle/rail interaction is to be accurately modeled. To accomplish this goal, interesting
geometric problems that are particular to railroad vehicle systems must be addressed
and solved. By considering these additional geometric variables, which are required
to describe the vehicle/rail interaction, multibody system formulations can be modified and improved for use in the analysis of detailed railroad vehicle models.
This book presents several computational multibody system formulations and
discusses their computer implementation. The computational algorithms based on
these general formulations can be used to develop general- and special-purpose
railroad vehicle computer programs for use in the analysis of railroad vehicle systems, including the study of derailment and accident scenarios, design issues, and
performance evaluation. The book focuses on the development of fully nonlinear
formulations, supported by an explanation of the limitations of the linearized formulations that are frequently used in the analysis of railroad vehicle systems.
This book is designed for an introductory course on railroad vehicle dynamics
that is suitable for senior undergraduate and first-year graduate students. The students
are expected to have knowledge of dynamics at the intermediate level and also have
knowledge of basic vector and matrix algebra. This book can also be used as a
reference by researchers and practicing engineers, who commonly use generalpurpose multibody system computer programs in the analysis, design, and performance evaluation of railroad vehicle systems. In this book, it is assumed that the
components of the railroad vehicle system that have distributed inertia are rigid
bodies; the generalization of the presented formulations to the case of flexible body
dynamics is straightforward, as described in previous publications by the authors of
this book.
The chapters of the book are organized to guide the reader from basic concepts
and definitions through a final understanding of the utility of fully nonlinear multibody system formulations in the analysis of railroad vehicle systems.
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Chapter 1 provides a brief introduction of basic concepts and definitions. The
motivation for using multibody system approaches in the analysis of railroad vehicles
is first discussed. The reader is then introduced to the subject of constrained dynamics, the geometry problem, contact theory, and general and specialized multibody
railroad vehicle formulations. Linearized vehicle models and motion stability and
scenarios are among the topics discussed in this chapter.
Chapter 2 reviews the analytical methods used to develop general multibody
system formulations. The generalized coordinates are defined, and methods for
describing the general displacement of a rigid body in space are presented. The
motion description is expressed in terms of a rotation matrix that defines the orientation of the rigid body in the global coordinate system. Newton-Euler equations
that can be used to obtain the equations of motion of the three-dimensional rigid
bodies are presented, and the reader learns how to use these equations to develop
general formulations that can be used to solve for the body accelerations. In particular, the augmented Lagrangian formulation and the embedding techniques are
discussed. The augmented Lagrangian formulation leads to a large system of equations that has a sparse matrix structure, while the embedding technique leads to a
minimum set of strongly coupled equations.
Chapter 3 discusses the geometry problem that is fundamental in railroad
vehicle dynamics. The theory of curves and surfaces and the local invariant geometric
properties are defined. Important definitions related to the rail and track geometry
are introduced, and the reader learns how to describe and construct a track with an
arbitrary shape using three input parameters that are used by the railroad industry.
The wheel geometry is also introduced in this chapter, and the equations used to
describe the wheel surface in terms of two geometric parameters are presented.
Contact mechanics is fundamental in the analysis of railroad vehicle systems,
and it is necessary to describe the vehicle/rail interaction. Chapter 4 reviews the
contact theories. In particular, Hertz contact theory is discussed in detail, since it is
widely used in the analysis of the wheel/rail contact. Creep forces are developed as
the result of the wheel/rail contact, and several creep-force theories are discussed.
Some of these theories are linear, while the others are nonlinear theories.
Chapter 5 shows how to model the wheel/rail contact in multibody system
formulations. Several formulations are presented. In some of these formulations, the
wheel and rail surfaces are assumed to remain rigid. The contact between the wheel
and the rail is described using kinematic algebraic constraint equations that do not
allow for wheel/rail separation or penetration. The methods that employ constraint
equations to describe the contact between the wheel and the rail are called constraint
contact formulations. Other multibody formulations presented in this chapter assume
that the wheel and rail surfaces can experience small local deformation in the contact
region. The normal force of the wheel/rail interaction is obtained by using a compliant force model. In this case, wheel/rail separation and penetration are allowed.
Methods that do not impose kinematic constraints to describe the contact are called
elastic contact formulations. Both the constraint and elastic contact formulations are
discussed in detail and compared in Chapter 5.
Chapter 6 discusses the computer implementation of the formulations presented
in this book as well as the formulation of special elements that are particular to
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railroad vehicle systems. The reader learns how to solve numerically the resulting
system of differential and algebraic equations that govern the motion of the multibody vehicle systems. Special elements, such as the magnetic levitation forces used
in magnetically levitated trains (Maglev), are also discussed. Numerical examples
are presented at the end of this chapter to compare the results obtained using the
different nonlinear formulations presented in Chapter 5.
General and specialized formulations are often used in the analysis of railroad
vehicle systems. General-purpose computer codes provide the flexibility and generality for building very detailed models and for exploiting advanced flexible body
capabilities in a straightforward manner. Special-purpose computer codes, on the
other hand, exploit the special features and characteristics of the railroad vehicle
systems. Chapter 7 introduces the trajectory coordinates, which can be used to
develop specialized formulations for railroad vehicle systems. The dynamic equations of motion are developed in terms of these trajectory coordinates. The resulting
equations are used to obtain reduced-order models that provide insight into the
stability and dynamics of railroad vehicles.
Many of the existing formulations used in railroad vehicle dynamics employ
linearization of the kinematic and dynamic equations. Chapter 8 examines the effect
of this linearization. Using results from the literature, it is shown that linearization
of the creep velocities can lead to significant errors in the prediction of the longitudinal and lateral forces. Such forces are important because they are factors in the
calculation of some of the derailment criteria used in the railroad industry. To help
in understanding the limits of linearization, the fully nonlinear equations are first
developed and used to obtain the linearized kinematic equations. The effect of this
linearization on the form of the Newton-Euler equations is also examined.
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Acknowledgments
The authors would like to acknowledge the contributions of many colleagues and
students to the development of this book. The materials presented in this book
summarize research that has been sponsored for several years by the Federal Railroad
Administration (FRA). The authors would also like to thank Dr. Magdy El-Sibaie
and Mr. Ali Tajaddini of FRA for their support and encouragement during this
project. Several researchers and engineers from Volpe National Transportation Systems Center, ENSCO Inc., Center for Automated Mechanics, National Transportation
Safety Board, and Northern Illinois University have been involved in this project.
The authors gratefully acknowledge the contributions of Professor Behrooz Fallahi,
Mr. Erik Curtis, Dr. Alan Kushner, Mr. Brian Marquis, Dr. Kevin Renze, Dr. Jalil
R. Sany, Mr. Amit Singh, and Dr. Brian Whitten. The chapters of this book were
reviewed by our Ph.D. students Cheta Rathod, Graham Sanborn, and Tariq Sinokrot,
whose efforts in proofreading the manuscript are very much appreciated. Thanks to
Mr. Chris Guenzler and Mr. Jason Heineman for providing some of the figures for
this book. Thanks also to our families for their patience and understanding during
the time of preparation of this book.
Ahmed A. Shabana
Chicago, Illinois
Khaled E. Zaazaa
Springfield, Virginia
Hiroyuki Sugiyama
Osaka, Japan
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1
Introduction
Railroad vehicles are among the most widely used methods of transporting passengers and goods. Trains have been used in commerce for more than a century, but
the last three decades, in particular, have seen significant progress in rail transportation technology. Some modern trains operate at high speeds to minimize cost and
transportation time. As train operating speeds increase, safety and comfort remain
paramount concerns. High-speed trains are designed to ensure safe operations by
attempting to identify and eliminate the causes of derailment. Comfort, on the other
hand, can be achieved by controlling undesirable vibration and noise sources. Modern trains are complex mechanical systems that can be better analyzed and designed
using modern computational mechanics techniques. Developing detailed computer
models of high-speed railroad vehicles is necessary to study vehicle performance,
to improve existing designs and develop new ones, and to develop safety guidelines
for different operating and loading conditions. Computer dynamic simulations have,
in fact, come to play an integral role in railroad vehicle performance, in safety and
accident evaluation, and in the design of new vehicles. Recent advances in the fields
of computational mechanics and numerical methods are directly applicable in the
nonlinear dynamic analysis of railroad vehicle systems.
The aim of this book is to develop computational methods for the dynamic
modeling of railroad vehicle systems as a tool for analyzing vibration and stability.
The emphasis will be on developing fully nonlinear formulations and computational
algorithms. Linearization techniques, which are often employed in developing algorithms for use in studying railroad vehicle system dynamics, are not adopted in this
book, although Chapter 8 does examine the effects of linearizing the kinematic and
dynamic equations of railroad vehicle systems.
This chapter introduces some of the topics, concepts, and definitions that are
discussed in the subsequent chapters of this book. First, the advantages of developing
and using general multibody system algorithms in the analysis of railroad vehicles are
summarized. In the second section of this chapter, the analytical methods that will
be used in developing the dynamic equations of motion for railroad vehicles are
described. The computational methods of constrained dynamics are highly recommended for use in developing detailed and accurate models for railroad vehicle
systems. Two problems distinguish railroad vehicle systems from many other multibody system applications: the geometry and contact problems. The surface geometry
of the wheel and rail, as well as the track geometric shape, enter into the formulation
of the equations of motion and in the formulation of the wheel/rail contact. These
two important geometry and contact problems are discussed in Sections 1.3 and 1.4.
In Section 1.5, the implementation of the wheel/rail contact element in general
multibody system algorithms is discussed. Several sets of coordinates can be used
to develop the equations of motion of railroad vehicle systems. Some of these
1
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2
Railroad Vehicle Dynamics: A Computational Approach
coordinates can be used to develop general-purpose computer formulations and codes,
while others can be used to develop specialized and less general formulations and
codes. This important topic of the coordinate selection is introduced in Section 1.6.
It is common in the literature to employ linearization techniques in developing the
dynamic equations of motion. However, as train operating speeds increase, one needs
to employ fully nonlinear formulations to accurately predict the vehicle dynamic
behavior. The issue of the linearization of the kinematic and dynamic equations is
introduced in Section 1.7. Sections 1.8 and 1.9 discuss railroad vehicle stability and
possible motion scenarios.
1.1 RAILROAD VEHICLES AND
MULTIBODY SYSTEM DYNAMICS
Multibody system dynamics is a branch of the general field of computational
mechanics that is concerned with developing and solving the nonlinear equations
that govern the motion of complex physical systems. The components of a multibody
system can experience large rotations and displacements. The motion of these components, however, is subjected to kinematic constraints that are the result of mechanical joints and specified motion trajectories. Multibody system techniques are general
and have been used in the analysis, design, and performance evaluation of numerous
applications including vehicles, machines, space structures, biomechanical and biological systems, robotics and mechanisms, as well as many other applications. The
equations of motion are developed in their most general form using the principle of
mechanics. These equations are implemented on digital computers in order to
develop programs that can automatically and systematically construct and numerically solve the equations of motion of a system that consists of an arbitrary number
of bodies and joints.
The dynamics of railroad vehicle systems, as will be shown in this book, can
be systematically described using computational multibody system algorithms.
There are several advantages for adopting multibody system methodologies in the
computer-aided analysis of railroad vehicle systems. Among these advantages, which
are discussed in this section, are their generality, their ability to systematically solve
nonlinear problems, and the straightforward implementation of special railroad vehicle elements.
1.1.1 GENERALITY
Multibody system formulations are designed to be general to facilitate development
of vehicle models that include significant details. General forcing functions and
motion constraints can be systematically introduced into the vehicle’s dynamic
equations. Development of accurate and realistic dynamic models of railroad vehicle
systems, such as the one shown in Figure 1.1, requires the inclusion of significant
details. For example, trains may consist of a large number of cars connected by
coupling elements. Each car, sometimes referred to as a vehicle, includes car body,
bogies, suspension elements, bushings, bearings, as well as other components. The
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Introduction
3
FIGURE 1.1 Railroad vehicles. (Courtesy of AMTRAK.)
bogies, such as the one shown in Figure 1.2, also represent complex systems that
include frames and wheelsets that can have independent motions. The wheelsets,
which can rotate freely about their own axes, are connected to the frame using
primary suspensions, while the frame is connected to the bolster using a pin joint,
and the bolster is connected to the car body using the secondary suspensions, as
shown in Figure 1.2. The motion of the train is produced as the result of the friction
between the rotating wheels and the rails. The forces of dynamic interaction between
the wheels and the rails significantly influence the dynamics and stability of railroad
vehicles. Clearly, a railroad vehicle system consists of a large number of interconnected components that experience independent relative motion. These components
are connected by force elements such as springs, dampers, and bushings as well as
joints that impose restrictions on the motion of the system. As will be shown
throughout this book, the dynamics of such a complex system can be described
using a system of differential and algebraic equations (DAE) that must be solved
simultaneously. The subject of multibody systems, a branch of the field of computational mechanics, is devoted, as previously mentioned, to the formulation and
numerical solution of the differential and algebraic equations of systems that consist
of interconnected bodies. Using multibody system techniques, the equations of
motion of railroad vehicle system models that include significant details can be
systematically developed and numerically solved.
Generality is one of the main advantages of using multibody system algorithms
that provide, in addition to the systematic inclusion of general forces and constraints,
the capabilities of modeling flexible bodies using the finite element method and other
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4
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.2 Example of a bogie.
structural analysis techniques (Seo et al., 2005; Shabana, 1997, 2005). While flexible
body dynamics will not be covered in this book, it is important to point out that, in the
case of high-speed trains, the effects on the vehicle system dynamics due to deformations of the car bodies, rails, wheelsets, and the pantograph/catenary systems can be
significant. By using a computational multibody system approach, the well-developed
and advanced flexible multibody system formulations can be exploited in a straightforward manner in the dynamic analysis of railroad vehicle systems (Seo et al., 2005).
1.1.2 NONLINEARITY
Multibody system algorithms are based on nonlinear formulations that can be used
to accurately model the complex nonlinear behavior of railroad vehicle systems. By
using these general nonlinear formulations, there is no need to resort to linearization
techniques to obtain a solution for the vehicle dynamic equations. As will be discussed in this book, the use of the linear theory can lead to erroneous results in
important simulation scenarios and can also lead to inaccurate prediction of the
vehicle critical speed. Many of the books that are devoted to railroad vehicle system
dynamics employ linearization techniques in formulating the dynamic equations of
motion. In contrast, this text is focused on developing nonlinear formulations and
describes the numerical algorithms that are used to solve the resulting system of
differential and algebraic equations of motion, thereby obtaining dynamics and
stability results that accurately represent the actual system behavior.
Nonlinearities in railroad vehicle systems can be geometric or material nonlinearities. Geometric nonlinearity is due to the large rotation of some components
of the vehicle system or due to large deformation of some of the elastic elements
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Introduction
5
FIGURE 1.3 Nonlinearity due to the track geometry. (Courtesy of AMTRAK.)
and components. For example, each wheelset can have an arbitrary large rotation
about its own axis. This large rigid body rotation introduces nonlinearities in the
kinematic and dynamic equations, as will be demonstrated in Chapter 2. On the
other hand, if the deformation of a suspension element is large, the use of a linear
force/displacement relationship does not lead to an accurate model. In the case of
large deformations, the forces depend nonlinearly on the displacements. Another
source of geometric nonlinearity is the track geometry. Figure 1.3 shows a train that
travels on a curved track. Curved-track geometry is a source of nonlinearity. Furthermore, the formulation of the problem of contact between the wheel and the rail must
take into account the geometry of the wheel and rail surfaces. The formulation of
the contact conditions, as shown in this book, requires the use of nonlinear kinematic
relationships expressed in terms of the wheel and rail geometric surface parameters.
Material nonlinearities arise when the force/displacement constitutive equations
are nonlinear. This can be the case when a component or an element experiences
plastic or viscoelastic behavior. For example, the dynamic interaction between the
wheel and the rail requires the calculation of creep forces (Johnson, 1985; Kalker,
1990). The general form of the creep-force/displacement relationship is expressed
in terms of nonlinear stiffness and viscoelastic coefficients. The calculation of these
creep forces is essential in the dynamic analysis of railroad vehicle systems, and the
constitutive laws used in the formulation of these forces are examples of material
nonlinearities. Other examples of material nonlinearities are the use of nonlinear
elastic, plastic, or viscoelastic coefficients in the formulation of the suspension,
bearing, and bushing forces. Both geometric and material nonlinearities can be
systematically incorporated into the multibody system formulations.
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6
1.1.3 IMPLEMENTATION
Railroad Vehicle Dynamics: A Computational Approach
OF
RAILROAD VEHICLE ELEMENTS
One important element that distinguishes railroad vehicles from other multibody
system applications is the vehicle/rail interaction. There are two types of elements
that are currently used in railroad vehicle systems. The first is the wheel/rail contact,
while the second is magnetic levitation (Maglev). The wheel/rail element and the
Maglev system are shown schematically in Figure 1.4. Generating train motion by
using wheels rolling and sliding on rails remains the most widely employed method.
As previously mentioned, the wheel/rail interaction is described in terms of creep
forces, as well as other forces and kinematic variables, and leads to the known hunting
phenomenon, which is a source of significant lateral and yaw oscillations that
contribute to vehicle instability, particularly at certain operating speeds. Multibody
system algorithms can also be used to study derailment scenarios and develop
derailment criteria. One derailment criterion used in the literature is to measure the
ratio between the lateral force L and the vertical force V acting on the wheel, as
shown in Figure 1.5a. When the lateral force exceeds a certain limit, the momentum
generated by this force can cause derailment. The L/V ratio can be predicted using
fully nonlinear models based on multibody system algorithms. In practice, however,
simplified approaches such as Nadal’s formula (Nadal, 1908) are often used to
determine the limit of L/V ratio. Nadal’s formula is based on a simple force balance
that can be used to determine the L/V ratio before a derailment occurs. As shown
in Figure 1.5a, the lateral and vertical forces L and V that apply on the rail are in
balance with the reaction forces N and F that apply on the wheel. Therefore, if the
wheel rotates relative to the rail in the sense shown in Figure 1.5, and if the friction
coefficient between the wheel and the rail is µ and the wheel flange angle is α, as
shown in Figure 1.5, then the L/V ratio is given by the following simple formula:
L
tan α − µ
=
V 1 + µ tan α
(1.1)
Because of the sense of the wheel rotation shown in Figure 1.5, the wheel has
a positive angle of attack (AOA), which is defined as the angle between the direction
FIGURE 1.4 Vehicle–track interaction.
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Introduction
7
FIGURE 1.5 Force balance in the case of a wheel climb.
of the forward velocity of the wheel and the longitudinal tangent to the rail at the
contact point. If the L/V ratio exceeds the right-hand side of Equation 1.1, wheel
climb occurs. It is important to point out that Nadal’s formula, as defined by Equation
1.1, depends only on the wheel flange maximum contact angle, and it is only valid
in the case of positive angle of attack. Therefore, one should in general use more
accurate methods that can correctly predict the vehicle derailment. In this book, the
general three-dimensional wheel/rail contact theory is discussed using several nonlinear multibody system contact formulations that can be used to study derailment
of detailed railroad vehicle models.
The problems that can be encountered when the L/V ratio exceeds a certain value
are not limited to wheel climbs. For instance, if the lateral force generated by the
second point of contact between the wheel flange and the rail is relatively high, this
force can cause lateral rail displacement. This rail displacement produces what is
known as gage widening that can lead to a wheel/rail separation, as shown in Figure
1.6. Furthermore, if the L/V ratio exceeds a certain limit, a rail rollover can occur,
as shown in Figure 1.7. Rail rollover is one of the most common sources of accidents,
especially when the vehicle travels over a spiral region of the track. In general, if the
L/V ratio is higher than the ratio D/H, where D and H are as shown in Figure 1.7,
the two forces L and V generate two opposite moments. If the moment generated
by the lateral force is higher than the moment generated by the vertical force, the
rail can rotate about its corner. Blader (1989) showed that if the contact point is
located at the gage point on the rail, the L/V ratio must be limited to the range 0.73
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8
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.6 Gage widening.
to 0.66, depending on the rail shapes. Clearly, as the contact point moves toward
the corner of the rail, the acceptable range of the L/V ratio is reduced. Other sources
of rail rollover are the wheel shape and the lubrication conditions of the rail and the
center plate. Therefore, it is difficult to determine the acceptable limit of the L/V
ratio by using simple models. For this reason, there is a need for computational
methods that can be used to develop detailed models to study the dynamics and
stability of railroad vehicle systems.
In Maglev trains, on the other hand, there is no contact between the vehicle and
the guide during the train motion, since magnetic forces are used to levitate the
vehicle. The use of Maglev trains is limited to special applications, and such trains
are still in the experimental stage. For this reason, most of the discussion in this
book is focused on developing formulations for wheel/rail contact in railroad vehicle
systems. Nonetheless, both wheel/rail contact and Maglev elements, as well as other
elements particular to railroad vehicles such as the pantograph/catenary systems
(Seo et al., 2005), can be systematically incorporated in multibody system algorithms, as will be demonstrated in this book.
FIGURE 1.7 Geometric parameters used to study rail rollover.
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Introduction
9
1.2 CONSTRAINED DYNAMICS
The equations of motion of mechanical systems can be formulated using the Newtonian or the Lagrangian approach. In the Newtonian approach, vector mechanics
is used to define the forces. If the system consists of bodies connected by mechanical
joints, free-body diagrams are constructed to show the joint reaction forces as well
as the inertia and applied forces. The Newtonian approach, in which the equilibrium
of each body is first studied separately, can be used for relatively simple systems,
but it is not suited for the analysis of complex systems such as railroad vehicles.
On the other hand, in the Lagrangian approach, which is based on D’Alembert’s
principle, scalar quantities such as the virtual work and kinetic and potential energies
can be used to develop the body equations of motion. There is no need in this case
to study the equilibrium of the bodies in the system separately. The Lagrangian
approach, which is based on the known Lagrange-D’Alembert equation, can be used
to systematically eliminate the reaction forces or to keep these forces in the final
form of the equations of motion, as discussed below.
Before using the general multibody system approaches to formulate the nonlinear
equations of motion of railroad vehicle systems, a set of coordinates called generalized coordinates that define the configuration of the system components must first
be introduced. Different sets of coordinate types can be used as generalized coordinates in the dynamic formulations. In general formulations of railroad vehicle
dynamics, two different sets of coordinates are often used: the absolute coordinates
and the trajectory coordinates. The trajectory coordinates, which are often used in
developing formulations tailored for railroad vehicle systems, are discussed in a later
section of this chapter. In the absolute-coordinate formulations, the configuration of
a rigid body in the multibody railroad vehicle system is defined using two sets of
coordinates. The first set consists of three absolute Cartesian coordinates that define
the global position vector of the origin of a selected body coordinate system, as
shown in Figure 1.8, while the second set consists of three independent rotation
parameters that define the orientation of the body coordinate system with respect to
the global frame of reference. Among the sets of rotation parameters that can be
used to describe the orientation of the moving body coordinate system are the
direction cosines, Euler angles, and Euler parameters. As shown in Chapter 2,
the absolute position vector of an arbitrary point on the body can be expressed in
terms of the generalized absolute Cartesian and rotation coordinates. Using the
absolute position vector, the absolute velocity and acceleration of an arbitrary point
on the body can be determined and used with the Newton-Euler equations to obtain
the body equations of motion. The motion of the components of the railroad vehicles
is subjected to constraints that result from mechanical joints and specified motion
trajectories. Thus, the formulation of the nonlinear algebraic equations that describe
the motion kinematic constraints becomes necessary in the Lagrangian formulation.
Chapter 2 presents the formulation of the kinematic constraints using a set of
nonlinear algebraic equations for several commonly used mechanical joints. These
constraint equations enter into the formulation of the dynamic equations of motion
of the three-dimensional, constrained, multibody vehicle systems. When algebraic
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10
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.8 Absolute coordinates.
constraint equations are present with the dynamic differential equations, one has a
mixed system of differential and algebraic equations (DAE) that must be solved
simultaneously.
Two approaches are commonly used to solve the resulting system of differential
and algebraic equations: the augmented formulation and the embedding technique.
In the augmented formulation, Lagrange multipliers are used to combine the constraint equations with the system differential equations of motion, leading to a larger
system of equations that has a sparse matrix structure. The embedding technique,
in contrast, uses the kinematic constraint equations to systematically eliminate some
of the coordinates, leading to a smaller system of equations that has a dense matrix
structure. Both the augmented formulation and the embedding technique are discussed in Chapter 2. The remainder of this section discusses the concept of the
system degrees of freedom and the relationship between the kinematic constraints,
dependent coordinates, and reaction forces.
Railroad vehicle systems consist of a large number of bodies that include
wheelsets, bogie frames, car bodies, suspension elements, and other components.
These bodies, as previously mentioned, are interconnected by mechanical joints and
are subjected to different types of forces and loading conditions. The coordinates
introduced to describe the motion of the bodies in the system and formulate the
dynamic equations of motion are called, as previously mentioned, the generalized
coordinates. If six coordinates (three translation and three rotation coordinates) are
used to describe the unconstrained motion of a rigid body in the system, the number
of the system generalized coordinates is equal to 6 multiplied by the number of bodies.
These generalized coordinates, however, are not independent, since they are related
by algebraic equations that represent mechanical joints and specified motion trajectories. The concept of the degrees of freedom is fundamental to the study of the
motion of the constrained dynamic systems. The degrees of freedom are defined as
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Introduction
11
the independent coordinates that are required to describe the configuration of the
system. The number of degrees of freedom, therefore, depends on the number and
types of mechanical joints, specified motion trajectories, and the number of coordinates used to describe the unconstrained motion of the bodies. Each algebraic
constraint equation can be used to eliminate one coordinate by writing this coordinate
in terms of the others. For a system that consists of nb rigid bodies subjected to nc
algebraic constraint equations, the number of the system degrees of freedom nd for
spatial problems, in which the unconstrained motion of a rigid body is described
using six coordinates (three translations and three rotations), is given by
nd = 6 × nb − nc
(1.2)
This equation is called the Kutzbach criterion. In this equation, 6 × nb is the
total number of coordinates required to describe the unconstrained motion of the
system, while the number of the constraint equations nc defines the number of
dependent coordinates, which must be equal to the number of linearly independent
constraint equations. It is shown in Chapter 2 that in the augmented formulation,
one obtains a system of equations that has dimension equal to 6 × nb plus the number
of constraint equations nc. This system can be solved for the body accelerations as
well as the constraint forces. That is, the number of independent constraint forces is
equal to the number of constraint equations, which is equal to the number of dependent
coordinates. Preventing the motion in one direction is physically equivalent to
introducing one reaction force, which is mathematically equivalent to introducing
an algebraic constraint equation and an additional dependent coordinate. In the
augmented formulation, the equations of motion are formulated in terms of redundant
coordinates and the constraint forces. Since the algebraic constraint equations are
not eliminated, one has a system of differential and algebraic equations that must
be solved simultaneously.
In the embedding technique, on the other hand, one obtains a number of acceleration equations equal to the number of the system degrees of freedom nd. The
algebraic constraint equations are used to systematically eliminate dependent coordinates and the associated constraint forces. In this case, the minimum number of
equations of motion is obtained, and the resulting system of differential equations
does not include constraint forces. These differential equations can be integrated
numerically using well-developed numerical differential equation solvers, since the
algebraic constraint equations are eliminated. The loss of the sparse matrix structure
is one of the main disadvantages of using the embedding techniques. Furthermore,
computer codes based on the embedding techniques tend to be less general and less
user friendly compared with the general-purpose computer codes that are based on
the augmented formulation.
1.3 GEOMETRY PROBLEM
The study of the geometry is necessary in the analysis of the wheel/rail contact in
railroad vehicle systems. In fact, the computer-aided analysis of railroad vehicle
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12
Railroad Vehicle Dynamics: A Computational Approach
systems is generally divided into two main stages. The first stage is a preprocessing
stage in which the track geometry and the wheel- and rail-surface profiles are defined.
In the second stage, the equations of motion of the multibody vehicle system are
numerically solved. In this second stage, the geometric parameters that define the
wheel and rail surfaces and the track geometry enter into the formulation of the contact
conditions and the system equations of motion. To develop general and detailed models
of railroad vehicle systems, the multibody system algorithms used in the second stage,
sometimes called the main processing stage, must be modified to include a wheel/rail
contact model. Three steps are employed in the computational algorithm used in the
main processing stage to obtain the numerical solution of the wheel/rail contact problem. First, the geometry of the contact surfaces of the wheels and the rails is used to
determine accurately the locations of the points of contact between the wheels and the
rails. Second, the kinematic variables are defined in terms of the geometric parameters
of the wheel and rail surfaces. These variables include normalized kinematic quantities
called creepages that measure the relative velocities between the wheels and the rails
at the contact points. In the third step, the dynamic or kinetic forces that act on the
wheels and the rails as the result of the contact are determined.
The accuracy of the numerical solution of the contact problem depends strongly
on the accurate prediction of the location of the contact points. The solution for the
contact locations requires an accurate representation of the geometry of the wheel and
the rail surfaces. This representation can be defined using local surface geometric
properties, such as the radii of curvature and the tangent and the normal vectors to the
surfaces. These geometric properties are not only important for determining the contact
locations, but they are also important (as described in Chapter 4) in determining the
forces that represent the interaction between the wheels and the rails. Therefore, basic
knowledge of differential geometry is necessary to understand the wheel/rail contact
problem. In particular, the theories of curves and surfaces are fundamental in the study
of the dynamic interaction between the wheel and the rail. For example, in the case
of curved tracks, the arc lengths of space curves are used to describe the distance
traveled by the vehicle and to define the orientation of a rail coordinate system at the
points of contact. The forces of the dynamic interaction between the wheels of the
vehicle and the track can be defined in this rail coordinate system.
1.3.1 DIFFERENTIAL GEOMETRY
The theories of curves and surfaces are covered in texts on the subject of differential
geometry. A curve is defined as a real vector function that can be uniquely expressed
in terms of one parameter, t. That is, the components of the vector function can be
determined once this parameter is specified. Using this definition, a curve over the
interval a ≤ t ≤ b can be written in the following form:
()
()
y t =  y1 t
()
y2 t
()
y3 t 
T
(1.3)
This equation, which is the parametric representation of a curve, can be used to
determine the location of a point on the curve for an arbitrary value of the parameter
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Introduction
13
FIGURE 1.9 Space curves.
t. Curves such as the one shown in Figure 1.9 can be parameterized using their arc
length s. It will be shown in Chapter 3 that when the arc length is used as a parameter,
the derivative of Equation 1.3 with respect to the arc length parameter defines a unit
tangent. Further differentiations lead to the definition of invariant local geometric
properties for the curve, such as the curvature and torsion.
While curves can be described using one parameter, the description of the
geometry of a surface requires the use of two independent parameters, as shown in
Figure 1.10. Using a Cartesian coordinate system, each point on the surface is assumed
to have a unique position vector x that can be defined in the three-dimensional space
in terms of the two independent parameters as follows:
x(s1, s2 ) =  x1 (s1, s2 )
x2 (s1, s2 )
x3 (s1, s2 ) 
T
(1.4)
where s1 and s2 are the parameters used to describe the surface geometry and are
called the surface parameters. In order to apply differential calculus, Equation 1.4
must satisfy certain differentiability requirements, which will be discussed in
Chapter 3. As in the case of curves, surfaces have invariant properties that can be
FIGURE 1.10 Geometry of surfaces.
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14
Railroad Vehicle Dynamics: A Computational Approach
used to uniquely define the surface geometry. Among these properties are the first
and second fundamental forms of the surface and the Gaussian curvature. Knowing
the surface geometry, one can define a tangent plane and a normal to this plane. At
the point of contact between the wheel and the rail, one must be able to define the
tangent plane and the normal in order to define the tangential creep forces and the
normal force that enter into the dynamic formulation of the equations of motion.
Furthermore, the geometric properties of the surfaces, such as the principal curvatures, are used to define the geometry and dimension of the contact area, as will be
discussed throughout this book.
1.3.2 RAIL
AND
WHEEL GEOMETRY
The dynamic behavior of railroad vehicles depends on the wheel and rail geometry,
and for this reason, it is important to accurately describe the wheel and rail geometry
to correctly predict the vehicle response. The method used for the description of the
surfaces of the wheels and rails should be general to allow representing arbitrary
geometry of the surfaces. It is also important to be able to describe mathematically
the wheel and rail profiles in a general form. For example, in the case of a straight
segment of a track, the surface of the rail can be obtained by translation of the profile
curve, as shown in Figure 1.11. This surface can be defined by the parametric
equations
u r =  s1r
s2r
f (s2r ) 
T
(1.5)
where sr1 is the distance along the rail (arc length) and is defined as the rail longitudinal surface parameter, and sr2 is the rail lateral surface parameter that is used
FIGURE 1.11 Rail surface.
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Introduction
15
FIGURE 1.12 Gage and super-elevation.
as an independent variable to describe the rail profile. The complete description of
the track geometry requires the use of several definitions that include, for example,
the gage and super-elevation, which are shown in Figure 1.12. The gage G is defined
as the lateral distance between two points on the inner faces of the right and left
rails, while the super-elevation h is defined, as shown in Figure 1.12, as the vertical
distance between the right and left rails. These definitions, among others, and their
use in formulating the geometric equations of the track are discussed in Chapter 3.
It will also be shown that the track geometry can be completely defined using three
inputs: the horizontal curvature, which can be defined by projecting the space curve
of the rail on the horizontal plane; the development angle, which defines the elevation
of the rail; and the bank angle, which defines the track super-elevation. Using these
three inputs, a track with a complex shape can be constructed in a straightforward
manner, as described in Chapters 3 and 6. An efficient and systematic description
of the track geometry is obtained using several standard track segment types shown
in Figure 1.13. These segments include tangent (straight), curved, tangent-to-curve
entry spiral, and curve-to-tangent exit spiral as well as other segment types, as
discussed in Chapter 3. Simple geometry is used to describe the shapes of these
segments to facilitate the development of an efficient mathematical formulation to
define tracks with complex geometry.
In general, as previously mentioned, the computer simulation of the nonlinear
dynamics of railroad vehicle systems consists of two main stages. In the first stage,
the track geometry and the wheel and rail profiles are defined, while in the second
stage, the equations of motion of the railroad vehicle are formulated and solved
using the geometry input obtained in the first stage. For the first stage, one often
FIGURE 1.13 Track segments.
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16
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.14 Wheel geometry.
develops a preprocessor computer code that can be used to define tracks with
arbitrary geometry. The track preprocessor code has input that is based on the
definitions and terminologies used by the railroad industry. The output of the track
preprocessor is a data file that is used as an input to the main processor computer
code that is used to solve the dynamic equations of the multibody railroad vehicle
system. Chapters 3 and 6 discuss the structure of a preprocessor computer code that
can be used to define the track geometry.
The wheel geometry also has a significant effect on the dynamics and stability
of railroad vehicles. The critical speed of a vehicle strongly depends on the shape
of the profiles of the wheels and rails. The surface of the wheel is a surface of
revolution obtained by a complete rotation of the curve that defines the wheel profile
about the wheel axis, as shown in Figure 1.14. Therefore, the surface of the wheel
can be defined mathematically in a selected Cartesian coordinate system by the
following equation:
 x 0 + g(s1w )sin s2w 


u w (s1w , s2w ) = 
y0 + s1w

 z − g(s w ) cos s w 
0
1
2


(1.6)
where s1w is the lateral surface parameter that represents the independent variable for
the wheel profile g(s1w), and s2w is an angular surface parameter that represents the
rotation of the wheel profile about its axis. The variables x0, y0, and z0 are constants.
Tapered wheels tend to self-center as compared with cylindrical wheels, reducing
the possibility of the contact between the wheel flange and the inner surface of the
rail. If cylindrical wheels are used, the flanges wear rapidly due to the repeated
contact with the inner side of the rail. Using a simplified model, it can also be shown
that, if the taper were to be in the opposite sense, the wheelset would be unstable
(Karnopp, 2004; Popp and Schiehlen, 1993). This unstable behavior is discussed in
greater detail in Section 1.8.
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Introduction
17
1.4 CONTACT THEORIES
The wheel/rail contact is another important problem that must be addressed in the
analysis of railroad vehicle systems. The contact analysis depends on the geometry
of the wheel and rail surfaces, and for this reason it is important to first understand
the geometry of the surfaces, as discussed in the preceding section. In railroad
dynamics, a standard procedure is followed to determine the contact forces for a
given wheel and rail configuration. First, the normal contact force is determined.
This is the force normal to the plane tangent to the wheel and rail surfaces at the
contact point. Second, using this normal force, the material and geometric properties
of the wheel and the rail and the relative velocities between the wheel and the rail,
the tangential creep force, and the spin moment can be determined. This section
introduces the wheel/rail contact theories that are the subject of more detailed
discussion in Chapter 4. In the discussion presented here and in Chapter 4, only the
contact between one wheel and one rail is considered in order to focus on the procedure
used for solving the contact problem. In the following section and in Chapter 5, the
use of the contact formulation in general multibody system algorithms is discussed.
Various contact models are used in different computer formulations to describe
the wheel/rail interaction. In general, the contact between two rigid bodies can be
at a single point or area, depending on the shape of the two bodies. These two types
of contact are known as nonconformal and conformal contact, respectively (Johnson,
1985). If the shape of the two bodies is such that the two bodies in the region of
contact fit exactly or even closely together, the contact is defined as a conformal
contact. If the two bodies, on the other hand, touch at a point or a line, the contact
is called nonconformal. If an external load is applied on each body, the two bodies
will deform at the contact point to form an area of contact. The contact area in the
case of the nonconformal contact is assumed to be small compared with the dimensions of the two bodies. The problem of nonconformal contact between two surfaces
was studied by Hertz (1882). Hertz assumed that the area of contact is elliptical. In
studying the wheel/rail dynamic interaction, the assumption of nonconformal contact
is justified because of the shape of the wheel and rail surfaces. The contact region
is assumed to be elliptical, and its dimension can be determined as described in
Chapter 4 using the geometry of the two surfaces. To this end, one has to determine
the principal curvatures of the two surfaces as well as the principal directions. Given
the configurations of the wheel and the rail, several methods can be used to determine
the normal contact forces. As will be discussed in the following section and in
Chapter 5, the methods for determining the normal contact forces can be based on the
assumption that the wheel and the rail surfaces are rigid or on the assumption that
the two surfaces can deform in the small region of contact.
1.4.1 CREEP FORCES
Due to the elasticity of the bodies and the external applied normal load, some points
on the surfaces in the contact region may slip while others may stick when the two
bodies move relative to each other. The difference between the tangential strains of
the bodies in the adhesion area leads to a small apparent slip. This slip is called
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18
Railroad Vehicle Dynamics: A Computational Approach
creepage and is defined using the relative velocity between the two bodies at the
contact point. Creepages generate tangential creep forces and creep spin moment,
which play a fundamental role in the dynamics and stability analysis of railroad
vehicles. The creep tangential forces and spin moment are generated during the
wheel/rail interaction, since the motion of the wheel relative to the rail is a combination of rolling and sliding. These creep tangential forces and spin moment have
a significant effect on the steering and stability of the railroad vehicle systems. For
this reason, the creep phenomenon cannot be ignored in the analysis of railroad
vehicle systems.
1.4.2 WHEEL/RAIL CREEP THEORIES
There are several contact theories that have been developed to determine the tangential
creep force and the spin moment for a given normal force and material and geometric
properties of the wheel and rails. Some of these theories are two-dimensional, while
others are three-dimensional; and some are based on linearized models, while the
others are based on nonlinear models. Some of the theories are based on closedform expression for the relationship between the creep forces and the creepages,
while other theories require numerical interpolation. Most of the creep-force models
that are in use are a function of the geometry of the contact area. To determine the
dimension and shape of the contact area for a given normal load, most of the models
employ Hertz’s contact theory and assume that the contact area has an elliptical
shape. Hertz’s contact theory can be used to determine the penetration of the two
surfaces in contact as well as the contact ellipse dimension. While Hertz’s theory is
developed for a static case and assumes frictionless contact, the use of the Hertz
theory to determine the shape and dimension of the contact area is widely accepted
by the railroad vehicle research community. Hertz’s contact theory as well as several
wheel/rail creep-force theories are discussed in more detail in Chapter 4.
1.5 GENERAL MULTIBODY RAILROAD
VEHICLE FORMULATIONS
By establishing a procedure for determining the normal and tangential creep forces
as well as the creep spin moment, such a procedure can be systematically implemented in a general multibody system algorithm that can be used in the analysis of
railroad vehicle models that include significant details. As previously mentioned,
the analysis of the wheel/rail interaction requires accurate prediction of the normal
contact forces. The normal contact force and the geometric and material properties
of the wheel and the rail are required for the evaluation of the creep tangential forces
and spin moment. There are two main computational approaches that can be used
in the multibody system formulations to predict the location of the contact points
on-line and determine the normal contact force when the wheel/rail interaction is
considered. In the first approach, the contact between the wheel and the rail is
described using kinematic constraint equations that are imposed at the position,
velocity, and acceleration levels. In this case, it is assumed that the wheel and rail
surfaces do not penetrate. The normal contact force can be determined as the reaction
45814_book.fm Page 19 Thursday, May 31, 2007 2:25 PM
Introduction
19
force due to imposing the contact constraint equations. When equality constraints
are used, it is assumed that there is no separation or penetration between the wheel
and the rail. In the second approach, which is based on an elastic force model, the
wheel and rail surfaces are allowed to have a small deformation in the contact region.
The normal contact force that results from the wheel/rail interaction is predicted
using a compliant force model with assumed stiffness and damping coefficients. In
this approach, the wheel has six degrees of freedom with respect to the rail, and
wheel/rail separation and penetration are possible.
Chapter 5 discusses four nonlinear dynamic formulations for the analysis of the
wheel/rail contact. Two of these formulations, which are based on the constraint
approach, assume that the wheel and rail surfaces are rigid and employ nonlinear
algebraic kinematic constraint equations to describe the contact between the wheel
and the rail. The other two formulations, which are based on the elastic approach,
assume that the wheel and rail surfaces can experience small local deformations, and
the contact force is modeled using a compliant force element. The constraint and
elastic approaches are conceptually different, and they lead to models with different
numbers of degrees of freedom. The basic features of the constraint and elastic
contact formulations are discussed in this section. A more detailed discussion is
provided in Chapter 5.
1.5.1 CONSTRAINT CONTACT FORMULATION
Figure 1.15 shows a wheel and a rail that are in contact. If no penetrations or
separations are allowed in the case of nonconformal contact, the wheel can have
five degrees of freedom with respect to the rail. These degrees of freedom are two
translations in the tangent plane at the contact point and three relative rotations. In
this case, it is assumed that there is no relative motion along the normal to the
tangent plane at the contact point. This implies that imposing the nonconformal
FIGURE 1.15 Multibody system contact formulation.
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20
Railroad Vehicle Dynamics: A Computational Approach
contact conditions eliminates one degree of freedom; this is the freedom of the
relative motion along the normal to the tangent plane at the contact point. Recall
that the general description of the geometry of the contact surfaces requires introducing two surface parameters for each body. That is, for each contact between a
wheel and a rail, one must define the geometry using four surface parameters. Since
four geometric parameters are introduced and one degree of freedom is eliminated,
imposing the nonconformal contact conditions requires, in general, the introduction
of five kinematic constraint equations that can be used to eliminate the four geometric
parameters and one degree of freedom for each contact. It is shown in Chapter 5
that the geometric surface parameters can be treated as nongeneralized coordinates
because, when the equations of motion are formulated in terms of the surface
parameters, there is no inertia or forces associated with these geometric parameters.
Two equivalent constraint formulations that employ two different solution procedures can be developed. The first method leads to a larger system of equations by
augmenting the dynamic equations of motion with all the contact constraint equations. In this augmented formulation (discussed previously in Section 1.2), the
surface parameters can be selected as degrees of freedom. In the second method, on
the other hand, an embedding procedure is used to obtain a reduced system of
equations from which the surface parameter accelerations are systematically eliminated (discussed previously in Section 1.2). In this second method, the surface
parameters cannot be selected as degrees of freedom.
1.5.2 ELASTIC CONTACT FORMULATION
In the formulations based on the elastic approach, the wheel has six degrees of
freedom with respect to the rail, and the normal contact force is defined as a function
of the penetration using Hertz’s contact theory or using assumed stiffness and
damping coefficients. Unlike the constraint contact formulation, the elastic contact
formulation allows for wheel/rail separation and penetration. From the solution of
the dynamic equations of motion, the position of the wheel with respect to the rail
can be determined and used to check whether or not the wheel and rail surfaces
penetrate. If penetration occurs, the penetration and its time rate can be used to
determine the normal force. Clearly, accurate prediction of the location of the contact
points is necessary for the robust implementation of the elastic contact formulations.
One of the elastic methods discussed in this book in Chapter 5 is based on a search
for the contact locations using discrete nodal points that are used to describe the
wheel and rail surface profiles. In the second elastic approach discussed in this book,
the contact points are determined by solving a set of nonlinear algebraic equations.
1.6 SPECIALIZED RAILROAD VEHICLE FORMULATIONS
Two different strategies are often adopted when developing computational tools for
a particular engineering application. The first strategy is based on developing algorithms that are tailored to that particular application. In this case, one can exploit
the particular features of the application to optimize the computational algorithm.
The disadvantage of using this approach is the difficulties that can be encountered
45814_book.fm Page 21 Thursday, May 31, 2007 2:25 PM
Introduction
21
when more-general scenarios or physical phenomena are considered. The second
strategy is based on modifying existing multibody system algorithms to provide the
capabilities for modeling that particular application. This second approach provides
the flexibility of dealing with general scenarios and also allows exploiting advanced
multibody capabilities that are already parts of the computational algorithms. Both
the general and specialized approaches are discussed in this book.
In this book, the absolute Cartesian coordinates are used to develop the general
computer formulations for the nonlinear railroad vehicle system. This motion
description leads to systematic and straightforward implementation of the contact
formulations in existing general-purpose multibody system computer algorithms.
This description also preserves the sparse matrix structure of the dynamic formulation and permits the exploitation of advanced and well-developed multibody system
dynamics capabilities such as body flexibility. However, the absolute Cartesian
coordinate description is not the only approach that has been used in the motion
description of railroad vehicle systems, as many specialized codes adopt other sets
of coordinates.
Another set of coordinates that is often used in the specialized railroad vehicle
formulations is trajectory coordinates. When these coordinates are used, the motion
of an arbitrary body in the railroad vehicle system is defined, as described in
Chapter 7, using coordinates that depend on the track geometry. A track coordinate
system X irY irZ ir, called the body-trajectory coordinate system, which follows the
motion of the body, is introduced in Figure 1.16; that is, each body i has its trajectory
coordinate system. The location of the origin and orientation of the body-trajectory
coordinate system can be uniquely defined by the arc length coordinate, which
represents the distance traveled along the track space curve. This trajectory coordinate system is used to define the configuration of the body in the global coordinate
system. To this end, another coordinate system, called the body coordinate system,
XirY irZ ir, is introduced for each body, as shown in Figure 1.16. The origin of the
body coordinate system is assumed to be attached to the body center of mass, and
the body coordinate system is selected such that it has no displacement in the
longitudinal direction of motion with respect to the trajectory coordinate system.
Since the geometry of the track is assumed to be known, the complete definition of
the trajectory coordinate system, which includes translation and orientation parameters, requires only one time-dependent coordinate: the distance traveled. On the
other hand, the description of the motion of the body with respect to the trajectory
coordinate system requires five time-dependent coordinates: two translations and
three angles defined with respect to the trajectory coordinate system. This leads to
the following set of trajectory coordinates of an arbitrary body i in the system:
pi = [s i
yir
z ir
ψ ir
φ ir
θ ir ]T
(1.7)
where si is the arc length coordinate along the track space curve, and y ir and z ir are,
respectively, the coordinates that define the location of the center of mass in the
lateral direction and in a direction normal to the plane that contains the track space
curve, as shown in Figure 1.16. These two coordinates are defined with respect to
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22
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.16 Trajectory coordinates.
the trajectory coordinate system whose location and orientation are defined by the
arc length si. The angles ψ ir, φ ir, and θ ir are, respectively, the yaw, roll, and pitch
angles that define the orientation of the body with respect to the trajectory coordinate
system. The use of the trajectory coordinates, compared with the absolute coordinates, has the advantage of simplifying the formulation of some railroad vehicle
constraints and of some forcing functions. The trajectory coordinates, on the other
hand, make the implementation of railroad vehicle system formulations in generalpurpose multibody system algorithms more difficult, can lead to the loss of the sparse
matrix structure of the dynamic equations, and make exploiting advanced flexible
body capabilities less straightforward.
Using the generalized trajectory coordinates, one can easily develop different
types of reduced-order vehicle models for different types of analysis, as discussed
in Chapter 7. Single-degree-of-freedom vehicle models can be developed using the
arc length coordinate, and these are often used to develop algorithms for the analysis
of the train longitudinal forces (Meng et al., 2005). Such an analysis is important
to avoid accidents and improve the operating efficiency for a particular train makeup.
The forces in the couplers that join vehicles can be examined to determine whether
or not these forces reach critical values that cause coupler failures.
A two-degree-of-freedom model can also be developed using the trajectory
coordinates to study the lateral stability of railroad vehicle systems, as discussed in
Chapter 7. The lateral instability caused by the hunting phenomenon is the result of
the coupling between the lateral and yaw displacements and the creep forces. For
this reason, a two-degree-of-freedom model that has the lateral displacement and the
yaw angle as independent coordinates can be used to examine the hunting instability.
In this case, one has the following set of trajectory coordinates for an arbitrary body i:
pi = [ yir
ψ ir ]T
(1.8)
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Introduction
23
The two-degree-of-freedom model can be developed by imposing constraints on
the other four trajectory coordinates. Since the trajectory coordinates are used as the
generalized coordinates, such constraint equations become linear functions in the
generalized trajectory coordinates, thereby leading to a simple procedure for eliminating the four dependent coordinates and the associated constraint forces from the
dynamic equations of motion. As a result, one obtains a minimum set of differential
equations of motion expressed in terms of the lateral and yaw trajectory coordinates.
Chapter 7 provides a more detailed discussion on the trajectory coordinates and their
use in formulating the equations of motion of railroad vehicle systems.
1.7 LINEARIZED RAILROAD VEHICLE MODELS
Some existing specialized railroad vehicle system formulations employ linearized
kinematic and dynamic equations. It is known that railroad vehicle models are
sensitive to such a linearization, and formulations that employ kinematic linearization can predict, particularly at high speeds, significantly different dynamic response
compared with models that are based on fully nonlinear kinematic and dynamic
equations (Shabana et al., 2006). To examine this problem analytically and numerically quantify the effect of the approximations used in the linearized railroad vehicle
models, the fully nonlinear kinematic and dynamic equations must first be obtained.
The linearized kinematic and dynamic equations used in some railroad vehicle
models can then be obtained from the fully nonlinear model to shed light on the
assumptions and approximations used in the linearized models. The assumptions of
small angles that are often made in developing railroad vehicle models and their effect
on the angular velocity, angular acceleration, and the inertia forces can be investigated. The creepage expressions that result from the use of the assumptions of small
angles can be obtained and compared with the fully nonlinear expressions.
As previously mentioned, among the parameters required to evaluate the force
of interaction between the wheel and the rail are the wheel and rail profile geometry
data, the materials of the wheel and the rail, and the creepages that depend on the
relative velocities between the wheel and the rail. To determine the longitudinal and
tangential creep forces as well as the spin moment, the creepages are multiplied by
very high creepage coefficients. Approximations used in the definition of the creepages can, therefore, lead to dynamic and stability results that differ from those
predicted using the nonlinear models, particularly at high speeds. A study that
compared different computer codes that are based on different formulations, some
of which employ linearized creepage expressions, showed that the difference in the
predicted critical speed between linearized railroad vehicle models and nonlinear
railroad vehicle models can be very significant, exceeding 21 m/sec for some models
(Iwnicki, 1999). This finding is significant, since the prediction of the critical speed
is one of the main objectives of using railroad vehicle codes and computer formulations. Inaccurate prediction of the critical speed can have serious consequences
and can negatively impact the accuracy of predicting derailment and accident scenarios and the evaluation of safety criteria.
Chapter 8 examines the approximations used in the linearized creepages. The
fully nonlinear expressions are first obtained and then used to derive the linearized
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24
Railroad Vehicle Dynamics: A Computational Approach
creepages. The basic assumptions used to derive the linearized creepages are summarized. It is shown that these assumptions are automatically satisfied if the roll
angle is assumed to be zero. The numerical results presented in the literature show
that the use of such assumptions can significantly influence the accuracy of the results
(Shabana et al., 2006). This explains the significant differences between the railroad
vehicle results predicted using computer codes that are based on fully nonlinear
formulations and those codes that employ kinematic linearization of the creepages.
In particular, the results presented in the literature clearly show that the linearization
of the kinematic creepage expressions can lead to significant errors in the values
predicted for the longitudinal and lateral creep forces as well as the spin moment
(Shabana et al., 2006). These results also show the errors in the lateral and vertical
forces that enter into the calculation of the L/V ratio that is used in some derailment
criteria.
1.8 MOTION STABILITY
One of the important characteristics of the motion of railroad vehicles is the hunting
phenomenon. Hunting is defined as the lateral motion of the wheelset with respect
to its initial (equilibrium) position. The wheelset consists of two wheels (right and
left) that are connected by an axle. In general, the wheel shape is conical, with the
largest diameter near the inner face of the rail, as shown in Figure 1.17. With this
shape, the wheelset would automatically tend to self-center during the motion, and
as a result there is less contact with the flange (Karnopp, 2004). A simple analysis
can be used to explain this behavior. To this end, assume that the wheelset shown
in Figure 1.17 travels over a track. Due to any excitation that is the result of track
deviations or perturbations, the wheelset moves laterally. Assume that, at the initial
(equilibrium) configuration, the wheelset has zero lateral position (y = 0) and the
right and left wheels have initial rolling radii Rr and Rl , respectively, as shown in
Figure 1.17. If the two wheels are identical and symmetrically located with respect
to the axle, the two rolling radii Rr and Rl in the initial symmetric configuration are
equal and are denoted as R0. The wheel conicity γ is defined by the slope of the
FIGURE 1.17 Wheelset rolling radii.
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Introduction
25
wheel profile, as shown in Figure 1.17. As the wheelset moves laterally, the change
∆R in the two rolling radii can be determined using the following equation:
∆R = yγ
(1.9)
It follows that the right and left wheels’ rolling radii are given at any instant of time by
Rr = R0 − yγ 

Rl = R0 + yγ 
(1.10)
If the wheelset is rotating with a constant angular velocity ω, the velocities Vr and
Vl of the right and left wheels are given, respectively, by
Vr = Rrω 

Vl = Rlω 
(1.11)
where Rr and Rl are as defined in Equation 1.10. The velocity of the center of the
wheelset is V and is given by
(
V = Vr + Vl
)
2 = R0ω
(1.12)
If the yaw angle ψ shown in Figure 1.18 is assumed small (tan ψ ≈ ψ), the rate of
change of the lateral motion can be determined as follows:
y =
dy dy dx
=
= ψ V = ψ R0ω
dt dx dt
FIGURE 1.18 Wheelset hunting motion.
(1.13)
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Railroad Vehicle Dynamics: A Computational Approach
In this simple analysis, the rate of change of the yaw angle ψ can be written as
(
)
ψ = Vr − Vl G = −2 yωγ G
(1.14)
where G is as shown in Figure 1.18. Differentiating Equation 1.13 with respect to
time and substituting for ψ from Equation 1.14, one obtains
 2 R ω 2γ 
y+ 0
y=0
 G 
(1.15)
This equation, which can be used to describe the lateral motion of the wheelset, is
similar to the equation that governs the motion of a simple mass-spring system. If
the coefficient of y in the preceding equation is positive, the solution of this equation
can be written in the following form:
y = A sin(ω nt + C )
(1.16)
where A and C are constants that can be determined using initial conditions, and ωn
is the system undamped natural frequency, which is given upon the use of Equation
1.15 by
ωn = V
2γ
R0G
(1.17)
The solution of Equation 1.16, which is obtained with the assumption that (ωn)2 is
positive, represents a sustained oscillation with constant amplitude. In this case, the
wheelset oscillates about its equilibrium position, as shown in Figure 1.19a. Clearly,
the coefficient of y in Equation 1.15 is positive if the conicity γ is positive, as shown
in Figure 1.17. The period of the oscillation is
λ=
2π 2π
=
ωn V
R0G
2γ
(1.18)
Equations 1.17 and 1.18 are known as Klingel’s formulas (Klingel, 1883).
In the case of a cylindrical wheel, on the other hand, the conicity is equal to
zero (γ = 0), and the coefficient of y in Equation 1.15 is equal to zero. The solution
of Equation 1.15 in this case is a straight line, as shown in Figure 1.19b, and the
wheelset motion due to a lateral disturbance is not oscillatory. In the case of a
negative conicity, the coefficient of y in Equation 1.15 is negative, and the solution
is an exponentially increasing function of time, as shown in Figure 1.19c. Therefore,
45814_book.fm Page 27 Thursday, May 31, 2007 2:25 PM
Introduction
27
(a)
(b)
(c)
FIGURE 1.19 Wheelset lateral motion.
based on the simple kinematics presented in this section, the wheel must have a
positive conicity to have a stable oscillatory solution.
It is important to point out that the simple analysis presented in this section is
based on purely kinematic considerations and does not take into account the effect
of any forces. In reality, the wheelset is subjected to friction forces due to the
difference in the velocities of the wheel and rail at the contact point. This velocity
difference is used to define the normalized relative velocities (creepages) that enter
into the calculation of the creep forces that act on the wheel. Because of these creep
forces, the wheelset lateral motion can be oscillatory about the equilibrium position
with amplitude that increases or decreases with time, even in the case of positive
conicity. It is also interesting to note by examining Equations 1.13 and 1.14 that, in
the case of oscillatory motion, ψ = 0 corresponds to y = 0; that is, the yaw angle is
maximum or minimum when the lateral displacement is zero and vice versa. This
implies that there is a phase angle of π/2 between the lateral displacement and the
yaw angle.
1.9 MOTION SCENARIOS
We conclude this chapter with a discussion on some important railroad vehicle
motion scenarios. These simulation scenarios are important in the design process of
railroad vehicles. Most standard regulatory codes, such as the Federal Railroad
Administration Code in the United States, require that a newly designed vehicle
must be tested for such motion scenarios before the vehicle is put into service.
During railroad vehicle operations, one or more of the following motion scenarios
are encountered (Blader, 1989): hunting, steady curving, spiral negotiation, twist
and roll, pitch and bounce, yaw and sway, dynamic curving, and response to discontinuities. In the remainder of this section, these motion scenarios are discussed
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28
Railroad Vehicle Dynamics: A Computational Approach
in greater detail. It is important, however, to point out that accurate computer
modeling of some of these scenarios requires the use of fully nonlinear dynamic
formulations and the use of a three-dimensional contact theory to describe the
wheel/rail dynamic interaction.
1.9.1 HUNTING
The hunting motion was discussed in the preceding section. As mentioned before,
a wheelset hunts due to its shape, and the result is a lateral oscillation coupled with
a yaw rotation. The resulting vibration must remain at a certain acceptable level in
order to achieve specific comfort and safety requirements. Above a certain operating
speed that depends on the railroad vehicle design, the vehicle may experience severe
hunting that can be a source of discomfort or even the cause of derailment. The
speed at which the railroad vehicle becomes unstable is called the critical speed.
Above the critical speed, the vehicle is subjected to significantly higher forces due
to the hunting phenomenon and the resulting impact between the wheel flange and
the rail. The impact between the wheel flange and the rail is known in the railroad
vehicle literature as the second point of contact, assuming that there is a first point
of contact between the wheel tread and the rail surface. Below the critical speed,
the second point of contact ensures vehicle stability and prevents derailments. However, if the L/V ratio increases beyond a certain limit due to severe hunting, derailment
can occur. It is important, however, to mention that some wheel profiles are designed
such that no second point of contact occurs, and the wheel will always have one
point of contact with the rail. In this case, the forces generated will be at only one point.
Regardless of the number of wheel contact points, it is important to know the critical
speed of the railroad vehicle. This speed can be determined using the results of
computer simulations or by experimental field testing.
In general, one expects to find two instability regions for a vehicle; the first one
occurs in a low-speed range and is associated with the instability of the car body,
while the second occurs at higher speed and is associated with the bogie hunting
motion. These two instability regions were identified by Matsudaira, who defined
these two regions as primary and secondary hunting (Matsudaira, 1960). It is important to point out that the first type of instability can be easily controlled by using
lateral dampers in the secondary suspension to reduce the amplitude of the lateral
motion of the car body. The second type of instability, which occurs at higher speeds,
is characteristic of the bogie system. This type of instability depends on the bogie
suspension design and the wheel geometry (Valtorta et al., 2001).
1.9.2 STEADY CURVING
When a vehicle travels over a constant curve (curve with a constant curvature), two
important forces must be studied. The first force is the lateral force, which must be
in balance with the centrifugal force due to the track curvature. The centrifugal force
tends to push the vehicle out of the curve toward what is known as the high rail.
Therefore, if the curve has a certain curvature and super-elevation, there is a balance
speed at which the component of the centrifugal force is equal to the lateral component
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Introduction
29
FIGURE 1.20 Balance speed.
of the gravity force, as shown in Figure 1.20. For instance, if the vehicle is traveling
with a speed equal to V over a track that has a radius of curvature R, then at the
balance speed, one must have the following relationship:
mV 2
cos φ = mg sin φ
R
(1.19)
where g is the gravity constant and m is the mass of the vehicle. In general, φ in
the preceding equation is equal to the track angle φtrack plus the roll angle of the car,
which is the result of the suspension elasticity. Assuming that φ is equal to φtrack and
for general track, one can use the following small angle approximation:
cos φ ≈ 1,
sin φ =
h
G
(1.20)
where h is the track super-elevation and G is the track gage. Therefore, the balance
speed is defined as follows:
V=
gRh
G
(1.21)
If the vehicle is traveling with a velocity below the balance speed, the vehicle
is said to have cant excess. Cant excess is defined as the amount of super-elevation
that needs to be reduced so that the current vehicle speed will be equal to the balance
speed. On the other hand, if the vehicle is traveling with a speed that is above the
balance speed, the vehicle is said to have cant deficiency. Cant deficiency is defined
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30
Railroad Vehicle Dynamics: A Computational Approach
as the amount of the super-elevation that is needed to be increased so that the vehicle
current speed will be equal to the balance speed. In most cases, the vehicle must be
tested in both cases: cant deficiency and excess. In the case of cant deficiency, the
vehicle has high lateral forces that can cause undesirable motion on the higher rail.
These lateral forces, if high enough, will produce a wheel climb that can lead to a
vehicle derailment.
Another type of force that the vehicle is subjected to is the longitudinal force.
When the wheelset negotiates a curved track, the outer rail has a larger radius of
curvature than the inner rail. This requires the outer wheel to travel larger distance
than the inner wheel. In this case, as the wheelset rotates with a constant angular
velocity, one of the wheels (outer or inner) or both wheels will slip. The slip can be
reduced if the rolling radii of the two wheels are allowed to vary during the wheel
motion. This change in the rolling radius is accomplished by using the conical
wheel profile. In the case of conical wheels, as the wheelset negotiates a curve, the
wheelset will move laterally in the direction of the outer rail. Consequently, the
outer wheel will have larger rolling radius and higher velocity in the longitudinal
direction as compared with the inner wheel. This reduces the slip and wear, and
leads to better curving behavior. Therefore, better curving behavior can be achieved by
increasing the wheel conicity. Some modern designs of wheel profiles are not conical.
These profiles are designed such that they consist of arcs that are developed based
on worn wheels. These profiles have shapes that lead to improvement in stability
and reduction of wear. In general, during curve negotiations, the wheelset tends to
self-steer over the track as the result of the torque generated by the longitudinal
forces acting on the right and left wheels. If the steering moment is not large enough
or if the vehicle suspension connections generate an opposite moment that is higher
than the moment generated by the longitudinal forces, the vehicle will fail to travel
over the curve. This scenario can happen if the vehicle has a very stiff suspension.
1.9.3 SPIRAL NEGOTIATION
Similar to the case of constant curve, the vehicle can be subjected to similar forces
during spiral negotiation. However, the spiral, unlike the constant curve, does not
have a constant curvature or super-elevation. Due to the change in the curvature and
super-elevation, spirals are the sections of track where a significant number of
derailments occur. In general, the vehicle travels with a constant speed over the
spiral region. This speed, for a certain portion of the spiral, can be above the balance
speed, leading to a sudden impact at a certain portion of the spiral between the wheel
flange and the higher rail. Furthermore, as the vehicle changes its direction over the
spiral area, twist is induced in the car body.
1.9.4 TWIST
AND
ROLL
The twist and roll motion can be the result of the response of the vehicle to periodic
track cross-level variation. Cross-level variation can be caused by staggered rails or
by the vehicle due to wheel lift. In general, it is important to examine the twist and
roll scenario, particularly in the case of freight cars. Statistics show that a good
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Introduction
31
percentage of freight car derailments are the result of cross-level variation, especially
in the case of cars with high centers of gravity. The car-body roll can reach a
dangerous level if the roll stiffness used between the car body and the center plate
and sidebars is low. The result in this case is an increase in the roll amplitude if the
vehicle speed and the track wavelength produce an excitation frequency that coincides with the natural frequency of the vehicle roll. Imposing a limit on the maximum
value of the motion amplitude that the vehicle can have is important, and such a
limit can be determined by simulations of the twist and roll motion scenarios.
1.9.5 PITCH
AND
BOUNCE
If the track has a vertical perturbation due to profile deviation, large pitch and bounce
oscillations of the vehicle can be generated. The pitch motion is defined as the rotation
about an axis across the track, while the bounce motion is defined as the vertical
motion of the vehicle. If the suspension damping in these motion modes is low,
oscillations that persist for a relatively long time occur as the result of track profile
deviations. Therefore, it is important to properly select the suspension parameters
to reduce the amplitude of these modes of vibration.
1.9.6 YAW
AND
SWAY
The yaw and sway is the response of the vehicle in the lateral and vertical directions
due to track perturbation. For example, hunting can cause such a motion scenario.
This type of motion leads to either high oscillations or high impact forces when the
wheel flange comes into contact with the rail. In most cases, the effect of this motion
scenario can be tested by using a track similar to the one shown in Figure 1.21. It
is important to examine this dynamic behavior, since it is one of the most common
sources of derailment.
1.9.7 DYNAMIC CURVING
During curve negotiation, high impulsive lateral forces may be produced between
the wheel and the rail as the result of rail irregularities. Computer simulations can
be performed to examine the response of the railroad vehicle when it negotiates a
curved track. In this case, profile and alignment deviations can be superimposed on
a constant curve to obtain the desired track configuration. The acceptable vehicle
FIGURE 1.21 Example of a track that can be used to test yaw and sway motion.
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 1.22 Example of a track that can be used for dynamic curving.
models should lead to forces that are below the margin that causes wheel-climb.
The dynamic-curving simulations must be performed for a wide range of the vehicle
speed to test the stability and check for cant deficiency and cant excess. Figure 1.22
depicts an example of a track that can be used to test the dynamic-curving scenarios.
1.9.8 RESPONSE
TO
DISCONTINUITIES
Track discontinuities are defined as abrupt changes in the rail position. These discontinuities can occur when there is an abrupt change in track stiffness, misalignment, a change in soil properties, etc. For example, bridge abutments and road
crossings can lead to track discontinuities. Such changes, if they are in the vertical
direction, can cause a significant pitch and bounce motion of the vehicle. If the
discontinuities are in the lateral direction, yaw and sway motion can be generated.
Discontinuities, in general, lead to high impulsive tangential and normal forces that
can cause derailment.
Turnouts (switches and crossings) are used to change the travel direction of the
train and are among several factors that can cause motion discontinuities (Kassa et
al., 2006; Schupp et al., 2004). In general, the turnout is constructed using a switch
panel (point section), a crossing panel (crossing section), and a closure panel (lead
rail) that connects the switch panel with the crossing panel, as shown in Figure 1.23.
For high-speed trains, a movable swing nose is used (Kono et al., 2005). As the
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Introduction
33
FIGURE 1.23 Rail turnouts.
wheel contact switches from the stock rail to the tongue rail in the switch panel or
the crossing panel, multiple wheel/rail contacts are possible, as shown in Figure 1.23.
These contacts lead to severe impact forces. Furthermore, as the tongue rail changes
its shape along the track, the wheel moves from the stock rail to the tongue rail.
This transition causes a large disturbance in the wheel motion, and if the tongue rail
is not close enough to the rail, derailment may even occur. The simulation of turnout
crossings is often required in accident investigations. In this case, a method that can
be used to describe the change of the rail profile as a function of the rail longitudinal
distance must be adopted. The general-purpose contact formulations presented in
Chapter 5 can be used in the simulation of such scenarios.
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2
Dynamic Formulations
Two approaches are often used to formulate the dynamic equations of motion of
mechanical systems: the Newtonian and the Lagrangian approaches. In the Newtonian approach, vector mechanics is used to develop the dynamic equations. If the
system consists of bodies connected by mechanical joints, free-body diagrams are
constructed to show the reaction joint forces as well as the inertia and applied forces.
The Newtonian approach, in which the equilibrium of each body is first studied
separately, can be used for relatively simple systems and is not suited for the analysis
of complex systems such as railroad vehicles. In the Lagrangian approach, which
is based on D’Alembert’s principle, on the other hand, scalar quantities such as the
virtual work and the kinetic and potential energies can be used to develop the body
equations of motion. In this case, there is no need to study the equilibrium of the
bodies in the system separately. The Lagrangian approach, which is based on the
known Lagrange-D’Alembert equation, can be used to systematically eliminate the
constraint forces or keep these forces in the final form of the equations of motion.
The concept of the generalized coordinates is fundamental in the Lagrangian
formulation of the equations of motion. Recall that the unconstrained motion of a
rigid body can be described using six coordinates; three coordinates are used to
describe the translation of a reference point on the body, and three coordinates are
used to define the orientation of the rigid body in space. As will be shown in later
sections of this chapter, the order of the finite rotation of rigid bodies is not commutative, and the components of the angular velocity vector of the rigid body are
not, in general, the time derivatives of a set of orientation coordinates. Two different
parameterizations of the finite rotation are commonly used in the analysis of multibody systems; the first is a set of three Euler angles, and the second is a set of four
Euler parameters. In the case of the Euler angle representation, the orientation of a
rigid body is defined using three successive rotations, while four parameters are used
in the Euler parameter representation to avoid singularities encountered when three
independent orientation parameters are used. In this book, the set of Cartesian translation and orientation coordinates is called the set of absolute generalized coordinates.
Other sets of generalized coordinates can also be selected, as will be discussed
elsewhere in this book.
This chapter presents general methods that can be used in the computer formulation of the equations of motion of multibody systems consisting of rigid bodies. The
generalized coordinates that define the global position of the origin and the orientation of a selected body reference frame are introduced and used to define the
relationships between the angular velocity vector and the time derivatives of the
generalized orientation coordinates. Expressions for the global velocity and acceleration vectors of an arbitrary point on the body are developed and used to derive
the equations of motion in the three-dimensional space. When the absolute Cartesian
35
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36
Railroad Vehicle Dynamics: A Computational Approach
generalized coordinates are used, one obtains the Newton-Euler equations of motion,
in which there is no inertia coupling between the translational and rotational coordinates of the rigid body. In railroad vehicle systems, the absolute Cartesian coordinates
that define the configurations of the vehicle components can be related because of
mechanical joints or specified motion trajectory constraints. This chapter discusses
the formulation of the algebraic joint constraint equations. Using these constraint
equations, two different methods for formulating the equations of motion can be
used. In the first method, the augmented formulation, the constraint equations augment the system differential equations of motion, leading to a large system that has
a sparse matrix structure. In the second method, the embedding technique, the
constraint equations are used to systematically eliminate the dependent variables
and the constraint forces, leading to a smaller system of dynamic equations expressed
in terms of the system degrees of freedom. Both the augmented formulation and the
embedding technique are discussed in this chapter. The trajectory coordinates, which
are used to develop special-purpose computational algorithms for railroad vehicle
systems, are briefly discussed in this chapter. A more detailed discussion of these
coordinates is presented in Chapter 7.
2.1 GENERAL DISPLACEMENT
In the three-dimensional analysis, the unconstrained motion of a rigid body is
described using six independent generalized coordinates: three independent coordinates define the translation of a selected reference point on the body, and three
coordinates define the body orientation. For an arbitrary rigid body i, as the one
shown in Figure 2.1, the translational motion can be defined using the global position
of the reference point O i that is fixed on the body, while the orientation of the body
FIGURE 2.1 Coordinates of rigid body i.
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Dynamic Formulations
37
can be defined using the direction cosines of the axes of the body coordinate system
XiY iZ i. Using this description, the global position vector of an arbitrary point on the
rigid body i can be written as
ri = R i + A i u i
(2.1)
where Ri is the global position vector of the origin of the body coordinate system
defined as
R i = [ R xi
Ryi
Rzi ]T
(2.2)
and Ai is a 3 × 3 rotation matrix that defines the orientation of the axes of the body
coordinate system with respect to the global coordinate system. The vector u i is the
position vector of the arbitrary point on the body with respect to the origin of the
body coordinate system and is defined in terms of its components as
u i = u xi
uyi
T
uzi  = [ x i
yi
z i ]T
(2.3)
In this equation, u xi = x i , uyi = y i , and uzi = z i are the local coordinates of the arbitrary
point defined in the body coordinate system. In rigid body dynamics, the local
position vector u i is assumed to be constant and does not depend on time. The
location of the origin of the body coordinate system (reference point) can be selected
arbitrarily. Nonetheless, it is advantageous to have a centroidal body coordinate
system, which has an origin that is rigidly attached to the center of mass of the body.
The use of such a centroidal body coordinate system leads to the well-known
Newton-Euler equations, which do not include inertia coupling between the translation and rotation of the body.
2.2 ROTATION MATRIX
This section discusses several methods for defining the orientation matrix Ai given
in Equation 2.1. In particular, three methods are addressed: the direction cosines,
Euler angles, and Euler parameters. The method of the direction cosines is rarely
used in computational dynamics, since it requires the use of nine parameters that
are not independent. In this case, six constraint equations must be imposed. The
method of Euler angles, which is widely used, employs three independent parameters
that can be used to define the body orientation in space. This method, however,
suffers from a singularity problem at certain configurations, and for this reason, the
method of Euler parameters, which employs four coordinates to define the body
orientation in space, is often used. In this case, one must impose one algebraic
constraint equation, since only three independent parameters are required to describe
general three-dimensional rotations.
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38
Railroad Vehicle Dynamics: A Computational Approach
2.2.1 DIRECTION COSINES
Let ii, ji, and ki be unit vectors along the Xi, Y i, and Z i axes of the body coordinate
system, respectively. The components of these three vectors can be defined in the
global coordinate system. These unit vectors can be written in terms of their components along the unit vectors i, j, and k along the X, Y, and Z axes of the global
coordinate system as follows:
i
i
i
ii = α11
i + α12
j + α13
k 


i
i
i
ji = α 21
i + α 22
j + α 23
k

i
i
i
i + α 32
j + α 33
k 
ki = α 31
(2.4)
The elements αij represent the components of the orthogonal unit vectors ii, ji, and
ki along the respective X, Y, and Z axes of the global coordinate system. Since the
unit vectors i, j, and k are also orthogonal, the elements αij can be defined using
the preceding equation as follows:
i
α11
= ii ⋅ i
i
α12
= ii ⋅ j
i
α13
= ii ⋅ k
i
α 21
= ji ⋅ i
i
α 22
= ji ⋅ j
i
α 23
= ji ⋅ k
i
α 31
= ki ⋅ i
i
α 32
= ki ⋅ j
i
α 33
= ki ⋅ k







(2.5)
where the nine scalar components αij (i, j = 1, 2, 3) are called the direction cosines.
Let ui be a vector whose components are defined in the body coordinate system
along the axes Xi, Y i, and Z i by the scalar coordinates xi, yi, and zi. This vector can
then be written as follows:
u i = x i ii + y i ji + z i k i
(2.6)
Substituting Equation 2.4 into Equation 2.6, one obtains
u i = u xi i + uyi j + uzi k
(2.7)
i i
i i
i i 
uxi = α11
x + α 21
y + α 31
z


i
i
u iy = α12
x i + α 22
yi + α 3i 2 z i 

i i
i
i i 
uzi = α13
x + α 23
yi + α 33
z 
(2.8)
where
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Dynamic Formulations
39
The preceding equations can also be written in a matrix form as follows:
ui = Aiui
(2.9)
where
u i = [u xi
uyi
uzi ]T
ui = [ x i
yi
z i ]T
and
i
α11

i
Ai = α12
 i
α13
i 
 ii ⋅ i
α 31

 i
i
α 32
 = i ⋅ j
i 
i
α 33
 i ⋅ k
i
α 21
i
α 22
i
α 23
ji ⋅ i
ji ⋅ j
ji ⋅ k
ki ⋅ i 

ki ⋅ j 
ki ⋅ k 
(2.10)
Since the components of the unit vectors i, j, and k along the X, Y, and Z axes of
the global coordinate system are simply defined by
i = [1
0
0]T ,
j = [0
0]T ,
1
k = [0
0
1]T ,
(2.11)
the matrix Ai given by Equation 2.10 can be rewritten simply as
A i = [ii
ji
ki ]
(2.12)
It is clear from this equation that the columns of the transformation matrix Ai are
ii, ji, and ki, which define unit vectors along the Xi, Y i, and Z i axes of the body
coordinate system, and the elements of the matrix Ai are the direction cosines that
define the components of these unit vectors in the global coordinate system.
The analysis presented in this section shows that the transformation matrix that
relates the components of vectors defined in the body coordinate system to their
global components can be constructed using the nine direction cosines αij (i, j =
1, 2, 3). However, these nine direction cosines are not independent due to the fact
that three independent parameters are sufficient to describe the orientation of the
rigid body in space. Since the direction cosines represent the components of three
orthogonal unit vectors defined in the global coordinate system, the following orthogonality condition must be satisfied by the elements of the transformation matrix Ai:
T
T
Ai Ai = Ai Ai = I
(2.13)
where I is a 3 × 3 identity matrix. The preceding equation leads to a total of six
independent constraint equations that impose the orthonormality (orthogonal unit
vectors) conditions on the vectors ii, ji, and ki, leading to only three independent
orientation parameters.
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Railroad Vehicle Dynamics: A Computational Approach
EXAMPLE 2.1
The Xi and Y i axes of the coordinate system of a rigid body i are given in the global
coordinate system by the vectors [0.0 1.0 1.0]T and [−1.0 1.0 −1.0]T, respectively.
Determine the transformation matrix that defines the orientation of body i in the global
system.
Solution. The unit vectors along the X i and Y i axes can be obtained as follows:
i =
i
 0.0   0.0 
  

1.0  =  0.7071
2
2 
(1.0) + (1.0)   
1.0   0.7071
1
and
j =
i
 −1.0   −0.5774 

 

1.0  =  0.5774 
2
2
2 
(−1.0) + (1.0) + (−1.0) 
 −1.0   −0.5774 
1
The unit vector along the Z i axis must be perpendicular to both vectors ii and ji, and
the direction of this vector is determined using the right-hand rule. This unit vector
can be determined using the following cross-product:
ki =
i i × ji
ii × ji
Note that, since the vectors ii and ji are unit vectors and mutually orthogonal, the scalar
ii × ji is equal to 1. It follows that the unit vector ki along the Z i axis is given by
 −0.8165 


k =  −0.4082 
 0.4082 
i
Using the orthogonal triad ii, ji, and ki, the transformation matrix of the coordinate
system of body i can be determined as
A i = [i i
ji
 0.0

−0.5774
−0.8165 
0.5774
 0.7071
−0.5774
−0..4082 
k i ] =  0.7071

0.4082 
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Dynamic Formulations
41
2.2.2 SIMPLE ROTATIONS
The transformation matrix as the result of simple finite rotations of the coordinate
system XiY iZ i about the axes of the global coordinate system XYZ can be obtained
as a special case of the direction cosines transformation matrix. For example, since
the columns of the transformation matrix Ai are unit vectors along the Xi, Y i, and Z i
axes of the body coordinate system, a simple rotation θz of the coordinate system
XiY iZ i about the global Z axis leads to the following global definition of the unit
vectors along the Xi, Y i, and Z i axes:
 cos θ z 


i =  sin θ z  ,
 0 


i
 − sin θ z 
0 


 
i
j =  cos θ z  , k =  0 


1
 0 
 
i
(2.14)
Using these three orthogonal unit vectors, the transformation matrix Ai as the result
of the rotation θz can be defined as
 cos θ z

A =  sin θ z
 0

i
− sin θ z
cos θ z
0
0

0
1 
(2.15)
Similarly, a rotation θy of the coordinate system XiY iZ i about the global Y axis leads
to the following rotation matrix:
 cos θ y

A = 0
 − sin θ
y

i
0
1
0
sin θ y 

0 
cos θ y 

(2.16)
while a rotation θx about the global X axis leads to the following rotation matrix:
1

A = 0
0

i
0
cos θ x
sin θ x
0 

− sin θ x 
cos θ x 
(2.17)
The transformation matrices given in Equations 2.15 to 2.17 for the simple
rotations can be used to develop the transformation matrix in the case of moregeneral three-dimensional rotations using the method of Euler angles.
2.2.3 EULER ANGLES
In the method of Euler angles, three simple successive rotations are used to define
the orientation of the rigid body in space. Using the three Euler angles associated
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42
Railroad Vehicle Dynamics: A Computational Approach
with three independent axes, the orientation matrix Ai given in Equation 2.1 can be
defined as the product of three simple rotation matrices as follows:
A i = A1i A 2i A 3i
(2.18)
where A ik (k = 1, 2, 3) are, respectively, the rotation matrices defined in terms of the
following three Euler angles:
θ i = [θ1i
θ 2i
θ 3i ]T
(2.19)
One can choose an appropriate sequence of the three successive rotations to
reach any orientation in space. For example, the orientation of the wheel shown in
Figure 2.2 can be conveniently described using the following three successive rotations:
θ i = ψ i
φi
θ i 
T
(2.20)
Let XYZ and XiY iZ i be, respectively, the global and the wheel-body coordinate
systems, which initially coincide. Rotation of the wheel-body coordinate system
XiY iZ i by an angle ψ i (yaw) about the Z i axis leads to the rotation matrix
 cos ψ i

A1i =  sin ψ i
 0

− sin ψ i
cos ψ i
0
0

0
1 
(2.21)
A second rotation φ i (roll) of the wheel coordinate system XiY iZ i about the Xi axis
leads to the rotation matrix
1

Ai2 =  0
0

FIGURE 2.2 Euler angles.
0
cos φ i
sin φ i
0 

− sin φ i 
cos φ i 
(2.22)
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Dynamic Formulations
43
Finally, the wheel coordinate system XiY iZ i is rotated by an angle θ i (pitch) about
the Y i axis. This rotation defines the transformation matrix
 cos θ i

A = 0
 − sin θ i

i
3
0
1
0
sin θ i 

0 
cos θ i 
(2.23)
Using the preceding three simple rotations, the orientation of the wheel can be
defined in the global coordinate system by substituting Equations 2.21 to 2.23 into
Equation 2.18 to obtain the following transformation matrix:
Ai =
 cos ψ i cos θ i − sin ψ i sin φ i sin θ i

i
i
i
i
i
sin ψ cos θ + cos ψ sin φ sin θ
i
i

− cos φ sin θ

(2.24)
− sin ψ i cos φ i
cos ψ i cos φ i
sin φ i
cos ψ i sin θ i + sin ψ i sin φ i cos θ i 

sin ψ i sin θ i − cos ψ i sin φ i cosθ i 

cos φ i cos θ i

As previously mentioned, the three column vectors of the matrix Ai define unit
vectors along the Xi, Y i, and Z i axes of the body coordinate system. The components
of these three unit vectors are defined in the global coordinate system. It is important
to point out that, in general, the order of the finite rotations in the three-dimensional
analysis is not commutative, that is, A1i A 2i A 3i ≠ A 3i A i2 A1i .
One of the major drawbacks of using three independent parameters such as Euler
angles is the existence of singular configurations that result when the three Euler angle
axes of rotation become dependent (Roberson and Schwertassek, 1988; Shabana,
2001). For the sequence of Euler angles used in this section, the singular configurations occur when the angle φ i is equal to ±π/2 (−π < φ i ≤ π). In such a case, the axes
of rotation associated with the angles ψ i and θ i are parallel, as shown in Figure 2.3,
and therefore, the angle ψ i cannot be distinguished from the angle θ i. A similar
singularity problem is encountered when using any known method that employs
three parameters to describe the orientation of the rigid body in the three-dimensional
FIGURE 2.3 Singular configuration of Euler angles.
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Railroad Vehicle Dynamics: A Computational Approach
space; therefore, all Euler angle representations suffer from the singularity problem.
As will be discussed in the next section, at the singular configurations, the time
derivatives of the three Euler angles cannot be defined in terms of the components
of the angular velocity vector.
EXAMPLE 2.2
Obtain the transformation matrix in terms of Euler angles if the sequence of rotation
is defined as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis,
and a rotation ψ i about the Z i axis.
Solution. The rotation matrix resulting from the angle φ i about the Z i axis is given as
 cos φ i

A1i =  sin φ i
 0

− sin φ i
0
cos φ
0

i
1 
0
The rotation matrix resulting from the angle θ i about the Xi axis is given as
1

A i2 =  0
 0


0
0
cos θ i
− sin θ i 
cos θ i 
sin θ i
Finally, the rotation matrix resulting from the angle ψ i about the Z i axis is given as
 cos ψ i

A i3 =  sin ψ i
 0

− sin ψ i
0
cos ψ
0
0
i

1 
Using the simple rotation matrices obtained in this example for the three successive
rotations, the rotation matrix that defines the orientation of the body in the global
coordinate system is given by
A i = A1i A i2 A i3
 cos ψ i cos φ i − cos θ i sin φ i sin ψ i

=  cos ψ i sin φ i + cos θ i cos φ i sin ψ i

sin θ i sin ψ i

− sin ψ i cos φ i − cos θ i sin φ i cos ψ i
sin θ i sin φ i 
− sin ψ i sin φ i + cos θ i cos φ i cos ψ i
− sin θ i cos φ i 
sin θ i cos ψ i

cos θ i
One can show that the preceding transformation matrix is orthogonal, that is, AiT Ai =
Ai AiT = I.


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45
2.2.4 EULER PARAMETERS
In order to avoid the singularity problem associated with the three-parameter representation, the four Euler parameters are often used in the computer-aided analysis
of multibody systems. As shown in Figure 2.4, the change in the orientation of an
arbitrary vector r i can be defined using the three components of a unit vector vi
along the instantaneous axis of rotation and the angle of rotation θ i about this axis.
Using these four parameters, the vector ri, shown in Figure 2.4, can be obtained by
transforming the vector r i using a rotation matrix that is a function of the rotation
θ i and the three components of the unit vector vi, as follows:
ri = A i r i
(2.25)
where the transformation matrix Ai is given as (Roberson and Schwertassek, 1988;
Shabana, 2005)
Ai = I + v i sin θ i + 2(v i )2 sin 2
θi
2
(2.26)
where I is a 3 × 3 identity matrix, and v i is a skew-symmetric matrix associated
with the unit vector vi and is defined as
 0

i
v =  v3i
− vi
 2
FIGURE 2.4 Rodriguez formula.
− v3i
0
v1i
v2i 

− v1i 
0 
(2.27)
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Railroad Vehicle Dynamics: A Computational Approach
where v1i , v2i , and v3i are the components of the unit vector vi. Equation 2.26 is known
as the Rodriguez formula. Since the components of the unit vector vi must satisfy
( v1i )2 + ( v2i )2 + ( v3i )2 = 1
(2.28)
the rotation matrix is function of only three independent parameters.
To demonstrate the use of Equation 2.26, a simple rotation about the global Z
axis is considered. In this case, vi = [0 0 1]T and the transformation matrix defined
by Equation 2.26 leads to the following simple rotation matrix:
1

i
A = 0
0

0
1
0
 cos θ i

=  sin θ i
 0

0  0
 
0  + sin θ i
1   0
0   cos θ i − 1
 
0
0 + 


0
0 
− sin θ i
0
0
0

0
1 
− sin θ i
cos θ i
0
0
cos θ i − 1
0
0

0
0 
(2.29)
This is the matrix previously obtained using the direction cosines.
The four Euler parameters are defined using the angle of rotation θ i and the
three components of the unit vector vi as follows:
θ 0i = cos
θi 

2

θi 
i
i
θ 3 = v3 sin 
2
θi
,
2
θ1i = v1i sin
θi
θ = v sin ,
2
i
2
i
2
(2.30)
Using these four parameters, the generalized orientation coordinates can be defined
as
θ i = [θ 0i
θ1i
θ 2i
θ 3i ]T
(2.31)
These four Euler parameters are not totally independent, since, as previously mentioned, the orientation of the body in space can be defined using only three independent coordinates. By using Equations 2.28 and 2.30, it can be shown that the
four Euler parameters must satisfy the following condition:
3
∑ (θ )
i 2
k
k =0
=1
(2.32)
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Dynamic Formulations
47
Substituting the expressions of the four Euler parameters given by Equation 2.30
into Equation 2.26, the orientation matrix Ai can be rewritten in terms of Euler
parameters as
Ai = I + 2θ is (θ 0i I + θ is )
(2.33)
where θ is = [θ1i θ 2i θ 3i ]T and θ is is the skew-symmetric matrix associated with the
vector θ is . The preceding equation can be written more explicitly as
1 − 2(θ 2i )2 − 2(θ 3i )2

Ai =  2(θ1iθ 2i + θ 0i θ 3i )
 2(θ iθ i − θ i θ i )
1 3
0 2

2(θ1iθ 2i − θ 0i θ 3i )
1 − 2(θ1i )2 − 2(θ 3i )2
2(θ 2iθ 3i + θ 0i θ1i )
2(θ1iθ 3i + θ 0i θ 2i ) 

2(θ 2i θ 3i − θ 0i θ1i )  (2.34)
1 − 2(θ1i )2 − 2(θ 2i )2 
Using Equation 2.32, Equation 2.34 can also be rewritten as
 2(θ 0i )2 + 2(θ1i )2 − 1

A =  2(θ1iθ 2i + θ 0i θ 3i )
 2(θ iθ i − θ i θ i )
1 3
0 2

2(θ1iθ 2i − θ 0i θ 3i )
2(θ 0i )2 + 2(θ 2i )2 − 1
2(θ 2iθ 3i + θ 0i θ1i )
i
2(θ1iθ 3i + θ 0i θ 2i ) 

2(θ 2i θ 3i − θ 0i θ1i )  (2.35)
2(θ 0i )2 + 2(θ 3i )2 − 1
Using the method of Euler parameters, the singularity problem associated with
the three-parameter representation is eliminated at the expense of adding an algebraic
constraint equation (Equation 2.32).
EXAMPLE 2.3
Use the Rodriguez formula to determine the transformation matrix of body i resulting
from a rotation θ i = π/3 about the vector [−2.0 1.0 5.0]T defined in the global coordinate
system. Determine the four Euler parameters associated with this rotation.
Solution. A unit vector along the axis of rotation is defined as
v =
i
 −2.0   −0.3651

 

1.0  =  0.1826 
2
2
2 
(−2.0) + (1.0) + (5.0) 
 5.0   0.9129 
1
Using the Rodriguez formula given by Equation 2.26, the transformation matrix is
obtained as
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Railroad Vehicle Dynamics: A Computational Approach
A i = I + v i sin θ i + 2(v i ) 2 sin 2
1

= 0
 0
0 
0
θi
2
1
0
 −0.8667

+2.0  −0.0667
 −0.3333

0.3651 sin
0
−0.3651
0
−0.0667
−0.3333 
−0.9667
0.1667  sin 2

−0.1667 
0.1667
 0.5667

−0.8239
−0.0086 
0.5167
 −0.3248
−0.2329
0.3996 
0.9167 
=  0.7572
0.1826 
−0.9129
0
 
0  +  0.9129
1   −0.1826

π
3
π
6

The four Euler parameters are determined using Equation 2.30 as
θ 0i = cos
θi
π
= cos = 0.8660
2
6
θ1i = v1i sin
π
θi
= −0.3651sin = −0.1826
2
6
θ 2i = v2i sin
θi
π
= 0.1826 sin = 0.0913
2
6
θ 3i = v3i sin
θi
π
= 0.9129 sin = 0.4564
2
6
The transformation matrix in terms of Euler parameters is determined using Equation
2.34 as
1 − 2(θ 2i ) 2 − 2(θ 3i ) 2

A i =  2(θ1iθ 2i + θ 0i θ 3i )
 2(θ1iθ 3i − θ 0i θ 2i )

 0.5667

=  0.7572
 −0.3248
2(θ1iθ 2i − θ 0i θ 3i )
1 − 2(θ1i ) 2 − 2(θ 3i ) 2
2(θ 2iθ 3i + θ 0i θ1i )
−0.8239
−0.0086 
0.5167
0.3996 
0.9167 
−0.2329
2(θ1iθ 3i + θ 0i θ 2i ) 

2(θ 2iθ 3i − θ 0i θ1i ) 
1 − 2(θ1i ) 2 − 2(θ 2i ) 2 

This transformation matrix is the same as the one previously obtained in this example
using the Rodriguez formula.
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Dynamic Formulations
49
2.3 VELOCITIES AND ACCELERATIONS
The global position vector of an arbitrary point on body i is defined, as previously
shown by Equation 2.1, as the sum of two displacement components. The first is
due to the translation of the origin of the body coordinate system, while the second
is due to the rotation of the body. Using this general displacement, one can derive
expressions for the absolute velocity and acceleration vectors of an arbitrary point
on the body.
2.3.1 VELOCITY VECTOR
The global velocity vector of an arbitrary point on the rigid body can be obtained
by differentiating Equation 2.1 with respect to time. This leads to
iui
r i = R i + A
(2.36)
In rigid body dynamics, the components of the vector u i are constant. The preceding
equation can also be written in the following form:
i + ω i × ui
r i = R
(2.37)
In this equation,
ω i = ω 1i
ω 2i
ω 3i 
T
(2.38)
is the absolute angular velocity vector defined in the global coordinate system. The
vector ui is defined as
ui = Aiui
(2.39)
·
The vector Ri is the global velocity of the reference point Oi, while the vector ωi ×
ui is the result of the rigid body rotation. Since the vectors ωi and ui can also be defined
in the body coordinate system, the absolute velocity vector of Equation 2.37 can be
written in the following alternative form:
i + Ai (ω i × u i )
r i = R
(2.40)
where ω i is the absolute angular velocity vector defined in the body coordinate
system and is related to the angular velocity defined in the global coordinate system
by the following transformation:
ω i = Ai ω i
(2.41)
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Using the orthogonality property of the orientation matrix given by Equation
2.13 and comparing Equations 2.36 and 2.40, one can show that
T
i
ω i = Ai A
(2.42)
where ω i is the skew-symmetric matrix associated with the vector ω i and is defined
as
 0

i
ω =  ω 3i
 −ω i
 2
ω 2i 

−ω 1i 
0 
−ω 3i
0
ω 1i
(2.43)
and ω 1i , ω 2i , and ω 3i are the components of the angular velocity vector defined in the
body coordinate system, that is,
ω i = ω 1i
ω 2i
ω 3i 
T
(2.44)
Similarly, one can write the following identity for the angular velocity defined
in the global coordinate system:
i Ai T
ω i = A
(2.45)
where
 0

ω =  ω 3i
 −ω i
 2
i
−ω 3i
0
ω 1i
ω 2i 

−ω 1i 
0 
(2.46)
It is important to note from Equation 2.45 that the components of the angular velocity
vector in the three-dimensional analysis cannot be defined, in general,
as the time
·
derivatives of the generalized orientation coordinates. That is, ωi ≠ θi, and the angular
velocity vector, therefore, cannot be integrated directly to obtain the orientation
coordinates.
2.3.2 ACCELERATION VECTOR
The absolute acceleration vector can be obtained by differentiating Equation 2.36
with respect to time. This yields
i + A
i u i
ri = R
(2.47)
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51
The acceleration vector can also be written in the following form:
i + α i × u i + ω i × (ω i × u i )
ri = R
(2.48)
where α i = [α1i α 2i α 3i ]T is the angular acceleration vector of body i defined in the
global coordinate system. The vector αi is the time derivative of the angular velocity
vector ωi. The acceleration term αi × ui on the right-hand side of Equation 2.48 is
ωi × ui) is called the normal component,
called the tangential component, while ωi × (ω
which is also known as the centripetal acceleration.
If the vectors αi, ωi, and ui are defined in the body coordinate system, Equation
2.48 can be written in the following alternative form:
{
(
i + Ai α i × u i + ω i × ω i × u i
ri = R
)}
(2.49)
where
α i = Ai α i
(2.50)
It is more convenient in some numerical algorithms to formulate the Newton-Euler
equations in terms of α i instead of αi, since the inertia coefficients associated with
α i are constant in the case of rigid body analysis.
2.3.3 GENERALIZED ORIENTATION COORDINATES
While the angular velocity vector cannot be directly integrated to obtain the orientation coordinates, the absolute angular velocity and acceleration vectors can be
written in terms of the time derivatives of the generalized coordinates. Using Equations 2.42 and 2.45, the angular velocity vector can be written in terms of the time
derivatives of the orientation generalized coordinates as follows:
ω i = Giθ i ,
ω i = Giθ i
(2.51)
where the coefficient matrices Gi and Gi are expressed in terms of the orientation
parameters θi. For example, the matrices Gi and Gi , in the case of the Euler angles
θi = [ψ i φ i θ i]T based on the sequence Z i,Xi,Y i, can be obtained as
0

G = 0
1

i
cos ψ i
sin ψ i
0
− sin ψ i cos φ i 

cos ψ i cos φ i  ,

sin φ i

 − cos φ i sin θ i

G =
sin φ i
 cos φ i cos θ i

i
cos θ i
0
sin θ i
0

1  (2.52)
0 
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Using Equations 2.51 and 2.52, the angular velocity vector ωi defined in the
global coordinate system can be written in the following vector form:
ω i = Giθ i = g1i ψ i + g i2 φ i + g 3i θ i
(2.53)
where
g1i =  0
0
g i2 =  cos ψ i
1
T
sin ψ i
g 3i =  − sin ψ i cos φ i
0 
T
cos ψ i cos φ i






T
i 
sin φ  

(2.54)
In Equation 2.53, the absolute angular velocity vector is written as a linear combination of three angular velocity vectors associated with ψ i , φ i , and θ i . The vectors
g ik (k = 1, 2, 3) represent unit vectors defined in the global coordinate system about
which the three successive rotations are performed. Similarly, the columns of the
matrix Gi define unit vectors in the body coordinate system along the axes about
which the Euler angles are performed. That is,
Gi = A i Gi
(2.55)
The matrices Gi and Gi that appear in Equation 2.51 will be repeatedly used in
this book, and these matrices also appear in the formulation of the generalized forces
and the Jacobian matrix of the kinematic constraint equations.
EXAMPLE 2.4
Determine the matrices Gi and G i in terms of Euler angles if the sequence of rotation
is defined as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis,
and a rotation ψ i about the Z i axis.
Solution. Using Equation 2.51, the absolute angular velocity vector defined in the
global coordinate system can be written as
ω i = G i θ i = g1i φ i + g i2 θ i + g 3i ψ i
where θi = [φ i θ i ψ i ]T and g ik (k = 1, 2, 3) are the columns of the matrix Gi that
represent unit vectors defined in the global coordinate system along which the rotations
φ i, θ i, and ψ i are performed, respectively. Therefore, the vector g1i is a unit vector along
the Z i axis of the body coordinate system at the initial configuration, that is,
g1i =  0
0
1
T
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53
The vector g i2 is a unit vector along the Xi axis of the body coordinate system after the
first rotation φ i is performed, that is,
g i2 =  cos φ i
0 
sin φ i
T
The vector g i3 is a unit vector along the Z i axis of the body coordinate system after the
first and second rotations φ i and θ i are performed. That is,
g i3 =  sin θ i sin φ i
cos θ i 
− sin θ i cos φ i
T
Accordingly, the matrix Gi is obtained as
0

i
G = 0
1

cos φ i
sin θ i sin φ i 
sin φ i
− sin θ i cos φ i 



cos θ i
0
On the other hand, the absolute angular velocity vector defined in the body coordinate
system can be written as
ω i = G i θ i = g1i φ i + g2i θ i + g3i ψ i
where gki (k = 1, 2, 3) are the columns of the matrix G i that represent unit vectors
defined in the body coordinate system along which the rotations φ i, θ i, and ψ i are
performed, respectively. Hence, the vectors g1i , g2i , and g3i are given as follows:
g1i =  sin θ i sin ψ i
cos θ i 
sin θ i cos ψ i
g2i =  cos ψ i
g3i =  0
0 
− sin ψ i
0
1
T
T
T
It follows that the matrix G i is defined as
 sin θ i sin ψ i

G i =  sin θ i cos ψ i
 cos θ i

0
cos ψ i
− sin ψ
0
i

0
1 
2.3.4 SINGULAR CONFIGURATION
As previously discussed, a singularity problem is encountered when Euler angles
are used in the description of the three-dimensional rotations. For example, the
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54
Railroad Vehicle Dynamics: A Computational Approach
matrices Gi and Gi given by Equation 2.52 in terms of Euler angles become singular
when the angle φ i is equal to ±π/2. In this case, the rotations ψ i and θ i are performed
about two parallel axes, and distinction between these two angles cannot be made,
as shown in Figure 2.3. As a consequence of the singularity of the matrices Gi
and Gi , the time derivatives of Euler angles cannot be expressed in terms of the
components of the angular velocity vector.
Euler parameters, on the other hand, do not lead to singular configurations. In
the case of Euler parameters, the matrices Gi and Gi are defined as (Nikravesh, 1988;
Shabana, 2001)
 −θ1i

Gi = 2  −θ 2i
 −θ i
 3
θ 0i
θ 3i
−θ 2i
−θ 3i
θ 0i
θ1i
θ 2i 

−θ1i  ,
θ 0i 
 −θ1i

Gi = 2  −θ 2i
 −θ i
 3
θ 0i
−θ 3i
θ 2i
θ 3i
θ 0i
−θ1i
−θ 2i 

θ1i  (2.56)
θ 0i 
One can show that the orientation matrix given by Equation 2.33 can be expressed
in terms of the preceding two matrices Gi and Gi as
Ai =
1 i iT
GG
4
(2.57)
Furthermore, the matrices Gi and Gi , which are linear in Euler parameters, satisfy
several interesting identities that can be utilized in developing the numerical algorithm. Some of these identities are summarized in Table 2.1.
EXAMPLE 2.5
Discuss the singularity problem of Euler angles if the sequence of rotation is defined
as follows: a rotation φ i about the Z i axis, a rotation θ i about the Xi axis, and a rotation
ψ i about the Z i axis.
Solution. Using the matrices Gi and G i obtained in Example 2.4 in terms of Euler
angles for the sequence Z i,Xi,Z i, one can show that the determinant of these matrices
is given as
det G i = det G i = − sin θ i
The singularity is encountered when the angle θ i is equal to zero or π (−π < θ i ≤ π).
In such a case, the axes of rotation of the angles φ i and ψ i are parallel; therefore, at
this singular configuration, one cannot distinguish between these two angles.
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Dynamic Formulations
55
TABLE 2.1
Identities of Euler Parameters
Absolute
 −θ1i

G = 2  −θ 2i
 −θ i
 3
i
θ 0i
θ 3i
−θ 2i
Local
θ 2i 

−θ1i 
θ 0i 
−θ 3i
θ 0i
θ1i
 −θ1i

G = 2  −θ 2i
 −θ i
 3
i
ω i = Gi θ i
i
α i = Gi θ
i
α i = Gi θ
Gi θ i = 0
Gi θ i = 0
i θ i = 0
G
Gi θ i = 0
T
Gi Gi = 4I
(
Gi Gi = 4 I4 − θ i θ i
−θ 2i 

θ1i 
θ 0i 
θ 3i
θ 0i
−θ1i
ω i = Gi θ i
T
Gi Gi = 4I
T
θ 0i
−θ 3i
θ 2i
T
)
T
(
Gi Gi = 4 I4 − θ i θ i
T
θ i = 41 Gi ω i
T
)
T
θ i = 41 Gi ω i
Ai = 41 Gi Gi
T
G i = Ai G i
T
θ i θ i = 0
3
∑ (θ )
i 2
k
=1
k =0
Using Equation 2.51, which is valid for any set of orientation parameters, the
global velocity vector defined by Equations 2.37 and 2.40 can, respectively, be
expressed in terms of the time derivatives of the generalized coordinates as
i − u iGiθ i
r i = R
(2.58)
i − Ai u iGiθ i
r i = R
(2.59)
and
where u i and u i are the skew symmetric matrices associated with the vectors ui and
u i , respectively. Furthermore, the absolute acceleration vector defined by Equations
2.48 and 2.49 in the global coordinate system can also be expressed in terms of the
time derivatives of the generalized coordinates as
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Railroad Vehicle Dynamics: A Computational Approach
i − u iGiθ
i + a i
ri = R
v
(2.60)
i − Ai u iGiθ
i + a i
ri = R
v
(2.61)
iθ i ]
iθ i = Ai [(ω i )2 u i − u iG
a iv = (ω i )2 u i − u iG
(2.62)
and
where
iθ i are identically zero when Euler parameters are
iθ i and G
One can show that G
used. However, this is not the case when Euler angles are used.
EXAMPLE 2.6
The orientation of a rigid body i is defined by the following Euler parameters:
θ i =  0.8660
0.2182 
−0.4364
0.1091
T
At this configuration, the absolute angular velocity vector in the global coordinate
system is given by
ω i = 10.0
1.0
3.0 
T
Find the time derivatives of Euler parameters.
Solution. The absolute angular velocity vector is defined in terms of the time derivatives
of Euler parameters using the equation ω i = G i θ i . However, since the matrix Gi given
by Equation 2.56 is a 3 × 4 matrix in the case of Euler parameters, one cannot directly
take an inverse of this matrix to write the time derivatives of the Euler parameters in
terms of the components of the angular velocity vector. Alternatively, one can premultiply both sides of the equation ω i = G i θ i by the transpose of the matrix Gi and
use the identity
T
(
T
Gi Gi = Gi Gi = 4 I4 − θ iθ i
T
)
presented in Table 2.1 (Nikravesh, 1988; Shabana, 2001). By doing this, one can show
that
(
)
T
T
T
G i ω i = G i G i θ i = 4 I 4 − θ i θ i θ i = 4θ i
T
where I4 is the 4 × 4 Tidentity matrix, and θ i θ i = 0 because of the Euler parameter
i
i
constraint equation, θ θ = 1. Using the preceding equation, the time derivatives of
Euler parameters can be determined as
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Dynamic Formulations
57
 −2θ1i
 i
1
1 2θ
θ i = G i ω i =  0 i
4
4  −2θ 3
 i
 2θ 2
2θ
T
 −0.2182

1 1.73320
= 
4  −0.4364

 −0.8728
−2θ 3i 
−2θ 2i
i
 ω 1 
−2θ 2i   i 
ω 2 
2θ1i   i 
 ω3
2θ 0i   
i
3
2θ 0i
−2θ1i
 −0.655 
 10.0  

0.8728  
  5.094 
=
1
0
.
  −0.494 
0.2182  
  3.0  

1.7320 
 −0.938 
−0.4364 
0.8728
0.4364
1.7320
−00.2182
Furthermore, using the equation ω i = G i θ i , one can show that
(
G i ω i = G i G i θ i = 4 I 4 − θ iθ i
T
T
T
)
θ i = 4θ i
This leads to
1 T
θ i = G i ω i
4
Therefore, the time derivatives of Euler parameters can be expressed in terms of the
global or local components of the angular velocity vectors as
4θ i = G i ω i = G i ω i .
T
T
Premultiplying both sides of this equation by the matrix Gi and using the identity
T
T
GiGi = GiGi = 4I ,
one can show that
T
T
G i G i ω i = G i G i ω i = 4ω i
from which
ωi =
1
4
T
GiGi ω i = Aiω i
T
1
where the transformation matrix Ai is given by A i = G i G i , as previously presented
4
in Equation 2.57.
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2.4 NEWTON-EULER EQUATIONS
The dynamic equations of motion of rigid body systems can be written in different
forms. One simple form of the equations of motion that is widely used is based on
Newton-Euler equations, which do not include inertia coupling between the translational and rotational displacements. To obtain the Newton-Euler equations of
motion, the origin of the body coordinate system has to be attached to the center of
mass of the body, which is the case with a centroidal body coordinate system.
Furthermore, the equations of motion associated with the rotation of the rigid body
must be expressed in terms of the angular velocity and acceleration vectors. NewtonEuler equations of motion are given in a matrix form as follows (Greenwood, 1988;
Shabana, 2001):
i  

0  R
Fei
=




i
i
i
i
i 
i
Iθθ   α   Me − ω × ( Iθθ ω ) 
 mi I

 0
(2.63)
where mi is the mass of the rigid body, I is a 3 × 3 identity matrix, Iθθi is the inertia
tensor defined with respect to the centroidal body coordinate system, Fei is the resultant of the external forces, and Mie is the resultant of the external moments defined
in the body coordinate system. The inertia tensor Iθθi is defined as
 I xxi

T
Iθθi = u i u i dm i =  I xyi
m
 i
 I xz
∫
I xyi
i
I yy
I yzi
I xzi 

I yzi 

I zzi 
(2.64)
where dmi = ρidVi, ρi is the material density, and dVi is the infinitesimal material
volume. Since u i = [ x i yi z i ]T , as given in Equation 2.3, the elements of the inertia
tensor can be explicitly written as (Greenwood, 1988; Shabana, 2001)
I xxi =
∫
I xyi = −
i
[( yi )2 + ( z i )2 ]dm i , I yy
=
m
∫
x i yi dm i ,
m
∫
I xzi = −
[( x i )2 + ( yi )2 ]dm i 
m



I yzi = − yi z i dm i

m
(2.65)
[( x i )2 + ( z i )2 ]dm i , I zzi =
m
∫
x i z i dm i ,
m
∫
∫
Table 2.2 shows the mass moments of inertia of some homogeneous solids when a
centroidal body coordinate system is used. In the preceding equation, I xxi, I yyi , and I zzi
are called the mass moments of inertia, while I xyi, I xzi, and I yzi are called the products
of inertia.
Equation 2.63 shows that there is no inertia coupling between the translational
and rotational coordinates in the Newton-Euler equations. Furthermore, one can
choose the orientation of the centroidal body coordinate system such that all products
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Dynamic Formulations
59
TABLE 2.2
Mass Moments of Inertia of Homogeneous Solids
Thin Circular Disk
I xx
1
I yy
4
m( a ) 2 ,
Sphere
1
I zz
2
m( a ) 2
I xx
I yy
Thin Ring
I xx
1
I yy
2
m( a ) 2 ,
I yy
1
12
m( a ) 2
I zz
I xx
83
I yy
320
I zz
1
12
1
12
m( a ) 2
m(l ) 2 ,
m( a ) 2 ,
I yy
m[( a ) 2
(b) 2 ]
m( a ) 2 ,
I zz
2
5
m (a )2
Cylinder
I zz
0
I xx
I yy
1
12
m[3( a ) 2
Thin Plate
I xx
5
Hemisphere
Slender Rod
I xx
2
I zz
( h) 2 ], I zz
1
m( a ) 2
2
Cone
1
12
m(b) 2
I xx
I zz
I yy
3
10
3
80
m[4( a ) 2
m( a ) 2
( h) 2 ],
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Railroad Vehicle Dynamics: A Computational Approach
of inertia, I xyi, I xzi, and I yzi, are equal to zero. In such a case, the axes of the body
coordinate system are called the principal axes, and the mass moments of inertia
associated with the principal axes are called the principal moments of inertia. The
principal moments of inertia and principal axes are determined by solving the
following eigenvalue problem for the inertia tensor:
 Iθθi − λ i I  a i = 0


(2.66)
In this equation, λi is the eigenvalue and ai is the eigenvector. The preceding equation
can be solved for three eigenvalues λki (k = 1, 2, 3), which define the principal
moments of inertia, and three eigenvectors a ik (k = 1, 2, 3), which define the principal
axes or directions associated with the three eigenvalues λki . Because Iθθi is a symmetric tensor, all λki are real and all a ik are mutually orthogonal. Using the fact that
the eigenvectors are orthogonal, the matrix of the principal moments of inertia is
given as
 λ1i

( Iθθi ) p = Ti Iθθi Ti =  0
0

T
0
λ2i
0
0

0
λ3i 
(2.67)
where the transformation matrix Ti can be defined using the normalized eigenvectors
a ik (k = 1, 2, 3) that represent unit vectors along the three mutually orthogonal
principal axes.
As previously pointed out, since the components of the angular velocity vector
are not, in general, the time derivatives of the generalized orientation coordinates,
the components of the angular velocity vector cannot be directly integrated to obtain
the generalized orientation coordinates. For this reason, it is important in constrained
multibody system applications to be able to write the Newton-Euler equations in
terms of the generalized coordinates and their time derivatives using Equation 2.51.
This leads to the generalized Newton-Euler equations, which can be written as
follows:
 miRR

 0
i  

(Qie ) R
0  R
=





i
i
i
i
mθθ   θ  (Qe )θ + (Qv )θ 
(2.68)
i
where the matrices m iRR and mθθ
are the generalized mass matrices associated with
i
the generalized coordinates R and θi, respectively. These matrices are written as
miRR = m i I ,
T
i
mθθ
= Gi Iθθi Gi
(2.69)
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Dynamic Formulations
61
The vector (Qiv )θ is quadratic in the generalized velocities and is given by
T
iθ i 
(Qiv )θ = −Gi ω i × ( Iθθi ω i ) + Iθθi G


(2.70)
iθ i is identically zero when Euler parameters are used as generalized
Recall that G
coordinates to describe the orientation of the body. The vectors (Qie ) R and (Qie )θ are,
respectively, the generalized external force vectors associated with the translational
and rotational coordinates and are defined as
(Qie ) R = Fei ,
T
(Qie )θ = Gi Mie
(2.71)
Note that the external force and moment vectors can be defined in either the
global or body coordinate system using the following transformation:
Fei = A i Fei ,
Mie = A i Mie
(2.72)
Furthermore, the generalized external force vector (Qie )θ can also be written using
Equations 2.13, 2.55, and 2.72 as
T
(Qie )θ = Gi Mie
(2.73)
The generalized Newton-Euler equations can be written in the following compact
matrix form:
i = Qie + Qiv
Mi q
(2.74)
where
 mi
Mi =  RR
 0
i 
(Qie ) R 
R
 0 
0 
i
i
i
,
q
,
Q
=
=

 i  , Qv =  i 


e
i
i
mθθ 
 (Qe )θ 
(Qv )θ 
θ 
(2.75)
For a given set of external forces and moments and a given set of initial coordinates
and velocities, the generalized Newton-Euler equations in the case of unconstrained
i and θ
i , which can be directly integrated forward in time
body i can be solved for R
to obtain the velocities and coordinates at the next time step. Alternatively, one can
use the original Newton-Euler equations defined by Equation 2.63 to solve
i and α i , which can be integrated to determine the velocities R i and ω i . Howfor R
ever, since the angular velocity vector ω i cannot be directly integrated to determine
the orientation coordinates, the time derivatives of the generalized orientation coordinates θ i can be determined using the equation ω i = Giθ i . The vector θ i can then
be integrated to determine the orientation coordinates.
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Railroad Vehicle Dynamics: A Computational Approach
2.5 JOINT CONSTRAINTS
Railroad vehicle systems consist of large number of bodies that include wheelsets,
frames, car bodies, and rails. These bodies are interconnected by mechanical joints
and force elements. In analytical dynamics, the method of treating mechanical
joints and specified motion trajectories that impose constraints on the motion of the
bodies is different from the method of treating the external forces. Constraint equations reduce the number of degrees of freedom, while external force elements do
not affect the number of the system degrees of freedom. The constraint conditions
can be, in most cases, formulated as a set of nonlinear algebraic equations that are
functions of the generalized coordinates and time. The use of the nonlinear algebraic
constraint equations with the system differential equations of motion leads to a
system of differential and algebraic equations that must be solved simultaneously.
In the case of constrained motion, two approaches can be used to formulate the
force-acceleration equations. In the first approach, the augmented formulation, the
equations of motion are formulated in terms of redundant coordinates that are related
by the constraint conditions. The constraint equations are added to the system
differential equations of motion, leading to a system that includes the constraint
forces. This augmented formulation defines a relatively large system of algebraic
equations that can be solved for the system accelerations and the constraint forces.
In the second approach, the embedding technique, the constraint forces are used to
systematically eliminate the dependent coordinates and the constraint forces, leading
to a reduced system of differential equations associated with the independent coordinates (degrees of freedom). The resulting system of equations of motion can be
solved for the accelerations associated with the independent coordinates. In both the
augmented formulation and the embedding technique, the formulation of the joint
constraint equations is necessary. For this reason, this section is devoted to the
formulations of the nonlinear algebraic constraint equations of some of the commonly
used mechanical joints. The augmented formulation is discussed in Section 2.6,
while the embedding technique is covered in Section 2.8.
2.5.1 SPHERICAL JOINT
The spherical joint, sometimes called a ball joint, allows three relative rotational
degrees of freedom between bodies i and j, as shown in Figure 2.5. The spherical joint
eliminates three relative translational degrees of freedom between bodies i and j.
The constraint equations of the spherical joint can be written as
C(q i , q j ) = rPi − rPj = 0
(2.76)
where point P is the joint definition point and
rPi = R i + A i u iP ,
as shown in Figure 2.5.
rPj = R j + A j u Pj
(2.77)
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63
FIGURE 2.5 Spherical joint.
2.5.2 REVOLUTE JOINT
The revolute joint allows one degree of freedom of relative rotation along the joint
axis between bodies i and j, as shown in Figure 2.6. The revolute joint, therefore,
eliminates five relative degrees of freedom between the bodies. Therefore, in addition
to the spherical joint constraints given by Equation 2.76, which constrains the relative
translational displacement, two degrees of relative rotations about two axes perpendicular to the joint axis must be eliminated. Let vi and vj be two vectors defined
along the joint axis on body i and body j, respectively, as shown in Figure 2.6. The
FIGURE 2.6 Revolute joint.
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Railroad Vehicle Dynamics: A Computational Approach
conditions of parallelism of the two vectors vi and vj throughout the motion can be
written using the following dot product (Shabana, 2001):
v1i ⋅ v j = 0 ,
v i2 ⋅ v j = 0
(2.78)
where (⋅⋅ ) denotes the dot product, and v1i and v i2 are two vectors defined on body i
that are orthogonal to vi. The vectors in the preceding equation can be written in
terms of their constant components defined in their respective body coordinate
systems as
v1i = A i v1i ,
v 2i = A i v 2i ,
vj = Aj vj
(2.79)
where Ai and Aj are the transformation matrices that define the orientation of body i
and body j, respectively, and v1i , v 2i , and v j are constant vectors that define, respectively, the components of the vectors v1i , v i2 , and vj in the body coordinate systems.
Using these definitions, the constraint equations of the revolute joint can be written as
rPi − rPj 


C(q i , q j ) =  v1i ⋅ v j  = 0
 vi ⋅ v j 
 2

(2.80)
2.5.3 CYLINDRICAL JOINT
The cylindrical joint allows relative translation and rotation between bodies i and j
along the joint axis, as shown in Figure 2.7. This joint, therefore, has two degrees
of freedom. One degree of freedom is the relative translation along the joint axis,
while the other degree of freedom is the rotation about this axis. Consequently, the
cylindrical joint eliminates four relative degrees of freedom, leading to four constraint equations. Let vi and vj be two vectors defined along the joint axis on body i
and body j, respectively, and let Pi and Pj be two points on body i and body j defined
along the joint axis, as shown in Figure 2.7. The conditions of parallelism of the
two vectors vi and vj throughout the motion can be written using the dot product as
v1i ⋅ v j = 0 ,
v i2 ⋅ v j = 0
(2.81)
Note that v1i and v i2 are two vectors orthogonal to vi. The relative translation
between the two bodies in directions perpendicular to the joint axis can be eliminated
using the conditions
v1i ⋅ rPij = 0 ,
v i2 ⋅ rPij = 0
(2.82)
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65
FIGURE 2.7 Cylindrical joint.
where
rPij = rPi − rPj
(2.83)
Therefore, the constraint equations of the cylindrical joint can be defined as
 v1i ⋅ v j 
 i j
v ⋅v
C(q i , q j ) =  2i ij  = 0
v ⋅ r 
 i1 Pij 
 v 2 ⋅ rP 
(2.84)
2.5.4 PRISMATIC JOINT
The prismatic joint allows one relative translation along the joint axis between bodies
i and j, as shown in Figure 2.8. This constraint can be obtained as a special case of
the cylindrical joint by eliminating the relative rotation along the joint axis. To this
end, two orthogonal vectors hi and hj that are, respectively, defined on body i and j
are introduced and used to define a constraint equation that eliminates the relative
rotation between the two bodies along the joint axis. Using these two vectors and
the constraint equations of the cylindrical joint, one can write the following five
constraint equations for the prismatic joint:
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 2.8 Prismatic joint.
 v1i ⋅ v j 
 i j
v 2 ⋅ v 
C(q i , q j ) =  v1i ⋅ rPij  = 0
 i ij 
 v 2 ⋅ rP 
 hi ⋅ h j 


(2.85)
The fifth equation in Equation 2.85 guarantees that there is no relative rotation
between the two bodies about the joint axis.
2.6 AUGMENTED FORMULATION
The system constraint equations that describe mechanical joints or specified motion
trajectories can be written in a general vector form as follows:
C(q, t ) = 0
(2.86)
where C is the vector of the system constraint equations, and q is the vector of the
system generalized coordinates. These constraint equations can be added to the
generalized Newton-Euler equations using the technique of Lagrange multipliers.
In such a case, the matrix equation of motion of the constrained multibody system
is given by
= Qe + Qv − CqT λ
Mq
(2.87)
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Dynamic Formulations
67
where M is the system mass matrix, Qv is the vector of inertia forces that are
quadratic in velocity, Qe is the vector of the generalized external forces, Cq is the
Jacobian matrix of the constraint equations, and λ is the vector of Lagrange multipliers that are used to define the generalized constraint forces as −CTq λ .
Equations 2.86 and 2.87 represent a system of differential and algebraic equations that can be written in the following augmented form (Shabana, 2001):
M

Cq
CTq   q
 Q + Qv 
  =  e

0   λ   Qd 
(2.88)
where Qd is the vector resulting from the differentiation of the system constraint
equations of Equation 2.86 twice with respect to time, that is,
(q, t ) = C q
C
q − Q d = 0
(2.89)
The number of equations in Equation 2.88 is equal to the total number of the
system generalized coordinates n plus the total number of constraint equations nc.
Note that the number of the unknown Lagrange multipliers is equal to the number
of constraint equations. As a result, the degrees of freedom of the system are equal
to n − nc. If the number of bodies in the system is nb, the vectors and matrices that
appear in Equation 2.88 can be written in a more explicit form as follows:
 M1

M=


 0
 q1 
 2
q
q =  ,
 
 n 
q b 
M







 










(2.90)
(Qie ) R 
 Ri 
 0 
0 
i
i
i
,
Q
,
Q
,
q
=
=
=



 i


e
v
i
i
i
mθθ

 (Qe )θ 
(Qv )θ 
θ 
(2.91)
2
 C1 
 
C2
C =  ,
 
 
Cnc 
 Q1e 
 Q1v 
0 
 2
 2

 , Q =  Qe  , Q =  Q v 
e
v
 
 



 n 

nb
M nb 
Qvb 
Qe 
 ∂C1
 1
 ∂q
 ∂C2

Cq =  ∂q1
 
 ∂Cnc
 ∂q1

∂C1
∂q2
∂C2
∂q2
∂Cnc
∂q
2
∂C1
∂qnb
∂C2
∂qnb
∂Cnc
∂qnb
in which
 mi
Mi =  RR
 0
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Railroad Vehicle Dynamics: A Computational Approach
A physical and straightforward interpretation of the augmented formulation of
Equation 2.88 is given in Section 2.9 using a simple planar example. Equation 2.88
can be solved for the accelerations and Lagrange multipliers, which can then be
used to determine the generalized constraint forces. Having obtained the acceleration
vectors, the independent accelerations can be identified and integrated forward in
time to determine the independent coordinates and velocities. The dependent coordinates and velocities can be determined using the constraint equations at the position
and velocity levels. The numerical algorithm used to solve the differential and
algebraic equations that result from the use of the augmented formulation is discussed in more detail in Chapter 6.
EXAMPLE 2.7
Determine the constraint Jacobian matrix Cq of a spherical joint that connects two
bodies i and j.
Solution. The constraint equations of the spherical joint are given by Equation 2.76.
Using Equation 2.77, these constraint equations can be written as
C(q i , q j ) = R i + A i u Pi − R j − A j u Pj = 0
Differentiating this equation with respect to time, one obtains
=R
i − A i u i G i θ i − R
j + A j u j G j θ j = 0
C
P
P
which can also be written as
= I
C

− A i u Pi G i
−I
A j u Pj G j  q ij = C qij q ij = 0
where I is a 3 × 3 identity matrix and qij = [qiT qjT ]T. It follows that the constraint
Jacobian matrix of the spherical joint between bodies i and j is given by
C qij =  I
− A i u Pi G i
−I
A j u Pj G j 
EXAMPLE 2.8
Determine the constraint Jacobian matrix Cq of a cylindrical joint that connects two
bodies i and j.
Solution. The constraint equations of a cylindrical joint that connects bodies i and j
are given by Equation 2.84 as
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69
 v1i ⋅ v j 
 i j
v ⋅v
C(q i , q j ) =  2i ij  = 0
 v1 ⋅ rP 
 i ij 
 v 2 ⋅ rP 
Note that these four equations can be written using the following two dot-product
equations:
CI = v ik ⋅ v j = 0,
CII = v ik ⋅ rPij = 0,
(k = 1, 2)
Differentiating the first equation CI with respect to time, one obtains
C I = v j ⋅ v ik + v ik ⋅ v j = 0
where
v ik = A i (ω i × vki ) = − A i v ki G i θ i
v j = A j (ω j × v j ) = − A j v j G j θ j
Using these equations, C I can be written as
C I =  0
− v j A i v ki G i
T

0
− v ik A j v j G j  q ij = CIqij q ij = 0
T

where 0 is a 1 × 3 null vector.
Similarly, differentiating the second equation CII with respect to time, one obtains
C II = v ik ⋅ rPij + rPij ⋅ v ik = 0
where
i − A i u i G i θ i − R
j + A j u j G j θ j = 0
rPij = R
P
P
v ik = A i (ω i × vki ) = − A i v ki G i θ i
Using these equations, C II can be written as
T
C II =  v ik

− v ik A i u Pi G i − rPij A i v ki G i
T
T
T
− v ik
v ik A j u Pj G j  q ij = CIIqij q ij = 0
T

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Railroad Vehicle Dynamics: A Computational Approach
Therefore, the constrain Jacobian matrix of the cylindrical joint between bodies i and
j can be written as
0

0
Cq =  T
 v1i
 iT
 v 2
ij
− v j A i v 1i G i
0
− v1i A j v j G j 
− v j A i v 2i G i
0
− v i2 A j v j G j 
− v1i A i u Pi G i − rPij A i v 1i G i
− v1i
− v i2 A i u Pi G i − rPij A i v 2i G i
− v i2
T
T
T
T
T
T
T
T


T
v1i A j u Pj G j 
T
v i2 A j u Pj G j 
T
T

where 0 is a 1 × 3 null vector.
2.7 TRAJECTORY COORDINATES
The analysis presented in this chapter has, so far, focused on using the absolute
Cartesian coordinates to derive the equations of motion. These coordinates are widely
used in developing general-purpose multibody system algorithms. However, it is
important to point out that other sets of coordinates can also be used to derive the
dynamic equations of motion of railroad vehicle systems. These sets of coordinates
can be used to develop specialized formulations that take advantage of particular
features of railroad vehicle systems. However, the relationship between any two sets
of coordinates can always be established using a proper coordinate transformation.
In this section, another set of coordinates, referred to in this book as the trajectory
coordinates, is introduced. More detailed discussion on the trajectory coordinates
and their use in developing special-purpose algorithms and computer codes for the
dynamic analysis of railroad vehicle systems is presented in Chapter 7. The kinematic
and dynamic equations can be formulated using the trajectory coordinates as an
alternative to the absolute Cartesian coordinates. In this case, the general displacement of body i shown in Figure 2.9 can be described using six trajectory coordinates:
the arc length coordinate si defined along the specified trajectory shown in the figure;
the lateral and vertical displacements of the body yir and zir relative to a trajectory
coordinate system that follows the body, as shown in Figure 2.9; and three rotation
angles of the body ψ tr (yaw), φ tr (roll), and θ tr (pitch) that define the orientation of
the body with respect to the trajectory coordinate system. These angles can be written
in a vector form as θir = [ψ ir φ ir θ ir]T. Given the parameter si, the location of the
origin and the orientation of the trajectory coordinate system that follows the motion
of the body can be uniquely defined, as shown in Chapter 3. The location of the
origin in the global system is defined by the vector Rti = Rti (si), while the orientation
of the trajectory coordinate system at this location can be defined using the three
Euler angles ψ ti(si), θ ti(si), and φ ti(si) about the three axes Z ti, Y ti, and X ti. (More
details on the choice of this sequence of rotations are presented in Chapter 3.)
Note that the three Euler angles that define the orientation of the trajectory coordinate
system depend solely on the arc length si. These Euler angles can be used to define
the following transformation matrix:
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Dynamic Formulations
71
FIGURE 2.9 Trajectory coordinates.
A ti =  i ti
jti
 cos ψ ti cos θ ti

=  sin ψ ti cos θ ti
 sin θ ti

k ti 
− sin ψ cos φ + cos ψ sin θ sin φ
ti
ti
ti
ti
cos ψ cos φ + sin ψ sin θ sin φ
ti
ti
ti
ti
− cos θ sin φ
ti
ti
ti
ti


cos ψ sin φ − sin ψ sin θ cos φ 
ti
ti

cos θ cos φ

(2.92)
− sin ψ sin φ − cos ψ sin θ cos φ
ti
ti
ti
ti
ti
ti
ti
ti
ti
ti
This transformation matrix defines the orientation of the trajectory coordinate system
with respect to the global coordinate system. If the space curve geometry is specified,
the angles ψ ti = ψ ti(si), θ ti = θ ti(si), and φ ti = φ ti(si) are known for a given value of
the parameter si. That is,
θ ti = [ψ ti (s i )
θ ti (s i )
φ ti (s i )]T
(2.93)
is a known vector for a given value of si. Using the trajectory coordinate system,
the global position vector of the center of mass of body i can be written as
R i = R ti + A ti u ir
(2.94)
where the vector u ir is the position vector of the center of mass of the body with
respect to the origin of the trajectory coordinate system. This vector is defined as
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u ir = [0
y ir
z ir ]T
(2.95)
where yir and zir are the center of mass coordinates in, respectively, the lateral
direction and the direction normal to the plane that contains the space curve. These
coordinates are defined in the trajectory coordinate system. Note that the location
of the center of mass in the longitudinal direction is assumed to be uniquely defined
using the arc length si and, therefore, the first element of the vector u ir is identically
zero.
2.7.1 VELOCITY
AND
ACCELERATION
Differentiating Equation 2.94 with respect to time, the absolute velocity vector of
the center of mass can be written as
i i
ti u ir + A ti u ir = Lp
R i = R ti + A
(2.96)
where pi is the vector of generalized coordinates of body i, defined as
pi = [s i
yir
z ir
ψ ir
φ ir
θ ir ]T
(2.97)

0

(2.98)
and the matrix Li is a 3 × 6 coefficient matrix given as
 ∂R ti ∂A ti

Li =  i + i u ir 
∂s

 ∂s
jti
k ti
where 0 is a 3 × 3 null matrix. Using Equation 2.96, the absolute acceleration of
the center of mass can be written as
i = Li p
i + γ iR
R
(2.99)
where γ iR includes all the terms that are quadratic in the generalized velocities.
Because the body has three relative rotational degrees of freedom with respect
to the trajectory coordinate system, the angular velocity of body i can be written in
the global coordinate system as
ω i = ω ti + ω ir
(2.100)
where ωti is the absolute angular velocity vector of the trajectory coordinate system,
and ωir is the angular velocity vector of the body with respect to the trajectory
coordinate system. This equation can be rewritten in terms of the generalized velocity
vector p i as
ω i = Gtiθ ti + AtiGir θ ir = Hi p i
(2.101)
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73
where
0

ti
G = 0
1

sin ψ ti
− cos ψ ti
0
0

ir
G = 0
1

cos ψ ir
sin ψ ir
0
− cos ψ ti cos θ ti 

− sin ψ ti cos θ ti  ,

− sin θ ti

(2.102)
− sin ψ ir cos φ ir 

cos ψ ir cos φ ir 

sin φ ir

and Hi is a 3 × 6 coefficient matrix given as

∂θ ti 
Hi =  Gti i 
∂s 

0
0

( AtiGir ) 

(2.103)
where 0 in this equation is a 3 × 1 zero vector.
The angular acceleration of body i in the global coordinate system can be
obtained by differentiating Equation 2.101 with respect to time as
i + γ αi
α i = Hi p
(2.104)
where γ αi includes all the terms that are quadratic in the generalized velocities. The
angular acceleration vector can also be written in the body coordinate system as
T
i + γ αi
α i = Ai α i = H i p
(2.105)
where
T
T
T
Hi = Ai Hi = Air Ati Hi ,
T
γ αi = Ai γ αi
(2.106)
and Air is the rotation matrix that defines the orientation of the body coordinate
system with respect to the trajectory coordinate system based on the Euler angle
sequence Z ir, Xir, Y ir. This matrix is given as
Air =
 cos ψ ir cos θ ir − sin ψ ir sin φ ir sin θ ir

ir
ir
ir
ir
ir
 sin ψ cos θ + cos ψ sin φ sin θ
ir
ir

− cos φ sin θ

(2.107)
− sin ψ ir cos φ ir
cos ψ ir sin θ ir + sin ψ ir sin φ ir cos θ ir 
cos ψ ir cos φ ir
sin ψ ir sin θ ir − cos ψ ir sin φ ir cos θ ir 
sin φ ir

cos φ ir cos θ ir


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Using Equations 2.99 and 2.105, one can write
i   Li  i  γ iR 
R
+  i 
 i  =  ip
 γ α 
 α  H 
(2.108)
The kinematic analysis presented in this section shows that the use of the
trajectory coordinate system requires information on the second derivatives of the
Euler angles ψ ti(si), θ ti(si), and φ ti(si) with respect to the arc length parameter si.
This is clear from the definition of the angular acceleration vector, which includes
the derivative of Hi of Equation 2.103 with respect to time. As is demonstrated in
Chapter 3, the Euler angles ψ ti(si), θ ti(si), and φ ti(si) are functions of the second
derivatives, since the orientation of the trajectory coordinate system requires knowledge of the curvature. For this reason, the use of the trajectory coordinates requires
fourth-order derivatives of the equation of the space curve that defines the track
centerline with respect to the arc length parameter si (Rathod and Shabana, 2007).
2.7.2 EQUATIONS
OF
MOTION
The nonlinear equations of motion for body i can be obtained by substituting
Equation 2.108 into the Newton-Euler equations of motion. For body i, this is defined
as
 mi I

 0
i  

Fei
0  R
=




i
i
i
i
i 
i
Iθθ   α   Me − ω × ( Iθθ ω ) 
(2.109)
Writing the absolute accelerations in terms of the trajectory accelerations using
Equation 2.108 and premultiplying by the transpose of the matrix of coefficients of
i in Equation 2.108, one obtains the following equations of motion expressed in
p
terms of the trajectory coordinates:
i = Qipe + Qipv
Mip p
(2.110)



T
T
Qipe = Li Fei + Hi Mie


T
T
Qipv = − m i Li γ iR − Hi [ Iθθi γ αi + ω i × ( Iθθi ω i )] 

(2.111)
where
T
T
Mip = m i Li Li + Hi Iθθi Hi
It is clear from this equation that, when the trajectory coordinates are used, the
resulting mass matrix does not take as simple a form as the one used in the NewtonEuler equations of Equation 2.109.
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75
As in the case of the absolute coordinates, an augmented formulation in terms
of the trajectory coordinates can be developed. In this case, the kinematic constraints
can be formulated in terms of the trajectory coordinates. The generalized forces
associated with these trajectory coordinates must also be evaluated. Using Equation
2.110, the augmented form of the equations of motion in the case of a system that
consists of nb bodies can be written in terms of the trajectory coordinates as
M p

 C p
T
T
CTp   p
 Q pe + Q pv 
  = 

0   λ   Qd

(2.112)
T
where p = [ p1 p2 … pnb ]T is the vector of system trajectory coordinates, Mp is
the system mass matrix associated with the trajectory coordinates, Qpe is the vector
of system applied forces associated with the trajectory coordinates, Qpv is the vector
of centrifugal and Coriolis forces, Cp is the Jacobian matrix of the kinematic constraint equations, λ is the vector of Lagrange multipliers, and Qd is the vector
resulting from the differentiation of the system constraint equations twice with
respect to time.
The use of the trajectory coordinates in developing specialized railroad vehicle
dynamic algorithms has the advantage of making the formulation of some kinematic
constraint equations easier. This is demonstrated by the following example.
EXAMPLE 2.9
Body i shown in Figure 2.9 is subjected to the following kinematic constraints:
1. The body is driven by a function f(t) along a space curve with predefined geometry.
2. The pitch rotation of the body is specified by a function g(t).
3. The body has no relative motion with respect to the trajectory coordinate system
along the Z ti axis.
Derive the equations of motion of the body i using the augmented formulation and the
trajectory coordinates.
Solution. Given that the motion of the body along the space curve is specified, one
has the following constraint equation expressed in terms of the function f(t):
C1 = s i − f (t ) = 0
The equation of the constraint on the pitch rotation of the body can be defined using
the function g(t) as
C2 = θ ir − g(t ) = 0
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Given that the vertical motion with respect to the trajectory coordinate system is
constrained, one has
C3 = z ir − d 0 = 0
where d0 is the constant Z ti coordinate of the origin of the body coordinate system with
respect to the trajectory coordinate system. Using the preceding three equations, the
vector of constraint equations can be written in the following form:
 s i − f (t ) 


C = θ ir − g(t )  = 0
 z ir − d 0 


Differentiating this equation twice with respect to time, the constraint equation at the
acceleration level is given as
 s i − f (t ) 
= θir − g(t )  = C p
C
p − Qd = 0


 z ir 


where the constraint Jacobian matrix Cp and the vector Qd are, respectively, given by
1

Cp = 0
 0
0
0
0
0
0
0
0
0
0
1
0
0
0

1 ,
0 
 f (t ) 


Qd =  g(t ) 
 0 
Note that because the constraint equations are linear in the generalized trajectory
coordinates, the constraint Jacobian matrix is constant, and thus the right-hand-side
vector of the constraint acceleration equation Qd depends on time only. Accordingly,
the augmented form of the equations of motion for this one-body system is given by
 M ip

 Cp
i 
C Tp   p
 Qipe + Qipv 
  = 

0   λ   Qd

2.8 EMBEDDING TECHNIQUE
In the augmented formulation discussed in the preceding two sections, the equations
of motion are expressed in terms of redundant coordinates that are not independent
because of the kinematic constraint relationships. For this reason, the constraint
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Dynamic Formulations
77
forces appear explicitly in the equations of motion. Using the embedding technique,
one can systematically eliminate the constraint forces to obtain a minimum set of the
equations of motion expressed in terms of the system degrees of freedom. To obtain
the minimum set of differential equations of motion and eliminate the constraint
forces from these equations, the constraint equations are used to define a velocity
transformation matrix that can be used to write the system velocities and accelerations in terms of the independent velocities and accelerations, respectively.
2.8.1 COORDINATE PARTITIONING
AND
VELOCITY TRANSFORMATION
Differentiating the constraint equations with respect to time, one obtains
C q q = −C t
(2.113)
This equation can also be written as
C qi q i + C q d q d = −C t
(2.114)
In this equation, the vectors qd and qi are, respectively, the vectors of dependent and
independent generalized coordinates that are written as
q = [q Td
q Ti ]T
(2.115)
Using Equations 2.114 and 2.115, the total vector of the system velocities can be
written in terms of the independent velocities as
q = B q i
(2.116)
It is important to point out that if the constraints are explicit functions of time
(rheonomic constraints), Equation 2.116 should be altered to include an additional
term that results from the differentiation of the constraint equations with respect to
time. In general, joint constraints are not explicit functions of time, while specified
motion trajectory constraints (driving constraints) can be explicit functions of time.
The matrix B that appears in Equation 2.116 is called the velocity transformation
matrix and is defined using Equation 2.114 as
 −C −1 C 
B =  q d qi 
I


(2.117)
In the case of complex railroad vehicle systems, the independent coordinates qi
can be identified using numerical procedures such as Gaussian elimination and LU
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factorization. The independent coordinates must be selected such that the constraint
Jacobian matrix C q d associated with the dependent coordinates is a nonsingular
square matrix (Garcia de Jalon and Bayo, 1994; Shabana, 2001; Wehage, 1980).
2.8.2 ELIMINATION
OF THE
CONSTRAINT FORCES
The velocity transformation matrix can be used to eliminate the generalized constraint forces −CTq λ that appear in the equations of motion (Equation 2.87). It can
be shown that
BT C Tq = 0
(2.118)
This result can be proved by substituting Equation 2.117 into Equation 2.118, leading
to
BT C Tq =  −C qTi C q−dT
C Tq 
I   Td  = −C Tqi C q−dT C qT d + C Tqi = 0
 C qi 
(2.119)
which shows that indeed BT C Tq = 0 .
2.8.3 REDUCED-ORDER MODEL
Using Equation 2.116, the system acceleration vector can be expressed in terms of
the independent accelerations as
= Bq
i + γ
q
(2.120)
where the vector γ includes all the terms that are quadratic in the velocities. This
vector is given by
 −C −1 [(C q ) q + 2Cq t q + Ctt ]
γ =  qd q q

0


(2.121)
Substituting Equation 2.120 into Equation 2.87, premultiplying by the transpose of
the velocity transformation matrix B, and using Equation 2.118, one obtains a
minimum set of dynamic differential equations of motion expressed in terms of the
system degrees of freedom as
i = (Qe )i + (Q v )i
Mi q
where
(2.122)
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79



(Qe )i = BT Qe


T
(Qv )i = B (Qv − Mγ ) 
Mi = BT MB
(2.123)
Note that Equation 2.122 does not include any constraint forces.
EXAMPLE 2.10
For the system of Example 2.9, use the embedding technique to derive the equations
of motion in terms of the degrees of freedom.
Solution. Since the system is subjected to three constraint equations associated with
the longitudinal and vertical displacements and the pitch rotation, the body has three
degrees of freedom. These degrees of freedom are the lateral displacement yir, the yaw
angle ψ ir, and the roll angle φ ir. Hence, the dependent and independent coordinates
can be, respectively, written as
p id = [s i
z ir
θ ir ]T ,
ψ ir
p ii = [ y ir
φ ir ]T
Using the preceding equations, the constraint Jacobian matrices are obtained in terms
of the independent and dependent coordinates as
1

C pd =  0
 0
0
0
0
1 ,
1

0 
0

C pi =  0
 0
0
0
0
0
0

0 
Note that the constraint Jacobian matrix associated with the dependent coordinates is
a nonsingular square matrix. Using the acceleration equation as previously obtained in
Example 2.9, the system accelerations can be expressed in terms of the independent
accelerations as
 s i   f (t ) 
 ir   ir 
 y   y 
 z ir   0 
p ii + γ i
p i =  ir  =  ir  = Bi ψ   ψ 
 φir   φir 
  

 θir   g(t ) 
ii = [ where p
y ir ψ ir φir ]T and
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0

1
0
Bi = 
0
0

 0
0
0
0
1
0
0
 f (t ) 


 0 
 0 
γi = 

 0 
 0 


 g(t ) 
0

0
0
,
0
1

0 
Using Equation 2.122, one has
ii = (Qipe ) i + (Qipv ) i
M ipi p
where
T
M ipi = Bi M ip Bi ,
T
(Qipe ) i = Bi Qipe ,
T
(Qipv ) i = Bi (Qipv − M ip γ i )
2.9 INTERPRETATION OF THE METHODS
The augmented formulation and the embedding technique discussed in this chapter
are the two basic approaches used to develop general-purpose computer multibody
system algorithms. It is clear that the augmented formulation leads to a larger system
of equations of motion that has a sparse matrix structure. This system of equations
is expressed in terms of redundant coordinates and, therefore, the constraint forces
appear explicitly in this system of equations. The use of Lagrange multipliers in the
augmented formulation leads to a symmetric coefficient matrix in the acceleration
equations. In the embedding technique, on the other hand, the dependent coordinates
and the constraint forces are systematically eliminated, leading to a minimum set
of equations of motion expressed in terms of the system independent coordinates
(degrees of freedom). In this section, a physical interpretation of the augmented
formulation and the embedding technique is presented using a simple example. To
this end, the planar wheel/rail example shown in Figure 2.10 is considered.
2.9.1 KINEMATIC
AND
DYNAMIC EQUATIONS
The wheel is assumed to have a radius r, mass mw, and mass moment of inertia I w
about its center of mass. The rail segment is assumed to be circular with radius R.
The wheel, which is subjected to an external force Fw = [Fxw Fyw ] T and a moment
M w, is assumed to roll without slipping on the rail and, therefore, there is no friction
force due to sliding. Since, in this example, pure rolling of the wheel is considered
and the rail is assumed to be fixed, the system has only one degree of freedom. It
is clear from Figure 2.10 that the tangent and normal vectors at the contact points
are given as follows:
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81
FIGURE 2.10 Planar wheel/rail example.
 − cos φ 
tr = 
,
 sin φ 
 − sin φ 
nr = 

 − cos φ 
(2.124)
The velocity on the wheel at the contact point can be written as
w + ω w × uw
rcw = R
c
(2.125)
In this equation,
0
 
ω =  0 ,
θ w 
 
w
 −r sin φ 


u =  −r cos φ 
 0 


w
c
(2.126)
where θ w is the angular velocity of the wheel. Using the preceding two equations,
one obtains
w

 w
w + ω w × u w =  Rx + rθ cos φ 
rcw = R
c
w
w
 Ry − rθ sin φ 
(2.127)
Since the condition of pure rolling is assumed, the absolute velocity of the wheel
at the contact point must be identically zero. That is, rcw = 0 . Therefore, Equation
2.127 leads to the following two constraint equations on the motion of the wheel:
R xw + rθ w cos φ = 0 

R yw − rθ w sin φ = 0 
(2.128)
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These two constraint equations that guarantee that the wheel will roll without
slipping reduce the number of independent velocities of the system to one. It is also
clear from Figure 2.10 that
(
)
(
Rxw = − R − r sin φ ,
)
Ryw = − R − r cos φ
Using these two equations and Equation 2.128, one obtains the following relationship
between the derivatives of the two angles θ w and φ:
(
)
rθ w = R − r φ
(2.129)
The constraints of Equation 2.128 can be used to identify two dependent velocities,
say R xw and R yw . Using the generalized coordinate partitioning method, the system
dependent velocities can be expressed in terms of the independent velocity θ w .
Figure 2.11 shows the free-body diagram of the wheel. In this figure, N = [Nx Ny]T
is the reaction force at the contact point. The free-body diagram of Figure 2.11 can
be used to write the following Newton-Euler equations for the planar wheel:
xw = Fxw + N x
mwR



w w
w
w
m Ry = Fy − m g + N y


I wθw = M w + N x r cos φ − N yr sin φ 
(2.130)
In this equation, g is the gravity constant. Equations 2.130 and 2.128 represent the
system differential and algebraic equations that include the differential equations of
motion and the algebraic constraint equations, respectively. We note that, for given
initial conditions for the coordinates and velocities, the three scalar equations in
xw , R
yw ,θw , Nx , and Ny . Therefore, one needs
Equation 2.130 have five unknowns: R
the two constraints of Equation 2.128 in order to have five equations that can be
solved for the three unknown accelerations and the two components of the reaction
FIGURE 2.11 Free-body diagram.
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83
force. We also note that the number of constraint equations is equal to the number
of dependent variables, and this number is also equal to the number of the unknown
independent constraint forces. In the remainder of this section, this simple planar
wheel/rail example is used to provide a physical interpretation of the augmented
formulation and the embedding technique.
2.9.2 AUGMENTED FORMULATION
In the augmented formulation, the constraint equations at the acceleration level are
combined with the system differential equations of motion to obtain a number of
equations equal to the number of the unknown accelerations and constraint forces.
Differentiating the velocity constraints of Equation 2.128 with respect to time, one
obtains
sin φ = 0 
xw + rθw cos φ − rθφ
R


cos φ = 0 
yw − rθw sin φ − rθφ
R

(2.131)
Combining Equations 2.130 and 2.131 and rearranging the terms in these equations
such that the unknowns are on the left-hand side, one obtains
xw − N x = Fxw
mwR


w w
w
w

m Ry − N y = Fy − m g

w w
w
I θ − N x r coos φ + N yr sin φ = M 

sin φ
xw + rθw cos φ = rθφ

R


cos φ
yw − rθw sin φ = rθφ
R

(2.132)
This equation can be written in the following matrix form:
mw

 0
 0

 1
 0

0
mw
0
0
1
0
0
Iw
r cos φ
−r sin φ
−1
0
−r cos φ
0
0
xw   F w

0  R
x

 w   w
w 
−1   Ry   Fy − m g 
r sin φ   θw  =  M w 



sin φ 
0   N x   rθφ
   
0   N y   rθφ
cos φ 
(2.133)
Equation 2.133 can be written in the form of Equation 2.88 as follows:
M

Cq
CTq   q
 Q + Qv 
  =  e

0   λ   Qd 
(2.134)
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Railroad Vehicle Dynamics: A Computational Approach
where, in this example, q = [Rxw Ryw θ w]T, Qv in this planar example is equal to zero,
and
mw

M= 0
 0

0
w
m
0
 Fxw

 w
w 
Qe =  Fy − m g  ,
 Mw 


0
1

0  , Cq = 
0
I w 
0
1







− N x  
λ=

 − N y  

r cos φ 

−r sin φ 
sin φ 
 rθφ
Qd = 
,
rθφ cos φ 
(2.135)
From the definition of λ in Equation 2.135, it is clear that Lagrange multipliers
can be interpreted in this example as the negative of the reaction forces. This simple
interpretation of Lagrange multipliers is not general, since in other constraint types,
the physical meaning of Lagrange multipliers can be different. It is clear that the
coefficient matrix in Equation 2.133 has a sparse matrix structure, with many elements equal to zero. This sparse matrix structure is exploited in most computational
algorithms used in general-purpose multibody computer codes.
2.9.3 EMBEDDING TECHNIQUE
In the embedding technique, a minimum set of equations of motion is obtained by
systematically eliminating the dependent accelerations and the constraint forces. The
constraints at the acceleration level (Equation 2.131) can be used to write two
dependent accelerations in terms of the independent acceleration. If Rxw and Ryw are
considered as the dependent accelerations and θw is considered as the independent
acceleration, it is clear from Equation 2.131 that
sin φ 
xw = −rθw cos φ + rθφ
R


w
w
Ry = rθ sin φ + rθφ cos φ 
(2.136)
This equation can be used to write all the system accelerations in terms of the
independent acceleration θw as follows:
sin φ 
xw   −r cos φ 
R
 rθφ
 w  

 w  =  Ry  =  r sin φ  θ + rθφ cos φ 
q
 θw   1 


0



  
(2.137)
This equation can be written in the form of Equation 2.120 as
= Bθw + γ
q
(2.138)
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Dynamic Formulations
85
where
sin φ 
 rθφ
 
γ = rθφ cos φ 


0


 −r cos φ 


B =  r sin φ  ,
 1 


(2.139)
The equations of motion of Equation 2.130 can also be written in the form of
Equation 2.87 as
= Qe + Qv − CqT λ
Mq
(2.140)
where the vectors and matrices in this equation are as defined in Equation 2.135.
Substituting Equation 2.138 into Equation 2.140 and premultiplying by BT, one
obtains the following equation of motion, which does not include the constraint
forces:
(I
w
)
(
)
+ m wr 2 θw = M w − rFxw cos φ + r Fyw − m w g sin φ
(2.141)
If the external forces and moments are assumed to be zero and if the wheel is
replaced by a cylinder with I w = mwr 2/2, Equation 2.141, upon the use of Equation
2.129, reduces to the following familiar equation of the single-degree-of-freedom
oscillatory system given by
3( R − r ) φ + g sin φ = 0
2
(2.142)
It is clear that Equation 2.141 does not have any constraint forces, since it can be
verified that BT C Tq = 0 in this example, as previously discussed in this chapter.
2.9.4 D’ALEMBERT’S PRINCIPLE
The concept used in the embedding technique was first introduced by D’Alembert,
who suggested treating the inertia forces in the same manner as the applied forces.
Using D’Alembert’s principle, the constraint forces can be eliminated from the
equations of motion, leading to a number of equations of motion equal to the number
of the system degrees of freedom. For example, if the inertia forces are treated as
the applied forces, one can take the moments of the two sets of forces about any
point and equate the moment of the inertia forces to the moment of the applied
forces. For example, using the free-body diagram in Figure 2.11, the moments of
the applied force about the contact point c can be written as
(
)
M w − rFxw cos φ + r Fyw − m w g sin φ
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86
Railroad Vehicle Dynamics: A Computational Approach
while the moment of the inertia forces about the same point is given by
xwr cos φ + m w R
ywr sin φ
I wθw − m w R
Substituting for the accelerations Rxw and Ryw from the constraint of Equation 2.136,
it is clear that the moment of the inertia forces about the contact point c can be
written as I w + m wr 2 θw . Equating the moments of the inertia forces to the moments
of the applied forces, one obtains Equation 2.141.
(
)
2.10 VIRTUAL WORK
While the concepts upon which the augmented formulation and the embedding
technique are based can be demonstrated using a simple example, as shown in the
preceding section, the principle of virtual work can be used to provide a systematic
and more general derivation of the methods. This principle will be used in Chapter 5
to provide a simple and straightforward derivation of the equations of motion when
the system is subjected to contact constraints that require the use of nongeneralized
surface parameters. A virtual change in the system coordinates δq is an infinitesimal
change that occurs without regard to time change. This virtual change in the coordinates can be used to develop the principle of virtual work in dynamics. To develop
this important principle, Equation 2.87 can be rewritten as
− Qe − Qv + CqT λ = 0
Mq
(2.143)
Multiplying this equation by the transpose of the vector δq, one obtains the following
scalar equation:
(
)
− Qe − Qv + CqT λ = 0
δ qT Mq
(2.144)
This equation is called the Lagrange-D’Alembert equation, and it is the basis for
the general development of the augmented formulation and the embedding technique.
The Lagrange-D’Alembert equation (Equation 2.144) can also be written as follows:
δ Wi = δ We + δ Wc
(2.145)
where δWi is the virtual work of the system inertia forces, δWe is the virtual work
of the system applied forces, and δWc is the virtual work of the system constraint
forces. These expressions of the virtual work are given as
(
)
− Qv ,
δ Wi = δ qT Mq
δ We = δ qT Qe ,
δ Wc = −δ qT CqT λ
(2.146)
Since the equilibrium of the system is considered, and since the reaction forces
acting on the interconnected bodies are equal in magnitude and opposite in direction,
the virtual work of the constraint forces is identically zero, that is,
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Dynamic Formulations
87
δ Wc = −δ qT CqT λ = 0
(2.147)
for kinematically admissible virtual displacements δq that satisfy the constraint
equations. In the case of specified motion trajectories, which are driving constraints,
the virtual work of the resulting constraint forces is also equal to zero, since the
coordinates associated with the specified motion are prescribed, and as a result, the
virtual changes in these prescribed coordinates are equal to zero. The result of
Equation 2.147 can also be proven mathematically by writing the virtual changes
in the system coordinates in terms of the virtual changes in the independent coordinates or degrees of freedom. This leads to the following relationship:
δ q = Bδ qi
(2.148)
In this equation, B is the velocity transformation matrix previously defined in this
chapter. Substituting Equation 2.148 into the expression of the virtual work of the
constraint forces and using the fact that BT C Tq = 0 as previously demonstrated, one
obtains the important result of Equation 2.147. Using the results of Equation 2.147,
Equation 2.145 can be written as
δ Wi = δ We
(2.149)
This is the principle of virtual work in dynamics, which states that, for a multibody
system, the virtual work of the system inertia forces is equal to the virtual work of
the system applied forces. Furthermore, the virtual work of the constraint forces is
identically zero. The results of the embedding technique can be obtained using
Equation 2.149 by substituting Equation 2.148 into Equation 2.149. This leads to a
minimum set of equations of motion from which the constraint forces are eliminated
(Rosenberg, 1977; Shabana, 2001).
It is important to point out that the condition of Equation 2.147 that is used to
obtain the principle of virtual work in dynamics given by Equation 2.149 is only
valid when the equilibrium of the entire system is considered. If the equilibrium of
individual bodies is considered, the virtual work of the constraint forces acting on
each body separately is not equal to zero. In this case, the virtual work principle for
body i in the system can be written as
δ Wii = δ Wei + δ Wci
(2.150)
In this equation, δWii, δWei, and δWci are, respectively, the virtual work of the inertia,
applied, and constraint forces of body i. Note that if body i is subjected to constraints,
then δWci ≠ 0, while
nb
∑δW
i
c
= 0,
i =1
where nb is the total number of bodies in the system.
(2.151)
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3
Rail and Wheel
Geometry
If the goal is to develop general and detailed models of railroad vehicle systems,
multibody system algorithms must be modified to include a wheel/rail contact model.
Three steps are employed in the computational algorithm used to obtain the numerical solution of the wheel/rail contact problem. The first is the geometry step, in
which the locations of the points of contact between the wheel and the rail are
determined. The second is the kinematic step, in which normalized kinematic quantities called creepages that measure the relative velocities between the wheel and
the rail at the contact points are determined. In the third step, called the dynamic or
kinetic step, the forces that act on the wheel and the rail as the result of the contact
are determined. The accuracy of the numerical solution of the contact problem
depends strongly on the accurate prediction of the location of the contact points.
The solution for the contact locations requires an accurate representation of the
geometry of the wheel and the rail surfaces. This representation can be defined using
local surface geometric properties such as the radii of curvature and the tangent and
normal vectors to the surfaces. These geometric properties are not only important
for determining the contact locations (geometry problem), but they are also important, as described in Chapter 4, in determining the forces that represent the dynamic
interaction between the wheel and the rail. Therefore, basic knowledge of differential
geometry is necessary to understand the wheel/rail contact problem. In particular,
the theories of curves and surfaces are fundamental in the study of the dynamic
interaction between the wheel and the rail. This chapter discusses topics in differential geometry that are repeatedly used in this book and that are used in the
geometric description of wheel and rail surfaces.
The spatial representation of the rail geometry in particular is crucial in investigating and predicting the railroad vehicle dynamic response. For example, the study
of the curving behavior of a vehicle is important in determining the safe speed of
operation. Furthermore, track irregularities that influence the dynamic response of the
vehicle can cause ride discomfort or even train derailments. The track geometry is
also fundamental in the formulation of the dynamic equations, which, as shown in
later chapters of this book, are expressed in terms of generalized coordinates that
depend on the track geometry. This chapter discusses the mathematical representation of the rail (or the track) geometry and introduces the definitions and terminologies that are used in the field of railroad vehicle dynamics and that will be used
repeatedly throughout this book. The parametric representation of the wheel surface
is discussed at the end of the chapter.
89
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90
Railroad Vehicle Dynamics: A Computational Approach
3.1 THEORY OF CURVES
To introduce the concept of a curve, a Cartesian coordinate system is used to define
the position vector of points in space in terms of three components as
y =  y1
y3 
y2
T
(3.1)
A curve is defined as a real vector function that can be uniquely expressed in
terms of one parameter, t. That is, the components of the vector function can be
determined once this parameter is specified. Using this definition, a curve defined
over the interval a ≤ t ≤ b can be written in the following form:
()
()
()
y t =  y1 t
y2 t
()
y3 t 
T
(3.2)
This equation, which is the parametric representation of a curve, can be used to
determine the location of a point on the curve for an arbitrary value of the parameter
t. The components yi (i = 1, 2, 3) must be differentiable over the interval on which
t is defined. An example of the preceding equation is the parameterized differentiable
curve y(t) = [rcost rsint αt]T, where r and α are constants. One can show that the
trace of this curve is a helix of pitch 2πα on the cylinder x2 + y2 = r 2 (Do Carmo,
1976).
3.1.1 ARC LENGTH
AND
TANGENT LINE
For a given t,
()
()
dy  dy1 t
=
dt  dt

dy2 t
dt
()
dy3 t 

dt 

T
(3.3)
is called the tangent line to the curve at t. In the study of the theory of curves, the
existence of the tangent line at every point is essential. If at a point t, dy(t)/dt = 0,
the point is called a singular point.
The arc length of a curve from point t0 to point t can be obtained using the
tangent line as follows:
t
s=
∫ dt dt
dy
(3.4)
t0
A curve can be parameterized by its arc length s. In this case, one can write y =
y(s). Using the parameterization in terms of the arc length, one can show using
Equation 3.4 that the tangent vector, t(s) = dy/ds, becomes a unit vector, that is,
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Rail and Wheel Geometry
91
()
dy
= t s =1
ds
(3.5)
It is important to recognize that while any parameter can be used to define the
curve, the tangent obtained by differentiation with respect to the parameter is a unit
vector only when the curve is parameterized by its arc length. Note that for an
arbitrary parameter t, one has
dy dy  ds 
=
dt ds  dt 
(3.6)
In the remainder of this section, we will assume that the curve is parameterized by
its arc length s.
3.1.2 CURVATURE
AND
TORSION
If a curve is parameterized by its arc length, the curvature vector is obtained by
differentiating the tangent vector. The curvature vector of the curve y(s) is defined as
()
y ′′ s =
d 2 y dt
=
ds 2 ds
(3.7)
Since the tangent obtained by differentiation with respect to the arc length is a unit
vector, the tangent and curvature vectors are orthogonal. This follows from differentiating the equation y′T y′ = 1, which leads to 2y′T y″ = 0, proving the orthogonality
of the tangent and curvature vectors. The curvature of the curve at a point s is defined
as
()
()
k s = y ′′ s = t ′(s )
(3.8)
Since the tangent vector t(s) has unit length, the curvature k(s) measures the rate of
change of orientation of the tangent vector. In other words, the curvature measures
how rapidly the curve pulls away from the tangent line. Because of the orthogonality
of the tangent and curvature vectors, a unit vector along the curvature vector defines
the unit normal to the curve nc (subscript c is used here to indicate a normal to a
curve, since n is used throughout this book as the normal to a surface). Therefore,
the unit normal to the curve at s is defined as
()
nc s =
( ) = t ′ (s )
k (s )
k (s )
y ′′ s
(3.9)
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Railroad Vehicle Dynamics: A Computational Approach
The plane formed by the unit tangent and normal vectors is called the osculating
plane at s. The radius of curvature at s is defined as R = 1/k(s).
The cross-product of the orthogonal tangent and normal unit vectors defines a
vector that is normal to the osculating plane. This new vector is called the binormal
vector at s and is given by
() ()
()
b s = t s × nc s
(3.10)
The three orthogonal unit vectors t, nc, and b form a coordinate system called the
Frenet frame. The orientation of this frame is defined by the matrix [t nc b]. The
t–b plane is called the rectifying plane, and the nc–b plane is called the normal plane.
Differentiating Equation 3.10 with respect to s and recognizing that the two
vectors t′(s) and nc(s) are parallel, one obtains
()
()
() ()
() ()
()
b′ s = t ′ s × nc s + t s × nc′ s = t s × nc′ s
(3.11)
That is, b′(s) is normal to t(s), and since b(s) is a unit vector, b′(s) is also orthogonal
to b(s). It follows that b′(s) is parallel to nc, and one can write b′(s) in the following
form:
()
() ()
b′ s = − τ s n c s
(3.12)
where τ is called the torsion. The curvature and torsion completely describe the
behavior of the curve in the neighborhood of s. In summary, we have the following
equations (Kreyszig, 1991):



n′c = − kt + τb 

b′ = − τn c

t ′ = kn c
(3.13)
These equations, which express the derivatives in terms of the tangent, normal, and
binormal unit vectors, are called the Serret-Frenet formulas.
3.2 GEOMETRY OF SURFACES
The geometry of a surface can be described using two independent parameters. Using
a system of Cartesian coordinates x1, x2, and x3, each point on the surface is assumed
to have a unique position vector x that can be defined in the three-dimensional space
in terms of these two independent parameters as follows:
x(s1, s2 ) =  x1 (s1, s2 )
x2 (s1, s2 )
x3 (s1, s2 ) 
T
(3.14)
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Rail and Wheel Geometry
93
where s1 and s2 are the parameters used to describe the surface geometry and are
called the parameters of the surface. If differential calculus is to be applied, Equation
3.14 must satisfy certain requirements of differentiability with respect to s1 and s2.
Furthermore, Equation 3.14 must satisfy the following conditions (Goetz, 1970;
Kreyszig, 1991):
(a) In a bounded s1–s2 domain, each point represented by Equation 3.14
corresponds to just one pair of s1 and s2 in this domain. That is, the
mapping of Equation 3.14 is one-to-one.
(b) In the bounded domain, the Jacobian matrix
 ∂x1
 ∂s
 1
∂x   ∂x2
=
∂s2   ∂s1
 ∂x3

 ∂s1
 ∂x
J=
 ∂s1
∂x1 
∂s2 
∂x2 

∂s2 
∂x3 

∂s2 
(3.15)
is of rank two.
Condition (b) is satisfied if
(
∂x s1, s2
∂s1
) × ∂x (s , s ) ≠ 0
1
2
∂s2
This condition implies that the two columns of the matrix of Equation 3.15 are
linearly independent. For example, assume that Equation 3.14 is used to represent
the surface shown in Figure 3.1. One can define a curve x(s1,s2c) for a constant
parameter s2 = s2c and another curve x(s1c,s2) for a constant parameter s1 = s1c.
Therefore,
∂x(s1 , s2c )
∂s1
represents the tangent vector to the curve x(s1,s2c) along the s1 coordinate line, and
∂x(s1c , s2 )
∂s2
represents the tangent vector to the curve x(s1c,s2) along the s2 coordinate line.
Therefore, if
(
∂x s1 , s2c
∂s1
) × ∂x ( s
1c
, s2
∂s2
)≠0
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94
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 3.1 Surface mapping.
for arbitrary s1c and s2c, the rank of the Jacobian matrix J is two, and Equation 3.14
represents a surface. The two tangents to the coordinate lines are not necessarily
orthogonal or unit vectors. In the wheel/rail contact problem, the tangent and the
normal vectors at the contact points on the surfaces enter into the formulation of
the kinematic and force relationships. For example, these vectors are used to determine the locations of the points of contact between the wheel and the rail. They are
also used to determine the principal curvatures and the principal directions of the
surface at a specified point. The principal curvatures and principal directions are
used to calculate the shape of the area of contact between the wheel and the rail, as
discussed in Chapter 4.
3.2.1 TANGENT PLANE
AND
NORMAL VECTOR
Equation 3.14 represents the surface as a function of the two parameters s1 and s2.
One can define a curve on the surface by assuming that these two parameters are
related and can be expressed as functions of a single parameter t, that is, s1 = s1(t)
and s2 = s2(t), where t is any arbitrary variable. Therefore, Equation 3.14 can be
written as follows:
(
)
x s1 (t ), s2 (t ) = y (t )
(3.16)
where y(t) is a regular curve on the surface and
dy ∂x ds1 ∂x ds2
=
+
≠0
dt ∂s1 dt ∂s2 dt
(3.17)
In writing this equation, it is assumed that ∂x/∂s1 × ∂x/∂s2 ≠ 0, ds1/dt ≠ 0, and ds2/dt
≠ 0. Equation 3.17 defines a tangent to the surface at point P that also belongs to
the curve y(t). This tangent is a linear combination of the two linearly independent
tangent vectors ∂x/∂s1 and ∂x/∂s2 at point P, as shown in Figure 3.2 and as also
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Rail and Wheel Geometry
95
FIGURE 3.2 Tangents to a surface.
demonstrated by Equation 3.17. Therefore, the plane that contains the two tangents
∂x/∂s1 and ∂x/∂s2 is the tangent plane to the surface. Consequently, a nonzero vector
dy/dt is tangent to the surface at point P if, and only if, it is parallel to the tangent
plane at this point. The normal to the surface at point P can be defined as the normal
to the tangent plane and can be written as follows:
n=
x ,1 × x ,2
x ,1 × x ,2
(3.18)
where x,1 = ∂x/∂s1 and x,2 = ∂x/∂s2.
As in the case of curves, the surface can be defined uniquely using certain local
quantities called the first and second fundamental forms. These fundamental forms,
which were introduced by Gauss and are discussed below, can be used to calculate
the curvatures and the principal directions of the surface at an arbitrary point.
3.2.2 FIRST FUNDAMENTAL FORM
The first fundamental form of a surface is defined as follows:
I = dx ⋅ dx = dx T dx
(3.19)
Since dx = x,1ds1 + x,2ds2, where x,1 = ∂x/∂s1 and x,2 = ∂x/∂s2, Equation 3.19 can be
written as follows:
I = ( x,1ds1 + x,2 ds2 )T ( x,1ds1 + x,2 ds2 )
(3.20)
This equation can also be written as
( )
I = E ds1
2
+ 2 Fds1ds2 + G ( ds2 )2
(3.21)
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Railroad Vehicle Dynamics: A Computational Approach
where
E = x,T1 x,1 , F = x,T1 x,2 , G = x,T2 x,2
(3.22)
From Equation 3.21, it is clear that the first fundamental form is a homogenous
function of second degree in ds1 and ds2 with coefficients E, F, and G. These
coefficients are called the coefficients of the first fundamental form. The first fundamental form can be used to measure distances, angles, and areas on the surface. For
example, consider the curve y(t) given by Equation 3.16. The length of this curve
over the domain a ≤ t ≤ b is given by
b
l=
∫
a
b
=
∫
a
dy
dt =
dt
b
∫
dx dx
dt =
⋅
dt dt
a
b
T
 ds1
ds2   ds1
ds2 
 x,1 dt + x,2 dt   x,1 dt + x,2 dt  dt
∫
a
2
2
 ds 
 ds 
ds ds
E  1  + 2 F 1 2 + G  2  dt
dt dt
 dt 
 dt 
(3.23)
Equation 3.23 shows that the arc length on the surface is the integral of the
square root of the first fundamental form. The coefficients of the first fundamental
form can also be used to determine the angle between the two tangents x,1 and x,2 as
cos α =
x ,T1x ,2
=
x ,1 x ,2
F
(3.24)
EG
This equation shows that the two tangents x,1 and x,2 to the surface at a point are
perpendicular if F = 0
3.2.3 SECOND FUNDAMENTAL FORM
The second fundamental form of a surface is defined as follows:
(
II = − dx ⋅ dn = − x,1ds1 + x,2 ds2
( )
= L ds1
2
) (n ds
T
,1
1
+ n,2 ds2
)
( )
+ 2 Mds1ds2 + N ds2
(3.25)
2
where
L = − x,T1 n,1 , M = −
(
)
1 T
x,1 n,2 + x,T2 n,1 , N = − x,T2 n,2
2
(3.26)
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Rail and Wheel Geometry
97
and n is the normal vector to the surface that is given by Equation 3.18. It is clear
from Equation 3.25 that the second fundamental form is also a homogenous function
of second degree in ds1 and ds2 with coefficients L, M, and N. These coefficients are
called the coefficients of the second fundamental form. Since x,1 and x,2 are perpendicular to n for all s1 and s2, one has the following identities:
(x n)
(x n)
(x n)
(x n)
T
,1
T
,1
T
,2
T
,2
= x,T11n + x,T1 n,1 = 0 


= x,T12 n + x,T1 n,2 = 0 
,2


T
T
= x,21n + x,2 n,1 = 0 
,1


= x,T22 n + x,T2 n,2 = 0 
,2

,1
(3.27)
These identities lead to
x,T11n = − x,T1 n,1 , x,T12 n = − x,T1 n,2 , x,T21n = − x,T2 n,1 , x,T22 n = − x,T2 n,2 (3.28)
Using the preceding equations, the coefficients of the second fundamental form can
be written as follows:
L = x,T11n, M = x,T12 n, N = x,T22 n
(3.29)
where x,ij = ∂2x/(∂si∂sj). Using the preceding equation, one can show that the second
fundamental form can be written as
II = d 2 x ⋅ n
(3.30)
where
( )
d 2 x = x,11 ds1
2
( )
+ 2 x,12 ds1ds2 + x,22 ds2
2
(3.31)
To explain the physical meaning of the second fundamental form, suppose that
a point P is on the surface x and a point Q is a point in the neighborhood of point
P on the surface x, as shown in Figure 3.3. The length of the components of the
vector u between points P and Q projected onto the normal n is defined as follows:
d = uT n
(3.32)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 3.3 Second fundamental form of a surface.
If point P is defined at x(s1, s2) and point Q is defined at x(s1 + ds1, s2 + ds2),
Equation 3.32 can be written as follows:
(
)
T
d = u T n = x(s1 + ds1 , s2 + ds2 ) − x(s1 , s2 ) n
((
) ( ))
2

1
=  dx + d 2x + O ds1 + ds1
2

= dx T n +
((
2
T

 n

(3.33)
) ( ))
2
1 2 T
d x n + O ds1 + ds1
2
2
Since dxTn = 0 by virtue that dx is parallel to the tangent plane, one has
d=
((
1 2 T
d x n + O ds1
2
) + ( ds ) ) = 21 II + O (( ds ) + ( ds ) )
2
2
1
2
2
1
1
(3.34)
It is clear from this equation that if higher-order terms are neglected, the second
fundamental form is twice the projection of the vector u onto n, and the absolute
value of the second fundamental form is twice the distance projected along the
normal from Q to the tangent plane at P. Clearly, if the surface is flat (planar), this
projected distance is zero for any arbitrary pairs of points. Equation 3.33 is called
the osculating paraboloid at P.
The coefficients of the second fundamental form can be used to determine the
nature of the surface in the neighborhood of P as follows:
The
The
The
The
surface
surface
surface
surface
is
is
is
is
called
called
called
called
elliptic at point P if LN − M2 > 0
hyperbolic at point P if LN − M2 < 0
parabolic at point P if LN − M2 = 0
planar at point P if L = N = M = 0
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Rail and Wheel Geometry
99
3.2.4 NORMAL CURVATURE
Let y = y(s1 (t ), s2 (t )) be a regular curve Ln defined on the surface x = x(s1, s2). The
normal curvature vector to Ln at point P denoted by Kn is defined as the projection
of the curvature vector K of the curve defined by y onto the normal n to the surface
at point P and is given by
(
)
Kn = K ⋅ n n
(3.35)
Note that Ln can be defined as the intersection of a plane that contains the tangent
to Ln and the normal vector, as shown in Figure 3.4. The component of the curvature
vector of Ln along the normal to the surface at P is called the normal curvature and
is defined as
kn = K ⋅ n
(3.36)
It is important to mention that, from Equation 3.36, the sign of kn depends on
the direction of the normal n. A positive sign of kn means that the normal vector n
is directed toward the curvature center. It is also clear from Equation 3.18 that the
sign of n depends on the manner in which the surface parameters are ordered and,
therefore, the choice of the order of the surface parameters must be taken into
consideration. In the wheel/rail contact problem, it is important to know the location
of the curvature center and whether it lies inside or outside the wheel or the rail, as
will be discussed in the following chapter. Recall that the curvature vector of the
curve Ln at a point P on the surface x is given by
K=
FIGURE 3.4 Surface curvature.
dt dt
=
ds dt
dx
dt
(3.37)
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Railroad Vehicle Dynamics: A Computational Approach
where t is the tangent vector to Ln at P, and s is the arc length parameter. Since t is
orthogonal to n, one has
( )
d T
t n =0
dt
which leads to
T
 dt 
T dn
 dt  n = −t dt
(3.38)
Substituting Equation 3.37 into Equation 3.36 and using Equation 3.38, one obtains
kn =
L ( ds1 dt )2 + 2 M ( ds1 dt )( ds2 dt ) + N ( ds2 dt )2
E ( ds1 dt )2 + 2 F ( ds1 dt )( ds2 dt ) + G ( ds2 dt )2
(3.39)
This equation shows that kn depends on the ratio (ds1/dt)/(ds2/dt), which is the
direction of the tangent line to Ln at point P, and also depends on the first and second
fundamental coefficients that are functions of the coordinates of point P. Therefore,
all the curves through point P that are tangent to the same line through P have the
same normal curvature. As the normal curvature to the curve Ln at point P depends
on the location of point P and the direction of the tangent line at point P, the normal
curvature can be written as
kn =
L ( ds1 )2 + 2 Mds1ds2 + N ( ds2 )2 II
= ,
I
E ( ds1 )2 + 2 Fds1ds2 + G ( ds2 )2
( ds1 )2 + ( ds2 )2 ≠ 0
(3.40)
The first fundamental form I is positive definite, since it is a measure of the square
of a distance. It follows from the preceding equation that the sign of kn depends on
the sign of the second fundamental form II. Clearly for a planar point, kn = 0 in all
directions, while for an elliptic point, kn ≠ 0 and has the same sign as ds1/ds2. In the
case of a hyperbolic point, kn can be positive, negative, or zero, depending on ds1/ds2,
while for a parabolic point, kn has the same sign and is zero for a direction for which
the second fundamental form II is equal to zero.
3.2.5 PRINCIPAL CURVATURES
AND
PRINCIPAL DIRECTIONS
The normal curvature is called the principal curvature if its value is maximum or
minimum. Therefore, the principal curvatures can be determined by solving the
following two equations:
∂kn
∂kn
=
=0
∂ ds1
∂ ds2
( )
( )
(3.41)
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Rail and Wheel Geometry
101
Using Equation 3.40, one can show that Equation 3.41 leads to two scalar equations
that can be written in the following matrix form:
 L − kn E

 M − kn F
M − kn F   ds1  0 
  =  
N − kn G   ds2  0 
(3.42)
This equation has a nontrivial solution if, and only if, the determinant of the coefficient matrix is equal to zero, that is,
 L − kn E
det 
 M − kn F
M − kn F 
=0
N − kn G 
(3.43)
− ( EN + GL − 2 FM ) kn + LN − M 2 = 0
(3.44)
or
( )
( EG − F 2 ) kn
2
The two roots of this quadratic equation determine the principal curvatures k1 and
k2 that can be used in Equation 3.42 to determine the principal directions. The mean
curvature Km at a point P is defined as the average of the principal curvatures:
Km =
(
1
k1 + k2
2
)
The Gaussian curvature KG at P is defined as KG = k1k2. The Gaussian curvature
is an invariant property of the surface. The principal curvatures and principal directions are used in Chapter 4 in Hertz contact theory to determine the dimension of
the area of contact between the wheel and the rail.
EXAMPLE 3.1
A surface is represented by
x = s1i + 2s1s2 j + s2 k
where i, j, and k are the unit vectors along the axes of a Cartesian coordinate system.
Determine the principal curvatures and the principal directions of this surface at point
(s1,s2) = (1,0).
Solution. Using the surface equation, one has
x,1 = i + 2s2 j,
x,2 = 2s1 j + k,
x,11 = 0,
x,12 = 2 j,
x,22 = 0
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Railroad Vehicle Dynamics: A Computational Approach
n=
x ,1 × x ,2
x ,1 × x ,2
=
( 2s i − j + 2s k )
( 2s ) + 1 + ( 2s )
2
1
2
2
2
1
The coefficients of the first fundamental form are
( ),
E = x T,1x ,1 = 1 + 4 s2
2
( )
F = x T,1x ,2 = 4 s1s2 , G = x T,2 x ,2 = 1 + 4 s1
2
The coefficients of the second fundamental form are
L = x T,11n = 0,
M = x T,12 n =
−2
( ) + ( 2s )
1 + 2s1
2
,
2
N = x T,22 n = 0
2
Because LN − M2 < 0 for all s1 and s2, the surface is hyperbolic. The principal curvatures
of this surface are the roots of Equation 3.44 and are given by
k1,2 =
− b ± b 2 − 4 ac
2a
where
a = EG − F 2 ,
b = − ( EN + GL − 2 FM ),
c = LN − M 2
At the given point
E = 1, F = 0, G = 5, L = 0, M = −2
5 , N = 0, a = 5, b = 0, c = −0.8
Therefore, the principal curvatures are given as
k1 = 0.4, k2 = −0.4
and the principal directions are
 1   1 

, 

 −0.4472   0.4472 
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Rail and Wheel Geometry
103
3.3 RAIL GEOMETRY
The dynamic behavior of railroad vehicles depends on the track geometry, and for
this reason, it is important to accurately describe the rail geometry to be able to
correctly predict the vehicle response. Using the methods of differential geometry
discussed in the preceding section, the rail surface can be defined in a parametric
form. The description of the surface geometry must be general to allow the use of
the parametric equations to represent arbitrary rail profiles that can be provided
analytically, in tabulated form, or from direct measurements. For example, in the
case of a straight segment of the track, the surface of the rail can be obtained by
translation of the profile curve, as shown in Figure 3.5. This surface can be defined
by the parametric equations

X = s1

Y = s2

Z = f (s2 ) 
(3.45)
where s1 is the distance along the rail (arc length) and is defined as the rail longitudinal surface parameter, and s2 is the rail lateral surface parameter that is used
as an independent variable to describe the rail profile. If the rail profile, for example,
has a sinusoidal shape, the function f(s2) can be defined analytically as follows:
 πs 
f (s2 ) = h cos  2 
 w 
(3.46)
where h is the height and w is the width of the rail head. The function f(s2) can also
represent measured rail profile. In this case, this function can be defined using a
spline function that depends on s2.
FIGURE 3.5 Rail surface and its coordinate system.
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 3.6 Coordinate systems used to define the track geometry.
In the case of curved rails, representation of the position and orientation of the
rail cross section (profile frame) as a function of the longitudinal surface parameter
s1 is required. To account for different possible scenarios, such as a variation in the
gage or relative rotations of the rails, it is necessary to define the surface of each
rail (right and left) in the track independently. It is also important to be able to
represent each rail as a separate body that can have independent motion and can be
subjected to independent loading conditions and kinematic constraints. Such a general representation can be achieved by introducing several coordinate systems, as
shown in Figure 3.6. First, a global coordinate system XYZ that is assumed to be
fixed in time is introduced. The track, as shown in Figure 3.6 can be defined by two
bodies l and r that represent, respectively, the left and right rails. The frames XrYrZr
and X lY lZ l are, respectively, the right and left rail body coordinate systems that can
have parallel axes and a common origin before displacement. When the right rail,
for example, has a contact with one of the wheels at point Pr, a profile frame for
this contact, XrpYrpZrp, is introduced, as shown in Figure 3.6 and Figure 3.7. The
profile frame XrpYrpZrp translates along the right rail space curve and rotates about
its origin. If the Xrp axis of the profile frame is assumed to be along the tangent to
the space curve, the location of the contact point Pr can be simply defined with
respect to the profile frame as
u rp = 0
s2
( )
f s2 
T
(3.47)
If the profile of the rail cross-section is changing along the space curve, Equation
3.47 takes the following general form:
u rp = 0
s2
(
)
f s1, s2 
T
(3.48)
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Rail and Wheel Geometry
105
FIGURE 3.7 Position of the contact point.
Therefore, the location of a point Pr on the rail surface in the body coordinate system
XrYrZr is given by the vector u r , as shown in Figure 3.7, as follows (Berzeri et al.,
2000):
u r = R rp + A rp u rp
(3.49)
where R rp is the vector that defines the location of the origin Orp in the body r
coordinate system, and A rp is the matrix that defines the orientation of the profile
frame with respect to the body coordinate system. Note that in Equation 3.49, R rp
and A rp depend only on the rail longitudinal surface parameter s1, while u rp depends
on the rail lateral surface parameter s2 and can also depend on the rail longitudinal
surface parameter s1 in the more general case, as previously discussed and shown
in Equation 3.48 and Figure 3.8. In this more general case, the rail profile changes
along the track, and Equation 3.49 can be written as
FIGURE 3.8 Contact points and surface parameters.
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Railroad Vehicle Dynamics: A Computational Approach
(
)
( )
( ) (
u r s1, s2 = R rp s1 + A rp s1 u rp s1, s2
)
(3.50)
Equation 3.49 or, equivalently, Equation 3.50 will be used in Chapter 5 to define
the conditions of contact between the wheel and the rail. Depending on how the
geometric surface parameters are treated, one can obtain different forms of the
equations of motion, as discussed in Chapter 5.
The description presented in this section also allows for the special case in which
the right and left rail can be treated as one body. In this case, one track frame X tY tZ t,
shown in Figure 3.6, can serve as the body coordinate system for both rails, while
the vector R rp and the matrix A rp can be defined as functions of the longitudinal
surface parameter of each rail space curve. In the method described in this book,
the right and left rail space curves are assumed to be independent, regardless of
whether or not the two rails are treated as one body or two separate bodies.
3.4 DEFINITIONS AND TERMINOLOGY
This section discusses some of the basic definitions and terminology used in railroad
vehicle dynamics, particularly at the preprocessing stage in which the track geometry
is defined.
Gage The gage G is defined as the lateral distance between two points on the heads
of the right and left rails, as shown in Figure 3.9. These two points are located at a
distance of 5/8 in. (14 mm) from the top of the rail head. In North America, the
standard gage value varies from 56 to 57.25 in.
Super-elevation The super-elevation h is defined as the vertical distance between
the right and left rail, as shown in Figure 3.9.
Curvature The curvature is different from zero in the case of curved track and is
defined as the value of the angle ψ required to obtain a 100-ft-length chord AB of
constant radius RH in the horizontal plane, as shown in Figure 3.10.
Grade The grade is defined as the ratio (percentage) between the vertical elevation
and the longitudinal distance.
FIGURE 3.9 Definition of the gage and super-elevation.
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Rail and Wheel Geometry
107
FIGURE 3.10 Definition of the curvature using 100-ft chord.
Cant angle The cant angle is defined as the rotation of each rail about its longitudinal axis, as shown in Figure 3.11.
FIGURE 3.11 Definition of the cant angle.
Profile The profile is defined as the vertical deviation of the rail space curve, as
shown in Figure 3.12.
Alignment The alignment is defined as the lateral deviation of the rail space curve,
as shown in Figure 3.12.
FIGURE 3.12 Rail deviations (profile and alignment).
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Railroad Vehicle Dynamics: A Computational Approach
3.5 GEOMETRIC DESCRIPTION OF THE TRACK
The coordinates of an arbitrary point on the rail surface can be determined in terms
of the coordinates of a point on the rail space curve and the orientation of a coordinate
system at this point. This section presents a computational method that can be used
to determine the locations of points on the rail space curve and the angles that define
the orientation of the profile frames at these points based on measured railroad
industry inputs. The method can be applied to the left rail space curve and its profile
frame X lpY lpZ lp, the right rail space curve and its profile frame XrpYrpZrp, or the track
centerline and the track frame X tY tZ t. Using the typical variables measured by the
railroad industry, one can develop a systematic procedure for constructing the track
geometry. First, the definitions used in the industry are presented, and analytical
interpretations of some of these definitions are provided before concluding this
section. For the sake of simplicity of the presentation in this section, some of the
conventionally used superscripts that indicate body numbers are dropped.
In the railroad industry, the geometry of the track is defined using the following
three variables (Berzeri et al., 2000; Dukkipati and Amyot, 1988; Rathod and Shabana,
2007):
1. Projection, which defines the planar curve obtained by projecting the
reference line on the horizontal plane
2. Development, which defines an elevation angle θ
3. Super-elevation, which defines a bank angle φ that represents the rotation
of the profile frame about the tangent to the reference line
For instance, the projection of segment AB of a space curve on the horizontal
plane can be represented by A0B0 (projection) as shown in Figure 3.13. Assume that
the arc length of the actual curve is denoted as s, while the arc length of the projected
curve is denoted as S. At point P0 on the horizontal plane, the radius of curvature
is RH. The following relationship between the horizontal curvature CH and the
FIGURE 3.13 Projection of the space curve on the horizontal plane.
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Rail and Wheel Geometry
109
FIGURE 3.14 Sign convention for positive rotations.
rotation angle ψ about the vertical axis holds (Berzeri et al., 2000; Rathod and
Shabana, 2007):
dψ = CH dS =
dS
RH
(3.51)
The relation between the arc length s and the projected arc length S is given by
ds =
dS
cos θ
(3.52)
where the development angle θ is as shown in Figure 3.13. Analytical derivation of
the preceding two equations is given later in this section.
In the railroad industry, the track is defined by providing: the horizontal curvature
CH as a function of the projected arc length S, the vertical development angle θ as
a function of the arc length s, and the bank angle φ as a function of the projected
arc length S. The signs of these quantities follow a standard convention. The curvature at an arbitrary point is positive if the y-coordinate of the center of the curvature
is positive with respect to the profile frame. In other words, the curvature is positive
if it is the result of a positive rotation ψ. The curvature angle ψ, the vertical
development angle θ, and the bank angle φ are positive when the rotations are in
the directions shown in Figure 3.14. Note that for the angles θ and φ, a sign
convention different from the right-hand rule is followed in order to be consistent
with the measurements made by the industry. As a result, the vertical development
angle is positive if, for a positive step increment ds, a positive increment dz is
obtained, as shown in Figure 3.13; and a positive bank angle would accommodate
a curve with a positive curvature. By using the functions CH (S), θ(s), and φ(S), the
reference line and the orientation of its coordinate system can be completely defined.
For instance, the rotation about the vertical axis can be obtained as
S1
∫
ψ = ψ 0 + CH ( S )  dS = ψ ( S )
(3.53)
S0
where ψ0 and S0 are the values of ψ and S at point A shown in Figure 3.13. Knowing
the three Euler angles ψ, θ, and φ that follow the sequence Z, -Y, -X, one can define
the following transformation matrix that defines the orientation of the profile frame:
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110
 cos ψ cos θ

A =  sin ψ cos θ
 sin θ

Railroad Vehicle Dynamics: A Computational Approach
− sin ψ sin φ − cos ψ sin θ cos φ 

cos ψ sin φ − sin ψ sin θ cos φ 

cos θ cos φ

− sin ψ cos φ + cos ψ sin θ sin φ
cos ψ cos φ + sin ψ sin θ sin φ
− cos θ sin φ
(3.54)
If x = [x y z]T is the vector that defines the coordinates of an arbitrary point on
the space curve, the unit tangent vector is defined as t = dx/ds, as was shown
previously. If the first column of the transformation matrix in Equation 3.54 is
considered as the unit tangent to the space curve at s, the location of an arbitrary
point on the space curve can be defined using the equation dx = tds, which, upon
integration and using Equation 3.52, leads to

x = x o + cos ψ S  dS = x S 

S0


S

y = yo + sin ψ S  dS = y S 

S0

s


z = zo + sin θ s  ds = z s

s0

S
∫
( )
( )
∫
( )
( )
∫
()
(3.55)
()
Furthermore, the projected arc length S can be defined using Equation 3.52 as
S1
∫
S = S0 + cos θ (s)  ds = S (s)
(3.56)
S0
The analysis presented in this section shows that if the inputs CH, θ, and φ are
given, the locations of the points on the space curve and the orientation of the profile
frame can be completely defined.
In the analysis presented in the preceding section, the tangent vector t to the
space curve is taken as the first column of the transformation matrix of Equation
3.54. A vector of length ds along the unit tangent can be written as
 cos ψ cos θ 


ds = ds  sin ψ cos θ 
 sin θ 


(3.57)
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Rail and Wheel Geometry
111
The projection of this vector on the horizontal plane is given by
(
)
t h = ds − ds T k k
(3.58)
where k = [0 0 1]T is a unit vector along the vertical axis defined in the global
coordinate system. Using the preceding two equations, one can show that
( )
2
t Th t h = ds cos 2 θ
(3.59)
This equation defines the length of th as the projected infinitesimal arc length dS =
ds cosθ, which shows that no approximation is made in the relationship given in
Equation 3.52 (Rathod and Shabana, 2006).
The vector th of Equation 3.58 that is the result of the projection of the curve
on the horizontal plane can be written in a more explicit form as:
 cos ψ cos θ 


t h =  sin ψ cos θ 


0


(3.60)
The unit tangent and the curvature vectors for the projection of the curve on the
horizontal plane are
 cos ψ 


t H =  sin ψ  ,
 0 


 − sin ψ 
∂ψ 

kH =
cos ψ 
∂S 


 0 
(3.61)
This equation shows that the horizontal curvature is ∂ψ/∂S, as given by Equation 3.51.
3.6 COMPUTER IMPLEMENTATION
The development presented in the preceding section shows that some of the basic
quantities that define the curved rail are expressed as functions of either the arc
length s or the projected arc length S. It is necessary at this point to express all
quantities in terms of a single parameter. Dukkipati and Amyot (1988) selected the
actual arc length s that represents the physical distance traveled by the wheelset.
This choice is more convenient when the equations of motion are formulated using
the trajectory coordinates. In this case, a coordinate system moving along the rail
is used in developing some specialized railroad vehicle formulations. In some of the
general multibody formulations presented in this book, however, there is no clear
advantage in using a certain independent parameter, and the projected arc length S
can be selected as well. However, to be consistent with the choice usually made in
the literature, the actual arc length s is used as the independent parameter, while the
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112
Railroad Vehicle Dynamics: A Computational Approach
value of the projected arc length S can be obtained by solving Equation 3.56. This
value can then be used in Equation 3.55, which depends explicitly on S.
3.6.1 TRACK SEGMENT TYPES
In general, it is not possible to obtain closed-form solutions of Equations 3.53, 3.55,
and 3.56 using the input quantities CH, φ, and θ. In reality, the rail is divided into
segments, and the input variables are assumed to vary linearly within each segment.
Only a small number of segment types is considered in railroad applications. Different segment types are described below (Berzeri et al., 2000).
Tangent: A tangent denotes a straight segment. Therefore, the horizontal
curvature CH is always equal to zero at all points of this segment.
Curve: A curve denotes a circular arc. Therefore, the horizontal curvature CH
is assumed constant along this segment.
Tangent-to-curve entry spiral: This segment connects a tangent to a curve.
Therefore, in this case, the horizontal curvature CH is equal to zero at the
beginning and is equal to the inverse of the radius of curvature at the end.
The variation from the initial value (zero) to the final value is assumed to
be linear with respect to the projected arc length S.
Curve-to-tangent exit spiral: This segment connects a curve to a tangent.
Therefore, in this case, the horizontal curvature CH is equal to the inverse
of the radius of the curve at the beginning and is equal to zero at the end.
The variation from the initial value to the final value is assumed to be linear
with respect to the projected arc length S.
Curve-to-curve spiral: This segment connects two curves. Therefore, the
horizontal curvature CH varies linearly with respect to the projected arc
length S between the values that it takes at the two ends of the segment.
In addition, each segment type described above can include a linear variation of the
vertical development angle or a linear variation of the bank angle.
3.6.2 LINEAR REPRESENTATION
OF THE
SEGMENTS
The segment definitions and assumptions described above can be used to obtain a
simple mathematical description of the rail. Given a segment whose end points are
A and B and using the assumption that the horizontal curvature varies linearly, the
horizontal curvature at any point within the segment can be defined as
CH =
C1 ( S − S0 ) − C0 ( S − S1 )
S1 − S0
(3.62)
where C0, C1, S0, and S1 are the values of the curvature and the projected arc length
at points A and B, respectively. Using Equation 3.53 and the linear form for the
horizontal curvature of Equation 3.62, one has
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Rail and Wheel Geometry
ψ =ψ0 +
113
1  C1
S − S0
S1 − S0  2
(
)
2
−
2
C0
C
S − S1  + 0 S1 − S0
2
 2
(
)
(
)
(3.63)
The angle θ is related to the vertical curvature Cv through the equation
dθ = Cv ds
(3.64)
Note that the assumption of linear relationship between θ and s implies that the
vertical curvature is assumed to be constant within each segment and equal to
Cv =
θ1 − θ 0
s1 − s0
(3.65)
where θ0 and θ1 are the values of the vertical development angle at the end points
A and B, respectively, and s0 and s1 are the values of the arc length at the same
points. Consequently, the angle θ is given by
θ = θ 0 + Cv (s − s0 )
(3.66)
Finally, the linearity of the bank angle function φ(S) implies that
φ=
φ1 ( S − S0 ) − φ0 ( S − S1)
S1 − S0
(3.67)
where φ0 and φ1 are the values of the bank angle at points A and B, respectively.
The linear relationship between the angle θ and the arc length s allows obtaining
a closed-form expression of the function S(s). Using Equations 3.52 and 3.64, it is
possible to write
dθ
C
= v
dS cos θ
(3.68)
This equation leads to (Berzeri et al. 2000)

1
1
 S0 + sin θ (s)  − sin θ 0
Cv
Cv
S (s ) = 
 S + (s − s ) cos θ
0
 0
if Cv ≠ 0
(3.69)
if Cv = 0
The value S obtained from this equation is then used in Equations 3.62, 3.63, and 3.67.
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114
Railroad Vehicle Dynamics: A Computational Approach
3.6.3 DERIVATIVES
OF THE
ANGLES
The formulation of the wheel/rail contact problem, as will be seen from the analysis
presented in Chapter 5, requires calculation of the tangent and normal vectors of
the surfaces as well as higher derivatives of these vectors with respect to the surface
parameters. Using some of the new finite element methods such as the absolute
nodal coordinate formulation (Berzeri et al., 2000; Shabana, 2005), these higher
vector derivatives can be evaluated without the need to evaluate higher derivatives
of the angles. In some specialized railroad vehicle formulations, however, due to the
nature of the coordinates used, higher derivatives of the angles must be evaluated
(Rathod and Shabana, 2006). In the remainder of this section, expressions for the
derivatives of the angles that are required in some specialized railroad vehicle
dynamic formulations are presented.
Differentiating Equation 3.63 with respect to s leads to

dCH
cos 2 θ − CH Cv sin θ 

dS

dCH
Cv sin θ cos θ − CH Cv2 cos θ 
ψ ′′′ = −3

dS
ψ ′ = CH cos θ , ψ ′′ =
(3.70)
In most railroad vehicle applications, the curvature CH and its derivatives, the angle θ
and the vertical curvature Cv , are assumed to be small. If these assumptions are used,
Equation 3.70 shows that the first derivative ψ ′ is small and is of the same order of
magnitude as CH , while ψ ″ and ψ are infinitesimal quantities of the third order.
In Equation 3.66, it is assumed that the development angle θ is a linear function
of the arc length s. Therefore, its first derivative with respect to s is constant and is
given by
θ ′ = Cv
(3.71)
It is clear from this equation that the higher derivatives θ ″ and θ are equal to zero.
In Equation 3.67, the bank angle is defined as a linear function of the projected
arc length S. Using Equations 3.52 and 3.64, the derivatives of φ with respect to s
can be written as
 φ −φ 
 φ −φ 
 φ −φ 
φ ′ =  1 0  cos θ , φ ′′ = −  1 0  Cv sin θ , φ ′′′ = −  1 0  Cv2 cos θ (3.72)
 S1 − S0 
 S1 − S0 
 S1 − S0 
If φ, θ, and Cv are assumed small, the first derivative φ′ is a small quantity of the
same order of magnitude as φ, while higher derivatives become of the third order.
The expressions of the derivatives of φ, θ, and ψ with respect to S and s are
presented in Table 3.1 and Table 3.2. In these tables, the exact expression is compared
with the expression obtained by retaining only first-order terms. The simplification
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Rail and Wheel Geometry
115
TABLE 3.1
Derivatives of the Orientation Angles with
Respect to S
Variable
Exact Expression
First-Order
Approximation
dψ/dS
d2ψ/dS2
d3ψ/dS3
dθ/dS
d2θ/dS2
d3θ/dS3
dφ/dS
d2φ/dS2
d3φ/dS3
CH
dCH /dS
0
Cv /cosθ
Cv2[sinθ/cos3θ]
Cv3[1 + 2sin2θ/cos5θ]
(φ1 − φ0)/(S1 − S0)
0
0
CH
dCH /dS
0
Cv
0
0
(φ1 − φ0)/(S1 − S0)
0
0
TABLE 3.2
Derivatives of the Orientation Angles with Respect to s
Variable
dψ/ds
d2ψ/ds2
d3ψ/ds3
dθ/ds
d2θ/ds2
d3θ/ds3
dφ/ds
d2φ/ds2
d3φ/ds3
Exact Expression
CH cosθ
[dCH /dS]cos2θ − CHCvsinθ
−3[dCH /dS]Cvsinθcosθ − CH Cv2cosθ
Cv
0
0
[(φ1 − φ0)/(S1 − S0)]cosθ
−[(φ1 − φ0)/(S1 − S0)]Cvsinθ
−[(φ1 − φ0)/(S1 − S0)]Cv2cosθ
First-Order
Approximation
CH
dCH /dS
0
Cv
0
0
(φ1 − φ0)/(S1 − S0)
0
0
is made by assuming that the quantities CH, Cv , φ, and θ and their derivatives are
small (Berzeri et al., 2000).
In some specialized railroad vehicle formulations, such as the one presented in
Chapter 7, it is necessary to differentiate the transformation matrix A of Equation 3.54
with respect to the surface parameter s. The first derivative of A can be obtained by
using the chain rule of differentiation as
A′ = Aψ ψ ′ + Aθθ ′ + Aϕφ ′
(3.73)
where Aα = ∂A/∂α for α = ψ, θ, φ, and a prime denotes a differentiation with respect
to the surface parameter s. The expression for the derivatives ψ ′, θ ′, and φ′ are given
in Table 3.2. Further differentiation with respect to the surface parameter leads to
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116
Railroad Vehicle Dynamics: A Computational Approach
A′′ = Aψ ψ ′′ + Aθθ ′′ + Aϕφ ′′ + Aψψ ψ ′ 2 + Aθθθ ′ 2 + Aφφφ ′ 2
+ 2 Aψθψ ′θ ′ + 2 Aψφψ ′φ ′ + 2 Aθφθ ′φ ′
(3.74)
In this equation, no simplifications are made. Clearly, further differentiation
would result in an even more complicated expression for the third derivative A″′.
Table 3.2 shows that when the input quantities CH, θ, and φ are small, the derivatives
of the orientation angles contain terms of the third order. It is possible to show that
by neglecting all the terms of the third order and higher, one obtains (Berzeri et al.,
2000)
A′′ = Aψ ψ ′′ + Aψψ ψ ′ 2 + Aθθθ ′ 2 + Aφφφ ′ 2 + 2 Aψθψ ′θ ′ + 2 Aψφψ ′φ ′ + 2Aθφθ ′φ ′ (3.75)
and
A′′′ = 3( Aψψ ψ ′ + Aψθθ ′ + Aψφφ ′)ψ ′′
(3.76)
Furthermore, by neglecting all the terms of the second order and higher, that is, by
retaining only first-order terms, one obtains the result
A′′ = Aψ ψ ′′ ,
A′′′ = 0
(3.77)
Depending on the simulation scenario, a first-order or second-order approach
may be justified, leading to a simplification of the equations of motion.
3.7 TRACK PREPROCESSOR
The computer simulation of the nonlinear dynamics of railroad vehicle systems
consists of two stages. In the first stage, the track geometry, based on the description
presented in the preceding sections, is defined. In the second stage, the equations
of motion of the railroad vehicle are solved using the track geometry input obtained
in the first stage. For the first stage, one often develops a preprocessor computer
code that can be used to define tracks with arbitrary geometry. The track preprocessor
code has input that is based on the definitions and terminology used by the railroad
industry. The output of the track preprocessor is a data file that is used as an input
to the main processor computer code used to solve the dynamic equations of the
multibody railroad vehicle system. In this section, the structure of the preprocessor
computer code that can be used to define the track geometry is discussed. The
preprocessor can be designed to read a simple and standard input and use the input
information to generate an output that can be used by the dynamic simulation code.
Based on the analysis presented in the preceding sections, one can recognize the
following basic tasks that can be performed by the track preprocessor (Berzeri et al.,
2000):
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Rail and Wheel Geometry
117
1. Read input information that defines the track segments.
2. Define the functions CH (S), θ(s), and φ(S) introduced in the preceding
sections. Use this information to determine the third Euler angle ψ as
described in the preceding section.
3. Calculate the coordinates x, y, and z of the nodes on the track centerline
by evaluating numerically the integrals presented in the preceding sections.
4. Define the space curves of the right and left rails using a finite number
of nodal points.
5. Create an output in a format that can be read by the dynamic simulation
code.
3.7.1 TRACK PREPROCESSOR INPUT
In order to define the input to the track preprocessor, each rail is divided into
segments. The input to the preprocessor provides information about the geometry
of each track segment. This information includes the length, the curvature, the superelevation, and the grade at the end points (nodes) of each segment. In the railroad
industry, this information is given in the form shown in Table 3.3. The first column
of this table contains the node number. The second column shows the distance of
each node from a selected origin, measured in feet along the track centerline. The
curvature is reported in the third column of Table 3.3, and generally in North America
its value is given using a 100-ft chord definition. Using Figure 3.10 and the information given in the third column of Table 3.3, the value of the curvature for a given
segment is obtained using the following equation:
CH =
sin(ψ 2)
50′
(3.78)
TABLE 3.3
Typical Railroad Input Entries Used to Define
a Curved Track
Node
No.
Distance
(ft)
Curvature
(deg)
Super-Elevation
(in.)
Grade
(%)
1
2
3
4
5
6
7
8
9
10
0
100
150
450
500
650
720
1020
1145
1195
0
0
5
5
−3
−3
7
7
0
0
0
0
1.5
1.5
−1
−1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
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118
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 3.15 Top view of the track given in Table 3.3.
The values of the super-elevation h for each node of the track are reported in the
fourth column of Table 3.3. To find the value of the bank angle φ, it is necessary to
know the value of the gage G. Using Figure 3.9, it is straightforward to obtain the
relationship
sin φ =
h
d
(3.79)
Finally, the grade is defined in the fifth column of Table 3.3.
The values reported in Table 3.3 correspond to the curved track shown in Figure
3.15. Based on the definitions given in the preceding sections, the first segment is
a tangent; the second segment is a tangent-to-curve entry spiral; the third segment
is a curve; the fourth segment is a curve-to-curve spiral; the fifth segment is a curve;
the sixth segment is again a curve-to-curve spiral; the seventh segment is a curve; the
eighth segment is a curve-to-tangent exit spiral; and the ninth segment is a tangent.
It is possible to determine the values of the functions CH , θ, and φ at any point
within the segment using Equations 3.62, 3.66, and 3.67 and a linear interpolation.
The value of the angle ψ is given by Equation 3.63. Finally, using Equation 3.69,
it is possible to write all these quantities as a function of the same variable s. The
values of these variables can be used with numerical integration to define the right
and left rail space curve, as described below.
3.7.2 NUMERICAL INTEGRATION
After obtaining the orientation angles as functions of the arc length s, the coordinates
defined by the integrals of Equation 3.55 can be evaluated. In principle, the results
of the integration define the coordinates of each point on the space curve with respect
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Rail and Wheel Geometry
119
FIGURE 3.16 Top view of a curve segment.
to the body coordinate system X lY lZ l for the left rail or XrYrZr for the right rail. Since
closed-form solutions of the coordinate integrals cannot be obtained in general,
numerical integration methods such as trapezoidal and Simpson rules or Gaussian
quadrature can be used (Atkinson, 1978).
In the case of curved tracks, the lengths of the segments of the right rail differ
from the lengths of the segments of the left rail. Since the input data are based on
the track centerline, the length of the segments of the right and left rails must be
adjusted during the process of the numerical integration. Figure 3.16 shows a segment of a curved track. In this figure, the value of the gage is exaggerated to show
the difference between the lengths of the arcs AB, ArBr, and AlBl as shown in Figure
3.17. It is clear from this figure that the right rail segment length can differ significantly from the left rail segment length. This difference, if not taken into consideration when the track geometry is defined, can have a significant effect on the
accuracy of the numerical solution when the equations of motion of the multibody
railroad vehicle system are integrated. A method for adjusting the length of the
segments of the right and left rails is described in Chapter 6.
FIGURE 3.17 Length of the right and left rail segments.
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120
Railroad Vehicle Dynamics: A Computational Approach
3.7.3 TRACK PREPROCESSOR OUTPUT
The output of the track preprocessor includes information that can be used to define
the space curve of the rails and the track centerline. The output of the track preprocessor has, as a minimum, the following information for each node:
1. The value of the arc length parameter s of the space curve at the node
2. The orientation angles θ, φ, and ψ of the space curve at each node
3. The location of each nodal point with respect to the initial reference frame
that will be used as the body rail or track frame in the multibody simulation
code
This information can be used to determine the properties of the space curve at
a specified arc length parameter s by using interpolation between the nodes during
the dynamic simulations. However, as will be shown in Chapter 5, some multibody
system formulations presented in this book require the evaluation of higher-order
derivatives of the tangents and the normal with respect to the arc length parameter
s in order to correctly define the contact constraints and forces. These higher derivatives, which can be up to the third derivative in some formulations, can be obtained
using a finite element formulation, as previously mentioned, or they can be expressed
in terms of the derivatives of Euler angles. Euler angle derivatives can be provided as
part of the track processor output, or they can be obtained during the dynamic
simulation by differentiation of the interpolation functions. If the higher derivatives
are to be included as part of the output, the output of the track processor can also
include the following information:
1. The longitudinal tangent vector given in Table 3.4 and its first and second
derivatives (Note that the longitudinal tangent is the first column of the
transformation matrix given by Equation 3.54.)
2. The first, second, and third derivatives of the orientation angles given in
Table 3.2.
TABLE 3.4
Track Longitudinal Tangent and Its Derivatives with Respect to s
Coordinate
t1
dt1/ds
d 2 t1/ds2
x
cosψ cosθ
−ψ ′sinψ cosθ − θ ′cosψ sinθ
y
sinψ cosθ
ψ ′cosψ cosθ − θ ′sinψ sinθ
z
sinθ
θ′cosθ
−ψ ″sinψ cosθ − (ψ ′)2cosψ cosθ
− θ ″cosψ sinθ − (θ ′)2cosψ cosθ
+ 2ψ ′θ ′sinψ sinθ
ψ ″cosψ cosθ − (ψ ′)2sinψ cosθ
− θ ″sinψ sinθ − (θ ′)2sinψ cosθ
− 2ψ ′θ ′cosψ sinθ
θ″cosθ − (θ′)2sinθ
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Rail and Wheel Geometry
121
Whether or not higher derivatives are included in the track preprocessor output,
this output can include, as previously mentioned, three different sets of data: one
set for the right rail, one set for the left rail, and one set for the track centerline.
3.7.4 USE OF THE PREPROCESSOR OUTPUT
SIMULATION
DURING
DYNAMIC
The output data of the track preprocessor are expressed in terms of the arc length
parameter of the space curve s = s1. The complete representation of the rail surface,
however, requires the use of two surface parameters, as described in Section 3.2. It
is assumed that the profile of the rail is known and can be described using the
function f(s2), where s2 is the second (lateral) rail surface parameter. If the rail profile
is constant along the track, Equation 3.50 can be written as a function of the two
surface parameters s1 and s2 as follows:
u r (s1, s2 ) = R rp (s1 ) + A rp (s1 )u rp (s2 )
(3.80)
Using the information included in the track preprocessor output, one can use linear
or cubical interpolation to define R rp (s1 ) and A rp (s1 ) and their derivatives with
respect to s1, as described in the preceding section. The vector u rp (s2 ) represents the
location of the contact point with respect to the profile coordinate system, as shown
in Figure 3.7, and is given as follows:
u rp (s2 ) = 0
s2
f (s2 ) 
T
(3.81)
Using Equation 3.80, one can define the two tangent vectors and the normal as
follows:

∂u rp dR rp dA rp (s1 ) rp
=
+
u (s2 ) 
ds1
ds1
∂s1


rp
rp
du (s2 )
∂u

t2 =
= A rp (s1 )

∂s2
ds2


n = t1 × t2



t1 =
(3.82)
where t1 and t2 are the two tangent vectors at a point on the rail that has the two
surface parameters s1 and s2 as coordinates. These two tangent vectors, which are
defined with respect to the rail coordinate system (body coordinate system) and are not
necessarily orthogonal, represent the tangent plane at this point. The normal vector n
is defined using the third equation in Equation 3.82 as the cross-product of the two
tangent vectors.
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122
Railroad Vehicle Dynamics: A Computational Approach
EXAMPLE 3.2
For the track described in Table 3.3, determine the location and orientation of the
profile frame at the nodes of the track centerline, assuming that the track has a gage
value equal to 1.42 m. Neglect the effect of the head width.
Solution. For an arbitrary node i, one can write
 0.0254 hi 
 G 
φi = sin −1 
i = 1, 2, … , 10
rad,
where h is the given super-elevation in inches at node i, and G is the gage. The angle
ψi can be written as
ψi =
π cri
180
where cri is the given curvature in degrees at the node. The angle θi can be defined as
 GRi 
 100 
θ i = sin −1 
i = 1, 2, …, 10
rad,
where GRi is the given grade percentage. The horizontal and vertical curvatures are
given by
CHi =
sin(ψ i 2)
Cv i =
θ i − θ i −1
si − si −1
i = 1, 2, … , 10
1/m,
50 × 0.3048
1/m,
i = 1, 2, … , 10
Because the grade is equal to zero for all nodes, one has
Si = Si + (si − si −1 ) cos θ i ,
ψ i = ψ i −1 +
CHi + CHi −1
2
i = 1, 2, … , 10
(S − S ),
i
i −1
i = 1, 2, …, 10
Finally, integrating Equation 3.55 numerically, the following are the locations and
orientations of the profile frame at each node:
45814_book.fm Page 123 Thursday, May 31, 2007 2:25 PM
Rail and Wheel Geometry
123
Node
no.
Distance, s
(m)
CH
(m−1)
θ
(deg)
φ
(deg)
ψ
(deg)
x
(m)
y
(m)
z
(m)
1
2
3
4
5
6
7
8
9
10
0
30.48
45.72
137.16
152.40
198.12
219.46
310.90
349.00
364.24
0.0
0.0
0.00286
0.00286
−0.00172
−0.00172
0.00401
0.00401
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.02932
0.02932
−0.01953
−0.01953
0.03909
0.03909
0.0
0.0
0.0
0.0
0.02181
0.28353
0.29225
0.21372
0.23813
0.60442
0.68073
0.68073
0.0
30.48
45.72
135.84
150.43
194.68
215.52
298.50
328.70
340.54
0.0
0.0
0.11
13.98
18.39
29.83
34.40
71.58
94.79
104.38
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3.8 WHEEL GEOMETRY
The surface of the wheel is a surface of revolution obtained by a complete rotation
of the curve that defines the wheel profile about the wheel axis, as shown in Figure
3.18. Therefore, the surface of the wheel can be defined mathematically by the
following equation:
 x0 + g ( s1 ) sin s2 


u( s1 , s2 ) = 
y0 + s1

 z − g ( s ) cos s 
1
2
 0
(3.83)
where s1 is the lateral surface parameter that represents the independent variable for
the wheel profile g(s1), and s2 is an angular surface parameter that represents the
rotation of the wheel profile about its axis. The variables x0, y0, and z0 are the position
coordinates of the origin of the profile coordinate system with respect to the wheelset
coordinate system or with respect to the wheel coordinate system if the two wheels
are not rigidly connected, as in the case of flexible axles. The values x0 and z0 can
be selected equal to zero, and y0 can be selected to be half of the back-to-back
distance between the two wheels of a wheelset.
FIGURE 3.18 Wheel surface.
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124
Railroad Vehicle Dynamics: A Computational Approach
Equation 3.83 shows that only one function, g(s1), is required to define the wheel
surface. It is convenient to use a general procedure based on a spline function
representation, as will be discussed in Chapter 6. In this case, measured data can be
used to define the wheel profile. Knowing the profile, one can determine the tangent
and normal vectors at an arbitrary point on the wheel surface. These vectors and
their derivatives with respect to the surface parameters are required to determine the
contact location and forces, as will be discussed in Chapter 5. The two tangent
vectors that can be used to represent the contact plane and the normal vector defined
with respect to the wheelset coordinate system can be written as follows:
t1 =
∂u
,
∂s1
t2 =
∂u
,
∂s2
n = t1 × t2
(3.84)
where u is the vector that defines the location of the contact point with respect to the
wheelset coordinate system and has the components given by Equation 3.83, t1 and
t2 are the two tangent vectors that define the tangent plane at the contact point,
and n is the normal vector.
EXAMPLE 3.3
For the wheel surface defined by Equation 3.83, determine the two tangent vectors and
the normal vector and their derivatives with respect to the two surface parameters at
an arbitrary point on the wheel surface.
Solution. The two tangent vectors and the normal vector are given as follows:
t1 = ∂u ∂s1
u
x
y
z
x0 + g(s1)sins2
y 0 + s1
z0 – g(s1)coss2
(dg(s1)/ds1)sins2
1.0
–(dg(s1)/ds1)coss2
t2 = ∂u ∂s2
g(s1)coss2
0.0
g(s1)sins2
n = t1 × t2
g(s1)sins2
–g(s1)(dg(s1)/ds1)
–g(s1)coss2
The derivatives of the tangent and normal vectors can be determined as follows:
First Derivative with Respect to s1
∂ t1 ∂s1
x
y
z
∂ t2 ∂s1
(d2 g(s1)/ds12)sins2
0.0
2
2
–(d g(s1)/ds1 )coss2
∂n ∂s1
(dg(s1)/ds1)coss2
(dg(s1)/ds1)sins2
0.0
–(dg(s1)/ds1)2 – g(s1)(d2 g(s1)/ds12)
–(dg(s1)/ds1)coss2
(dg(s1)/ds1)sins2
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Rail and Wheel Geometry
125
First Derivative with Respect to s2
∂ t1 ∂s2
x
y
z
∂ t2 ∂s2
(dg(s1)/ds1)coss2
0.0
(dg(s1)/ds1)sins2
∂n ∂s2
–g(s1)sins2
0.0
g(s1)coss2
g(s1)coss2
0.0
g(s1)sins2
Second Derivative with Respect to s1
∂2 t1 ∂s1
x
2
∂2 t2 ∂s1
2
∂2 n ∂s1
y
(d3 g(s1)/ds13)sins2
0.0
(d2 g(s1)/ds12)coss2
0.0
z
–(d3 g(s1)/ds13)coss2
(d2 g(s1)/ds12)sins2
2
(d2 g(s1)/ds12)sins2
–3(dg(s1)/ds1)(d2 g(s1)/ds12)
–g(s1)(d3 g(s1)/ds13)
–(d2 g(s1)/ds12)coss2
Second Derivative with Respect to s2
∂2 t1 ∂s2
2
x
y
z
∂2 t2 ∂s2
∂2 n ∂s22
–g(s1)coss2
0.0
–g(s1)sins2
–g(s1)sins2
0.0
g(s1)coss2
2
–(dg(s1)/ds1)sins2
0.0
(dg(s1)/ds1)coss2
Second Derivative with Respect to s1 and s2
∂2 t1 ∂s1∂s2
x
y
z
(d2g(s1)/ds12)coss2
0.0
(d2g(s1)/ds12)sins2
∂2 t2 ∂s1∂s2
–(dg(s1)/ds1)sins2
0.0
(dg(s1)/ds1)coss2
∂2 n ∂s1∂s2
(dg(s1)/ds1)coss2
0.0
(dg(s1)/ds1)sins2
45814_book.fm Page 126 Thursday, May 31, 2007 2:25 PM
45814_book.fm Page 127 Thursday, May 31, 2007 2:25 PM
4
Contact and Creep-Force
Models
The formulation of the contact forces that describe the dynamic interaction between
the wheel and the rail is one of the fundamental problems that must be addressed
in developing a multibody system formulation for railroad vehicle systems. Various
contact models are used in different computer formulations to describe the wheel/rail
interaction. In this chapter, the rigid body contact is discussed, with particular
emphasis on Hertz theory and the formulations of the wheel/rail creep forces. The
definitions and concepts introduced in this chapter will be used in the formulation
of the normal and tangential forces that describe the wheel/rail interaction.
The contact between two rigid bodies can be at a single point or area, depending
on the shape of the two bodies. These two types of contact are known as nonconformal or conformal contact, respectively. If the shape of the two bodies is such that
the two bodies in the region of contact fit exactly or even closely together, the contact
is defined as a conformal contact. If the two bodies, on the other hand, touch at a
point or a line, the contact is called nonconformal. If an external load is applied on
each body, the two bodies will deform at the contact point to form an area of contact.
The contact area in the case of the nonconformal contact is small as compared with
the dimensions of the two bodies. In 1882, Heinrich Hertz presented a contact theory
that accounts for the shape of the surfaces in the neighborhood of the contact area
(Hertz, 1882). Hertz assumed that the area of contact is elliptical. In wheel/rail
dynamics, the assumption of nonconformal contact is often used, since the shapes
of the wheel and rail surfaces are significantly different. In this case, the use of Hertz
theory to examine the contact geometry and the maximum stresses can be justified.
Due to the elasticity of the bodies and the externally applied normal load, some
points on the surfaces in the contact region may slip while others may stick when
the two bodies move relative to each other. The difference between the tangential
strains of the bodies in the adhesion area leads to a small apparent slip. This slip is
called creepage and is defined using the kinematics of the two bodies. Creepages
generate tangential creep forces and creep spin moment; all will sometimes be
referred to collectively in this book as the creep forces. For example, in the case of
the wheel/rail contact, tangential forces and spin moment are generated, since the
motion of the wheel relative to the rail is a combination of rolling and sliding. These
creep forces and moment have a significant effect on the steering and stability of
railroad vehicle systems.
In this chapter, Hertz theory, which is often used in the study of the wheel/rail
contact problem, is first discussed. The creepages, the normalized relative velocities
that enter into the calculation of the creep forces, are then defined. It is shown that
most creep-force models are expressed in terms of the creepages. Different creep-force
127
45814_book.fm Page 128 Thursday, May 31, 2007 2:25 PM
128
Railroad Vehicle Dynamics: A Computational Approach
theories can be used; some of these theories are based on linear models, while the
others employ nonlinear force-creepage relationships. The chapter concludes with
a discussion of some of the creep theories that are used in railroad vehicle system
formulations.
4.1 HERTZ THEORY
In 1882, Hertz introduced a contact theory that accounts for the shape of the surfaces
in the neighborhood of the area of contact between nonconformal bodies (Hertz,
1882). In this theory, it is assumed that the contact area is, in general, elliptical. This
assumption was based on an observation of interference fringes at the contact of
two glass lenses (Johnson, 1985). In the Hertz theory, it is further assumed that each
body is an elastic half-space loaded over a small elliptical region. The assumption
of elastic half-space implies that the concentrated contact stresses can be treated
separately from the general distribution of the stresses in the two bodies due to their
shape and the way in which they are supported. This assumption is valid when the
dimensions of the contact area are small compared with the dimensions of the two
bodies and the relative radii of curvature of the two surfaces. Hertz theory is based
on the assumption of small deformation of two elastic bodies due to the static
compression, and it neglects the effect of the friction forces. For the wheel/rail
contact problem, Hertz theory is the most commonly used theory to determine the
shape of the contact area and the local deformation of the wheel and rail surfaces
at the contact region. The assumptions used in Hertz theory can then be summarized
as follows:
1. The surfaces of the bodies are continuous and nonconformal.
2. The strains are small.
3. The stress resulting from the contact force vanishes at a distance far from
the contact area.
4. The surfaces are frictionless.
5. The bodies are elastic, and no plastic deformation occurs in the contact area.
In the case of wheel/rail contact, most researchers assume that these assumptions
are valid.
4.1.1 GEOMETRY
AND
KINEMATICS
Following these assumptions, Hertz assumed that if two bodies i and j are in contact
as shown in Figure 4.1, the shape of the surface of each body in the region close to
the origin can be defined as follows:
z i = Ai ( x i )2 + Bi ( y i )2 + C i ( x i y i ) + 

z j = A j ( x j )2 + B j ( y j )2 + C j ( x j y j ) + 
(4.1)
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Contact and Creep-Force Models
129
FIGURE 4.1 Two bodies in contact.
where Ak, Bk, and C k (k = i, j) are constants that depend on the body geometry, and
x k and y k (k = i, j) are the axes of the two bodies at the contact plane. The gap
between the two surfaces at the origin can then be defined as
h = zi − z j
(4.2)
This equation, upon the use of Equation 4.1, leads to
h = A( x )2 + B( y )2 + Cxy + (4.3)
In this equation, A, B, and C are constants. The axes x k and y k (k = i, j) and x and
y can be chosen such that the terms x ky k (k = i, j) and xy are zero. In this case,
Equations 4.1 and 4.3 can be written, after neglecting higher-order terms, as follows:




 1

1
j 2 
j
j 2
z = −
(x ) +
(y )  
2 R2j
 2 R1j

zi =
1
1
( x i )2 +
( y i )2
i
2 R1
2 R2i
(4.4)
and
h = A( x ) 2 + B( y ) 2 =
1
1
( x )2 +
( y )2
2 R1
2 R2
(4.5)
where R1k and R2k (k = i, j) are, respectively, the principal radii of curvature of the
surfaces of bodies i and j at the origin; and R1 and R2 are the principal relative radii
45814_book.fm Page 130 Thursday, May 31, 2007 2:25 PM
130
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 4.2 Contact plane.
of curvature. That is, ∂2 z k /∂x k = 1/R1k and ∂2 z k /∂y k = 1/R2k . Note that the axes x i and
x j are not in general parallel, and their orientation can differ by an angle ψ. These
two axes can differ from the common x-axis by arbitrary angles α and β, as shown
in Figure 4.2a. Using Equation 4.2 and the transformation of the axes x i and x j to
the x-axis, the term C in Equation 4.3 is defined as follows:
2
C=
2
1 1
1
1 1
1
− j  sin 2β −  i − i  sin 2α
j

2  R1 R2 
2  R1 R2 
(4.6)
Therefore, Equation 4.5 is satisfied if C in Equation 4.6 is equal to zero, and this
leads to
1 1
1
1 1
1
− j  sin 2α =  i − i  sin 2β

j
2  R1 R2 
2  R1 R2 
Based on the preceding equation, one can draw the triangle shown in Figure 4.2b.
This triangle can be used to define the following equation:
B− A=
1 1
1
1 1
1
− i  cos 2α +  j − j  cos 2β

i
2  R1 R2 
2  R1 R2 
(4.7)
2
2
 1
1  1
1
1  1
1  1
1
=
− i  +  j − j  + 2  i − i   j − j  cos 2ψ

i
2  R1 R2   R1 R2 
 R1 R2   R1 R2 
and
A+ B =
1 1
1
1
1
+ i + j + j
i

2  R1 R2 R1 R2 
(4.8)
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Contact and Creep-Force Models
131
EXAMPLE 4.1
To show how Equation 4.5 can be obtained from Equation 4.4, one may write these
equations in the following form:
T
T
z i = u i Ci u i ,
z j = u j C ju j ,
h = u T Cu
where
xi 
ui =  i  ,
y 
x j 
x
, u= 
j
y
 y
 
uj = 
and
1
 i
1 R1
Ci = 
2
0

 1
 j
 , C j = − 1  R1
2
1 
0
i 
R2 



1

 , C = 1  R1
2
1 
0
j 
R2 

0 

0 
0 

1 

R2 
It is also clear that
u i = Ai u i ,
u j = A ju j
where
 cos α
− sin α 
 sin α
cos α 
Ai = 
 cos β
sin β 
 − sin β
cos β 
Aj = 
,

and
x j 
x i 
j
,
u
=
 j
i
y 
y 
ui = 
It follows that
T
T
h = z i − z j = u i Ci u i − u j C j u j
T
T
= u T Ai Ci Ai u − u T A j C j A j u
(
T
T
)
= u T Ai Ci Ai − A j C j A j u
45814_book.fm Page 132 Thursday, May 31, 2007 2:25 PM
132
Railroad Vehicle Dynamics: A Computational Approach
which can be written as
h = z i − z j = u T Cu = A( x ) 2 + B( y) 2 + Cxy
where
( )
T
T
C = Cij = A C A − A C A
i
i
i
j
j
j
is a symmetric matrix whose elements are defined using the preceding equation as
C11 =
1 1
1
1
1
2 
2
2
2
 i cos α + i sin α + j cos β + j sin β 
2  R1
R2
R1
R2

C12 = C21 =
C22 =

 1
1  1
1
1
 j − j  sin 2β −  i − i  sin 2α 
4  R1 R2 
 R1 R2 

1
1
1
1 1
2
2
2
2 
 i sin α + i cos α + j sin β + j cos β 
2  R1
R2
R1
R2

This leads to A = C11 , B = C22 , and C = C12 + C21 = 2C12 . Note that the coefficient
given by Equation 4.6 is obtained as C = 2C12 . If the x and y axes are chosen to be the
principal directions, the matrix C must be diagonal, and its diagonal elements are the
principal values. For this condition to be satisfied, one must have C = 2C12 = 0, that is,
C = 2C12 =
1 1
1
1 1
1
−
sin 2β −  i − i  sin 2α = 0
2  R1j R2j 
2  R1 R2 
which is the same as equating the expression of Equation 4.6 to zero.
It is clear from Equation 4.5 that the contours of the gap h between the two
bodies are ellipses with axes ratio equal to ( A/B). To determine the size of the
contact ellipse, assume that an external normal load Fn is applied to press the two
bodies against each other. Due to the pressure applied on the bodies, the surfaces
of body i and j will be displaced vertically by the distance ui and u j, respectively,
as shown in Figure 4.3. If the two points Pi and Pj coincide, the total deformation
can be given as follows:
ui + u j + h = δ i + δ j
(4.9)
where h is given by Equation 4.2. Therefore, if the two bodies are deformed and
the total deformation is given as δ = δ i + δ j, the following equation, after substituting
h from Equation 4.5 into Equation 4.9, must be satisfied:
u i + u j = δ − A( x )2 − B( y)2
(4.10)
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Contact and Creep-Force Models
133
FIGURE 4.3 Contact bodies under externally applied normal load.
4.1.2 CONTACT PRESSURE
Since the contact area is assumed small compared with the dimensions of the two
bodies, one can consider the two bodies in contact as semi-infinite. The contact
pressure can be assumed to satisfy the following requirements for the equilibrium
of the two bodies (Goldsmith, 1960):
1. The total applied force Fn must be equal to the total resisting force generated by the vertical component of the pressure p in the contact area, that is,
Fn =
∫∫ pdxdy .
2. The components of displacement vanish at infinity, that is, the displacement at a distance away from the contact region can be neglected.
3. The normal stresses outside the contact region are assumed to be zero.
4. The normal stresses acting on the two bodies are in balance within the
contact region.
5. The shear stresses τxz and τyz along the surfaces of the bodies are zeros.
These conditions can be satisfied by assuming that the pressure p is a quadratic
function of x and y; therefore, the pressure distribution in the contact area is assumed
to take the following form:
2
p = p0
 x
 y
1−   −  
 a
 b
2
(4.11)
In this equation, p0 is a constant, and a and b are the lengths of the ellipse semiaxes. As shown in Appendix A, the pressure p produces a displacement u given by
(Goldsmith, 1960; Johnson, 1985; Love, 1944)
45814_book.fm Page 134 Thursday, May 31, 2007 2:25 PM
134
Railroad Vehicle Dynamics: A Computational Approach
u=
1− ν 2
( L − M ( x ) 2 − N ( y) 2 )
πE
(4.12)
where E is Young’s modulus of elasticity, ν is the Poisson’s ratio, and
π p0 ab
M=
2
N=
L=
π p0 ab
2
π p0 ab
2
∞
∫
0
∞
∫
0
∞
∫
0



(a 2 + w)3 (b 2 + w) w


2

dw
π p0 b  a 
= 2 2   Ee − Ke  
 
(a 2 + w)(b 2 + w)3 w e a  b 


dw

= π p0 bKe
2
2

(a + w)(b + w) w

dw
=
π p0 b
Ke − Ee
e 2a 2
(
)
(4.13)
where Ee and Ke are the complete elliptical integrals of argument e = 1 − b 2 /a 2 , b <
a, and are given in Appendix B. Using Equation 4.12, the displacement ui + u j can
be written as
u i + u j = ( L − M ( x )2 − N ( y)2 ) /π E ij
(4.14)
1 1 − (ν i )2 1 − (ν j )2
=
+
E ij
Ei
Ej
(4.15)
where
Since the pressure distribution is semi-ellipsoidal, the total normal load Fn is
given by
Fn =
2
p0π ab
3
(4.16)
Using Equations 4.11 and 4.16, one obtains (Goldsmith, 1960; Hertz, 1882; Love,
1944)
2
p=
 x  y
3Fn
1−   +  
2π ab
 a  b
2
(4.17)
Using Equations 4.10 and 4.14, ui + u j can be written as follows:
δ − Ax 2 − By 2 =
L − M ( x ) 2 − N ( y) 2
π E ij
(4.18)
45814_book.fm Page 135 Thursday, May 31, 2007 2:25 PM
Contact and Creep-Force Models
135
Equating the coefficients in Equation 4.18, one has
δ=
A=
B=
L
pb
= 0ij Ke
ij
πE
E
(4.19)
(
M
pb
= ij 02 2 Ke − Ee
ij
πE
E ea
N
pb
= ij 02 2
ij
πE
E ea
)
(4.20)
 a  2

  Ee − Ke 
 b 

(4.21)
Using these equations, one obtains
B R1 (a /b)2 Ee − Ke
=
=
A R2
Ke − Ee
1
AB =
2
1
p b
= 0
R1R2 E ij a 2e 2
((a/b) E − K )(
2
e
e





Ke − Ee 

(4.22)
)
The contact ellipse semi-axes are defined as follows:
(
(
)
)
(4.23)
(
(
)
)
(4.24)
a = m 3π Fn K1 + K 2 4 K3
b = n 3π Fn K1 + K 2 4 K3
13
13
where K1 and K2 are constants that depend on the material properties of the two
bodies and are given as follows:
K1 =
( )
1− νi
π Ei
2
,
K2 =
( )
1− ν j
2
πE j
(4.25)
In Equations 4.23 and 4.24, K3 is a constant that depends on the geometric properties
of the two bodies and is defined as follows:
K3 = A + B =
1 1
1
1
1
+
+
+
2  R1i R2i R1j R2j 
(4.26)
45814_book.fm Page 136 Thursday, May 31, 2007 2:25 PM
136
Railroad Vehicle Dynamics: A Computational Approach
TABLE 4.1
Hertz Coefficients m and n
θ
(deg)
m
n
θ
(deg)
m
n
0.5
1
1.5
2
3
4
6
8
61.4
36.89
27.48
22.26
16.5
13.31
9.79
7.86
0.1018
0.1314
0.1522
0.1691
0.1964
0.2188
0.2552
0.285
10
20
30
35
40
45
50
55
6.604
3.813
2.731
2.397
2.136
1.926
1.754
1.611
0.3112
0.4125
0.493
0.530
0.567
0.604
0.641
0.678
θ
(deg)
m
n
60
65
70
75
80
85
90
1.486
1.378
1.284
1.202
1.128
1.061
1.0
0.717
0.759
0.802
0.846
0.893
0.944
1.0
Source: Hertz, H., Über die berührung fester elastische Körper und über die Harte,
Verhandlungen des Vereins zur Beförderung des Gewerbefleisses, Leipzig, Nov. 1882.
The coefficients m and n in Equations 4.23 and 4.24 are given by Hertz in Table 4.1
as functions of the angular parameter θ for the values of θ between 0° and 180°
(Hertz, 1882), where θ is defined as
(
θ = cos−1 K 4 /K3
)
(4.27)
where
K4 = B − A
=
2
2
(4.28)
 1
1  1
1  1
1
1  1
1
+
cos
+
−
−
−
2
−
2
ψ
 Ri Ri   R j R j 
2  R1i R2i   R1j R2j 
1
2
1
2
In the computer implementation of this formulation, the coefficients m and n
can be interpolated for a given value of the angle θ using the entries of Table 4.1.
These coefficients are needed to calculate the semi-axes a and b. One can use linear
or cubic spline interpolation to determine these coefficients using the values of Table
4.1. An alternative approach to the numerical interpolation is to develop closed-form
expressions for the coefficients m and n as functions of θ. The following closedform equations were proposed by Berzeri (Shabana et al., 2001):
m = Am tan(θ − π 2) +




+ Dn sin θ 

Bm
+ Dm
θ Cm
1
n=
+ Bnθ Cn
An tan(θ − π 2) + 1
(4.29)
45814_book.fm Page 137 Thursday, May 31, 2007 2:25 PM
Contact and Creep-Force Models
137
TABLE 4.2
Coefficients Used for the Closed-Form
Functions m and n
Coeff.
Value
Coeff.
Value
Am
Bm
Cm
Dm
−1.086419052477
−0.106496432832
1.350000000000
1.057885958251
An
Bn
Cn
Dn
−0.773444080706
0.256695354565
0.200000000000
−0.280958376499
Source: Shabana, A.A., Berzeri, M., and Sany, J.R., ASME
Journal of Dynamic Systems, Measurement, and Control, 123,
168, 2001. With permission.
where the value of θ is given in radians, and the coefficients Ak, Bk, Ck, and Dk (k =
m,n) are given in Table 4.2 (Shabana et al., 2001). Equation 4.29 provides a good
approximation for m and n. Note also that the first equation of Equation 4.29 captures
the asymptotic behavior of the function m when θ approaches zero.
Using Equations 4.19 and 4.24, the Hertz force law can be defined as follows:
Fn = K hδ 3 2 =
(
4β
3 K1 + K 2
)
A+ B
δ3 2
where β is a constant and is given in Table 4.3 (Goldsmith, 1960).
TABLE 4.3
Hertz Coefficient β
for Hertz Force
A/B
β
1.0
0.7041
0.4903
0.3333
0.2174
0.1325
0.0718
0.0311
0.00765
0.3180
0.3215
0.3322
0.3505
0.3819
0.4300
0.5132
0.6662
1.1450
(4.30)
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Railroad Vehicle Dynamics: A Computational Approach
4.1.3 COMPUTER IMPLEMENTATION
In the case of the wheel/rail contact problem, the wheel (body i) and the rail (body j)
have the radii of curvature shown in Figure 4.4. In this figure, R1i and R1j are the
principal rolling radii of the wheel and the rail, respectively, and R2i and R2j are the
principal transverse radii of curvature of the wheel and the rail, respectively. The
radius of curvature of a body is considered to be positive if the corresponding center
of curvature is within the body, as shown in Figure 4.4 (Garg and Dukkipati, 1988).
Knowing the normal contact force or the deformation of the two bodies (penetration), the rolling radii and the principal transverse radii of the wheel and the rail
(which can be determined from the geometry, as described in the preceding chapter),
and the coefficients m and n (which can be determined using interpolation or the
closed-form equations), one can determine the contact ellipse semi-axes using Equations 4.24 and 4.25. It is important to distinguish between the longitudinal and the
lateral semi-axes, since the creep forces are functions of the dimensions of the contact
ellipse, as will be shown in the next section. In general, one can determine the
direction of the contact ellipse based on the radii of curvature for the two bodies in
contact. The following rule can be applied: if
1
1
1
1
+ j ≥ i + j,
i
R1 R1 R2 R2
then the transverse semi-axis of the contact ellipse (in the y direction) is greater than
or equal to the longitudinal semi-axis. Conversely, if
1
1
1
1
+ j ≤ i + j,
i
R1 R1 R2 R2
then the transverse semi-axis (in the y direction) is less than or equal to the longitudinal semi-axis. Using Equations 4.24 and 4.26 and Table 4.1, it is clear that the
larger dimension of the contact ellipse is associated with the coefficient m.
FIGURE 4.4 Wheel and rail radii of curvature.
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Contact and Creep-Force Models
139
EXAMPLE 4.2
Each of the two wheels of a single wheelset traveling on a tangent track is assumed
at a given time to have a normal load equal to 45 kN. The right wheel rolling radius
is assumed at this time to be 0.457 m. The wheel profile is assumed to be conical, and
the transverse radius of curvature of the rail is equal to 0.254 at this configuration.
Assuming the wheelset has zero yaw rotation, determine the dimensions of the contact
ellipse and the maximum contact pressure if the wheel and the rail are made of steel.
Solution. Since the wheel and the rail are made of steel, one can assume the following
material properties for the wheel and the rail: Poisson’s ratio ν is 0.28, and Young’s
modulus of elasticity is 2.1 × 1011 N/m2. Since the wheel is conical, R2w is equal to ∞,
and since the rail is tangent, the longitudinal rolling radius R1r is equal to ∞. Equations
4.23 and 4.26 lead to
K1 =
K2 =
K3 =
1 1

2  R1w
+
( )
1− νw
2
π Ew
( )
1− νr
2
π Er
1
R2w
+
=
1
R1r
=
+
1 − (0.28) 2
2.1 × π × 1011
1 − (0.28) 2
2.1 × π × 1011
= 1.396 × 10 −12
m 2 /N
= 1.396 × 10 −12
m2/ N
1 
1 1
1 
−1
 = 2  0.457 − 0 + 0 + 0.254  = 3.064 m


R2r 
Since the wheelset has zero yaw rotation, then ψ = 0, and one has
K4 =
=
1  1
2  R1w
1 
−
1 
2
2
 1
 1
1 
1  1
1 
+  r − r  + 2  w − w   r − r  cos 2 ψ
w 
 R1 R2   R1 R2 
R2   R1 R2 
2
2
 
 1

1 
1 
− 0 +  0 −
+ 2
−0 0−
cos(0)
 
 0.457  
0.254 
2  0.457
0.254 
= 0.875
1
m -1
Using Equation 4.27, the argument for the m and n coefficients can be evaluated as
(
)
θ = cos−1 K 4 /K3 = 1.281 rad
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Railroad Vehicle Dynamics: A Computational Approach
Using the closed-form equations given in Equation 4.29 and Table 4.2,
m = 1.306,
n = 0.813
From Equations 4.24 and 4.25
(
)
13
= 0.599 × 10 −2 m
(
)
13
= 0.373 × 10 −2 m
a = m 3π Fn K1 + K 2 4 K3 
b = n 3π Fn K1 + K 2 4 K3 
It is clear that
1
1
1
1
+
<
+
R1w R1r R2w R2r
Therefore, the longitudinal contact ellipse semi-axis dimension is equal to a, and the
transverse semi-axis dimension is equal to b. The maximum contact pressure can be
determined from Equation 4.17 as follows:
p0 =
3
Fn = 96.075 MPa
2π ab
The contact area is equal to πab = 0.703 × 10−4 m2.
4.2 CREEP PHENOMENON
The relative motion between two bodies i and j that are in contact can be the result
of rolling and sliding motion. In the general case of rolling and sliding, the two
bodies have different velocities vi and vj at the contact point and different angular
velocities ω i and ω j. The relative angular velocity along the normal to the surfaces
at the contact point is called the spin. If the velocities vi and vj at the contact point
are not equal, the rolling motion is accompanied by sliding. If ω i and ω j are not
equal, the motion is accompanied by rolling and/or spin. When rolling occurs without
sliding or spin, the motion is considered to be pure rolling. In the case of the contact
of two elastic bodies subjected to external applied normal load, some contact points
on the contact surface may slip, while other points may stick. The difference between
the tangential strains of two bodies in the adhesion area leads to a small slip that is
called creepage. The creepage is, therefore, due to a combination of elastic deformation and friction. This phenomenon was recognized in 1926 by Carter (Carter,
1926).
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Contact and Creep-Force Models
141
FIGURE 4.5 Wheel rolling over a rail.
Consider the case of a wheel rolling on a rail, as shown in Figure 4.5. Let t1r
and t r2 , as shown in Figure 4.6, be the unit orthogonal tangents to the rail at the
contact point in the longitudinal and lateral directions, and let nr be the unit normal
to the surfaces at the contact point, that is,
nr =
t1r × t 2r
t1r × t 2r
(4.31)
The absolute velocity of the wheel w is assumed to be vw. The magnitude of the
wheel velocity along the longitudinal tangent defined at the contact point is given by
T
V = v w t1r
(4.32)
The creepages are normalized relative velocities that are defined as follows:
(v w − v r )T t1r 

V

w
r T r 
(v − v ) t 2 
ζy =

V

w
r T r 
(ω − ω ) n 
ϕ=
V

ζx =
FIGURE 4.6 Contact frame.
(4.33)
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Railroad Vehicle Dynamics: A Computational Approach
These creepage expressions are fundamental in the kinematic and force analysis of
the wheel/rail interaction. In the case of pure rolling with no lateral oscillations, one
can show that
ζx = ζy = 0
(4.34)
In the multibody system formulations, the global velocity of an arbitrary point
on an arbitrary rigid body i can be defined, as described in Chapter 2, as follows:
i + ω i × ui
r i = R
(4.35)
where the vector Ri is the global velocity vector of the origin of the body coordinate
system, ui is the local position vector of the arbitrary point on body i defined in the
global frame, and ω i is the absolute angular velocity vector of the body coordinate
system defined in the global coordinate system. This angular velocity vector is given
as
ω i = ω xi
ω iy
ω zi 
T
(4.36)
If a wheel w is in contact with a rail r at point P whose global position is defined
using the coordinates of the two bodies by the two vectors rPw and rPr , respectively,
the global velocity vector of the contact point can be defined in terms of the
coordinates of the two bodies as follows:
w + ω w × uw 
rPw = R
P

r + ω r × ur 
rPr = R
P 
(4.37)
The velocities of Equations 4.36 and 4.37 can be used in Equation 4.33 to define
the creepages in terms of the generalized coordinates and velocities of the two bodies
as follows:
ζx =
(rPw − rPr ) ⋅ t1r
,
V
ζy =
(rPw − rPr ) ⋅ t r2
,
V
ϕ=
(ω w − ω r ) ⋅ n r
V
(4.38)
These definitions of the creepage are the general expressions used in the nonlinear
analysis of multibody railroad vehicle systems. Note that these expressions are
functions of the geometry of the rail at the contact point. These creepage expressions,
however, are linearized and simplified in some specialized railroad vehicle system
formulations. The effect of this linearization is discussed in more detail in Chapter 8.
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Contact and Creep-Force Models
143
EXAMPLE 4.3
A wheelset, denoted as body w, is traveling on a tangent track as shown in Figure 4.7.
At the initial configuration, the location of the right wheel contact point with respect
to the wheelset coordinate system is given by u rw = [0 –a –r0]T, where a is the lateral
position of the right wheel geometric center and is assumed to be equal to 0.75 m, and
r0 is the initial rolling radius and is assumed to be equal to 0.457 m. The wheelset is
assumed to have a forward velocity of 10 m/s. The orientation of the wheelset is defined
with respect to a global frame by the Euler angles θw = [ψ w φ w θ w]T = 0 at the initial
configuration. The global angular velocity vector of the wheelset is given by ωw =
[ω xw ω yw ω zw ]T . It is assumed that the angle between the lateral tangent and the wheelset
axis at the contact point is defined by the contact angle δk for contact k (k = r, l), as
shown in Figure 4.7 (subscripts r and l are used here to denote right and left contact,
respectively). Determine the creepage values for the right wheel contact at this configuration assuming that δr = 0.025 rad.
Solution. The orientation of the wheelset can be defined with respect to the global
coordinate system using the transformation matrix given in Chapter 2. The sequence
of rotation is assumed as follows: a rotation ψ w about the Zw axis, a rotation φ w about
the Xw axis, and a rotation θ w about the Yw axis. Accordingly, the rotation matrix of
the wheelset w is given as follows:
Aw =
 cos ψ w cos θ w − sin ψ w sin φ w sin θ w

w
w
w
w
w
 sin ψ cos θ + cos ψ sin φ sin θ
w
w

− cos φ sin θ
− sin ψ cos φ
w
cos ψ cos φ
w
sin φ
w
w
cos ψ sin θ + sin ψ sin φ cos θ
w
sin ψ sin θ − cos ψ sin φ cos θ
w
w
w
1

A = I = 0
 0
FIGURE 4.7 Wheel set frame.
w
w
0
0
1
0
0

1 
w
w
cos φ cos θ
w
At the initial configuration, one has
w
w
w
w
w




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144
Railroad Vehicle Dynamics: A Computational Approach
The global position vector of the center of mass of the wheelset at the given configuration can be written as
R w =  Rxw
Ryw
T
Rzw  =  0
0
0.457 
T
The global velocity of the right wheel geometric center is given by
w + ω w × (Aw u w )
r0w = R
0
where u 0w = [0 –a 0]T = [0 –0.75 0]T is the local position vector of the geometric
center of the right wheel defined in the body coordinate system. Assuming pure rolling
at the initial configuration, the angular velocity can be written as

V

r0
ω w = 0

T
0  =  0
21.882

0 
T
Thus,
10 
10 


 
r =  0  + 0 =  0 
 0 
 0 
w
0
The global velocity vector of the contact point is
w + ω w × (Aw u w ) =  0
rPw = R
P

0
0 
T
The tangents and the normal can be defined in the global coordinate system as follows:
 t1r
1

n r  = A w  0
t r2
 0
0 
0
cos δ r
− sin δ r

sin δ r 
cos δ r 
Note that at the initial configuration, Aw = I. Assuming that the rail is fixed, the creepage
expressions are defined as follows:
ζx =
rPw ⋅ t1r
V
,
ζy =
rPw ⋅ t r2
V
,
ϕ=
ω w ⋅ nr
V
w ⋅ t r . Using the data and the results of this example, one can show that
where V = R
1
the creepages are given by
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Contact and Creep-Force Models
ζx =
145
0
10
= 0, ζy =
0
10
=0
and
ϕ=
V
 r  sin δ r
0
V
=
sin δ r
r0
=
sin(0.025)
0.457
= 0.0547
As a result, the longitudinal and lateral creepages are zero, while the spin creepage is
not equal to zero. This spin creepage is called geometric spin and is attributed to the
component of the wheel angular velocity along the spin axis. This component is defined,
as shown in the preceding equation, by ϕ = (sin δr)/r0, which is function of the geometric
parameters δr and r0 only.
4.3 WHEEL/RAIL CONTACT APPROACHES
When a wheel travels on a rail, the creepages at the contact point generate tangential
forces that play an important role in the steering and stability of railroad vehicles.
According to Hertz contact theory, the contact area, as described in Section 4.1, is
assumed to be elliptical. Due to the compressive forces in the contact region, the
wheel and the rail will deform. The deformations can be defined in the coordinate
system XrcYrcZrc, whose origin is defined at the contact point, as shown in Figure
4.5. If the two bodies move relative to each other in the presence of Coulomb’s
friction, tangential forces are developed in the contact region. Tangential forces are
also developed if the bodies spin relative to each other.
In general, the contact area is divided into two regions; the adhesion region,
where the surface particles of the bodies do not slide relative to each other, and the
slip region, where there is sliding. Based on the assumptions stated in Section 4.1,
Hertz theory does not consider the shear traction Ft = [Ftx Fty]T between the two
is defined in the global coordinate
bodies in the contact region. The true slip w
system as the relative velocity on the tangent plane. In most railroad vehicle formulations, for simplicity, the true slip is determined based on pure rigid body kinematics. In this case, this slip component is defined as follows:
( )
(
T
)
x , y = r wr − r wr n r n r
w
(4.39)
In this equation, nr is the normal to the surface at the contact point, and r wr is the
relative velocity at the contact point defined as
r wr = r w − r r
(4.40)
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Railroad Vehicle Dynamics: A Computational Approach
The components of the slip given by Equation 4.39 can be defined in a coordinate
system defined at the contact point using the following projection:
 ⋅ t1r   rPwr ⋅ t1r 
x, y =  w1  =  w
w
=

2
⋅ t 2r  rPwr ⋅ t r2 
w
w
( )
(4.41)
where n r ⋅ t1r = n r ⋅ t 2r = 0 . The preceding tangential slip defined on the contact plane
can be used to define the tangential traction Ft using Coulomb’s law as
T
Ft =  Ftx
Fty  ≤ µ p;
µ pw
Ft = −
;
w
= 0 (adhesion area ) 
w




≠ 0 (slip area )
w


(4.42)
where µ is the friction coefficient and p is the contact pressure. Therefore, the
longitudinal and lateral creep forces and the spin creep moment can be expressed
as follows:
Fx =
∫∫
Fy =
∫∫
M=
∫∫





Fty dx dy




( xFty − yFtx ) dx dy 

Ftx dx dy
(4.43)
The creep forces and the spin moment depend on the creepages, the contactellipse dimension, and the normal force. The relationship between the longitudinal,
lateral, and spin creepages and the longitudinal and lateral forces and spin moments
are governed by the creep-force law (Kalker, 1990). Some creep-force law classifications are discussed below (Garg and Dukkipati, 1988).
4.3.1 EXACT THEORY
OF
ROLLING CONTACT
In the exact theory of rolling contact, the constitutive law is obtained by deriving
the traction-displacement relationship using the general elasticity theory. In this case,
the wheel and the rail near the contact region are treated as elastic half-space, an
assumption that is employed in most railroad vehicle formulations (Kalker, 1979).
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Contact and Creep-Force Models
4.3.2 SIMPLIFIED THEORY
OF
147
ROLLING CONTACT
In the simplified theory of rolling contact, the traction-displacement constitutive law
takes a simple form, which is given by (Kalker, 1973):
u wr = u w − u r = [ L x Ftx
Ly Fty ]T
(4.44)
where uw and ur are the displacements of the two bodies at the contact point (both
components are defined in the same coordinate system), and Lx and Ly are, respectively, the compliant parameters in the longitudinal and lateral directions. These
compliant coefficients depend on the material and geometric parameters of the two
bodies in the contact region (Kalker, 1973).
4.3.3 DYNAMIC
AND
QUASI-STATIC THEORY
If the inertial effects are included in the formulation of the rolling contact theory,
the theory is called dynamic. On the other hand, if the inertial effect is neglected, the
theory is called quasi-static. The inertial effect becomes significant if the speed is
higher than 500 km/h (Kalker, 1979). Therefore, in the railroad vehicle dynamics,
the quasi-static theory is used instead of the dynamic theory.
4.3.4 THREE-
AND
TWO-DIMENSIONAL THEORY
If the displacement-force relationship depends on all three coordinates x, y, and z,
the theory is said to be three-dimensional theory. On the other hand, if the displacement-force relationship is independent of the lateral coordinate, the theory is called
two-dimensional. Therefore, in the two-dimensional theories, the effect of the spin
creepage is not considered, and this limits the use of such two-dimensional theories
in railroad vehicle dynamics formulations.
4.4 CREEP-FORCE THEORIES
Several creep-force theories have been developed and applied to solve the wheel/rail
contact problem. Kalker has provided surveys of wheel/rail contact theories (Kalker,
1979, 1980, 1991). This section presents brief descriptions of some of these theories.
It is important, however, to mention that some of the theories presented in this section
are no longer being used in wheel/rail contact computer codes that are based on
nonlinear three-dimensional multibody system formulations. They can be used,
however, for linear or special models.
4.4.1 CARTER’S THEORY
In 1926, Carter introduced the first two-dimensional creep theory (Carter, 1926). In
this theory, Carter introduced a closed-form relationship between the longitudinal
creepage and the tangential force. In this theory, Carter approximated the shape of
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148
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 4.8 Contact area according to Carter.
the contact area by a two-dimensional uniform rectangular strip. It was assumed
that the wheel and rail surfaces can be represented, respectively, by cylinder and
thick plate. The radius of the wheel is assumed to be larger than the circumferential
length of the contact area and, therefore, the problem can be treated as an infinite
elastic medium bounded by a plane with a local pressure distribution and tangential
traction in the contact area. Carter showed that the difference between the circumferential velocity of the driven wheel and the translation velocity of the wheel at the
contact point is not equal to zero when acceleration or braking couple is suddenly
applied to the wheel. This difference increases in absolute value with increasing
couple until a Coulomb maximal value is reached. The behavior in the contact region,
based on Carter’s theory, can be described using a simple model of a wheel traveling
in the positive direction of the x-axis. Let A be the point at which the first contact
occurs, and A′ be the point of departure, as shown in Figure 4.8. Let ABA′ represent
the curve of the limiting value of the tangential traction, and let ADCA′ be the actual
curve of the tangential traction that starts at A and never exceeds the limiting curve
ABA′, as shown in Figure 4.8. Beyond point C, the surfaces slip with limiting
tangential traction, as the pressure between the surfaces is insufficient to prevent the
movement. Carter’s closed-form solution for the relation between longitudinal creepage and tangential force is given by (Carter, 1926):
f=
(
) R lN 

2 ( λ + 2G )
 1−
πG λ + G
w


1− q 
q
(4.45)
where ƒ is the tractive force per unit creepage in the longitudinal direction; q is the
ratio F/Fx, where F is the total tractive effort of the wheel and Fx is the tangential
force in the longitudinal direction; G and λ are, respectively, the material modulus
of rigidity and Lame’s constant; Rw is the wheel rolling radius; l is the equivalent
length of the contact in the transverse direction of the rail and is equal to (4b/3),
where b is the contact ellipse semi-axis dimension in the lateral direction; and N is
the total normal force. Equation 4.45 can be written as follows (Kalker, 1991):
− kζ x + 0.25k 2ζ x ζ x
Ft

=
µ N l −sign(ζ )
x

k ζx ≤ 2
k ζx > 2
(4.46)
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Contact and Creep-Force Models
149
FIGURE 4.9 Comparison between Carter’s law and linear law.
where Ft is the tangential force per unit lateral length exerted at the contact on the
wheel; µ is the coefficient of friction; Nl is the total normal force per unit lateral
length exerted on the wheel; ζx is the creepage and is equal to 2(Vt − Vc)/(Vt + Vc),
where subscripts t and c refer, respectively, to tangential forward and circumferential
velocities; and k is Carter’s creepage coefficient, which is equal to 4Rw /µa, where
Rw is the wheel rolling radius and a is semi-axis dimension of the contact ellipse in
the rolling direction (longitudinal). Figure 4.9 shows the difference between Carter’s
law and the linear law. In general, Carter’s theory is capable of predicting the
frictional losses in a locomotive driving wheel. It is clear from Equation 4.46 that
the tractive force per unit creepage depends on the tractive effort.
4.4.2 JOHNSON
AND
VERMEULEN’S THEORY
In 1958, Johnson (Johnson, 1958a, 1958b) extended Carter’s theory to the threedimensional case of two spheres without spin. Later, Vermeulen and Johnson
extended the method to smooth half-spaces without spin (Vermeulen and Johnson,
1964). In this theory, the contact surface between the two rolling bodies transmitting
a tangential force is asymmetrically divided into two regions: the slip region and
the stick or no-slip region. The adhesion area was assumed to be elliptical. The area
of adhesion is assumed to touch the leading edge of the contact ellipse, as shown
in Figure 4.10. According to Johnson and Vermeulen, the total resulting tangential
force F = [Fx Fy]T can be determined as follows:

 1  3 

 1 − τ  − 1 ζ i + η j for τ ≤ 3,
τ
(
1
/
)
F

 3 

=
µN 
−(1 / τ ) ζ i + η j for τ > 3,

(
(
)
)
(4.47)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 4.10 Contact area according to Johnson and Vermeulen.
where ζ is the normalized longitudinal creepage, defined as (πabGζx)/µNφ; η is the
normalized lateral creepage, defined as (πabGζy)/µNψ ; τ = ζ 2 + η 2 ; i = [1 0]T; j =
[0 1]T; N is the normal force; G is the modulus of rigidity; ζx is longitudinal creepage;
ζy is the lateral creepage; a and b are the dimensions of the contact ellipse semiaxes in the rolling and lateral directions, respectively; µ is the coefficient of friction;
and
φ = Be − ν ( De − Ce ) 

ψ = Be − ν (a / b)2 Ce 
for a ≤ b, e =
1 − a /b
φ = (b / a)  De − ν ( De − Ce )  



ψ = (b / a)  De − νCe 

for a ≥ b, e =
1 − b /a
( )
( )
2
2











In this equation, Be, De, and Ce are the complete elliptical integrals of argument e,
and ν is the Poisson’s ratio.
Vermeulen and Johnson compared the results obtained from Equation 4.47 with
results of measurements. The difference between the formula and measurement results
was attributed to the assumption of an elliptical no-slip region. It is important to
note that this theory is valid in the case of no spin. In railroad dynamics, since the
spin effect is important, the Vermeulen and Johnson theory is rarely used in the
computer formulation of the creep forces.
4.4.3 KALKER’S LINEAR THEORY
Kalker suggested that, for very small creepages, the area of slip is very small and
its effect can be neglected (Kalker, 1967). Therefore, the adhesion area can be
assumed to be equal to the area of contact. Kalker described the behavior of
the contact point as follows. Along a line parallel to the direction of rolling, the
particle starts to penetrate, and as the slip is very small (no-slip condition), traction
starts to build up. As the particle leaves the contact area, the traction becomes zero.
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Contact and Creep-Force Models
151
Therefore, the true slip given by Equation 4.39 vanishes everywhere in the contact
area, that is,
( )
x, y = 0
w
(4.48)
Integrating this equation with respect to x
∫ w ( x, y ) dx = u( x, y)
(4.49)
where u(x,y) is the tangential displacement difference and is determined by assuming
that the traction is continuous at the leading edge of the contact area. Using the
traction-displacement relationship based on the general elasticity theory and integrating the traction over the contact area (Equation 4.43), the linear relation between
the creepages and the creep forces and moment are obtained as
c11
 Fx 

 
 Fy  = −Gab  0

M 
 
 0
0
 ζ x 
 
abc23  ζ y 

abc33   ϕ 
0
c22
− abc23
(4.50)
where ζx, ζy , ϕ are the longitudinal, lateral, and spin creepages, respectively; a is
the contact ellipse semi-axis dimension in the rolling direction; b is the contact
ellipse semi-axis dimension in the lateral direction; G is the modulus of rigidity;
and cij are creepage coefficients that depend only on Poisson’s ratio and the ratio of
the semi-axis of the contact ellipse, as shown in Table 4.4 (Kalker, 1990). Kalker
also introduced the following definitions:
G=
1 1
1 
+ r ,

w
2 G
G 
ν 1  νw νr 
= 
+

G 2  G w Gr 
(4.51)
where G is an average shear modulus of rigidity of the wheel w and the rail r, and
ν is a combined Poisson’s ratio of the wheel and the rail.
In 1984, Kalker calculated the creepage and spin coefficients when the relative
slip is small but the contact area is not necessarily elliptic. These calculations were
made with the aid of the program CONTACT. The error was found to be less than
5%, which led to the conclusion that the coefficients given in Table 4.4 are accurate
enough for the analysis of the wheel/rail contact problem. The linear theory is
extensively used in railroad vehicle dynamics. Haque et al. (Haque et al., 1979)
developed a program that can be used to interpolate for the creep coefficients. In
this program, linear interpolation is used to obtain the coefficients using the data
given in Table 4.4.
45814_book.fm Page 152 Thursday, May 31, 2007 2:25 PM
152
Railroad Vehicle Dynamics: A Computational Approach
TABLE 4.4
Kalker’s Creepage and Spin Coefficients
c11
c23 = −c32
c22
c33
g
ν=0
0.25
0.5
ν=0
0.25
0.5
ν=0
0.25
0.5
ν=0
0.25
0.5
(a/b)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2.51
2.59
2.68
2.78
2.88
2.98
3.09
3.19
3.29
3.31
3.37
3.44
3.53
3.62
6.72
3.81
3.91
4.01
4.85
4.81
4.8
4.82
4.83
4.91
4.97
5.05
5.12
2.51
2.59
2.68
2.78
2.88
2.98
3.09
3.19
3.29
2.52
2.63
2.75
2.88
3.01
3.14
3.28
3.41
3.54
2.53
2.66
2.81
2.98
3.14
33.1
3.48
3.65
3.82
0.334
0.483
0.607
0.720
0.827
0.930
1.03
1.13
1.23
0.473
0.603
0.715
0.823
0.929
1.03
1.14
1.25
1.36
0.731
0.809
0.889
0.977
1.07
1.18
1.29
1.40
1.51
6.42
3.46
2.49
2.02
1.74
1.56
1.43
1.34
1.27
8.28
4.27
2.96
2.32
1.93
1.68
1.50
1.37
1.27
11.7
5.66
3.72
2.77
2.22
1.86
1.60
1.42
1.27
(b/a)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
3.4
3.51
3.65
3.82
4.06
4.37
4.84
5.57
6.96
10.7
4.12
4.22
4.36
4.54
4.78
5.10
5.57
6.34
7.78
11.7
5.2
5.3
5.42
5.58
5.8
6.11
6.57
7.34
8.82
12.9
3.40
3.51
3.65
3.82
4.06
4.37
4.84
5.57
6.96
10.7
3.67
3.81
3.99
4.21
4.50
4.90
5.48
6.40
8.14
12.8
3.98
4.16
4.39
4.67
5.04
5.56
6.31
7.51
9.79
16.0
1.33
1.44
1.58
1.76
2.01
2.35
2.88
3.79
5.72
12.2
1.47
1.57
1.75
1.95
2.23
2.62
3.24
4.32
6.63
14.6
1.63
1.77
1.94
2.18
2.50
2.96
3.70
5.01
7.89
18.0
1.21
1.16
1.10
1.05
1.01
0.958
0.912
0.868
0.828
0.795
1.19
1.11
1.04
0.965
0.892
0.819
0.747
0.674
0.601
0.526
1.16
1.06
0.954
0.852
0.751
0.650
0.549
0.446
0.341
0.228
Note: g = 0, c11 = π2/4(1 – ν); c22 = π2/4; c23 = –c32 = π g {1 + ν(0.5Λ + ln 4 – 5)}/3(1 – ν); Λ =
ln(16/g2); and c33 = π2/16(1 – ν)g.
Source: Kalker, J.J., Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht, Netherlands, 1990. With permission.
EXAMPLE 4.4
At a certain configuration, a wheel has contact ellipse semi-axis dimensions a = 6.523 ×
10−3 m and b = 2.298 × 10−3 m. Assume that the wheel and the rail have Poisson’s
ratio equal to 0.24 and modulus of rigidity equal to 8 × 1010 N/m2. At this configuration,
the longitudinal creepage is equal to zero, the lateral creepage is 0.01, and the spin
creepage is 0.1016 m−1 Determine the creep forces.
Solution. The contact ellipse semi-axis ratio a/b = 2.838. Using Kalker’s table, the
creep coefficients are determined as
c11 = 5.8468,
c22 = 5.8192,
c23 = 3.6574,
c33 = 0.72518
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Contact and Creep-Force Models
153
Using Equation 4.51, the tangential creep forces and moment can be determined as
 c11
 Fx 

 
 Fy  = −Gab  0

 M 
0
 ζ x 
0


 


abc23  ζ y  = −  71511.292 N 

 −168.4938 N ⋅ m 
abc33   ϕ 
0
0
c22
− abc23
4.4.4 HEURISTIC NONLINEAR CREEP-FORCE MODEL
In general, the wheel/rail contact is a highly nonlinear problem. Using linear models
for the force-creepage relationship can lead to errors. Two sources of the nonlinearity
in the wheel/rail contact problem are:
1. Nonlinear wheel/rail geometric functions
2. Adhesion limits on the force-creepage relationship
The creep-force models discussed thus far in this section are not based on fully
nonlinear theory and employ simplifying assumptions. In Johnson and Vermeulen’s
theory, the spin creepage is ignored. This assumption limits the application of the
theory to the case of pure longitudinal and lateral creepages. The effect of the spin
creepage is important, especially in the case of flange contact. Kalker’s linear theory
is limited to the case of small creepages. White et al. (1978) and Shen et al. (1983)
suggested a new approximate heuristic nonlinear theory based on Kalker’s linear
theory. In this theory, the saturation law of Johnson and Vermeulen is used, and the
effect of spin creepage on the creep forces is considered. The longitudinal and lateral
creep forces are first calculated using Kalker’s linear theory, defined in Equation
4.50, as follows:
c11
 FxK 
 K  = −Gab 
 0
 Fy 
ζ 
 x
 ζ y 
abc23   
ϕ 
0
0
c22
(4.52)
where FxK and FyK are the forces evaluated using Kalker’s linear theory. The resultant
creep force obtained from the linear model is calculated as follows:
FL =
(F ) + (F )
K
x
2
K
y
2
(4.53)
According to Coulomb friction law, the magnitude of the resultant creep force
cannot exceed the pure slip value µN. Using Johnson and Vermeulen’s theory, the
resultant force FL is limited by the nonlinear value FL as follows (Shen et al., 1983):
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154
Railroad Vehicle Dynamics: A Computational Approach
2
3


 µ N  FL  − 1  FL  + 1  FL   ,




27  µ N  
FL = 
 µ N  3  µ N 


 µ N ,
FL ≤ 3µ N
(4.54)
FL > 3µ N
Therefore, the creep-force reduction coefficient is defined as follows:
ε=
FL
FL
(4.55)
The nonlinear creep-force model is given by
 Fx 
 Fx 
 =ε 
 Fy 
 Fy 
(4.56)
It is important to note that the spin-creepage contribution to the lateral creep
forces is included in computing the creep force given by Equation 4.52. This theory
gives more realistic values for creep forces outside the linear range than Kalker’s
linear theory (Shen et al., 1983). Shen et al. (1983) showed that the results obtained
from the heuristic model are in good agreement with Kalker’s simplified theory.
However, in the case of high values of spin, the heuristic theory leads to unsatisfactory results (Kalker, 1991).
4.4.5 POLACH NONLINEAR CREEP-FORCE MODEL
In 1999, Polach introduced an algorithm for the computation of the wheel/rail creep
forces (Polach, 1999). Polach assumed that the shape of the contact area is elliptic.
In Hertz contact theory, the maximum stress distribution is equal to σ. Therefore,
the maximum tangential stress at any arbitrary point is
τ max = µσ
(4.57)
where µ is the coefficient of friction and is assumed to be constant in the whole
contact area. Assuming that the relative displacement between the bodies in the
adhesion area increases linearly from one side on the edge of the contact A to the
other side C, as shown in Figure 4.11, the tangential stress increases linearly with
the distance from the leading edge. As the tangential stress reaches its maximum
value, sliding takes place. Due to the longitudinal and lateral creepage ζx and ζy,
respectively, the tangential force is given by
Fpx = −

2µ N  ε
+ tan −1 ε 

2
π  1+ ε

(4.58)
45814_book.fm Page 155 Thursday, May 31, 2007 2:25 PM
Contact and Creep-Force Models
155
FIGURE 4.11 Normal and tangential stress distribution for Polach theory. (Courtesy of
Polach, O., Vehicle System Dynamics Supplement, 33, 728, 1999. With permission.)
where N is the normal contact force, and ε is the gradient of the tangential stress in
the area of adhesion and is given by
ε=
1 Gπ abCh
νc
4 µN
(4.59)
where G is the modulus of rigidity, and Ch is a constant that depends on Kalker’s
coefficients as follows:
 ζ   ζ 
Ch =  c11 x  +  c22 y 
ν 
ν

2
2
(4.60)
and ν is the magnitude of the creepage and is given by
ν = ζ x2 + ζ y2
(4.61)
Using a modified lateral creepage that accounts for the effect of the spin creepage φ,
Equation 4.61 can be written as:
ν c = ζ x2 + ζ yc2
(4.62)
where the modified lateral creepage ζyc is given by
ζ y
; ζ y + ϕa ≤ ζ y

ζ yc = 
ζ y + ϕ a ; ζ y + ϕ a > ζ y

(4.63)
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156
Railroad Vehicle Dynamics: A Computational Approach
Assuming that the moment effect caused either by the spin creepage or by the
lateral creepage is small compared with other moments acting on the system, the
lateral tangential force that accounts for the effect of the spin is given by
Fpy = −
9
aµ NK 1 + 6.3(1 − e − a b ) 
16
(4.64)
where K is a constant defined as
 δ3 δ2 1 1
(1 − δ 2 )3
K = εy  −
+
−
2 6  3
 3
(4.65)
and δ is given by
δ=
(ε y )2 − 1
(4.66)
(ε y )2 + 1
where εy is the gradient of the tangential stress and is given by
εy =

c23ζ yc
8 Gb ab

3 µ N  1 + 6.3 1 − e − a b

(
)



(4.67)
where c23 is Kalker’s coefficient. Finally, the creep forces are given as follows:




ζy
ϕ
Fy = Fpx + Fpy 
νc
νc 
Fx = Fpx
ζx
νc
(4.68)
Polach’s algorithm yields accurate prediction of the tangential contact forces,
and it has been implemented in some computer codes developed recently for the
dynamic analysis of railroad vehicle systems. A FORTRAN code based on this
theory has been published in the literature (Polach, 1999).
4.4.6 SIMPLIFIED THEORY
As briefly discussed in the previous section, one can approximate the relationship
between the tangential surface traction Ft = [Ftx Fty]T and the tangential surface
displacement uwr = uw − ur using compliant parameters Lx and Ly as follows:
u wr = u w − u r = [ L x Ftx
Ly Fty ]T
(4.69)
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Contact and Creep-Force Models
157
In such a case, the complex expressions that result from the use of the general
elasticity theory can be simplified by using the compliant parameters. In the case
of steady-state rolling, the slip inside the contact region is assumed to be zero
(Kalker, 1979), that is,
ζ − ϕ y 
∂u wr
x, y = V  x
=0
w
−V
∂x
ζ y + ϕ x 
( )
(4.70)
Substituting Equation 4.69 into Equation 4.70, one has
ζ − ϕ y 
 Lx (∂Ftx /∂x ) 
x, y = V  x
w
−V 
=0
ζ y + ϕ x 
 L y (∂Fty /∂x ) 
( )
(4.71)
Integrating the preceding equation with respect to x, the tangential traction can be
obtained as
 x (ζ x − ϕ y) + k ( y) 

Lx
 Ftx  

 =
x (ζ y + 12 ϕ x ) + l ( y) 
 Fty 


Ly


(4.72)
where k(y) and l(y) are the integration constants that are arbitrary functions of y.
These integration constants are determined by using the condition that the tangential
traction is equal to zero at the leading edge of the contact ellipse. This leads to the
following equation:


( x − a( y))(ζ x − ϕ y)


Lx
 Ftx  

 =
2
2
1
 Fty  ζ y ( x − a( y)) + 2 ϕ ( x − a( y) ) 


Ly


(4.73)
where a(y) is given by a( y) = a 1 − ( y /b)2 . Substituting Equation 4.73 into Equation
4.43, one obtains
Fx =
∫ ∫
Fy =
∫ ∫
b
a( y)
−b
− a( y)
b
−b
Ftx dx dy = −
8a 2bζ x
3Lx
8a 2bζ y π a 3bϕ
−
Fty dx dy = −
3L y
4 Ly
− a( y)
a( y)







(4.74)
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158
Railroad Vehicle Dynamics: A Computational Approach
where the spin moment is neglected in the preceding equation. On the other hand,
Kalker’s linear theory, obtained using the general elasticity theory, as shown in Equation 4.50, leads to the following longitudinal and lateral force-creepage relationship
Fx = −Gabc11ζ x
Fy = −Gabc22ζ y − G (ab)1.5 c23ϕ




(4.75)
Note that cij are creepage coefficients given by Table 4.4. By equating Equations 4.74
and 4.75, the compliant parameters Lx and Ly are given by
Lx =
8a
3Gc11
π a2
8a
L y1 =
, Ly2 =
3Gc22
4G abc23







(4.76)
where two different definitions Ly1 and Ly2 are given for Ly so that the creep forces
obtained using the simplified theory will be equal to those obtained using Kalker’s
linear theory. As a result, while only two compliant parameters are originally defined
in the constitutive model given by Equation 4.69, the three compliant parameters
Lx, Ly1, and Ly2 are required, since Ly1 and Ly2 differ significantly. For this reason,
the following single compliant parameter L is introduced (Jacobson and Kalker,
2001):
L=
Lx ζ x + L y1 ζ y + L y 2 ab ϕ
(ζ x )2 + (ζ y )2 + abϕ 2
(4.77)
where
L = Lx ;
ζy = ϕ = 0
L = L y1;
ζx = ϕ = 0
L = Ly2;
ζx = ζy = 0






(4.78)
It is important to note that the analytical solution based on the simplified theory
can be obtained when the effect of spin moment is ignored, while a numerical method
is required for a more general case of wheel/rail contact problems. A program called
FASTSIM that is based on the simplified theory was developed by Kalker (Kalker,
1982) and has been widely used in railroad vehicle computer programs. In this
program, the contact surface is discretized into several strips, and the creep forces
45814_book.fm Page 159 Thursday, May 31, 2007 2:25 PM
Contact and Creep-Force Models
159
are calculated by incrementing the tangential tractions from one strip to another.
The complete algorithm of FASTSIM can be found in the literature (Kalker, 1982).
4.4.7 KALKER’S USETAB
USETAB is one of the available codes that can be used to accurately predict the
wheel/rail creep forces. Since this program is designed for a general wheel/rail
contact problem, the coefficient table becomes very large to account for all of the
possible contact geometry. USETAB has two sets of tables. In the first set, the
creepage coefficients given in the literature (Kalker, 1990) are listed. In the second
set, the Hertzian creep-force law constants that were generated by numerous runs
of the CON93 program developed by Kalker are listed. The inputs to USETAB are
the contact ellipse semi-axis dimensions, the creepages, the equivalent modulus of
rigidity, the Poisson’s ratio, and the coefficient of friction. The output of USETAB
is the creep forces and spin moment defined in the contact coordinate system. The
estimated error resulting from the use of USETAB is approximately 1.5% compared
with the fully nonlinear theory, while this error for FASTSIM, which is the code
currently used in many programs for the calculation of the creep forces, is approximately 15%.
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5
Multibody Contact
Formulations
As discussed in the preceding chapter, if one wants to evaluate the tangential creep
forces and spin moment, one must know the normal contact force as well as other
wheel/rail material and geometric parameters. There are two main approaches that can
be used in the multibody system formulations to determine the normal contact force
when the wheel/rail interaction is considered. In the first approach, the contact
between the wheel and the rail is described using kinematic constraint equations.
The normal contact force can be determined as the reaction force due to the imposition of the contact constraint equations. When equality constraints are used, it is
assumed that there is no penetration or separation between the wheel and the rail.
In the second approach, a compliant force with assumed stiffness and damping
coefficients is used to describe the wheel/rail contact. In this approach, the wheel
has six degrees of freedom with respect to the rail and the wheel/rail separation is
possible.
In this chapter, four nonlinear dynamic formulations for the analysis of the
wheel/rail contact are discussed. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and
the rail (constraint approach), while in the other two formulations, the contact force
is modeled using a compliant force element (elastic approach). In the formulations
based on the elastic approach, as previously mentioned, the wheel has six degrees
of freedom with respect to the rail, and the normal contact force is defined as a
function of the penetration using Hertz’s contact theory or using assumed stiffness
and damping coefficients. One of the elastic methods discussed in this chapter is
based on a search for the contact locations using discrete nodal points. In the second
elastic approach discussed in this chapter, the contact points are determined by
solving a set of algebraic equations. In both elastic methods, the contact points are
determined on-line instead of using a look-up table, which is a commonly used
method for determining the location of the contact points in some specialized railroad
vehicle dynamic algorithms.
In the formulations based on the constraint approach, on the other hand, the
case of a nonconformal contact is assumed, and nonlinear kinematic constraint
equations are used to impose the contact conditions. This approach leads to a model
in which the wheel has five degrees of freedom with respect to the rail. In the
constraint approach, unlike the elastic approach, wheel penetration and lift are not
permitted, and the normal contact forces are calculated as constraint forces. Two
equivalent constraint formulations that employ two different solution procedures are
discussed in this chapter. The first method leads to a larger system of equations by
adding all the contact constraint equations to the dynamic equations of motion, while
161
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162
Railroad Vehicle Dynamics: A Computational Approach
in the second method, an embedding procedure is used to obtain a reduced system
of equations from which the surface parameter accelerations are systematically
eliminated. In the constraint formulations presented in this chapter, three-dimensional contact conditions are imposed. That is, the contact conditions are expressed
in terms of four geometric parameters that define the wheel and rail surfaces. In some
other constraint formulations that exist in the literature, the contact is treated as a
two-dimensional problem and only two geometric parameters are used to describe
the wheel and rail profiles. This planar contact approach, however, has been used
for the most part with the trajectory coordinates. As will be discussed in Chapter 7,
the use of the trajectory coordinates requires a higher order of differentiability and
smoothness as compared with the use of the absolute Cartesian coordinates employed
in this chapter. Furthermore, the trajectory coordinates, as previously mentioned, are
more suited for the development of specialized railroad vehicle dynamic formulations. Nonetheless, the planar contact formulation is discussed in the last section of
this chapter for completeness.
The focus in this chapter will be mainly on the formulation of the multibody
system equations of motion. It will be shown how the wheel/rail contact model can
be systematically included in the multibody system formulations when the constraint
and elastic approaches are used. Computational algorithms and solution procedures
that can be used with the formulations presented in this chapter are discussed in
Chapter 6.
5.1 PARAMETERIZATION OF WHEEL AND RAIL SURFACES
To accurately determine the location of the point of contact between two bodies, a
complete parameterization of the surfaces must be used. In general, as discussed in
Chapter 3, a set of four surface parameters can be used to describe the geometry of
the two surfaces in contact, as shown in Figure 5.1. The surface parameters can be
written in a vector form as
s = s1i s2i s1j s2j 
T
(5.1)
where superscripts i and j denote bodies i and j, respectively. Using these parameters,
the location of the contact point P can be defined, respectively, in the coordinate
systems of bodies i and j as
u
i
P
(s , s )
i
1
i
2
(
(
(
)
)
)
 x i s1i , s2i 


=  y i s1i , s2i  ,


 i i i 
z
s
,
s
1 2 


u
j
P
(s , s )
j
1
j
2
(
(
(
)
)
)
 x j s1j , s2j 


=  y j s1j , s2j 


 j j j 
z
s
,
s
1 2 


(5.2)
The tangents to the surface at the contact point are defined in the body i
coordinate system as
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163
FIGURE 5.1 Schematic representation of two bodies in contact.
t1i =
∂u iP
,
∂s1i
t2i =
∂u iP
∂s2i
(5.3)
and the normal vector as
n i = t1i × t2i
(5.4)
The parameterization used in Equation 5.2, as well as the tangent and normal
vectors defined in Equations 5.3 and 5.4, can be used to describe the geometry of
the wheel and rail surfaces as described below.
5.1.1 TRACK GEOMETRY
It was shown in Chapter 3 that the surface geometry of the rail r can be described
in the most general form using the two surface parameters s1r and s2r , where s1r represents the rail arc length and s2r is the surface parameter that defines the rail profile,
as shown in Figure 5.2. The surface parameters s1r and s2r are defined in a profile
coordinate system XrpYrpZrp, as shown in Figure 5.3. The location of the origin and
the orientation of the profile coordinate system, defined respectively by the vector
R rp and the transformation matrix A rp, can be uniquely determined using the surface
parameter s1r . Using this description, the global position vector of an arbitrary point
on the surface of the rail r can be written as follows:
rr = R r + A r (R rp + A rp u rp )
(5.5)
where Rr is the global position vector of the origin of the rail body coordinate system
XrYrZr, Ar is the transformation matrix that defines the orientation of the rail coordinate
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FIGURE 5.2 Surface parameters.
system, and u rp is the local position vector that defines the location of the arbitrary
point on the rail surface with respect to the profile coordinate system. The location
and orientation of the rail profile coordinate system depends only on the distance
traveled along the track s1r , while the local position of an arbitrary point on the rail
surface at any section depends on the value of s2r , that is,
R rp = R rp (s1r ), A rp = A rp (s1r ), u rp = 0
where f is the function that defines the rail profile.
FIGURE 5.3 Track geometry.
s2r
f (s2r ) 
T
(5.6)
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165
FIGURE 5.4 Wheel geometry.
5.1.2 WHEEL GEOMETRY
The geometry of the wheel surface can be described using the two surface parameters
s1w and s2w . These surface parameters are defined in a wheelset coordinate system
XwYwZw. The surface parameter s1w defines the wheel profile, and s2w is an angular
parameter that represents the rotation of the contact point, as shown in Figure 5.2.
The location of the origin and the orientation of the wheel coordinate system are
defined, respectively, by the vector Rw and the transformation matrix Aw. Using this
description, the global position vector of an arbitrary point on the surface of the
wheel w can be written as follows:
r w = R w + Aw u w
(5.7)
where u w is the local position vector that defines the location of the arbitrary point
on the wheel surface with respect to the wheel coordinate system. For example, in
the case of the right wheel of a wheelset, this vector is defined as
( )
u w =  g s1w sin s2w

− L + s1w
( )
− g s1w cos s2w 

T
(5.8)
where g is the function that defines the wheel profile, and L is the distance between
the origin of the wheelset coordinate system and point Q of the wheel, as shown in
Figure 5.2 and Figure 5.4.
5.2 CONSTRAINT CONTACT FORMULATIONS
In this chapter, two constraint contact formulations and two elastic contact formulations that can be used to study the wheel/rail dynamic interaction are discussed.
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Some of these three-dimensional contact formulations differ conceptually in the way
the geometric surface parameters are treated. As shown in the preceding section, the
geometry of the surfaces of the wheel and rail can be described using the surface
parameters that define the location of the contact points. In the constraint formulations, the contact between two surfaces can be described using a set of nonlinear
algebraic equations that must be imposed at the position, velocity, and acceleration
levels. One may choose not to eliminate the surface parameters, leading to an
augmented form of the equations of motion expressed in terms of Lagrange multipliers associated with the kinematic contact constraint equations. This augmented
constraint contact formulation will be referred to in this chapter as ACCF. Alternatively, one may choose to systematically eliminate the surface parameters using the
contact constraints, leading to an embedding formulation of the equations of motion
that does not explicitly include the surface parameter accelerations. This embedded
constraint contact formulation will be referred to in this chapter as ECCF. In both
formulations, the nonlinear algebraic contact constraint equations are necessary in
the formulation of the dynamic equations of motion. This is despite the fact that the
two formulations (augmented and embedded) lead to significantly different forms
of the dynamic equations.
5.2.1 CONTACT CONSTRAINTS
When two rigid bodies come into contact as shown in Figure 5.1, two types of
nonconformal kinematic contact conditions need to be satisfied. First, two points
(contact points) on the two surfaces must coincide; and second, the two surfaces
must have the same tangent planes at the contact point. These two conditions define
the following five constraint equations that are required to describe the nonconformal
contact between the wheel and rail:
k
rPw − rPr 


C k (q w , q r , s wk , s rk ) =  t1w ⋅ n r  = 0
 t w ⋅ nr 

 2
(5.9)
or, equivalently,
k
 t1r ⋅ (rPw − rPr ) 
 r
w
r 
 t 2 ⋅ (rP − rP ) 
C k (q w , q r , s wk , s rk ) = n r ⋅ (rPw − rPr )  = 0


w
r
 t1 ⋅ n

 t w ⋅ nr 
2


(5.10)
where superscript k denotes the contact number; superscripts r and w denote rail
and wheel, respectively; qw and qr are, respectively, the generalized coordinates of
the wheel and rail; and t1, t2, and n are the two tangents and the normal to the surface
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167
at the contact point. In Equation 5.10, the three relative displacement constraints
(point constraints defined by the first three scalar equations) are defined in a rail
coordinate system at the contact point. The first three point constraints of Equation 5.9,
on the other hand, are defined in the global coordinate system.
5.2.2 CONSTRAINED DYNAMIC EQUATIONS
As discussed in Chapter 2, the principle of virtual work in dynamics for a system
that consists of interconnected bodies states that the virtual work of the system’s
inertia forces is equal to the virtual work of the system’s externally applied forces
(Shabana, 2001). This principle can be stated mathematically as follows:
δ Wi = δ We
(5.11)
where δWi is the virtual work of the system inertia forces, and δWe is the virtual
work of the applied forces. In the preceding equation, the virtual work of the
constraint forces is not included, and it is identically equal to zero, since the dynamic
equilibrium of the multibody system is considered. The virtual work of the inertia
forces and the virtual work of the applied forces can, in general, be written for a
multibody system as
(
)
− Qv ,
δ Wi = δ qT Mq
δ We = δ qT Qe
(5.12)
where q is the vector of the system generalized coordinates, M is the system mass
matrix, Qv is the vector of centrifugal and Coriolis inertia forces, and Qe is the vector
of the system applied forces. Therefore, the principle of virtual work of Equation 5.11
leads to
(
)
− Q = 0
δ qT Mq
(5.13)
where Q = Qv + Qe. In Equation 5.13, the elements of the vector of generalized
coordinates q are not independent because of the kinematic constraint equations.
The vector of the kinematic algebraic constraint equations, which include the contact
constraints and other constraints that describe mechanical joints and specified motion
trajectories, can be written in a vector form as
( )
C q, s = 0
(5.14)
where s represents the vectors of surface parameters that describe the wheel and rail
surface geometry and that are used to formulate the contact constraints as previously
described in this section. Taking a virtual change in the constraint equations leads to
δ C = Cqδ q + C sδ s = 0
(5.15)
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In this equation, Cq and Cs are the constraint Jacobian matrices resulting from the
differentiation of the constraint equations with respect to the vectors q and s,
respectively. It follows from the preceding equation that
(
)
λ T Cqδ q + C sδ s = 0
(5.16)
for any vector λ . The vector λ is called the vector of Lagrange multipliers. Adding
Equations 5.13 and 5.16, one obtains
(
)
+ CTq λ − Q + δ s T CTs λ = 0
δ qT Mq
(5.17)
It is important to note from this equation that there are no inertia and generalized forces
associated with the surface parameters used to describe the wheel/rail surface geometry, and for this reason, these parameters are treated as nongeneralized coordinates.
Differentiating Equation 5.14 twice with respect to time, the constraint equations
at the acceleration level can be written as
+ C ss = Q d
Cq q
(5.18)
where Qd is a quadratic velocity vector resulting from the differentiation of the
constraint equations twice with respect to time. Equations 5.17 and 5.18 are the basis
for the two constraint formulations that will be discussed in the following two sections.
5.3 AUGMENTED CONSTRAINT CONTACT
FORMULATION (ACCF)
In the constraint contact formulations, the system differential equations of motion
and the wheel/rail algebraic contact constraint equations are solved simultaneously
for the system generalized coordinates and the system surface parameters in order
to correctly account for the kinematic and dynamic couplings between the wheel/rail
generalized coordinates and the nongeneralized surface parameters. One can combine the vector of generalized and nongeneralized coordinates in one vector, which
can be written as follows:
p = q T
s T 
T
(5.19)
In terms of this vector, Equation 5.17 can be written as follows:
(
)
δ pT H + CTp λ = 0
(5.20)
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169
In this equation,
− Q 
Mq
H=
,
 0 
C p = C q
C s 
(5.21)
Because of the algebraic kinematic constraints, the elements of the vector p are
not independent. The number of dependent coordinates is equal to the number of
kinematic constraint equations, nc. If the total number of coordinates including the
nongeneralized surface parameters is n, the number of independent coordinates
(degrees of freedom) is nd = n – nc. One can then select a set of independent
coordinates pi, which may include nongeneralized surface parameters, and write the
total vector of coordinates p as follows:
p = p Ti
p Td 
T
(5.22)
In this equation, pd is the vector of dependent coordinates that has dimension nc.
Using this coordinate partitioning, Equation 5.20 can be written as
δ pTi

 Hi + CTp λ 
i
=0
δ pTd  
T
 H d + C pd λ 
(5.23)
In this equation, subscripts i and d refer to vectors and matrices associated with
independent and dependent coordinates, respectively, and CPi and CPd are the constraint Jacobian matrices associated with the independent and dependent coordinates,
respectively. Since the number of constraint equations that are assumed to be linearly
independent is equal to the number of dependent coordinates, the dependent coordinates can be selected such that the constraint Jacobian matrix CPd is a square
nonsingular matrix. The preceding equation can also be written in the following
form:
(
)
(
)
δ pTi Hi + CTpi λ + δ pTd H d + CTpd λ = 0
(5.24)
Using the fact that the constraint Jacobian matrix CPd is a square nonsingular
matrix, one can choose to define the arbitrary vector λ as the solution of the following
system of algebraic equations:
H d + CTpd λ = 0
(5.25)
Furthermore, since the elements of the vector pi are assumed to be independent, the
first part of Equation 5.24 leads to
Hi + CTpi λ = 0
(5.26)
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Combining the preceding two equations, one obtains
H + CTp λ = 0
(5.27)
Using the definitions of the matrices and vectors in this equation, which are given
previously in this section, it is clear that Equation 5.27 can be written as
M

0
C
 q
0
0
Cs
CqT   q
  Q 

 
T  
C s   s  =  0 
0   λ  Qd 

(5.28)
The solution of this equation contains not only the generalized accelerations and
Lagrange multipliers, but also the second time derivatives of the nongeneralized
surface parameters. These augmented equations of motion are solved for the generalized and nongeneralized accelerations and Lagrange multipliers. Equation 5.28
ensures that the generalized and nongeneralized accelerations automatically satisfy
the contact constraints at the acceleration level. Having obtained the acceleration
vectors, the independent accelerations, which may include the generalized and nongeneralized variables, can be identified using the constraint Jacobian matrix. These
independent accelerations can be integrated forward in time to determine the independent coordinates and velocities. The dependent generalized and nongeneralized
coordinates can be determined by solving the constraint equations at the position
level using an iterative Newton-Raphson solution procedure. The constraint equations at the velocity level can be solved to determine the dependent generalized and
nongeneralized velocities. A more detailed discussion on the numerical algorithm
used with the augmented constraint contact formulation is presented in Chapter 6.
It is important to emphasize again that in the ACCF method, generalized and
nongeneralized coordinates (surface parameters) can be selected as the independent
variables. The state equations associated with these independent variables are identified and integrated forward in time. This is one of the important differences between
the ACCF method and the ECCF method, discussed in the following section, where
the surface parameters are systematically eliminated and thus cannot be selected as
independent coordinates or degrees of freedom in the ECCF method.
It is clear that the solution of Equation 5.28 must satisfy the following equation:
CTs λ = 0
(5.29)
We also note that only the contact constraints or specified motion trajectories can
be functions of the nongeneralized surface parameters; other joint constraints do not
depend on these geometric parameters. Each contact introduces five independent
contact constraints and five Lagrange multipliers associated with these constraints.
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171
These five constraint equations and five Lagrange multipliers must satisfy the preceding equation. Since there are only four surface parameters, the preceding equation
for each contact represents a system of four scalar equations in five unknowns. If
the constraint equations are linearly independent, the rank of the constraint Jacobian
matrix associated with the surface parameters for each contact will be four. Therefore, Equation 5.29, when applied to each contact, implies that there is only one
independent Lagrange multiplier; that is, the five contact constraints, which require
introducing four nongeneralized surface parameters and eliminate only one degree
of freedom, can be used to determine only one independent constraint (reaction)
force. This force can be used to determine the normal contact force that enters in
the formulation of the tangential creep forces.
5.4 EMBEDDED CONSTRAINT CONTACT
FORMULATION (ECCF)
In the embedded constraint contact formulation (ECCF), the surface parameters
(nongeneralized coordinates) are systematically eliminated from the equations of
motion. This leads to a smaller system of dynamic equations of motion, from which
the contact constraint forces (except for the normal contact force) are eliminated.
Before providing the details of this elimination, further discussion of the relationship
between the dependent coordinates and the constraint forces is needed. To this end,
Equation 5.29 for a contact k is written as follows:
C ks λ k = 0
T
(5.30)
where
T
s k =  s1wk s2wk s1rk s2rk  ,
λ k =  λ1k λ2k λ3k λ4k λ5k 
T
(5.31)
As previously pointed out, the five Lagrange multipliers associated with each
contact are not totally independent. Because of the dimension and rank of the
T
coefficient matrix C ks in Equation 5.30, only one Lagrange multiplier is independent,
as discussed in the preceding section. This result is consistent with the fact that the
contact constraints eliminate one degree of freedom, and the normal contact force
can be expressed in terms of one Lagrange multiplier only. In the embedded constraint contact formulation discussed in this section, the final form of the dynamic
equations of motion includes only one Lagrange multiplier associated with each
wheel/rail contact. To arrive at this form, the four surface parameters associated with
each contact are treated as dependent coordinates, and these geometric parameters
do not explicitly appear in the final form of the equations of motion, as is the case
in the augmented constraint contact formulation.
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5.4.1 POSITION ANALYSIS
To systematically eliminate the four surface parameters associated with each contact,
four contact constraint equations are selected such that their Jacobian matrix associated with the four surface parameters is a nonsingular square matrix. Since the
degree of freedom eliminated by the contact constraints can be (without loss of
generality) the relative motion along the normal to the surface at the contact point,
the four constraint equations are chosen from Equation 5.10 by excluding the third
equation that defines the relative motion along the normal to the tangent surface at
the contact point. This third equation will be added to the system differential equations of motion in order to determine the normal constraint contact force. To this
end, the vector of constraint equations associated with contact k is written in the
following partitioned form:
T
C k (q w , qr , s wk , s rk ) = C dk

T
C nk 

T
(5.32)
where
k
 t1r ⋅ (rPw − rPr ) 
 r

t 2 ⋅ (rPw − rPr ) 
dk
w
r
wk
rk

C (q , q , s , s ) =
=0
 t w ⋅ nr 
1


w
r
 t 2 ⋅ n

(5.33)
C nk (q w , q r , s wk , s rk ) = n rk ⋅ (rPwk − rPrk ) = 0
(5.34)
Assuming that the Jacobian matrix associated with the constraints of Equation
5.33 is nonsingular, these four constraint equations can be used to express the four
surface parameters in terms of the generalized coordinates of the wheel and the rail.
However, since the wheel/rail generalized coordinates must satisfy the fifth contact
constraint (Equation 5.34) as well as other kinematic constraints imposed on the
motion of the multibody vehicle system, the position analysis involves, at the position
level, two coupled solution stages, each of which requires the use of the iterative
Newton-Raphson procedure. For a given set of wheel/rail generalized coordinates,
Equation 5.33 is solved iteratively using a Newton-Raphson algorithm to determine
the surface parameters that enter into the formulations of Equation 5.34. Using these
calculated surface parameters, Equation 5.34 and other nonlinear kinematic constraints imposed on the motion of the multibody system are also iteratively solved
using a Newton-Raphson algorithm to determine the system generalized coordinates.
This iterative process continues until convergence is achieved for both stages. This
is discussed in greater detail in Chapter 6.
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5.4.2 EQUATIONS
OF
173
MOTION
The four contact constraints of Equation 5.33 do not prevent the penetration along
the normal to the surfaces at the contact point. Consequently, the fifth constraint of
Equation 5.34 must be used to ensure that such a penetration does not occur. The
embedded form of the dynamic equations of motion are developed by substituting
Equation 5.32 into Equation 5.17 to obtain the following variational equation of
motion:
(
T
T
)
(
T
T
)
+ Cqd λ d + Cqn λ n − Q + δ s T C ds λ d + C ns λ n = 0
δ qT Mq
(5.35)
where C ds and C ns are the Jacobian matrices associated with the contact constraints
of the types presented in Equations 5.33 and 5.34, respectively; and λd and λn are
the Lagrange multipliers associated with these constraint equations. Note that the
virtual change in the system surface parameters can be expressed in terms of the
virtual change in the wheel/rail coordinates using Equation 5.33, as follows:
δ s = Bδ q
(5.36)
where B = −(C ds )−1Cqd is the velocity transformation matrix associated with the contact constraints. Substituting Equation 5.36 into Equation 5.35, the principle of
virtual work for this system can then be written in terms of the virtual changes in
the generalized coordinates only as
(
)
T
 
+ Cqn + C ns B λ n − Q = 0
δ qT  Mq


(5.37)
It is important to note in Equation 5.37 that the generalized contact forces
associated with the contact constraints of Equation 5.33 are systematically eliminated
from the equations of motion using the following identity:
( )
T
T
T

Cqd λ d + BT C sd λ d = Cqd λ d +  − C sd

−1
T
T

Cqd  C sd λ d = 0

(5.38)
Furthermore, the acceleration kinematic equation
s = Bq
+ Bq
(5.39)
which is the result of Equation 5.36, can be used to eliminate s from the second
derivative of the constraint of Equation 5.34 with respect to time. Following a
procedure that can be considered as a special case of the procedure used in the
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preceding section to derive the augmented constraint contact formulation, one can
show by using Equations 5.37 and 5.39 that the system equations of motion, after
eliminating the nongeneralized surface parameters, can be written as
M

J
  Q 
JT   q
  = 

0   λ  Qdn 
(5.40)
where J, an implicit function of s, is the Jacobian matrix of the constraints of Equation
5.34 and other constraints imposed on the motion of the multibody system; λ is the
vector of the system Lagrange multipliers, which include only Lagrange multipliers
associated with the constraints of Equation 5.34 and other noncontact constraints
imposed on the motion of the system; and Qdn is a quadratic velocity vector that
results from the differentiation of the constraint equations twice with respect to time.
This vector is a function of the surface parameters s and their first time derivatives.
It is clear from the analysis presented in this section that, in the embedded
constraint contact formulations (ECCF), the surface parameters are treated as dependent variables and are systematically eliminated from the equations of motion. For
this reason, only wheel/rail accelerations are integrated forward in time. It should
also be noted that only one Lagrange multiplier associated with the contact constraint
of Equation 5.34 is used in Equation 5.40. This Lagrange multiplier defines the
normal contact force used to define the longitudinal, lateral, and spin creep forces.
Furthermore, since the surface parameters and the dependent Lagrange multipliers
are systematically eliminated without making any assumptions, the embedded constraint contact formulation (ECCF), as defined by Equation 5.40, and the augmented
constraint contact formulation (ACCF), as defined by Equation 5.28, are in principle
equivalent, despite the fact that these two formulations require the use of different
solution procedures and numerical algorithms, as will be discussed in Chapter 6.
5.5 ELASTIC CONTACT FORMULATION-ALGEBRAIC
EQUATIONS (ECF-A)
The two constraint contact formulations discussed in the preceding two sections do
not allow penetration or separation between the two bodies in contact. Imposing the
contact conditions eliminates one degree of relative motion between the wheel and
the rail. In the elastic approach, on the other hand, no kinematic contact constraints
are imposed; the wheel has six degrees of freedom with respect to the rail; and small
penetrations at the contact points are allowed. In the elastic contact formulations, a
compliant force element that consists of stiffness and damping forces is used to
determine the normal contact force. The location of the points of contact can be
determined using look-up tables, or they can be determined on-line using a discrete
nodal search or by solving a set of algebraic equations. In this and the following
section, two elastic contact formulations are discussed. In the first, called ECF-A,
the locations of the contact points are determined by solving a set of algebraic
equations. In the second elastic contact method, called ECF-N, the locations of the
contact points are determined using a nodal search.
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175
In the first elastic method discussed in this section, the location of the contact
points is determined by first solving a set of algebraic equations. For each contact,
four algebraic equations (Equation 5.33) are solved to determine the four parameters
that describe the geometry of the wheel and the rail surfaces. After determining the
four surface parameters that satisfy the algebraic nonlinear equations, the distance
between the surfaces along the normal is evaluated using a fifth equation (Equation
5.34) to determine whether or not there is a contact. The method presented in this
section allows for the definition of the wheel and rail surfaces using spline function
representations, thereby enabling the use of measured profile data for the wheel and
the rail.
It is important to point out that, while the algebraic equations used in this method
are the same as the contact constraints presented in the preceding sections, the
method presented in this section (ECF-A) cannot be considered as a constraint contact
formulation because the algebraic equations: (a) are not imposed at the velocity and
acceleration levels, (b) allow for penetrations and separations, and (c) do not introduce a Lagrange multiplier, as is used in the constraint method, to determine the
normal contact force. In the elastic methods discussed in this section, the normal
contact force is determined using a compliant force model that has stiffness and
damping coefficients.
To determine the location of the contact point using the elastic contact approach,
one can define the following four algebraic equations (Escalona, 2002; Pombo and
Ambrosio, 2003; Shabana et al., 2005):
t1r ⋅ r wr = 0 

t r2 ⋅ r wr = 0 

t1w ⋅ n r = 0 

t 2w ⋅ n r = 0 
(5.41)
where t1i and t i2 (i = w, r) are, respectively, the tangents to the wheel and rail surfaces
at the potential contact point; rwr = rw − rr is the vector that defines the relative
position of the point on the wheel with respect to the point on the rail; and nr is the
normal to the rail surface. Note that the first two equations in the preceding equation
are the same as the first two equations in Equation 5.10, while the last two equations
in the preceding equation are the same as the last two equations in Equation 5.10.
Because the tangent and normal vectors are functions of the surface parameters,
and assuming that the generalized coordinates of the wheel and the rail are known,
one can rewrite the preceding set of algebraic equations in a vector form as
E ( s) = 0
(5.42)
where E is the vector of nonlinear algebraic equations that can be solved using an
iterative Newton-Raphson algorithm for the surface parameters that define potential
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Railroad Vehicle Dynamics: A Computational Approach
nonconformal contact points. This requires evaluating the Jacobian matrix of the
algebraic equations and iteratively solving the following system for each contact to
determine Newton differences associated with the surface parameters:
 r w
 t1 ⋅ t1


 t r2 ⋅ t1w

 w
 ∂t1 ⋅ n r
 ∂s1w
 w
 ∂t 2 ⋅ n r
 ∂s w
 1
t1r ⋅ t 2w
∂t1r wr r r
⋅ r − t1 ⋅ t1
∂s1r
t r2 ⋅ t 2w
∂t r2 wr r r
⋅ r − t 2 ⋅ t1
∂s1r
∂t1w r
⋅n
∂s2w
∂n r w
⋅ t1
∂s1r
∂t 2w r
⋅n
∂s2w
∂n r w
⋅ t2
∂s1r
∂t1r wr r r 
⋅ r – t1 ⋅ t 2 
∂s2r

 t1r ⋅ r wr 
∂t r2 wr r r  s1w 
⋅ r – t2 ⋅ t2   w 
 r wr 
r
∂s2
  s2 
t2 ⋅ r 
−
=
  s r 
r
 t w ⋅ nr 
∂n w
1

 1w r 


⋅
t
1
  s2r 
∂s2r
 t 2 ⋅ n 

r
∂n w

(5.43)
⋅ t2

∂s2r

In this equation, s1w , s2w , s1r , and s2r are the Newton differences. Convergence is
achieved when the norm of the violation of the algebraic equations or the norm of
the Newton differences is less than a specified tolerance.
Having determined the vector of the surface parameters, the penetration can be
calculated using the third equation of Equation 5.10 as
δ = r wr ⋅ n r
(5.44)
If the surfaces penetrate, normal contact forces can be calculated using Hertz's
contact theory, while the creep forces can be calculated using one of the creep-force
models discussed in Chapter 4. The generalized normal and creep forces associated
with the system generalized coordinates are determined and introduced to the
multibody system dynamic equations of motion as generalized external forces. In
the evaluation of the normal contact force, as an alternative to the use of the Hertzian
component that is a function of the indentation, a damping force proportional to the
time derivative of indentation can also be included. An expression of the normal
force that can be used is given by (Shabana et al., 2004)
F = Fh + Fd = − K hδ 3 2 − Cδ δ
(5.45)
where δ is the indentation, Fh is the Hertzian (elastic) contact force, Fd is the damping
force, Kh is the Hertzian constant that depends on the surface curvatures and the
elastic properties, and C is a damping constant. The velocity of indentation δ is
evaluated as the dot product of the relative velocity vector between the contact points
on the wheel and rail and the normal vector to the surface at the contact point. The
reason for including the factor δ in the damping force is to guarantee that the contact
force is zero when the indentation is zero.
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177
5.6 ELASTIC CONTACT FORMULATION-NODAL
SEARCH (ECF-N)
As an alternative for using the ECF-A method that employs algebraic equations to
define the location of the contact points, one can define the profile of the wheel and
rail using discrete nodal points (Shabana et al., 2004, 2005). The distance between
these nodes can be calculated to determine the points on the wheel and the rail that
may come into contact. The use of this method has the advantage that it does not
require a certain degree of smoothness of the surfaces. It has, however, the disadvantage that the change in the lateral surface parameters of the wheel and rail is
not smooth, since the contact is assumed to occur at discrete nodal points. When
the contact jumps from one node to the neighboring one, a small jump in the relative
velocity between the wheel and the rail at the contact point is expected. This small
jump leads, in some examples, to discontinuity of the creepages. Since the creepage
coefficients that enter into the calculation of the creepage forces are very high, the
discontinuous change in the contact locations resulting from the use of the nodal
search leads to high impulsive forces. This problem, however, can be solved by
using interpolation to determine the contact points instead of using discrete points.
As discussed in the literature (Shabana et al., 2004), the nodal search for the
contact points consists of the following three steps:
1. Calculation of the rail arc length s1r traveled by the wheel. The parameter
s1r defines the rail cross-section in which the points of contact are located.
2. Calculation of the wheel angular parameter s2w . The parameter s2w defines
the wheel diametric section in which the points of contact are located.
3. Search for the contact points. In this phase, the rail parameter s2r and the
wheel parameter s1w of the points of contact are determined. This phase of
the search starts once the sections of the wheel and rail in which the
contact points are located have been determined. The exact position of
the contact points is determined in this phase of the search.
To determine the arc length traveled by the wheel, a selected point Q on the
center of the wheel, as shown in Figure 5.4, is used first to determine the rail space
curve parameter s1r . It is assumed that the rate of change of the rail parameter s1r is
equal to the projection of the velocity of this point on the tangent along the longitudinal rail direction, that is,
T
s1r = rQw ⋅ t1r
(5.46)
where rQw is the global velocity vector of point Q, and t1r is the longitudinal tangent
to the rail. The preceding differential equation is solved simultaneously with the
differential and algebraic equations of the multibody system to determine s1r , which
is used for the search for the point of contact between the wheel and the rail. Clearly,
one needs to introduce a number of arc length first-order differential equations
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Railroad Vehicle Dynamics: A Computational Approach
(Equation 5.46) equal to the number of wheels in the dynamic model. Note that the
ECF-A method does not require introducing these first-order differential equations,
since the arc length traveled by the wheel is readily available from the solution of
the algebraic equations (Equation 5.42).
To determine the points of contact between the wheel and the rail, the global
position vectors of the nodal points that define the wheel and rail profiles must be
determined (Shabana et al., 2004). The distance between the points on the wheel
and the points on the rails can be calculated and used with a user-specified tolerance
criterion to determine the points of contact. Since this search can lead to a large
number of contact points, an optimized procedure that improves the computational
efficiency can be adopted. The contact points can be grouped in batches. A batch is
a collection of a sequence of pairs of points on the wheel and rail that have
penetration. While the algorithm can allow for an arbitrary number of contact
batches, a limit on the number of contact batches can be imposed in the numerical
implementation. The two points (one on the wheel and one on the rail) that lead to
the maximum indentation can be selected as the points of contact for any given
batch. The number of points of contact between the wheel and the rail is equal to
the number of the contact batches. That is, the algorithm used to search for the
contact points can be designed to allow for multiple contacts between the wheel and
the rail. Once the contact points are determined, an elastic force model similar to the
one presented for the ECF-A method can be used.
Before concluding this section, it is important to mention that Equation 5.46 is
not the only method that can be used to determine the rail longitudinal surface
parameter s1r . Another possible method is to use the wheel absolute Cartesian coordinates and solve a system of nonlinear algebraic equations instead of differential
equations to determine s1r . The use of a simple first-order differential equation,
however, can be more efficient as compared with the solution of nonlinear algebraic
equations, which can be prone to convergence problems.
5.7 COMPARISON OF DIFFERENT
CONTACT FORMULATIONS
As discussed previously in this chapter, there are two different approaches that are
commonly used to solve the multibody contact problems: the constraint and elastic
contact approaches. These two approaches lead to different mathematical models
for determining the normal contact force. In the constraint approach, the nonconformal contact conditions (Equations 5.9 or 5.10) are imposed on the motion of the
system, and the normal contact force is predicted as a constraint force obtained using
the technique of Lagrange multipliers associated with the contact constraints (Equations 5.28 and 5.40). In this case, no wheel/rail separations or penetrations are
allowed. The study of the wheel/rail separation scenarios using the constraint
approach requires a special mathematical treatment, since in this case one must deal
with a system with variable kinematic structure. This may require the use of a
generalized impulse-momentum approach, as discussed in the literature (Khulief
and Shabana, 1986).
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Multibody Contact Formulations
179
On the other hand, in the elastic approach, no contact constraint conditions are
imposed. The normal contact force is defined using a compliant force model, which
is a function of the indentation between the two contact surfaces (Equation 5.45)
and the assumed stiffness and damping coefficients. In other words, the normal
contact force in the elastic approach is defined as an external force by allowing small
elastic deformation of the contact surfaces. Using this approach, the wheel/rail
separation scenarios can be easily examined, since the system degrees of freedom
do not change due to contact or separation. For this reason, the elastic approach can
be used effectively in modeling many simulation scenarios such as derailment and
flange contact. It is well known, however, that the use of the compliant contact force
model in the elastic approach introduces high frequencies in the solution of the system
equations of motion. This problem is not generally encountered when the constraint
formulations are used.
Using the constraint approach, different contact formulations that require the
use of different numerical solution procedures can be developed. In this chapter,
two general multibody system constraint contact formulations are discussed. In the
augmented constraint contact formulation (ACCF), the equations of motion are
expressed explicitly in terms of the nongeneralized surface parameters. The second
time derivatives of the nongeneralized surface parameters can be chosen as independent accelerations that can be integrated forward in time using direct numerical
integration methods. In the embedded constraint contact formulation (ECCF), on
the other hand, the nongeneralized surface parameters are systematically eliminated
from the equations of motion using the velocity transformation given by Equation
5.36. The surface parameters in the ECCF are always considered as dependent
coordinates that are determined by solving Equation 5.33.
Two elastic contact formulations are also discussed in this chapter. The main
difference between the two formulations is the method used to determine the location
of the contact points. The form of the equations of motion remains the same in both
formulations. In the first elastic method (ECF-A), algebraic equations are solved
using an iterative Newton-Raphson procedure to determine the locations of the contact
points. These algebraic equations are not imposed as constraints at the position,
velocity, or acceleration levels; no Lagrange multipliers are introduced; and no
degrees of freedom are eliminated. As demonstrated in this chapter, this method
requires a certain degree of smoothness in the definition of the wheel and rail
surfaces. This degree of smoothness is not required when using the second elastic
formulation, ECF-N, which is based on nodal search techniques. The ECF-A method,
however, leads to smoother solution as compared with the ECF-N, which employs
discrete wheel/rail profile points. Table 5.1 provides a comparison of the four contact
formulations discussed in this chapter. Computational algorithms based on the four
contact formulations are discussed in more detail in Chapter 6, where examples are
also presented to compare the results obtained using different formulations.
5.8 PLANAR CONTACT
In three-dimensional problems, the location of the points of contact between the
wheel and the rail can be determined by solving a two-dimensional contact problem.
Elastic
Constraint
Embedded constraint
contact formulation
(ECCF)
Elastic contact
formulation-algebraic
equations (ECF-A)
Elastic contact
formulation-nodal
search (ECF-N)
Method
Augmented constraint
contact formulation
(ACCF)
Wheel
Separation
Requires
special
mathematical
treatment
Yes
Normal
Contact
Force
Lagrange
multiplier
(constraint
force)
Compliant
force
No
restriction
Smooth
E(s) = 0
Nodal search
Smooth
Contact
Surface
Smooth
C(q,s) = 0
(two stages)
Contact
Search
C(q,s) = 0
M

 J
M

0
C
 q
CTq   q
  Q 
   
CTs   s  =  0 
0   λ  Qd 

= Q
Mq
  Q 
JT   q
  = 

0   λ  Qdn 
0
Cs
0
Equations of Motion
180
Approaches
TABLE 5.1
Contact Formulations
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Multibody Contact Formulations
181
This is the case of planar contact analysis, in which the kinematic coupling between
some of the geometric surface parameters is neglected. The use of this planar contact
method leads to approximate prediction of the locations of the contact point, as will
be later explained in this section. In the planar contact formulation, the contact conditions are formulated in terms of only two surface parameters that define the wheel
and rail profiles. The other two surface parameters, the rail arc length and the
rotational wheel parameter, are assumed to be known. If a constraint contact formulation is used in this case of planar contact, one needs only three contact constraint
equations to determine the two unknown profile surface parameters and also eliminate one degree of freedom. The same three algebraic equations used as constraints
in the constraint contact formulation can also be used as the basis for developing
an elastic contact formulation by following the procedure described in Section 5.5.
In this section, the formulation of the planar contact conditions is discussed.
5.8.1 INTERMEDIATE WHEEL COORDINATE SYSTEM
As discussed in Chapter 2, two different sets of coordinates are often used to
formulate the kinematic and dynamic equations of the railroad vehicle systems:
absolute Cartesian coordinates and trajectory coordinates. Recall that the vector of
trajectory coordinates is given for the wheel as
p w = [s w
y wr
z wr
ψ wr
φ wr
θ wr ]T
(5.47)
where sw is the rail arc length that defines the location of the origin of the wheel
trajectory coordinate system introduced in Chapter 2; ywr and zwr define the relative
position of the wheel center of mass with respect to the wheel trajectory coordinate
system; and ψ wr, φ wr, and θ wr are the Euler angles that define the orientation of
the wheel with respect to the trajectory coordinate system. The sequence of rotations
used for Euler angles is Zw, Xw, Yw, as discussed in Chapter 2. If the equations of
motion of the railroad vehicle system are formulated using the trajectory coordinates,
all the elements of the vector of Equation 5.47 are known at a given instant of time.
On the other hand, if the equations are formulated in terms of the absolute Cartesian
coordinates, one can always develop the relationships between the absolute coordinates and the trajectory coordinates and use these relationships to determine the
elements of the vector pw of Equation 5.47. These relationships are discussed in
greater detail in Chapter 7. Therefore, in this section, it is assumed that the elements
of the vector pw of Equation 5.47 can be determined regardless of the set of coordinates used in the formulation of the dynamic equations of the railroad vehicle
system.
In the planar contact formulation, one introduces an intermediate wheel coordinate system that does not experience the pitch rotation θ wr of the wheel about its
Yw axis. The orientation of this intermediate wheel coordinate system can be defined
in the wheel trajectory coordinate system using the two Euler angles ψ wr and φ wr.
The transformation matrix that defines the orientation of the intermediate wheel
coordinate system in the global coordinate system can then be written as follows:
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Railroad Vehicle Dynamics: A Computational Approach
A wi = Atw A z A x =  a1wi
a 2wi
a 3wi 
(5.48)
where Atw is the transformation matrix that defines the orientation of the wheel
trajectory coordinate system introduced in Chapter 2 with respect to the global
coordinate system, and
 cos ψ wr

A z =  sin ψ wr
 0

− sin ψ wr
cos ψ wr
0
1
0


0  , Ax = 0
0
1 

0


− sin φ 
cos φ wr 
0
cos φ
sin φ wr
wr
wr
(5.49)
It follows that the transformation matrix that defines the orientation of the intermediate wheel coordinate system is given by
 cos ψ wr

A = A  sin ψ wr
 0

wi
tw
− sin ψ wr cos φ wr
cos ψ wr cos φ wr
sin φ wr
sin ψ wr sin φ wr 

− cos ψ wr sin φ wr 

cos φ wr

(5.50)
At this point, it is important to point out that, if the absolute Cartesian coordinates
are used, one can simply obtain the transformation matrix Awi using one simple
matrix multiplication. Let Aw be the transformation matrix that defines the orientation
of the wheel coordinate system in the global coordinate system. The matrix Awi can
then be obtained from Aw as
A wi = A w A Ty
(5.51)
where the matrix Ay accounts for the pitch rotation about the wheel Y w axis and is
given by
 cos θ wr

Ay =  0
 − sin θ wr

0
1
0
sin θ wr 

0 
cos θ wr 
(5.52)
The columns of the transformation matrix Awi of Equations 5.50 or 5.51 define unit
vectors along the axes of the intermediate wheel coordinate system.
5.8.2 DISTANCE TRAVELED
It is important to differentiate between the coordinate sw that defines the position of
the origin of the wheel trajectory coordinate system and the arc length s1r of the rail
space curve at which the contact occurs. In some railroad vehicle formulations, track
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Multibody Contact Formulations
183
coordinate systems that travel with a constant velocity are introduced to define the
configuration of the vehicle components. The distance traveled by the wheel s1r is
determined as the product of this constant velocity and time. In addition to the
unnecessary complications that arise from using such a motion description, the use
of a track coordinate system that travels with a constant velocity has a major
disadvantage when simulations of braking are considered (Handoko, 2006). Since
the rails can have deviations, it is important to use an accurate method that is
independent of s w or the component forward velocity to determine the coordinate s1r .
In this section, the discussion focuses on two planar contact methods that can
be used to predict the surface parameter s1r at which the contact occurs. In the first
method, the kinematic coupling between the parameter s1r and the two parameters s2r
and s1w is neglected. In the second method, which is briefly discussed before concluding this section, the kinematic coupling between the parameter s1r and the two
parameters s2r and s1w is considered. This second planar contact method is more accurate, since the three parameters s1r , s2r , and s1w are determined simultaneously by
solving a set of nonlinear algebraic equations. Recall that the location of an arbitrary
point on the rail space curve can be defined in the global coordinate system as
( )
r0r = R r + A r u r0 s1r
(5.53)
In this equation, s1r is the arc length of the rail space curve, Rr is the global position
vector of the rail body coordinate system, and u r0 (s1r ) is the vector that defines the
position of the arbitrary point on the rail space curve with respect to the rail body
coordinate system, as shown in Figure 5.5. To determine the arc length s1r at which
FIGURE 5.5 Planer contact condition.
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Railroad Vehicle Dynamics: A Computational Approach
the vector r0r of the preceding equation has a zero component along the Xwi axis of
the wheel, one must solve the following equation for the arc length s1r :
T
(
)
a1wi ror − R w = 0
(5.54)
In this equation, Rw is the vector that defines the global position vector of the origin
of the wheel body coordinate system, and a1wi is a unit vector along the Xwi axis of
the intermediate wheel coordinate system. The vector a1wi is the first column of the
transformation matrix Awi defined by Equation 5.50 or, alternatively, by Equation
5.51. Using Equation 5.53, the preceding equation can be written as
a1wi
T
(R
r
)
+ Ar u r0 − R w = 0
(5.55)
In the case of arbitrary curved track, this equation is a nonlinear function of the rail
arc length s1r . Assuming that the wheel and the rail generalized coordinates are
known, the preceding equation can be solved iteratively using a Newton-Raphson
procedure. In this procedure, the following algebraic equation must be iteratively
solved:
(a
wi T
1
)
Ar t1r ∆s1r = − a1wi
T
(R
r
+ Ar u ro − R w
)
(5.56)
where ∆s1r is the Newton difference, and t1r = ∂u ro ∂s1r is the longitudinal tangent
of the rail space curve defined in the rail body coordinate system. The preceding
scalar equation can be solved for ∆s1r as follows:
{
 a1wi T R r + Ar u ro − R w
∆s1r = − 
T
a1wi Ar t1r

} 

(5.57)
Convergence is achieved if the Newton difference is smaller than a specified tolerance or if Equation 5.55 is satisfied. Note that the procedure described in this section
for determining s1r is an alternative for using Equation 5.46, which requires introducing a first-order differential equation.
5.8.3 PROFILE PARAMETERS
Knowing the coordinate s1r , the planar contact conditions can be imposed to determine the location of the point of contact between the wheel and the rail. It is assumed
that the profile of the wheel is defined using the parameter s1w , while the profile of
the rail is defined using the parameter s2r . Clearly, when the wheel comes into contact
with the rail, one degree of freedom is eliminated. By introducing the intermediate
wheel coordinate system, one can impose only three constraint equations, which can
be used to eliminate one degree of freedom and determine the profile surface
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Multibody Contact Formulations
185
parameters s1w and s2r . The first two conditions ensure that the distance between the
wheel and the rail at the contact point in the plane Y wiZ wi is equal to zero. The third
condition ensures that the normal to the rail at the contact point is perpendicular to
the tangent to the wheel at that point. This third condition is required when the
assumption of nonconformal contact is used. These three contact conditions can be
written as follows:
(
(
)
)
t r2 R r + Ar u r − R w − Awi u wi = 0 


rT
r
r r
w
wi wi
n R + A u − R − A u = 0

rT w

n t1 = 0

T
(5.58)
In this equation, t r2 and nr are the lateral tangent and normal to the rail profile at the
contact point, respectively; and t1w is the tangent to the wheel at the contact point.
If the generalized coordinates of the wheel and rail are given and s1r is assumed to
be known, u r in the preceding equation is a function of s2r only while u wi is a function
of s1w only. Therefore, two of the nonlinear equations in the preceding equation can
be used to determine the profile parameters s1w and s2r , while the remaining equation
can be used to either eliminate one degree of freedom by expressing one coordinate
in terms of the other wheel coordinates or to introduce a compliant force model to
describe the wheel/rail interaction, as described in Section 5.5. The first and third
equations in Equation 5.58 are the two equations that are often used to solve for the
profile surface parameters s1w and s2r , while the second equation is the one that is
used to eliminate the degree of freedom that represents the relative motion along
the normal to the rail at the contact point. This second equation can alternatively be
used to define a penetration that enters into the formulation of a compliant force
model in an elastic contact approach. As previously pointed out, the constraint
approach requires imposing the constraints at the position, velocity, and acceleration
levels, and as a result, there are Lagrange multipliers associated with these constraints
when the augmented formulation is used. An embedded constraint contact formulation can also be used in the case of planar contact by systematically eliminating
the surface parameters, as described in Section 5.4.
5.8.4 COUPLING
BETWEEN THE
SURFACE PARAMETERS
It is clear from the analysis presented thus far in this section that, when the planar
contact is used, the search for the two profile surface parameters is independent of
the other two parameters s2w and s1r . In the planar contact analysis, the parameter s2w
is assumed to be equal to the pitch angle θ wr, while the parameter s1r can be determined using an independent search, as described in this section. These two parameters s2w and s1r are assumed to be fixed while searching for the profile parameters.
This approach is different from the three-dimensional contact analysis in which the
four surface parameters are solved for simultaneously using nonlinear coupled algebraic equations, as previously described in this chapter. It is important to point out
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Railroad Vehicle Dynamics: A Computational Approach
that the planar contact method described here does not guarantee that the two points
on the wheel and the rail coincide; that is, these two designated contact points do
not have the same global position vector. These two points can differ by a longitudinal
shift. To ensure that there is no longitudinal shift between the two points on the
wheel and the rail, Equation 5.55 must be replaced by the following equation:
t1r
T
(R
r
)
+ Ar u r − R w − Awi u wi = 0
(5.59)
In this equation, t1r is the longitudinal tangent at the potential contact point. Therefore, to guarantee that there is no longitudinal shift, the following three equations
must be solved simultaneously to determine s1w , s1r , and s2r :
t1r
T
t r2
T
(R + A u − R
(R + A u − R
r
r
r
r
r
r
T
n r t1w = 0
)
)
− Awi u wi = 0 


w
− Awi u wi = 0 



w
(5.60)
The penetration can be calculated using the following equation:
δ = nr
T
(R
r
+ Ar u r − R w − Awi u wi
)
(5.61)
This modified planar contact formulation, which is a special case of the threedimensional formulation given by Equation 5.41, ensures that — in the case of a
yaw angle, variable vehicle velocity, or in the case of a curved track — there is no
longitudinal shift between the two points of contact on the wheel and the rail. If a
contact constraint formulation is used by replacing Equation 5.61 with a constraint
equation, the two contact points coincide, and both will have the same global position
vector.
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Implementation and
Special Elements
The preceding chapter presented several three-dimensional formulations for solving
the wheel/rail contact problem. These formulations, as demonstrated in this chapter,
can be implemented in general multibody system computer algorithms that are
designed to solve a system of differential and algebraic equations. In railroad dynamics, the multibody vehicle model can have an arbitrary number of bodies whose
motions are subjected to kinematic constraints in addition to the wheel/rail contact
elements. Therefore, it is important to discuss the general computational multibody
algorithms before discussing the implementation of the special wheel/rail contact
elements in these algorithms. Consequently, the first section of this chapter focuses
on the structure of the multibody system computational algorithms and the numerical
procedures that can be used to develop a general-purpose multibody system computer
code.
Sections 6.2 and 6.3 describe the implementation of the wheel/rail contact
elements in multibody system algorithms. This implementation is based on the
constraint and elastic contact formulations presented in Chapter 5 for determining
the wheel/rail contact locations and forces. Section 6.4 outlines the procedure used
to calculate the creep forces, and Section 6.5 explains the need for the use of higher
derivatives with respect to the surface parameters in the contact formulations. Sections 6.6 and 6.7 explain the structure of the track preprocessor based on the
geometric description presented in Chapter 3. Section 6.7 focuses on the computer
description of rail irregularities, such as deviations, as well as the use of measured
track data. Section 6.8 develops some force elements that are frequently used in
railroad vehicle dynamic models. The use of some of these elements is not limited
to railroad vehicle system applications; indeed, such force elements can also be found
in other multibody system applications. Magnetically levitated vehicles (Maglev),
which are supported by magnetic forces, are another example of guided trains.
Maglev vehicles are still in an experimental stage, but they are in service in some
countries that have the potential for developing very high speed trains. Section 6.9
gives a brief introduction to the Maglev systems. One of the important problems
encountered in the computer-aided analysis of large-scale railroad vehicle models
is the accurate determination of the initial static equilibrium configuration. For this
reason, computational approaches that can be used to determine the initial static
equilibrium configuration of railroad vehicle models are discussed in Section 6.10.
This chapter concludes with numerical examples of railroad systems that demonstrate the computer implementation and the use of the multibody wheel/rail contact
formulations discussed in Chapter 5.
187
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Railroad Vehicle Dynamics: A Computational Approach
6.1 GENERAL MULTIBODY SYSTEM ALGORITHMS
Multibody system computational algorithms are designed to solve the differential
and algebraic equations (DAE) that govern the constrained motion of physical
systems. The capability of solving differential and algebraic equations is one of the
main features that distinguish the general computational multibody system algorithms from other algorithms that are used to develop special-purpose computer
codes. This section reviews the basic equations used in the multibody system formulations and explains the use of these equations to develop a computational algorithm for solving the differential and algebraic equations of the multibody system.
6.1.1 CONSTRAINED DYNAMICS
Two basic methods were introduced in Chapter 2 for the computer formulation of
the dynamic equations of motion of the multibody railroad vehicle systems. In the
first method, the augmented formulation, the equations of motion are formulated in
terms of redundant coordinates and Lagrange multipliers; in the second method, the
embedding technique, the kinematic constraint equations are systematically eliminated from the dynamic equations of motion, which are expressed in terms of the
system degrees of freedom. In the case of the augmented formulation, the equations
of motion of the system (as discussed in Chapter 2) can be written in the following
matrix form (Shabana, 2005):
M

Cq
CTq   q
  Q 
  =  
0   λ  Qd 
(6.1)
where M is the system symmetric mass matrix, Cq is the Jacobian matrix of the
is the system generalized accelerations vector, λ is
system constraint equations, q
the vector of Lagrange multipliers associated with the system constraints, Q is the
generalized force vector, and Qd is the vector that results from the differentiation of
the kinematic constraint equations twice with respect to time and absorbs the quadratic terms in the first time derivatives of the generalized coordinates. In Equation
6.1, the mass matrix M and the constraint Jacobian matrix Cq are functions of the
system coordinates q; and the vectors Q and Qd can be functions of the system
generalized coordinates and velocities q and q as well as time. Therefore, given the
initial conditions, which are defined by the initial values of the vectors q and q at
time t = 0, Equation 6.1 can be constructed and solved for the unknown accelerations
and the vector of Lagrange multipliers λ. Since the system is subjected to conq
straint conditions, the coordinates and velocity vectors q and q must satisfy the
constraint equations at the position and velocity levels. That is, the coordinate vector
must satisfy the following constraint conditions at any time t, including t = 0:
C(q, t ) = 0
(6.2)
Differentiating this equation with respect to time, it is clear that the velocity vector
must satisfy the following conditions at any time t, including t = 0:
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189
C q q = −C t
(6.3)
where Ct = ∂C/∂t is the vector of the partial derivatives of the constraints of Equation
6.2 with respect to time. The vector Ct is identically equal to zero if all the constraint
equations are not explicit functions of time. Note that Equation 6.1 ensures that the
constraint equations are satisfied at the acceleration level. Methods that can be used
to avoid violations of the kinematic constraints at the position and velocity levels
will be discussed later in this section.
The solution of Equation 6.1 defines the vector of the generalized accelerations
at a given time t. This vector can be integrated numerically to define the system
q
generalized velocities and coordinates q and q at t = t + ∆t, where ∆t is the integration
time step. To enable the use of well-developed integration methods designed to solve
nonlinear first-order differential equations, the following state vector and its time
derivatives are defined:
q 
y =  ,
q 
q 
y =  

q
(6.4)
Given the vector y at time t, it is clear that the solution of Equation 6.1 can be
used to define the vector y at time t, which is required to advance the integration to
determine y at time t + ∆t. The vector y can be used with standard numerical
integration methods designed to solve first-order differential equations to determine
the system generalized velocities and coordinates at time t = t + ∆t. These coordinates
at time
and velocities can be used in Equation 6.1 to evaluate the accelerations q
t + ∆t. This process continues until the end of the simulation is reached. It is
important, however, to point out that, due to the numerical error that results from
the numerical integration of the accelerations, the coordinates and velocities do not
necessarily satisfy the constraint equations at the position and velocity levels, respectively. The violations in the constraint equations at the position and velocity levels
can be significant, particularly for long simulations of systems that consist of many
interconnected bodies that are subjected to severe loading conditions, such as in the
case of railroad vehicles. Robust multibody system algorithms are designed to avoid
the violation of the kinematic constraint equations and to ensure that these constraint
equations are satisfied at both the position and velocity levels. Two techniques are
often used to ensure that the numerical solution of the multibody system equations
satisfies the constraint equations. These two techniques are based on the penalty and
constraint stabilization methods, and the generalized coordinates partitioning
method. Both of these different techniques are discussed below.
6.1.2 PENALTY
AND
CONSTRAINT STABILIZATION METHODS
Among the methods used in the multibody system literature to avoid violations of
the constraint equations are the penalty and the constraint stabilization methods.
The penalty method is based on the concept of introducing a force function that
reduces the violation in the constraint. For example, if C(q,t) = 0 is the vector of
the constraint functions, one can define a force vector that takes the following form:
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Railroad Vehicle Dynamics: A Computational Approach
, where K and D are matrices of stiffness and damping coefficients,
F = −KC − DC
respectively. Note that by a proper selection of the matrices K and D, the force F
when added to the vector Q of Equation 6.1 will introduce a restoring force if the
constraint equations are violated at the position or velocity level. The results obtained
by the penalty method have been compared with the results of other methods that
are used to ensure that the constraint equations are not violated (Ozaki and Shabana,
2003a, 2003b).
In the constraint stabilization method, which borrows a concept from the vibration and feedback control theories (Nikravesh, 1988; Shabana, 1996a), velocity- and
coordinate-dependent terms are introduced to satisfy the constraint equations. For
example, a simple constraint on a coordinate y that requires this coordinate to remain
zero all the time can be written at the acceleration level as y = 0 . The numerical
integration of this acceleration equation, due to numerical errors, does not, in general,
guarantee that y = y = 0 at an arbitrary time t. This system can be stabilized by
introducing damping and stiffness terms and then writing the constraint equation at
the acceleration level as y + 2α y + β 2 y = 0. In this equation, 2α y and β 2y are the
terms introduced to achieve stability, and α and β are assumed coefficients that can
be selected in order for y and y to approach zero with a minimum level of oscillations.
This simple concept can be generalized and applied to the constraint equations. Recall
that if one has the correct solution, the constraint equations and their derivatives can
be written as follows:
( )
( )
( )
q, t = C
q, t = 0
C q, t = C
(6.5)
Following the argument made for the single equation, the constraints can be
stabilized by writing the constraint equations at the acceleration level instead of
q, t = 0 as (Baumgarte, 1972)
C
( )
+ 2α C
+ β 2C = 0
C
(6.6)
= Q de
Cq q
(6.7)
− β 2C
Qde = Qd − 2α C
(6.8)
This equation can be written as
In this equation,
Using Equation 6.7, the augmented form of the equations of motion (Equation 6.1)
can be modified and written as
M

Cq
CTq   q
 

Q
  = 
2 
0   λ  Qd − 2α C − β C 
(6.9)
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191
The difference between the penalty method and the constraint stabilization
method is clear from this equation. In the penalty method, the generalized force
vector Q is changed, while in the constraint stabilization method discussed in this
section, the constraint equations are changed. Based on the constraint stabilization
method discussed in this section, the following simple algorithm for solving the
dynamic equations of motion of the multibody system can be proposed:
1. At the initial time t = t0, the initial conditions defined by the initial values
of q and q can be used to evaluate the mass matrix M, the Jacobian matrix
of the constraints Cq, the vector of applied forces Q, and the vector Qde of
Equation 6.8.
2. By selecting the coefficients α and β, Equation 6.9 can be constructed
and the vector of Lagrange muland solved for the acceleration vector q
tipliers λ at the specified time t. Lagrange multipliers can be used to define
the generalized constraint forces.
can be integrated numerically to determine the
3. The acceleration vector q
coordinate and velocity vectors q and q at time t = t + ∆t.
4. If the end of the simulation time is reached, stop; otherwise, go to Step
2 of the algorithm and continue until the end of the simulation time is
reached.
The accuracy of the solution obtained using this algorithm depends, as documented in the multibody system dynamics literature, on the values of the coefficients
α and β. In general, if α and β are not equal to zero, the solution of the system
predicted by the algorithm outlined above oscillates around the correct value. In
general, α and β can be selected to take values between 1 and 10. The coefficients
α and β can also be chosen such that the system is critically damped (Nikravesh,
1988).
6.1.3 GENERALIZED COORDINATES PARTITIONING
Numerical problems can be encountered in many applications when the penalty or
the constraint stabilization method is used. Another alternative that can be used to
develop a robust numerical algorithm that ensures that the constraints are not violated
is the method of the generalized coordinates partitioning (Wehage, 1980). This
method, which is widely used in multibody system algorithms and computer codes,
ensures that the constraints are satisfied at the position, velocity, and acceleration
levels. In the generalized coordinate partitioning method, a set of independent coordinates is identified and used to define the associated differential equations that can
be integrated to determine the independent velocities and coordinates. Knowing the
independent generalized coordinates (degrees of freedom), one can determine the
dependent generalized coordinates using the kinematic constraint equations. Knowing the independent velocities, the constraint equations at the velocity level can be
used to determine the dependent velocities. In order to explain this procedure,
consider a virtual change of the system generalized coordinates δq. In this case, the
constraints of Equation 6.2 lead to
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Railroad Vehicle Dynamics: A Computational Approach
Cqδ q = 0
(6.10)
In this equation, the constraint Jacobian matrix Cq has the number of rows equal to
the number of the constraint functions nc, and the number of columns is equal to
the number of the system generalized coordinates n. If the constraint equations are
linearly independent, the Jacobian matrix Cq has a full row rank. In this case, one
can identify a set of independent coordinates equal to (n − nc) and write the total
vector of the system generalized coordinates in the following partitioned form:
q 
q= i
q d 
(6.11)
where qi and qd are the vectors of independent and dependent coordinates, respectively. Note that the number of dependent coordinates is equal to the number of
constraint equations. Therefore, if the independent coordinates (degrees of freedom)
are known, the nonlinear constraint equations of Equation 6.2 can be solved for the
dependent coordinates using an iterative Newton-Raphson algorithm. By applying
this method, it is guaranteed that the constraint equations are satisfied at the position
level.
Using this coordinate partitioning, Equation 6.10 can be written as follows:
C qi δ q i + C q d δ q d = 0
(6.12)
Since the number of the dependent coordinates is equal to the number of the system
constraint equations, the Jacobian matrix C q d associated with these coordinates is a
square matrix with dimension nc × nc, while the Jacobian matrix C qi associated with
the independent coordinates has dimension nc × (n − nc). If the constraint equations
are independent, the system degrees of freedom can be selected such that the matrix
C q d is nonsingular. In this case, the virtual change in the dependent coordinates can
be expressed in terms of the virtual change of the independent coordinates as follows
(Shabana, 2005):
δ qd = Cq−1d (Cqi δ qi )
(6.13)
Differentiating the constraint equations with respect to time and using a similar
procedure, one can obtain the following relationship for the dependent and independent velocities:
C qi q i + C q d q d = −C t
(6.14)
The dependent velocities can then be written in terms of the independent velocities
as follows:
q d = −C q−1d (C qi q i + C t )
(6.15)
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193
When the generalized coordinate partitioning method is used, one needs to integrate
only the state equations associated with the independent coordinates. Therefore, the
following state vector and its derivative are formed:
q 
y =  i ,
q i 
q 
y =  i 
i 
q
(6.16)
can be determined for a given q and q
Note that the total acceleration vector q
using Equation 6.1. The independent accelerations can be selected and the vector y
can be formed. By numerically integrating y , one obtains the independent coordinates and velocity vectors qi and q i , respectively. Knowing the independent coordinates
qi, the dependent coordinates can be determined as previously mentioned by solving
the nonlinear constraint equations of Equation 6.2 using an iterative Newton-Raphson
algorithm. Similarly, knowing the independent velocities q i , the dependent velocities
can be determined using the linear velocity relationship of Equation 6.15. Therefore,
by using the generalized coordinate partitioning method, it is guaranteed that the
constraint equations are satisfied at the position and velocity levels. Furthermore,
by determining the accelerations as the solution of Equation 6.1, it is also guaranteed
that the constraint equations are satisfied at the acceleration level.
In summary, the following computational algorithm based on the generalized
coordinate partitioning method can be proposed:
1. At the initial time t = t0, the initial conditions defined by the initial value
of q and q can be used to evaluate the mass matrix M, the Jacobian matrix
of the constraints Cq, the vector of applied forces Q, and the vector Qd
that appear in Equation 6.1. It is assumed that the initial conditions satisfy
the kinematic constraint conditions imposed on the motion of the system.
2. Equation 6.1 can be constructed and solved for the acceleration vector q
and the vector of Lagrange multipliers λ at the specified time t. Lagrange
multipliers can be used to define the generalized constraint forces.
i associated with the system degrees of
3. The independent accelerations q
freedom are identified and used to define the vector y of Equation 6.16.
This vector can be integrated numerically to determine the independent
coordinate and velocity vectors qi and q i at time t = t + ∆t.
4. Knowing the independent coordinates qi, the nonlinear constraint equations of Equation 6.2 can be solved for the dependent coordinates qd using
an iterative Newton-Raphson algorithm.
5. Knowing all the coordinates q and the independent velocities q i from the
numerical integration of y , the dependent velocities can be determined
by solving the system of linear equations of Equation 6.15.
6. At this stage, all the coordinates and velocities are known. If the end of
the simulation time is reached, stop; otherwise, go to Step 2 of the
algorithm and continue until the end of the simulation time is reached.
It is clear that this algorithm is more computationally intensive than the algorithm
based on the penalty or the constraint stabilization methods.
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Railroad Vehicle Dynamics: A Computational Approach
6.1.4 IDENTIFICATION
OF THE INDEPENDENT
COORDINATES
In the generalized coordinate partitioning method, the coordinates are partitioned as
independent and dependent. Therefore, identifying the independent coordinates is
required in order to use the generalized coordinate partitioning method. For large
and complex systems that include many bodies and joints, the identification of the
independent coordinates by inspection can be a very difficult or even impossible
task. In this case, one must resort to numerical methods to select the system degrees
of freedom. There are several methods that can be used to determine the independent
coordinates using the Jacobian matrix of the kinematic constraints. For example, the
Gaussian procedure with full or partial pivoting can be used to identify the independent and dependent coordinates by performing on the rectangular constraint
Jacobian matrix a sequence of elementary operations that defines a nonsingular
square submatrix. The columns of this nonsingular matrix are associated with the
dependent coordinates, while the remaining columns are associated with the independent coordinates. For example, if a system has n coordinates and nc constraint
equations, the constraint Jacobian matrix Cq is rectangular and has the dimension
nc × n. Each column in this matrix is associated with an element of the vector of
generalized coordinates q. Performing the Gaussian procedure, which consists of
elementary row and column operations on the rectangular Jacobian matrix Cq,
changes the order of the elements in the vector q. The Gaussian procedure eventually
leads to an identity nonsingular submatrix that has dimension nc × nc associated with
a reordered set of coordinates, the dependent coordinates qd; while the remaining
n − nc columns are associated with another set of reordered coordinates, the independent coordinates qi. For example, if a system has four constraint equations
(nc = 4) and six coordinates (n = 6), the number of independent coordinates is 2.
The coordinates are initially arranged in the following order [1 2 3 4 5 6]. After the
Gaussian procedure is performed on the constraint Jacobian matrix, the first four
columns of the resulting matrix will form an nc × nc = 4 × 4 identity matrix. Assume
that the Gauss procedure results in the following reordering of the columns and
coordinates [1 4 5 6 2 3]. Then, the system dependent coordinates are the coordinates
1, 4, 5, and 6, while the system independent coordinates are the coordinates 2 and 3.
6.2 NUMERICAL ALGORITHMS —
CONSTRAINT FORMULATIONS
Several methods for the wheel/rail contact problem were presented in the preceding
chapter. Some of these methods are based on the constraint contact formulation,
while the others are based on the elastic contact formulation. In particular, the
following two constraint methods were discussed:
•
•
Augmented constraint contact formulation (ACCF)
Embedded constraint contact formulation (ECCF)
These methods require the use of different solution procedures and numerical algorithms to determine the location of the contact points and normal forces that act at
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195
these points. In this section, the implementation of these two methods in a general
multibody system computational algorithm is discussed. It is important, however,
to point out that, while these methods use different schemes to determine the location
of the contact points and normal forces, all these methods can employ the same
procedure to determine the tangential contact forces, as will be discussed in Section
6.4. Furthermore, the rail and wheel geometric description discussed in Chapter 3
is the same for all these methods.
6.2.1 AUGMENTED CONSTRAINT CONTACT FORMULATION (ACCF)
In this method, as explained in Chapter 5, five scalar contact constraint equations
are used to describe the wheel/rail interaction. These five constraint equations for
an arbitrary contact k are
C p  rPw − rPr 
  

C k (q w , q r , s w , s r ) =  C1  =  t1w ⋅ n r  = 0
 C   t w ⋅ nr 

 2  2
(6.17)
where superscripts r and w denote rail and wheel, respectively; qw and qr are,
respectively, the generalized coordinates of the wheel and rail; sw and sr are the
vectors of surface parameters; and t1, t2, and n are the two tangents and normal to
the surface at the contact point, respectively. In addition to the contact constraint
equations, railroad vehicle models may include mechanical joints and specified
motion trajectories. In this case, the total vector of the system constraint equations
can be written as
(
)
C q, s, t = 0
(6.18)
where q is the vector of the system generalized coordinates, s is the vector of the
system surface parameters, and t is time. As discussed in Chapter 5, after differentiating the preceding equation twice with respect to time, the constraint equations
at the acceleration level can be written as
+ C ss = Q d
Cq q
(6.19)
where Cq and Cs are the sub-Jacobians of the constraint equations associated,
respectively, with the generalized coordinates q and the surface parameters s; and
Qd is a vector that absorbs quadratic terms in the first derivatives of the generalized
coordinates and the surface parameters
The augmented form of the equations of motion (Equation 6.1) can then be
modified to account for the contact conditions as follows:
M

0
C
 q
0
0
Cs
CqT   q
  Q 

 
T  
C s   s  =  0 
0   λ  Qd 

(6.20)
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Railroad Vehicle Dynamics: A Computational Approach
and s as well as the
These equations can be solved to determine the accelerations q
vector of Lagrange multipliers λ. In the constraint contact formulations, the constraint equations including the contact constraints must be satisfied at the position,
velocity, and acceleration levels.
When the constraint contact formulation is used, it is necessary to determine the
sub-Jacobians Cq and Cs as well as the quadratic velocity terms that result from
differentiating the constraint equations twice with respect to time. For example, the
first vector constraint equation (three scalar equations) in Equation 6.17 can be written
more explicitly as follows:
C p (q, s) = R w + Aw u w − R r − Ar u r
(6.21)
where Rw and Rr are the global position vectors of the origins of the body coordinate
systems of the wheel and the rail, respectively; Aw and Ar are the transformation
matrices that define the orientation of the wheel and the rail body coordinate systems,
respectively; and u w and u r are the local position vectors of the contact point with
respect to the wheel and the rail coordinate systems, respectively. Recall that Rw,
Rr, Aw, and Ar in the preceding equation are functions of the body generalized
coordinates q only, while u w and u r are functions of the wheel and the rail surface
parameters. The sub-Jacobian matrices associated with the preceding constraint
equations can be written as
(C )
=  I
( )

∂u w
=  Aw w
∂s

p q
Cp
s
− Aw u wG w
− Ar
Ar u r Gr  





−I
∂u r 

∂s r 
(6.22)
where G is the matrix that relates the angular velocity vector defined in the body
coordinate system to the time derivatives of the orientation coordinates, as discussed
in Chapter 2. Note that in the preceding equation
∂u i  ∂u i
=
∂s i  ∂s1i
∂u i   i
 = t1
∂s2i  
t2i  ,
i = w, r
(6.23)
where t1i and t2i are the vectors tangent to the surface at the contact point, as shown
in Chapter 3. A similar procedure can be applied to the last two constraint equations
of Equation 6.17. One can verify that the sub-Jacobian matrices associated with
these two constraint equations are given by
(C )
= 0

(C )
∂ tl w
= [n A A
∂s w
l q
l s
rT
− n r Ar A w tl wG w
T
rT
w
T
− tl w A w Ar n r Gr  


,
r
w T w T r ∂n

tl A A
]

∂s r
0
T
T
l = 1, 2 (6.24)
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197
Similarly, the quadratic velocity vector Qd that results from the differentiation
of the constraint equations twice with respect to time can be evaluated. A simple
procedure that can be used to determine Qd is to differentiate the constraint equations
twice with respect to time. The differentiation leads to two types of terms: the first
type includes second derivatives of the generalized coordinates and surface parameters, while the second type includes terms that are quadratic in the velocities. These
quadratic velocity terms can be grouped together and moved to the right-hand side of
the equation to define the vector Qd. This process can be demonstrated using the
constraint vector of Equation 6.21. To this end, one can write the second time
derivative of this constraint equation in the following form:
 w
(q, s) = R
w − A w u wG wθ
w + A w ∂u s w + Qw
C
p
p
 ∂s w 


( )
 r
r − Ar ∂u sr − Qr
+ Ar u r Gr θ
p
 ∂s r 


( )
d
r
−R
(6.25)
d
=0
In this equation,


(Q ) = (G θ ) × u + ω × (ω × u ) + 2ω ×  A ∂∂us s 
i
p
i i
i
i
i
i
i
i
i
i
i
d
2
2
∂2u i
∂2u i

 ∂2u i
+ A  i 2 s1i + 2 i i s1i s2i + i 2 s2i  ,
∂
s
s
s
∂
∂
s
∂

1 2
2
 1
( )
i
( )
(6.26)
i = w, r
The quadratic velocity vector (Qp)d can then be defined as follows:
(Q )
p d
( ) + (Q )
= − Qwp
d
r
p
(6.27)
d
and the other vectors that appear in Equation 6.26 are defined as follows:
ω i = Giθ i ,
u i = Ai u i ,
i = w, r
(6.28)
where ωi (i = w, r) is the angular velocity vector defined in the global coordinate
system. Following a similar procedure, one can show that the quadratic velocity
vectors that result from the differentiation of the last two constraints of Equation 6.17
twice with respect to time are given by
(Q )
l d
(
)
T
T
T
= − n r blw + 2tlw n r + t lw br ,
l = 1, 2
(6.29)
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Railroad Vehicle Dynamics: A Computational Approach
In this equation,
 ∂t w  
tlw = ω w × t lw + A w  lw  s w 
 ∂s  

r 


∂
n
n r = ω r × n r + Ar  r  s r 
 ∂s 

(6.30)
where l = 1, 2, and the vectors bw and br are given, respectively, by
w
wθ w × t w + ω w × ω w × t w + 2ω w ×  A w ∂ tl s w 
blw = G
l
l


w
∂s


(
)
(
 ∂2 t w
+ A w  wl 2 s1w
 ∂s1
( )
2
)
2

∂2 t w
∂2 t w
+ 2 w l w s1w s2w + wl 2 s2w 
∂s1 ∂s2
∂s2

( )
(6.31)
and
r
r θ r × n r + ω r × ω r × n r + 2ω r ×  Ar ∂n s r 
br = G


r
∂s


(
)
(
 ∂2n r
+ Ar  r 2 s1r
 ∂s1
( )
2
)
( )
∂2n r
∂2n r
+ 2 r r s1r s2r + r 2 s2r
∂s1 ∂s2
∂s2
2



(6.32)
It is clear from the formulation of the quadratic velocity vector that the constraint
contact formulation requires the evaluation of the third derivatives of vectors with
respect to the surface parameters. This is clear, since, for example
∂2 tmw
∂slw
2
=
∂2  ∂u w 
,
2 
w 
∂slw  ∂sm 
m = 1, 2; l = 1, 2
(6.33)
and
∂2n r
∂slr
2
=
(
∂2 t1r × t r2
∂slr
2
),
l = 1, 2
(6.34)
where
tlr =
∂u r
,
∂slr
l = 1, 2
(6.35)
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199
This section focuses on the contact constraints, but railroad vehicle models may
include other joints that are standard elements of the constraint library of generalpurpose multibody computer codes. The formulation of the constraint equations
associated with these joints, as well as their Jacobian matrices and quadratic velocity
vectors, can be found in the literature (Shabana, 2001).
The constraint Jacobian matrix and the quadratic velocity vector that result from
the differentiation of the constraint equations twice with respect to time can be used
with the system mass matrix and the vector of applied forces to construct Equation
6.20. This equation can be solved for the accelerations and the vector of Lagrange
multipliers. Lagrange multipliers can be used to predict the constraint forces, including the normal contact constraint forces. As discussed in Chapter 5, it is clear that
the solution of Equation 6.20 must satisfy the following equation:
CTs λ = 0
(6.36)
For an arbitrary contact, the preceding equation has four scalar equations, since there
are four surface parameters associated with each contact, while the vector of
Lagrange multipliers associated with this contact has five elements. The preceding
equation, therefore, can be used to define one independent Lagrange multiplier; other
Lagrange multipliers can be expressed in terms of this independent multiplier. That
is, there is only one independent constraint force associated with the degree of
freedom eliminated by the five contact constraint equations. This independent constraint force can be used to define the normal contact force. In the numerical
algorithm described in this section, the normal contact force obtained by solving
Equation 6.20 is stored and used in the next time step to calculate the tangential
creep forces and spin moment.
Another important issue that must be considered when developing a computational algorithm to solve the wheel/rail contact problem is the accurate determination
of the initial values of the surface parameters and their time derivatives. In the
augmented constraint contact formulation, surface parameters can be selected with
other independent coordinates as the optimum set of the system degrees of freedom.
In this case, it is important to provide accurate values of these parameters and their
time derivatives, since other dependent coordinates are expressed in terms of the
degrees of freedom. While the initial values of the surface parameters (contact
location) can be easily determined using a CAD program, determining the initial
values of the first derivatives of the surface parameters is not as simple. In order to
determine the derivatives of the surface parameters at the initial configuration, one
can use the available known initial values of the time derivatives of the coordinates
(velocities). Using the contact constraint equations and the procedure described in
Chapter 5, the initial values of the surface parameters can be determined using the
following equation:
s 0 = Bq 0
(6.37)
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Railroad Vehicle Dynamics: A Computational Approach
where B = −(C ds )−1Cqd is the velocity transformation matrix, which can be obtained
using the contact constraints.
Based on the discussion presented in this section, a computational multibody
system algorithm can be developed based on the augmented constraint contact
formulation (ACCF). This algorithm is summarized as follows:
1. At the initial configuration, t = t0, accurate values of the surface parameters
s, the coordinates q, and the velocities q are provided.
2. Using the initial values, the matrix B of Equation 6.37 is evaluated and
used to determine the initial values of the time derivatives of the surface
parameters s .
3. If the generalized coordinate partitioning method is used, the constraint
Jacobian matrix [Cq Cs] is evaluated. A set of independent coordinates
or degrees of freedom is identified based on the numerical structure of
the constraint Jacobian matrix. Note that surface parameters may be
selected as degrees of freedom. If the penalty method or the constraint
stabilization method is used, this step is skipped.
4. Given the independent coordinates, the nonlinear constraint equations are
solved iteratively using a Newton-Raphson algorithm to determine the
dependent coordinates and surface parameters. This step guarantees that
the constraint equations are satisfied at the position level. If the penalty
method or the constraint stabilization method is used, this step is skipped.
5. Given the independent velocities, which may include time derivatives of
the surface parameters, the constraint equations at the velocity level
C q q + C s s = −C t is solved for the dependent velocities using the generalized coordinate partitioning method, as previously described in this
chapter. This step guarantees that the constraint equations are satisfied at
the velocity level. If the penalty method or the constraint stabilization
method is used, this step is skipped.
6. Using the system coordinates and velocities that satisfy the constraint
equations at the position and velocity levels, the mass matrix M, the vector
of applied forces Q, the constraint Jacobian matrix, and the quadratic
velocity vector that arise from differentiating the constraint equations
twice with respect to time are evaluated and used to construct the augmented form of the equations of motion (Equation 6.20). The vector Q
includes the tangential creep forces, which are calculated using the procedure described in Section 6.4.
7. Equation 6.20 is solved for the generalized coordinate and surface parameter accelerations as well as the vector of Lagrange multipliers. Lagrange
multipliers associated with the contact constraints are used to determine
the normal contact force, which is stored to be used in the next time step
to determine the tangential creep forces and moments.
8. The independent accelerations, which may include second derivatives of
the surface parameters, are identified and used to define the state equations
associated with the system degrees of freedom. These independent state
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201
equations are integrated forward in time to determine the independent
coordinates and velocities at time t = t + ∆t.
9. If the end of the simulation time is reached, stop; otherwise, go to Step
4 and continue until the end of the simulation is reached.
It is clear from this algorithm that the use of the generalized coordinate partitioning method requires adopting a more sophisticated procedure as compared with
the use of the penalty method or the constraint stabilization method. In the generalized coordinate partitioning method, the independent variables must be identified,
and only the state equations associated with these independent variables are integrated forward in time to determine the independent coordinates and velocities. One
of the main features that distinguish the augmented constraint contact formulation
from the embedded constraint contact formulation is the fact that, in the augmented
constraint contact formulation, the system degrees of freedom or the independent
coordinates can include surface parameters, since these degrees of freedom are
identified based on the numerical properties of the constraint Jacobian matrix. In
fact, for the three-degree-of-freedom unsuspended wheelset with a prescribed forward velocity, the optimum set of degrees of freedom is found to include only surface
parameters, and no other generalized coordinates are selected by the algorithm
outlined above among the three degrees of freedom.
It is also clear from this algorithm that an approximate value of the normal
contact force is used to determine the tangential creep forces and spin moment at a
given time step. The normal force is stored from the previous time step to avoid the
use of an iterative procedure performed on Equation 6.20 to determine the accelerations, Lagrange multipliers, and the creep forces and moments. The creep forces
and moments are a function of the normal force, and all these forces enter into the
formulation of Equation 6.20. Since the relationship between these forces is nonlinear, an iterative Newton-Raphson algorithm can be used to determine these forces
and, at the same time, solve for the accelerations. However, numerical experimentation with several railroad vehicle models showed that storing the values of the
normal forces from the previous time step and using these values to avoid applying
the iterative Newton-Raphson procedure on Equation 6.20 lead to accurate results.
Comparison was also made with the results obtained using the elastic contact formulations, which do not require the use of an iterative procedure to determine the
normal and creep forces (Shabana et al., 2005). This comparison showed very good
agreement between the results of the algorithm outlined above and the results
obtained using the elastic contact formulations.
6.2.2 EMBEDDED CONSTRAINT CONTACT FORMULATION (ECCF)
This method, as discussed in the preceding chapter, is conceptually the same as the
augmented constraint contact formulation. The contact constraints are imposed at
the position, velocity, and acceleration levels. The basic difference between the two
methods is that, in the embedded constraint contract formulation, the surface parameters are systematically eliminated. This difference, which necessitates the use of a
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Railroad Vehicle Dynamics: A Computational Approach
different numerical solution procedure, forces the numerical algorithm to select the
system degrees of freedom from the elements of the generalized coordinate vector
q. In this case, as discussed in the preceding chapter, one constraint between the
wheel and the rail is added to the system differential equations of motion in order
to determine the normal force. Recall from the analysis presented in the preceding
chapter that this equation is given for a contact k by
C nk (q w , q r , s wk , s rk ) = n rk ⋅ (rPwk − rPrk ) = 0
(6.38)
where all the variables that appear in this equation are as defined in Chapter 5. This
equation is added to the system equations of motion, and the Lagrange multiplier
associated with this constraint is used to determine the normal contact force. The
other four contact constraint equations that are used to eliminate the surface parameters (as presented in Chapter 5) are
k
 t1r ⋅ (rPw − rPr ) 
 r

t ⋅ (rw − rr )
C dk (q w , q r , s wk , s rk ) =  2 w P r P  = 0
 t ⋅n

1


w
r
 t 2 ⋅ n

(6.39)
Using the preceding two equations, it was shown in the preceding chapter that
the augmented form of the system equations of motion can be written as follows:
M

J
  Q 
JT   q
  = 

0   λ  Qdn 
(6.40)
where J, an implicit function of the vector of the system surface parameters s, is
the Jacobian matrix of the constraint equations, including the constraint of Equation
6.38 and other constraints imposed on the motion of the multibody system; λ is the
vector of the system Lagrange multipliers that include only Lagrange multipliers
associated with the constraints of Equation 6.38 and other noncontact constraints
imposed on the motion of the system; and Qdn is a quadratic velocity vector that
results from the differentiation of the constraint equations twice with respect to time.
This vector is a function of the surface parameters s and their first time derivatives.
Recall that for a virtual change in the system variables, Equation 6.38 yields
δ C nk = Cqnkδ q + Csnkδ s k = 0
(6.41)
The Jacobian matrices that appear in this equation can be written as follows:
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T
Cqnk =  n r

 T
C =  n r t1w

nk
s
nr
T
( − A u G )
w
rT w
2
n t
w
P
w
wr T
P
r
−nr
 ∂n r 
 ∂s r 
 1
203
{n ( A u G ) + r (− A n G )} 
T
rT
wr T
P
r
r
r
P
 ∂n r  
 ∂s r  
 2  
r
wr T
P
k
r
r
r




(6.42)
k
where rPwr = rPw − rPr and all the other symbols that appear in this equation are again
as defined in Chapter 5. Equation 6.39 can be used to write the virtual changes of
the surface parameters in terms of the virtual changes in the generalized coordinates.
Substituting the results into Equation 6.41, the virtual changes in the surface parameters can be eliminated. Similarly, by differentiating Equation 6.39 once and twice
with respect to time, one can write the first and second time derivatives of the surface
parameters in terms of the first and second time derivatives of the generalized
coordinates. As demonstrated in Chapter 5, these two relationships can be written,
respectively, as follows:
s = Bq ,
s = Bq
+ Bq
(6.43)
In particular, one can show that the second relationship in this equation can be
obtained from the second derivative of the constraints of Equation 6.39, which can
be written as follows:
k
dk
C
 t1r ⋅ rPwr + t1r ⋅ rPwr + 2 t1r ⋅ rPwr 
r wr r wr

r + 2 t2r ⋅ rPwr 
t ⋅ r + t ⋅ =  2w P r 2w P r
=0
 t ⋅ n + t ⋅ n
w r 
1
1 + 2 t 1 ⋅ n
 w r w r

+ 2 t2w ⋅ n r 
 t 2 ⋅ n + t 2 ⋅ n
(6.44)
The preceding equation can also be used to determine the quadratic velocity vector,
as described in the case of the augmented constraint contact formulation. Given the
generalized coordinates q, Equation 6.39 can be solved using an iterative NewtonRaphson procedure to determine the surface parameters. Given the vector of generalized velocities q , the constraints of Equation 6.39 at the velocity level can be
solved to determine the time derivatives of the surface parameters. This is equivalent
to using the linear system of algebraic equations in the velocities given by the first
equation in Equation 6.43. The second equation in Equation 6.43 is also a linear
system of algebraic equations in the accelerations. This equation can be used to
eliminate the second time derivatives of the surface parameters from the second time
derivative of the constraint of Equation 6.38, which is given by
r + 2 n r ⋅ rPwr = 0
rPwr + rPwr ⋅ n
C nk = n r ⋅ (6.45)
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Railroad Vehicle Dynamics: A Computational Approach
As mentioned in Chapter 5, the use of this procedure implies that the geometric
surface parameters are always treated as dependent variables and are systematically
eliminated from the equations of motion. For this reason, only independent generalized accelerations are integrated forward in time. It is, perhaps, important at this
point to reiterate that only one Lagrange multiplier associated with the contact
constraint of Equation 6.38 is used in Equation 6.40. This Lagrange multiplier defines
the normal contact force used to define the longitudinal, lateral, and spin creep
forces. One can then outline the following numerical solution procedure based on
the embedded constraint contact formulation:
1. At the initial configuration, t = t0, accurate values of the surface parameters
s, the coordinates q, and the velocities q are provided. Note that the
surface parameters in the embedded constraint contact formulation cannot
be selected as degrees of freedom.
2. Using the initial values of the coordinates, the matrix B of Equation 6.37
is evaluated and used to determine the initial values of the time derivatives
of the surface parameters s .
3. Given the values of the generalized coordinates q, Equation 6.39 can be
solved using an iterative Newton-Raphson algorithm to determine the
surface parameters. The surface parameters can be substituted into Equation 6.38, thereby writing this equation solely in terms of the generalized
coordinates q.
4. If the generalized coordinate partitioning method is used, and because the
surface parameters are eliminated from the constraint equations, the constraint Jacobian matrix J is evaluated. A set of independent coordinates
or degrees of freedom is identified based on the numerical structure of
the constraint Jacobian matrix. Note that in the embedded constraint
contact formulation, the surface parameters cannot be selected as degrees
of freedom. If the penalty method or the constraint stabilization method
is used, this step is skipped.
5. Given the independent coordinates qi, the nonlinear constraint equations
are solved iteratively using a Newton-Raphson algorithm to determine the
dependent coordinates qd. Using qi and qd, Equation 6.39 is solved iteratively for the surface parameters. This two-stage iterative procedure continues until the obtained generalized coordinates and surface parameters
satisfy the constraint equations. This step guarantees that the constraint
equations are satisfied at the position level. If the penalty method or the
constraint stabilization method is used, this step is skipped.
6. Given the independent velocities q i that do not include time derivatives
of the surface parameters, the constraint equation at the velocity level is
solved for the dependent generalized velocities q d using the generalized
coordinate partitioning method, as previously described in this chapter.
7. Using the vector q = [q Ti q Td ]T , the first equation in Equation 6.43 can be
solved for the time derivatives of the surface parameters This step and the
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205
previous step guarantee that the constraint equations are satisfied at the
velocity level. If the penalty method or the constraint stabilization method
is used, these two steps are skipped.
8. Using the second equation in Equation 6.43, the second derivatives of the
surface parameters can be substituted into Equation 6.45, thereby writing
.
this equation solely in terms of the vector of generalized accelerations q
9. Using the system coordinates and velocities that satisfy the constraint
equations at the position and velocity levels, the mass matrix M, the vector
of applied forces Q, the constraint Jacobian matrix, and the quadratic
velocity vector that arise from differentiating the constraint equations
twice with respect to time are evaluated and used to construct the augmented form of the equations of motion (Equation 6.40).
as well as the
10. Equation 6.40 is solved for the generalized accelerations q
vector of Lagrange multipliers. Lagrange multipliers associated with the
contact constraints are used to determine the normal contact force, which
is stored to be used in the next time step to determine the tangential creep
forces and moments.
11. The independent accelerations that do not include second derivatives of
the surface parameters are identified and used to define the state equations
associated with the system degrees of freedom. These independent state
equations are integrated forward in time to determine the independent
coordinates and velocities at time t = t + ∆t.
If the end of the simulation time is reached, stop; otherwise, go to Step 3 and
continue until the end of the simulation is reached.
Clearly, the main difference between this algorithm and the algorithm based on
the augmented constraint contact formulation is the systematic elimination of the
surface parameters. By doing so, the surface parameters are always treated as
dependent variables.
6.3 NUMERICAL ALGORITHMS — ELASTIC FORMULATIONS
In the preceding chapter, two elastic contact formulations were discussed. In these
elastic formulations, the contact conditions are not imposed as constraints and,
therefore, no degrees of freedom are eliminated as the result of the wheel/rail
dynamic interaction. The elastic contact formulations that allow for wheel/rail separation and penetration require the use of simpler numerical algorithms as compared
with the constraint contact formulations. The following two methods were discussed
in detail in Chapter 5:
•
•
Elastic contact formulation using algebraic equations (ECF-A)
Elastic contact formulation using nodal search (ECF-N)
This section discusses the computer implementation of these two methods.
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Railroad Vehicle Dynamics: A Computational Approach
6.3.1 ELASTIC CONTACT FORMULATION USING ALGEBRAIC
EQUATIONS (ECF-A)
This method, as discussed in the preceding chapter, is not considered as a constraint
method, since no Lagrange multipliers are associated with the contact conditions
and no degrees of freedom are eliminated as the result of the wheel/rail interaction.
In this case, the system equations of motion take a form similar to Equation 6.1. As
described in Chapter 5, the location of the contact point is determined by solving
the nonlinear algebraic equations of Equation 6.39 for the four surface parameters
that define the geometry of the wheel and rail surfaces. To this end, the formulation
of the following equation is required for each contact when the iterative NewtonRaphson algorithm is used:
 r w
 t1 ⋅ t1


 t r2 ⋅ t1w

 w
 ∂t1 ⋅ n r
 ∂s1w
 w
 ∂t 2 ⋅ n r
 ∂s w
 1
t1r ⋅ t 2w
∂t1r wr r r
⋅ r − t1 ⋅ t1
∂s1r
t r2 ⋅ t 2w
∂t r2 wr r r
⋅ r – t 2 ⋅ t1
∂s1r
∂t1w r
⋅n
∂s2w
∂n r w
⋅ t1
∂s1r
∂t 2w r
⋅n
∂s2w
∂n r w
⋅ t2
∂s1r
∂t1r wr r r 
⋅ r – t1 ⋅ t 2 
∂s2r

 t1r ⋅ r wr 
∂t r2 wr r r  s1w 
⋅ r − t2 ⋅ t2   w 
 r wr 
r
∂s2
  s2 
t2 ⋅ r 
−
=
  s r 
r
 t w ⋅ nr 
∂n w
1

 1w r 


⋅
t
1
  s2r 
∂s2r
 t 2 ⋅ n 

r
∂n w

⋅ t2

∂s2r

(6.46)
The coefficient matrix in this equation is the Jacobian matrix of the algebraic
equations of Equation 6.39, while the vector on the right-hand side represents the
values of the algebraic equations for given values of the surface parameters. Note that
in this section, the phrase algebraic equations is adopted instead of the phrase
constraint equations to make clear that this method is not a constraint method that
requires certain constraint functions be imposed at the velocity and acceleration
levels and, therefore, that no degrees of freedom are eliminated and no Lagrange
multipliers are introduced. The preceding equation is iteratively solved until convergence is achieved. Convergence is achieved when the norm of the violation of
the algebraic equations or the norm of the Newton differences is less than a specified
tolerance. The speed of convergence depends on the initial guess used for the surface
parameters.
Having determined the vector of the surface parameters that define the location
of the contact points on the wheel and the rail, the value of the penetration can be
calculated using the following equation:
δ = r wr ⋅ n r
(6.47)
The normal contact forces can be calculated using Hertz's contact theory, while the
creep forces can be calculated using one of the creep-force models discussed in
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207
Chapter 4. As presented in Chapter 5, the following expression of the normal force
can be used (Shabana et al., 2004):
F = Fh + Fd = − K hδ 3 2 − Cδ δ
(6.48)
Using the elastic contact formulation based on the algebraic equations (ECF-A),
the following numerical procedure, which can be implemented in a general multibody
system computational algorithm, can be summarized:
1. At the initial time t = t0, the initial values of the generalized coordinate
and velocity vectors q and q , respectively, are provided. An initial guess
of the surface parameters must also be provided.
2. The independent generalized coordinates are identified, and the constraint
equations and the generalized coordinate partitioning method are used to
adjust the dependent generalized coordinates. This ensures that the constraint equations, which do not include any contact constraints, are satisfied
at the position level. If the penalty or the constraint stabilization method
is used, this step is skipped.
3. The vector of generalized coordinates is used to solve Equation 6.39 for
the surface parameters using a Newton-Raphson algorithm that requires
formulating the matrix equation given in Equation 6.46. This second
Newton-Raphson solution stage, in the case of the generalized coordinate
partitioning method, is independent of the first Newton-Raphson stage
discussed in Step 2 of the algorithm. This is an important difference
between the position analysis performed in this method and the position
analysis performed in the embedded constraint contact formulation discussed in the preceding section. Furthermore, the surface parameters must
be determined regardless of whether or not the generalized coordinate
partitioning method is used.
4. The independent generalized velocities are used to determine the dependent generalized velocities by solving the system of linear equations
C q q = −C t using the generalized coordinate partitioning method. This
step guarantees that the constraint equations are satisfied at the velocity
level. Note that in this step, the time derivatives of the surface parameters
are not required. If a penalty method or the constraint stabilization method
is used, this step is skipped.
5. Equation 6.47 and the value of the surface parameters determined in Step 3
are used to determine the value of the penetration for each contact. If the
value of the penetration is positive, determine the normal contact force using
Equation 6.48; otherwise, skip the calculations of the normal force.
6. In the case of positive penetration, the normal force, the material properties, and the geometry of the wheel and the rail surfaces can be used to
determine the creep forces and moments that can enter into the formulation
of the dynamic equations as external forces.
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Railroad Vehicle Dynamics: A Computational Approach
7. Since all the coordinates and velocities are known, the mass matrix M, the
vector of applied forces Q, and the quadratic velocity vector Qd of Equation 6.1 can be evaluated. Equation 6.1 can then be constructed and solved
and the vector of Lagrange multipliers λ .
for the acceleration vector q
8. Lagrange multipliers can be used to determine the generalized constraint
forces, while, in the case of the generalized coordinate partitioning
method, the independent accelerations can be identified and integrated
forward in time to determine the independent coordinates and velocities. In
the case where a penalty method or the constraint stabilization method is
used, all the accelerations are integrated to determine all the generalized
coordinates and velocities.
9. If the end of the simulation time is reached, stop; otherwise, go to Step 2
and continue until the end of the simulation time is reached.
While a set of algebraic equations must be solved in the ECF-A method, the
computational algorithm outlined above makes clear the basic differences between
this method, which does not eliminate degrees of freedom due to the contact, and
the ECCF method discussed in the preceding section. It is also clear that, in the
elastic contact formulation, there is no need to store the normal contact force from
the previous time step for the creep-force calculations, as is the case in the constraint
contact formulation. Obviously, one can always use a hybrid method in which the
surface parameters or, equivalently, the locations of the contact points are determined
using a constraint method, while the normal force is determined using an elastic
approach. The use of this hybrid method, however, can be questioned, since in the
case of wheel/rail separation, the motions of the wheel and the rail should be totally
independent.
6.3.2 ELASTIC CONTACT FORMULATION USING NODAL
SEARCH (ECF-N)
This formulation uses a method similar to the one used in the ECF-A method for
calculating the normal force. However, the two methods differ in the way the location
of the contact point is determined. In the ECF-N method, the contact location is
determined based on a search using the relative distance between nodal points that
represent the wheel and the rail profiles instead of solving a set of algebraic equations, as in ECF-A. Therefore, the numerical procedure used for the ECF-N method
is similar to the numerical procedure used for the ECF-A method. The only difference
is replacing the solution of the algebraic equations with an optimized search procedure to determine the two nodal points on the wheel and the rail surfaces that have
the maximum indentation. As mentioned in Chapter 5, the nodal search calculation
is performed in a two-dimensional plane for simplicity. Furthermore, since the
longitudinal distance traveled by the wheel at the contact point needs to be determined, an additional differential equation can be introduced for each contact to
determine the distance traveled on the rail, and this information is then used in
constructing the projection plane used for the nodal search. This first-order differential equation is given by
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s1r = rQwT ⋅ t1r
209
(6.49)
where s1r is the rail longitudinal surface parameter, rQw is the absolute velocity of a
point on the wheel, and t1r is the tangent to the rail in the longitudinal direction. The
following numerical algorithm based on the ECF-N can be proposed:
1. At the initial time t = t0, the initial values of the generalized coordinate
and velocity vectors q and q , respectively, are provided. An initial guess
of the surface parameters must also be provided.
2. The independent generalized coordinates are identified, and the constraint
equations and the generalized coordinate partitioning method are used to
adjust the dependent generalized coordinates. This ensures that the constraint equations, which do not include any contact constraints, are satisfied at the position level. If a penalty method or the constraint stabilization
method is used, this step is skipped.
3. The vector of generalized coordinates and the distance traveled by the wheel
s1r are used to search for the location of the contact points. The result of
this search can be used to determine the rail and wheel profile parameters
s2r and s1w at the points of contact.
4. The independent generalized velocities are used to determine the dependent generalized velocities by solving the system of linear equations
C q q = −C t using the generalized coordinate partitioning method. This
step guarantees that the constraint equations are satisfied at the velocity
level. Note that, in this step, the time derivatives of the surface parameters
are not required. If the penalty method or the constraint stabilization
method is used, this step is skipped.
5. Equation 6.47 and the value of the surface parameters determined in Step 3
are used to determine the value of the penetration for each contact. If the
value of the penetration is positive, the normal contact force is calculated
using Equation 6.48; otherwise, the calculations of the normal force are
skipped.
6. In the case of positive penetration, the normal force, the material properties, and the geometry of the wheel and the rail surfaces can be used to
determine the creep forces and moments that can enter into the formulation
of the dynamic equations as external forces.
7. Since all the coordinates and velocities are known, the mass matrix M, the
vector of applied forces Q, and the quadratic velocity vector Qd of Equation 6.1 can be evaluated. Equation 6.1 can then be constructed and solved
and the vector of Lagrange multipliers λ.
for the acceleration vector q
8. Lagrange multipliers can be used to determine the generalized constraint
forces, while, in the case of the generalized coordinate partitioning method,
the independent accelerations can be identified and integrated forward in
time to determine the independent coordinates and velocities. In cases
where the penalty method or the constraint stabilization method is used,
all the accelerations are integrated to determine all the generalized coordinates and velocities.
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Railroad Vehicle Dynamics: A Computational Approach
9. With the accelerations, the first-order differential equations defined by
Equation 6.49 are integrated to determine the distance traveled by the
wheels. The number of these first-order differential equations is equal to
the number of wheel/rail contacts. The numerical solution of these equations defines the parameter s1r used in Step 3.
10. If the end of the simulation time is reached, stop; otherwise, go to Step 2
and continue until the end of the simulation time is reached.
Numerical experimentation showed that Equation 6.49 provides an accurate
estimate for the parameter s1r . While this equation represents an efficient method to
solve for the parameter s1r , this is not the only method that can be adopted. Clearly,
if the generalized coordinates are known, one can write the relationship between
these coordinates and any other set of coordinates such as the trajectory coordinates
discussed in Chapters 2 and 7. This is a nonlinear relationship that can be used with
a Newton-Raphson algorithm to solve for the trajectory coordinates that include the
distance traveled by the wheel. This approach, however, requires the solution of at
least three nonlinear algebraic equations for each contact. Equation 6.49, therefore,
represents a more efficient and robust method for obtaining the parameter s1r .
6.4 CALCULATION OF THE CREEP FORCES
In all the computational algorithms discussed so far in this chapter, one method can
be used for the calculation of the creep forces. Despite the fact that different methods
can be used to determine the normal forces, the same methods of calculating the
principal curvatures, the normalized velocity creepages, and the dimension of the
contact ellipse that enter into the calculation of the creep forces can be employed
for all contact formulations and can be common in all computational algorithms
discussed in this chapter. The calculated creep forces can be introduced to the
dynamic equations of motion as generalized applied forces using the vector Q of
Equations 6.1, 6.20, or 6.40, depending on the contact formulation used to describe
the wheel/rail interaction. The main steps for calculation of the creep forces can be
summarized as follows:
1. Knowing the location of the point of contact, the velocity of the contact
point on the wheel and the rail can be determined (as discussed in
Chapter 2) using the following equation:
i + ω i × ui ,
r i = R
i = w, r
(6.50)
where Ri is the global position vector of the origin of the body coordinate
system, ωi is the absolute angular velocity vector of the body defined in
the global coordinate system, and ui is the position of the contact point
with respect to the origin of the body coordinate system defined in the
global system.
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211
2. Knowing the location of the contact point, the vectors tangent and normal
to the surfaces at the contact points can be determined. The longitudinal
and lateral tangent vectors to the rail, t1r and t r2 , respectively, and the
normal vector nr at the contact point can be determined using the procedure described in Chapter 3.
3. Using the absolute velocities of the wheel and the rail and the tangent
and normal vectors, the normalized velocity creepages can be calculated
as described in Chapter 4 using the following equations:
ζx =
(r w − r r ) ⋅ t1r
,
V
ζy =
(r w − r r ) ⋅ t r2
,
V
ϕ=
(ω w − ω r ) ⋅ n r
V
(6.51)
where V is the forward velocity of the wheel.
4. Using the methods of differential geometry discussed in Chapter 3, the
principal radii of curvature of the wheel and rail surfaces at the contact
point can be determined.
5. Using the normal force, the wheel and the rail material properties, and
the principal curvatures of the wheel and rail surfaces at the contact point,
Hertz contact theory can be used to determine the dimension of the semiaxes of the contact ellipse.
6. The information obtained in the previous steps enters into the formulation
of the creep forces. The creep forces can be determined using any of the
creep theories discussed in Chapter 4.
As discussed in Chapter 4, there are several creep theories in the literature, some
of which are linear while the others are nonlinear, and some are two dimensional
while the others are three dimensional.
6.5 HIGHER DERIVATIVES AND
SMOOTHNESS TECHNIQUE
The solution of the contact problem using the methods discussed in this chapter and
the preceding chapter requires, particularly in the case of the constraint formulations,
evaluating higher derivatives of position vectors of points on the wheel and rail
surfaces with respect to their geometric surface parameters. For example, the calculations of the dimensions of the contact ellipse semi-axes, which are necessary
for all methods, require evaluating the principal curvatures and principal directions
at the contact points, as explained in Chapter 4. The evaluation of the principal
curvatures and principal directions requires the evaluation of second derivatives of
vectors with respect to the surface parameters. Furthermore, determining the location
of the contact points requires evaluating the tangent vectors and their derivatives
with respect to the surface parameters, as discussed in this book. Similarly, one can
show that when the constraint contact formulations are used, third derivatives of
position vectors with respect to the surface parameters need to be evaluated. It is
necessary to use an accurate procedure for evaluating these derivatives to correctly
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Railroad Vehicle Dynamics: A Computational Approach
determine the location of the contact points and forces. Note that in the computer
implementation of the formulation discussed in this book, the wheel and rail profiles
are described using spline functions defined using discrete nodal points. Accurate
evaluation of the higher derivatives of the profile functions with respect to the surface
parameters is required and, therefore, lower-order polynomials must be excluded.
Note that the use of a cubic spline leads to a linear second derivative and to a constant
third derivative within each interval. This approximation can be acceptable when a
sufficiently large number of nodes (interpolation points) is used. To obtain a better
description of the third derivatives, a spline of order higher than three can be used.
The use of even-order splines, however, is in general not recommended because of
several drawbacks ranging from the lack of symmetry in the boundary conditions
to poor convergence properties (Ahlberg et al., 1967). Consequently, the best choice
after a cubic spline is a fifth-order spline.
One can still use the cubic spline and develop a procedure that leads to a better
approximation of the second derivatives. For example, one can use a three-layer spline
by building second and third spline functions on top of the original cubic spline
(Shabana et al., 2001). As shown in Figure 6.1, a first cubic spline function S1(x) is
obtained using the nodes (xi,yi), (i = 1,…,n) and the derivatives y1′ and y n′ at the end
points. As a result, the spline S1(x) gives a good representation of the original function
and its derivatives. This good approximation is used to generate a new set of data
points, ( xi , yi′), (i = 1, …, n). The second cubic spline S2 is obtained using this new
set of data points plus the derivatives y1′′ and y n′′ at the end points. Following the same
procedure, it is possible to define a third spline S3 that yields a better approximation
for the third derivatives. This method clearly does not affect the order of convergence
of the higher derivatives, as this depends on the first spline S1(x). As is the case in
many practical problems, the reference function is not known in closed form, and the
main concern is the continuity of the higher derivatives. The three-layer spline proposed in this section defines smooth second and third derivatives starting with a few
interpolating points. The best choice for the number of the spline layers used to
represent the profiles depends on the number of data points available. If only a small
number of measurement points is available, then it might be necessary to use a twolayer or three-layer spline representation to obtain a better approximation for the
higher derivatives. The procedure described in this section can be used during the
dynamic simulations to evaluate the tangents and their derivatives. Cubic spline
subroutines are well documented and are available in the literature (Press et al., 1992).
The spline representation for the profiles of the wheel and the rail allows the
use of data obtained from direct measurements. However, these data points will
contain all the irregularities coming from the manufacturing process and the noise
from measurement and other sources. In the case of using the assumptions of
nonconformal contact, it is reasonable to assume that the irregularities and the
measurement noise do not have an effect on the contact force on a macro scale.
However, the presence of small irregularities can have an adverse effect on the
calculations of the higher derivatives of the curve, leading to a high level of numerical
noise that can be the source of convergence problems (Shabana et al., 2001). Therefore, it is necessary to use smoothing techniques with the data obtained from
measurements before using these data in a spline function representation. An efficient
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213
FIGURE 6.1 Scheme for the generation of two- and three-layer spline. (From Shabama A.A.,
Berzeri, M., and Sany, J.R., 2001, Numberical procedure for the simulation of wheel/rail
contact dynamics, ASME Journal of Dynamic Systems, Measurement, and Control, 123,
169–178. With permission.
way to obtain a smooth curve based on measured data is to use one of the smoothing
cubic spline functions presented in the literature (Shikin and Plis, 1995). These spline
functions are defined in terms of continuous cubic polynomials that have continuous
first and second derivatives in the domain of definition [a,b] = [x1,xn], and they
minimize the function
b
n
∫
j( f ) = [ f ′′( x )]2 dx +
∑ ρ1 [ f (x ) − y ]
2
i
i =1
a
i
(6.52)
i
where (xi,yi) are the coordinates of node i, and ρi ≥ 0 are given weight functions (i =
1, …, n). The resulting spline will pass through the points (xi,yi + δi), where δi decreases as ρi increases. If the weight coefficients are chosen to be all equal to
zero, then the smoothing spline becomes an interpolating spline. One possible
algorithm that utilizes the spline smoothing procedure presented by Shikin and Plis
(1995) can be summarized as follows:
1. Make change of coordinates by using a cumulative chordal distance sj
calculated from the original measured points Pi (xi,yi), (i = 1, …, n) as
j
sj =
∑P
P.
i −1 i
i=2
2. From the two sets (si,xi) and (si,yi), the curves x = x(s) and y = y(s) are
obtained using the smoothing code given by Shikin and Plis (1995).
3. After this process, the new xi and yi values are combined to form a new
curve y(x).
4. A third smoothing spline with smaller weighting coefficients is applied
as a final refinement. This spline function generates the data points that
will be used in the actual dynamic simulation.
This representation has two advantages. The first is that the curves obtained are
typically smoother than the curve y(x), and give a better description for the arches
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Railroad Vehicle Dynamics: A Computational Approach
of the original curve y(x), particularly when a curve has very high values of the first
derivative or, in the case a cubic polynomial, would not provide a good fit. The
second advantage is that, using two smoothing splines, it is possible to correct
measurement errors that can be present not only in the coordinates yi, but also in
the coordinate xi.
6.6 TRACK PREPROCESSOR
Chapter 3 presents a detailed description of the geometry and the main parameters
and segments used to define a track. This description can be used to numerically
construct the track geometry in a form that can be utilized by computational multibody system algorithms. Two steps are followed to efficiently utilize the track
geometry data. In the first step, the preprocessor stage, the track is numerically
described based on specific inputs that are used and provided by the railroad industry,
as discussed in Chapter 3. The track preprocessor, based on these inputs, defines the
track geometry using a number of nodal points that are specified by the user. The
preprocessor defines the locations of the nodes and three Euler angles that define
the orientation of a coordinate system attached to these nodes. The preprocessor
then generates a data file that is used as an input to the main processor used in the
dynamic analysis. The distance traveled by a wheel can be predicted using the contact
formulations discussed in this book. This defines the longitudinal arc length parameter that can be used, in turn, to define the segment of the rail within which the
contact point lies. Using the arc length and knowing the two nodes that define this
track segment, an interpolation scheme can be adopted to determine the geometric
parameters that enter into the formulation of the contact conditions. These geometric
parameters include the tangents and normal vectors as well as their higher derivatives, depending on the contact formulation used. It is important to point out that
the track data file generated by the track preprocessor can be very large, particularly
in cases where simulations are performed over a long distance.
As discussed in Chapter 3, the input data to the track preprocessor include
variables such as the track gage, the horizontal curvatures, the super-elevation, the
grade, and the right and left rail cant angles. Specifically, the output of the track
preprocessor has, as a minimum, the following information for each node:
1. The value of the arc length parameter s of the space curve at the node
2. The orientation angles θ, φ, and ψ that define the orientation of a coordinate system at the nodes on the space curve
3. The location of each nodal point with respect to the initial reference frame
that will be used as the body frame of the rail or the track in the multibody
system simulation code
Within a segment, a polynomial whose coefficients are functions of the nodal
variables given in the track preprocessor output can be formulated in the main
processor to define the position vector and its derivatives between the two nodes of
this segment. As another alternative, one can write the tangent and normal vectors
as well as their derivatives explicitly in terms of Euler angles and their derivatives.
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215
Euler angle derivatives can be provided as part of the track preprocessor output, or
they can be obtained during the dynamic simulation by differentiation of the interpolation functions. If the higher derivatives of Euler angles are to be included as
part of the output, the output of the track preprocessor can also include the following
information:
1. The longitudinal tangent vector given in Table 3.4 of Chapter 3 and its
first and second derivatives
2. The first, second, and third derivatives of the orientation angles given in
Table 3.2 of Chapter 3
The disadvantage of using this alternative, in which higher derivatives of Euler angles
are provided as part of the output of the track preprocessor, is the large size of the
data file.
In general, the output of the track preprocessor must have three different sets of
data: one set for the right rail, one set for the left rail, and one set for the track
centerline. If higher derivatives are to be part of the track preprocessor output, the
computational algorithm presented in Chapter 3 can be slightly modified. The computational algorithm used in the track preprocessor to define the track geometry in
terms of higher derivatives can then be summarized as follows:
1. Read the basic track input data, which includes the node numbers, the
distance of the nodes from a specified origin, the curvature, the superelevation, the gage, and the grade.
2. Determine the functions CH(S), θ(s), and φ(S) by the following respective
equations:
CH =
C1 ( S − S0 ) − C 0 ( S − S1 )
,
S1 − S0
θ = θ 0 + Cv (s − s0 ),
and
φ=
φ1 ( S − S0 ) − φ0 ( S − S1)
,
S1 − S0
where all the symbols used in these equations are as defined in Chapter 3.
Use this information to determine the third Euler angle ψ from the equation
ψ =ψ0 +
1  C1
S − S0
S1 − S0  2
(
)
2
−
2
C0
C
S − S1  + 0 S1 − S0 ,
2
 2
which also was obtained in Chapter 3.
(
)
(
)
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Railroad Vehicle Dynamics: A Computational Approach
3. Calculate the coordinates x, y, and z of the nodes on the track centerline
by evaluating numerically the integrals presented in Chapter 3 and given
by the following equations:
S
∫
( )
( )
( )
( )
()
()
x = x 0 + cos ψ S  dS = x S ,
S0
S
∫
y = y0 + sin ψ S  dS = y S ,
S0
and
s
∫
z = z0 + sin θ s  ds = z s .
s0
4. Define the space curves of the right and left rails using a finite number
of nodal points.
5. Generate the derivatives of the longitudinal tangent and its derivatives up
to the third derivatives using the information in Table 3.4 of Chapter 3.
6. Create an output in a format that has the track node locations and the
tangent vector and its derivatives as a function of the track arc length s.
6.6.1 CHANGE
IN THE
LENGTH DUE
TO
CURVATURE
The input data to the track preprocessor describe mainly the track centerline. Given
the gage and other input parameters, the geometry of the right and left rails can be
constructed. In the case of tangent tracks with no deviations, the length of the right
and left rails is the same as the length of the track centerline. In the case of a curved
track, on the other hand, the arc lengths of the space curves of the track centerline
and the right and left rails differ. As described in Chapter 3, the location of the nodal
points on the space curves are obtained using numerical integration. The results of
the numerical integration define the coordinates of each point of the track centerline
with respect to the body coordinate system: O lX lY lZ l for the left rail or OrXrYrZr for
the right rail. To account for the change in the arc length of the right and left rails
due to the curvature, another term in addition to the coordinates of the track centerline
needs to be calculated using numerical integration. Figure 6.2 shows a curve segment
of a track. The value of the gage is exaggerated in this figure to illustrate the
difference between the lengths of the arcs AB, ArBr, and AlB l. The length of the arc
ArBr can be calculated by considering the fact that, at an arbitrary point on this arc,
the horizontal radius of curvature will be equal to the radius RH and half the horizontal
distance ∆R between the rails. The value of the horizontal distance can be calculated
as shown in Figure 3.9 of Chapter 3 as
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217
FIGURE 6.2 Top view of a curve segment.
∆R = d cos φ
(6.53)
where the distance d between the rails is given as the sum of the gage G and the
width of the rails w. From this equation, it is clear that the infinitesimal arc length
dSr for the right space curve is given by


 1

1
dS r =  RH + d cos φ  dψ =  1 + CH d cos φ  dS
2


 2

(6.54)
When no irregularities are considered, the rail profile frames and the track frame
corresponding to the same cross-section have the same orientation angles. Therefore,
the equation
S1
∫
S = S0 + cos θ (s)  ds = S (s)
S0
obtained in Chapter 3 also applies to both dsr and dSr, so that one obtains
ds r =

dS r  1
= 1 + CH d cos φ  ds
cos θ  2

(6.55)
The difference between dsr and ds can be used to determine the total length lr of
the segment (arc) ArBr that differs from the length l of AB by the quantity
s1
∫
1
∆l =
CH d cos φ ds
2
s0
(6.56)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.3 Lengths of the curve segments.
The same difference, with opposite sign, applies to the length ll of the segment AlB l.
Therefore, the length l r and l l of the right and left segments, respectively, can be
calculated as
l l = l − ∆l , l r = l + ∆l
(6.57)
Figure 6.3 shows the difference between the length l of the arc AB and the length
l r of the arc ArB r. The arc length s measured on the track centerline and the arc
length sr measured on the space curve of the right rail are also shown. The output
of the track preprocessor must show the value of the arc length s for the right and
left rails, which is updated according to Equation 6.57.
6.6.2 USE
PREPROCESSOR OUTPUT
DYNAMIC SIMULATION
OF THE
DURING
The output data of the track preprocessor are expressed in terms of the arc length
parameter of the space curve s = s1. However, the complete representation of the
rail surface requires the use of two surface parameters, as described in Section 3.2
of Chapter 3. It is assumed that the profile of the rail is known and can be described
using the function f(s2), where s2 is the second (lateral) rail surface parameter. (The
superscript r that refers to the rail is dropped here for simplicity.) If the rail profile
remains the same along the track, and if s1 and s2 are the parameters that define the
location of a contact point, the location of the contact point on the rail with respect
to the rail body coordinate system can be written as a function of the two surface
parameters s1 and s2 (as shown in Chapter 3) as follows:
u r (s1, s2 ) = R rp (s1 ) + A rp (s1 )u rp (s2 )
(6.58)
where R rp is the vector that defines the location of the origin Orp of the profile
coordinate system in the body r coordinate system, and A rp is the matrix that defines
the orientation of the profile frame with respect to the body frame, as discussed in
Chapter 3. Using the information included in the track preprocessor output, one can
use linear or cubic interpolation to define R rp (s1 ) and A rp (s1 ) and their derivatives
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219
with respect to s1, as previously described in this chapter. The vector u rp (s2 ) , which
represents the location of the contact point with respect to the profile coordinate
system (as shown in Chapter 3) is given as follows:
u rp (s2 ) = 0
s2
f (s2 ) 
T
(6.59)
Using Equation 6.58, one can define the two tangent vectors and the normal
vector as follows:

∂u rp dR rp dA rp (s1 ) rp
=
+
u (s2 ) 
ds1
ds1
∂s1


rp
rp
du (s2 )
∂u

rp
= A (s1 )
t2 =

∂s2
ds2


n = t1 × t2



t1 =
(6.60)
where t1 and t2 are the two tangent vectors at a point on the rail that has the two
surface parameters s1 and s2 as coordinates. These two tangent vectors, which are
defined with respect to the rail coordinate system (body coordinate system) and are
not necessarily orthogonal, represent the tangent plane at this point. The normal
vector n is defined using the third equation in Equation 6.60 as the cross-product of
the two tangent vectors. As previously explained, the tangent and normal vectors
and their derivatives enter into the formulation of the contact forces and kinematic
constraints.
6.7 DEVIATIONS AND MEASURED DATA
Thus far the discussion has been focused on the analytical description of the track
geometry. In reality, tracks can have irregularities known as deviations. It is important
to include the effect of the deviations in the description of the track geometry to
accurately examine the response of railroad vehicle systems. As discussed in
Chapter 1, new vehicles are tested using specific simulation scenarios that involve
certain standard track irregularities designed to excite a certain mode of motion.
This section presents a brief description of track deviation functions that can be used
in railroad vehicle dynamic simulations, and the implementation of these functions
in the track preprocessor is discussed. Another important requirement in railroad
dynamics is to examine the response of a vehicle over real tracks that are defined
using field measured data. A track preprocessor must be designed to read these data
and use this information to construct the track geometry. This section proposes a
procedure for the use of field measured data that can be implemented in the track
preprocessor.
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Railroad Vehicle Dynamics: A Computational Approach
6.7.1 TRACK DEVIATIONS
Track deviations are defined either in the vertical or lateral direction. The vertical
deviations are known as profile deviations, while the lateral deviations are known
as alignment deviations. Profile and alignment deviations can have arbitrary shapes.
In railroad dynamics, the analytical track deviations are defined using a set of
functions that are shown in Table 6.1 and that are defined based on irregularities
TABLE 6.1
Track Deviation Functions
Deviation
Shape
Function
Cusp
y
Bump
y
Jog
y
Plateau
y
Trough
y
Ae
Ae
ks
1 2 ks
2
Aks
(1 4k 2 s 2 )
A2
1
Ak
8
ks
1/ k
2
s2
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221
TABLE 6.1 (continued)
Track Deviation Functions
y
Sinusoid
y
Damped Sinusoid
A sin ks
Ae
ks
cos ks
Source: Hamid, A. et al., Analytical Description of Track Geometry Variations, FRA report
DOT/FRA/ORD-83/03.1, USDOT, Federal Railroad Administration, Washington, DC, 1983.
observed from the measured track data (Hamid et al., 1983). Clearly, some of these
functions are exponential, which means that their starting and ending points are at
infinity. One can assume that these functions are applied on the positive section of
its abscissa. Therefore, the starting point of the function will be at s = 0 in Table 6.1,
and it will increase along the track, assuming that the end of the track is at the point
of infinity. In general, these functions can be used either in the vertical or the lateral
direction. Since it is easier to define these functions with respect to the local space
curve frame, and since the track preprocessor defines the locations of the nodes with
respect to the body coordinate system, the track deviations must also be defined
with respect to the body coordinate system. For a given deviation function, one can
write a vector u that defines the coordinates of a point on the space curve with
respect to the track frame, as shown in Figure 6.4. If the deviation is in the lateral
direction, the vector u takes the following form:
u = 0
f (s r )
0 
T
(6.61)
where ƒ(sr) is a function selected from Table 6.1. The coordinates of the rail space
curve can be changed using the definition of the vector u . This vector can be defined
with respect to the body coordinate system as follows:
u = A rp u
(6.62)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.4 Rail deviation.
where A rp, as discussed in Chapter 3, is the orientation matrix of the rail profile
coordinate system with respect to the body coordinate system. The change in the
orientation of the space curve due to the deviation can be determined as follows:
∆β = tan −1
df (s r )
ds r
(6.63)
where β = ψ or θ, depending on whether the deviation is lateral or vertical, respectively. Therefore, one can implement the following numerical procedure in the track
preprocessor to account for the effect of the deviations:
1. After generating the track nodes, the numbers of the nodes at which the
track deviation starts and ends are determined.
2. The space curve orientation matrix A rp at the nodes that represent the
domain of the deviation is evaluated.
3. Depending on the deviation type, the vector u is evaluated.
4. Evaluate the vector u and define the change in the angle according to
Equations 6.62 and 6.63.
5. Superimpose the vector u on the track nodal position vectors.
6. Based on the deviation type, update the track nodal rotations in the
deviation region.
Using this simple algorithm, the track preprocessor can be designed to construct
track geometry with arbitrary deviations.
6.7.2 MEASURED TRACK DATA
Measured track data are important in examining the vehicle dynamic response. In
most accident investigations, dynamic simulations based on field measured track
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223
data must be performed to identify the cause of the accident. Measured track data
can be provided in different formats. In general, measured track data include the
point locations along the track, the gage, the curvature, the super-elevation, right
and left rail profile deviations, and right and left rail alignment deviations. A simple
and straightforward procedure for implementing the field measured data in a track
preprocessor is to use the measured data, including the deviations, to define the
standard inputs to the track preprocessor. In this case, the gage is variable and must
be defined at each node. For simplicity, it is recommended to use a fixed mean value
for the gage and use the actual gage to define alignment deviation. In this case, the
alignment due to the gage variation can be equally divided between the right and
left rails. Furthermore, since the measured track data are often provided as raw data,
it is recommended to use a smoothing process similar to the one previously discussed
in this chapter. In some standards, the gage and super-elevation in the measured data
can also be filtered using a moving-average window. The width of this window
depends on the format of the data.
6.7.3 TRACK QUALITY
AND
CLASSES
Tracks are classified based on the maximum allowable deviations (irregularities).
As previously mentioned in this section, the track irregularities can be classified into
two categories: alignment and profile. In some cases, due to load changes across the
ties, the track experiences a change in the cross level known as warp (twist), which
is more accurately defined as the rate of the change in cross level. In general, there
are nine classes of track in the U.S. (Track Safety Standards Part 213 subpart A-F
and G). Each class is defined by a gage limit and by maximum allowable deviations
for alignment and profile. Based on the vehicle type and design, each class corresponds to a maximum allowable vehicle speed. Table 6.2 shows the maximum
allowable speed for each class of track. In general, lower classes are used for freight
trains, but in some cases a passenger train can be operated on tracks of lower classes.
TABLE 6.2
Track Classes (Track Safety Standards
Part 213, Subpart A to F and G)
Maximum Allowable Speed (mph)
Track
Class
Freight Train
Passenger Train
1
2
3
4
5
6
7
8
9
10
25
40
60
80
…
…
…
…
15
30
60
80
90
110
125
160
200
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Railroad Vehicle Dynamics: A Computational Approach
TABLE 6.3
Gage Limit for Each Track Class (Track Safety
Standards Part 213, Subpart A to F and G)
Track
Class
Minimum Gage Limit
(in.)
Maximum Gage Limit
(in.)
1
2
3
4
5
6
7
8
9
56
56
56
56
56
56
56
56
56.25
58
57.75
57.75
57.50
57.50
57.25
57.25
57.25
57.25
The data of Table 6.2 show that higher track classes correspond to higher track
quality that allows trains to operate at higher speeds. For track classes 6 to 9, freight
trains are allowed to travel at passenger train speeds if specific requirements are met
(Track Safety Standards Part 213 subpart A-F and G). Note that the gage depends
on the maximum allowable track alignment irregularities. Table 6.3 shows the gage
limit for each class of track. In general, the change in the gage is limited to a
maximum value of 0.5 in. within 31 ft for track classes 6 and higher. Table 6.4 and
Table 6.5 show the maximum allowable alignment and profile deviations for track
classes 6 and higher, respectively. In Table 6.4 and Table 6.5, δ1, δ2, and δ3 are the
maximum allowable deviation from uniform profile on either rail at the middle of
31-, 62-, and 124-ft chords, respectively.
TABLE 6.4
Alignment Limit for Each Track Class (Track Safety
Standards Part 213, Subpart G)
Three or More
Nonoverlapping Deviations
Single Deviation
Track
Class
δ1 (in.)
δ2 (in.)
δ3 (in.)
δ1 (in.)
δ2 (in.)
δ3 (in.)
6
7
8
9
0.5
0.5
0.5
0.5
0.75
0.5
0.5
0.5
1.5
1.25
0.75
0.75
0.375
0.375
0.375
0.375
0.5
0.375
0.375
0.375
1.0
0.875
0.5
0.5
Note: δ1, δ2, and δ3 are the maximum allowable deviation in alignment on
either rail at the middle of 31-, 62-, and 124-ft chords, respectively.
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225
TABLE 6.5
Profile Limit for Each Track Class (Track Safety
Standards Part 213, Subpart G)
Three or More
Nonoverlapping Deviations
Single Deviation
Track
Class
δ1 (in.)
δ2 (in.)
δ3 (in.)
δ1 (in.)
δ2 (in.)
δ3 (in.)
6
7
8
9
1.0
1.0
0.75
0.5
1.0
1.0
1.0
0.75
1.75
1.50
1.25
1.25
0.75
0.75
0.50
0.375
0.75
0.75
0.75
0.50
1.25
1.0
0.875
0.875
Note: δ1, δ2, and δ3 are the maximum allowable deviation from uniform
profile on either rail at the middle of 31-, 62-, and 124-ft chords, respectively.
As mentioned previously, a track can have certain deviations that can be represented by using closed-form functions. These functions are cusp, bump, jog, plateau,
trough, sinusoid, and damped sinusoid. These functions, which are sometimes called
signatures, can be used to represent track alignment or profile deviations. Hamid
(Hamid et al., 1983) defined the parameters that can be used to represent these
functions based on actual measurements. The values of these parameters are given
in Table 6.6. Hamid et al. (1983) also provided a list of the possible occurrences for
each of these track deviation functions, a summary of which is given in Table 6.7.
In general, a single cusp occurs as a profile irregularity due to the joint between
welded rails. Bumps can be alignment or profile irregularities and can be found
simultaneously on both rails. In general, a bump has a smoother shape and covers
a longer distance than a cusp. In the case of bridges, bumps can be considered as
profile irregularities, while spirals can have alignment bumps at a certain distance
from their start. A jog can occur due to variations in the track stiffness, as in the
case of the connection between soft track and bridge. Plateau is caused by variations
of track stiffness or the wear of the high rail. In some cases, a combination of cusp
and plateau can be found prior to the spiral. Such a combination is dangerous, since
it causes a rapid change in the wheel load. Trough is the result of poor drainage or
localized soft subgrade. A sinusoid is found in the case of bridge or reverse curves
where no tangent segment is used to separate between the two curves. A damped
sinusoidal occurs usually on a single rail, and it is usually the result of a significant
change in the track stiffness due to switches, grade crossings, and curves.
6.8 SPECIAL ELEMENTS
In addition to the geometry and contact problems, there are some special elements
that distinguish railroad vehicles from other multibody system applications. This
section discusses the formulations of several force elements that are often used in
railroad vehicle models. Among these elements are the translational and rotational
0.8–1.4
0.8–1.4
…
0.8–1.3
…
…
0.5–1.0
Cusp
Bump
Jog
Plateau
Trough
Sinusoid
Damped sinusoid
0.5–0.3
0.5–2.8
0.5–3.3
1.2–1.6
1.4–2.2
0.8–1.2
1.0–2.2
0.011–0.103
0.009–0.083
0.006–0.025
0.025–0.027
0.013–0.029
0.033–0.020
0.013–0.015
0.9–3.0
1.0–3.0
1.6–2.8
0.6–1.0
…
…
0.9–1.2
0.031–0.095
0.017–0.031
0.020–0.05
0.026–0.04
…
…
0.051–0.061
Cross Level
k (ft−1)
A (in.)
Range of Values
Alignment
k (ft−1)
A (in.)
0.9–3.0
0.5–4.0
0.5–5.0
0.9–3.0
0.7–2.0
1.0–1.5
…
A (in.)
0.016–0.095
0.013–0.065
0.008–0.045
0.009–0.033
0.020–0.025
0.020–0.025
…
Profile
k (ft−1)
Source: Hamid, A. et al., Analytical Description of Track Geometry Variations, FRA report DOT/FRA/ORD-83/03.1, USDOT,
Federal Railroad Administration, Washington, DC, 1983.
0.016–0.061
0.031–0.040
…
0.029–0.08
…
…
…
Gage
A (in.)
k (ft−1)
226
Deviation
Function
TABLE 6.6
Parameters of Analytical Representations of Track Irregularities
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227
TABLE 6.7
Occurrence of Track Deviation Functions
Deviation
Function
Occurrence
Cusp
Joints, turnouts, interlocking, sun kinks, buffer rail, insulated joints in
continuous welded rail (CWR), splice bar joint in CWR, piers at bridge
Bump
Soft spots, washouts mud spots, fouled ballast, joist, spirals, grade crossings,
bridges, overpasses, loose bolts, turnouts, interlinking
Jog
Spirals, bridges, crossings, interlocking, fill-cut transitions
Plateau
Bridges, grade crossing, areas of spot maintenance
Trough
Soft spots, soft and unstable subgrades, spirals
Sinusoid
Spirals, soft spots, bridges
Damped sinusoid
Spirals, turnouts, localized soft spot
Source: Hamid, A. et al., Analytical Description of Track Geometry Variations, FRA report
DOT/FRA/ORD-83/03.1, USDOT, Federal Railroad Administration, Washington, DC, 1983.
spring-damper-actuator elements, the series spring-damper element, and the bushing
element. Railroad vehicles, for example, have two types of suspensions, primary
and secondary suspensions. The primary suspension is used to connect bodies such
as bogie frames or equalizer bars to the wheelset. The secondary suspension is used
to connect the car body to the bogie or trucks. These suspensions are a combination
of translational spring-damper-actuator elements, rotational (torsional) spring damper
elements, friction elements, etc. In some vehicles, the car can be connected to the
bogie by a joint such as a pin (revolute) joint or a cylindrical joint. The wheels are
mounted on the frames using bearing elements. Figure 6.5 shows examples of
elements that can be used in modeling railroad vehicles. Chapter 2 presents several
joints that can be used in the modeling of railroad vehicles. This section, on the
other hand, discusses the formulations of several force elements that can be used in
modeling the suspension of a railroad vehicle.
6.8.1 TRANSLATIONAL SPRING-DAMPER-ACTUATOR ELEMENT
This element can consist of a spring, a damper, and an actuator. The coefficients
used in this element formulation to define the force can be linear or nonlinear
functions of the relative motion and velocity of the two bodies connected by this
element. Consider a spring-damper-actuator element that connects two bodies i and
j as shown in Figure 6.6. The force element is connected to body i at point Pi that
has a local position vector u iP with respect to the body coordinate system. It is also
connected to body j at point Pj that has a local position vector u Pj with respect to
the body j coordinate system, as shown in Figure 6.6. The spring constant is assumed
to be k, the damping coefficient is c, and the actuator force acting along a line
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.5 Elements of railroad vehicle models.
FIGURE 6.6 Translational spring-damper-actuator element.
connecting points Pi and Pj is fa. The spring has an undeformed length equal to l0.
Therefore, the force acting along a line connecting points Pi and Pj can be written
as follows:
(
)
Fs = k l − l0 + cl + f a
(6.64)
where l and l are the deformed spring length and its time derivative, respectively.
The length l is given by
T
l = rPij = rPij rPij
(6.65)
where rPij is the position vector of point P i with respect to point P j and is given by
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229
rPij = rPi − rPj = R i + A i u iP − R j − A j u Pj
(6.66)
where Ri and R j are the global position vectors of the coordinate systems of bodies
i and j, respectively, and Ai and Aj are the transformation matrices that define the
orientation of the coordinate systems of the two bodies. The virtual work of the
force Fs of Equation 6.64 is given by
δ W = − Fsδ l
(6.67)
where δl is the virtual change in the spring length given by Equation 6.65. This
virtual change can be written as follows:
T
δl =
rPij δ rPij
(r r )
ij
p
T
ij
p
1 T
= rPij δ rPij
l
(6.68)
T
=
rPij  i
δ R − u iPGiδθ i − δ R j + u PjG jδθ j 
l 
where u iP and u Pj are the skew symmetric matrices associated with the vectors A i u iP
and A j u Pj , respectively; θ i and θ j are the orientation parameters of the two bodies;
and Gi and G j are the matrices that relate the angular velocity vector to the time
derivatives of the orientation parameters of the two bodies. Therefore, the virtual
work of Equation 6.67 can be written as follows:
δ W = − Fsr̂Pij δ R i − u iPGiδθ i − δ R j + u PjG jδθ j 
T
(6.69)
where r̂Pij = rPij l is a unit vector along the line that connects points P i and P j. Equation
6.69 can also be written in the following form:
T
T
T
T
δ W = QiR δ R i + Qθi δθ i + Q Rj δ R j + Qθj δθ j
(6.70)
where QiR , Qθi , Q Rj , and Qθj are the generalized forces associated with the coordinates of bodies i and j, respectively, and are given by



i
i T i T ij
Qθ = FsG u P rˆP 


Q Rj = FsrˆPij

T
T
Qθj = − FsG j u Pj rˆPij 
QiR = − FsrˆPij
(6.71)
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Railroad Vehicle Dynamics: A Computational Approach
Therefore, the generalized spring-damper-actuator force vectors acting on bodies i
and j, respectively, can be defined using the preceding equations as follows:
 Qi 
Qi =  Ri  ,
 Qθ 
Q j 
Q j =  Rj 
 Qθ 
(6.72)
These vectors can be introduced to the right-hand side of Equation 6.1 as generalized
forces associated with the generalized coordinates of the two bodies.
6.8.2 ROTATIONAL SPRING-DAMPER-ACTUATOR ELEMENT
While the translational spring-damper-actuator element produces rectilinear force,
the rotational spring-damper-actuator element produces torque. The coefficients that
enter into the formulation of the rotational spring-damper-actuator element can be
linear or nonlinear functions of the relative rotations and relative angular velocities
of the two bodies connected by this element. Consider a rotational spring-damperactuator element that connects two bodies i and j as shown in Figure 6.7. The spring
constant is assumed to be kθ , the damping coefficient is cθ , and the actuator torque
acting in the direction of the rotation axis defined by the unit vector n ij is Tθ.
Therefore, the torque exerted on body i as the result of the rotation θ ij with respect
to body j about the spring axis is
(
)
Ts = kθ θ ij − θ 0ij + cθθ ij + Tθ
(6.73)
In this equation, θ oij is the relative rotation between the two bodies about the axis of
rotation when the spring is undeformed. The virtual work of the torque Ts is given by
δ W = −Tsδθ ij
FIGURE 6.7 Rotational spring-damper-actuator element.
(6.74)
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231
In the analysis presented in this section, it is assumed that the relative rotation
θ ij between bodies i and j is small. In this case, it can be shown that the relation
between the infinitesimal virtual rotation vector δβ i about the axes of the global
Cartesian coordinate system and the virtual change in the generalized orientation
coordinates δθi are related as follows (Shabana, 2001):
δ β i = Giδθ i
(6.75)
where Gi is the matrix that relates the angular velocity vector to the time rate of the
orientation coordinates of the body, as discussed in Chapter 2. Using the preceding
equation, one can write the virtual change of the relative rotation δθ ij between bodies
i and j about the axis nij as
T
T
δθ ij = n ij (δ β i − δ β j ) = n ij (G jδθ j − G jδθ j )
(6.76)
It follows that the virtual work of Equation 6.74 can be written as follows:
T
δ W = −Ts n ij (G jδθ j − G jδθ j )
(6.77)
This equation can be written compactly as
T
T
δ W = Qθi δθ j + Qθj δθ j
(6.78)
One can, therefore, use the preceding two equations to define the generalized
forces associated with the two bodies that result from the rotational spring-damperactuator element as


0
Qi = 
,
i T ij
 −TsG n 
 0 
Qj = 

j T ij
TsG n 
(6.79)
These forces can be used in Equation 6.1 as generalized applied forces. Note that
this force element has no components associated with the translation coordinates of
the two bodies.
6.8.3 SERIES SPRING-DAMPER ELEMENT
The series spring-damper element is widely used in railroad vehicle system applications. In this element, the spring and the damper are connected in series, as shown
in Figure 6.8. In the case of the series connection, the force in the spring and the
damper is the same and is denoted as Fs. One can therefore write the following
equation for the force Fs:
Fs = kδ s = cδd
(6.80)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.8 Series spring-damper element.
where δs is the deformation in the spring, and δd is the relative velocity between the
two ends of the damper defined by points P1 and P2, as shown in Figure 6.8. The
total relative displacement between the two end points P1 and P3 of the series springdamper element is denoted as δt and can be written as
δt = δs + δd
(6.81)
δt = δs + δd
(6.82)
It follows that
It is clear from Equation 6.80 that Fs = kδs . Using this equation with Equations
6.80 and 6.82, one obtains the following differential equation:
dFs k
dδ
+ Fs = k t
dt c
dt
(6.83)
Since the positions and velocities of the end points of the series spring-damper
element are functions of the generalized coordinates and velocities of the two bodies
connected by this element, the solution of Equation 6.83 can be obtained on-line to
determine the spring force Fs. Following a procedure similar to the one used for the
translation spring-damper-actuator element, the generalized forces due to the series
spring-damper element can be formulated.
6.8.4 BUSHING ELEMENT
The bushing element is commonly used in vehicle applications, including railroad
vehicles. In most applications, the bushing element is made of rubber and it produces
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233
FIGURE 6.9 Bushing element.
stiffness and damping forces in different directions. In the formulation presented in
this section, it is assumed that the bushing element can have different values of
linear or nonlinear stiffness and damping coefficients in different directions. In the
model developed in this section, a bushing coordinate system that is assumed to be
rigidly attached to one of the bodies connected by the bushing element (say body j)
is introduced, as shown in Figure 6.9. This coordinate system can be defined in the
initial configuration based on data provided by the user for the location of two points
on the body. Using the position vectors u Pj1 and u Pj2 of two points P1j and P2j on body j,
one axis of the coordinate system of the bushing can be defined as follows:
n
j
(u
=
j
P1
− u Pj2
u −u
j
P1
)
(6.84)
j
P2
where n j is one of the bushing axes defined in the body j coordinate system, as
shown in Figure 6.9. This axis can be defined in the global coordinate system as nj =
A j n j , where Aj is the transformation matrix that defines the orientation of the
coordinate system of body j. The axis n j can be used to define the directional
properties of the bushing element. Using this axis, one can construct the other two
axes of the bushing coordinate system and define its orientation with respect to the
body j coordinate system using the following transformation matrix:
A bj =  t1j
t2j
n j 
(6.85)
where t1j and t2j are the two unit vectors that complete the three orthogonal axes of
the bushing coordinate system. Assuming that body j is a rigid body, the bushing
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Railroad Vehicle Dynamics: A Computational Approach
coordinate system defined with respect to the global frame can be determined as
follows:
A bj = A j A bj
(6.86)
If the bushing (a) is assumed to be connected to body i at point Pi, which is
defined by the position vector u iP with respect to the body i coordinate system, and
(b) is connected to body j at point P1j , which is defined by the position vector u Pj1
with respect to the body j coordinate system, then one can define the following
vector of relative position between the two points Pi and P1j :
rij = R i + A i u iP − R j − A j u Pj1
(6.87)
i u i − R j − A
ju j
r ij = R i + A
P
P1
(6.88)
It follows that
Using Equation 6.86, the preceding two equations, and assuming that points P i
and P1j initially coincide, one can define the following bushing deformation and rate
of deformation vectors in the bushing coordinate system:
T
δ bij = Abj r ij
T
δ bij = A bj r ij ,
(6.89)
The rotational deformation of the bushing is defined by the relative orientation
of the bushing coordinate systems of bodies i and j as follows:
T
A bij = A bj A bi
(6.90)
where Abj is the orientation matrix of the bushing coordinate system of body j (as
previously defined by Equation 6.86), while Abi is the orientation matrix of the bushing
coordinate system of body i that is defined as A bi = A i A bi . Note that the local
orientation matrix of the bushing coordinate system of body i is defined at the initial
configuration as
T
A bi = Ai A bj
(6.91)
t =0
Assuming that the relative rotations between the two bodies connected by the
bushing element are small, the relative rotation matrix Abij can be used to extract
three relative rotations, θ xbij , θ ybij , and θbij
z , defined in the bushing coordinate system,
i.e.,
θ bij = θ xbij
θ ybij
T
θ zbij  .
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235
The relative angular velocity between the two bodies defined in the bushing coordinate system can also be determined as follows:
T
(
ω bij = A bj ω i − ω j
)
(6.92)
where ω i and ω j are the absolute angular velocity vectors of bodies i and j, respectively, defined in the global coordinate system.
The bushing stiffness and damping coefficients are often determined using experimental testing, and these coefficients are defined generally in the bushing coordinate
system. In the multibody formulations, the bushing coefficients are defined using
stiffness and damping matrices such as
 k xx

Kr =  kyx

 kzx
k xy
kyy
kzy
k xz 

kyz  ,

kzz 
c xx

C r =  c yx

 c zx
c xy
c yy
c zy
c xz 

c yz 

c zz 
(6.93)
where Kr and Cr are the translational stiffness and damping matrices, respectively,
defined with respect to the bushing coordinate system. The rotational stiffness matrix
Kθ and damping matrix Cθ can be written in a similar form. In most cases, the
diagonal form of the stiffness and damping matrices is used for simplicity. In terms
of these matrices, the force vector defined in the bushing coordinate system can be
written as follows:
 FRb  K r
 b = 
 Mθ   0
0   δ bij  Cr
 + 
Kθ   θ bij   0
0   δ bij 


Cθ  ω bij 
(6.94)
This force vector can be defined in the global coordinate system as
 FRb   A bj FRb 
 b  =  bj b 
 Mθ   A Mθ 
(6.95)
Using the principle of virtual work, the generalized bushing forces associated
with the generalized coordinates of the two bodies can be obtained as follows:



i
iT iT b
iT
b
Qθ = G u P FR − G Mθ 

Q Rj = FRb


T
T
T
Qθj = −G j u Pj1 FRb + G j Mθb 

QiR = − FRb
(6.96)
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Railroad Vehicle Dynamics: A Computational Approach
These generalized forces are introduced to the system equations of motion using the
vector Q of Equation 6.1.
6.9 MAGLEV FORCES
The contact between the wheels and the rails is one of the main factors that limit
increasing the speed of railroad vehicle systems. The vehicle critical speed depends
on the creep forces, which significantly influence the hunting motion. On the other
hand, the noncontact driving concept used in magnetically levitated (Maglev) trains
eliminates this restriction and allows for increasing the train operational speed. It is
known that magnetic suspensions require less power than conventional railroad
vehicle systems, and since there is no contact or friction in the magnetic levitation,
the noise and wear can be reduced, resulting in improved level of comfort, less maintenance, and environmentally acceptable mass-transportation systems (Dukkipati,
2000). There are two different types of magnetic suspension systems currently used
in Maglev vehicles: electrodynamic suspension (EDS) and electromagnetic suspension (EMS), as shown in Figure 6.10. In this section, the fundamental principles of
the magnetic levitation system are briefly reviewed, and the modeling issues that
arise in developing multibody Maglev vehicle models are discussed.
6.9.1 ELECTRODYNAMIC SUSPENSION (EDS)
In the electrodynamic suspension system, the vehicle is lifted by the magnetic forces
that act on the vehicle and the guideway, as shown in Figure 6.10a. Since the electromagnetic field is developed as the vehicle moves, the flux produced by the onboard
coils induces currents in the passive coils or nonmagnetic sheets on the guideway.
As a result, the induced currents produce a magnetic flux that opposes the magnetic
flux of the onboard electromagnet, producing repulsive forces between the vehicle
and the guideway. Since the repulsive forces are produced as the vehicle moves
above the passive coil on the guideway, the vehicle cannot be lifted unless a certain
speed is achieved and, therefore, the electrodynamic suspension system requires
FIGURE 6.10 Maglev suspensions.
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237
FIGURE 6.11 Electromagnetic suspension.
wheels at standstill and at low speed. For this reason, this suspension system is not
suitable for an urban mass transportation that requires frequent stops at several
stations.
6.9.2 ELECTROMAGNETIC SUSPENSION (EMS)
In the electromagnetic suspension system, the vehicle is lifted by the forces of
attraction produced between the electromagnet built on the vehicle and the ferromagnet on the guideway, as shown in Figure 6.10b. The iron core of a magnetic
circuit is excited by a current-carrying coil, and as a result, the core on the vehicle
is attracted to the ferromagnetic rail. As will be explained later in this section, the
electromagnetic suspension is statically unstable in the sense that the attractive force
increases as the air gap between the pole face of the electromagnet and the ferromagnet decreases. For this reason, an appropriate levitation control needs to be
developed and implemented to achieve statically and dynamically stable suspension
characteristics. Since the levitation force is independent of the vehicle speed, the
vehicle can be lifted at low speed and at standstill when the electromagnetic suspension system is used. This suspension system, therefore, is suitable for urban mass
transportation systems that require low operating speed and frequent stops at several
stations. Note also that eddy currents that tend to decrease energy efficiency are
induced in the case of high-speed operation.
6.9.3 MODELING
OF
ELECTROMAGNETIC SUSPENSIONS
The levitation forces of electromagnetic suspension systems are discussed here in
an effort to develop a model for these forces that can be implemented in the
computational multibody system algorithms. As shown in Figure 6.11, let the length
of the pole face be a, the width be b, the permeability of the free space be µ0, and
the flux density across the air gap be Ba. The force of attraction between the pole
surface and the ferromagnet is defined by (Sinha, 1987)
Fz =
( Ba )2 A
µ0
(6.97)
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Railroad Vehicle Dynamics: A Computational Approach
where A = ab. The flux density can be expressed in terms of the reluctance of the
mutual flux RM as
Ba =
Fmmf
ARM
(6.98)
where Fmmf is a magnetomotive force, which can be defined as
Fmmf = N i(t )
(6.99)
In the preceding equation, N is the number of turns of the coil, and i(t) is the current
in the coil. Substituting Equation 6.98 into Equation 6.97 and using Equation 6.99,
one obtains
Fz =
1  N i(t ) 
µ0 A  RM 
2
(6.100)
The reluctance of the mutual flux, denoted as RM, consists of the reluctance of the
air gap, electromagnet, and the ferromagnet, which can be respectively defined as
(Dukkipati, 2000)
RM (d z ) =
2d z
w + 2h + 2a w + 2a
+
+
µ0 ab
µc ab
µ pa1b
(6.101)
where dz is the distance between the pole face and the ferromagnet, i.e., dz = dz(t);
a1 is the thickness of the ferromagnet; w and h are the dimensions shown in Figure
6.11; and µc and µp are, respectively, the permeability of the electromagnet and the
ferromagnet. Assuming that µi ≡ µc ≈ µp and a1 ≈ a, the preceding equation can be
reduced to
RM (d z ) (
2
d z (t ) + r0
µ0 A
)
(6.102)
where r0 is given as
r0 =
µ0
(2a + w + h)
µi
(6.103)
Substituting Equation 6.102 into Equation 6.100, one has
2
µ N 2 A  i(t ) 
Fz (d z , i) = 0
≤ ( Fz )max
4  d z (t ) + r0 
(6.104)
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239
FIGURE 6.12 Characteristics of the levitation force.
It can be seen from the preceding equation that the attractive (levitation) force
increases as the current increases, as shown in Figure 6.12a. On the other hand,
since the attractive (levitation) force is defined as a reciprocal function of the air
gap distance dz(t), the levitation force decreases as the air gap increases, as shown
in Figure 6.12b, leading to inherently unstable suspension characteristics that are a
function of the air gap. For this reason, the electromagnetic suspension requires an
appropriate feedback control system to achieve stable suspension characteristics. It
should also be noted at this point that the flux density across the air gap Ba has
saturation characteristic, and therefore, the preceding attractive force should be less
than the maximum force (Fz)max that is given by
( Fz )max =
( Ba )2max A
µ0
(6.105)
where (Ba)max is the maximum flux density that is given as (Dukkipati, 2000)
( Ba ) max =
PM
B0
PM + PL
(6.106)
and B0 in this equation is in the range of 1.5 ≈ 2.0 Wb/m2; PM is the permeance of
mutual flux, given as PM = 1/RM; and PL is the permeance of the leakage flux given as
b
 πa 
PL = µ0 h  + ln  1 +
 2 w  
w
(6.107)
Having obtained the electromagnetic force, one can determine the generalized
electromagnetic forces of body j associated with the reference coordinates Rj and
the orientation coordinates θ j as
T
j
j
(Qmf
) R = Fmfj , (Qmf
)θ = G j (u Pj × Fmfj )
(6.108)
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Railroad Vehicle Dynamics: A Computational Approach
where Fmfj is the vector of the electromagnetic force defined in the global coordinate
system, Gj is the matrix that relates the global angular velocity of body j to the time
derivatives of the orientation coordinates, and u Pj is the vector of the point of application of the force defined in the global coordinate system. Since it is convenient
to define the magnetic force with respect to the body coordinate system, the following
transformation is used:
Fmfj = A j Fmfj , u Pj = A j u Pj
(6.109)
where Fmfj and u Pj are, respectively, the electromagnetic force vector and the vector
of the point of application of the force defined in the body coordinate system, and
are given as
Fmfj = [0
0
Fz j (d zj , i)]T , u Pj = [ x Pj
y Pj
j
z P ]T
(6.110)
The generalized electromagnetic forces of Equation 6.108 can simply be added
to the generalized forces on the right-hand side of the multibody system equations
of motion as
M

Cq
CTq   q
 Qe + Qv + Qmf 
  = 

0   λ  
Qd

(6.111)
where M is the system mass matrix; Qv is the vector of inertia forces that are
quadratic in the velocities; Qe is the vector of the generalized external forces; Cq is
the Jacobian matrix of the constraint equations; λ is the vector of Lagrange multipliers; and Qd is the vector resulting from the differentiation of the system constraint
equations twice with respect to time.
6.9.4 MULTIBODY SYSTEM ELECTROMECHANICAL EQUATIONS
Recall that the electromagnetic force given by Equation 6.104 is function of the
current i(t) as well as the air gap dz (t) defined by the system generalized coordinates
q. As a result, one needs to solve the preceding multibody equations simultaneously
with electronic circuit equations expressed for i(t) as
v = Ri(t ) +
d
( L (t )i(t ))
dt
(6.112)
= Ri(t ) + L (t )i(t ) + i(t ) L (t )
where v is a supplied power voltage, and L is the inductance of the electromagnetic
suspension defined by
 1
1 
L (t ) = N 2 
+
 RL RM 
(6.113)
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241
where the reluctance of the leakage flux RL = 1/PL can be obtained using Equation
6.107, while the reluctance of the mutual flux RM is given by Equation 6.102. As a
result, the total inductance L(t), which can be written as a function of the air gap
dz(t), is obtained as
L (d z ) = LL +
µ0 N 2 A
2(d z (t ) + r0 )
(6.114)
where LL = N 2PL = N 2/RL. To achieve stable electromagnetic suspension characteristics, the supplied power voltage is controlled by the position feedback such that
the air gap remains constant (Sinha, 1987). In such a case, the power voltage is
defined as a function of the air gap dz and its velocity as
v ( d z , dz ) = v0 + f ( d z , dz )
(6.115)
where ν0 is the power voltage at the static equilibrium configuration given for a gap
distance in operation, and f is a control function that depends on dz or dz . Substituting
Equation 6.114 into Equation 6.112, one has the following equation for i(t ) :

µ0 N 2 A dz (t )
µ0 N 2 A   LL + 2(d (t ) + r )  i (t ) = − Ri(t ) + 2(d (t ) + r )2 i(t ) + v

0 
z
z
0
(6.116)
Using the preceding equation, the current for all the electromagnetic suspensions
in the multibody vehicle can be written in the following vector form:
i(t ) = Z (i, q, q )
(6.117)
where i = [i1, …, inm]T and nm is the number of the electromagnetic suspensions,
and Z is a vector function that depends on the currents as well as the system
generalized coordinates and velocities. Using Equations 6.111 and 6.117, the equations of motion of an electromagnetic-type Maglev vehicle can be written as
M

C q

 0
CTq
0
0
0 q
 Q(ii, q, q ) 
  

0   λ  =  Qd (q, q ) 

I   i   Z(i, q, q ) 
(6.118)
where I is an identify matrix, and Q = Qe + Qv + Qmf . The preceding augmented
multibody electromechanical equations of motion can be used to determine variable
derivatives that can be integrated forward in time to determine the generalized
coordinates q and the velocities q as well as the current i. A numerical algorithm
similar to the one discussed at the beginning of this chapter can be used to obtain
the solution and check on the violation of the kinematic constraints.
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Railroad Vehicle Dynamics: A Computational Approach
6.10 STATIC ANALYSIS
One of the important problems encountered in the computer-aided analysis of railroad vehicle models is the accurate determination of the initial static equilibrium
configuration. When the contact constraint formulations are used to model the
wheel/rail interaction, the resulting nonlinear static equilibrium equations are functions of the generalized coordinates and the surface parameters. If the elastic contact
formulations are used, on the other hand, the form of equations of motion remains
the same as the standard form of the multibody equations of motion, since no
additional constraints associated with the wheel/rail contact are introduced. In this
case, the geometric surface parameters are used only to determine on-line the
coordinates of the contact points that enter into the calculations of the generalized
contact forces. Using the form of the equations of motion in the case of the elastic
contact formulations, the static equilibrium equations, assuming that the velocities
and accelerations are zero, can be written as follows:
CT λ − Q(q, t ) 
g ( z) =  q
=0
 C(q, t ) 
(6.119)
where the generalized force vector Q includes the elastic contact forces, and the vector
z = [qT λT]T consists of the unknown generalized coordinates q and the Lagrange
multipliers λ associated with the joint and specified motion trajectory constraints.
The preceding equation, whose solution is discussed in the literature (Shabana, 2001,
2005) is not expressed explicitly in terms of the surface parameters, and there are
no Lagrange multipliers associated with the wheel/rail contact forces. This, however,
is not the case when the constraint contact formulations are used. This section
discusses the use of the augmented and embedded constraint contact formulations
in the static equilibrium analysis of railroad vehicle systems. The numerical algorithms used to solve the resulting nonlinear algebraic equations are also summarized.
To improve the convergence of the iterative Newton-Raphson algorithm used to
solve the nonlinear algebraic static equilibrium equations, the line search method
and the continuation method are often used. These two methods are discussed before
concluding this section.
6.10.1 AUGMENTED CONSTRAINT CONTACT FORMULATION
Using the principle of virtual work, the variational form of the static equilibrium
equation of a railroad vehicle system subjected to contact constraints can be written
as follows:
δ qT (CTq λ − Q) + δ s T CTs λ = 0
(6.120)
where the vectors q and s are, respectively, the vectors of the system generalized
coordinates and surface parameters; C(q,s,t) is the vector of the system constraint
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243
functions that include the contact constraints as well as mechanical joints and
specified motion trajectories; Q is the vector of the system generalized forces; Cq
and Cs are, respectively, the constraint Jacobian matrices resulting from the differentiation with respect to the vectors q and s; and λ is the vector of Lagrange
multipliers associated with joint and contact constraints. Equation 6.120 is a special
case of the variational form of the dynamic equations of motion obtained in
Chapter 5, when the velocities and accelerations are assumed to be zero. Using the
constraint equations C(q,s,t) = 0 and following a procedure similar to the one used
to derive the augmented form of the dynamic equations of motion, one obtains the
following nonlinear algebraic static equilibrium equations:
CTq λ − Q(q, t ) 


g ( z) = 
CTs λ
=0
 C(q, s, t ) 


(6.121)
These equations can be solved using an iterative Newton-Raphson procedure to
determine the vector of unknown variables z = [qT sT λT ]T. In a Newton-Raphson
algorithm, the following equation is iteratively solved for the Newton differences:
K∆z ( k ) = − g ( k )
(6.122)
where ∆z(k) is the vector of Newton differences and
(CTq λ − Q)q

K =  (CTs λ )q

Cq

(CTq λ )s
(CTs λ )s
Cs
CqT 

CTs 

0 
(6.123)
is the consistent Jacobian matrix defined as K = ∂g/∂z. In Equation 6.123, the
subscripts indicate differentiation with respect to the vectors q and s. Using Newton
differences at iteration k, the solution vector z is updated as follows:
z ( k +1) = z ( k ) + ∆z ( k )
(6.124)
where the index k denotes the iteration number. Recall, as discussed in Chapter 5,
that the generalized force vector Q is a function of the generalized coordinates q
only. The analytical expressions of the Jacobian matrices Cq and Cs in Equation
6.123 in the case of the contact constraints were presented previously in Section 6.2.
A computational algorithm for the static equilibrium analysis of railroad vehicle
systems based on the augmented constraint contact formulation can be summarized
as follows:
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Railroad Vehicle Dynamics: A Computational Approach
1. An initial estimate of the surface parameters s = s0 and the generalized
coordinates q = q0 is made.
2. Using the first equation in Equation 6.121, the vector of Lagrange multipliers λ can be determined by solving the equation CqCTq λ = CqQ(q, t ),
where the coefficient matrix C q C Tq is a nonsingular square matrix.
3. The residual vector g(z) for the given solution vector z is evaluated. If
the norm of the residual vector is less than the specified tolerance, convergence is achieved and the solution for the static equilibrium analysis
is successfully obtained and is defined by the vector z. If convergence is
not achieved, go to Step 4.
4. Evaluate the Jacobian matrix K given by Equation 6.123 and solve the
system of algebraic equations of Equation 6.122 for Newton differences
∆z(k). Use the Newton differences to update the vector z as described by
Equation 6.124 and go to Step 3.
6.10.2 EMBEDDED CONSTRAINT CONTACT FORMULATION
The embedded constraint contact formulation can be used to systematically eliminate
the nongeneralized surface parameters from the static equilibrium equations, as
demonstrated previously in the more general case of dynamics. It can be shown that
the use of the embedded constraint contact formulation leads to the following
variational static equilibrium equation:
δ qT ( J T λ − Q) = 0
(6.125)
where the matrix J, which is an implicit function of s, is the embedded contact
constraint Jacobian matrix; λ is the vector of the Lagrange multipliers that include
only independent Lagrange multipliers associated with the normal contact forces;
and other noncontact constraints imposed on the motion of the system that are defined
by the vector Cn(q, s(q), t) = 0. Recall that, in the case of the embedded constraint
contact formulation, the surface parameters s are expressed as functions of the vector
of generalized coordinates q. It can be shown in this case that the static equilibrium
equations are defined as
 J T λ − Q(q, t ) 
g ( z) =  n
=0
 C (q, s(q), t ) 
(6.126)
This system of nonlinear algebraic equations can be solved for the vector of
unknowns z = [qT λT]T using a Newton-Raphson algorithm. In this case, the consistent Jacobian matrix K associated with Equation 6.126 is given as follows:
( J T λ − Q)q + ( J T λ )s B
K=
J

JT 

0 
(6.127)
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245
where B, as discussed in Chapter 5, is the velocity transformation matrix that results
from the use of the embedded constraint contact formulation and that relates the
virtual change in the surface parameters to the virtual change in the generalized
coordinates as δs = Bδq. This equation clearly shows that the change in the generalized coordinates q leads to change in the dependent surface parameters s; therefore,
the surface parameters that are not included in the solution vector z must be iteratively
updated at each Newton step by solving, for given values of the generalized coordinates q, the following nonlinear contact constraint algebraic equations for all
contacts in the system:
 t1r ⋅ (rPw − rPr ) 
 r

t ⋅ (rw − rr )
C d (q w , q r , s w , s r ) =  2 w P r P  = 0
 t ⋅n

1


w
r
 t 2 ⋅ n

(6.128)
A computational algorithm for the static equilibrium analysis based on the
embedded constraint contact formulation can be summarized as follows:
1. An initial estimate of the generalized coordinates q = q0 is made.
2. Using the vector q, Equation 6.128 is solved using an iterative NewtonRaphson algorithm to determine the surface parameters s. The surface
parameters are required to evaluate the contact constraint equations and
the constraint Jacobian matrix.
3. Using the first equation of Equation 6.126, Lagrange multipliers λ are
determined by solving the algebraic equations JJT λ = JQ(q,t).
4. The residual vector g(z) for the given vector z is evaluated. If the norm
of the residual vector is less than the specified tolerance, convergence is
achieved and the static equilibrium configuration is defined by the vector
z. Otherwise, proceed to Step 5.
5. Evaluate the Jacobian matrix K defined in Equation 6.127 and use this
matrix to solve the system of algebraic equations of Equation 6.122 for
Newton differences ∆z(k). Use the Newton differences to update the vector
z as described by Equation 6.124 and go to Step 4.
6.10.3 LINE SEARCH METHOD
In most railroad vehicle system applications, the static equilibrium equations given
by Equation 6.121 or Equation 6.126 are highly nonlinear as the result of the
nonlinearity of the kinematic constraint equations and the forcing functions. For this
reason, difficulties can be encountered in achieving convergence using the conventional Newton-Raphson procedures. The line search method is an approach that can
be used to improve the stability and convergence of the Newton-Raphson procedure
by employing a backtracking algorithm. In this method, the solution vector is not
necessarily updated using the full Newton step to ensure that the norm of the vector g
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Railroad Vehicle Dynamics: A Computational Approach
is continuously reduced at every step. In the line search method, the solution vector is
updated as follows:
z( k +1) = z( k ) + η ∆z( k ) ,
0 <η ≤1
(6.129)
where η is a scalar parameter that can be selected to ensure that g(z(k+1)) is sufficiently
smaller than g(z(k)). Note that according to the preceding equation, while the full
Newton step is not used, the updating is made in the direction of this Newton step.
The parameter η can be selected to minimize the following function:
f (η) =
(
1
g z( k +1)
2
) g (z )
T
( k +1)
(6.130)
Using Equations 6.129 and 6.130, an appropriate line search parameter η can then
be determined such that the Newton step is in a descent direction by imposing the
following minimization condition:
f′ =
df
=0
dη
(6.131)
Equation 6.131 leads to a nonlinear scalar equation that can be solved for η at
every Newton step. To avoid the use of an iterative solution to solve for η, the
function f can be approximated as a quadratic function, and the line search parameter
η can be obtained in the following closed form (Press et al., 1992):
η=−
f0′
2( f1 − f0 − f0′)
(6.132)
where f0 = f(0) and f1 = f(1). Note that f1, here, is the function in the case of a full
Newton step. By using the line search method, one can avoid using a full Newton
step if such a full step will not lead to a reduction of the norm of the vector g.
6.10.4 CONTINUATION METHOD
If the initial estimates of the Newton-Raphson procedure are far from the exact
solution, difficulties are encountered in achieving convergence. For this reason, it is
recommended in some cases to use the continuation method (Eich-Soellner and
Führer, 1998) to incrementally decrease the residual vector g until the final solution
is obtained. In the continuation method, instead of solving g(z) = 0 using a NewtonRaphson procedure, the following equations are incrementally solved for k = 1, …, n
until g(z) = 0 is achieved:
h k = g(z k ) − α k g 0 = 0,
0 ≤ αk < 1
(6.133)
where g0 is the initial residual vector obtained using the initial estimates that the
user provides; and α, a scalar function of k, is a parameter used to split the final
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247
equation g(z) = 0 into the equation h(z) = 0 at the intermediate stages. That is, α = 0
leads to the equations that must be finally solved, while α = 1 is a trivial case, since
h = 0 is automatically satisfied at the first iteration. In the continuation method, the
parameter α is consecutively reduced from 1 to 0 until the final solution that satisfies
g(z) = 0 is obtained. If a one-stage Newton-Raphson procedure is split into n
intermediate stages, the choice of α is simply given as
k
αk = 1− ,
n
k = 1, …, n
(6.134)
Since the solution to the equations at stage k is used as the initial estimate for the
next stage k+1, convergence can be achieved more efficiently. The use of the continuation method is recommended when the initial estimate is not close to the exact
solution.
6.11 NUMERICAL COMPARATIVE STUDY
A numerical study is presented in this section to compare the results of the four
contact formulations discussed in this chapter and in Chapter 5. Two numerical
examples are used in this comparative study: a simple suspended wheelset and a
complete vehicle. In these two examples, different contact methods are used and
compared to show that, although the methods are conceptually different, they lead
to results that are in good agreement. The numerical results are obtained using the
computer program SAMS2000/Rail.
6.11.1 SIMPLE SUSPENDED WHEELSET
As previously pointed out, while the two methods ACCF and ECCF are in principle
the same, the use of the surface parameters as degrees of freedom in the ACCF
method requires the use of a solution algorithm that is different from the one used
with the ECCF method. To compare the two methods, the suspended wheelset model
shown in Figure 6.13 and Figure 6.14 is used. The data for this wheelset are shown
FIGURE 6.13 Top view of the suspended wheelset.
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.14 Geometry of wheelset and rails.
TABLE 6.8
Suspended Wheelset Model
Variable
Description
Value
mw
–
Ixxw
–
Iyyw
–w
Izz
W
kx
ky
cx
cy
2b
2a
Wheelset mass
Inertia moment
Inertia moment
Inertia moment
Applied vertical load
Stiffness for longitudinal springs
Stiffness for lateral springs
Damping coefficient for longitudinal springs
Damping coefficient for lateral springs
Distance between longitudinal springs
Gage distance
1568 kg
656 kg·m2
168 kg·m2
656 kg·m2
98,000 N
1.35E+5 N/m
2.5E+5 N/m
0 N/m·sec
0 N/m·sec
1.8 m
1435 mm
in Table 6.8. The wheelset is assumed to have a constant forward velocity of 40 m/sec.
The left rail is assumed to be perfectly straight, while the right rail has a deviation
that defines a sinusoidal gage reduction of 10 mm between two points located at
distances 20 and 30 m on the rail. Figure 6.15 and Figure 6.16 show, respectively,
the lateral and yaw displacements of the wheelset as a function of the distance traveled.
The results presented in these two figures show that the motion of the wheelset is
stable, and the lateral and yaw displacements approach zero as the distance traveled
increases. Figure 6.17 shows the normal contact force acting on the right wheel,
while Figure 6.18 shows the normal force acting on the left wheel. The results
presented in these figures show a very good agreement between the solutions obtained
using the ACCF and ECCF methods. Recall that, since the surface parameters in
the ECCF method are eliminated, these parameters cannot in this case be selected
by the multibody computer code as independent variables.
6.11.2 COMPLETE VEHICLE MODEL
In this example, we consider the complete vehicle model shown in Figure 6.19 to
compare the results obtained using the embedded constraint contact method (ECCF)
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249
FIGURE 6.15 Lateral displacement of the wheelset (— ACCF, –•– ECCF).
FIGURE 6.16 Yaw angle of the wheelset (— ACCF, –•– ECCF).
and the elastic contact formulations (ECF-A) method that employs algebraic equations to determine the location of the contact point. The vehicle, consisting of two
trucks and a car body, is assumed to travel on a tangent track. Each truck consists of
two wheelsets, two equalizer bars, a frame, and a bolster, as shown in Figure 6.20.
The front wheelsets of the leading and trailing trucks are assumed to have a constant
forward velocity (V = 20 m/sec). The track is assumed perfectly tangent with two
vertical bumps, as shown in Figure 6.21. The dimensions and inertia properties of
this model are the same as presented by Zaazaa (2003). Figure 6.22 shows the lateral
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.17 Normal contact forces of the right wheel (— ACCF, –•– ECCF).
FIGURE 6.18 Normal contact forces of the left wheel (— ACCF, –•– ECCF).
displacement of the rear trailing wheelset, while Figure 6.23 shows the vertical
displacement. Figure 6.24 shows the normal forces acting on the right wheel of the
leading wheelset of the leading truck. It can be seen that there is in general a good
agreement between the constraint and the elastic formulations.
Numerical experimentation showed that the ECF-A method, which determines
the location of the contact points by solving algebraic equations, generally leads to
smoother results compared with the ECF-N method, which uses nodal search to
determine the contact points. The use of the nodal search in the ECF-N method can
lead, in some examples, to nonsmooth change in the lateral surface parameters, and
as a consequence, the contact and creep forces can be impulsive because of the very
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251
FIGURE 6.19 Vehicle model.
FIGURE 6.20 Components of the truck.
high creep coefficients. Nonetheless, the ECF-N method has the advantage in that
it can be used in the simulations of models that have surface irregularities, since in
this method smoothness of the surfaces is not required as is the case with other
methods.
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 6.21 Track with vertical bump (---- Left rail, — Right rail).
FIGURE 6.22 Lateral displacement of the rear wheelset of the trailing truck (— ECCF,
ECF-A).
–•–
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253
FIGURE 6.23 Vertical displacement of the rear wheelset of the trailing truck (— ECCF,
–•– ECF-A).
FIGURE 6.24 Normal contact force of the right wheel of the front wheelset of the leading
truck (— ECCF, –•– ECF-A).
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7
Specialized Railroad
Vehicle Formulations
The computer formulations presented in this book are based, for the most part, on
the use of absolute Cartesian coordinates. Using this motion description allows
systematic and straightforward implementation of the railroad vehicle contact formulations in existing general-purpose multibody system computer algorithms. This
description also preserves the sparse matrix structure of the dynamic formulation
and allows for exploiting advanced and well-developed multibody system dynamics
capabilities such as body flexibility. The absolute Cartesian coordinate description,
however, is not the only approach that has been used in the motion description of
railroad vehicle systems. Another set of coordinates that can be used is the trajectory
coordinates, briefly introduced in Chapter 2. It was also shown in Chapter 2 that
the unconstrained motion of a body can be described using the six absolute Cartesian
coordinates; three coordinates are used to describe the translational displacement of
a reference point on the body, and three coordinates are used to describe the orientation of the body. This set of coordinates is sufficient to describe the general threedimensional motion of an arbitrary rigid body in the multibody vehicle system. The
trajectory coordinates, as shown in Chapter 2, serve as another alternative set of
generalized coordinates. The use of the trajectory coordinates, as compared with the
absolute coordinates, has the advantage of making the formulation of some railroad
vehicle constraints and forcing functions easier. The trajectory coordinates, on the
other hand, make implementing railroad vehicle formulations in general-purpose
multibody system algorithms more difficult, can lead to the loss of the sparse matrix
structure of the dynamic equations, and make exploiting advanced flexible body
capabilities less straightforward. Furthermore, if the rails are allowed to have an
arbitrary displacement, the use of the trajectory coordinates leads to very complex
kinematic and dynamic equations.
When the trajectory coordinates are used, a body trajectory coordinate system
that follows the body motion is introduced for each component in the railroad vehicle
system. The location of the origin of the body trajectory coordinate system is defined
by the arc length measured along the track centerline curve, which is defined as part
of the assumed known geometry of the problem, as discussed in Chapter 3. The
orientation of the trajectory coordinate system is defined using three Euler angles
that are also functions of the arc length coordinate. Knowing the arc length that
defines the origin and orientation of the body trajectory coordinate system, the
location of the center of mass of a body in a railroad vehicle system can be uniquely
defined in the track coordinate system by the lateral and vertical displacements of
the body relative to the body trajectory coordinate system. The motion description
255
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Railroad Vehicle Dynamics: A Computational Approach
is completed by introducing three rotations (roll, pitch, and yaw) that define the
orientation of the body with respect to its trajectory coordinate system. Six trajectory
coordinates are, therefore, required to describe the unconstrained motion of the body.
Note that instead of using only one body coordinate system when the absolute
Cartesian coordinates are used, the use of the trajectory coordinates requires introducing two coordinate systems. The first coordinate system is the body trajectory
coordinate system defined by the geometry and the distance traveled by the body,
and the second coordinate system is the centroidal body coordinate system. If the
rails are fixed, the six trajectory coordinates are sufficient to define the configuration
of the body in the global system.
The velocity and acceleration of the center of mass and the angular velocity and
angular acceleration of the body can be expressed in terms of the generalized
trajectory coordinates and their time derivatives. These kinematic equations can be
used with the Newton-Euler equations to obtain the equations of motion of an
arbitrary body in the railroad vehicle system, as will be demonstrated in this chapter.
7.1 GENERAL DISPLACEMENT
In the formulation presented in this chapter, the motion of an arbitrary body i in
the railroad vehicle system is first defined with respect to the body trajectory coordinate system. The location of the origin and orientation of the body trajectory
coordinate system is defined from the assumed known geometry and the distance
traveled by the body. To define the configuration of body i in the global coordinate
system, a second coordinate system is introduced. The origin of this body
coordinate system is assumed to be attached to the body center of mass. The
body coordinate system is selected such that it has no displacement in the longitudinal direction of motion with respect to the trajectory coordinate system. Therefore,
two translational coordinates and three angles are required to completely define the
location and orientation of the body coordinate system with respect to the trajectory
coordinate system. Since the geometry of the track is assumed to be known, we note
that the definition of the trajectory coordinate system requires only one timedependent coordinate (distance traveled), while the description of the motion of the
body with respect to the trajectory coordinate system requires five time-dependent
coordinates: two translations and three angles.
7.1.1 TRAJECTORY COORDINATE SYSTEM
As shown in Chapter 3, the space curve that defines the centerline of the track can
be described in terms of its arc length si. The distance traveled by body i along this
space curve is defined by the time-dependent arc length parameter. Given the parameter si, the location of the origin of a trajectory coordinate system X tiY tiZ ti that follows
the motion of body i can be uniquely defined by the vector Rti = Rti(si), as shown
in Figure 7.1. The orientation of the trajectory coordinate system at this location
with respect to the track coordinate system can be defined, as described in Chapter 3,
using the three Euler angles ψ ti, θ ti, and φ ti about the three axes Z ti, –Y ti, and –X ti,
respectively. These three Euler angles can be uniquely defined in terms of the arc
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Specialized Railroad Vehicle Formulations
257
FIGURE 7.1 Trajectory coordinates.
length si, and their sequence is selected to be consistent with what is used by the
railroad industry. The transformation matrix, expressed in terms of these Euler
angles, that defines the orientation of the trajectory coordinate system can be written
as follows:
A ti =  i ti
jti
 cos ψ ti cos θ ti

=  sin ψ ti cos θ ti
 sin θ ti

k ti 
(7.1)
− sin ψ cos φ + cos ψ sin θ sin φ
ti
ti
ti
ti
cos ψ cos φ + sin ψ sin θ sin φ
ti
ti
ti
ti
− cos θ sin φ
ti
ti
ti


cos ψ sin φ − sin ψ sin θ cos φ 
ti
ti

cos θ cos φ

− sin ψ sin φ − cos ψ sin θ cos φ
ti
ti
ti
ti
ti
ti
ti
ti
ti
ti
ti
Note that if the space curve geometry is specified, one has ψ ti = ψ ti(si), θ ti =
θ ti(si), and φ ti = φ ti(si). Since the trajectory coordinate system follows the body motion,
its velocity and acceleration depend on the forward velocity and acceleration of the
body. Using the Euler angle description, the angular velocity vector of the trajectory
coordinate system defined in the global coordinate system can be written in terms
of the time derivatives of Euler angles as
ω ti = Gtiθ ti
(7.2)
where
θ ti = [ψ ti
θ ti
φ ti ]T
(7.3)
and
0

ti
G = 0
1

sin ψ ti
− cos ψ ti
0
− cos ψ ti cos θ ti 

− sin ψ ti cos θ ti 

− sin θ ti

(7.4)
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Railroad Vehicle Dynamics: A Computational Approach
7.1.2 BODY COORDINATE SYSTEM
Using the trajectory coordinate system, whose location and orientation can be
uniquely defined by the generalized arc length coordinate si, the global position
vector of the center of mass of body i can be written as
R i = R ti + A ti u ir
(7.5)
where u ir is the position vector of the center of mass defined with respect to the
trajectory coordinate system. This vector is defined as
u ir = [0
y ir
z ir ]T
(7.6)
where yir and zir are, respectively, the center-of-mass coordinates in the lateral
direction and in a direction normal to the plane that contains the track space curve.
These coordinates are defined in and with respect to the origin of the trajectory
coordinate system, as shown in Figure 7.1. The location of the center of mass in the
longitudinal direction is assumed to be uniquely defined using the arc length si, and
for this reason, the first element of the vector u ir is selected to be zero.
To define the configuration of the body in the global system, a rotation matrix
Air is introduced. This matrix defines the orientation of the centroidal body coordinate
system X irY irZ ir with respect to the trajectory coordinate system using the three Euler
angles ψ ir, φ ir, and θ ir. These three Euler angles are defined with respect to the
trajectory coordinate system. This sequence of Euler angles was introduced in
Chapter 2 and leads to the following transformation matrix that is used to define the
orientation of the body coordinate system relative to the trajectory coordinate system:
Air =
 cos ψ ir cos θ ir − sin ψ ir sin φ ir sin θ ir

ir
ir
ir
ir
ir
 sin ψ cos θ + cos ψ sin φ sin θ

− cos φ ir sin θ ir

− sin ψ ir cos φ ir
cos ψ ir sin θ ir + sin ψ ir sin φ ir cos θ ir 
cos ψ cos φ
sin ψ ir sin θ ir − cos ψ ir sin φ ir cos θ ir 
ir
ir
sin φ ir

cos φ ir cos θ ir


(7.7)
Using the sequence of Euler angles chosen for ψ ir, φ ir, and θ ir, one can show
that the angular velocity of the body coordinate system defined in the global coordinate system can be written in terms of the time derivatives of the orientation
parameters as
ω ir = AtiGir θ ir
(7.8)
where
θ ir = [ψ ir
φ ir
θ ir ]T
(7.9)
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Specialized Railroad Vehicle Formulations
259
and
0

ir
G = 0
1

− sin ψ ir cos φ ir 

cos ψ ir cos φ ir 

sin φ ir

cos ψ ir
sin ψ ir
0
(7.10)
It is important to note that the matrix Gir becomes singular when cosφ ir = 0,
that is, when the roll angle approaches π/2. Such a singular configuration is not
encountered in most railroad vehicle simulation scenarios. The Euler angle singularities are avoided in the formulation presented in this chapter by judicious selection
of the sequence of rotations used in the definition of these angles.
7.1.3 GENERALIZED TRAJECTORY COORDINATES
As previously mentioned, the motion of the body with respect to the trajectory
coordinate system can be described using the translations yir and zir and the three
relative rotations ψ ir, φ ir, and θ ir. On the other hand, the motion of the trajectory
coordinate system can be uniquely defined in the track coordinate system using the
generalized arc length coordinate si. It follows that a total of six coordinates is
required to describe the general three-dimensional motion of the body in the global
coordinate system in terms of the generalized trajectory coordinates of body i, which
are defined as follows:
pi = [s i
yir
z ir
ψ ir
φ ir
θ ir ]T
(7.11)
Using these trajectory coordinates, the global position vector of an arbitrary point
on the body can be written as follows:
ri = R i + A i u i
(7.12)
where Ri is the position vector of the center of mass of body i defined by Equation
7.5, and Ai is the rotation matrix that defines the orientation of the body coordinate
system with respect to the global coordinate system. This matrix is defined by the
following successive rotations:
A i = A ti A ir
(7.13)
Recall that Ati is the rotation matrix that defines the orientation of the trajectory
coordinate system with respect to the global coordinate system, while Air is the
rotation matrix that defines the orientation of the body coordinate system with respect
to the trajectory coordinate system. The vector u i in Equation 7.12 is the position
vector of the arbitrary point on the body defined in the body coordinate system. This
vector is defined as
ui = [ x i
yi
z i ]T
(7.14)
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Therefore, Equation 7.12 can also be written using Equations 7.5 and 7.13 in the
following alternative form:
ri = R ti + A ti u ir + A ti A ir u i
(7.15)
This equation is expressed in terms of the six generalized trajectory coordinates
defined by Equation 7.11, since Rti = Rti(si), Ati = Ati(si), u ir = u ir ( y ir , z ir ) , and Air =
Air(ψ ir, φ ir, θ ir).
7.2 VELOCITY AND ACCELERATION
Using the expression for the general displacement given by Equation 7.15, the
velocity and acceleration equations of body i in terms of the trajectory coordinates
and their derivatives can be obtained. The velocity and acceleration equations will
be used with the Newton-Euler equations to obtain the nonlinear equations of motion
expressed in terms of the trajectory coordinates.
7.2.1 VELOCITY
OF THE
CENTER
OF
MASS
Differentiating the absolute position vector of the origin of the centroidal body
coordinate system defined by Equation 7.5 with respect to time, the absolute velocity
vector of the center of mass can be written as
ti u ir + A ti u ir
R i = R ti + A
(7.16)
The preceding equation can be expressed in terms of the generalized trajectory
velocities as
∂R ti i ∂A ti ir i
i i
s + i u s + A ti u ir = Lp
R i =
∂s
∂s i
(7.17)
where the vector ∂Rti/∂si is tangent to the space curve and can be determined from
the geometry data for a given si. Since this tangent vector is obtained by differentiation with respect to the arc length parameter si, it is a unit vector denoted as ∂Rti/∂si =
iti. The coefficient matrix Li in Equation 7.17 is a 3 × 6 coefficient matrix given as


∂A ti
Li =  iti + i u ir 
∂s


jti
k ti

0m 

(7.18)
The vectors jti and kti are, respectively, unit vectors along the Y ti and Z ti axes of the
trajectory coordinate system, that is, they are, respectively, the second and third
columns of the rotation matrix of Equation 7.1; 0m is a 3 × 3 null matrix, and p i is
the generalized velocity vector defined as
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p i = [si
y ir
zir
261
ψ ir
φ ir
θ ir ]T
(7.19)
Since the three Euler angles θti given by Equation 7.3 are functions of the arc length
si, the term ∂Ati/∂si can be written as
∂Ati ∂Ati ∂ψ ti ∂Ati ∂θ ti ∂Ati ∂φ ti
=
+ ti
+ ti
∂φ ∂s i
∂s i
∂ψ ti ∂s i
∂θ ∂s i
(7.20)
The angular velocity vector ω ti of the trajectory coordinate system with respect
to the global coordinate system can be written as

∂θ ti 
ω ti = Gtiθ ti =  Gti i  si
∂s 

(7.21)
where the matrix Gti is given by Equation 7.4 and
∂θ ti  ∂ψ ti
=
∂s i  ∂s i
∂φ ti 

∂s i 
∂θ ti
∂s i
T
(7.22)
Another route can be used, as discussed in Chapter 2, to define the angular
velocity vector ωti of the trajectory coordinate system. Since Ati is an orthogonal
matrix, one has
T
T
Ati Ati = Ati Ati = I
(7.23)
By differentiating this equation with respect to time, one can show that
ti AtiT = Atii AtiT si
ω ti = A
s
(7.24)
where ω ti is the skew symmetric matrix associated with the angular velocity ω ti.
7.2.2 ACCELERATION
OF THE
CENTER
OF
MASS
The absolute acceleration of the center of mass of the body can be obtained by
differentiating Equation 7.17 with respect to time. This leads to
i = Li p
i + γ iR
R
(7.25)
where the coefficient matrix Li is given by Equation 7.18, and the vector γ iR includes
all the terms that are quadratic in the generalized velocities. This vector is given as
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Railroad Vehicle Dynamics: A Computational Approach
 ∂ iti ∂2 Ati 
ti u ir
γ iR = L i p i =  i + i 2 u ir  (si )2 + 2 A
∂
s
∂
s


(7.26)
where ∂i ti/∂si = ∂2Rti/∂si2. A preprocessor that describes the space curve geometry
can be developed, as discussed in Chapters 3 and 6. This preprocessor can define
the vector Rti and the Euler angles ψ ti, θ ti, and φ ti and their derivatives for a given
value of the arc length si. The output generated by the preprocessor is used to define
the coefficients that appear in Equations 7.18 and 7.26.
7.2.3 ANGULAR VELOCITY
AND
ACCELERATION
Since a body has three relative rotational degrees of freedom with respect to the
trajectory coordinate system, the absolute angular velocity of body i can be written
in the global coordinate system as
ω i = ω ti + ω ir
(7.27)
where ω ti is the angular velocity vector of the trajectory coordinate system, and ω ir
is the angular velocity vector of the body coordinate system with respect to the
trajectory coordinate system. Using Equations 7.8 and 7.21, the preceding equation
can be written in terms of the generalized velocity vector p i as
ω i = Gtiθ ti + AtiGir θ ir = Hi p i
(7.28)
where Hi is a 3 × 6 coefficient matrix defined as

∂θ ti 
Hi =  Gti i 
∂s 

0v

( AtiGir ) 

0v
(7.29)
and 0v is a three-dimensional zero vector. The absolute angular velocity vector given
by Equations 7.27 or 7.28 can also be defined in the body coordinate system as
T
ω i = Ai ω i = Hi p i
(7.30)
where
T
T
T
Hi = A i Hi = A ir A ti Hi
(7.31)
Substituting Equation 7.29 into Equation 7.31, one obtains
 T
∂θ ti 
Hi =  Air Gti i 
∂s 

0v
0v

Gir 

(7.32)
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Specialized Railroad Vehicle Formulations
263
where
Gti = A ti Gti ,
Gir = A ir Gir
(7.33)
and

sin θ ti

Gti =  − cos θ ti sin φ ti
 cos θ ti cos φ ti

0
− cos φ ti
− sin φ ti
−1

0 ,
0 
 − cos φ ir sin θ ir

sin φ ir
Gir = 
 cos φ ir cos θ ir

cos θ ir
0
sin θ ir
0

1
0 
(7.34)
The absolute angular acceleration of the body defined in the global coordinate
system can be obtained by differentiating Equation 7.28 with respect to time. This
leads to
i + γ αi
α i = Hi p
(7.35)
where γ αi includes all the terms that are quadratic in the generalized velocities. This
vector is given by

∂2θ ti ∂Gti ∂θ ti  i 2  ti ir ∂Ati ir i  ir
γ αi =  Gti i 2 + i
(s ) +  A G + i G s  θ
∂s
∂s ∂s i 
∂s



(7.36)
Note that the term ∂Gti/∂si can be written as
∂Gti ∂Gti ∂ψ ti ∂Gti ∂θ ti ∂Gti ∂φ ti
=
+ ti
+ ti
∂φ ∂s i
∂ψ ti ∂s i
∂θ ∂s i
∂s i
(7.37)
The absolute angular acceleration vector can also be defined in the body coordinate system as
T
i + γ αi
α i = Ai α i = H i q
(7.38)
where the quadratic velocity terms are defined in the body coordinate system as
T
γ αi = Ai γ αi
(7.39)
The definition of the absolute angular acceleration vector in the body coordinate
system will be used in the following section with the Newton-Euler equations to
develop the dynamic equations in terms of the trajectory coordinates.
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7.3 EQUATIONS OF MOTION
Using Equations 7.25 and 7.38, one can write
i   Li  i  γ iR 
R
+  i 
 i  =  ip
 γ α 
 α  H 
(7.40)
This equation can also be rewritten in the following form:
i + γ i
a ip = Bi p
(7.41)
iT α iT ]T , γ i = [ γ iT γ iT ]T , and Bi is a velocity transformation matrix,
where a ip = [R
α
R
which is defined as
 ti ∂Ati ir 
 i + i u 
∂s
 Li  

i
B =  i = 
ti
 H    ir T ti ∂θ 
A G


∂s i 

jti
0v
kti
0v

0m 


Gir 


(7.42)
Having obtained the absolute Cartesian and the angular accelerations of the body
expressed in terms of the time derivatives of the generalized trajectory coordinates,
one can use Newton-Euler equations to derive the equations of motion in terms of
the generalized trajectory coordinates pi. Recall from Chapter 2 that Newton-Euler
equations of motion are given as
 mi I

 0
i  

Fei
0  R
= i
i
i  i 
i
i 
Iθθ   α   Me − ω × ( Iθθ ω ) 
(7.43)
where mi is the mass of the rigid body, I is a 3 × 3 identity matrix, Iθθi is the inertia
tensor defined with respect to the centroidal body coordinate system, Fei is the resultant of the external forces defined in the global coordinate system, and Mie is the
resultant of the external moments defined in the body coordinate system. Substituting
Equation 7.40 into Newton-Euler equations defined by Equation 7.43 and premultiplying by the transpose of the velocity transformation matrix Bi of Equation 7.42,
one obtains the equations of motion expressed in terms of the generalized trajectory
coordinates as
i = Qipe + Qipv
Mip p
(7.44)
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265
where
T
T





Iθθi γ αi + ω i × ( Iθθi ω i ) 

Mip = m i Li Li + Hi Iθθi Hi
T
T
Qipe = Li Fei + Hi Mie
T
Qipv = − m i Li γ iR − Hi
T
{
(7.45)
}
One can show by further examination of Equation 7.44 that the generalized mass
matrix Mip in the case of curved tracks includes inertia coupling between the translational and rotational trajectory coordinates. The Newton-Euler equations (Equation
7.43), on the other hand, do not include inertia coupling between the Cartesian
translation and rotational coordinates.
Another very important difference between the absolute-Cartesian coordinate
based and trajectory coordinate based formulations is the required degree of
differentiability with respect to the geometric surface parameters. It is clear from
the analysis presented in this section that the evaluation of the angular acceleration
vectors in terms of the trajectory coordinates requires the second derivatives of the
track Euler angles with respect to the longitudinal surface parameter (arc length). The
Euler angles of the track are functions of the curvature, which is obtained using the
second derivative of the position vector. Therefore, the use of the trajectory coordinates without imposing any contact constraint conditions requires the use of fourth
derivatives with respect to the geometric parameters (Rathod and Shabana, 2007).
Recall that when the contact constraint conditions are imposed in the absoluteCartesian coordinate based formulations, only third derivatives with respect to the
geometric parameters need to be evaluated. When an elastic contact approach is
used, a lower degree of differentiation is required when the absolute Cartesian
coordinates are used. For this reason, the use of the trajectory coordinates in railroad
vehicle system dynamics requires a higher degree of smoothness to avoid numerical
problems, particularly in the case of curved track simulations.
7.4 TRAJECTORY COORDINATE CONSTRAINTS
One of the important advantages of using the generalized trajectory coordinates in
railroad vehicle system dynamics is the simplicity of imposing kinematic constraints
on the motion of the body with respect to the track. For example, in railroad vehicle
system applications, a wheelset is often assumed to be driven using a specified
forward velocity V along the track. In such a case, the constant forward velocity
constraint can be defined in terms of the trajectory coordinate si as
C = s i − V t − s0i = 0
(7.46)
where s0i is the initial arc length coordinate. Using the preceding constraint equation,
the origin of the trajectory coordinate system is constrained to move along the track
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centerline with a constant velocity si = V . It is important to note that such a constraint
is linear when it is expressed in terms of the trajectory coordinates. Similarly,
kinematic constraints can also be imposed on the orientation of the body with respect
to the track coordinate system. For example, if it is required to constrain the yaw
motion of a body with respect to the trajectory coordinate system, the following
simple linear constraint equation is defined in terms of the yaw angle with respect
to the trajectory coordinate system:
C = ψ ir − ψ 0ir = 0
(7.47)
where ψ 0ir is assumed to be the yaw angle of the body with respect to the trajectory
coordinate system at the initial configuration. Using the preceding constraint equation,
the yaw angle of the body always remains the same as the initial yaw angle.
The simple trajectory coordinate constraint equations, as the ones given by
Equations 7.46 and 7.47, can be written in the following general form and can be
used for all six of the generalized trajectory coordinates:
C = pli − f (t ) = 0
(7.48)
where pli denotes the lth component of the trajectory coordinate vector pi defined
by Equation 7.11, and f(t) is an explicit function of time if the driving constraint is
simple, as shown by the examples of Equations 7.46 and 7.47. Note that the preceding
equation is linear in the generalized trajectory coordinates, and as a result, the
constraint Jacobian matrix becomes constant, and the quadratic velocity vectors that
arise when the constraints are imposed at the acceleration level are always equal to
zero. As will be discussed later in this section, imposing the trajectory coordinate
constraints when the dynamic equations are formulated in terms of the absolute
Cartesian coordinates leads to more complex and nonlinear expressions.
7.4.1 NUMERICAL EXAMPLE
To demonstrate the use of the trajectory coordinate constraints discussed in this
section, a numerical example of a single wheelset is considered, as shown in Figure
7.1. The wheelset mass used in this example is assumed to be 1568 kg, and the
moments of inertia are assumed to be Iyy = 168 kg⋅m2 and Ixx = Izz = 656 kg⋅m2. The
products of inertia are assumed to be equal to zero. The wheels are assumed to be
profiled with approximate conicity of 1/40, while the rail profile is assumed to be
of the American Railway Engineering Association (AREA) type. The track preprocessor discussed in Chapters 3 and 6 is used to generate the track data that enter
into the formulation of the velocity transformation matrix and the quadratic velocity
vectors defined by Equation 7.40. The track is assumed to consist of a 30 m tangent
segment followed by a 30 m spiral segment used to define a 1.5° constant curve
segment, as shown in Figure 7.2. A 1.25 mm super-elevation is assumed on the right
and left rails with respect to the center reference space curve. The longitudinal
motion of the wheelset is specified by imposing a constraint on the trajectory
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Specialized Railroad Vehicle Formulations
267
FIGURE 7.2 Curved track.
coordinate si. Using this constraint, the wheelset is assumed to travel with a constant
forward velocity V = 5 m/sec along the tangent to the track centerline. This speed
is above the balance speed for this curve. The embedded constraint contact formulation is used to model the tread contact between the wheel and rail, while the elastic
approach is employed to model the flange contact. (For more details on the contact
formulations, see Chapter 5.) The numerical results are obtained using the computer
program SAMS2000/Rail.
Figure 7.3 shows the arc length coordinate si as a function of time. The results
presented in this figure demonstrate that the wheelset is traveling along the track
FIGURE 7.3 Traveled distance of the wheelset.
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FIGURE 7.4 Normal contact forces on the right wheel (-- tread contact, -●- flange contact).
with the specified constant velocity of 5 m/sec. Figure 7.4 shows the vertical contact
forces on the right wheel. It can be seen from this figure that the wheel flange comes
in contact when the wheelset is in the spiral region. At the beginning of the simulation, the right wheel impacts the rail due to the yaw rotation. The wheel then
rebounces and continues with a steady-state flange contact. The lateral displacement
of the wheelset with respect to the trajectory coordinate system is presented in Figure
7.5. The results presented in this figure show that the wheelset starts moving laterally
once it enters into the spiral region, and the lateral displacement approaches a
constant value when the wheelset has the flange contact and is in the constant curve
region.
FIGURE 7.5 Lateral displacement with respect to the track coordinate system.
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7.4.2 USE
OF THE
269
CARTESIAN COORDINATES
As demonstrated in this section, the use of the generalized trajectory coordinates
makes it easy to impose simple constraints on the motion of the vehicle system with
respect to the track. In fact, the use of the trajectory coordinates leads to simple
linear constraint equations. When the absolute Cartesian coordinates are used, on
the other hand, the trajectory coordinate constraint equations given by Equation 7.48
become highly nonlinear functions of the absolute Cartesian coordinates. This is
mainly due to the nature of the nonlinear relationship between the absolute Cartesian
and the trajectory coordinates. To demonstrate the complexity of these constraint
equations when the absolute Cartesian generalized coordinates are used, the constraints on the translational trajectory coordinates piR = [s i yir z ir ]T are first considered. In this case, the trajectory coordinate constraint can be written as
Cl = pli − f (t ) = 0 ,
l = 1, 2, 3
(7.49)
where pli denotes the lth component of the trajectory coordinate vector p iR . In an
absolute Cartesian coordinate formulation, the trajectory coordinate pli must be
expressed in terms of the absolute coordinates. Using Equation 7.5, one has the
following relationship between the two sets of translational coordinates:
g (s i , y ir , z ir ) = R i − R ti − A ti u ir = 0
(7.50)
This equation can be used to determine the trajectory coordinates si, yir, and zir for
given translational Cartesian coordinates R i = [ Rxi Ryi Rzi ]T . Note that since the preceding equation is a nonlinear function of the trajectory coordinates si, yir, and zir, an
iterative procedure such as the Newton-Raphson method must be used to determine
the three translational trajectory coordinates. Therefore, imposing the constraints at
the position level in an absolute coordinate formulation requires the solution of a
nonlinear system of algebraic equations, a step that is not required when the equations are formulated in terms of the trajectory coordinates.
Differentiating Equation 7.50 with respect to time, the time derivatives of the
trajectory coordinates si, yir, and zir can be written in terms of the time derivatives
of the Cartesian coordinates Ri as
i
p iR = Ψ i R
(7.51)
where p iR = [si y ir zir ]T , and Ψi is the 3 × 3 velocity transformation matrix defined
as

∂Ati 
Ψ =  iti + i u ir 
∂s


i
ti
j

k 

ti
−1
(7.52)
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Differentiating Equation 7.51 with respect to time, one obtains
i + Ψ
iR
i
iR = Ψ i R
p
(7.53)
iR
i includes all the terms that are quadratic in the generalized velocities.
where Ψ
Using Equation 7.51, the Jacobian matrix of the constraints of the type given by
Equation 7.49 can be obtained. This Jacobian matrix associated with the Cartesian
coordinates Ri is written for a constraint on a trajectory coordinate pli , l = 1, 2, 3, as
∂Cl  i
= ψ l1
∂R i 
ψ li3 
ψ li2
(7.54)
i
where ψ lm
refers to the l-th row and m-th column component of the matrix Ψi. Using
Equation 7.53, the quadratic velocity term that appears in the constraint equation at
the acceleration level is given by
(Q )
d l
iR
i) +
= − (Ψ
l
∂2 f (t )
∂t 2
(7.55)
iR
i.
iR
i ) refers to the l-th component of the quadratic velocity vector Ψ
where (Ψ
l
Accordingly, the constraint equation at the acceleration level is written as
3
∑ψ
m =1
i
lm
( )
mi = Qd
R
l
(7.56)
It is clear from the preceding equations that the constraint Jacobian matrix is a
nonlinear function of the generalized Cartesian coordinates Ri. Furthermore, the
constraint quadratic velocity term (Qd)l is a highly nonlinear function when the
absolute Cartesian coordinates are used. Recall that the constraint Jacobian matrix
for these simple trajectory constraints is constant and that the quadratic velocity
term (Qd)l is zero when the generalized trajectory coordinates are used.
Complexities are also encountered when imposing constraints on the orientation
of the body with respect to the track coordinate system in an absolute-Cartesian
coordinate based formulation. In this case, the trajectory coordinate constraint equations can be written as follows:
Cl = θ lir − θ 0irl = 0
(7.57)
where θ lir refers to the l-th component of the orientation trajectory coordinates
θ ir = [ψ ir φ ir θ ir ]T , and θ 0lir is the value of this coordinate at the initial configuration.
Clearly, the preceding equation leads to a simple constraint when the trajectory coordinates are used. To demonstrate that this is not the case when the absolute coordinates
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271
are used, we first write the matrix that defines the orientation of the body coordinate
system with respect to the trajectory coordinate system as follows:
T
A ir = A ti A i
(7.58)
Using this equation and assuming that the matrix Ai is known in terms of the absolute
orientation coordinates, one can extract the Euler angles ψ ir, φ ir, and θ ir for the given
absolute Cartesian rotation coordinates θi and the arc length si, which is used to
determine Ati. The arc length si can be uniquely determined for given values of the
absolute generalized coordinates Ri, as previously demonstrated in this section by
solving a set of nonlinear algebraic equations (Equation 7.50). Using Equation 7.27,
the angular velocity of the body with respect to the trajectory coordinate system can
be written as
ω ir = ω i − ω ti
(7.59)

∂θ ti 
θ ir = ( AtiGir )−1  Giθ i − Gti i si 
∂s 

(7.60)
This leads to
where ω i = Giθ i , ω ti = Gtiθ ti , and ω ir = AtiGir θ ir . Note that since the time derivative
of the arc length si is given by the first equation of Equation 7.51, the preceding
equation can be rewritten in terms of the time derivatives of the absolute generalized
Cartesian coordinates as
θ it = Φ i q i
(7.61)
i θ i ]T , and Φi is a velocity transformation matrix defined as
where q i = [R

∂θ ti i
Φ i = ( AtiGir )−1  −Gti i ψ 11
∂s

−Gti
∂θ ti i
ψ 12
∂s i
−Gti
∂θ ti i
ψ 13
∂s i

Gi  (7.62)

Differentiating Equation 7.61 with respect to time, one obtains
it = Φ i q
i + θ i q i
θ
(7.63)
i q i includes all the terms that are quadratic in the first derivatives of the
where Φ
generalized absolute Cartesian coordinates.
Using Equation 7.61, the constraint Jacobian matrix associated with the Cartesian coordinates can be written as
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Railroad Vehicle Dynamics: A Computational Approach
∂Cl  i
= Φ l1
∂qi 
Φ li2
Φ lnii 
q 
…
(7.64)
where the index l refers to the constrained rotation coordinate (see Equation 7.57);
Φilm refers to the l-th row and m-th column component of the matrix Φi; and nqi is
the number of absolute Cartesian generalized coordinates of body i, where nqi = 7 if
four Euler parameters are used to describe the orientation of the body coordinate
system in the absolute Cartesian coordinate formulation. Using Equation 7.63, the
quadratic velocity term that appears in the constraint equation at the acceleration
level is given as
(Q )
d l
i q i )l
= −(Φ
(7.65)
i q i . The constraint equa i q i )l refers to the l-th component of the vector Φ
where (Φ
tion at the acceleration level can then be written as
nqi
∑Φ
m =1
( )
q = Qd
i
i
lm m
l
(7.66)
which clearly shows the complexity of the trajectory constraints at the acceleration
level when the absolute Cartesian coordinates are used.
7.5 SINGLE-DEGREE-OF-FREEDOM MODEL
In the analysis of the train longitudinal force interactions, one-degree-of-freedom
vehicle models have frequently been used. To develop these models, the following
simplifying assumptions are often made:
1. The center of mass of the vehicles follows the track geometric centerline,
and the vehicle has no translation degrees of freedom with respect to the
trajectory coordinate system.
2. The vehicle has no rotation degrees of freedom with respect to the trajectory coordinate system.
Using these simplifying assumptions, one can write the equations of motion of each
vehicle in the train in terms of the arc length coordinate si only. In this special case,
the global position vector of the center of mass of body i can be written using
Equation 7.5 as
R i = R ti + A ti u ir0
(7.67)
where the vector u ir0 is the local position vector of the center of mass with respect
to origin of the trajectory coordinate system. This vector is constant because of
assumption 1 and is given as
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273
FIGURE 7.6 Single-degree-of-freedom model.
u ir0 = [0
y0ir
z0ir ]T
(7.68)
where y0ir and z0ir are, respectively, the constant coordinates of the center of mass in
the lateral direction and in a direction normal to the plane that contains the space
curve, as shown in Figure 7.6. Differentiating Equation 7.67 with respect to time,
the absolute velocity vector of the center of mass can be written as
ti u ir = Li si
R i = R ti + A
0
(7.69)
where
Li = iti +
∂A ti ir
u0
∂s i
(7.70)
and iti = ∂R ti/ ∂s i . Furthermore, since there are no relative rotations with respect to
the trajectory coordinate system (see assumption 2), the angular velocity of body i
defined in the body coordinate system can be written as
ω i = ω ti = Gtiθ ti = Hi si
(7.71)
where
Hi = Gti
∂θ ti
∂s i
(7.72)
and

sin θ ti

Gti =  − cos θ ti sin φ ti
 cos θ ti cos φ ti

0
− cos φ ti
− sin φ ti
−1

0
0 
(7.73)
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Railroad Vehicle Dynamics: A Computational Approach
θti/∂si is as previously defined by Equation 7.22. Using
In Equation 7.72, ∂θ
Equations 7.69 and 7.71, the acceleration vector can be written as follows:
i   Li  i  γ iR 
R
s + i 
 i  =  i   γ α 
 α  H 
(7.74)




 ti ∂2θ ti ∂Gti ∂θ ti  i 2 
i
+ i
(s ) 
γα = G
2
∂s ∂s i 

∂s i

(7.75)
where
 ∂ iti ∂2 Ati ir  i 2
u o  (s )
γ iR =  i +
2

 ∂s
∂s i
Substituting Equation 7.74 into the Newton-Euler equations and premultiplying by
the transpose of the coefficient of si , one obtains the equation of motion of the onedegree-of-freedom vehicle model in terms of the generalized arc length coordinate
si as
i
i
+ Q pv
M pi st = Q pe
(7.76)



i
iT i
iT
i
Q pe = L Fe + H Me


i
i iT i
iT
i i
i
i
i
Q pv = − m L γ R − H [ Iθθ γ α + ω × ( Iθθ ω )] 

(7.77)
where
T
T
M ip = m i Li Li + Hi Iθθi Hi
EXAMPLE 7.1
Derive the equations of motion of an arbitrary body i in a train if the track geometry
is a function of the angle ψ ti, while the other two angles, θ ti and φ ti, are zero everywhere.
Solution. Since the shape of the space curve can be uniquely described by the angle
ψ ti, the coefficient matrix Li given by Equation 7.70 is defined as
 cos ψ ti 


∂A ir ∂ψ
Li = i ti +
uo
= (1 − yoirψ sti )  sin ψ ti 
∂ψ ti
∂s i
 0 


ti
ti
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275
where ψ sti = ∂ψ ti / ∂s i and
 cos ψ ti 


i ti =  sin ψ ti 
 0 


and
 − sin ψ ti

=  cos ψ ti
∂ψ ti
 0

∂A ti
− cos ψ ti
0
− sin ψ ti
0
0

0 
The coefficient matrix H i given by Equation 7.72 is defined as
0
 
H =G
= 0
∂s i  ti 
ψ s 
i
ti
∂θ ti
where
0

G = 0
 1
ti
0
−1
−1
0
0 
0

Substituting the preceding equations into Equation 7.77, one obtains
T
T
M pi = m i Li Li + H i Iθθi H
= m i (1 − yoirψ sti ) 2 + I zzi (ψ sti ) 2
The quadratic velocity vector can also be written as
 ∂ i ti ∂ 2 Ati ir  i 2
u o  (s )
γ iR =  i +
∂s i 2
 ∂s


 sin ψ ti 
 cos ψ ti  
 ti

ir
ti 
ti 
ir
ti 
= − ψ s (1 − yo ψ s )  − cos ψ  + yo ψ ss  sin ψ ti   (si ) 2

 0 
 0 





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where
∂ 2 A ti
∂s i 2
=
∂A ti ∂ 2ψ ti
∂ψ ti ∂s i 2
+
∂ 2 A ti  ∂ψ ti 
2
∂ψ ti 2  ∂s i 
and
 0 
 ti ∂ 2θ ti ∂G ti ∂θ ti  i 2   i 2
γ = G
(s ) =  0  (s )
+
∂s i 2
∂s i ∂s i 

ψ ssti 
i
α
where ∂G ti / ∂s i = 0, and ψ ssti = ∂ 2ψ ti / ∂s i 2 . Using the preceding equations, the generalized force vectors defined in Equation 7.77 are written as
T
T
i
Q pe
= Li Fei + H i M ie = (1 − y0irψ sti )( Fexi cos ψ ti + Feyi sin ψ ti ) + ψ sti M ezi
and
i
Q pv
= − m i Li γ iR − H i  Iθθi γ αi + ω i × ( Iθθi ω i )  =  m i y0ir (1 − y0irψ sti ) − I zziψ sti  ψ ssti (si ) 2
T
T


where ω i × ( Iθθi ω i ) = 0, Fexi , and Feyi are components of the vector Fei that are defined
in the global coordinate system, and M ezi is the z component of the moment vector M ie
defined in the body coordinate system. Accordingly, the equation of motion of body i
traveling on a curve defined by the angle ψ ti can be written as
[m i (1 − y0irψ sti ) 2 + I zzi (ψ sti ) 2 ] s i = (1 − y0irψ sti )( Fexi cos ψ ti + Feyi sin ψ ti ) + ψ sti M ezi
+  m i y0ir (1 − y0irψ sti ) − I zziψ sti  ψ ssti (si ) 2
If the lateral displacement y0ir of the center of mass of the body with respect to the
origin of the trajectory coordinate system is assumed to be zero, the preceding equation
reduces to
[m i + I zzi (ψ sti ) 2 ] s i = Fexi cos ψ ti + Feyi sin ψ ti + ψ sti M ezi − I zziψ stiψ ssti (si ) 2
Furthermore, in the special case of a tangent track where the angle ψ ti and its derivatives
are equal to zero, the preceding equation further reduces to
m i s i = Fexi
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277
7.6 TWO-DEGREE-OF-FREEDOM MODEL
One of the important problems in railroad vehicle system dynamics is the hunting
of the wheelsets due to the coupling between the lateral and yaw motions that result
from the creep forces. For this reason, two-degree-of-freedom wheelset models that
have the lateral displacement and the yaw angle as independent coordinates are often
used to examine the fundamental dynamic characteristics of railroad vehicle systems.
Such a two-degree-of-freedom model can be obtained from the general trajectory
coordinate formulation presented in this chapter by imposing constraints on the other
four trajectory coordinates: the arc length coordinate si, the vertical displacement
zir, the roll angle φ ir, and the pitch angle θ ir. Assuming that the wheelset has a constant
forward velocity V and an angular velocity Ω = V/r0, where r0 is the nominal wheelset
rolling radius, one has the following trajectory coordinate constraint equations:



ir
ir
C2 = θ − Ω t − θ 0 = 0 
C1 = s i − Vt − s0ti = 0
(7.78)
If the wheels are always in contact with the rails, the vertical displacement of the
wheelset and the roll angle become functions of the lateral and yaw displacements
only and can be written as follows:
C3 = z ir − f ( yir , ψ ir ) = 0 

C4 = φ ir − g( yir , ψ ir ) = 0 
(7.79)
It is assumed in some specialized railroad dynamic formulations that the vertical
and roll displacements are dependent on the lateral displacement only, since the
dependence of the vertical displacement and roll angle on the yaw is of the second
order (Garg and Dukkipati, 1988). Using this assumption, the preceding equation
reduces to
C3 = z ir − f ( yir ) = 0 

C4 = φ ir − g( yir ) = 0 
(7.80)
One can develop a look-up table to determine zir and φ ir for a given coordinate
y , which is the generalized coordinate that can be determined from the solution of
the dynamic differential equations of motion. Since the wheels and rails are profiled,
the look-up table can be used to determine the contact points for a given value of the
lateral displacement yir (Cooperrider et al., 1975). Recall that in the general constraint
contact formulation, zir and φ ir can also be selected as the dependent coordinates,
since this contact formulation leads to two contact constraint equations imposed on
the right and left wheels. These two constraint equations can be used to eliminate
ir
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two degrees of freedom. However, in general, the vertical displacement and roll
angle are dependent not only on the lateral displacement and yaw angle, but also
on the arc length and pitch angle, since the global position vector at the contact
point P is a function of all the trajectory coordinates, as is clear from Equation 7.15,
and the fact that the nonconformal contact constraints are functions of the global
position vector rPi . These contact constraint equations presented in the preceding
chapters for a contact k are repeated here for convenience:
k
rPi − rPj 


C k (p i , p j , s ik , s jk ) =  t1i ⋅ n j  = 0
 ti ⋅ n j 

 2
(7.81)
where pi and pj are, respectively, the generalized trajectory coordinates of the
wheelset (body i) and the rail (body j), and sik and sjk are, respectively, the nongeneralized surface parameters of bodies i and j used to describe the wheel and rail profiles
for a contact k. The use of the preceding nonconformal contact constraint equations
leads to a more general procedure for determining the contact points between the
wheels and rails for the two-degree-of-freedom model discussed in this section.
Nonetheless, the simplifying assumptions described by Equation 7.80 are often made
in the literature to simplify the dynamic formulation.
The four constraint equations defined by Equations 7.78 and 7.80 can be written
in a vector form as
 s i − Vt − s0i 
 ir

θ − Ω t − θ 0ir 
C =  ir
=0
 z − f ( yir ) 
 ir

 φ − g( yir ) 
(7.82)
Differentiating this equation twice with respect to time, the constraint equations at
the acceleration level are given as


si


ir
θ

= 
C
=0
 z ir − f y yir − fd 


φir − g y yir − gd 
(7.83)
∂2 f
∂f
f ( y ir ) = ir yir + ir 2 ( y ir )2 = f y yir + f d
∂y
∂y
(7.84)
where
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279
where f y = ∂f / ∂yir and fd = (∂2 f / ∂yir 2 )( y ir )2 . Similar relationships are obtained for
the function g as
g( y ir ) =
∂g ir ∂2 g ir 2
y + ir 2 ( y ) = gy yir + gd
∂y
∂y ir
(7.85)
where g y = ∂g/ ∂yir and gd = (∂2 g /∂yir 2 )( y ir )2 .
As previously discussed in Chapter 2, one can systematically eliminate the
dependent coordinates and the constraint forces using the embedding technique.
Since four trajectory coordinates that include the arc length, vertical displacement,
and roll and pitch angles are constrained, the independent degrees of freedom are
reduced to two. Hence, the dependent and independent coordinates can be written,
respectively, as
pid = [s i
z ir
φ ir
θ ir ]T , pii = [ yir
ψ ir ]T
(7.86)
Using the constraint equations at the acceleration level, the system accelerations can
be written in terms of the independent accelerations yir and ψ ir as

0
 si  

 ir   ir
y   y

  yir + fd 
z ir   f y i

=  ir  = 
p
ir

ψ   ψ

ir
 φir   g y
+
g
d
   y
ir

0
 θ  
(7.87)
This equation can also be written in the following form:
i = Bi p
ii + γ i
p
(7.88)
ii = [ where p
yir ψ ir ]T and
0

1
 fy
Bi = 
0

 gy
 0
0

0
0
,
1

0
0 
0
 
0
f 
γi =  d 
0
g 
 d
 0 
(7.89)
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Substituting Equation 7.88 into Equation 7.44 and premultiplying by the transpose
of Bi, one obtains two equations of motion expressed in terms of the two independent
coordinates yir and ψ ir as
ii = (Qipe )i + (Qipv )i
Mipi p
(7.90)
where
T
T
T
Mipi = Bi MipBi , (Qipe )i = Bi Qipe , (Qipv )i = Bi (Qipv − Mip γ i )
(7.91)
The generalized mass matrix Mip and the generalized force vectors Qipe and Qipv are
as previously defined by Equation 7.45.
7.7 LINEAR HUNTING STABILITY ANALYSIS
As demonstrated in the preceding section, the dynamic equations of the two-degreeof-freedom wheelset models that have the lateral displacement and the yaw angle
as independent coordinates can be systematically obtained using the trajectory coordinate formulation (Equations 7.90 and 7.91). These equations, in their most general
form, are nonlinear. Nonetheless, by using linearization and simplifying assumptions, one can obtain in a straightforward manner, as a special case of the general
nonlinear formulation, a linear two-degree-of-freedom wheelset model that has been
used in the literature in the hunting stability analysis. In the linearized and simplified
wheelset model, a wheel with conical profile and a knife-edged rail are used.
Furthermore, a tangent track is assumed for simplicity, and as a consequence of this
assumption, the orientation matrix of the track coordinate system is the identity
matrix. Therefore, the global position vector of the contact point P on the wheelset
w can be defined as a special case of Equation 7.15 as follows:
rPwk = R w + A w u wk
P
(7.92)
where superscript w refers to the wheelset and superscript k denotes the contact
number (here k = 1 is used for the right wheel and k = 2 for the left wheel); Rw is
the vector of reference coordinates defined by R w = [s w y wr z wr ]T ; u wk
P is the local
position vector of the contact point P with respect to the origin of the wheelset
coordinate system; and Aw is the matrix that defines the orientation of the wheelset
in terms of the three Euler angles θ w = [ψ wr φ wr θ wr ]T , as shown in Figure 7.7.
Using the assumptions of a tangent track, the velocity transformation matrix Bw,
defined using Equation 7.42 in the case of a general space curve, reduces to
 Lw   I
Bw =  w  = 
H  0
0 

Gw 
(7.93)
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281
FIGURE 7.7 Single wheelset.
where I and 0 are, respectively, 3 × 3 identity and null matrices. The quadratic
T
T
velocity vector γ w = [ γ wR γ αw ]T defined by Equations 7.26 and 7.39 also reduces to
 γ w   0v 
γ w =  wR  =  w wr 
 γ α  G θ 
(7.94)
where 0v is a 3 × 1 zero vector. Substituting Equations 7.93 and 7.94 into Equations
7.44 and 7.45, the equations of motion of the wheelset traveling on a tangent track
can be obtained as
w = Qwpe + Qwpv
Mwp p
(7.95)
 Few  
Qwpe =  T

w
w
G Me  


0v



w ww
w
w w
Iθθ G θ + ω × ( Iθθ ω ) 



(7.96)
where
m wI
M wp = 
 0

Qwpv =  wT
 −G


,
wT w w
G Iθθ G 
0
{
}
Since a tangent track is used, the three translational coordinates sw, ywr, and zwr
represent the absolute Cartesian coordinates, and the orientation of the track coordinate system coincides with that of the global coordinate system. In this special
case, the preceding equations are the same as the standard rigid body Newton-Euler
equations of motion discussed in Chapter 2. Therefore, the use of the geometric
profile assumptions for the wheel and the rail stated previously in this section, the
assumption of the tangent track, and the trajectory coordinate constraints given by
Equation 7.82 (with the preceding equations) lead to a special form of Equations
7.90 and 7.91 for the two-degree-of-freedom wheelset model, which can be written as
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Railroad Vehicle Dynamics: A Computational Approach
wi = (Qwpe )i + (Qwpv )i
Mwpi p
(7.97)




w
wT w
(Q pe )i = B Q pe



w
wT
w
w ω
(Q pv )i = B (Q pv − M p γ ) 

(7.98)
iw = [ where p
y wr ψ wr ]T and
T
M wpi = B w M wp B w
T
T
In this equation, Jijw = G w Iθθw G w , M iw = Gw Mwe , and
0

1
 fy
Bw = 
0

 gy
 0
0

0
0
,
1

0
0 
0
 
0
f 
γw =  d 
0
g 
 d
 0 
(7.99)
As discussed in Chapter 4, the use of Kalker’s linear creep theory leads to the
following expression for the longitudinal, lateral, and spin creep forces:
Fx = − f33ζ x
Fy = − f11ζ y − f12ϕ
M z = f12ζ y − f22ϕ






(7.100)
where f11, f12, f22, and f33 are the creep coefficients that depend on the dimensions of
the contact ellipse and Poisson’s ratio; and ζx , ζy, and ϕ are the longitudinal, lateral,
and spin creepages. Since the creep forces are defined in the contact coordinate
system for given values of the creepages, an appropriate coordinate transformation
must be applied when these forces are used in the equations of motion. Using the
assumption that the lateral displacement and the yaw angle of the wheelset are small
with respect to the equilibrium configuration, Equations 7.97 and 7.98 can be
simplified and written as
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283
w
w
w
0   y wr   Fey + Fez f y + M x g y 
=





I zzw  ψ wr   M zw − I yywφ wrθ wr 
mw

 0
(7.101)
where Feyw and M zw are the resultants of the lateral force and the yaw moment due to
the creep forces applied at the right and left wheels, while the vertical force Fezw is
due to the effect of the gravitational force, and M xw is associated with the roll motion.
In the simple model discussed in this section, the pith velocity θ wr is assumed to be
constrained such that θ wr = V /r0 for a constant forward velocity V and nominal rolling
radius r0.
In defining the creepages at the contact points, it is convenient to introduce the
intermediate body coordinate system Awi that does not rotate with the wheelset about
its Yw-axis. Using this coordinate system, the global position vector at the kth contact
point P can be written as
rPwk = R w + A wi u wik
P
(7.102)
where the transformation matrix Awi that defines the orientation of the wheelset
intermediate body coordinate system is given as (see Chapter 8 for more details)
A wi
 cosψ wr

=  sin ψ wr
 0

− sin ψ wr cos φ wr
cosψ wr cos φ wr
sin φ wr
sin ψ wr sin φ wr 

cosψ wr sin φ wr 

cos φ wr

(7.103)
and the local position vector u wik
P of point P at the contact point k defined in the
intermediate body coordinate system is given as
u wik
P = [0
ak
−r k ]T
(7.104)
In this equation, ak is the lateral distance of the contact point with respect to the
center of the wheelset, while rk is the rolling radius at the contact point. In the case
of conical wheels, these variables are defined for the right (k = 1) and left (k = 2)
contact points as
a1 = − a,
a 2 = a,
r 1 = r0 − γ y wr 

r 2 = r0 + γ y wr 
(7.105)
where a is half the gage, and γ is the conicity of the wheel, as shown in Figure 7.7.
Differentiating the global position vector given by Equation 7.102 with respect to
time, the absolute velocity vector at the contact point k is defined in the wheelset
intermediate coordinate system as
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284
Railroad Vehicle Dynamics: A Computational Approach
T
w + ω wi × u wik
rPwk = A wi R
P
(7.106)
where ω wi is the absolute angular velocity vector of the wheelset defined in the
intermediate coordinate system, and it is given by (see details in Chapter 8)
θ wr + ψ wr sin φ wr
ω wi = [φ wr
ψ wr cos φ wr ]T
(7.107)
As will be shown in Chapter 8, the use of the simplifying assumption,
ψ wr sin φ wr = 0 and cos φ wr = 1 leads to the following linearized angular velocity:
ω wi [φ wr
θ wr
ψ wr ]T
(7.108)
Furthermore, using the assumption of infinitesimal lateral and vertical displacements
and yaw and roll angles in the transformation matrix Awi, the velocity vector given
by Equation 7.106 can be written as
rPwk
Vxwik  
  − a kψ wr − r kθ wr 
V
 wik   wr

 
= Vy  =  y − Vψ wr  + 
r kφ wr

k wr
wr
V wik  

 
z
a
φ
 
 z  

(7.109)
Using this equation, the longitudinal, lateral, and spin creepages are obtained as



ζ yk = Vywik /V


ϕ k = ω wi ⋅ n wik /V 
ζ xk = Vxwik /V
(7.110)
where n wik is a unit normal at the contact point defined in the wheelset intermediate
coordinate system and, therefore, this unit normal is a function of the contact angle
δ, which is equal to the conicity γ in the case of conical wheels. Substituting Equation
7.109 into Equation 7.110 and using Equation 7.105, the following linearized creepage expressions are obtained for the right wheel (k = 1):
ζ 1x =
a wr γ wr
ψ + y
V
r0
ζ 1y =
1 wr r0 − γ y wr wr
y +
φ − ψ wr
V
V
ϕ1 =
ψ wr + γ θ wr
V










(7.111)
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285
and for the left wheel (k = 2),
ζ x2 = −
a wr γ wr
ψ − y
V
r0
ζ y2 =
1 wr r0 + γ y wr wr
φ − ψ wr
y +
V
V
ϕ2 =
ψ wr − γ θ wr
V










(7.112)
In deriving these equations, n wi1 = [0 γ 1]T and n wi 2 = [0 −γ 1]T are used, and the
assumption θ wr = V /r0 is used. Substituting Equations 7.111 and 7.112 into Equation
7.100, the creep forces for the right and left wheels can be obtained. For small lateral
and yaw displacements, it is further assumed that the creep coefficients that depend
on the dimensions of the contact ellipse are equal for the right and left contacts.
This leads to the following simplified creep forces associated with the wheelset ywr
and ψ wr coordinates:
2
Feyw =
∑
(− f11ζ yk − f12ϕ k )
k =1
2
M zw = a(− f33ζ 1x + f33ζ x2 ) +
∑( f ζ
k
12 y
k =1
− f22ϕ k )







(7.113)
Note that the longitudinal creep forces applied at the right and left wheels create a
yaw moment. Substituting the linearized creepage expressions defined by Equations
7.111 and 7.112 into the preceding equation, one obtains the following expression
for the creep force and moment:
Feyw = −
M zw =
2 f11  r0γ  wr
2f
1+
y + 2 f11ψ wr − 12 ψ wr
V
V 
a 

a
2 f12  r0γ  wr
2f
γ
1+
y − 2 f12ψ wr − 22 ψ wr − 2af33  ψ wr + y wr 


V 
a 
V
r0

V



 (7.114)



In deriving these force and moment equations, the following relation is used to
express the time rate of the roll angle φ wr in terms of the lateral velocity y wr in the
creepage expressions defined by Equations 7.111 and 7.112:
γ
φ wr = y wr
a
(7.115)
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 7.8 Suspended wheelset.
Substituting Equation 7.114 into Equation 7.101, one obtains the following linearized
equations of motion of the two-degree-of-freedom single-wheelset model:
 w
m


 0
2 f11  r0γ 
  wr  
1+
y  
0   V 
a 




+
  2f  rγ 
γV

I zzw  ψ wr   − 12  1 + 0  + I ywy

r0 a
a 
  V 

 kg1
+
 2af33γ

 r0


  y wr 


2( f22 + a 2 f33 )   wr 
 ψ 
V
 

2 f12
V
(7.116)


−2 f11   y wr   0 
 

= 

  
2 f12 + kg 2  ψ wr   0 
  

where kg1 and kg2 are the gravitational stiffnesses. For a suspended wheelset, such
as the one shown in Figure 7.8, the wheelset is connected to the frame by the springs
and dampers in the longitudinal and lateral directions. In this case, the preceding
equations of motion can be modified, leading to
 w
m


 0
  wr   2 f11  1 + r0γ  + 2c
y  
y
0   V 
a 
+

  2f  rγ 
γV

I zzw  ψ wr   − 12  1 + 0  + I yyw

r0 a
V
a


 

 2k y + kg1
+
 2af γ
33

 r0


  y wr 


2

 wr 
2( f22 + a f33 )
2
+ 2cx b  ψ 
V
 

2 f12
V
  wr   
  y  0 

= 

  
2
wr
2 f12 + 2k x b + kg 2  ψ   0 
 
  
(7.117)
−2 f11
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287
where kx, ky and cx, cy are, respectively, the stiffness and damping coefficients of the
spring and dampers shown in Figure 7.8. In the remainder of this section, two simple
models are considered. In the first model, an unsuspended wheelset (no primary
suspension) is used; in the second model, a suspended wheelset connected to a frame
by a primary suspension is used.
7.7.1 MODEL 1
In this model, a single wheelset without primary suspension is considered; this is
the case of an unsuspended wheelset. It is known that, if the effects of gravity and
the spin creepage are neglected, such a model is unstable for any value of the forward
velocity, and its hunting frequency can be evaluated using Klingel's formula (Klingel,
1883):
f=
V
2π
γ
r0 a
(7.118)
This formula can be derived from purely kinematic considerations, as demonstrated
in Chapter 1. The eigenvalue analysis of the wheelset without a primary suspension
yields a solution that is very close to Klingel's solution at low speeds, as demonstrated
by the results presented in Figure 7.9.
FIGURE 7.9 Hunting frequencies (— Klingel’s formula, ---- Linear analysis). (From Valtorta,
D., Zaazaa, K.E., Shabana, A.A., and Sany, J.R., 2001, A study of the lateral stability of
railroad vehicles using a nonlinear constrained multibody formulation, in Proceedings of 2001
ASME International Mechanical Engineering Congress and Exposition, New York. With
permission.)
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Railroad Vehicle Dynamics: A Computational Approach
TABLE 7.1
Suspended Wheelset Model
Variable
Description
Value
mw
–
Ixxw
–
Iyyw
–w
Izz
kx
ky
cx
cy
2b
2a
r0
γ
Wheelset mass
Inertia moment
Inertia moment
Inertia moment
Longitudinal spring constant
Lateral spring constant
Longitudinal damping coefficient
Lateral damping coefficient
Distance between the longitudinal springs
Gage distance
Nominal rolling radius
Conicity
1568 kg
656 kg·m2
168 kg·m2
656 kg·m2
1.35E+5 N/m
2.50E+5 N/m
0 N/m·s
0 N/m·s
1.8 m
1.435 m
0.4566 m
1/40
7.7.2 MODEL 2
In this model, the wheelset is connected to a frame with primary suspensions, as
shown in Figure 7.8. This system can become unstable if the wheelset forward
velocity exceeds a certain speed, called the critical speed. Using the model data
presented in Table 7.1, the effect of the primary suspension and conicity on the
critical speed can be investigated. Figure 7.10 shows the root loci for different values
FIGURE 7.10 Root loci. (From Valtorta, D., Zaazaa, K.E., Shabana, A.A., and Sany, J.R.,
2001, A study of the lateral stability of railroad vehicles using a nonlinear constrained
multibody formulation, in Proceedings of 2001 ASME International Mechanical Engineering
Congress and Exposition, New York. With permission.)
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289
FIGURE 7.11 Effect of the conicity on the degree of stability. (From Valtorta, D., Zaazaa,
K.E., Shabana, A.A., and Sany, J.R., 2001, A study of the lateral stability of railroad vehicles
using a nonlinear constrained multibody formulation, in Proceedings of 2001 ASME International Mechanical Engineering Congress and Exposition, New York. With permission.)
of the speed V. It can be seen from the results presented in this figure that, as the
forward speed increases, the real part of the eigenvalues becomes positive (instability
region). The conicity γ of the wheel profile has an effect on the critical speed, as
demonstrated by the results presented in Figure 7.11. In linear control theory, the
ratio α /ω defines the degree of stability, where α and ω are the real and the imaginary
parts of the eigenvalue, respectively. The critical speed is reached when this ratio
approaches zero. It is clear that, by increasing the conicity, the wheel/rail force
introduces more energy into the system, thereby causing the instability. This instability phenomenon can be easily observed at a low speed by choosing low values
for the stiffness of the primary suspension. For low speeds, the hunting frequency
can still be predicted using Equation 7.118. The linear theory can be considered
only as a first step in the study of the stability. In fact, when the amplitude of the
lateral motion increases in the case of higher values of the conicity, one must use
other techniques such as the quasi-linearization technique (Cooperrider et al., 1976),
which gives better results for large lateral displacements, or use nonlinear dynamic
simulations based on the methods discussed throughout this book.
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8
Creepage Linearization
It is known that railroad vehicle dynamic models that employ kinematic linearization
can predict significantly different dynamic responses compared with models that are
based on fully nonlinear kinematic and dynamic equations. This is particularly true
for simulations at high speeds. To analytically examine this problem and study
the effect of the approximations used in the linearized railroad vehicle models, the
fully nonlinear kinematic and dynamic equations of a wheelset — obtained using
the methods discussed in preceding chapters — are summarized in this chapter. The
linearized kinematic and dynamic equations used in some railroad vehicle models
are obtained from the fully nonlinear model to shed light on the assumptions and
approximations used in the linearized models. The assumptions of small angles that
are often made in developing railroad vehicle models and their effect on the angular
velocity, angular acceleration, and the inertia forces are investigated. The velocity
creepage expressions that result from the use of the assumptions of small angles are
obtained and compared with the fully nonlinear expressions. Newton-Euler equations
for the wheelset are presented, and their dependence on Euler angles and their time
derivatives is discussed. The effect of the linearization assumptions on the form of
Newton-Euler equations is examined. A suspended wheelset model is used as an
example to obtain the numerical results required to quantify the effect of the linearization. The results presented in this chapter show that linearization of the creepages
can lead to significant errors in the values predicted for the longitudinal and tangential forces as well as the spin moment. There are also significant differences between
the nonlinear and the linearized models in the prediction of the lateral and vertical
forces used to evaluate the L/V ratios.
8.1 BACKGROUND
It is known that the dynamics of railroad vehicle systems can be very sensitive to
changes in the system parameters. It is also known that railroad vehicle models that
are based on linearized kinematic equations can predict a dynamic response that is
significantly different from that predicted using the fully nonlinear models. A study
that compared different computer codes that are based on different formulations, some
of which employ linearized creepage expressions, showed that the difference in the
predicted critical speed between linearized and nonlinear railroad vehicle models can
be very significant, exceeding 21 m/sec for some models (Iwnicki, 1999). The findings
presented by Iwnicki (1999) and summarized in Table 8.1 for two different vehicle
models are significant, since the prediction of the critical speed is one of the main
objectives of using railroad vehicle codes and computer formulations. Inaccurate
prediction of critical speed can have serious consequences and can negatively impact
the accuracy of predicting derailment scenarios and the evaluation of safety criteria.
291
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Railroad Vehicle Dynamics: A Computational Approach
TABLE 8.1
Critical Speed as Calculated by Five Computer Codes for Two
Vehicle Models (m/sec)
Vehicle1
Vehicle2
VAMPIRE™
GENSYS™
SIMPAK™
ADAMS™
NUCARS™
74
58
77.05
70.5
70
80
72
75
79
79
Source: Iwnicki, S., The Manchester Benchmarks for Rail Vehicle Simulation, Supplement to Vehicle System Dynamics, Taylor & Francis, London, 1999.
The dynamics and stability of railroad vehicles strongly depend on the wheel/rail
interaction. Among the parameters required to evaluate the force of interaction
between the wheel and the rail are the wheel and rail profile geometry data, the
materials of the wheel and the rail, and the creepages that depend on the relative
velocities between the wheel and the rail. The creepages are multiplied by very high
creepage coefficients to determine the longitudinal and tangential creep forces as
well as the spin moment. Approximations used in the definition of the creepages
can lead to dynamic and stability results that differ from those predicted using the
nonlinear model, particularly at high speeds. Figure 8.1 shows the suspended
wheelset used as the study model in this chapter. The inertia and material properties
of this model are presented in Table 8.2. The wheelset is assumed to travel with a
constant forward velocity of 69 m/sec on a tangent track with a lateral deviation, as
shown in Figure 8.1. Figure 8.2 shows the lateral displacement of the suspended
wheelset obtained using linearized and fully nonlinear models. It is clear from the
results presented in this figure that the two solutions have significant amplitude
differences. Figure 8.3 shows the spin creepage at the right wheel obtained using
the two models. It is clear that, while the spin creepage is small in this example,
the error in the linearized model exceeds 100%. The numerical results presented in
this chapter for the fully nonlinear model are obtained using the constraint contact
formulation discussed in Chapter 5 and implemented in the SAMS2000/Rail computer program.
In this chapter, the assumptions and approximations used in linear railroad vehicle
models are explained and examined by first presenting the fully nonlinear equations.
The assumptions of small angles are then used to obtain the linearized equations.
The focus will be on identifying the terms that are neglected in the linearized
kinematic creepage expressions that contribute to the significant differences in the
amplitude of the lateral displacement in some simulation scenarios. This study will
shed light on the sources of the differences in the critical speeds predicted by
computer codes that are developed based on linearized and fully nonlinear models.
In Section 8.2, the general expression of the angular velocity is presented, since
understanding its derivation and the definition of its components in different coordinate systems plays a central role in the analysis presented in this chapter. In
Section 8.3, the sequence of Euler angles commonly employed in railroad vehicle
dynamic formulations is reviewed and used to obtain the absolute angular velocity
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293
FIGURE 8.1 Suspended wheelset and the track deviation.
vector defined in different coordinate systems. The angular velocity vector is defined
in the wheelset coordinate system, the global coordinate system, and an intermediate
wheelset coordinate system. It is shown that, regardless of the coordinate system
used, the angular velocity is a nonlinear function of Euler angles, this despite the fact
that the expression of the absolute angular velocity vector becomes relatively simpler
when it is defined in the wheelset intermediate coordinate system. In Section 8.4,
the nonlinear creepage expressions are presented and used to obtain the linearized
longitudinal and tangential creepages that are used in railroad vehicle formulations.
In Section 8.5, the spin creepage is discussed, and the approximations used to obtain
the simplified spin creepage used in some codes are examined. In Section 8.6, the
effect of the assumptions used in the linearized models on the dynamic NewtonEuler equations of motion is investigated. The results presented in this section show
that the linearization of the kinematic creepage expressions can lead to significant
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Railroad Vehicle Dynamics: A Computational Approach
TABLE 8.2
Data Used in the Suspended Wheelset Model
Variable
Description
Value
mw
Ixx
Iyy
Izz
W
kx
ky
cx
cy
2b
2a
Wheelset mass
Inertia moment
Inertia moment
Inertia moment
Applied vertical load
Stiffness for longitudinal springs
Stiffness for lateral springs
Damping coefficient for longitudinal spring
Damping coefficient for lateral spring
Distance between longitudinal springs
Gage value
1568 kg
656 kg·m2
168 kg·m2
656 kg·m2
98000 N
1.35 × 105 N/m
2.5 × 105 N/m
0 N/m·s
0 N/m·s
1.8 m
1432 mm
errors in the values predicted for the longitudinal and lateral creep forces as well as
the spin moment. The errors in the lateral and vertical forces that enter into the
calculations of the L/V ratio as the result of using the linearized model are found
to be significant, and these are reported in Section 8.6. These results shed light on
the significant effect of the linearization on the L/V ratio that is often used in
derailment criteria (Shabana et al., 2006). In Section 8.7, summary and conclusions
drawn from this study are presented.
FIGURE 8.2 Lateral displacement of the wheelset (— Nonlinear, --- Linearized).
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Creepage Linearization
295
FIGURE 8.3 Spin creepage for the right wheel (— Nonlinear, --- Linearized).
8.2 TRANSFORMATION AND ANGULAR VELOCITY
In this section, the general equations of the absolute angular velocity and angular
acceleration vectors are reviewed. A sequence of Euler angles is then chosen in the
following section to define the absolute angular velocity in terms of the derivatives
of the orientation parameters. The angular velocity is also defined in the intermediate
wheelset coordinate system that is often used in railroad vehicle computer codes. It
is shown in this chapter that, regardless of the coordinate system used to define the
components of the angular velocity, these components are not the time derivatives
of Euler angles. Nonetheless, the components of the angular velocity vector defined
in any coordinate system can be written as linear functions of the time derivatives
of Euler angles.
8.2.1 MATRIX IDENTITIES
Some of the basic matrix and vector identities that will be used in the definition of
the angular velocity are first reviewed. While some of these identities were presented
in Chapter 2, they are reviewed here for convenience. Recall that if A is an orthogonal
matrix, one has
A T A = AA T = I
(8.1)
This matrix identity can be used to define the components of the angular velocity
vector in different coordinate systems. For instance, differentiating the matrix product ATA with respect to time and using the preceding equation, one obtains
T A + AT A
=0
A
(8.2)
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Railroad Vehicle Dynamics: A Computational Approach
This equation shows that
(
= −A
T A = − AT A
AT A
)
T
(8.3)
A matrix that is equal to the negative of its transpose is a skew symmetric matrix.
is a skew symmetric matrix denoted as ω , i.e.,
Therefore, A T A
 0

A A = ω =  ω3
 −ω 2

T
ω2 

−ω 1 
0 
−ω 3
0
ω1
(8.4)
In this chapter, a bar over a vector or a matrix is used to indicate a vector or a
matrix whose components are defined in a body or local coordinate system. Similarly,
one can differentiate the second part of Equation 8.1 with respect to time and follow
T is
a procedure similar to the one used to obtain Equation 8.4 to show that AA
another skew symmetric matrix, that is,
 0

T
AA = ω =  ω 3
 −ω 2

−ω 3
0
ω1
ω2 

−ω 1 
0 
(8.5)
It is clear from Equations 8.4 and 8.5 that ω = Aω AT .
Another important identity is associated with skew symmetric matrices and the
cross-product. The cross-product of two vectors a = [a1 a2 a3]T and b = [b1 b2 b3]T
can always be written as
= −ba
a × b = ab
(8.6)
where a and b are skew symmetric matrices defined as
 0

a =  a3
 − a2

− a3
0
a1
a2 

− a1  ,
0 
 0
b =  b
 3
 −b2

−b3
0
b1
b2 

−b1 
0 
(8.7)
The simple matrix and vector identities introduced in this section will be used
to define the general expression of the absolute velocity vector expressed in terms
of the global and local components of the angular velocity.
8.2.2 DEFINITION
OF THE
ANGULAR VELOCITY
The position vector of an arbitrary point on the wheelset can be defined in the global
coordinate system (as described in Chapter 2) as follows:
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297
rw = R w + A w u w
(8.8)
where Rw is the global position vector of the center of mass of the wheelset, Aw is
the orthogonal transformation matrix that defines the orientation of the wheelset
coordinate system, u w is the local position vector of the arbitrary point with respect to
the wheelset coordinate system, and superscript w refers here to the wheelset.
Differentiating the preceding equation with respect to time, the absolute velocity
vector of the arbitrary point on the wheelset can be written as
w uw
r w = R w + A
(8.9)
Using Equations 8.4, 8.5, and 8.6, the preceding equation can be written in the
following two equivalent forms:
(
)
w + Aw ω w × u w 
r w = R


w + ω w × uw

r w = R

(8.10)
where ω w and ω w are called the angular velocity vectors defined, respectively, in the
wheelset coordinate system and the global coordinate system, and
uw = Aw uw
(8.11)
Note that in the preceding two equations, the cross-product can be replaced by
a skew symmetric matrix multiplied by a vector, as shown by Equations 8.6 and 8.7.
It is important in the analysis presented in this chapter to note the following:
1. As shown in Chapter 2 of this book, the transformation matrix is, in
general, a highly nonlinear function of the coordinates used to describe
the orientations of the bodies in space. This fact is clear, particularly when
three independent Euler angles are used to describe the finite rotations. It
follows from Equations 8.4, 8.5, and 8.10 that the angular velocity vectors
can be highly nonlinear functions of the orientation coordinates. This
comment, however, does not apply in the case of Euler parameters, since
in this particular case, the angular velocity vectors are linear functions of
the four Euler parameters. However, if a dependent Euler parameter is
eliminated in order to use three independent parameters, the angular
velocity vectors again become highly nonlinear functions of the remaining
Euler parameters.
2. While the angular velocity vectors are, in general, highly nonlinear functions of the orientation coordinates, Equations 8.4 and 8.5 show that the
angular velocity vectors are linear functions of the time derivatives of
the orientation coordinates. This is always the case, since the definition
of the angular velocity vectors requires one differentiation with respect
to time.
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Railroad Vehicle Dynamics: A Computational Approach
Using these two remarks, it is clear that the angular velocity vector defined in
the wheelset and the global coordinate systems can always be written, as described
in Chapter 2, as
w
ω w = G wθ 

w
ω w = G wθ 
(8.12)
where Gw and Gw are two matrices that can be nonlinear functions of the orientation
coordinates θw.
8.3 EULER ANGLES
It is clear from the analysis presented in the preceding section that the angular
velocity vector is not, in general, equal to the derivatives of the orientation parameters, and the form of the angular velocity vector depends on the orientation parameters
used. If Euler angles are used, the expression of the angular velocity depends on
the sequence of rotations used to define the Euler angle transformation matrix. In
railroad vehicle dynamics, it is convenient to use the following sequence: a rotation
ψ about the wheel Zw axis, followed by a rotation φ about the Xw axis, followed by
a third rotation θ about the Yw axis. These three angles in the order given are the
yaw, roll, and pitch angles. By using this sequence, the singularity associated with
Euler angles can be avoided, since, in railroad vehicle applications, the yaw and the
roll angles remain small.
Using the sequence of rotations introduced in this section, the simple rotation
matrices associated with the three Euler angles are given by
 cos ψ

A z =  sin ψ
 0

1

Ax = 0
0

− sin ψ
cos ψ
0
0 

− sin φ  ,
cos φ 
0
cos φ
sin φ
 cos θ

Ay =  0
 − sin θ

0

0 ,
1 
0
1
0
(8.13)
sin θ 

0 
cos θ 
The product of these three matrices defines the following wheelset transformation
matrix
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299
Aw = Az Ax A y
 cos ψ cos θ − sin ψ sin φ sin θ

= sin ψ cos θ + cos ψ sin φ sin θ

− cos φ sin θ

cosψ sin θ + sin ψ sin φ cos θ 

sin ψ sin θ − cos ψ sin φ cos θ 

cos φ cos θ

− sin ψ cos φ
cos ψ cos φ
sin φ
(8.14)
in terms of the Euler angles
θ w = ψ
φ
θ 
T
(8.15)
As pointed out in Chapter 2, the matrices Gw and Gw that are used to define the
angular velocity vectors can be determined by differentiation of Equation 8.14 with
respect to time and substituting, respectively, into Equations 8.4 and 8.5. A second,
simpler approach that can be used to determine the matrices Gw and Gw, which is
also discussed in Chapter 2, is to recognize the columns of these matrices as unit
vectors along which the three Euler angle rotations are performed. The columns of
the matrix Gw are unit vectors, defined in the wheelset coordinate system, along
which the three Euler rotations are performed. The columns of the matrix Gw are
the same unit vectors defined in the global coordinate system. This second approach
for determining the matrices Gw and Gw is the one that is used in this chapter, since
it clearly shows the dependence of the expressions of the angular velocities on the
sequence of rotations used in defining Euler angles. Nonetheless, the final obtained
results should be the same as the results that can be obtained using Equations 8.4
and 8.5. Using this second approach, it can be shown that the matrices Gw and Gw
are given by
 − cos φ sin θ

G =  sin φ
 cos φ cos θ

w
cos θ
0
sin θ
0

1 ,
0 
0

G = 0

1
w
cos ψ
sin ψ
0
− sin ψ cos φ 

cos ψ cos φ  (8.16)

sin φ

Using these two matrices, one obtains the following expressions for the absolute
angular velocity vector defined, respectively, in the wheelset and the global coordinate systems:
 −ψ cos φ sin θ + φ cos θ 


ω =
ψ sin φ + θ
,
 ψ cos φ cos θ + φ sin θ 


w
φ cos ψ − θ sin ψ cos φ 


ω = φ sin ψ + θ cos ψ cos φ  (8.17)


ψ + θ sin φ


w
This equation shows that the components of the angular velocity vector are not, in
general, the derivatives of Euler angles. Therefore, the derivatives of Euler angles
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Railroad Vehicle Dynamics: A Computational Approach
do not represent the components of the angular velocity vectors; these angle derivatives only enter into the formulation of the components of the angular velocity. It
is also important to point out that, in the fully nonlinear models, the pitch velocity is
predicted by integration of the pitch acceleration, while in some linearized railroad
vehicle models the pitch velocity is obtained using the assumption that the wheelset
forward velocity is predefined. The use of this assumption makes it difficult to study
braking, traction, and skidding scenarios (Handoko, 2006).
The angular velocity vector can be defined in a coordinate system that does not
rotate with the wheelset about its Yw axis with the angle θ. This coordinate system,
which was briefly discussed in Chapter 5, will be referred to in this chapter as the
wheelset intermediate coordinate system. Using this intermediate coordinate system,
this vector of absolute angular velocity ωwi can be obtained from ω w and ωw and the
simple rotation matrices defined by Equation 8.13 using one of the following relationships:
ω wi = A yω w ,
ω wi = A x T A z T ω w
(8.18)
This defines the vector ω wi as
ω wi = φ
θ + ψ sin φ
ψ cos φ 
T
(8.19)
This equation defines the matrix Gwi as
 0

G =  sin φ
 cos φ

wi
1
0
0
0

1
0 
(8.20)
Note that while this matrix takes a simpler form compared with the matrices Gw
and Gw of Equation 8.16, this matrix is not an identity matrix. Therefore, in the
wheelset intermediate coordinate systems, the components of the angular velocity
vectors are still not the derivatives of Euler angles. The matrix Gwi is required to
formulate the wheelset dynamic equations of motion using the intermediate wheelset
coordinate system, as will be demonstrated later in this chapter.
8.4 LINEARIZATION ASSUMPTIONS
As mentioned in the preceding section, and as is clear from Equation 8.19, the
components of the angular velocity vector ω wi defined in the intermediate wheelset
coordinate system are again not the time derivatives of Euler angles. Nonetheless,
using the two assumptions
ψ sin φ = 0 ,
cos φ = 1
(8.21)
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301
which imply that φ = 0, it is easy to see that, in this special case,
ω wi ≈ φ
θ
ψ 
T
(8.22)
If the assumption of Equation 8.22 is used, the following definitions for the absolute
angular velocity defined in the wheelset and the global coordinate systems can be
obtained:
(ω )
 −ψ sin θ + φ cos θ 


= Ay ω = 
θ

 ψ cos θ + φ sin θ 


(ω )
φ cos ψ − θ sin ψ cos φ − ψ sin ψ cos φ 


= A z A x ω wi = φ sin ψ + θ cos ψ cos φ − ψ cos ψ sin φ 


ψ cos φ + θ sin φ


T
w
l
w
l
wi
(8.23)
where the subscript l refers to the linearized expressions. In obtaining the preceding
equation, no linearization assumptions are made in the transformation matrix that
defines the orientation of the intermediate wheelset coordinate system with respect
to the global coordinate system. Only linearization assumptions are used in the
definition of the angular velocity in the wheelset coordinate system (Equation 8.22).
The motivation for not applying the same assumptions to the transformation is to
make clear the difference between the expressions of the exact global angular
velocity and the global angular velocity obtained using Equation 8.22. The error in
the angular velocity as the result of the linearization used in Equation 8.22 can be
determined by comparing the expressions presented in the preceding equation with
the expressions given by Equation 8.17. For example, the error in the angular velocity
defined in the global system is given by
∆ω w = ψ  − sin ψ cos φ
− cos ψ sin φ
( cos φ − 1)
T
(8.24)
As previously pointed out, both assumptions of Equation 8.21 are automatically
satisfied if the roll angle φ is assumed to be equal to zero.
8.5 LONGITUDINAL AND LATERAL CREEPAGES
In this and the following section, the effect of the linearization assumptions discussed
in the preceding section on the expressions of the longitudinal, lateral, and spin
creepages is discussed. As discussed in Chapter 4, the longitudinal, lateral, and spin
creepages are defined using the following general expressions:
ζx
(r
=
w
c
− rcr
V
)
T
t1r
, ζy
(r
=
w
c
− rcr
V
)
T
t r2
,
(ω
ϕ=
w
− ωr
V
)
T
nr
(8.25)
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Railroad Vehicle Dynamics: A Computational Approach
where V is the nominal velocity of the wheel, subscript c refers to the contact point,
t1r and t r2 are the tangents at the contact point, nr is the normal at the contact point,
and superscript r refers to the rail. The tangent and normal vectors in the preceding
equation are functions of the wheel and rail geometry and their coordinates. Details
on the formulations of these vectors, in terms of the wheel and rail coordinates and
geometrical surface parameters when multibody system algorithms are used, are
provided in the preceding chapters.
The absolute velocity vector of the contact point on the wheelset can be written
as follows (See Equation 8.10):
(
)
w + Aw ω w × u w 
rcw = R
c 

w + ω w × uw

rcw = R
c

(8.26)
or alternatively as
(
w + A wi ω wi × u wi
rcw = R
c
)
(8.27)
where
w
A wi = A z A x , u wi
c = A y uc = 
0
ak
−rk 
T
(8.28)
In this equation, subscript k = R or L, with R referring to the right wheel and L
referring to the left wheel; a is the lateral position of the contact point; and r is the
instantaneous rolling radius. The lateral position a and the instantaneous rolling
radius r can be defined using the multibody system formulations discussed in the
preceding chapters of this book. In these formulations, the locations of the contact
points that are defined on-line can be used to determine both a and r. The
cross-product expression on the right-hand side of Equation 8.27 is given in a more
explicit form without using the linearization assumptions as
(
)
 −ψ ak cos φ + rk sin φ − θrk 


φrk
ω wi × u cwi = 



φ ak


(8.29)
It follows that the second term on the right-hand side of Equation 8.27 is explicitly
given by
(
)
A wi ω wi × u cwi =
(
(
)
)
(
(
)
)
)
 −ψ ak cos φ + rk sin φ cos ψ − θrk cos ψ − φ rk cos φ − ak sin φ sin ψ 

 (8.30)
 −ψ ak cos φ + rk sin φ sin ψ − θrk sin ψ − φ rk cos φ − ak sin φ cosψ 


φ rk sin φ + ak cos φ


(
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303
The use of this equation, with Equations 8.27 and 8.25, leads to the exact expression
for the longitudinal and lateral creepages.
Note also that if the assumptions of Equation 8.21 are used, one has
 −ψ ak − θrk 


ω ×u = 
φrk

 φak



wi
wi
c
(8.31)
and
A
wi
(ω
wi
×u
wi
c
)
 −ψ ak cos ψ − θrk cos ψ − φrk sin ψ 


=  −ψ ak sin ψ − θrk sin ψ − φrk cos ψ 


φak


(8.32)
In some railroad vehicle codes, when the creepage expressions are defined, the
pitch velocity θ is assumed to be known and is given by θ = V /r0 , where r0 is the
nominal wheelset rolling radius. The wheelset forward velocity V is assumed to
be predefined in some of these codes. In nonlinear multibody railroad vehicle codes,
on the other hand, there are no assumptions made with regard to the value of the
pitch velocity θ; therefore, the angular acceleration is not assumed to be constant.
For this reason, these general multibody system codes can be effectively used in the
analysis of braking, traction, and skidding scenarios. The assumption used in some
specialized railroad vehicle codes is that θ = V /r0 implies that the effect of the oscillations or perturbations in the angular velocity on the creepage expressions is negligible. Figure 8.4 and Figure 8.5 show a comparison between the right wheel
FIGURE 8.4 Longitudinal creepage of the right wheel (— Nonlinear, --- Linearized).
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Railroad Vehicle Dynamics: A Computational Approach
FIGURE 8.5 Lateral creepage of the right wheel (— Nonlinear, --- Linearized).
longitudinal and lateral creepages obtained using the linear and nonlinear models.
The linearized creepage expression is obtained by using the assumptions of Equations
8.21 and 8.30, in addition to assuming an infinitesimal yaw rotation (φ sin ψ = 0,
θ sin φ = 0, and ψ sin ψ = 0). These results are obtained using the suspended wheelset
model previously described in this chapter.
It is also important to reemphasize that the longitudinal and lateral creepage
expressions presented in Equation 8.25 are based on velocity components defined
along the axes of a coordinate system defined at the contact point. These expressions
are different from the velocity components defined along the axes of the wheelset
coordinate system. For instance, the component of the velocity of the contact point
on the wheel along the intermediate Xwi axis can be defined using Equation 8.29 as
follows:
(
)
Vxcwi = Vxwi − ψ ak cos φ + rk sin φ − θrk
(8.33)
w i wi , with iwi as a unit vector that defines the longitudinal axis of the
where Vxwi = R
wheelset intermediate coordinate system. Using the assumptions of Equation 8.21,
the preceding equation reduces to the following familiar form that is used in the
numerator of the longitudinal creepage expression used in some railroad vehicle
codes that are based on simplified formulations:
T
Vxcwi = Vxwi − ψ ak − θrk
(8.34)
Note that this simplified expression for the velocity component is not a function of
the parameters that define the coordinate system at the contact point and does not
depend on the yaw and roll angles of the wheelset.
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Creepage Linearization
305
8.6 SPIN CREEPAGE
A numerical comparison between the spin creepages obtained using the linearized
and the fully nonlinear models was presented in Figure 8.3, which showed that the
two solutions differ significantly. To examine the approximation used in the linearized expression of the spin creepage, the following scalar component is evaluated:
T
T
ω w n r = ω wi n wi
(8.35)
where nwi is the normal at the contact point defined in the wheelset intermediate
coordinate system. Since the locations of the contact points can be determined online using the multibody system formulations discussed in the preceding chapters,
one can define the tangent (lateral) at the contact point and use this tangent vector
with the wheelset axis to define the contact angle δ. The components of the normal
nwi in the preceding equation can be written in terms of the contact angle δ. For the
right and left wheels, the normal to the surface at the contact point can be defined
in the wheelset intermediate coordinate system as
n wi
R = 
0
T
n wi
L = 
0
cos δ R  ,
sin δ R
cos δ L 
− sin δ L
T
(8.36)
Using this equation and Equation 8.19, one has
(ω n ) = (ω
(ω n ) = (ω
wT
r
wi T
n wi
R
wT
r
) = (θ + ψ sin φ ) sin δ + (ψ cosφ ) cosδ 

) = − (θ + ψ sin φ ) sin δ + (ψ cosφ ) cosδ 
wi T
n
wi
L
R
R
R
L
L
(8.37)
L
Using the assumptions of Equation 8.21, the preceding equations reduce to
(ω n ) = (ω
(ω n ) = (ω
wT
r
wi T
n wi
R
wT
r
L
) = θ sin δ + ψ cosδ 

sin δ + ψ cos δ 
=
−
θ
)

R
R
wi T
n
wi
R
L
L
(8.38)
L
For fixed rails, the nonlinear expressions for the spin creepage are then given by
(ϕ )
(ϕ )
R
L
=
1
V
{(θ + ψ sin φ ) sin δ + (ψ cosφ ) cosδ }
R
{(
)
R
(
)
1
=
− θ + ψ sin φ sin δ L + ψ cos φ cos δ L
V
}






(8.39)
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306
Railroad Vehicle Dynamics: A Computational Approach
The linearized expressions, on the other hand, are given by
(ϕ )
(ϕ )
R
L
=
{
1 θ sin δ R + ψ cos δ R
V
}
{
1
=
−θ sin δ L + ψ co
os δ L
V
}






(8.40)
Note that Equation 8.39 can also be written as follows:
(ϕ )
(ϕ )
{
)}




1

=
−θ sin δ L + ψ cos δ L + φ 
V

=
R
(
1 θ sin δ R + ψ cos δ R − φ
V
{
L
(8.41)
)}
(
The spin creepage results presented in Figure 8.3 in the case of the linearized
model are obtained using Equation 8.40. The error in the spin creepages as the result
of using the linearized expression can be obtained using the preceding two equations
as
εR =
{
(
)
1
ψ sin φ sin δ R + ψ 1 − cos φ cos δ R
V
} 

1

εL =
−ψ siin φ sin δ L + ψ 1 − cos φ cos δ L 
V

{
(
)
}
(8.42)
Note that the linearized expressions of Equation 8.40 can be obtained by simply
using the assumptions of Equation 8.21 in the expression of the angular velocity of
Equation 8.19.
8.7 NEWTON-EULER EQUATIONS
The Newton-Euler equations can be expressed in terms of the absolute accelerations
of the center of mass, and the angular velocity and acceleration vectors defined in
the body coordinate system as follows:
w = F w
m wR



I wα w = M w − ω w × I wω w 

(
)
(8.43)
where mw is the total mass of the wheelset, Rw is the position vector of the center
of mass of the wheelset defined in the global coordinate system, Fw is the vector of
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Creepage Linearization
307
the resultant forces acting on the wheelset center of mass, Mw is the vector of Cartesian moments defined in the wheelset coordinate system, I w is the inertia tensor
defined in the wheelset coordinate system, and α w is the absolute angular acceleration vector defined in the wheelset coordinate system. The vector Fw is assumed to
contain the contact forces that can be introduced using the constraint contact formulation or the elastic contact formulation, as discussed in Chapters 5 and 6. Since
the angular acceleration defined in the wheelset coordinate system is used, the inertia
tensor I w is a constant symmetric matrix, and the preceding equation can be solved
for α w as
αw = I w
−1
{M
w
(
− ω w × I wω w
)}
(8.44)
It is important to point out that linearizing the angular velocity vector and the
use of small angle assumptions discussed in the preceding sections of this chapter
will, in turn, have an effect on the accuracy of the calculation of the angular acceleration. Furthermore, before using direct numerical integration methods to determine
the angles and velocities, one must first use the angular accelerations to determine the
second time derivatives of the angles. In the general fully nonlinear model, the angular
acceleration vector can be expressed in terms of the derivatives of Euler angles as
follows:
wθ w
w + G
α w = G wθ
(8.45)
Since Euler angles ψ and φ are assumed to be small, the singularity associated with
Euler angles is unlikely to be encountered, and the preceding equation can be solved
for the second time derivatives of Euler angles as follows:
(
wθ w
w = G w−1 α w − G
θ
)
(8.46)
These second derivatives of the angles can be integrated using direct numerical
integration methods to obtain the angles and their derivatives. Alternatively, one can
directly integrate the angular acceleration to obtain the angular velocity, which can
be used to determine the time derivatives of Euler angles. The time derivatives of
Euler angles can be integrated to determine the angles. Equation 8.46, however, is
often used in multibody system computer simulations, since the system is assumed
to be subjected to kinematic constraints that must be imposed at the position, velocity,
and acceleration levels. The same expression and dimensions for the constraint
Jacobian matrix can be used at all these levels if the equations of motion are
expressed in terms of the second derivatives of the angles instead of the angular
acceleration vector.
If Euler equations are defined in the intermediate coordinate system of the
wheelset, one then has
(
I wiα wi = M wi − ω wi × I wiω wi
)
(8.47)
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308
Railroad Vehicle Dynamics: A Computational Approach
where
I wi = A y T I w A y
(8.48)
If the products of inertia of the wheelset are equal to zero, one can show that the
preceding equation yields I wi = I w = constant, and the Euler equation leads to
α wi = I wi
−1
{M
wi
(
− ω wi × I wiω wi
)}
(8.49)
The second derivatives of the angles can be obtained using the following relationship
(
wiθ w
w = G wi −1 α wi − G
θ
)
(8.50)
Using Equation 8.20, the effect of the linearization in the preceding two equations
can be examined. In particular, the preceding equation clearly shows the relationship
between the second time derivatives of the angles and the angular acceleration vector.
Figure 8.6, Figure 8.7, Figure 8.8, and Figure 8.9 show a comparison between
the contact forces and spin moment predicted using the linearized and fully nonlinear
models. These figures show the normal force, the creep tangential longitudinal and
lateral contact forces, as well as the spin moment. The forces presented in these
figures are defined in a contact frame, which is a coordinate system defined by the
tangents and the normal at the contact point on the right wheel. These results show
significant differences between the values of the forces predicted using the linearized
and the fully nonlinear models. Figure 8.10 and Figure 8.11 show the errors in the
lateral and vertical forces defined in the track frame. These forces enter into the
calculation of the L/V ratio that is used in derailment criteria.
FIGURE 8.6 Normal contact force of the right wheel (— Nonlinear, --- Linearized).
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Creepage Linearization
309
FIGURE 8.7 Longitudinal contact force of the right wheel (— Nonlinear, --- Linearized).
FIGURE 8.8 Tangential contact force of the right wheel (— Nonlinear, --- Linearized).
8.8 CONCLUDING REMARKS
This chapter examined the effect of the approximations used in the linearized velocity
creepage expressions. The fully nonlinear expressions were obtained and then used to
derive the linearized creepage expressions that are used in some specialized railroad
vehicle formulations. The basic assumptions used to derive the linearized creepage
expressions are summarized by Equations 8.21 and 8.22. The numerical results
presented in this chapter show that the use of such assumptions, which are automatically satisfied if the roll angle is assumed to be zero, can significantly influence the
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310
Railroad Vehicle Dynamics: A Computational Approach
FIGURE 8.9 Spin moment of the right wheel (— Nonlinear, --- Linearized).
dynamics of the wheelset. This explains the significant differences between (a) the
railroad vehicle results predicted using computer codes that are based on fully
nonlinear formulations and (b) those codes that employ kinematic linearization of
the creepages. In particular, the results presented in this chapter clearly show that
the linearization of the kinematic creepage expressions can lead to significant errors
in the values predicted for the longitudinal and lateral creep forces as well as the
spin moment (Shabana et al., 2006). These results also show the errors in the lateral
and vertical forces defined in the track frame. These forces enter into the calculation
of the L/V ratio that is used in derailment criteria.
FIGURE 8.10 Error in the vertical contact force.
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Creepage Linearization
311
FIGURE 8.11 Error in the lateral contact force.
The results presented in this chapter, while identifying some of the serious
problems that can arise from the use of linearized railroad vehicle models, can also
be used to define the corrections that can be implemented in computer codes that
employ linearized formulations. These corrections, however, must be carefully
implemented to be consistent with the assumptions employed in defining the kinematic variables used to formulate the dynamic equations of motion.
45814_book.fm Page 312 Thursday, May 31, 2007 2:25 PM
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APPENDIX A
Contact Equations
This appendix provides derivations of some of the equations used in the Hertz contact
theory presented in Chapter 4. For an elastic half-space, the stress distribution and
the displacement can be determined by using the methods described by Boussinesq
and Cerruti (Boussinesq, 1885; Cerruti, 1882; Love, 1944). Consider an elastic halfspace as shown in Figure A.1. Assuming that the area due to the load is defined by
S and that the coordinates of an arbitrary point within this area are defined by C(ξ,η),
if A is a point within the solid, then
(ξ − x ) + (η − y )
2
CA = δ =
2
+ z2
(A.1)
It is assumed that the distribution of the normal pressure and the tangential
tractions on the area S are defined, respectively, by p, τx, and τy , which are functions
of ξ and η. Therefore, one can define the following potential functions:
Χ=
∫∫ ( )
S
ϒ=
∫∫ ( )
S
Ζ=

τ x ξ , η β d ξ dη 



τ y ξ , η β d ξ dη 



p ξ , η β d ξ dη 

(A.2)
∫∫ ( )
S
and

τ x ξ ,η ln(δ + z )dξ dη 

S


∂ϒ
ϒ=
=
τ y ξ ,η ln(δ + z )dξ dη 
∂z

S


∂Ζ
=
p ξ , η ln(δ + z )dξ dη 
Ζ=
∂z

S
Χ=
∂Χ
=
∂z
∫∫ ( )
∫∫ ( )
(A.3)
∫∫ ( )
313
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314
Railroad Vehicle Dynamics: A Computational Approach
FIGURE A.1 Contact area on an elastic half-space.
where β is defined as β = zln(δ + z) − δ, and δ is given by Equation A.1. Furthermore,
assume that ψ is defined as follows:
ψ=
∂Χ ∂ϒ ∂Ζ
+
+
∂x
∂y ∂z
(A.4)
ψ=
∂Χ ∂ϒ ∂Ζ
+
+
∂x ∂y ∂z
(A.5)
and
One can then show that the elastic displacement of point A can be expressed in
terms of the functions given in Equations A.2 through A.5 as follows (Love, 1944):
1  ∂Χ ∂Ζ
∂ψ  
∂ψ
2
−z
+ 2ν
−


4π G  ∂z
∂x  
∂x
∂x

1  ∂ϒ ∂Ζ
∂ψ
∂ψ  
2
−
+ 2ν
−z
uy =

4π G  ∂z ∂y
∂y
∂y  

1  ∂Ζ
∂ψ 

+ 1 − 2ν ψ − z
uz =
2

∂z 
4π G  ∂z

ux =
(
(A.6)
)
where G and ν are, respectively, the material modulus of rigidity and Poisson’s ratio.
45814_book.fm Page 315 Thursday, May 31, 2007 2:25 PM
Contact Equations
315
If a purely normal pressure p(ξ,η) is applied on a frictionless contact, the
functions Χ , Χ , ϒ , and ϒ are equal to zero. If these assumptions are used in Equations A.4, A.5, and A.6, one obtains
ψ =Ζ =
∫∫ p (ξ,η) ln (δ + z ) dξdη
(A.7)
S
ψ=
∂ψ
=
∂z
∫∫ p (ξ,η) δ dξdη
1
ux = −
1 
∂ψ
∂ψ 
(1 − 2ν )
+z

4π G 
∂x
∂x 
uy = −
1 
∂ψ
∂ψ 
(1 − 2ν )
+z

4π G 
∂y
∂y 
uz =
(A.8)
S











∂ψ 
1 
2(1 − ν )ψ − z

∂z 
4π G 
(A.9)
For a point on the surface (z = 0), Equation A.9 leads to
ux
uy
uz
z=0
z=0
z=0
=−
1 − 2ν  ∂ψ 
4π G  ∂x  z = 0
=−
1 − 2ν  ∂ψ 
4π G  ∂y  z = 0
=
1 − ν  ∂ψ 
2π G  ∂z  z = 0











(A.10)
If the contact area is assumed elliptical, as in Hertz theory, the applied normal
pressure can be written as follows:
  x 2  y 2 
p ( x, y ) = p0 1 −   +   
 b 
  a


n
(A.11)
and the contact area is defined as follows:
2
2
 x
 y
 a  +  b  − 1 = 0
(A.12)
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316
Railroad Vehicle Dynamics: A Computational Approach
Using the potential theory, the Boussinesq function (Equation A.5) for a general
point in the solid is given by
ψ ( x , y, z ) =
∞
Γ (n + 1)Γ (1 2)
p0 ab
Γ (n + 3 2)
∫

z2 
x2
y2
 1 − a2 + w − b2 + w − w 


n +1 2
(a 2 + w)(b 2 + w) w
γ1
dw (A.13)
where Γ is the gamma function, and γ1 is the positive root of the equation
x2
y2
z2
+ 2
+ =1
a +γ b +γ γ
(A.14)
2
If n in Equation A.11 is equal to
leads to
1
ψ ( x , y, z ) = π p0 ab
2
1
2
∞
∫
,
as in the case of Hertz theory, Equation A.13

x2
y2
z2 
1
−
−
−
 a2 + w b2 + w w 


(a 2 + w)(b 2 + w) w
γ1
(A.15)
dw
Note that, for a point on the surface, z = 0, and Equation A.15 can be written as
follows:
1
ψ ( x , y, 0) = π p0 ab
2
∞
∫
0

x2
y2 
 1 − a2 + w − b2 + w 


(a 2 + w)(b 2 + w) w
(A.16)
dw
Using Equations A.16 and A.10, the displacement along the z-axis within the
loaded region is given by
uz =
1− ν 2
( L − Mx 2 − Ny 2 )
πE
(A.17)
where E is the Young’s modulus of elasticity and
M=
π p0 ab
2
π p0 ab
N=
2
π p0 ab
L=
2
∞
∫
0
dw
(a 2 + w)3 (b 2 + w) w
∞
∫
0
dw
(a 2 + w)(b 2 + w)3 w
∞
∫
0
dw
(a + w)(b 2 + w) w
2
=
π p0 b
 Ke − Ee 
e 2a 2 
=
2

π p0 b  a 


−
E
K
e
e
e 2a 2  b 


= π p0 bKe













(A.18)
45814_book.fm Page 317 Thursday, May 31, 2007 2:25 PM
Contact Equations
317
where Ee and Ke are complete elliptical integrals of argument e = 1 − a 2 / b 2 for b > a,
as shown in Appendix B. Note that, since the pressure distribution is semi-ellipsoidal,
the total normal load Fn is given by
Fn =
2
p0π ab
3
(A.19)
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APPENDIX B
Elliptical Integrals
The complete elliptical integrals presented in Chapter 4 are given as follows:
π 2
Be =
∫
0
π 2
cos 2 w
1 − e 2 sin 2 w
dw =
∫
0
cos 2 w
cos2 w − g 2 sin 2 w
π 2
Ce =
∫
−
dw
3
sin 2 w cos2 w(1 − e 2 sin 2 w) 2 dw
0
π 2
∫
De =
0
sin 2 w
1 − e 2 sin 2 w
dw
π 2
Ee =
∫
1 − e 2 sin 2 w dw
0
π 2
Ke =
∫
0
1
1 − e sin 2 w
2
dw
where e = 1 − a 2 /b 2 , b > a and g = a/b
It is clear that the elliptical integrals are related to each other as follows:
Ke = 2 De − e 2Ce
Ee = (2 − e 2 ) De − e 2Ce
Be = De − e 2Ce
De = ( Ke − Ce ) /e 2
Be = Ke − De
Ce = ( De − Be ) /e 2
The values of the elliptical integral as a function of g are given in Table B.1.
319
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320
Railroad Vehicle Dynamics: A Computational Approach
TABLE B.1
Complete Elliptical Integrals
g
Be
Ce
De
Ee
Ke
e2
0
1.0
–2 + ln(4/g)
–1 + ln(4/g)
1.0
+ln(4/g)
1.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.9964
0.9889
0.9794
0.9686
0.9570
0.9451
0.9328
0.9205
0.9081
0.8959
0.8838
0.8719
0.8603
0.8488
0.8376
0.8267
0.8159
0.8055
0.7953
2.3973
1.7352
1.3684
1.1239
0.9463
0.8105
0.7036
0.6170
0.5460
0.4863
0.4360
0.3930
0.3555
0.3235
0.2955
0.2706
0.2494
0.2289
0.2123
3.3877
2.7067
2.3170
2.0475
1.8442
1.6827
1.5502
1.4388
1.3435
1.2606
1.1879
1.1234
1.0656
1.0138
0.9669
0.9241
0.8851
0.8490
0.8160
1.0049
1.0160
1.0315
1.0505
1.0723
1.0965
1.1227
1.1507
1.1802
1.2111
1.2432
1.2763
1.3105
1.3456
1.3815
1.4181
1.4554
1.4933
1.5318
4.3841
3.6956
3.2964
3.0161
2.8012
2.6278
2.4830
2.3593
2.2516
2.1565
2.0717
1.9953
1.9259
1.8626
1.8045
1.7508
1.7010
1.6545
1.6113
0.9975
0.9900
0.9775
0.9600
0.9375
0.9100
0.8775
0.8400
0.7975
0.7500
0.6975
0.6400
0.5775
0.5100
0.4375
0.3600
0.2775
0.1900
0.0975
1.0
π
0.7864= --4
π
0.19635 = -----16
π
0.7864 = --4
π
1.5708 = --2
π
1.5708 = --2
0.00
45814_book.fm Page 321 Thursday, May 31, 2007 2:25 PM
References
Ahlberg, J.H., Nilson, E.N., and Walsh, J.L., 1967, The Theory of Splines and Their Applications, Academic Press, New York.
Ahmadian, M. and Yang, S., 1998a, Hopf bifurcation and hunting behavior in a rail wheelset
with flange contact, Nonlinear Dynamics, 15, 15–30.
Ahmadian, M. and Yang, S., 1998b, Effect of system nonlinearities on locomotive bogie
hunting stability, Vehicle System Dynamics, 29, 365–384.
Andersson, C. and Abrahamsson, T., 2002, Simulation of interaction between a train in general
motion and a track, Vehicle System Dynamics, 38, 33–455.
Arnold, M., 1996, Numerical problems in the dynamical simulation of wheel-rail systems,
Zeitschrift Fur Angewandte Mathematik Und Mechanik, 76, 151–154.
Atkinson, K.E., 1978, An Introduction to Numerical Analysis, John Wiley & Sons, New York.
Bailey, J.R. and Wormley, D.N., 1992, A comparison of analytical and experimental performance
data for a two-axle freight car, ASME Journal of Dynamic Systems, Measurement, and
Control, 114, 141–147.
Baumgarte, J., 1972, Stabilization of constraints and integrals of motion, Computer Methods
in Applied Mechanics and Engineering, 1, 1–16.
Bavinck, H. and Dieterman, H.H., 1996, Wave propagation in a finite cascade of masses and
springs representing a train, Vehicle System Dynamics, 26, 45–60.
Berg, M., 1998, A non-linear rubber spring model for rail vehicle dynamics analysis, Vehicle
System Dynamics, 30, 197–212.
Berghuvud, A., 2002, Freight car curving performance in braked conditions, IMechE Journal
of Rail and Rapid Transit, 216, 23–29.
Berzeri, M., Sany, J.R., and Shabana, A.A., 2000, Curved Track Modeling Using the Absolute
Nodal Coordinate Formulation, technical report MBS00-4-UIC, Department of
Mechanical Engineering, University of Illinois at Chicago, Chicago.
Bing, A.J., Shaun, R.B., and Henderson, B., 1996, Design Data on Suspension Systems of
Selected Rail Passenger Cars, FRA report DOT/FRA/ORD-96-01, USDOT, Federal
Railroad Administration, Washington, DC.
Blader, F.B., 1989, A Review of Literature and Methodologies in the Study of Derailments
Caused by Excessive Forces at the Wheel/Rail Interface, AAR technical report R-717,
AAR Technical Center, Chicago.
Blader, F. and Klauser, P., 1989, User’s Manual for NUCARS, Version 1.0, technical report
R-734, Association of American Railroads, Washington, DC.
Boocock, D., 1969, The steady-state motion of railway vehicles on curved track, Journal of
Mechanical Engineering Science, 2.
Boussinesq, J., 1885, Application des Potentials à l’étude de l’équilibre et du mouvement des
solides élastiques. Gauthier-Villars, Paris.
Bracciali, A., Cascini, G., and Ciuffi, R., 1998, Time domain model of the vertical dynamics
of a railway track up to 5 kHz, Vehicle System Dynamics, 30, 1–15.
Carter, F.W., 1926, On the action of a locomotive driving wheel, Proceedings of the Royal
Society of London A, 112, 151–157.
Cerruti, V., 1882, Acc. Lincei, Mem. Fis. Mat., Rome.
321
45814_book.fm Page 322 Thursday, May 31, 2007 2:25 PM
322
Railroad Vehicle Dynamics: A Computational Approach
Cooperrider, N.K., 1991, Ride Quality Assessment for a 6-Axle Locomotive and a Heavy
Truck, ASME paper, RTD, 4, 153–160.
Cooperrider, N.K., Hedrick, J.K., Law, E.H., and Malstrom, C.W., 1976, The application of
quasi-linearization techniques to the prediction of nonlinear railway vehicle response,
in Proceedings of the IUTAM Symposium, Delft, Netherlands, pp. 314–325.
Cooperrider, N.K., Law, E.H., Hull, R., Kadala, P.S., and Tuten, P.S., 1975, Analytical and
Experimental Determination of Nonlinear Wheel/Rail Geometric Constraints, Report
FRA-OR&D 76-244 (PB-252 290) USDOT, Federal Railroad Administration, Washington, D.C.
Datoussaid, S., Verlinden, O., Wenderloot, L., and Conti, C., 1998, Computer-aided analysis
of urban railway vehicles, Vehicle System Dynamics, 30, 213–227.
De Pater, A.D., 1974, The propagation of waves in a semi-infinite chain of material points
and springs representing a long train, Vehicle System Dynamics, 3, 123–140.
De Pater, A.D., 1988, The geometrical contact between track and wheel-set, Vehicle System
Dynamics, 17, 127–140.
Diana, G., Fossati, F., and Resta, F., 1998, High speed railway: collecting pantographs active
control and overhead lines diagnostic solutions, Vehicle System Dynamics, 30, 69–84.
Dietz, S., Netter, H., and Sachau, D., 1988, Fatigue life prediction of a railway bogie under
dynamic loads through simulation, Vehicle System Dynamics, 29, 385–402.
Diomin, Y.V., 1994, Stabilization of high-speed railway vehicles, Vehicle System Dynamics,
23, 107–114.
Do Carmo, M.P., 1976, Differential Geometry of Curves and Surfaces, Prentice Hall, New
York.
Duffek, W., 1982, Contact geometry in wheel rail vehicles, in Proceedings of Contact Mechanics and Wear of Rail/Wheel Systems, University of Waterloo, Vancouver, pp. 161–181.
Dukkipati, R.V., 1998, Dynamics of wheelset on roller rig, Vehicle System Dynamics, 30,
409–430.
Dukkipati, R.V., 2000, Vehicle Dynamics, CRC Press, New Delhi.
Dukkipati, R.V. and Amyot, J.R., 1988, Computer-Aided Simulation in Railway Dynamics,
Marcel-Dekker, New York.
Dukkipati, R.V. and Dong, R., 1999, Impact loads due to wheel flats and shells, Vehicle System
Dynamics, 31, 1–22.
Dukkipati, R.V., Swamy, S.N., and Osman, M.O.M., 1991, Independently rotating wheel
systems for railway vehicle: a survey of the state of the art, Archive of Transport, 3,
297–330.
Durali, M. and Shadmehri, B., 2003, Nonlinear analysis of train derailment in severe braking,
ASME Journal of Dynamic Systems, Measurement, and Control, 125, 48–53.
Eich-Soellner, E. and Führer, C., 1998, Numerical Methods in Multibody Dynamics, Teubner,
B.G. GmbH, Stuttgart.
Elkins, J.A., 1992, Prediction of wheel rail interaction: the state-of-the-art, Vehicle System
Dynamics, 20, 1–27.
Elkins, J.A. and Eikhoff, B.M., 1982, Advances in nonlinear wheel/rail force prediction
methods and their validation, ASME Journal of Dynamic Systems, Measurement, and
Control, 104, 133–142.
Elkins, J.A. and Gostling, R.J., 1977, A general quasi-static curving theory for railway
vehicles, in Proceedings of 2nd IUTAM Symposium, Vienna, pp. 388–406.
Elkins, J.A. and Carter, A., 1993, Testing and analysis techniques for safety assessment of
rail vehicles: the state-of-the-art, Vehicle System Dynamics, 22, 185–208.
Endlicher, K.O. and Lugner, P., 1990, Computer simulation of the dynamical curving behavior
of a railway bogie, Vehicle System Dynamics, 19, 71–95.
45814_book.fm Page 323 Thursday, May 31, 2007 2:25 PM
References
323
Escalona, J.L., Elastic Contact, personal communication, 2002.
Fisette, P. and Samin, C., 1991, Lateral dynamics of a light railway vehicle with independent
wheels, in Proceedings of the 12th IAVSD Symposium on the Dynamics of Vehicles
on Road and Tracks, Lyon, pp. 157–171.
Fries, R.H., Cooperrider, N.K., and Law, E.H., 1981, Experimental investigation of freight
car lateral dynamics, ASME Journal of Dynamic Systems, Measurement, and Control,
103, 201–210.
Garcia de Jalon, J. and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody
Systems: The Real-Time Challenge, Springer-Verlag, Heidelberg.
Garg, V.K. and Dukkipati, R.V., 1988, Dynamics of Railway Vehicle Systems, Academic Press,
New York.
Gilchrist, A.O., 1998, The long road to solution of the railway hunting and curving problems,
IMechE Journal of Rail and Rapid Transit, 212, 219–226.
Gilchrist, A.O. and Brickle, B.V., 1976, A re-examination of the proneness to derailment of
a railway wheel-set, Journal of Mechanical Engineering Science, 18, 131–141.
Gladwell, G.M.L., 1980, Contact Problem in the Classical Theory of Elasticity, Alphen and
den Rijn, Sijthogg and Noordhof, the Netherlands.
Goetz, A., 1970, Introduction to Differential Geometry, Addison Wesley Publishing Company.
Goldsmith, W., 1960, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward
Arnold Ltd., London.
Grassie, S.L., 1993, Dynamic modeling of the track, in Rail Quality and Maintenance for
Modern Railway Operation, Kalker, J.J., Cannon, D., and Orringer, O., Eds., Kluwer,
Dordrecht, Netherlands.
Greenwood, D.T., 1988, Principle of Dynamics, 2nd ed., Prentice Hall, New York.
Hamid, A., Rasmussen, K., Baluja, M., and Yang, T.-L., 1983, Analytical Description of Track
Geometry Variations, FRA report DOT/FRA/ORD-83/03.1, USDOT, Federal Railroad Administration, Washington, DC.
Handoko, Y.A., 2006, Investigation of the Dynamics of Railway Bogies Subjected to Traction/Braking Torque, Ph.D. dissertation, Central Queensland University, Australia.
Handoko, Y. and Dhanasekar, M., 2006, An inertial reference frame method for the simulation
of the effect of longitudinal force to the dynamics of railway wheelsets, Nonlinear
Dynamics, 45, 399–425.
Handoko, Y., Xia, F., and Dhanasekar, M., 2004, Effect of asymmetric brake shoe force
application on wagon curving performance, Vehicle System Dynamic Supplement, 41,
113–122.
Haque, I., Law, E.H., and Cooperrider, N.K., 1979, User’s Manual for Program for Calculation
of Kalker’s Linear Creep Coefficients, FRA report FRA/ORD-78/71, USDOT, Federal
Railroad Administration, Washington, DC.
Hedrick, J.K. and Arslan, A.V., 1979, Nonlinear analysis of rail vehicle forced lateral response
and stability, ASME, Journal of Dynamics Systems, Measurement, and Control, 101,
230–237.
Heller, R. and Cooperrider, N.K., 1977, Users’ Manual for Asymmetric Wheel/Rail Contact
Characterization Program, FRA report FRA/ ORD-78/05, USDOT, Federal Railroad
Administration, Washington, DC.
Hesser, K., 1998, Progress in railway mechanical engineering: passenger and commuter rail
vehicles and components, in Proceedings of the 1998 ASME International Mechanical
Engineering Congress & Exposition (Rail Transportation), Anaheim, CA.
Hertz, H., 1882, Über die berührung fester elastische Körper und über die Harte, Verhandlungen des Vereins zur Beförderung des Gewerbefleisses, Leipzig, Nov. 1882.
Huston, R.L., 1990, Multibody Dynamics, Butterworth-Heinemann, Oxford, U.K.
45814_book.fm Page 324 Thursday, May 31, 2007 2:25 PM
324
Railroad Vehicle Dynamics: A Computational Approach
Huston, R.L., 1993, Flexibility effects on multibody systems, in Proceedings of the NATOAdvanced Study Institute on Computer Aided Analysis of Rigid and Flexible Mechanical Systems, Troia, Portugal.
Iwnicki, S., 1999, The Manchester Benchmarks for Rail Vehicle Simulation, Supplement to
Vehicle System Dynamics, Taylor & Francis, London.
Iwnicki, S.D. and Wickens, A.H., 1998, Validation of a MATLAB railway vehicle simulation
using a scale roller rig, Vehicle System Dynamics, 30, 257–270.
Jacobson, B. and Kalker, J.J., 2001, Rolling Contact Phenomena, Springer, Wien.
Jahnke, E., and Emde, F., 1943, Tables of Functions, Dover Publications, Mineola, NY.
Jaschinski, A., 1990, On the Application of Similarity Laws to a Scaled Railway Bogie Model,
Ph.D. thesis, Delft University of Technology, Delft, Netherlands.
Jeambey, J., 1998, Improving high speed stability of freight cars with hydraulic dampers, in
Proceedings of the 1998 ASME International Mechanical Engineering Congress &
Exposition (Rail Transportation), Anaheim, CA.
Johnson, K.L., 1958a, The effect of spin upon the rolling motion of an elastic sphere upon a
plane, ASME Journal of Applied Mechanics, 25, 332–338.
Johnson, K.L., 1958b, The effect of a tangential contact force upon the rolling motion of an
elastic sphere on a plane, ASME Journal of Applied Mechanics, 25, 339–346.
Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, U.K.
Kalker J.J., 1991, Wheel-rail rolling contact theory, Wear, 144, 243–261.
Kalker, J.J., 1967, On the Rolling Contact of Two Elastic Bodies in the Presence of Dry
Friction, Ph.D. thesis, Delft University of Technology, Delft, Netherlands.
Kalker, J.J., 1973, Simplified Theory of Rolling Contact, progress report 1, Delft University
of Technology, Delft, Netherlands, pp. 1–10.
Kalker, J.J., 1979, Survey of wheel-rail rolling contact theory, Vehicle System Dynamics, 5,
317–358.
Kalker, J.J., 1982, A fast algorithm for the simplified theory of rolling contact, Vehicle System
Dynamics, 11, 1–13.
Kalker, J.J., 1986, Wheel-rail wear calculations with the program contact, in Proceedings of
International Symposium on Contact Mechanics and Wear of Rail-Wheel Systems II,
Gladwell, G.M.L., Ghonem, H., and Kalonusek, J. (eds.), University of Waterloo
Press, Waterloo, Ontario, pp. 3–26.
Kalker, J.J., 1989, Review of wheel-rail rolling contact theories, in The General Problem of
Rolling Contact, Browne, L. and Tsai, N.T., Eds., 1989 ASME Winter Annual Meeting, Chicago, AMD, 40, 77–92.
Kalker, J.J., 1990, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht,
Netherlands.
Kalker, J.J., 1994, Consideration of rail corrugation, Vehicle System Dynamics, 23, 1994, 3–28.
Kalker, J.J., 1996, Discretely supported rails subjected to transient loads, Vehicle System Dynamics, 25, 71–88.
Karnopp, D., 2004, Vehicle Stability, Marcel Dekker, New York.
Kassa, E., Andersson, C., and Nielsen, J., 2006, Simulation of dynamic interaction between
train and railway turnout, Vehicle System Dynamics, 44, 247–258.
Kerr, A.D., 2000, On the determination of the rail support modulus k, International Journal
of Solids and Structures, 37, 4335–4351.
Kerr, A.D. and El-Sibaie, M.A., 1987, On the new equations for the lateral dynamics of rail-tie
structure, ASME Journal of Dynamic Systems, Measurement, and Control, 109, 180–185.
Khulief, Y.A. and Shabana, A., 1986, Dynamics of multi-body systems with variable kinematic
structure, ASME paper 85-DET-83, ASME Journal of Mechanisms, Transmissions,
and Automation in Design, 108, 167–175.
45814_book.fm Page 325 Thursday, May 31, 2007 2:25 PM
References
325
Kik, W., 1992, Comparison of the behavior of different wheelset-track models, Vehicle System
Dynamics, 20, 325–339.
Kik, W. and Piotrowski, J., 1996, A fast approximation method to calculate normal load at
contact between wheel and rail and creep forces during rolling, in Procedures of 2nd
Mini Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Zabory, I.,
Ed., TU, Budapest.
Klingel, W., 1883, Über den Lauf von Eisenbahnwagen auf Gerarder Bahn, Organ für die
Fortschritte des Eisenbahnwesens, 20, 113–123.
Knothe, K. and Bohm, F., 1999, History of stability of railway and road vehicle, Vehicle
System Dynamics, 31, 283–323.
Knothe, K.L. and Grassie, S.L., 1993, Modeling of railway track and vehicle/track interaction
at high frequencies, Vehicle System Dynamics, 22, 209–262.
Knothe, K. and Stichel, S., 1994, Direct covariance analysis for the calculation of creepages
and creep-forces for various bogie on straight track with random irregularities, Vehicle
System Dynamics, 23, 237–251.
Knudsen, C., Feldberg, R., and Jaschinski, A., 1991, Non-linear dynamic phenomenon in the
behavior of a railway wheelset model, Nonlinear Dynamics, 2, 389–404.
Kono, H., Suda, Y., Yamaguchi, M., Yamashita, H., Yanobu, Y., and Tsuda, K., 2005, Dynamic
analysis of the vehicle running on turnout at high speed considering longitudinal
variation of rail profiles, in Proceedings of IDETC/CIE 2005 Conference, ASME
2005 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, September 24–28, 2005, Long Beach, CA.
Kreyszig, E., 1991, Differential Geometry, Dover, Mineola, NY.
Kwak, B.M., 1991, Complementarity problem formulation of three-dimensional friction contact, ASME Journal of Applied Mechanics, 58, 134–140.
Law, E.H. and Cooperrider, N.K., 1974, A survey of railway vehicle dynamics research, ASME
Journal of Dynamic Systems, Measurement, and Control, 96, 132–146.
Lee, T.W. and Wang, A.C., 1983, On the dynamics of intermittent motion mechanisms, part
I: dynamic model and response, ASME Journal of Mechanisms, Transmissions, and
Automation in Design, 105, 534–540.
Lieh, J. and Haque, I., 1991, A study of the parametrically excited behavior of passenger and
freight railway vehicles using linear models, ASME Journal of Dynamic Systems,
Measurement, and Control, 113, 336–338.
Litvin, F.L., 1994, Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, NJ.
Love, A.E.H., 1944, A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover,
Mineola, NY.
Luo, Y., Yin, H., and Hua, C., 1996, The dynamic response of railway ballast to the action
of trains moving at different speeds, IMechE Journal of Rail and Rapid Transit, 210,
95–101.
Mace, S., Pena, R., Wilson, N., and DiBrito, D., 1996, Effects of wheel-rail contact geometry
on wheelset steering forces, Wear, 191, 204–209.
Malvezzi, M., Presciani, P., Allotta, B., and Toni, P., 2003, Probabilistic analysis of braking
performance in railways, IMechE Journal of Rail and Rapid Transit, 217, 149–165.
Matsudaira, T., 1960, ORE report RP2/SVA-C9, UIC paper awarded prize in the competition
sponsored by Office of Research and Experiment (ORE) of the International Union
of Railways (UIC), Utrecht, Netherlands.
Matsumoto, A., Sato, A., Ohno, Y., Suda, H., Nishimura, Y., Tanimoto, R., and Oka, M., 1999,
Compatibility of curving performance and hunting stability of railway bogie, Vehicle
System Dynamics Supplement, 33, 740–748.
45814_book.fm Page 326 Thursday, May 31, 2007 2:25 PM
326
Railroad Vehicle Dynamics: A Computational Approach
Mayville, R., Rancatone, R., and Tegeler, L., 1998, Investigation and simulation of lateral
buckling in trains, in Proceedings of the 1998 ASME International Mechanical Engineering Congress & Exposition (Rail Transportation), Anaheim, CA.
McClanachan, M., Handoko, Y., Dhanasekar, M., Skerman, D., and Davey, J., 2004, Modeling
freight wagon dynamics, Vehicle System Dynamics Supplement, 41, 438–447.
McPhee, J.J. and Anderson, R.J., 1996, A model reduction procedure for the dynamic analysis
of rail vehicles subjected to linear creep forces, Vehicle System Dynamics, 25,
349–367.
Meijaard, J.P., 1991, Dynamics of Mechanical Systems: Algorithms for a Numerical Investigation of the Behaviour of Non-Linear Discrete Models, Ph.D. thesis, Delft University of Technology, Delft, Netherlands.
Meijaard, J.P. and De Pater, A.D., 1989, Railway vehicle systems dynamics and chaotic
vibrations, International Journal of Non-Linear Mechanics, 24, 1–17.
Melzer, F., 1994, Symbolic computations in flexible multibody systems, in Proceedings of
NATO: Advanced Study Institute on Computer Aided Analysis of Rigid and Flexible
Mechanical Systems, Vol. II, Troia, Portugal, pp. 365–381.
Meng, Q., Heineman, J., and Shabana, A.A., 2005, A longitudinal force model for multibody
railroad vehicle system applications, in Proceedings of ASME International Design
Engineering Technical Conferences and Computer and Information in Engineering
Conference, DETC2005/ MSNDC-84058, September 24–28, Long Beach, CA.
Miyamoto, T., Suda, Y., and Ishida, H., 2006, Dynamics simulation of railway vehicles on
vibrated track upon seismicity, in Proceedings of the Third Asian Conference on
Multibody Dynamics, August 1–4, Tokyo, Japan.
Murase, S., Tanaka, A., Takahashi, Y., and Sekine, Y., 1988, Rail vehicle dynamics of pitch
caused by phase difference inputs, in Proceedings of the ASME Rail Transportation
Winter Conference, Chicago, pp. 45–51.
Nadal, M.J., 1908, Locomotives a Vapeur, in Collection Encyclopédie Scientifique, Bibliothèque de Mécanique Appliquée et Génie, Vol. 186, Paris, France.
Nagase, K., Wakabayashi, Y., and Sakahara, H., 2002, A study of the phenomenon of wheel
climb derailment: results of basic experiment using model bogies, IMechE Journal
of Rail and Rapid Transit, 216, 237–247.
Nakanishi, T., Yin, X.G., and Shabana, A.A., 1996, Dynamics of multibody tracked vehicles
using experimentally identified modal parameters, ASME Journal of Dynamic Systems, Measurement, and Control, 118, 449–507.
Netter, H., Schupp, G., Rulka, W., and Schroeder, K., 1998, New aspects of contact modeling
and validation within multibody system simulation of railway vehicles, Vehicle System
Dynamics Supplement, 28, 246–269.
Newland, D.E. and Cassidy, R.J., 1975, Suspension and structure: some fundamental design
considerations for railway vehicles, IMechE Railway Vehicles Suspension and Structure, March 1975, 4, 4–26.
Nielsen, J.C.O., 1994 Dynamic interaction between wheel and track: a parametric search
towards an optimal design of rail structures, Vehicle System Dynamics, 23, 115–132.
Nikravesh, P.E., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice Hall, Englewood Cliffs, NJ.
O’Connor, D.N., Eppinger, S.D., Seering, W.P., and Wormley, D.N., 1997, Active control of
a high-speed pantograph, ASME Journal of Dynamic Systems, Measurement, and
Control, 119, 1–4.
O’Donnell, W., 1993, Experimental and Numerical Investigation of the Pitch and Bounce
Response of a Railroad Vehicle, M.S. thesis, Department of Mechanical Engineering,
University of Illinois at Chicago, Chicago.
45814_book.fm Page 327 Thursday, May 31, 2007 2:25 PM
References
327
Ozaki, T. and Shabana, A.A., 2003a, Treatment of constraints in complex multibody systems,
part I: methods of constrained dynamics, Journal of Multi-Scale Computational
Engineering, 1, 235–252.
Ozaki, T. and Shabana, A.A., 2003b, Treatment of constraints in complex multibody systems,
part II: application to tracked vehicles, Journal of Multi-Scale Computational Engineering, 1, 253–276.
Pascal, J.P., 1993, About multi-Hertzian-contact hypothesis and equivalent conicity in the
case of S1002 and UIC60 analytical wheel/rail profiles, Vehicle System Dynamics,
22, 57–78.
Pascal, J.P. and Sauvage, G., 1993, The available methods to calculate the wheel/rail forces in
non-Hertzian contact patches and rail damaging, Vehicle System Dynamics, 22, 263–275.
Pereira, M. and Ambrosio, J., Eds., 1994, Computer-Aided Analysis of Rigid and Flexible
Mechanical Systems, Kluwer, Dordrecht, Netherlands.
Pfeiffer, F. and Glocker, C., 1996, Multibody Dynamics with Unilateral Contacts, John Wiley
& Sons, New York.
Poetsch, G., Evans, J., Meisinger, R., Kortum, W., Baldauf, W., Veitl, A., and Wallaschek, J.,
1997, Pantograph/catenary dynamics and control, Vehicle System Dynamics, 28,
159–195.
Polach, O., 1999, A fast wheel-rail forces calculation computer code, Vehicle System Dynamics
Supplement, 33, 728–739.
Polach, O., 2001, Influence of locomotive tractive effort on the forces between wheel and
rail, Vehicle System Dynamics Supplement, 35, 7–22.
Polach, O., 2005, Creep forces in simulation of traction vehicles running on adhesion limit,
Wear, 258, 992–1000.
Pombo, J. and Ambrosio, J., 2003, A wheel/rail contact model for rail guided vehicles
dynamics, in Proceedings of ECCOMAS Thematic Conference on Advances in Computational Multibody Dynamics, July 1–4, 2003, Lisbon, Portugal.
Popp, K. and Schiehlen, W., 1993, Fahrzeugdynamik, Teubner, Stuttgart.
Press, W.H., Teukolsky, S.A., Vetterling, W.T, and Flannery, B.P., 1992, Numerical Recipes
in Fortran, 2nd ed., Cambridge University Press, Cambridge, U.K.
Profilidis, V.A., 2000, Railway Engineering, Cambridge University Press, Cambridge, U.K.
Rabinowicz, E., 1995, Friction and Wear of Material, John Wiley & Sons, New York.
Raymond, G.P. and Cai, Z., Dynamic track support loading from heavier and faster train sets,
Transportation Research Record, 1381, 53–59.
Rathod, C. and Shabana, A.A., 2006, Rail geometry and Euler angles, ASME Journal of
Computational and Nonlinear Dynamics, 1, 264–268.
Rathod, C. and Shabana, A.A., 2007, Geometry and differentiability requirements in multibody railroad vehicle dynamic formulations, Nonlinear Dynamics, Vol. 47, 249–261.
Ripke, B. and Knote, K., 1995, Simulation of high frequency wagon-track interaction, Vehicle
System Dynamics Supplement, 24, 72–85.
Roberson, R.E. and Schwertassek, R., 1988, Dynamics of Multibody Systems, Springer-Verlag,
Heidelberg.
Rosenberg, R.M., 1977, Analytical Dynamics of Discrete Systems, Plenum Press, New York.
Samin, J.C. and Neuve, L., 1984, A multibody approach for dynamic investigation of rolling
systems, Ingenieur-Archiv, 54, 1–15.
Schmid, R., Endlicher, K.O., and Lugner, P., 1994, Computer simulation of the dynamical
behavior of a railway bogie passing a switch, Vehicle System Dynamics, 23, 481–499.
Schupp, G., 1996, Different Contact Models for Wheel-Rail Systems: A Comparison, internal
report IB 515-96-22, Institute for Robotics and System Dynamics, Oberpfaffenhofen,
Germany.
45814_book.fm Page 328 Thursday, May 31, 2007 2:25 PM
328
Railroad Vehicle Dynamics: A Computational Approach
Schupp, G., 2003, Simulation of railway vehicles: necessities and application, Mechanics
Based Design of Structures and Machines, 31, 297–314.
Schupp, G., Weidemann, C., and Mauer, L., 2004, Modelling the contact between wheel and
rail within multibody system simulation, Vehicle System Dynamics, 41, 349–364.
Schwab, A.L., 2002, Dynamics of Flexible Multibody Systems, Ph.D. thesis, Delft University
of Technology, Delft, Netherlands.
Schwab, A.L. and Meijaard, J.P., 2002, Two special finite elements for modelling rolling
contact in a multibody environment, in Proceedings of the First Asian Conference
on Multibody Dynamics, Shimizu, N. et al., Eds., July 31–August 2, 2002, ACMD’02,
Iwaki, Fukushima, Japan, The Japan Society of Mechanical Engineering, pp. 386–391.
Schwab, A.L. and Meijaard, J.P., 2003, Dynamics of flexible multibody systems with nonholonomic constraints: a finite element approach. Multibody System Dynamics, 10,
107–123.
Schwartz, B., 1988, An analytical study of the bounce motion of a freight car model in
response to profile irregularities, in Proceedings of the ASME Rail Transportation
Conference, Chicago.
Schwarz, B., 1991, Wheel Climb and Rollover Potential due to Excess Elevation and Curvature, ASME paper, RTD, Vol. 4, pp. 83–91.
Schwertassek, R. and Wallrap, O., 1999, Dynamik flexibler Mehrkorper Systeme, Vieweg,
Germany.
Seering, W., Armbruster, K., Vesely, C., and Wormley, D., 1991, Experimental and analytical
study of pantograph dynamics, ASME Journal of Dynamic Systems, Measurement,
and Control, 113, 591–605.
Seo, J.H., Sugiyama, H., and Shabana, A.A., 2005, Three dimensional large deformation
analysis of the multibody pantograph/catenary systems, Nonlinear Dynamics, 42,
199–215.
Shabana, A.A., 1986, Dynamics of inertia-variant flexible systems using experimentally
identified parameters, ASME Journal of Mechanisms, Transmissions, and Automation
in Design, 108, 358–366.
Shabana, A.A., 1996a, Theory of Vibration: An Introduction, 2nd ed., Springer Verlag, Heidelberg.
Shabana, A.A., 1996b, An Absolute Nodal Coordinate Formulation for the Large Rotation
and Deformation Analysis of Flexible Bodies, technical report MBS96-1-UIC,
Department of Mechanical Engineering, University of Illinois at Chicago.
Shabana, A.A., 1997, Flexible multibody dynamics: review of past and recent development,
Multibody System Dynamics, 1, 189–222.
Shabana, A.A., 2001, Computational Dynamics, 2nd ed., Wiley, New York.
Shabana, A.A., 2005, Dynamics of Multibody Systems, 3rd ed., Cambridge University Press,
Cambridge, U.K.
Shabana, A.A., Berzeri, M., and Sany, J.R., 2001, Numerical procedure for the simulation of
wheel/rail contact dynamics, ASME Journal of Dynamic Systems, Measurement, and
Control, 123, 168–178.
Shabana, A.A., Hussien, H.A., and Escalona, J.L., 1998, Application of the absolute nodal
coordinate formulation to large rotation and large deformation problems, ASME
Journal of Mechanical Design, 120, 188–195.
Shabana, A.A. and Sany, J.R., 2001, An augmented formulation for mechanical systems with
non-generalized coordinates: application to rigid body contact problems, Nonlinear
Dynamics, 24, 183–204.
Shabana, A.A., Tobaa, M., Sugiyama, H., and Zaazaa, K.E., 2005, On the computer formulations of the wheel/rail contact, Nonlinear Dynamics, 40, 169–193.
45814_book.fm Page 329 Thursday, May 31, 2007 2:25 PM
References
329
Shabana, A.A., Tobaa, M., Marquis, B., and El-Sibaie, 2006, Effect of the linearization of
the kinematic equations in railroad vehicle system dynamics, ASME Journal of
Computational and Nonlinear Dynamics, 1, 25–34.
Shabana, A.A., Zaazaa, K.E., Escalona, L.J, and Sany, J.R., 2004, Development of elastic
force model for wheel/rail contact problems, Journal of Sound and Vibration, 269,
295–325.
Sharma, V., Sneed, W., and Punwani, S., 1984, Freight equipment environmental sampling
test-description and results, in Proceedings of the ASME Rail Transportation Spring
Conference, Chicago.
Shen, G., Ayasse, J.B., Chollet, H., and Pratt, I., 2003, A unique design method for wheel
profiles by considering the contact angle function, IMechE Journal of Rail and Rapid
Transit, 217, Part F, 25–30.
Shen, G. and Goodall, R., 1997, Active yaw relaxation for improved bogie performance,
Vehicle System Dynamics, 28, 273–289.
Shen, G. and Pratt, I., 2001, The development of railway dynamics modelling and simulation
package to cater for current industrial trends, IMechE Journal of Rail and Rapid
Transit, 215, 167–178.
Shen, Z.Y., Hedrick, J.K., and Elkins, J.A., 1983, A comparison of alternative creep-force
models for rail vehicle dynamic analysis, Vehicle System Dynamics, 12, 79–87.
Shikin, E.V. and Plis, A.I., 1995, Handbook on Splines for User, CRC Press, Boca Raton, FL.
Simeon, B., Fuhrer, C., and Rentrop, P., 1991, Differential algebraic equations in vehicle
system dynamics, Survey on Mathematics in Industry, 1, 1–37.
Sinha, P.K., 1987, Electromagnetic Suspension: Dynamics & Control, P. Peregrinus, London.
Smith, C.A.M. and Bowler, E.H., 1981, A curved track simulator, in Proceedings of the XV
Pan American Railway Congress, Mexico, D.F.
Stichel, S., 1999, On freight wagon dynamics and track deterioration, IMechE Journal of Rail
and Rapid Transit, 213, 243–254.
Suda, Y. and Anderson, R.J., 1994, High speed stability and curving performance of longitudinal unsymmetric trucks with semi-active control, Vehicle System Dynamics, 23,
29–52.
Sugiyama, H. and Shabana, A.A., 2006, Trajectory coordinate constraints in multibody railroad vehicle systems, in Proceedings of the Third Asian Conference on Multibody
Dynamics, August 1–4, Tokyo, Japan.
Sun, Y.Q. and Dhanasekar, M., 2001, A dynamic model for the vertical interaction of the rail
track and wagon system, International Journal of Solids and Structures, 39, 1337–1359.
Sweet, L.M. and Karmel, A., 1981, Evaluation of time-duration dependent wheel load criteria
for wheel climb derailment, ASME Journal of Dynamic Systems, Measurement, and
Control, 103, 219–227.
Sweet, L.M. and Garrison-Phelan, P., 1980, Identification of wheel/rail creep coefficients from
steady state and dynamic wheelset experiments, in The General Problem of Rolling
Contact, Browne, L. and Tsai, N.T., Eds., ASME 1980 Winter Annual Meeting,
Chicago, AMD, 40, 77–92.
Szolc, T., 1998, Simulation of bending-torsional-lateral vibrations of the railway wheelsettrack system in the medium frequency range, Vehicle System Dynamics, 30, 473–508.
Tse, Y. and Martin, G., 1975, Flexible Body Railroad Freight Car, Vol. I, technical report
R-199, Association of American Railroads, Washington, DC.
Tuten, J.M., Law, E.H., and Coperrider, N.K., 1979, Lateral stability of freight cars with axles
having different wheel profiles and asymmetric loading, ASME Journal of Engineering for Industry, 101, 1–16.
45814_book.fm Page 330 Thursday, May 31, 2007 2:25 PM
330
Railroad Vehicle Dynamics: A Computational Approach
U.S. Department of Transportation, Track Safety Standards Part 213 Subpart G, Federal
Railroad Administration, Office of Safety, October 2004.
U.S. Department of Transportation, Track Safety Standards Part 213 Subpart A to F, Federal
Railroad Administration, Office of Safety, December 2005.
Valtorta, D., Zaazaa, K.E., Shabana, A.A., and Sany, J.R., 2001, A study of the lateral stability
of railroad vehicles using a nonlinear constrained multibody formulation, in Proceedings of 2001 ASME International Mechanical Engineering Congress and Exposition,
New York.
Vermeulen, P.J. and Johnson, K.L., 1964, Contact of nonspherical bodies transmitting tangential forces, ASME Journal of Applied Mechanics, 31, 338–340.
Wang, W., Suda, Y., Komine, H., Sato, Y., Nakai, T., and Shimokawai, Y., 2006, Transition
curve negotiating simulation of railway vehicle for air suspension control to prevent
wheel load reduction, in Proceedings of the Third Asian Conference on Multibody
Dynamics, August 1–4, Tokyo.
Warren, P., Samavedam, G., Tsai, T., and Gomes, J., 1998, Passenger vehicle dynamic
performance: tests and analysis correlation, in Proceedings of the 1998 ASME International Mechanical Engineering Congress & Exposition (Rail Transportation),
Anaheim, CA.
Wehage, R.A., 1980, Generalized Coordinate Partitioning in Dynamic Analysis of Mechanical
Systems, Ph.D. thesis, University of Iowa, Iowa City.
Weidemann, C., Mauer, L., and Arnold, M., 2006, Improving the calculation speed for time
domain integration of complex railway vehicles, in Proceedings of the Third Asian
Conference on Multibody Dynamics, August 1 – 4, Tokyo.
Weinstock, H., 1989, Wheel climb derailment criteria for evaluation of rail vehicle safety, in
The General Problem of Rolling Contact, Browne, L. and Tsai, N.T., Eds., paper
84-WA/RT-1, 1989 ASME Winter Annual Meeting, Chicago, AMD, 40, 77–92.
White, R.C., Limbert, J.K. Hedrick, and Cooperrider, N.K., 1978, “Guideway-Suspension
Tradeoffs in Rail Vehicle Systems,” Report DOT-OS-50107, U.S. Department of
Transportation, Washington D.C.
Wickens, A.H., 1965, The dynamic stability of railway vehicle wheelsets and bogies having
profiled wheels, International Journal of Solids and Structures, 1, 319–341.
Wickens, A.H., 1986, Non-linear dynamics of railway vehicles, Vehicle System Dynamics,
15, 289–301.
Wickens, A.H., 1988, Stability optimization of multi-axle railway vehicles possessing perfect
steering, ASME Journal of Dynamic Systems, Measurement, and Control, 110, 1–7.
Wickens, A.H., 1991, Steering and stability of bogie: vehicle dynamics and suspension design,
IMechE Journal of Rail and Rapid Transit, 205, 109–122.
Wickens, A.H., 1996, Static and dynamic instabilities of bogie railway vehicles with linkage
steered wheelsets, Vehicle System Dynamics, 26, 1–16.
Wickens, A., 1998, Keynote address, Vehicle System Dynamics, 30, 193–195.
Wickens, A.H., 2003, Fundamental of Rail Vehicle Dynamics, Swets & Zetlinger, the Netherlands.
Wu, T.X. and Brennan, M.J., 1998, Basic analytical study of pantograph-catenary system
dynamics, Vehicle System Dynamics, 30, 443–456.
Wu, W.X., Brickle, B.V., Smith, J.H., and Luo, R.K., 1998, An investigation into stick-slip
vibrations on vehicle/track systems, Vehicle System Dynamics, 30, 229–236.
Yabuno, H., Okamoto, T., and Aoshima, N., 2001, Stabilization control for the hunting motion,
Vehicle System Dynamics Supplement, 35, 41–55.
Yang, G., 1994, Aspects in Modeling a Railway Vehicle on an Arbitrary Track, ASME paper,
RTD, 8, 31–36.
45814_book.fm Page 331 Thursday, May 31, 2007 2:25 PM
References
331
Zaazaa, K.E., 2003, Elastic Force Model for Wheel/Rail Contact in Multibody Railroad
Vehicle Systems, Ph.D. thesis, Department of Mechanical Engineering, University of
Illinois at Chicago.
Zhang, Y., El-Sibaie, M., and Lee, S., 2004, FRA Track Quality Indices and Distribution
Characteristics, presented at AREMA 2004 Annual Conference, Nashville, TN.
Zobinski, K., 1999, The importance of imaginary forces and kinematic-type nonlinearities
for the description of railway vehicle dynamics, IMechE Journal of Rail and Rapid
Transit, 213, 199–210.
Zobory, I., 1997, Prediction of wheel/rail profile wear, Vehicle System Dynamics, 28, 221–259.
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45814_Idx.fm Page 333 Tuesday, June 5, 2007 2:24 PM
Index
A
Absolute acceleration vector, 51, 55
Absolute angular velocity vector, 51, 292, 299,
301
Absolute coordinates, 9, 10, 21, 70
Absolute nodal coordinate formulation, 114
Acceleration
of center of mass, 261–262
Acceleration kinematic equation, 173
Acceleration vector, 50–51, 51, 68, 188, 274
Adhesion region, of contact area, 145
Algebraic equations, 174, 175, 205, 206–208, 269
elastic contact formulations with, 174–176
static equilibrium, 243
Alignment, 107
limit by track classes, 224
Alignment irregularities, 223
Alignment variation, 32
Angle derivatives, 114–116
Angle of attack (AOA), 6
Angular acceleration vector, 73, 262–263, 307
Angular velocity
creepage linearization, 295
matrix identities and, 295–296
Angular velocity vector, 51, 72, 140, 257,
262–263, 293
in creep theory, 142
defining in body coordinate system, 262
as linear functions of time derivatives, 297
and linearization assumptions, 300
with trajectory coordinate system, 271
of wheel, 81
Applied forces
vector of, 199
virtual work of, 87, 167
Arc length, 216, 255
in curve theory, 90–91
as function of time, 267
and projected arc length, 109
Augmented constraint contact formulation
(ACCF), 166, 168–171, 179, 180, 194,
195–201
in static analysis, 242–244
surface parameters in, 199
Augmented formulation, 10, 36, 62, 66–68, 76,
188
examples, 68–70
interpretation of methods, 83–84
B
Backtracking algorithm, 245
Balance speed, 28, 29
Ball joint, 62
Binormal vector, 92
Body coordinate system, 21
angular velocity vector in, 262, 273
and general displacement, 258–259
Body trajectory coordinate system, 21, 255
Bogie frames, 3, 4, 10
Bolster, 3
Bounce, 27, 31
Bump deviations, 220, 225, 226, 227
Bushing element, 227, 232–236
rotational deformation of, 234
Bushing stiffness, 235
C
Cant angle, defined, 107
Cant deficiency, 29–30
Carter's creepage coefficient, 149
Carter's theory
comparison with linear law, 149
of creep forces, 147–149
Cartesian coordinates, 255, 270
complex expressions using, 266
constraint Jacobian matrix with, 271–272
Jacobian matrix associated with, 270
time derivatives of, 271
use with trajectory coordinate constraints,
269–272
vs. trajectory coordinate-based formulations,
265
Center of mass, 256, 272
absolute accelerations of, 306
absolute velocity vector of, 273
acceleration of, 261–262
global position vector of, 258
velocity of, 260–261
333
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Railroad Vehicle Dynamics: A Computational Approach
Centripetal acceleration, 51
Chain rule of differentiation, 115
Closed-form functions, 140
in Hertz contact theory, 136–137
Closure panel, 32
Coefficients, of second fundamental form, 97
Complete vehicle model, 248–253
Cone, mass moments of inertia, 59
Conformal contact, 17, 127
Conicity. See Wheel conicity
Constant curve, 28
Constant forward velocity constraint, 265
Constrained dynamics, xii, 1, 9–11, 167–168,
188–189
degrees of freedom in, 10
Constraint approach, to wheel/rail contact, 161
Constraint contact formulations, 19–20, 165–166
constrained dynamic equations, 167–168
contact constraints, 166–167
Constraint equations, 67, 206
and degrees of freedom, 62
Constraint forces, elimination through embedding
techniques, 78
Constraint formulations
augmented constraint contact formulation
(ACCF), 195–201
embedded constraint contact formulation
(ECCF), 201–205
numerical algorithms, 194–195
Constraint Jacobian matrix, 68, 76, 79, 199, 204
with Cartesian coordinates, 271–272
Constraint library, 199
Constraint stabilization method, 189–191, 200
difference from penalty method, 191
Contact angle, and force balance, 7
Contact area
adhesion region and slip region, 145
assumption of elliptical shape, 145
on elastic half-space, 314
elliptical shape of, 128
per Carter, 148
per Johnson/Vermeulen, 150
shape of, 128
Contact bodies, 129
under externally applied normal load, 133
schematic representation, 163
Contact-constraint conditions, 179
Contact constraints, 166–167, 173
Contact ellipse semi-axes, 135, 139, 152, 282
Contact equations, 313–317
Contact formulations, comparison of, 178–179, 180
Contact frame, 141
Contact models, 127–128
and Hertz theory, 128–140
wheel/rail contact approaches, 145–147
Contact plane, 130
Contact point, 177
absolute velocity vector of, 302
determining location of, 249
global velocity vector of, 144
location of, 104, 105
and surface parameters, 105
tangents to surface at, 162–163
velocities at, 140
Contact pressure
in Hertz theory, 133–137
maximum, 139
CONTACT program, 151
Contact theories, 1, 17, 128
creep forces, 17–18
three-dimensional, 28
wheel/rail creep theories, 18
Continuation method, 242
in static analysis, 246–247
Coordinate partitioning, 169, 192
Coordinate systems, in track geometry, 104
Coulomb's friction, 145, 153
Coupling. See also Kinematic coupling
between surface parameters, 185–187
Creep-force models, 127–128
Carter's theory, 147–149
and Hertz theory, 128–140
heuristic nonlinear model, 153–157
Johnson and Vermeulen's theory, 149–150
Kalker's linear theory, 150–153
Kalker's USETAB, 159
Polach nonlinear model, 154–156
simplified theory, 156–159
theories of, 147
Creep-force reduction coefficient, 154
Creep forces, 5, 17–18, 18, 22, 127, 140–142, 206,
207, 209, 277
calculation of, 210–211
example, 143–145
longitudinal and lateral, 146
simplified, 285
Creep spin moment, 18, 127
Creepage, 127, 140
for right wheel contact, 143–145
small area of slip, 150
Creepage coefficients, 292
Kalker's, 152
Creepage linearization, 291
background, 291–295
Euler angles and, 298–300
linearization assumptions, 300–301
longitudinal and lateral creepages, 301–304
Newton-Euler equations and, 306–309
spin creepage, 305–306
transformation and angular velocity, 295–298
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Index
Critical damping, 191
Critical speed, 4, 28
comparative computer code calculations, 297
consequences of inaccurate predictions, 291
role of wheel/rail contact in, 236
and wheel conicity, 289
Cross-level variation, 30, 32
as cause of freight car derailments, 31
Crossing panel, 32
Crossings, 32
Cubic splines, 212
Curvature, 13, 269
in curve theory, 91–92
defined, 106, 107
radius of, 92
surface, 99
Curve segment
lengths of, 218
Curve smoothing, 213. See also Smoothness
technique
Curve theory, 12, 89, 90
arc length and tangent line in, 90–91
curvature and torsion in, 91–92
Curve-to-curve spiral, 112
Curve-to-tangent exit spiral, 15, 112
Curves
parametric representation, 12, 90
rail segment type, 112
Cusp deviations, 220, 225, 226, 227
Cylinder, mass moments of inertia, 59
Cylindrical joints, 65
constraint Jacobian matrix, 68
joint constraints, 64–65
Cylindrical wheels, conicity in, 26
D
D'Alembert's principle, 35
interpretation of methods, 85–86
Damped sinusoid deviations, 221, 225, 226, 227
Damping coefficients, 235
Damping matrix, 235
Deformation, 133, 138. See also Local
deformation
Degrees of freedom, 11, 77, 161, 174, 202. See
also Independent generalized coordinates
in constrained dynamic systems, 10
for cylindrical joints, 64
eliminating with two-degree-of-freedom
model, 277–278
elimination by imposing nonconformal
contact conditions, 20
and number of equations of motion, 85
reduction by constraint equations, 62
335
single-degree-of-freedom model, 272–276
with trajectory coordinate system, 272
Dependent coordinates, 11, 77
Derailment, 1
due to cross-level variation, 31
L/V ratio and, 308, 310
Derailment criteria, 6
Development angle, 15, 108, 109, 114
Deviations, 219
and measured data, 219
track deviations, 220–222
Differential and algebraic equations (DAE), 3, 62,
188
Differential geometry, 12–14
Direction cosines, 9, 37, 38–39
Discontinuities, response to, 27, 32–33
Driving constraints, 77
Dynamic curving, 27, 31–32
Dynamic equations
linearization of, 23
Dynamic formulations, 35–36
augmented formulation, 66–70
embedding technique, 76–80
general displacement, 36–37
interpretation of methods, 80–86
joint constraints, 62–66
Newton-Euler equations, 58–61
rotation matrix, 37–48
trajectory coordinates, 70–76
velocities and accelerations, 49–57
virtual work, 86–87
E
Elastic approach, 175
to wheel/rail contact, 161
Elastic bodies, 128
in Hertz theory, 128
Elastic contact formulations, 20, 165, 179, 180,
249
using algebraic equations, 174–176, 205,
206–208
using nodal search, 177–178, 205, 208–210
Elastic deformation, 140. See also Deformation
Elastic force model, 19
Elastic formulations, numerical algorithms, 205
Elastic half-space, 146
contact area, 314
Electrodynamic suspension (EDS), 231–232
Electromagnetic suspension (EMS), 236, 237
inductance of, 240
modeling of, 237–240
Elliptic surface, 98
assumptions in Hertz contact theory, 145
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Railroad Vehicle Dynamics: A Computational Approach
Elliptical contact area, in Hertz theory, 128
Elliptical integrals, 319–320
Embedded constraint contact formulation
(ECCF), 166, 171, 174, 179, 180, 185, 194,
201–205, 204, 248
equations of motion, 173–174
position analysis, 172
in static analysis, 244–245
surface parameter elimination in, 201
Embedding technique, 10, 11, 36, 62, 76–78, 77,
162, 188
and elimination of constraint forces, 78
reduced-order model, 78–79
Equality constraints, 161
Equations of motion, 15, 75, 116, 180
for arbitrary body, 274–276
for ECCF, 173–174
as generalized trajectory coordinates, 264
Newton-Euler, 264
specialized formulations, 264–265
Euler angles, 9, 37, 41–44, 70, 74, 120, 214, 256,
292, 299
creepage linearization and, 298–300
singular configuration, 43
time derivatives, 257
Euler parameters, 9, 37, 45–47
identities of, 55
Even-order splines, 212
External force vectors, generalized, 61
F
FASTSIM program, 158
Federal Railroad Administration Code, 27
Free body diagram, 82
First derivatives, 114
in wheel geometry, 124–125
First fundamental form, 14
in surface geometry, 95–96
Flange contact, 268
Flexible bodies, modeling with finite element
method, 3
Force balance, in wheel climb, 7
Force of attraction, in Maglev trains, 237
Force vector, 188, 235
Forward velocity, 303
Fourth-order derivatives, 74
Frenet frame, 92
Frictionless surfaces, in Hertz theory, 128
G
Gage, 15
defined, 106
Gage limits, by track class, 224
Gage variation, 32
Gage widening, 7, 8
Gap contours, in Hertz theory, 132
Gaussian curvature, 14, 101
Gaussian elimination, 77
Gaussian procedure, with full or partial pivoting,
194
General displacement, 36–37, 256
body coordinate system, 258–259
generalized trajectory coordinates, 259–260
trajectory coordinate system, 256–257
General elasticity theory, 146
Generalized coordinates, 9, 10, 35, 168, 191, 200,
202, 204
partitioning method, 201
using Euler parameters, 46
Generalized coordinates partitioning, 189,
191–193
Generalized external force vectors, 61, 240
Generalized force vectors, 280
Generalized impulse-momentum approach, 178
Generalized mass matrix, 265, 280
Generalized Newton-Euler equations, 60
Generalized orientation coordinates, 51–52
Generalized trajectory coordinates, 259–260, 269
Generalized velocity vector, 72, 209
Geometric profile assumptions, 281
Geometry problems, 1, 11–12
differential geometry, 12–14
rail and wheel geometry, 14–16
Global position vector, 37, 71, 144, 163, 297
of arbitrary body point, 259
of body center of mass, 258
of center of mass, 259
Global velocity, 144
Grade, defined, 106
H
Hemisphere, mass moments of inertia, 59
Hertz coefficients, 136
for Hertz force, 137
Hertz contact theory, 128, 145, 206
computational example, 139–140
computer implementation, 138–140
contact pressure in, 133–137
and creep-force models, 18
geometry and kinematics, 128–130, 132
Hertz force law, 137
Heuristic nonlinear creep-force model, 153–154
High rail, 28
High-speed trains, 1
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Index
Higher-order derivatives, 120, 203, 216
of constraint equations, 197
or profile functions, 212
of position vectors, 211
and smoothness technique, 211–214
with trajectory coordinate-based
formulations, 265
Horizontal curvature, 15
relationship to rotation angle about vertical
axis, 109
Hunting, 24, 27, 28
and two-degree-of-freedom model, 277
Hunting frequency, 287
Hunting phenomenon, 6
effects of wheel/rail contact on, 236
and lateral instability, 22
and motion stability, 24
Hybrid methods, 208
Hyperbolic surfaces, 98
I
Implementation, 187
Independent generalized coordinates, 77, 191,
identification of194
Independent generalized velocities, 207
Independent velocities, 200
Inertia
mass moments of, 58, 59, 86
principal moments of, 60
products of, 58
vector of quadratic, 67
virtual work of, 87, 167
Inertia coupling, between translational and
orational displacement, 58
Initial static equilibrium configuration, 242
Instability phenomenon, 289
Intermediate wheel coordinate system, 181–182,
184, 293, 300
Euler equations in, 307
transformation matrix defining, 182
J
Jacobian matrix, 93, 169, 188, 196, 202
of kinematic constraint equations, 52, 270
Jog deviations, 220, 225, 226, 227
Johnson and Vermeulen's theory
contact area in, 150
of creep forces, 149–150
Joint constraints, 62
cylindrical joints, 64–65
prismatic joints, 65–66
337
revolute joints, 63–64
spherical joint, 62, 63
K
Kalker's coefficient, 156
Kalker's linear theory, 153, 158, 282
of creep forces, 150–152
creepage and spin coefficients, 152
example, 152–153
Kalker's USETAB, 159
Kinematic constraint equations, 18, 25, 74, 162,
266
avoiding violation of, 189
interpretation of methods, 80–83
linearization of, 23
with trajectory coordinates, 265
Kinematic coupling, 183
between surface parameters, 185–186
Kinematic linearization, 291
Kinematic step, in rail/wheel geometry, 89
Kinematic variables, 12
Kinetic forces, 12
Klingel's formulas, 26, 287
Kutzbach criterion, 11
L
L/V ratio, 6
in derailment criteria, 310
problems in wheel climbs, 7
use in derailment criteria, 308
Lagrange-D'Alembert equation, 9, 35, 86
Lagrange multipliers, 10, 66, 84, 168, 170, 171,
200, 202, 205, 208, 209, 240
vector of, 67
Lagrangian formulation, 9, 35
Lame's constant, 148
Lateral contact force, error in, 294, 311
Lateral creepages, 301–304, 304
nonlinear vs. linearized formulations, 304
Lateral force, 282, 283, 284
ratio to vertical force, 6
Lateral instability, and hunting phenomenon, 22
Lateral motion, 26
Lateral oscillations, 6
Lateral stability, 288
Levitation force, characteristics of, 239
Line search method, 242
in static analysis, 245–246
Linear constraints, 266
Linear hunting stability analysis, 280–287
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Linearization techniques, 1
assumptions, 300–301
Linearized creepage expressions, 23, 284, 306
Linearized kinematic/dynamic equations, 23
Linearized vehicle models, xii, 23–24
Local deformation, in contact theory, 128
Longitudinal contact force, nonlinear vs.
linearized formulations, 309
Longitudinal creep forces, 285, 301–304, 304
nonlinear vs. linearized formulations, 303
Longitudinal force, 30, 282, 284
Longitudinal shift, 186
Longitudinal tangent vector, 120
LU factorization, 78
M
Maglev forces, 236
electrodynamic suspension (EDS), 231–232
electromagnetic suspension (EMS), 237
modeling of electromagnetic suspensions,
237–240
multibody system electromechanical
equations, 240–241
Magnetic levitation forces, 6
Magnetomotive force, 238
Mass matrix, 208
Mass moments of inertia, 58, 59
Material modulus of rigidity, 148
Material nonlinearities, 4
Matrix identities, 295–296
Measured data, 222–223
deviations and, 219
Measurement errors, correcting with smoothing
splines, 214
Motion amplitude, limiting maximum value of, 31
Motion scenarios, 27–28
dynamic curving, 31–32
hunting, 28
pitch and bounce, 31
response to discontinuities, 32–33
spiral negotiation, 30
steady curving, 28–30
twist and roll, 30–31
yaw and sway, 31
Motion stability, 24–27
Motion trajectories, 62
Multibody contact formulations, 161–162, 187
augmented constraint contact formulation
(ACCF), 168–171
comparison of, 178–179
constraint contact formulations, 165–168
elastic contact formulation-algebraic
equations (ECF-A), 174–176
elastic contact formulation-nodal search
(ECF-N), 177–178
embedded constraint contact formulation
(ECCF), 171–174
parameterization of wheel and rail surfaces,
162–165
planar contact, 179–186
Multibody railroad vehicle formulations
constraint contact formulation, 19–20
elastic contact formulation, 20
general, 18–19
Multibody system algorithms, 188
constrained dynamics, 188–189
constraint stabilization method, 189–191
generalized coordinates partitioning, 191–193
identification of independent coordinates, 194
for Maglev forces, 240–241
penalty method, 189–191
Multibody system contact formulation, 19
Multibody system dynamics, 2
generality, 2–4
implementation of railroad vehicle elements
into, 6–8
nonlinearity, 4–5
N
Nadal's formula, 6
Natural frequency, 26
Newton differences, vector of, 243
Newton-Euler equations, xii, 9, 58–61, 260, 264,
291, 293
creepage linearization and, 306–309
effect of linearization on, xiii
generalized form, 60
of motion, 36
Newton-Raphson algorithm, 170, 175, 179, 184,
201, 203, 204, 210, 245, 246, 247
in static analysis, 242
Newtonian approach, 9, 35
Nodal points, relative distance between, 208
Nodal search, 174, 179, 205, 208–210
elastic contact formulation using, 177–178
Nonconformal contact, 17, 127, 166
in Hertz theory, 128
Noncontact driving concept, 236
Nongeneralized coordinates, 168
surface parameters as, 171
Nonlinear constrained multibody formulation, 288
Nonlinear creep-force model, 154, 293
Nonlinear dynamic analysis, 1, 161
Nonlinear equations of motion, 74, 292
for spin creepage, 305
vs. linearized, 291
45814_Idx.fm Page 339 Tuesday, June 5, 2007 2:24 PM
Index
Nonlinear force-creepage relationships, 128
Nonlinearity
of multibody system dynamics, 4–5
sources in wheel/rail contact problem, 153
Normal component, 51, 176
Normal contact force, 201, 206, 268
estimating for wheel/rail interactions, 161
nonlinear vs. linearized formulations, 308
right wheel of front wheelset, 253
Normal curvature, 99–100
Normal plane, 92
Normal stress distribution, 133, 155
Normal vector, 89, 121, 124
in surface geometry, 94–95
Normalized relative velocities, creepage as, 141
Numerical algorithms
augmented constraint contact formulation
(ACCF), 195–201
constraint formulations, 194–195
elastic contact formulation using algebraic
equations (ECF-A), 206–208
elastic contact formulation using nodal search
(ECF-N), 208–210
elastic formulations, 205
embedded constraint contact formulation
(ECCF), 201–205
Numerical comparative study, 247
complete vehicle model, 248–253
simple suspended wheelset, 247–248
O
Orientation angles
derivatives of, 115
first, second, and third derivatives of, 120
Orthagonality condition, 39
Oscillating plane, 92
Oscillation period, 26
P
Pantograph/catenary systems, 8
Parabolic surfaces, 98
Parameterization
track geometry, 163–164
of wheel and rail surfaces, 162–163
wheel geometry, 165
Penalty method, 189–191, 200
comparison with generalized coordinate
partitioning method, 201
difference from constraint stabilization
method, 191
339
Penetration, 205, 207
calculating for ECF-A, 176, 186
determining with surface parameters, 209
Pitch, 27, 31, 256
Pitch angles, 22
Pitch motion, 31, 75
Pitch rotation, 181
Pitch velocity, 300, 303
Planar contact, 162, 179–181
condition of, 183
intermediate wheel coordinate system,
181–182
modified formulation, 186
Planar surfaces, 98
Plateau deviations, 220, 225, 226, 227
Poisson's ratio, 134, 282
Polach nonlinear creep-force model, 154–156
Position analysis, 172
Pressure distribution
in Hertz theory, 133
semi-ellipsoidal, 134
Primary hunting, 28
Primary suspensions, 3, 227
Principal axes, 60
Principal curvatures, 14, 100–101
Principal directions, 100–101
Principal moments of inertia, 60
Principal relative radii of curvature, 129–130
Principal transverse radii, 138
Prismatic joints, joint constraints, 65–66
Products of inertia, 58
Profile, 223
defined, 107
Profile deviations, 220
Profile frame
location and orientation example, 122–123
transformation matrix defining orientation of,
109–110
Profile irregularities, 223
Profile limits, by track class, 225
Projection, of space curve on horizontal plane, 108
Pure rolling, 81
Q
Quadratic velocity vector, 197, 198, 203, 270
Quasi-linearization technique, 289
R
Radius of curvature, 92, 129–130, 138
Rail arc length, calculating during nodal search,
177
45814_Idx.fm Page 340 Tuesday, June 5, 2007 2:24 PM
Rail deviations, 107, 222
Rail elevation, 15
Rail geometry, 14–16, 89, 103–106, 207
computer implementation, 111–116
and curve theory, 90–92
and surface geometry, 92–102definitions and
terminology, 106–107
and track geometry, 108–111
track preprocessor and, 116–123
Rail lateral surface parameter, 14, 103
Rail longitudinal surface parameter, 14, 103
Rail profile geometry, 23
Rail radii of curvature, 138
Rail rollover, 7
geometric parameters in, 8
Rail segments, length of, 119
Rail space curve, changing coordinates of, 221
Rail surface
and coordinate system, 103
parameterization of, 162–163
track geometry, 163–164
Rail turnouts, 32, 33
Railroad input entries, defining curved track, 117
Railroad vehicle elements, 228
implementation in multibody-system
dynamics, 6–8
Railroad vehicles, 1–2, 3
and constrained dynamics, 9–11
contact theories, 17–18
general multibody formulations, 18–20
geometry problem, 11–17
linearized models, 23–24
motion scenarios, 27–33
motion stability and, 24–27
and multibody system dynamics, 2–8
specialized formulations, 20–23
Rectifying plane, 92
Reduced-order model, 78–79
Redundant coordinates, 80
Relative angular velocity, 140
Revolute joints, joint constraints, 63–64
Rheonomic constraints, 77
Right rail, data sets for, 121
Right wheel contact, creepage values for, 143–145
Rigid bodies
contact among, 127
coordinates of, 36
equations of motion for multibody systems of,
35
global position vector of, 37
orientation of, 36–37, 56–57
translational motion of, 36
velocity vector, 49
Rodriguez formula, 45, 46
Roll angles, 22, 256
Rolling contact, 141
exact theory of, 146
simplified theory of, 147
Rolling radii, 138
Root loci, 288
Rotation matrix, 37, 41, 258, 259, 298
direction cosines, 38–40
Euler parameters, 45–48
simple rotations, 41
Euler angles, 41–44, 42
Rotation parameters, 9
Rotational spring-damper-actuator element, 225,
227, 230–231
S
Scalar equations, 196
Scalar quantities, and Lagrangian approach, 35
Second derivatives, 211
in wheel geometry, 125
Second fundamental form, 14
in surface geometry, 96–98
of surfaces, 98
Second-order derivatives, 74
Second point of contact, 28
Secondary hunting, 28
Secondary suspensions, 3, 227
Series spring-damper element, 227, 231–232
Shear stresses, in Hertz theory, 133
Signatures, 225
Simple rotations, 41
Simple suspended wheelset, 247–248
Simplified creep-force theory, 156–159
Single-degree-of-freedom model, 272–274
Singular configuration, 53–54, 55–56
of Euler angles, 43
Singular points, 90
Singularity problem
avoiding with Euler parameters, 45, 47
with Euler angles, 44, 53–54
Sinusoid deviations, 221, 225, 226, 227
Skew-symmetric matrix, using Euler parameters,
45
Slender rod, mass moments of inertia, 59
Slip region
of contact area, 145
vanishing for small creepage forces, 151
Smoothing splines, 214
Smoothness technique, higher derivatives and,
211–214
Space curves, 13
Special elements, 187, 225–227
45814_Idx.fm Page 341 Tuesday, June 5, 2007 2:24 PM
Index
bushing element, 232–236
rotational spring-damper-actuator element,
230–231
series spring-damper element, 231–232
translational spring-damper-actuator element,
227–230
Specialized railroad vehicle formulations, 20–23,
255–256
equations of motion, 264–265
general displacement, 256–260
linear hunting stability analysis, 280–289
single-degree-of-freedom model, 272–276
trajectory coordinate constraints, 265–272
two-degree-of-freedom model, 277–280
velocity and acceleration, 260–263
Sphere, mass moments of inertia, 59
Spherical joints, 63
constraint Jacobian matrix, 68
joint constraints, 62
Spin coefficients, 140, 151
Kalker's, 152
Spin creep moment, 146, 282, 284
Spin creepage, 301, 305–306
nonlinear vs. linearized formulations, 295
Spin moment, 17
nonlinear vs. linearized formulations, 310
Spiral negotiation, 27, 30
Spline representations, 213
Stability, and wheel geometry, 16
Static analysis, 242
augmented constraint contact formulation,
242–244
continuation method, 246–247
embedded constraint contact formulation in,
244–245
line search method, 245–246
Static compression, 128
Static equlibrium, algebraic equations, 243
Steady curving, 27, 28–30
Stock rail, 33
Stress distribution, normal and tangential, 155
Successive rotations, and Euler angles, 41–44
Super-elevation, 15, 108, 117
defined, 106
Surface curvature, 99
Surface gap, 129
Surface geometry, 13, 89, 92–94
first fundamental form, 95–96
normal curvature in, 99–100
principal curvatures and principal directions
in, 100–102
second fundamental form, 96–98
tangent plane and normal vector in, 94–95
Surface mapping, 94
Surface parameters, 13, 93, 163, 164, 171
341
in ACCF, 199
coupling between, 185–186
as dependent variables, 204, 205
determining from iterative Newton-Raphson
procedure, 203
determining value of penetration using, 209
eliminating from equations of motion, 171,
185, 201, 204, 244
eliminating in elastic contact formulations,
166
as independent variables, 170
second derivatives of, 205
selecting as degrees of freedom, 200
Surfaces theory, 12
Suspended wheelset, 247–248, 286, 287, 292
model, 288
model data, 294
and track deviation, 293
Suspension elements, 10
Sway, 27, 31
Switch panel, 32
Switches, 32
Symmetric coefficient matrix, 80
System accelerations, in two-degree-of-freedom
model, 279
System mass matrix, 67, 199
System symmetric mass matrix, 188
T
Tangent, rail segment type, 112
Tangent line, in curve theory, 90–91
Tangent plane
in surface geometry, 94–95
at wheel contact point, 124
Tangent-to-curve entry spiral, 15, 112
Tangent vectors, 121
Tangential component, 51
Tangential contact force, nonlinear vs. linearized
formulations, 309
Tangential creep forces, 127, 201
Tangential forces, 145
Tangential stress distribution, 155
Thin circular disk, mass moments of inertia, 59
Thin plate, mass moments of inertia, 59
Thin ring, mass moments of inertia, 59
Three-dimensional contact theory, 28, 147, 162,
166
Three-dimensional rotations, and singularity
problems, 53–54
Three-layer spline, generation of, 212
Time derivatives
of absolute generalized Cartesian coordinates,
271
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342
Railroad Vehicle Dynamics: A Computational Approach
and absolute vectors, 51
angular velocity vectors as functions of, 297
of Euler angles, 54, 257
of Euler parameters, 56
of nongeneralized surface parameters, 170
of orientation parameters, 258
Tongue rail, 33
Torsion, 13
in curve theory, 91–92
Total applied force, in Hertz theory, 133
Track centerline, 121
coordinates of, 117
Track classes, 223–225
alignment limit by, 224
gage limit by, 224
profile limit by, 225
Track deviation functions, 219, 220–221
Track deviations, 220–222
and measured track data, 222–223
and track quality/classes, 223–225
Track geometry, 15, 164
coordinate systems in, 104
descriptive, 108–111
Track irregularities, parameters of, 226
Track perturbation, 31
Track preprocessor, 116–117, 214–216
input, 117–118
length change due to curvature, 216–218
numerical integration, 118–119
output, 120–121
use of output during dynamic simulation,
121–123, 218–219
Track quality, 223–225
Track Safety Standards, 223
Track segments, 15
linear representation, 112–113
Track super-elevation, 15
Traction displacement relationship, 146, 150–151
Trajectory coordinate constraints, 265–266, 269,
270
numerical example, 266–268
and use of Cartesian coordinates, 269–272
Trajectory coordinates, 21, 22, 36, 70–72, 75, 162,
181, 210
advantages and disadvantages, 255
constraints on, 265–272
and general displacement, 256–257
generalized, 259–260
velocity and acceleration, 72–74
Transformation matrix, 258, 297, 298–299
and orthagonality condition, 39
and simple rotations, 41
using Euler angles, 43
using Euler parameters, 45
Translational spring-damper-actuator element,
225, 227–230, 228
Translational stiffness, 235
Traveled distance, of wheelset, 267
Tread contact, 268
and force balance, 7
Trough deviations, 220, 225, 226, 227
Turnouts, 32
Twist and roll, 27, 30–31
Two-degree-of-freedom model, 27–280, 281, 286
Two-dimensional contact theory, 147, 179
Two-layer spline, generation of, 212
U
Unit vectors, 38
and direction cosines, 38
in simple rotations, 41
USETAB software, 159
V
Vehicle dynamic response, 222
Vehicle-track interaction, 6
Vehicles, defined for railroad, 2–3
Velocities and accelerations, 260
acceleration vector, 50–51
angular, 262–263
center of mass acceleration, 261–262
center of mass velocity, 260–261
generalized orientation coordinates, 51–53
singular configuration, 53–57, 55–56
trajectory coordinates, 72–74
velocity vector, 49–50
Velocity, of center of mass, 260–261
Velocity transformation matrix, 77, 264, 269, 280
Velocity vector, 49–50
with single-degree-of-freedom model, 273
Vertical bump, track with, 252
Vertical contact forces, 268
error in, 294, 310
Vertical displacement, of trailing wheelset, 250,
253
Vertical force, ratio of lateral force to, 6
Vertical motion, constraints on, 76
Virtual work, 86–87, 167, 230, 242
W
Wheel angular parameter, calculating for nodal
search, 177
Wheel conicity, 24, 283
effect on degree of stability, 289
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Index
Wheel geometry, 14–16, 16, 23, 89, 123–125,
165, 207
computer implementation, 111–116
and curve theory, 90–92
definitions and terminology, 106–107
first derivatives, 124–125
second derivatives, 125
and surface geometry, 92–102
Wheel profile, 184
Wheel radii of curvature, 138
Wheel/rail contact, 6
approaches to, 145–147
dependence of stability on, 292
dynamic and quasi-static theory of, 147
elastic vs. constraint approaches to, 161
exact theory of rolling, 146
model of, 12, 89
points of, 178
simplified theory of, 147
three- and two-dimensional theory, 147
vs. Maglev forces, 236
Wheel/rail creep theories, 18
Wheel/rail separation, 7, 178, 205
Wheel rolling radius, 148
Wheel separation, 180
343
Wheel surface
mathematical definition, 123
parameterization of, 162–163
track geometry, 163–164
Wheel velocity, at contact point, 81
Wheelset
lateral displacement, 249
single, 281
traveled distance, 267
Wheelset frame, 143
Wheelset hunting motion, 25
Wheelset rolling radii, 24, 25
Wheelsets, 10
lateral motion in, 25
Y
Yaw, 22, 27, 31, 256, 277
Yaw angle, 25
rate of change, 26
of wheelset, 249
Yaw moment, 282, 283, 285
Yaw oscillations, 6
Young's modulus of elasticity, 134, 316
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