2. The Dynamical System
Mechatronical Modelling
Tamás Dr. Szabó
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Mechatronikai modellezés
by Tamás Dr. Szabó
Publication date 2014
Copyright © 2014 Miskolci Egyetem
A tananyag a TÁMOP-4.1.2.A/1-11/1-2011-0042 azonosító számú „ Mechatronikai mérnök MSc tananyagfejlesztés ” projekt keretében készült. A tananyagfejlesztés az Európai Unió támogatásával és az Európai Szociális Alap társfinanszírozásával valósult meg.
Kézirat lezárva: 2014 február
A kiadásért felel a(z): Miskolci Egyetem
Felelős szerkesztő: Miskolci Egyetem
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Table of Contents
3.5. System Optimization (cp. Chapter Identification and Optimisation) ...................... 12
5. Analysis of Typical Problems of System Dynamics by Means of State Equations ............ 32
3.1. Solution of homogenous Systems, Fundamental Matrix ........................................ 53
1.6.3. Stability Area of Runge-Kutta Methods of Order 1≤p≤4 ........................... 94
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Mechatronical Modelling
1.7.1. 3. Order Runge-Kutta, Two Calculations of an Integration Step ............... 95
1.14. Comparison of Methods for Numerical Solution of Initial Value Problems (IVP) 112
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Mechatronical Modelling
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List of Figures
5.5. Eigenvalues of the quarter-car where 1500 Ns/m ≤ d
≤ 6500 Ns/m . ....................................... 68
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Mechatronical Modelling
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List of Tables
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Chapter 1. Introduction
This manuscript contains descriptions of modelling and simulation methods and aims at understanding the characteristics of complex (and other) dynamical systems in order to influence them effectively.
Consequently, it is required to know how to apply methods of modelling and analysis of such systems, their subsystems and components reasonably. This lecture deals with:
1.
mathematical formulations of dynamical systems,
2.
modelling techniques,
3.
simulation,
4.
numerical methods to integrate differential equations and to solve non-linear equation systems,
5.
identification and estimation of system parameters,
6.
stabilizing characteristics of dynamical systems.
In this context it is important to emphasize that the subject systems are not part of one particular discipline (e.g. mechanics), but that in order to describe such systems academic collaborations between different methods of
branches of study have to be made, cp. [System descriptions in different fields of study .].
Figure 1.1. System descriptions in different fields of study [1].
1. Introductive Examples
The following examples will be covered in lecture in more detail.
Example 1.1: ABS – Magnetic Valve, [ABS- magnetic valves, interfaces.]
System: ABS
Subsystem (components): Magnetic valve
Potential objectives:
1.
Describe the time-dependent process x(t) of the armature .
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2.
Where and how one can exert influence on the dynamic behaviour (e.g. a faster build-up of x to 90% of the total travel)?
3.
What other objectives of non-dynamic nature exist in this context?
4.
What other opportunities exist to influence the dynamic characteristics in a positive way (what is the meaning of “positive” in this context?)?
The construction of the ABS involves different disciplines of engineering, e.g.:
1.
mechanics (brake mechanism, driving dynamics),
2.
hydraulics (valves, pumps, pipes, ...),
3.
electrotechnology (controller, power electronics, …),
4.
control theory (ABS-algorithm).
Figure 1.2. ABS- magnetic valves, interfaces.
System: Percussion hammer
Subsystem (component): Crank mechanism
Potential objectives:
1.
Maximize the impulse velocity of the percussion.
2.
Minimize the vibrations of the hand grip while keeping the percussion power at a constant level.
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3.
How must the components of the crank mechanism be designed in order to achieve good results of drilling?
4.
Is there a potential risk of breakage caused by heavy dynamic (working voltage) loadings on the components of the crank mechanism?
Figure 1.3. Above: Percussion hammer [2]; Below: Simulation model of a percussion
hammer.
2. The Dynamical System
Examples of the colloquial usage of the term system are solar system, transportation system, computer system, cardiovascular system, nervous system, etc.
In this context and in the fields of engineering and science, the term system is used in connection with complex and in transparent operations and processes. In general, we can observe a collaboration of different disciplines.
2.1. Definitions
A system is a set of elements (parts, components, objects),
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1.
Which mutually influence each other (interaction),
2.
Which are subjected to external influences and affection (input) and
3.
Which effect to the external (output).
Annotation 1.1:
Science provides access to various definitions of the term system, e.g.:
An object is identified as a system as long as all particular elements and all attributes with their interdependencies (to outside the system also) are seen to be components of that whole in some logical sense.
Additionally, there are axioms.
Elements are e.g. modules, components, objects, fractions.
Attributes are properties, qualities, features, characteristics and also interfaces between the system and its environment. The term state characterizes the constitution of the system at any given time.
A system is a well-defined assembly which consists of interacting entities. This assembly is delimited by a cladding plain or boundary (a set of system elements with interfaces to outside the system). The cladding plain provides for an interface between the system and its environment. The relationships between attributes and states, transported by the interfaces mentioned above, are variables which describe the peculiar characteristics of the system.
A block is a way of illustrating a system. The borderline represents the cladding plain.
Figure 1.4. Representation of a system as a block.
Interfaces are
1.
input quantities ,
2.
output quantities and
3.
interfering quantities , if applicable.
Closed systems are self-contained and maintain no connections with the environment, i.e. outside events have no influence on the system. The total state of the system is defined by n state quantities .
Definition 1.4:
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State variables are those variables in a system, which completely describe the system’s behaviour. State variables are time-dependent. The term system state refers to the entirety of all values of the state variables.
Annotation 1.2:
1.
State variables are “internal variables” of a system.
2.
There is no definite choice of state variables. But characteristics can be assigned to the chosen state variables, such as uniqueness, independence and freedom of redundancy. From the latter characteristic it follows that the choice of state variables is absolutely arbitrary; however, the values are fixed.
3.
In practice, one is not interested in the entirety of all state variables, but only chooses the variables needed for the application; the so-called output quantities. Output quantities can also result from a combination of state variables.
Table 1.1. Examples of state variables in different disciplines.
Electrotechnology Mechanics Process engineering Environmental engineering
Current Position Temperature Population
Voltage Velocity Mass fraction Environmental state
Load Acceleration CO
2 production
Kinetic energy Ozone value
Potential energy
Example 1.3: Motor Vehicle System
In a motor vehicle system, the variables which emanate from the driver are e.g. the steering wheel angle and the brake and acceleration pedal travels which represent the input quantities. The variables describing the motion of the vehicle are defined as the state variables. Combinations or subsets of the quantities mentioned above are
Figure 1.5. System of a motor vehicle [5].
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Introduction
Figure 1.6. Degrees of freedom of the spatial twin-track model [6].
Example 1.4: Framework System
Elements of the system are specific bars of the framework. Internal interactions between the bars are nodal forces. Input quantities are external loads and reaction forces in the bearings can be defined as output quantities.
Example 1.5: Road Traffic System
Specific motor vehicles are the system elements in a delimited road traffic network. The driver and his motor vehicle can represent the subsystems. Internal interactions can be comparative distances and relative velocities of the vehicles to each other.
Definition 1.5:
A system with more than one input quantity and more than one output quantity is defined as a multi-quantities system (multi-input-multi-output system “MIMO”).
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Table 1.2. Examples of system elements from different disciplines.
Electrotechnology Mechanics Process engineering
Resistance
Capacitor
Coil
Mass
Spring
Damper
Vessel
Valve
Pipe line
Population development
Birth
Death
Disease
Transistor
Amplifier
Filter
Beam
Bearing
Guidance
Dead stop
Stirrer vessel
Filter
Reactor
Consumption
Position actuator
Force actuator
The form of interaction between the system elements defines the system structure. A system can be structured into subsystems (components).
The structuring process into subsystems has the advantage that we obtain a better overview of the behaviour of the system and that the subsystems can be constructed by different people. The structuring process should be carried out in such way that the subsystems can be reintegrated into the total system. This makes the clear definition of the interfaces inevitable, which also represents one of the major problems of industrial treatment of complex dynamical systems.
The system boundary (cladding plain) is arbitrary. But in general, limitations are introduced where a clear distinction to the environment exists, with a few uniquely defined relationships that can be observed and where the system is reactionless to its environment.
Definition 1.6:
The motion of dynamical systems is a change of the variables of the system with respect to time. Mechanical motion (changes referring to place) represents a special case of all general types of motions.
Furthermore, the term motion appears in literature a number of times, [7]:
System parameters are quantities which remain constant for the system during the observation period. Examples are given by natural constants, spring constants and damping factors, resistance, etc.
Environmental influences are quantities which affect the system from the outside, but which are not influenced by the system in return (reactionless). Examples are changes in temperature in big rooms, forced movements, etc.
Initial values of the state quantities.
Rate of change of a state quantity refers to the rate at which the value of the state quantity changes in course of time. Examples are given by velocity, acceleration, rate of birth and death of a population, mass flows, pressure changes, etc. According to the first example (velocity), state variables can be both rate of change of state quantities and state quantities themselves. This becomes important in mechanics.
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Introduction
In mathematics, dynamical systems are described by differential equations, algebraic equations of n-th order and
2.2. System Classification
Definition 1.7:
A system is
1.
static, if the conditions inside the system boundary (cladding plain) do not change in runtime. The model of a static system bases upon algebraic equations.
2.
dynamic, if the present state of definitely depends on its initial state and the input quantity in the time slice , with
.
Consequently, dynamical systems always comprise of storage elements like for e.g. energy, mass, information, etc. The characteristics of dynamical elements are therefore described by differential equations or differential-algebraic-equations.
3.
a system with concentrated parameters, if system quantities such as masses, capacitors, springs, damper, etc. can be represented by the integral of particular constants; i.e. if the system quantities are not locally dispersed
(e.g. spring-mass-pendulum). In dynamics, such systems are illustrated by common differential equations. If such a simplification is not permitted (e.g. if we have to take the local dispersion of the elements into consideration), the system is referred to as a system with distributed parameters. In this case system description takes place by means of partial differential equation (e.g. oscillating beam).
4.
time invariant, if its parameters remain constant in runtime. A system is time variant when the parameters do change in course of time (e.g. rockets which change their masses due to fuel consumption or plastic parts whose modulus of elasticity constantly drops due to increasing temperature)
5.
causal, if its output signal at an arbitrary point in time only depends on the value of the input signal at that time and before it (referred to as principle of causality = principle of cause and effect). This lecture exclusively deals with causal systems.
6.
deterministic, if the system’s behaviour can be predicted with hundred percent certainty using equations. If a system is stochastic, a system’s behaviour can only be predicted by means of probability calculation and statistics.
This lecture concentrates on the dynamic behaviour of (mainly technical) systems. In this context, the term dynamics must be defined in more detail. In historical context, the term dynamics (Greek: dynamis = force) originally derives from the study of changes in motions as a result of forces acting on the mechanical systems, and vice versa (inverse dynamics). In general, dynamics refers to the analysis and description of changes in time and, thereby, the phenomena which appear and how they proceed, under influence of general stimuli and incidents (e.g. forces, mass flows, voltages, death, birth, etc.). In fact, dynamics means motion, change.
3. System Models
A model is a simplified version of the complex reality which aims at analysing specific target functions. Models are usually not unique. Models are for e.g. the basis of system simulation.
There is no model which represents an exact copy of a real world system. Behaviour of the real system and its model always diverge. This divergence is called modelling mistakes (error).
3.1. Model Types
Different types of investigation (especially in case of different depths of investigation) require different types of models:
1.
object models: copy of an object on a smaller scale; e.g. reduced version of vehicle and aeroplane models for researches on wind tunnels,
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2.
conceptual models;
3.
mathematical and physical models.
This lecture mainly concentrates on the mathematical and physical model types.
Example 1.6:
1.
Scale reduced models of vehicles, aeroplanes and ships allow us to conduct experiments on their dynamical characteristics in wind channels and water tunnels. Here, similarity laws of fluid mechanics are used: Even if the fluid or the measurements do diverge, at a constant Reynolds number, surge flow similarity will always be observed.
2.
Analogous computer models, with electrical circuits which serve as reference systems. This procedure is based on analogies between mechanical and electrical systems, e.g. let the differential equations of single mass pendulums in mechanics be with deflection , mass , damper , rigidity and imposed force .
1.
If we equate the stored loads of a capacitor with the deflection , the inductance with the mass , the resistance with the damper , the capacity with the reciprocal spring stiffness and the adjacent/ fitting voltage with the imposed force . Subsequently, we obtain the equation:
1.
Hardware-in-the-Loop simulation (HIL) with mathematical models which are coupled with subsystems of the target system, e.g. electrical controllers. In fact, the physical component is not a model, but an essential part of the target system.
3.2. System Analysis
The analysis of a target system aims at identifying input and output characteristics of the system in order to understand the system and, where necessary, to develop methods to exert an influence on it.
Basically, there exists the opportunity to stimulate the system, which means to memorise the input quantities for the time elapsed and to measure the output quantities.
This procedure frequently proves to be difficult because
1.
in many cases the target system does not exist (innovation of new products) or does not exist anymore
(reconstruction of casualties).
2.
geometric admeasurements of the system are either too small or too large to perform measurements
3.
adequate sensors to measure the characteristics of the system are not available or are too expensive
4.
appropriate experiments take too long because the system is too languid
5.
the experiment with its target system is too dangerous, too expensive or not justifiable for other reasons (biomechanics, crash tests, threshold driving tests)
6.
the system is unique and should not be, is not allowed to be or cannot be stimulated (economic systems, biology, weather, …)
3.3. System Simulation
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Introduction
target system (on grounds which were mentioned above or for other practical reasons). Instead, it is applied to an appropriate model.
This procedure is also called system simulation, which de facto means that experiments are conducted on the
shows a basic schema on which such an analysis can be regulated:
Figure 1.7. Principle process of system simulation.
The process of simulation generally refers to the conduction of experiments on the basis of mathematical models of dynamical system instead of real world systems.
Simulations have the following advantages, i.e.
1.
the understanding of the functions of the system can be deepened
2.
in some cases, system simulation is faster and cheaper than conducting and evaluating experiments
3.
simulations are reproducible and are normally also comparable
In order to conduct experiments on mathematical simulation models, first of all equations have to be formulated and solved. To set up an equation, fundamental physical laws from different disciplines are required, e.g.
1.
Euler’s Equations and Newton’s Law,
2.
the Law of Conservation of Energy,
3.
the Material Law,
4.
d’ Alembert’s Principle,
5.
the Maxwell Equations
6.
Navier-Stokes Equations
Equations]-[Numerical Methods with Dynamical Systems].
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In order to develop a simulation system (which means: formulation and solving of equations), the model requires basic simplifications, e.g.:
1.
substitution of the elastic bodies with distributed masses by rigid bodies with concentrated mass quantities
(mass and moment of inertia),
2.
replacement of non-linear material laws by linear ones,
3.
use of simple, linear kinetic relations instead of the actual non-linear ones,
4.
Negligence or highly simplified description of friction effects.
3.4. System Identification
(cp. Chapter [ Identification and Optimisation])
A systematic, computerised method to approximately identify system parameters is the process of system
identification (or parameter identification), [Process of parameter identification.].
An appropriate system model and available measurements of the input and output characteristics of the target system are prerequisites. In the following, we will identify a system model with a physical system model, which was constructed on the basis of physical laws. In situations involving complex issues (e.g. description of the flow ratio of air in the passenger compartment of a motor vehicle in order to interpret the climate control), we often employ other descriptions, e.g. neural networks which extensively abandon physical approaches.
On the basis of measurements the unknown parameters are varied in such a way that we obtain the best possible conformity between the real system and the model.
Adequate mathematical optimisation methods are used in order to identify the unknown parameters (cp. Chapter
[ Identification and Optimisation]).
This method is particularly is used when parameters are difficult to measure, e.g.
1.
friction factors and material damping,
2.
heat transfer coefficients,
3.
flow resistance, etc.
Further physical quantities such as masses, spring stiffness, geometric data, material properties (elasticity modules, viscosity, etc.) are taken from tables, data sheets, CAD drawings and models.
In general, as many parameters as possible are measured directly and the rest will be identified by parameter identification.
Parameter identification can be applied with limited success. In most cases, it depends on the appropriate choice of the models and the availability and quality of the measurements.
Figure 1.8. Process of parameter identification.
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3.5. System Optimization (cp. Chapter [ Identification and
Optimisation])
System optimization aims at creating a plan of a new system with characteristics which are optimised in respect
to fixed criteria (cp. [Process of system optimization.]).
A mathematical model will be parameterised in such a way that the parameterised model definitely satisfies the criteria of the target behaviour. The target behaviour will be described by target solutions of given input and output signals.
Deviations between the actual behaviour of the model and the target behaviour will be determined by simulating with test signals. A performance function is defined as an admeasurement of the difference between the actual solution and the target solution, which is identified as target function too.
In fact, we can observe similarities between the process of system identification and the process of parameter identification. Both aims at defining the parameter set which guarantees an approximate conformity of the actual characteristics with the measured value
(system optimization).
(parameter identification) and the target value
The process of system optimization will be discussed in Chapter [ Identification and Optimisation].
Figure 1.9. Process of system optimization.
3.6. Model Schemes and Block Diagrams
Model schemes are detailed descriptions of systems which (e.g. in mechanics) are strongly oriented on representation. The way of representing the model plan depends on the correlation between the problem and the discipline of engineering. Because of the increasing importance of interdisciplinary applications
(“mechatronics”), these schemes should, if possible, be substituted by universal diagrams (e.g. block diagrams).
However, one faces considerable difficulties by doing so with complex systems (especially those deriving from the field of mechatronics).
Block diagrams are descriptive representations of model equations. This consequently means that they provide an allegorical representation of response relationships between two quantities (signals) of a system.
Following assignments are essential:
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1.
Signal ↔ arrow of action
2.
operation ↔ block
A block diagram consists of
1.
the blocks,
2.
directed arrows of actions (lines of actions) representing outgoing and arriving signals to the blocks. The arrowhead, directing to the particular direction of action, indicate whether it deals with input or output signals,
3.
the title of input and output quantities,
4.
summation points represented by little circles,
5.
signal branch points represented by little points.
Further details of the structure and the elements of block diagrams are standardized in DIN19226.
Figure 1.10. Example of a block diagram (single mass pendulum).
Particularly in the fields of control theory, block diagrams are wide spread. Draft programmes and simulation programmes support the graphical direct input of block diagrams. System equations will be put together by the
Block diagrams can also be divided hierarchically.
Example 1.7: Wheel Suspension [Simple model of a wheel suspension.]
Figure 1.11. Simple model of a wheel suspension.
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Model Equations
Newton’s equation for the physical construction:
By means of the equilibrium condition where we obtain:
Introducing a new model variable, we obtain the linear equation of motion,
and out of it the block diagram in [Block diagram for suspension, possible refinement levels.].
Figure 1.12. Block diagram for suspension, possible refinement levels.
Example 1.8: Electrical Circuits , [Electrical low pass-filter.]
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Figure 1.13. Electrical low pass-filter.
Introduction
Model Equations
Kirchhoff’s loop rule:
Capacitor:
Connection between current and load: .
As a result, we obtain: , , thus,
Block Diagram
Figure 1.14. Block diagram for electrical low-pass filter, possible refinement levels.
Signal Flow Diagrams are mainly preferred in English-speaking countries. They have a very simple structure and can be easily drawn. The disadvantage is that the representation of these diagrams is highly limited to linear correlations.
Elements of signal flow diagrams are
1.
connections with signed amplification factor
2.
coupling points represented by system variables
Figure 1.15. Signal flow diagrams for a) suspension and b) low-pass filter.
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4. Task of System Dynamics
Task 1: Modelling
Formulation of mathematical relationships which describe the system behaviour: Modelling is always connected with abstraction and idealisation.
Examples: mass point, rigid body, mass-free spring, “free of time lag” position actuator …
Depending on the given task, the same system requires different modelling depths and therefore also different idealisations.
Example 1.9: Vehicle Model
1.
Analysis of driving dynamics: Complex multibody system (degree of freedom, e.g. roll, pitch, yaw, translation). The drive train and with it also the longitudinal dynamics are frequently neglected,
(substituted by a default longitudinal vehicle motion)
2.
Analysis of jerking: Rigid body (structure) with four wheels, suspension is neglected, but a detailed description of the drive train.
3.
Vehicle simulation: Detailed model of the whole vehicle,
4.
Crash simulation: Detailed FEM-model of the car body and the load-bearing components.
Task 2: Model Research
Research on system characteristics
Examples: stability, response time, functionality survey …
Task 3: Design of Controlling Inputs
The inputs of the system must be designed in such a way that the desired target system characteristics are achieved.
Examples:
1.
In order to set weld points correctly, motor currents of welding robots have to be regulated
2.
In order to achieve a sufficient impulse velocity of the percussion piston, the drive piston of a hammer drill must be set in motion.
Task 4: Simulation of System Behaviour
Conducting experiments in reality is frequently not possible due to various reasons (e.g. new vehicle constructions (no prototypes), a high level of danger (crash tests, possible environmental damage), long-term duration (population development)),
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5. Summary
The systems discussed in modelling and simulation originally stem from different fields: Technical areas:
1.
mechanics
2.
electrotechnology
3.
hydraulics
4.
pneumatics
5.
informatics
6.
etc.
Non-technical areas:
1.
social sciences
2.
business administration
3.
economics
4.
biology
5.
mathematics
6.
meteorology
7.
etc.
In system dynamics, the systems will be analysed with similar methods and standardized criteria.
Consequences:
1.
System dynamics is interdisciplinary.
2.
Similarities between systems from different disciplines are utilized in order to make use of uniformed analysis techniques.
3.
In order to represent uniformed systems from different disciplines, general description techniques are preferred.
4.
But: In course of time different disciplines (e.g. mechanics, electrotechnology) developed own description techniques (and software packages) which are irreconcilable to each other: Exemplifying a complex mechanical system which is difficult to be represented as a block diagram. Therefore, different forms of description on mathematical system level have to be retained in order to link these systems.
5.
We need description languages which are easy to learn and which are unique.
6.
Trend: Substitution of mathematical description forms by diagrams: a.
block diagrams, b.
program plan, c.
Bond-graph, Rosenberg and Karnopp [8]
d.
Multiport and Cut Set Method e.
etc.
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The knowledge of basic equations and solution formalism is fundamental, to guarantee a thorough understanding of the application possibilities and the limits of programmes for system analysis.
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Chapter 2. Mathematical Description of Dynamical Systems
1. Differential Equations
The time-dependent behaviour of a dynamical system can be described by differential equations (DE) or, in more general cases, by differential-algebraic equations (DAE). Our approach is limited to such systems which can only be analysed by differential equations. Systems, which are only describable by differential-algebraic
detail.
Definition 2.1:
An ordinary differential equation is a relation which contains a function of a conditional equation with only one independent variable, and one or more of its derivatives with respect to that variable.
Systems with concentrated parameters (e.g. multi-body-systems) are described by ODEs.
Example 2.1: DE (motion equation) of a Mathematical Pendulum, [Mathematical pendulum.]
Figure 2.1. Mathematical pendulum.
From the principle of conservation of angular momentum with mounting point 0 we first obtain by substitution:
(2.1)
(2.2)
equation (s. passage [State Equations]) of this system.
Definition 2.2:
A partial differential equation involves a relation which contains a function (of a conditional equation) with several independent variables and its partial derivatives with respect to those variables.
Systems with (locally) distributed parameters (e.g. continuous mechanical systems) are described by partial DE.
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Mathematical Description of
Dynamical Systems
Example 2.2: Longitudinal Oscillations of a Continuous Beam, [Longitudinal beam oscillations.]
Figure 2.2. Longitudinal beam oscillations.
Because the relevant physical characteristics of the beam (rigidity, mass and, if necessary, variable cross section) are distributed over the whole beam length, a partial differential equation is necessary to describe the dynamics of the system.
The equation to describe the longitudinal waves of a slim beam is
With
: density
: elasticity modulus
: beam cross section
: beam length
: total mass
Next to the derivative of the spatiotemporal deflection with respect to time, derivatives with respect to the position coordinate x also appear. The exact solution of such equations would go beyond the scope of this lecture and will therefore not be discussed for the time being.
Annotations 2.1:
We will approach a system with distributed parameters with an appropriate spacious discretisation of a system
instead of a partial differential equation, we obtain a system with ODEs.
Figure 2.3. Discrete model of the description of longitudinal oscillations.
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Mathematical Description of
Dynamical Systems with and
In matrix form we obtain:
.
The Matrix is called the stiffness matrix of a discrete system.
This lecture mainly deals with systems with concentrated parameters. In this case, the dynamical system can be described by ODEs:
(2.3) or resolved in respect to ,
Annotation 2.2:
1.
In some cases the transition from implicit to explicit equation form is not possible analytically.
(2.4)
2.
In practice, there exist some systems which can only be described by intercoupled differential and algebraic equations. Examples are given by multibody systems with kinematical loops.
3.
The highest number of derivatives a differential equation contains is called its order.
4.
To solve a differential equation of -th order, we have to determine all the continuously differentiable functions which together with their derivatives satisfy the DE.
5.
In system dynamics, the unknown function depends on time. The derivative of is therefore, in the following, marked by a dot over the variable ( ) instead of an inverted comma ( ).
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Mathematical Description of
Dynamical Systems
(2.5)
(2.6)
Figure 2.4. Simple frequency response system with concentrated parameters
(multybody system).
2. State Equations
By introducing the state variables and the substitutions
(2.7)
equation (2.4) can be transformed to DE of first order.
Example 2.4: One Dimensional Oscillator
The motion equation of a one-dimensional oscillator with the undamped eigenfrequency the external excitation is as follows:
, the damping and
By substitution
,
, we obtain a DE-system of first order:
,
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Mathematical Description of
Dynamical Systems
It is useful to generally use this transformation in order to carry further research methods on DE systems of first order.
Definition 2.3
The general state equation for a finite dimensional, non-linear and time continuous dynamical system with state variables and output quantities is:
State equation
(2.8)
Output equation
(2.9)
Where
– state vector
– output vector
– control vector
non-linear - respectively - vector function
Figure 2.5. Block diagram of the non-linear state equation.
Example 2.5: Point Mass
Principle of linear momentum:
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Substitution:
Mathematical Description of
Dynamical Systems
State equation:
Substitution:
State equation:
Here, the force excitation represents the control factor .
Example 2.7: Mathematical Pendulum, [Mathematical pendulum.]
State quantities:
(position)
(velocity)
State equation:
Introduction 2.8: Logistical Growth at Constant Crop, [7]
State quantity: (crop, e.g. fishes)
Control factor: (removal, e.g. fishing)
State equation: : capacity limit
Figure 2.6. Block diagram: Logistical growth at constant crop.
Annotation 2.3:
1.
The non linearity of the state equation is predetermined by the structure of the dynamical system, e.g. the powers of the state variables or non-linear characteristic lines.
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Mathematical Description of
Dynamical Systems
2.
Analytical solutions of non-linear state equations are not possible for most parts. A numerical approach is normally necessary (simulation).
3. Stationary Solutions and Equilibrium Positions
Definition 2.4:
The solution that
of a dynamical system is called a stationary solution, if for the excitation it is found
(2.10) i.e.
.
Where represents an equilibrium position of the free (non excited, uncontrolled) system.
Annotation 2.4:
Numerical Solution of Non-linear Sysem Equations].
Example 2.9: Mass Point
and i.e. an equilibrium position is achieved when the excitation force vanishes and the point originally rests in equilibrium.
Example 2.10: Spring-Mass Oscillator with Viscous Damping
, where
The quantity is the tendency by which the mass drops as a result of only the gravitational force.
Example 2.11: Mathematical Pendulum
i.e.
This example shows that the equilibrium positions have different characteristics, if for e.g. the equilibrium positions and are compared to each other.
Example 2.12: Logistical Growth at Constant Crop
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Mathematical Description of
Dynamical Systems
The equilibrium positions are defined as follows:
Especially when we obtain:
.
4. Linear State Equations
In reality, we are more often than not confronted with non-linear systems. In non trivial cases such systems mostly imply considerably difficulties. This is why one is interested in investing high efforts to approximate non-linear relations by linear ones and to linearize such equations. Then we obtain linear approximations which reflect the behaviour of such systems, except for certain errors. But these approximations are significantly easier, i.e. in the first place manageable and resolvable. Generally, we observe that linear approximation relations are valid only to a certain extend, and that the error grows with an increasing degree as much as the scope of validity is transgressed.
.
The principal of linearization can easily be described by a sufficiently smooth function of a variable
Figure 2.7. Linearisation of a scalar function of a variable.
This function can be approximated by a straight line in the environment of the point. The environment of the point shall be analysed with respect to its system behaviour.
(2.11)
By the use of coordinates transformation
(2.12) we finally obtain the homogenous and linear relation
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Mathematical Description of
Dynamical Systems
(2.13)
A general method to linearize functions (more than one variable) is the Taylor series expansion (will be discussed later in more detail).
Example 2.13: Sinus Function
The function above and obtain
should be linearized around where . We proceed as described
,we obtain a linear relation with the constant proportionality factor After transformation
.
Example 2.14: Mathematical Pendulum
For small angle we obtain a linear pendulum equation:
In order to specify the term “linear” we use the following
Definition 2.5:
A function is linear, if the following characteristics are applicable:
This definition is also applicable, if and are vectors and is a vector function.
Description of additive characteristics in a block diagram:
Figure 2.8. Block diagram of additivity of a linear relation.
: .
(2.14)
(2.15)
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Mathematical Description of
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Example 2.15: Non-linear Functions
Figure 2.9. Non-linear force characteristic (in combination with play).
The linearization of the state equations can be accomplished in the proximity of any time-dependent target processes and (not necessarily constant!). Depending on the problem, it however makes sense to linearize in the proximity of reference processes which are in equilibrium positions i.e. steady solutions of a system.
,
with and , with and
,
which are each identified as typical quantities of the system. By inserting this relation in the nonlinear state equations and developing the functions and around the points second element in a Taylor series, we obtain:
and up to the
(2.16)
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Mathematical Description of
Dynamical Systems
As the reference variables must also satisfy the basic relations (2.8) and (2.9), it follows
and .
We take this into consideration in (2.16) and (2.17), and obtain the linear state equation
After introducing the Jacobian matrices: the linear equations
In long form we obtain:
(2.17)
(2.20)
(2.21)
(2.22)
(2.18)
(2.19)
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Mathematical Description of
Dynamical Systems
(2.23)
Instead of writing linear dynamical system:
and , we use and again and, at last, obtain the famous state equations of a with the matrices
– system matrix,
– input matrix,
– observance matrix (also: measuring matrix or output matrix),
– straight-way matrix.
(2.24)
(2.25)
We obtain a more compact form, if we build the matrices
:
and into an block matrix
(2.26)
By means of (2.24) and (2.25) we verify the typical characteristics of a linear dynamical system:
1.
Proportionality: With is true .
2.
Superposition:
Figure 2.10. Block diagram of linear state equations.
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Mathematical Description of
Dynamical Systems
Annotation 2.5:
A time-variant system is identified as such, if the matrices and do explicitly depend on time.
A time-invariant system is identified as such, if the matrices and are constant matrices.
A time-invariant system is often the case when
Example 2.16: Mathematical Pendulum.
.
(2.27)
(2.28)
Example 2.17: Undamped Logistical Growth
By linearizing around the equilibrium point , we obtain:
, and approximated around :
.
Example 2.18: Linearization to a Periodic Solution
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Mathematical Description of
Dynamical Systems with
.
The Jacobian matrix results in
,
.
Consequently, the linear system equations forms to
5. Analysis of Typical Problems of System Dynamics by Means of State Equations
Search for the function of on the control vector .
, i.e. the time-dependent characteristics of the state parameters, depending
We have to differentiate between three cases:
1.
Non-linear State Equations
In this case, the solution of the state equations is either impossible or only approximately determinable.
1.
Linear time-variant System
There is mostly only a formal approach to the solution. The difficulties are very similar to those occurring in the case 1.
1.
Linear time-invariant State Equations
Only in this case we have an explicit solution. But difficulties also occur with systems of higher order.
Because the explicit solution of the state equations is only seldom possible, research on the system has to be done in another way and, thus, we have to concentrate on the following particular questions:
Problem II: Stability (Chapter [ Identification and Optimisation])
A technical system is neither allowed to run away nor to explode. The state vector must therefore be finite. In this context, one deals with the question for which parameter values there can be “stability” in a broad sense of the definition :
for alternatively, when a system becomes unstable , i.e.
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Mathematical Description of
Dynamical Systems
for .
The question of stability must also be answered without solving the equation. We might face difficulties by doing this with non-linear and time variant systems whereas it is easy to make a stability statement for linear systems.
Problem III: Controllability (not be covered in the following)
The user must be interested to design a simple control of the system. In this context, the question emerges whether a system is controllable or not, i.e. whether the control value can be chosen in such a way that the system can be transferred from an arbitrary state into a target state linear systems, whereas we still have great difficulties with non-linear systems.
. This problem is solvable for
Definition 2.6:
A system of n-th order is completely controllable, if for any initial condition be assigned an input function defined at a finite point in time such that the solution trajectory with
and within the time interval
,which started in , satisfies the value .
and anystate , there can
,
Kalman’s criteria (Kern (2002)) can be provided in order to control this characteristic.
Problem IV: Optimisation (not be covered in the following)
The problem of controlling a system from an initial state to a given final state can usually be solved through different control programs. Here, the control should be chosen in such a way that we reach a process which is as cheap and as fast as possible. The outcomes are optimality criteria on which the optimal control should be based upon.
Problem V: Control (s. lecture: Automatic Control)
There exist two ways in which to determine the control method:
1.
as a function of time:
2.
as a function of state:
The first case refers to control in narrow sense whereas in the second case we speak of feedback, and will be produced by a controller. The determination of a controller which is as simple as possible and optimal in certain sense is the main problem of automatic control.
A simulation (from the Latin term simulatio = feint) is an imitation of a real system. The act of simulating is not based on the analysis of the real system but alternatively a model of the system. In literature, the term simulation
(in narrow sense) refers to the solution (mostly numerical) and interpretation of system equations.
Problem VII: Identification (Chapter [ Identification and Optimisation])
The theoretical approach of system analysis (deductive modelling; deductive = inference from the general to the special) is mostly insufficient because the system parameters are either difficult to determine or completely unknown. In these cases it is necessary either to experimentally identify the whole structure or the parameters of the mathematical model of the analysed system.
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Chapter 3. Differential-Algebraic-
Equation-Systems and Multiport
Method
1. Differential-Algebraic-Equation-Systems (DAE-
Systems)
Hitherto, system equations were always explicitly given, which means that the rate of change of the state vector could be calculated by a clear calculation rule consisting of the state and the state quantity .
In many applications, system equations according to (2.3) are only existent in implicit form:
(3.1)
DAE- systems represent an important special case: Here, the state vector x is composed of two partial vectors
and
, ref. Section [Equation Structure]:
(3.2) where refers to the vectors of the variables, whose derivatives are also part of the system equation.
The state vector summarizes all state quantities, whose derivatives do not appear.
A special case which often appears is the following system:
(3.3)
These special forms of differential-algebraic equations are also referred to as Hessenberg form.
If is solvable for , then can be inserted and transferred into an ODE-system (ODE =
Ordinary Differential Equation)
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Frequently this is exactly not the case: It could be, for example, that does not depend on . In this case can of course not be eliminated by . Here,
can only be eliminated by differentiating (3.3) once or several
times in respect to time.
Example 3.1: Non-linear Simple Pendulum as DAE-System
A mass point , which is only movable on the -plane, is suspended without friction and rotates at point with a massless beam with fixed length .
The distance between the mass point to point will be in the horizontal and in the vertical direction.
Figure 3.1. Non-linear simple pendulum.
The principal of linear momentum in and direction is as follows:
With the auxiliary quantity one obtains the following equation of motion:
One obtains a differential equation system of first order through substitution of :
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(3.4)
(3.5)
(3.6)
One then adds the kinematic constraint and subsequently the DAE results in:
Differential-Algebraic-Equation-
Systems and Multiport Method
(3.7)
(3.8)
(3.9)
Thus, one obtains the five equations (3.9) for the five variables
On separation of the state quantities one yields
and .
(3.10)
equations, we obtain:
(3.11)
Further differentiation of this equation yields:
(3.12)
By differentiating a third time with respect to time, one further obtains
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and therefore the ODE
Differential-Algebraic-Equation-
Systems and Multiport Method
(3.13)
(3.14)
The number of differentiations which are needed to transfer a DAE into its proper ODE is also called the Index of the DAE.
Definition 3.1:
The Index of the DAE is the number of differentiations that are needed to transfer a DAE into an ODE.
Annotation 3.1:
1.
The Index describes so-to-say the “distance” between a DAE and its illustration as ODE. The index of the mathematical pendulum therefore is 3.
2.
An ODE has therefore the index 0.
3.
DAE’s with an index higher than 1 are also referred to as DAE’s with a higher index.
4.
The index of a DAE can change along the solution (local index).
5.
The obstacles which occur during the numerical solution increase according to the level of the index.
The numerical approach, especially of DAE with higher index is often very difficult and complex. Thus, one
and, therefore, to eliminate from the remaining equations.
In this case we obtain the ODE:
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Furthermore, in this illustration the state quantities and are not dependent from one another and can (apart from their algebraic sign) autonomously be calculated
This state space representation can therefore also be called non-minimal.
Annotation 3.2:
This example seems to be more complex when illustrated in the descriptor form or even in the non-minimal
special state space representations are frequently picked out because
1.
the representation of complex systems in minimal coordinates is sometimes very laborious
2.
the equations in minimal coordinates are sometimes very complex and error-prone.
A great disadvantage of non-minimal illustrations is that the auxiliary condition (in the mathematical pendulum for e.g. ) is only considered in the differentiated form. The initial values of the (non-minimal) state equations have necessarily been chosen is such a way that they satisfy the kinematic auxiliary condition.
Furthermore, during the numerical solution of non-minimal differential equations the auxiliary conditions normally drift away. This makes a stability procedure necessary, which corrects the state quantities in the course of time so that the algebraic auxiliary conditions are again exactly satisfied.
The ability to efficiently and accurately solve this kind of equation is a requirement for the application of the methods described in the following section.
2. The “Cut-Set“ or “Multiport“-Method
2.1. Basic Idea
In contrast to block diagrams, the following model equations do not represent the structure of a model
connected to and interact with other components and the surroundings using appropriate interfaces (so-called
“Cuts”).
Figure 3.2. Simulation on the basis of state equations.
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Figure 3.3. Object-oriented modelling/simulation.
Each “Cut“ is therefore composed of two variables:
Figure 3.4. Block diagram.
With: across: a quantity will be measured between two interfaces; comparable with e.g. a voltage (= potential difference of an electrical circuit)
Figure 3.5. Difference measurement (across).
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Differential-Algebraic-Equation-
Systems and Multiport Method through: a quantity will be measured by the installation of an apparatus within the structure; comparable with the measurement of electricity in an electrical circuit.
Figure 3.6. Flow measurement (through).
Table 3.1. Classification of across and through qualities in various fields.
Field mechanics electrotechnology hydraulics/pneumatics
Across velocity voltage (potential difference) compression
Through
Force
Electricity
Circulatory thermo dynamics temperature entropy current
There is only one way to link objects on the basis of the “Cut-Set“ method:
Figure 3.7. Linkage of objects on the basis of the Cut-Set method.
To link blocks in an intersection the following two rules must be applied:
1.
all across variables are identical:
2.
the sum of through variables disappears: .
Note:
The connection is based on a real existing connection between the physical components.
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The Assembly to a Total System:
Single components are connected by means of “Cuts” and convey motion and force without power (apart from dissipative and driving effects which act as drains or sources)
Advantages:
1.
Here, the blocks and connections directly reflect the physical components and their connections.
2.
The mathematical model can be directly obtained by means of this illustration.
2.2. Equation Structure
A general block is based on the following two groups of equation systems:
1.
Ordinary differential equation
1.
Algebraic equation
Where:
(3.15)
(3.16)
1.
Variables whose time derivatives are defined in block ( = differential)
2.
Variables whose time derivatives do not appear in block ( = algebraic)
Input and Output Variables:
Variables whose time derivatives are defined in block can basically be regarded as input variables.
Some of these variables are defined as function of the rest by means of the equation (3.15).
Example of Algebraic Equation (Mathematical Pendulum):
Figure 3.8. Mathematical pendulum example.
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Hence we obtain:
: input or
: output
: input
: output.
with
Annotation 3.3:
Graphic presentation:
We also use a kind of block diagram of the system elements here. However, the blocks do not represent mathematical operations in this case (like in the classical kind of block diagram), but in each case they represent the physical object.
Figure 3.9. Graphical representation.
Sign convention:
Figure 3.10. Sign convention.
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Through variables always lead from cut to block. To fix the positive sign, we draw a bar over the cut-circle to indicate the direction of the positive flow.
2.3. Examples
2.3.1. Electrical Components
Example 3.2: Ohmic Resistance
Figure 3.11. Ohmic resistance.
Physical laws: junction rule
Ohm’s rule
Example 3.3: Inductive Reactance
Figure 3.12. Inductive reactance.
Physical laws: junction rule
Inductance
Example 3.4: Capacity
Figure 3.13. Capacity.
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Physical laws:
junction rule
capacity
Annotation 3.4:
Examples (a) to (c) represent so-called „two-ports“(widely used objects in electrical engineering).
Example 3.5: Simple Electrical Network
Figure 3.14. Simple electrical network.
Equation Structure:
Voltage source:
Ohmic resistance:
Inductive reactance:
Capacitance
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(algebraic)
(algebraic)
(algebraic)
(differential)
(differential)
Differential-Algebraic-Equation-
Systems and Multiport Method
Differential quantities contained in the equation system:
Algebraic quantities contained in the equation system:
From this follows the differential-algebraic system of equations:
2.3.2. Mechanical Systems
In mechanical systems, multi-ports can be taken as “kinetostatic” transmission elements which transfer motion and forces. Hence, the “cut” will be broadened.
Position, velocity and acceleration are kinematical quantities (across) and force is a static quantity (through).
Example 3.6: Spring
Figure 3.15. Spring.
Positive forces convey positive power with positive displacement of the junction to the surrounding.
From the equilibrium of forces follows, [Spring.]:
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The algebraic system of equations for the spring is:
Figure 3.16. Multiport model spring.
In this case the differential system of equations is inapplicable.
Example 3.7: Mass
Figure 3.17. Mass.
The differential equation of mass is as follows:
Figure 3.18. Multiport model mass
One also alludes to “Terminal” or “one-Port”.
Example 3.8: Spring-mass system.
Figure 3.19. Spring-mass system.
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One obtains the algebraic equation:
The differential equation is:
The three equations at hand contain four variables can be provided. A fixed restraint would be
. This means that one variable is arbitrary, i.e.
, a harmonic excitement .
Annotation 3.5:
Mechanical, electromechanical, hydraulic and similar components have a “natural” transmission direction with across and through transfers being in the reversed direction.
Figure 3.20. Multiport.
The “Cuts” are arranged with the incoming across on the left hand side and outgoing across on the right hand side. The connection between incoming and outgoing across variables are described by means of the matrix .
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Differential-Algebraic-Equation-
Systems and Multiport Method
(3.17)
Because of the absence of power loss in the multiport, it follows:
In matrix notation, incoming and outgoing across and through become vectors:
(3.18)
Hence, the equations (3.17) and (3.18) can be described compactly:
Equation (3.19) inserted in (3.20) results to:
By conversion , it follows:
As this is applicable for any the vector , it therefore yields
.
Thus, the transpose of conveys the through variable: In reverse direction to the across transmission.
(3.19)
(3.20)
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Chapter 4. Solution of State Space
Equations
1. Existence and Uniqueness of Solutions of Ordinary
Differential Equations
The following section deals with the solution of differential state equations. For simplifying the notation, we first consider a single equation of the form:
(4.1)
, we obtain the following solution If the function is furthermore independent from and continuous at
(4.2) according to the fundamental rule of analysis. To clearly define of the function (initial value)
, we need further information, e.g. the value
at the initial point. Hence, the solution is as follows:
(4.3)
The ODE (4.1) in combination with the initial condition
is said to be an initial value problem.
The application of ODE in dynamic problems only then makes sense, if
1.
the existence of a solution is generally guaranteed.
2.
there is only one solution to a given problem.
Both are provided by the following theorem for a wide range of problems:
Picard- Lindelöf theorem:
(4.4)
If is continuous at all points of an area:
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Solution of State Space Equations
, and there exist a constant (Lipschitz constant) with
(4.5) for all .
Then for every initial condition there exists exactly one solution with being continuous and differentiable for all .
for the initial value problem,
Thus when the general constraints are satisfied, exactly one solution of the state equation is guaranteed.
Annotation 4.1:
1.
The requirements of the theorem are especially then satisfied, if with respect to , then one can set :
in has a limited partial derivative
(4.6)
1.
This theorem can be generalized to ODE systems without any difficulties.
2.
Only the existence and the uniqueness of a solution are guaranteed. An analytical approach cannot be specified except in special cases. An important special case is represented by linear ODE. These can be specified by the form
(4.7) with a real factor , which is independent from , and an excitation vector uniqueness of a solution in this case always guaranteed?
. Why is the existence and the
This theorem is also applicable to the ODE of -th order described above; we obtain the following theorem:
Theorem 4.1:
The complete solution of a ODE of order
(4.8) contains arbitrary parameters (solution diversity). The total solution is comprehended as the general solution of the ODE. If one sets the parameters by choosing appropriate additional conditions, one then obtains a particular solution.
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Solution of State Space Equations
Although a general solution for non-linear differential equations is not impossible, it is at least very difficult.
Nevertheless it is possible to specify solutions in particular cases. In all other cases the only alternative is the numerical approach.
Example 4.1: Uninterrupted Logistic Growth
System Equation:
By separating the variables we obtain:
.
Both sides of the equation can now be integrated separately and first one obtains
By raising to the power of one obtains
With the initial condition we finally obtain
and therefore
.
The solution trajectory for
.
Figure 4.1. Solution of the equation for logistic growth (qualitative).
2. Solution Design in the Phase Plane
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Solution of State Space Equations
The behavior of dynamical systems can be illustrated by applying the time dependent progress of the state variable in a two-dimensional diagram over time.
One obtains a better overview of the system behavior by eliminating the time from the state equations:
This can be demonstrated by the following examples of a system with two state quantities:
(4.9)
(4.10)
(4.11)
By forming a quotient,
(4.12) time is eliminated formally and we obtain a differential equation with which is dependent from . The
space is called phase space in general case and phase plane in the case of a system with two degrees of freedom. In the phase space, the point dependent solution of the system.
passes through a trajectory which depicts the process of the time
Example 4.2: Linear Gyroscopic Pendulum, Mathematical Pendulum
State equations
(4.13)
Division one equation by the other one obtains an equation of the phase curves
(4.14) with an integration constant which is at first arbitrary and which finally results from the determination of appropriate initial conditions.
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(4.15) with and substituted for the initial condition on the right hand side.
In this case the phase curves are therefore ellipses with the semi-major axis and , with
Figure 4.2. Phase curves of the mathematical pendulum.
3. Solution Methods for Linear State Equations
3.1. Solution of homogenous Systems, Fundamental Matrix
Linear state equations can be solved to complete solutions in a relatively simple way. In order to avoid certain technical difficulties, the following section additionally assumes that the system matrix is only featured with simple eigenvalues, i.e. every eigenvalue appears only once.
The homogenous state equation
(4.16) with the solution
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(4.17) results in
one divides by and subsequently obtain the linear homogenous equation system:
(4.18)
Because of
(4.19)
Equation (4.19) defines a eigenvalue problem with the eigenvalue
and the eigenvectors
, which can be fixed except for one arbitrary scalar factor. The eigenvectors will be standardized in terms of a vector norm:
.
In the following one assumes that standardization in terms of Euclid took place:
.
The total solution can be represented by a linear combination of eigenvectors:
is as follows: The initial condition with
By summing up the eigenvectors in successive columns to a modal matrix the eigenvalues to a diagonal matrix
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(4.20)
(4.21)
(4.22)
and the constant to a column vector
Solution of State Space Equations we obtain the following statement
(4.23)
(4.24)
(4.25) for the general solution of .
The exponential function of the statement , will first be expanded from scalar independencies to a function of diagonal matrices. Obviously, this satisfies the definition of the Taylor series:
(4.26)
Furthermore it is true that the inverse form of the modal matrix exists (because of the independence of the eigenvector):
and
(4.27)
Hence, we obtain the total solution
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Solution of State Space Equations
The bracketed term represents the fundamental matrix:
To broaden the definition of the diagonal matrix it is also true
(4.28)
(4.29)
(4.30)
The fundamental matrix describes the transition from a state from an initial state , one also denotes with .
to a state . In a more general approach
The matrix is endowed with some important characteristics which widely coincide with scalar exponential functions:
1.
Differentiability:
2.
Regularity:
3.
Calculation rules:
.
.
,
Annotation 4.2:
(4.31)
, where the function will be applied element wise on the elements of the diagonal matrix . Further methods of calculation are based on the Taylor series and the theorems by Cayley and Hamilton.
Example 4.3: One-dimensional Oscillator
We have a system
,
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Solution of State Space Equations which describes the forced oscillations of a one-dimensional oscillator. We have to calculate the general solution of a homogenous equation.
The state equations are
and or in matrix form
(4.32)
The system matrix defines the eigenvalues and and the eigenvectors
.
and
We obtain a modal matrix
(4.33) and the fundamental matrix
(4.34)
The general solution of the homogenous equation is therefore with respect to (4.28):
with an arbitrary initial vector .
3.2. Solution of the inhomogeneous State Equation
If a system is influenced by an outside excitement , we obtain the following state equation:
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(4.35)
Solution of State Space Equations
(4.36) hereafter
will be assumed to be constant. The complete solution of (4.36) consists of the homogenous in
(4.37) and the inhomogeneous part
of the outside excitement, i.e. this yields:
(4.37)
For the particular solution, we can apply the method of variation of constants for the equation. This makes use of the following basic approach:
(4.38) with a “varied“ constant .
After differentiating (4.38) with respect to time we first obtain:
(4.39)
Because it yields:
Equation (4.40) can directly be integrated with respect to and we subsequently obtain
(4.40)
(4.41)
and one obtains a general complete solution: As one only needs any particular solution, one can equate
(4.42)
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The first summand on the right hand side refers to the free motion of the system with given initial conditions, while the second summand refers to the forced motion expressed by the control vector . It is very important for this application that the first summand converges to zero with time. This is the case if the free system
of the system is only described by the second summand.
Example 4.4: One-dimensional oscillator with excitement
(4.43)
and one obtains One equates the initial condition to zero. Thus, it yields
(4.44)
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Chapter 5. State Space Equations with Normal Coordinates
1. Normal Coordinates
Through appropriate transformation one can frequently simplify linear state equations. Obviously, one practically restricts oneself to linear transformation. Otherwise, one would have to transfer equations from linear to non-linear forms which would rather complicate the calculation.
A linear transformation of a state vector to a state vector can generally be described by the following statement
(5.1) with as a matrix. Furthermore, there must be a clear inverse transformation:
It follows that the matrix has to be regular, i.e. it satisfies:
(5.2)
One applies the transformation (5.1) to the linear state equation:
Thus, one first obtains
(5.3)
(5.4)
(5.5) and then, after left-hand multiplication with , we obtain the following state equation with
(5.6)
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(5.7) and
(5.8)
transformation derives from the fact that the matrices and possess identical eigenvalues.
Theorem 5.1:
The eigenvalues of a similar system are invariant against linear and regular transformations.
The goal of a transformation is such that after transformation, the solution and the interpretation of the system equations are simplified, i.e. the matrix should have a more “convenient” structure than the initial matrix .
The term “convenient” depends on the respective problem.
A sought after characteristic is that the couplings between the equations is as low as possible. If all eigenvalues of are different from each other, it would be possible to completely decouple all system equations from each other, i.e. the new system matrix has a diagonal structure.
Theorem 5.2:
If all eigenvalues of the with a regular transformation matrix . Then the system matrix with eigenvalues arranged diagonally.
matrix are different from each other, one has a linear transformation
would have a diagonal characteristic
This yields
(5.9)
A matrix with these characteristics is referred to as diagonalizable.
One obtains the transformation matrix by inserting the eigenvectors in columns horizontally, i.e. the matrix
is equal to the modal matrix
introduced in Section [ Solution of homogenous Systems, Fundamental
The transformed uncontrolled system each other
consists of n differential equations which are independent from
(5.10)
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Coordinates and the n-dimensional total motion z(t) is composed of single motions
(5.11)
The coordinates in (5.11) are referred to as normal coordinates, this yields
(5.12) which means that the vectors are standing vertically (normal) on top of each other.
In a system based on normal coordinates each eigenvalue affects exactly one coordinate, i.e. time-dependent changes of each main coordinate depend on the time-dependent change of the residual main coordinates.
The connection between the original coordinates (state vectors) and the normal coordinates is given by transformation:
(5.13)
Annotation 5.1:
Note that if complex eigenvalues (and therefore also complex eigenvectors) appear equations with complex components ensue, which elude themselves from direct precise analysis. This disadvantage can be eliminated if one takes into account that in real matrices complex eigenvalues are generally real or they generally appear as a pair wise conjugated complex:
(5.14)
The adequate equations would be
(5.15)
(5.16) with the solutions
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(5.17)
After addition and subtraction of (5.15) and (5.16) we obtain
(5.18)
(5.19) and with the new (real!) coordinates
(5.20) with the (real) equations
(5.21)
(5.22)
These two equations of first order can be summed up to a single equation of second order by differentiating
(5.21) with respect to time and by subsequently inserting (5.22)
.
This represents an equation of a linear homogenous oscillation in the normal form:
Summary
Interpretation of the Solution of Linear Systems by means of the Eigenvalues
The system matrix A generally possesses real eigenvalues:
(5.23)
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(5.24) and pairs of conjugated complex eigenvalues
(5.25)
Hence, the first equations are equal to the real eigenvalues and can be completely uncoupled. The residual equations are in each case pair wise coupled and can be substituted by equations of second order by eliminating each second coordinate. Each main coordinate therefore describes either a non-periodic motion or a
(damped or excited) oscillation.
Thus, one can make a complete statement about the possible motions of a linear dynamical system by means of the eigenvalues. The eigenvalues are material to the system behavior.
In graphical representation, the position of the eigenvalues, the root loci (sing. root locus), are drawn into the
root loci provide concise information about the change of the eigenvalues with respect to parameter change (root locus curves).
The root locus diagram also directly provides information regarding the stability of the solutions of the linear system. All stable solutions have their eigenvalues on the left hand side of the imagined axis, i.e. the eigenvalues have negative real parts, which guarantees for the decrease of the solution curve against zero, cp. equation
Apart from the stability aspect, the root locus diagram also provides information of the long-term behavior of
(autonomous) systems. This behavior will be determined by such solutions which experience the slightest damping (if the solutions are stimulated by initial conditions). The solutions that are affected by the slightest damping are those that are closest to the imaginary axis on the negative side. The eigenvalues of these solutions are referred to as dominant eigenvalues and the appropriate solutions of the state equation which are referred to as dominant solution.
Connection between the Eigenvalues with Eigenfrequency, Damping, Mass and Time Constant of a System
Each real eigenvalue has a so-called time constant with
Thus, the time response of the state variables of the non-excited system contains a part in the form with as an eigenvector which belongs to . Where
behaviour with real eigenvalues.].
the solution component decays, cp. [Decaying
Figure 5.1. Decaying behaviour with real eigenvalues.
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In [Decaying behaviour with real eigenvalues.] the so-called half-life is drawn in
.
Conversely, one can also calculate a real eigenvalue out of a measured half-life with a complex eigenvalue pair the system has a eigenfrequency with the appropriate damping degree
.
In this case, the time processes of the state variable contains a part in the form with the appropriate eigenvector belonging to the part decays where , thus . The quotient of two successive amplitudes is
[ Decaying behaviour with complex eigenvalues. ] represents a corresponding process.
Figure 5.2. Decaying behaviour with complex eigenvalues.
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One can calculate two conjugated complex eigenvalues of the system matrix from an eigenfrequency of the system and the corresponding damping degree:
Figure 5.3. Behaviour of linear systems in dependence of the position of the complex plane.
Example 5.1: Quarter-Car (cp. [9])
Figure 5.4. Quarter-car.
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Subject of the following example is the represented model of a quarter-car, [Quarter-car.].
The unsprang mass of a wheel (rim + hoop + brake + part of the suspension) is summed up to a single substituted mass which is reinforced by a spring on the street. The spring represents the vertical stiffness of the wheel. The damping of the wheel will be neglected.
Between the wheel mass and a further substituted mass (pro rata body, assembly, engine, gearbox, etc.) is a spring strut (structure spring and damper). Under the assumption of linear characteristic lines of the spring and damping elements of the wheel theorems:
and the assembly , the motion equations result from the impulse
As static equilibrium position one obtain
, as well as
One chooses the following state vector of the equilibrium position (static position) of the linearized system therefore, the following equations ensue
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The structural acceleration (comfort degree) and the dynamical wheel load (degree of driving security) are our initial quantities: with as static wheel load therefore,
In matrix form we obtain
To analyze the system behavior one chooses the following data
plane. Hereafter, we draw the root locus curves where .
Obviously, the four eigenvalues move towards the imagined axis with increasing damping. Finally they reach real values, which means that the system is asymptotically damped for sufficient large .
Figure 5.5. Eigenvalues of the quarter-car where 1500 Ns/m ≤ d
A
≤ 6500 Ns/m .
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2. Eigenbehaviour of Systems with Multiple
Eigenvalues
2.1. Effect of Multiple Eigenvalues
The previous considerations premise that the system matrix possesses linear independent eigenvectors.
This is surely the case if all eigenvectors reach values which are different from each other. If multiple eigenvalues appear, the diagonalisability cannot be guaranteed anymore. This will be exemplified by systems of second order:
The matrices
(5.27) both possess the double eigenvalue . If was diagonalizable, there would be a transformation matrix with
(5.28)
With
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(5.29)
(5.30) already implies the statements
, which means that the matrix is written on the form
(5.30) and is therefore non-regular. There also does not exist a transformation matrix which can be used to transfer to , which means that and are not similar. One finds the physical explanation of this behavior when one interprets the corresponding solutions of the equations of state. The equations belonging to the system matrix ,
,
The following system equations
.
with the solutions
belong to the system matrix .
Both solutions refer to completely different physical relations. This becomes more obvious by choosing
In this case both solutions yield
.
and respectively.
This means that the first system is in rest position, while the second one moves with constant velocity .
Basically, in this case it deals with different motions and it becomes obvious that through simple coordinate transformation of the rest position one cannot derive an even motion.
On the other hand, it shows that a system of second order with a double eigenvalue can be retransformed to one
of the two basic forms (5.27) through similarity transformation.
2.2. Jordan’s Normal Form
The diagonalization of the system matrix by means of the similarity transformation is only possible if all eigenvectors do exist.
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But if multiple eigenvalues with the multiplicity appear, this so-called multiple eigenvalue will have number of independent eigenvectors, whereby is the rank drop or damage that can be determined from the relation
The matrix is not diagonalizable in this case, but there is a transformation matrix anyway, with
(5.31)
(5.32)
The matrices
(5.33) are referred to as Jordan blocks. These differ from the diagonal structure in that they have “ones” in the first upper secondary diagonal. The number of Jordan blocks with identical eigenvalue is equal to the number of
exactly d i
Jordan blocks with on the diagonal.
Let the system equation be in normal coordinates.
One examines on the first Jordan block with length in order to solve the problem:
(5.34)
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If one starts with the last equation, one will successively obtain: etc.
On the whole one obtains in matrix form
Similarly, one can operate on the remaining Jordan blocks and obtain in normal coordinates and in original state coordinates.
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(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
State Space Equations with Normal
Coordinates
can be calculated in successive rows by the transformation of The transformation matrices
Starting with the first Jordan block again one obtains
(5.40)
(5.41)
The vector is also referred to as eigenvector of the eigenvalue , like before. The vectors also referred to the appropriate main vectors.
Example 5.2
The matrix
are has the eigenvalues and . Thus, the multiplicities of the eigenvalues are eigenvector of the first eigenvalue result from equation
and . The
to .
The residual eigenvectors and respectively main vectors remain the conditional equation
.
But the matrix on the left-hand side has rank two. Hence, it has a rank drop of 1, which means that one can calculate only one single linear independent eigenvector. The Jordan matrix results to with the Jordan blocks and
The system equation in normal coordinates is
.
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.
From the first Jordan block the following solution ensures
and, therefore, .
The second Jordan block leads to an equation chain and therefore
The first column of the transformation matrix is the first eigenvector . The residual row vectors result from to
and .
Thus, one obtains the transformation matrix and the solution of the initial equation
.
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Chapter 6. Numerical Methods with
Dynamical Systems
1. Introduction
1.1. Taylor Expansion
Developments of sufficiently smooth functions in the Taylor series will play an essential role in the following sections. Thus, the first part contains definitions and applications of the Taylor series.
A Taylor series is defined as a power series which is, in a certain radius around a specific point, a perfect representation of the function. Due to this, the comparison of the Taylor series with the numerical approximation-polynomial provides statements of the accuracy of the numerical method. This error is referred to as discretization error or truncation error.
Furthermore, the Taylor series allows for the derivation of new numerical methods by disregarding all elements of the series apart from the first ones. The resultant polynomial is called discontinued Taylor series.
Taylor Series of a scalar-valued Function
Definition 6.1
A function power series of
is said to be analytical at point
.
Theorem 6.1:
, if in a surrounding of , can be developed into a
A necessary condition for to be an analytical function is the stiffness of all derivatives of in proximity to .
Theorem 6.2:
, and its
. Then there would be a power series development of Suppose is an analytical function at point
(6.1) with , where would be the -thderivative of with respect to .
Example 6.1: Exponential function
Expansion of around with .
Interesting are the developments around .
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Example 6.2: Taylor expansion around the Zero Point (x=0)
Taylor Series of a Function with two Variables
(6.2) where
The appropriate process is applicable to the residual partial derivatives.
1.2. Numerical Algorithms
A numerical algorithm is a set of instruction to solve a mathematical problem by exclusively using the arithmetical basic operation. A numerical algorithm is not only applicable if the problem cannot be solved analytically, but also if a numerical approach is more appropriate for practical reasons.
The choice of algorithm mainly depends on the factors of speed (computing time) and accuracy. Here, the advancement of computer technology affects the speed-argument. Thus, this argument becomes less important in less complex problems and frequently steps back behind the premise of accuracy.
Example 6.3: Numerical Calculation of the Square Root
We are searching for the approximation of the solution of the following equation
(6.3)
A possibility to solve the problem is offered by Newton’s approximation method with the following calculation specification:
(6.4)
The application from (6.3) to (6.4) leads to the algorithm
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Systems and to the following approximation values (round up to 4 positions)
1.3. Rounding Error and Error Propagation
There is a basic distinction between various error sources:
1.
Input error, which means that the existing values (e.g. measured and estimation values), which are incorporated into the calculation, are already erroneous.
2.
Errors which came into being during the substitution of the real problem by an approximation method are referred to as truncation error.
3.
Errors which came into being during the process of round off are called round off error.
Input errors are basically not completely avoidable. Errors which emerged during the process of approximation depend on the utilized methods and will be discussed later on in detail.
Round off errors emerge in numerical calculations when infinitive decimal fractions, which appear in course of the calculations, are approximated by finite decimal fractions. With the use of computers, one refers to them as machine numbers. The use of machine numbers involves great difficulties because they are not compatible with basic arithmetic operations. This means that the result of a computing operation might not necessarily be a machine number again. Neither the associative law nor the distributive law are applicable to machine numbers.
The emerging errors can propagate and fortify in course of the calculus. Thereby, grave deviations from the true result might occur.
Example 6.4: Round off errors
The calculation of the term is precisely be analysed. It yields:
.
Once we substitute by the approximation , we obtain 4 (analytically identical) expressions with the following values: 0.005044, 0.005035, 0.005088, 0.020000.
The result which is exactly rounded off to its fourth decimal place is 0.005051. But the deviation is, as one can see, considerable. Because of the analytical equation of the results, we would expect identical values.
Furthermore, the following identities hold:
One obtains the following numerical values for the last two expressions of a 6 digit calculation:
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Obviously, the subtraction of nearly similar large numbers involves considerable errors which sometimes can be avoided by a different choice of algorithm.
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Chapter 7. Numerical Methods for
Initial Value Problems
1. Numerical Solution of Initial Value Problems
The solutions for the ordinary differential equations one has been dealing with till now must, at least for the non-linear problems, be solved by a numerical method solution using a digital computer.
In order to derive appropriate methods, one shall consider the following numerical method used to solve a scalar differential equation of this kind
(7.1) with the initial condition
(7.2)
The methods derived in the following sections can be extended to a differential equation of higher order by means of the previously introduced substitution method. The application of this method on systems of differential equation is also unproblematic.
The solution of initial value problems, in numerical methods, allow for the determination of solutions a series of discrete points in time (grid points) with
for
(7.3) where is the time increment which can generally change for every step. For the sake of simplicity , one shall refer to it as constant time increment h from now on.
1.1. Explicit Euler Method (Forward Euler Method)
difference term:
(7.4)
Figure 7.1. Forward Euler Method.
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Solving for , this yields an approximate formula for the value of at position
Thus, the solutions can be recursively calculated as follows:
(7.5)
(7.6)
Hence, an easily applicable and understandable method is made available. But the practical application of this method faces various difficulties:
1.
The error which emerges within the process of discretization (substitution of the differential by the difference quotient) strongly depends on the time increments .
2.
In some cases one can observe a trend of numerical instability. An analysis of this effect is based on the socalled test equation
(7.7)
The exact solution
(7.8) is always positive and converges with to zero. The least prerequisite for a useful approximation method is that it also converges to zero and always remains positive.
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If we apply the forward Euler Law on equation (7.7), we will obtain a solution at position
(7.9) which means that the approximation solution remains positive and it only disappears where converges to infinity, if it satisfies that .
If it yields , the approximation solution indeed disappears where converges to infinity. The approximation solution oscillates around the zero point value. The solution increases with each integration section and changes the algebraic sign at the same time where . One only obtains a meaningful numerical solution when . This characteristic of the Euler method is referred to as numerical instability.
Example 7.1: Numerical instability of the Euler method
The solution of the following equation should numerically be calculated.
The application of the Euler method in the interval with various increments reveals the results
the increment.
Furthermore, the solution oscillates for
and it finally diverges for
Figure 7.2. Application of forward Euler method.
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Figure 7.3. Forward Euler method; instability.
In order to analyze the characteristics of oscillating solutions, one frequently allows to be specified as a complex number, see the following section.
1.2. Numerical Stability and Stability Domain
For the stability analysis of a numerical integration method the linear test equation
(7.10) is applied. The exact solution of this initial value problem
(7.11) always remains positive and converges towards zero for . If one tries to solve the test equation by an integration method, the least requirement to obtain a useful approximate solution is that it also converges towards zero and remains ever positive.
Definition 7.1
A numerical integration method is called (numerically) stable in a domain of the complex number field, if the series time for
of the approximate solutions decreases absolutely at the points of
(according to the behaviour of the exact solution).
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Let’s analyze the explicit Euler’s method. By inserting the test equation (7.10) in (7.5) we obtain
According to the stability condition this method is numerically stable if
This is only valid for
(7.12)
(7.13)
(7.14)
I.e., must be inside a circle of radius (the so-called unit circle) around the origin. Thus, must be inside the circle of radius around the point , such that the explicit Euler’s method remains numerically
Figure 7.4. Stability domain of the explicit Euler’s method.
Definition 7.2
A numerical integration method for differential equations is A-stable or Absolute-stable if and only if its stability domain covers at least the complete left z-half plane.
Definition 7.3
A numerical integration method for differential equations is F-stable or faithfully-stable if and only if its stability domain is exclusively the left z-half plane.
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1.3. Modified Euler Method (Trapezoidal Rule)
The modified Euler Method results from the application of the Trapezoidal Rule on the integration of the function .
Figure 7.5. Derivation of the Trapezoidal Rule.
Trapezoidal Rule:
(7.15) applied on , we obtain with and :
(7.16) and therefore as a calculation rule:
(7.17)
In contrast to the Forward Euler Method, in this case the resulting calculation rule is not explicitly solved for the searched quantity . Thus this method is said to be an implicit integration method. The question emerges,
in practice.
Case 1: f is linear in x
Obviously, (7.17) can easily be solved for
.
Example 7.2:
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We obtain a calculation rule:
This special case provides the option to explicitly solve for . We obtain the following formula
Case 2: f is a non-linear function of x.
.
It is only possible in special cases to explicitly solve for . Instead, we must generally resort to numerical methods for the solution of non-linear algebraic equation systems. In this case, we have to have a good initial value for for an iterative solution. Thus, we apply an iterative method. One possibility is the method of
successive substitution and fixed-point iteration respectively (Section [ Fixed Point Iteration]):
(7.18) where identifies the value of iteration where and represents an appropriate initial value for iteration. We chose , as an initial value. Thus, the first iteration increment is identical with the formula of the explicit Euler Rule. As soon as the difference of the successive two iterations is , we cancel the iteration. Here, is a given accuracy bound which should be chosen near to the machine accuracy. However,
Point Iteration]) is applied alternatively as a rule.
Figure 7.6. Stability domain after applying the fixed-point iteration.
To sum up, we obtain the following calculation rule:
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(7.19) etc. until convergence arrives.
The question is why the modified Euler method has a higher accuracy and an improved stability behavior among numerical integration methods.
. In this case, we obtain the In order to answer that question, we have to consult the test equation following calculation rule
(7.20)
If we develop the coefficient of the factor in front of into a Taylor series, we obtain
(7.21)
If we apply the same analysis for the forward Euler-method, we obtain
as represented above. If we compare the same results (7.21) and (7.22) with the exact result
we obtain the following cancellation error for the modified Euler formula
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(7.22)
(7.23)
(7.24)
Numerical Methods for Initial Value
Problems and for the explicit Euler formula as
(7.25)
Thus, the Euler formula precisely describes the exact solution locally up to two elements of second order
(method of second order). In contrast to that, the forward Euler method only represents an approximation up to the first element of first order (method of first order).
A further important advantage of the Trapezoidal Rule is its stability behavior. One obtains the following equation for integration section:
(7.26)
, if it yields and the absolute value of the approximation solution exactly coincides where
(7.27)
But this is the case where all rule is -stable as well as -stable.
, i.e. the trapezoidal rule is stable for arbitrary step sizes. The trapezoidal
1.4. Implicit Euler method
We obtain the implicit Euler method by substituting the forward difference quotient by the backward quotient in the explicit Euler’s process.
Figure 7.7. The implicit Euler method.
The application of this method too needs an iteration to calculate .
1.5. Summary of Euler’s Method
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1.
The explicit Euler’s method represents a favored method which is very easily applicable and easy to program, especially in real-time operation. The local discretization error behaves like where
, the global error behaves like . A great disadvantage is the instability, which can, as the case may be, be eliminated by the choice of extreme short increments.
2.
The modified Euler method (Trapezoidal Rule) is -stable, the local discretization error behaves like , the global error like . A disadvantage (which is similar to all implicit methods) is the necessity to solve a non-linear equation system within each integration section.
3.
In respect to the discretization error, the implicit Euler method behaves like the explicit method, but in contrast to the explicit version, the implicit one has the advantage of being -stable.
1.6. General One-Step Procedures
The Euler methods discussed above are the simplest methods of the great class of one-step procedures. Here, approximations will only be calculated at the increments .
solely from the approximation at the points and
One-step procedures can generally be written on the form
(7.28) with the so-called process regulation .
Example 7.3
For the explicit Euler method it is true:
.
To measure the quality of one-step procedures, we utilize the following terms:
Initial value problem
.
This means that
is the local error which emerges in this step of the method defined in (7.28).
This procedure is consistent, if it yields
, it is of order where
.
Annotations 7.1:
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If the approximation of with the one-step procedure is calculated, satisfies a Lipschitz condition in respect to and the one-step procedure is consistent of
, the global error is true for
order with .
Expressed in words this means that the consistency order of the global error is always one order lower than the local error.
1.6.1. Classical Runge-Kutta Methods
A basic disadvantage of Euler’s method is the low accuracy achieved. This demands very short integration increments and leads to high computing times and an accumulation of round-off errors during the calculation.
Very early endeavors have been made to increase the degree of accuracy. An option is to calculate the righthand side of differential equations at additional interpolation points.
order to calculate the function value where :
in
(7.29)
We obtain the Runga-Kutta-method if we approximately integrate the right-hand side of (7.29)
Runga-Kutta Method of Second Order
The following statement results from the application of the Trapezoidal Rule in the approximation integration of
(7.30)
The value for is unknown, thus the term
Thus, we obtain the following formula
will be approximated by the explicit Euler method.
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Problems which will usually be formulated in the following calculation formula:
(7.31)
(7.32)
The Runga-Kutta method at hand is also referred to as a predictor-corrector method on the basis of the Euler method. In this context, the explicit Euler method plays the role of the predictor whereas the Trapezoidal Rule inherits the role of the corrector.
To determine the accuracy, with which this method discretizes the differential equation, we develop in the surrounding of with the Taylor series
(7.33) at the same time, the partial derivatives of f will be shortened by:
etc.
For means of comparison, we develop (7.31) in an appropriate Taylor series
(7.34)
proportional to .
Hereafter, the remaining Runga-Kutta methods will only be specified by the error order and the calculation formula without derivation.
Runge-Kutta methods of third order
Calculation formula:
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Local discretization error: .
Runge-Kutta Method of Fourth Order
This method is based on Simpson’s -Rule and yields
(7.35)
(7.36)
The local error order of this method is .
This version of the Runge-Kutta method is also referred to as classical Runge-Kutta method:
Stability Observation
In order to analyze the Runga-Kutta method, we consider again the test equation with a real at first
(7.37)
For a given value we obtain an approximation value where
(7.38)
(7.39)
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It is obvious that the factor just contains the first five summands of the expansion power series where . A comparison with the real value of the exponential function reveals that the deviation from the true value increases as and grows and that instability is observable when
section while the true solution decreases .
We receive similar results when we make the same analysis with further Runge-Kutta methods.
Similar to Euler method, we can more generally consider the case that is a complex number. Here, we also analyze of the behavior of oscillating solutions. In this case, we do not obtain an interval of the real axis as a
Figure 7.8. Instability area in the Runga-Kutta method of 4
th
order.
1.6.2. General Runga-Kutta Methods
The calculation formula of an ODE discussed in section [Classical Runge-Kutta Methods]
can be generally represented for a method with function evaluations ( -stepped method)
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(7.40)
(7.41)
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(7.42)
The values can be interpreted as an approximation of the solution with one integration section in the following points in time:
Figure 7.9. Transition points in the Runge-Kutta method.
We choose the coefficients to approximate as much as possible. A clear representation of the coefficients
is frequently based on the following schema (Butcher schema)
(7.43)
It is real for
(7.44)
If
(7.45) the variables can be directly calculated from already known quantities, i.e. we refer to an explicit method
Example 7.4: Butcher Schema for Classical Runge-Kutta Methods
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Annotations 7.2:
1.
order where . The question emerges whether there also exists a method where whether there is always, at least, one method where . One must negate this in both cases.
and
2.
Table 7.1. Accumulation of the amount of transition values and achieved order.
Number of 1 interim values
2 3 4 5 6 7 8 9 10
Achievabl e order
1 2 3 4 4 5 6 6 7 7 i.e. the method of order represents a kind of optimum in respect to this observation.
1.6.3. Stability Area of Runge-
Kutta Methods of Order 1≤p≤4
In this case we can prove that the area of absolute stability of a method of order results from
(7.46)
This particularly means that all methods of order p have the same stability area.
The stability border results from the complex solution of the statement with arbitrary of the interval .
The stability areas of the Runge-Kutta method up to
are represented in [Stability area of the Runge-Kutta
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Figure 7.10. Stability area of the Runge-Kutta method of order 1-4.
1.7. Step Size Control
Errors are inevitable if iterative methods are applied. The emerging error depends on the used step size.
Practically however, mostly a desired accuracy is merely known. Thus, one would like to provide the accuracy instead of the step size as input of the numerical integration method.
For this purpose, the algorithm is supposed to choose a step size automatically in every iteration step, such that the local discretization error lies below a desired value . Thus, the error which is made in an iteration step must be estimated in advance.
Therefore, two different possibilities exist. On the one hand, an iteration step can be performed twice by using different step sizes. From the difference of the calculated solutions the local error can be estimated. On the other hand, two different solutions can also be obtained by using two different integration algorithms (embedded methods).
1.7.1. 3. Order Runge-Kutta, Two Calculations of an Integration Step
For the local discretization error of the Runge-Kutta method of third order with an integration interval of length
it yields
(7.47) where the constant depends from the envisioned differential equation. If one integrates the same interval in two integration steps of length , the following approximation statement of the discretization error at the end of the interval is true
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(7.48)
If one subtracts the equation (7.47) from (7.48), we obtain
(7.49)
and subsequently repeating the calculation with the integration increment . If the specific results are referred to as and respectively, we obtain in this case
(7.50)
By replacing equation (7.49) with (7.50) and after solving for
, it yields
(7.51)
If
is known, we can calculate an estimated value by means of equation (7.47)
(7.52)
1.7.2. Embedded Methods
It is not efficient to calculate the same integration step twice in succession in order to estimate the local integration error. For this reason, embedded methods are more advantageous. Hereby, two different algorithms are applied which use the greatest possible number of steps. The Runge-Kutta-Fehlberg algorithm
(RKF4/5) is a possible choice for an embedded algorithm. The Butcher scheme has the following form:
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(7.53)
Thereby, is a Runge-Kutta solution of fourth order and is a Runge-Kutta solution of order five.
Therefore,
The relative error can be obtained by
(7.54)
(7.55) with .
The objective is to choose a new step size, such that the relative error lies near the relative tolerance.
We thus want the following equation to be fulfilled:
(7.56)
Therefore, the proposal for the choice of the step size is
(7.57)
If the error is too great, the step size is decreased, if the error is too small, the step is increased. Notice, that hereby, steps are never repeated, even if the error is much too great. Therefore, this type of step size control is called optimistic. In contrast, a conservative step size control repeats a step with a new step size if the estimated error is greater than the tolerance.
The idea of step size control can also be understood as a (feedback) control problem. In this case, the above
given formula equals a P-controller ([Step size control depicted as a (feedback) control problem]).
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Figure 7.11. Step size control depicted as a (feedback) control problem
Kjell Gustafsson developed a PI-controller in order to control the step size. The corresponding formula is:
(7.58)
1.8. Linear Multi-Step Methods
The previous integration methods had all in common that the approximation value directly results from its predecessor value . Within one integration step, we exclusively use the information of the last step. Methods with these characteristics are referred to as one-step procedures.
The question emerges whether one can achieve an enhanced accuracy by consulting the calculated values of the previous steps . These methods are called multi-step procedures.
(compare the derivation of the Runge-Kutta method in section [General One-Step Procedures])
(7.59)
The integrand will subsequently be integrated approximately by an interpolative quadrature formula with grid points
Simpson’s Rule:
. If and the grids equidistant and the increment , we then obtain for e.g The
(7.60)
If , on the other hand, we act on the original differential equation
(7.61)
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Problems by approximating the derivation directly on the basis of a numerical differential formula, as e.g.
(7.62) we obtain the following procedure
Both methods are examples of linear multi-step procedures. This will lead us to the following definition.
(7.63)
Definition 7.4
A linear multi-step procedure with n increments (also: linear -step procedure), which aims at the determination of the approximations for the solution defined by the specification of initial values
, with the initial value problem (7.1), (7.2) is
(7.64) and the calculation rule (difference equation)
(7.65) with
(7.66)
Annotations 7.3:
1.
function (method function) depends linearly from the function values
(7.67)
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1.
The condition
guarantees that the implicit differential equation (7.65) holds an exact solution, at least
for sufficient small increments .
2.
By means of the condition , the step number is exactly determined.
3.
is true for explicit linear multi-step procedures.
Example 7.5:
1.
By inserting and
in (7.65), we obtain an implicit procedure
(7.68) which is referred to as Adams-Moulton formula.
1.
If we proceed like in , but insert , we obtain the following procedure
(7.69)
This procedure constitutes the class of Adams-Bashford formulae.
1.
obtain a category of implicit integration procedures
(7.70) with representatives which are referred to as Backward Difference Formulae or, abbreviated, BDF formulae, cp.
problems and in the solution of DAEs.
An example of a favored multi-step procedure is the method of Adams-Bashford, which is based on the following formula:
(7.71)
The local method error yields
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(7.72)
In addition to explicit linear multi-step methods, one often utilizes implicit procedures in practice. Reasons are:
1.
they are more accurate when compared to explicit procedures,
2.
they have considerably better stability characteristics and
3.
they have easy strategies to estimate errors and to control step sizes.
In order to calculate a good starting value, which is necessary for the solution of the non-linear conditional equation where , we use a multi-step procedure with the form of a predictor-corrector method. One example
is the procedure by Adams-Moulton which is defined by the predictor (7.38) and the corrector
(7.73)
Both formulae can also be used independently from each other.
In this case, the first formula is an explicit method (right hand side does not depend on ), whereas the second formula is an implicit method (right hand side depends on ). The first formula predominantly aims at achieving a good starting value for the iterative solution of the second. The second formula describes an implicit method which must be solved iteratively. In practice, we often use one or two iteration steps. The method error for the Adam-Moulton procedure is as follows
(7.74)
In most cases, a predictor-corrector procedure consists of an explicit method (the predictor) and an implicit method (the corrector) with an error order which is at least equal to the predictor.
An error estimation (e.g. required for the increment control) results from additional computing with a double increment, as described within the scope of the Runge-Kutta methods:
(7.75)
procedures: In order to start the calculation we already need solution values at the four sampling points
. These points must be calculated by means of another method (e.g. Runge-Kutta procedure). A further disadvantage is the complicated way of increment variations in the integration process (in contrast to one-step procedures).
An advantage is that the calculation of a new solution value merely requires one analysis of the differential equation. In contrast to that, Runge-Kutta methods of similar error order basically require analysis of differential equation at several points.
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Stability analysis can also be made for multi-step procedures. But to deal with it here would go beyond the scope of this lecture.
1.9. Activation of Linear Multi-Step Procedures
Because only the starting value is given in a initial value problem, we must determine further initial values with a sufficient accuracy in order to start the linear multi-step procedure. As a general rule, there are to basic strategies:
1.
We use an one-step procedure with a automatic step size control and calculate approximates with given accuracy for the starting values at points . Here, the following problems might occur: a.
The grid points which are automatically determined by the step size control are not necessarily equidistant. In this case, we must calculate the required values at equidistant sampling points by interpolating the calculated sampling points.
b.
Even if the increment is constant, it does not have to be identical to the increments which are needed by the linear multi-step procedure to comply with the accuracy requirements. Therefore, we advantageously use a one-step procedure which has the same consistency order like the multi-step procedure.
2.
We use a family of multi-step procedures with ascending consistency order. We start with a method of lower order and successively increase the order. As discussed in , the grid points might also not be equidistant.
1.10. System of Differential Equations
The numerical integration of a system of differential equation ensues analogically to the procedure in scalar differential equation. This will be exemplified by the Runge-Kutta procedure. In this case, the application of
(7.59) to the equation system yields
(7.76) the calculation rule
(7.77) i.e. the scalar quantities must be substituted by vectors or, expressed differently, the numerical integration procedure will simply be applied equation wise.
1.11. BDF Methods
The multi-step procedures discussed before are based on numerical solutions of the integral equations
of numerical differentiation.
In order to determine an approximate value of for points
, we define an interpolation polynomial by the
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(7.78)
The polynomial can be written as follows
(7.79) with the so-called backward differences
(7.80)
The unknown value x n+1 will be determined in such a way that the polynomial satisfies the differential equation
(7.81)
at one point of equation (7.78).
When we obtain explicit formulae, namely when
Midpoint Method (cp. exercise). The formulae when
the explicit Euler method and when
are instable.
When , we obtain implicit formulae, the so-called BDF methods.
the
(7.82) with the coefficient
(7.83)
With
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(7.84) we obtain
(7.85) thus,
(7.86)
Where , we obtain for e.g.
(7.87)
The formulae (7.86) are stable when
; when , there are instable.
1.12. Remarks on Stiff Differential Equations
The following example will help us to understand the phenomenon which is referred to as stiff differential equations in literature. Let us first of all envision the following example:
Example 7.6:
We have the following equations
(7.88) with the constants where .
The exact general solution of this equation is
Both solutions converge to zero as
(7.89)
. Using the explicit Euler method, we obtain the numerical solution
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(7.90)
These approximate solutions evidently only converge to zero, if
(7.91)
Here, must be decreased in such way that
(7.92)
In this special technical process, which is described by this equation, suppose that .
In the analytical approach, the contribution of the term is negligibly small when compared to the contribution of . But in the numerical approach, the solution component of becomes relevant for the choice of the
minimal values of h due to eq (7.92).
Suppose that and , it would be . If the second solution component does not exist,
would be sufficient. Hence, although is more or less relevant to solve the problem, the factor determines the choice of the integration increment. This characteristic of the differential equation system in numerical integration is referred to as stiff. If the system at hand is stiff, not only the -stability of the integration method has to be considered but the damping behavior as well. For this purpose the term L-stability is introduced.
Definition 7.5
A numerical integration method for differential equations is called L-stable if and only if it is -stable and the following additionally holds:
(7.93)
Annotation 7.4:
1.
All -stable methods are -stable, but never -stable.
2.
While the solution portions including high damping are only weakly numerically damped when using stable methods, these portions are strongly numerically damped when using -stable integration methods.
A principle difference between the characteristic of implicit and explicit integration methods will be discussed in the following.
Example 7.7: Integration of the Equation of Motion for Single Mass Pendulum [9]
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The motion of a linear not stimulated single mass pendulum will be described by the state equation
(7.94)
For simplification, we write
Hence, the equations (7.94) yields
(7.95)
Application of the explicit Euler method on (7.95) results in
(7.96)
In the damped case , the pendulum oscillates with constant amplitudes and the frequency
The amplitude results from the initial condition.
The parameters with the increments
Figure 7.12. Integration of one-mass pendulum by means of the explicit Euler method.
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It becomes obvious that the oscillating amplitude does not remain constant (in contrast to reality), but that it grows with increasing increment.
In order to describe this characteristic, we calculate the total energy of a pendulum:
By means of the solutions by the Euler method, we obtain
(7.97)
(7.97) shows obviously that the total energy of the numerical solution increases with the factor
in every step.
With the time period
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Since at (reversal point), the amplitude increases for each oscillation with the factor
.
Suppose this consideration would be true for the implicit Euler method, we would first obtain
If we insert equation (7.98) in (7.99), we obtain
(7.98)
(7.99) or after transformation and, subsequently, after calculation with the modified damping
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Suppose that the same parameter values of the explicit Euler method would also be real in this case. Hence, we
Figure 7.13. Integration of one-mass oscillator by means of the implicit Euler method.
Obviously, the oscillating amplitude decreases. After energy observation, analogue to the explicit method, we now obtain a decrease in energy for each oscillating period with the factor
Figure 7.14. Energy behaviour of a plane oscillator.
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This example suggests an essential general characteristic of integration methods, which will be discussed later:
1.
Explicit methods insert additional (only numerical qualified) excitement into the system.
2.
Implicit methods lead to an additional (numerical qualified) damping of the system.
In fact, this behavior also reoccurs in complex applications in practice.
Trapezoidal Rule, with a local consistency order of and respectively, as an integrated part of -stable methods. According to the theorem by Dahlquist, there is however no “better” method.
Theorem 7.1 (Dahlquist):
1.
Explicit multi-step methods are never -stable.
2.
The order of an -stable implicit multi-step method is at the most .
3.
The Trapezoidal Rule is an -stable method of second order with the lowest error constants.
Therefore, we introduce a further definition, which even though weakens the -stability approach, can be expediently be utilized in many applications.
Definition 7.6:
A method is -stable if a stability area comprises a sector of the following form
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It is referred to as -stable, if
Figure 7.15. A(0)-stability.
-stable where
(7.100)
The BDF-methods are e.g. -stable with the opening angle
([ Opening angle α for different k]).
Table 7.2. Opening angle α for different k
1 2 3
90 90 86
1.13. Implicit Runge-Kutta Methods
4
73,3
5
51,8
6
17,8
sequentially calculated, but that with each integration step we have to solve a non-linear equation system.
Example 7.8: Implicit Euler Method
The implicit Euler method is obviously a single-step implicit Runge-Kutta method.
The Trapezoidal Rule can be regarded as a two-step implicit Runge-Kutta method.
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Butcher’s scheme is according to (7.43) as follows
Implicit Euler method
Trapezoidal Rule
A great disadvantage of implicit Runge-Kutta methods is the necessity to solve a non-linear equation system for the variables (with representing the dimensions of the equation system which must be solved). On
the other hand, the additional constants in (7.43) can thereunto be used in order to
1.
achieve a higher consistency order with identical number of steps .
2.
seriously improve the stability of the numerical method, which is “almost” -stable.
Example 7.9: The Midpoint Method
The “Midpoint Method” is obviously a single-step method which, however has a consistency order of 2.
Within the framework of this lecture, we will not cater to the construction and the application of implicit Runge-
Kutta methods in detail. Detailled data is given by Jumann, M. (2004) for example.
1.14. Comparison of Methods for Numerical Solution of Initial
Value Problems (IVP)
The described methods can be divided into the following three classes
1.
Euler method,
2.
Runge-Kutta method,
3.
Multi-step procedure,
4.
BDF method
Furthermore, there exist other classes (e.g. extrapolation technique) which will not be discussed in this lecture.
Runge-Kutta methods and multi-step procedures allow for error estimation and increment step control.
Additionally, multi-step procedures are open to variations of the method order. Commercially distributed
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Problems program libraries (e.g. IMSL, NAG) offer sophisticated computer programs for all classes. Furthermore, there are programs with published source code which are freely available (e.g. Shampine and Gordon, 1983).
Advantages and Disadvantages of this Application:
In respect to the analyses of the differential equation, multi-step procedures call for the least effort. An explicit multi-step procedure often requires only a single analysis per integration step of the so-called right hand side.
Implicit methods, on the other hand, further require the appropriate evaluations needed for the (“few”) iterations. Compared to this is the effort required for the increment control of the multi-step procedure. Multistep procedures include relatively high efforts in terms of calculation control (overhead). To sum up, most of the advantages result from a complex structure of the differential equation as a result of which each analysis implies comparatively high calculating efforts.
Runge-Kutta methods of lower order prove to be more advantageous in cases of simply built differential equations or a discontinuity on the right-hand side. On the other hand, they become more disadvantageous in the case of complex differential equations. Furthermore, Runge-Kutta methods allow for very simple step increment regulations.
In the last years, Euler methods went through a kind of renaissance in the field of real time applications.
Nevertheless, the potentially negative stability characteristics of the explicit Euler method should be considered.
Thus, Euler methods can be recommended in case of stability problems, which can also cause problems because of the necessary iterations in real time applications. Furthermore, Euler methods require for very short integration steps (increments) because of their low consistency order.
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Chapter 8. Integration of
Discontinuous Systems and DAEs
1. Integration of Discontinuous Systems
Step increment regulations in integration methods fail at discontinuous differential equations. Above all, the existence of a definite solution is also not guaranteed.
Physical reasons for discontinuous differential equations are e.g.
1.
Impact: Velocity jump
2.
Fixed friction: sign inversion of the friction force
3.
Halt-slide-transition: Change of the dimension of the state equation
In these cases we frequently make an alternative description of the physical process.
Example Percussion hammer: Modelling of the impact between the percussion piston and the tool by means of an extremely stiff spring/ damping combination.
A better solution would be to exactly identify the point in time where a discontinuity occurs and then abort the integration at exactly this point. Subsequently, the integration will be restarted at exactly this point with a changed differential equation.
Example 8.1: Bouncing Ball
Figure 8.1. Bouncing ball.
Discontinuity appears where .
Here we can observe a partial elastic impact with the coefficient of restitution , i.e. it yields
Figure 8.2. Bouncing ball (simulation result).
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2. Differential Algebraic Equations (DAEs)
Especially mechatronic systems or systems which incorporate components from different disciplines are based
If equation (8.1) can clearly be solved for , we can reproduce a statement
(8.1) and the methods represented in the previous sections can be applied again. But this is impossible in most cases
such
(8.2)
This special form of DAE is referred to as semi-explicit.
In this statement
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(8.3) is a system of differential equations while
(8.4) represents a system of purely algebraic equations.
If (8.4) can explicitly be solved for
, then
can be eliminated from (8.3) and can be inserted back into (8.4),
which would then lead to a simple system of differential equations.
But this is generally not the case. It is sometimes possible that
does not even exist in (8.4) In this case, the
system of DAE must be solved in another way. For that purpose, we first of all define the index of the DAE.
Definition 8.1:
The (differentiation) index of a DAE is the minimal number of differentiations of the equation
in order to transform (8.3) and (8.4) into a regular differential equation system.
Index 1:
(8.5)
(8.6)
Differentiation of (8.6) with respect to time yields:
(8.7)
Inserting (8.5) in (8.7) and solving for
yields
(8.8)
Note that must not be singular to guarantee, that it actually deals with a system of index .
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Index 2:
In this case does not depend from or, at least, the matrix has a rank drop of .
Consider the following system
We assume that does not depend on
Differentiating with respect to time; then
, further differentiation with respect to time yields Where
If the inverse of
exists, (8.11) can be solved for
and we obtain
Index 3:
Analogous to index systems, we obtain the same statement (8.13) after double differentiation.
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(8.9)
(8.10)
(8.11)
(8.12)
(8.13)
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Systems and DAEs
Further differentiation gives
(8.14)
In this case, we obtain an equation system with the form
(8.15) where the function depends on the index of the system.
The systems which occur in mechatronics are normally index -problems.
In many cases it is more efficient to solve the DAE directly instead of reducing it to an ODE: This is especially true for DAEs of index . Therefore, in many simulation programs the index is reduced to and afterwards the resulting DAE is solved. In most cases either implicit Runge-Kutta- or BDF-methods are applied to solve DAEs.
At first, the solution of DAEs with the help of BDFs is presented.
In case of ordinary differential equations their solution is obtained by solving Eq. (7.86) for
purpose, the last points
.
. For this
as well as are necessary, whereby can be easily determined by evaluating
equations. Additionally, a nonlinear system of equations has to be solved due to the implicit calculation rule.
However, this enormous calculation cost can be avoided by solving Eq. (7.86) for
and inserting it into Eq.
(8.1). In this case, merely the nonlinear system of equations
(8.16) has to be solved in order to obtain . Therefore, only the solution of a nonlinear system of equations of dimension of the state vector is necessary by inserting the integration rule into the model equations.
The frequently used method for the solution of DAEs is DASSL. This method uses the BDF formulas of order to , as well as a step size control. DASSL is very well suited for the solution of stiff DAEs and is therefore surely capable of solving non-stiff problems. In this case however, DASSL is relatively inefficient due to the bad error coefficients of the BDFs. If a problem is not stiff, it is more efficient to use an implicit Runge-Kutta method. The most popular implicit Runge-Kutta method for solving ODEs and DAEs is the Radau method of fifth order. This method has the Butcher table
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The solution of DAEs with the help of implicit Runge-Kutta methods resembles the solution of ODEs. Merely the steps are calculated with the help of the DAE formulation
(8.17) where are the entries of the Butcher scheme. Consequently, using fifth order Radau method, the solution of a nonlinear system of equations of dimension is necessary. Thus, the cost for one integration step is slightly higher than that of DASSL. However, the error coefficient of Radau’s method is much smaller than the coefficients of the BDF methods, for which reason Radau’s method allows the choice of greater step sizes. This, especially for non-stiff systems Radau’s method is better suited. Additionally, Runge-Kutta methods can be more easily started, as described above.
Notice that beside the initial values for also initial values for have to be provided in order to apply DASSL
determination of consistent initial values is closely related to index reduction, but is not discussed in this lecture.
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Chapter 9. Numerical Solution of
Non-linear Sysem Equations
1. Nonlinear Equations
The need to solve non-linear system of equations frequently occurs in mechatronic systems, e.g.
1.
In the determination of equilibrium positions,
2.
In the solution of kinematic ties,
3.
In the estimation of system parameters.
We consider a non-linear equation system to discuss the appropriate solution methods
(9.1)
Definition 9.1:
of .
Example 9.1:
Solving non-linear equation systems frequently causes problems in practice.
The reasons are
1.
generally, there are no analytical solution, i.e. a numerical approximation (iteration) is necessary,
2.
the convergence of an approximation procedure can generally not be guaranteed,
3.
there are generally no applicable methods to determine all zero points.
Example 9.2: Planar Four-bar Mechanism, [Planar fourbar mechanism.]
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Figure 9.1. Planar fourbar mechanism.
With a given set of measurements, the angle relative to (dependent on) , should be calculated.
Loop-closure (closed-loop) condition: where we obtain
In this case, there exists an analytical (but not unique) solution with
The solution is not unique, see [Solutions of the planar four-bar mechanism.]:
Figure 9.2. Solutions of the planar four-bar mechanism.
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Example 9.3: Five-link Wheel Suspension, [Five-link wheel suspension.]
Figure 9.3. Five-link wheel suspension.
freedom which is described by the spring deflection .
We have to calculate the remaining coordinated and of the wheel carriage in the vehicle fixed coordinate system as well as the Bryant angle of the wheel carriage.
To determine the geometry, the vector c will be introduced at the origin of the vehicle fixed coordinate system
which has the following coordinates
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(9.2)
Furthermore, the vectors
(9.3) and of the wheel carriage and the vehicle fixed coupling point of the guide are given by the construction data.
(9.4)
Assuming that the guides have identical lengths, we obtain the following constraints
These are the 5 equations of the variables
(9.5)
and
In written form we obtain
(9.6)
method.
Figure 9.4. Vectors of the five-point wheel suspension.
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Figure 9.5. Wheel centre trajectory of the five-point real axle wheel suspension.
2. Solution with Numerical Integration
This method is occasionally used in practice, if the non-linear equation system
of the dynamical system defines the rest position
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(9.7)
Numerical Solution of Non-linear
Sysem Equations
(9.8) and this rest position is asymptotically stable. Starting from an initial value , we will integrate numerically until the fluctuations of remain within the limits of tolerance.
A great advantage of this method is its simplicity due to the fact that only the numerical integration of the state equations are required, which are frequently implemented a priori anyway.
Disadvantages are:
1.
the solution must be in an asymptotically stable rest position; instable rest positions cannot be found
2.
sometimes, the convergence is very slow (“creeping” into the rest position)
3.
the initial velocity must be in the proximity of the rest position
3. Fixed Point Iteration
Again, the non-linear equation system is given by
(9.9)
(9.10)
Now we shall try to solve equations (9.10) by iteration
(9.11)
The iteration will be continued, until the following condition for a given accuracy bound is satisfied.
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(9.12)
Numerical Solution of Non-linear
Sysem Equations
The iteration only converges for the fixed point if the starting value is close enough to and all eigenvalues of Jacob’s matrix
(9.13) can be assigned to the unit circle of the complex number field.
Figure 9.6. Convergent fix-point iteration; gradient in fix point.
Figure 9.7. Divergent fix point iteration
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4. Newton-Raphson Iteration
The Newton-Raphson iteration is a widely spread method. The basic principle is based on the linearization of
at the respective starting point and the substituted calculation of the zero point of the linearized function.
The basic principle is highly sensitive in the one-dimensional case.
1.
Determination of the starting value
2.
Calculation of the zero point of function , which is linearized where
1.
Verification of the convergence condition if
Bisection of until
Figure 9.8. Newton-Raphson iteration steps.
The extension to the n-dimensional case is obvious
1.
Determination of the starting value
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2.
Calculation of the zero point of the function , which is linearized where : where, now, it yields
1.
Verification of the break condition
e.g. (Euclid’s norm) where
Bisection of until
The Newton-Raphson iteration converges exactly for the zero point , if the initial value is close enough to
and the Jacobi matrix is regular.
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Chapter 10. Identification and
Optimisation
An important problem of modelling and simulation of dynamical systems is the identification or, at least, the estimation of model parameters. We envision a system with an output vector and an input vector and the vector of the determined parameter.
Figure 10.1. Parameter determined system.
Examples of parameters are:
1.
spring stiffness
2.
aerodynamic drag coefficient
3.
masses, moment of inertia
4.
amplitude interval
5.
damping and friction constants, etc.
Parameter identification aims at choosing the parameters p in such a way that the solution extensively identical with the measured solution , i.e.
of the model is
(10.1)
We assume that at least the structure of the model is known a priori or at least predefined. In general, if the appropriate model is first generated by measurements, we refer to it as model identification.
Example 10.1:
1.
In linear systems: Determination of the transfer function on the basis of measurements
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(10.2)
1.
In non-linear systems: “Acquisition (learning)” by neural networks
Here: Envision the model with given (or assumed) basic structure
Wanted: best approximation of the model parameters in reality
1. Linear Compensation Problem
This section defines a linear approach to parameter identification. The initial vector and, therefore, the solution
will first of all be developed into a Taylor series around the parameter vector , with incorporating the first estimation values.
(10.3)
In the case of linearization, terms of second and higher order can be neglected, and it yields in the first approximation
(10.4) with time-dependent coefficients .
To determine the coefficients , first small parameter variations will be introduced
(10.5)
. The new output vector will Finally, the simulation will be carried out with the new parameters be
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(10.6)
From we find
(10.7)
Thus, the -th column of matrix becomes
(10.8) i.e.
(10.9)
The matrix statement where
is also referred to as sensitivity matrix for the parameter . We therefore obtain the following
To determine the vector we conduct measurements at various points in time
(10.10)
(10.11)
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From this we obtain the reference (target) values of the output vector with respect to the chosen measuring points
(10.12)
The vector must be chosen in such a way that the equation systems are satisfied at the same time.
We sum up
(10.13)
(10.14)
Figure 10.2. Target system.
Number of rows from to is
... Number of readings
... Number of outputs
where
Where , we obtain an over-determined system which cannot be solved directly and exactly. In order to obtain the best solution, we use the concept of pseudo-inverse matrix, i.e. we chose in such a way that
(10.15) i.e.
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, the problem is equivalent to Since is also true for
Acting on the minimization of in respect to , we find
(10.16)
(10.17)
(10.18)
Figure 10.3. Geometric illustration of the pseudoinverse.
2. Non-linear Parameter Dependency
1.
Develop a simulator model (cp. [Simulator model])
Figure 10.4. Simulator model
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1.
Scan the solution of each component
for a given time point (see [Scan of y(t)])
⇒ Add up values
(10.19)
1.
Build function
should conduct a simulation with model 1 for given parameters . It should discretize the solution in terms of scan . and it should build the following value:
(10.20)
Figure 10.5. Scan of y(t)
1.
Pass on to an optimization method with the task:
Optimize the target function under the secondary (side) conditions
⇒ Non-linear problem with one-sided secondary (side) conditions where
(10.21)
… cost function,
vector of the model parameters.
In order to search for an optimal parameter vector, we can use a multitude of methods. These embark on different strategies which in each case select the better parameters. The efforts invested per iteration are equal to a complete simulation process and are, therefore, very high. This is why great importance should be attached to the quality of the used methods.
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Basically, we differentiate between the following important categories of optimization methods:
1.
Newton’s methods and variations: These methods choose zero points of the first derivations of the target functions by means of Newton’s methods.
2.
Gradient’s methods: These methods choose along the direction of the steepest descent (gradient direction) for a better value of the parameter vector .
3.
Genetic algorithm: These methods choose better parameter values by means of biological optimization principles of mutation and selection on the basis of mathematical algorithm.
3. Stability of Dynamical Systems
Terms and Definitions
It is not sufficient enough to acquire knowledge of steady state solutions in order to characterize the dynamical behavior of systems. Additional information about the behavior of solutions near the steady state solutions is necessary.
In the following we will cover
1.
the stability of equilibrium positions .
2.
the stability of motions .
In order to achieve a more precise definition of the term stability, we first of all need a unique admeasurement to measure the deviation of the real solution trajectory from the target state. In this context, the definition of the norm will be introduced.
Definition 10.1:
A norm is an admeasurement of the quantitative value of the state quantity with the following characteristics:
(10.22)
(10.23)
(10.24)
(10.25)
Examples for norms are:
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(10.26)
(10.27)
(10.28)
(10.29)
Definition 10.2:
The undisturbed solution exists
of the state equation is referred to as stable, if for each
, which is generally dependent on , such that for each solution of with for all , the condition is satisfied.
Definition 10.3:
there
(10.30)
(10.31)
is referred to as steady state solution or equilibrium point if .
The stability of a steady state solution is defined in relation to the stability of an arbitrary solution.
The solution is moreover asymptotically stable if additionally a fixed exists so that is true for all solutions with
(10.32)
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4. Stability Criteria for Linear Systems
The following statements are true for linear systems (theorems by Ljapunov):
Theorem 10.1:
(10.33)
The equilibrium state of the linear time invariant system
(10.34) is
1.
asymptotically stable if all eigenvalues of have negative real parts
2.
instable if at least one eigenvalue of possesses a positive real part.
,
In order to evaluate the stability by means of the eigenvalues, the calculation of the eigenvalues is made necessary, which is possibly complicated. Eigenvalues, for example, can result from the characteristic equation of :
(10.35)
This can only be accomplished numerically. But the effects of parameter variances on the stability behavior are frequently interesting when constructing systems, which makes the existence of analytical expressions beneficial. Thus it is interesting to find those methods which make it possible to draw conclusions from the structure alone or, more precisely, from the coefficients of the characteristic equations regarding the stability of the solution.
4.1. Stability Criteria based on Stodola
Theorem 10.2:
negative real parts.
(10.36)
Thereby, without loss of generality, it is true that .
Conversely, this theorem states that a system can definitely not be asymptotically stable if a change of sign occurs in the characteristic equation.
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This criterion is not sufficient when
(Quadratic equation)
if we consider the solution of the quadratic equation:
(Solution)
For systems of higher order (10.36) has only one necessary condition.
Example 10.2:
4.2. Hurwitz Criteria
(Hurwitz, 1895)
Definition 10.4:
The matrix built by the coefficients of the characteristic polynomial of matrix
(10.37)
(10.38)
(10.39)
(10.40) is referred to as Hurwitz Matrix.
The main section determinants of H i
are called Hurwitz determinants.
The construction of the Hurwitz matrix is as follows:
1.
Index will be increased by in each column.
2.
Index will be decreased by in each row.
3.
and
Theorem 10.3:
where .
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The zero points of the characteristic equation have exclusively negative real parts if it yields for all Hurwitz determinants :
(10.41)
Example 10.3: n=4
(10.42)
(10.43)
Annotation 10.1:
1.
The Hurwitz condition represents a necessary condition and a sufficient condition for stability
2.
From follows that is also true
3.
If we, conversely, assume that determinant (Cremer’s theorem).
is true, we only have to evaluate each second characteristic Hurwitz
The latter annotation finally leads to the criterion by Lienard-Chipard:
It is a necessary and sufficient for an asymptotical stability of (10.34) that the condition
is satisfied. Thus, the calculation of each second Hurwitz determinant is rendered unnecessary.
(10.44)
4.3. Stability Criteria based on Routh
The criteria by Hurwitz (particularly the criteria based on Lienard-Chipard) is appropriate to calculate the stability area analytically, depending on the system parameters. The criterion based on Routh is more qualified for a numerical stability check for systems of higher order.
are positive. The numbers will be calculated by means of the Routh scheme.
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The coefficients of the characteristic polynomial will be inserted in the first two rows. Further quantities of the schema will be calculated row-wise. In doing so, the elements of the -th row with the auxiliary quantity result from the values of the the recursive formula
-th and -th row of the adjacent row on the right hand side in respect to
The calculation scheme is cancelled where
with the -th row and at least where .
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Chapter 11. Modelling of Mechanical
Systems
Mechanical Systems are characterized by elements (bodies) with elasticity and mass inertia. Additionally, as a general rule there are the effects by motion resistances (friction) and viscosity (damping) as well as the excitement from external forces.
1. Fundamental Terms
1.1. Modelling: Mass, Elasticity and Damping
Mass:
The mass depends from the density and the geometric dimension of the component. In respect to the axioms of classical mechanics it yields:
1.
Masses are always positive:
2.
The mass of a body is time-invariant, .
3.
Masses can be divided and summated .
The modelling of masses orients on the specific objective. Mass models in increasing order of complexity are point masses, rigid bodies, elastic masses, masses of general deformable media. The prevailing mechanical laws become more complex in the respective order.
Elastic elements:
Through an appropriate construction it is made possible that the elasticity of the components becomes very large. In this case we refer to it as spring elements (e.g. leaf, coil and torsion spring)
Damping and frictional elements:
Reasons for damping and friction can be:
1.
Material damping in the components.
2.
Friction between components which move relatively to each other.
3.
Constructively designed damping elements.
4.
Components which are moved in liquids.
External forces:
External forces emerge due to
1.
Effects from force fields (gravitation, magnetism …).
2.
Drive elements (servo motor, combustion motor …).
3.
Given motions (due to bearing).
Modelling:
The characteristics of a mechanical system must be described by (possibly easy) idealized model. Here we have to differentiate between models with distributed and models with concentrated parameters.
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Models with distributed parameters: Continuum mechanics (keywords: Elastic bodies, beams, distributed forces
...)
Models with concentrated parameters: Stereo mechanics (keywords: mass free spring, mass free damping, single motion ...)
In this lecture we will restrict ourselves to the detailed discussion on systems with concentrated parameters. In the following, we will cover modelling of mechanical systems. Thus, we will concentrate on one fundamental modelling technique, the so-called multi-body systems.
Multi-body systems
Definition 11.1:
A multi-body system consists of a set of mass-afflicted, rigid bodies which are bound to each other by couple elements like springs, dampers, bearings and guides. By means of these coupling elements single bodies are affected at discrete points by single forces and single moments. Besides, spaciously and laminarly distributed forces take effects on the bodies.
Figure 11.1. Examples for elements of a multi-body system.
The method of multi-body systems base upon the fact that mechanical characteristics (inertia, elasticity, damping, friction, force …) are assigned to discrete elements. Relatively simple motion equations (ordinary differential equations) are achieved through discretization. Apart from multi-body systems we also use further idealizations like Finite-Elements Systems and Continuous Systems in mechanics. But these would go beyond the scope of this lecture and will therefore not be covered here.
1.2. Forces, System Boundary, Method of Sections
Definition 11.2:
Force is referred to as that interaction between system elements which initiate acceleration.
Definition 11.3:
Forces which affect the system from the outside are called external forces. Forces which emerge inside the system are referred to as internal forces. The distinction between external and internal forces depends on the specific system boundary.
Figure 11.2. Method of section in mechanics.
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System 1 (load) is effected by external forces and F21. In system 2 (chassis and wheel), , F12, F3 and
F4 are external forces; whereby F32, F23, F42 and F24 represent internal forces of the system. Internal forces have to be contra wise identical and, thus, cancel out each other (F32-F23=0 and F42-F24=0). They therefore do not appear outside the system.
If we summed up system 1 and 2 into a total system, forces F12 and F21 would also change into internal forces.
1.3. Constraints
Most mechanical systems are subject to constraints, which, in the general case, depend on position coordinates, velocity coordinates and time. Constraints are described by algebraic equations.
Definition 11.4:
A constraint is a rheonom, if time explicitly appears in the constraint equation; otherwise it is referred to as skleronom.
Constraints give rise to constraint or reaction forces which would not appear if these constraints do not exist.
Figure 11.3. Examples for constraints.
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perpendicularly to the area generated by the constraints. This is the starting point for d’Alembert’s Principle.
1.4. Virtual Displacements
Definition 11.5:
A virtual displacement of a mechanical system is a shift of the position of the system as a result of (imagined) arbitrary displacements of the body. But this has to be compatible with the constraint. The virtual displacement which elicits a change in the size of location is referred to by the symbol .
Example 11.1: Sphere Pendulum, [Sphere pendulum. ]
The motion of a spatial mathematical pendulum (spherical pendulum) is described by means of the Cartesian coordinates:
.
Note that the mass of the pendulum only moves on the surface of the sphere. This will be described by the following equation:
Thus, the constraint of the (Cartesian) position coordinates of the pendulum mass is defined.
If we virtually displace the pendulum mass with
(11.1)
,
impossible by definition.
changes by the virtual value If changes by ,
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(11.2)
Thereby we use the following calculation rule (not proved here)
Statements (11.1) indicate that the coordinates
hold redundant information on the position of the mass
angle
and . An additional constraint of the form (11.1), of course, does not appear.
The new coordinates already imply a constraint in motion which was considered before in their special choice.
Less than two coordinates are not sufficient enough for a unique position description. We therefore refer to and as minimal coordinates or generalized coordinates. The constraint only appears as an explicit connection between the position vector and the new generalized coordinates.
(11.3)
Figure 11.4. Sphere pendulum.
; the mass pendulum does not leave the surface of
the sphere.
In a virtual displacements, time is imagined to be fixed; in contrast to a real displacement which elapses in course of an finite time interval . Thus, we envision a snap shot of the system. This can be graphically
represented by a mass point on a moving, skew plane, [Virtual Displacement.].
Figure 11.5. Virtual Displacement.
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1.5. Kinematics
1.5.1. Coordinate Systems and Coordinates
Definition 11.6:
A coordinate system is a set of three vectors which are orthogonal to each other. These build the basis of the vectors which are represented in the imaginable space . This vector space base will be added to one origin in an Euclidian space. We refer to the new defined coordinate system as
(11.4)
Definition 11.7:
A coordinate system is called inertial system if the basis vectors are time-constant. is body fixed if it is fixed to one body point (e.g. the midpoint of the mass) and if the coordinates of the body points have the same coordinates with respect to this coordinate system.
Annotation 11.1:
The position and the orientation of a rigid body can be described by six coordinates, e.g. the specification of the translator coordinates and of the mass midpoint and by three Bryant angles system.
towards the inertial
The rotation of the body fixed coordinate system with respect to the inertial system is only then defined if an arbitrary vector can be defined in both the coordinate systems.
This connection is described by a matrix product:
.
Note that and refer to the same vector. The elements of the transformation matrix (rotary matrix) must be independently calculated from the considered case.
Example 11.2:
If a rigid body rotates around a fixed -axis around the angle (Fig. 11.6), we obtain the following rotary matrix
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Thus, it yields
.
The respective rotations around the and axis respectively will be described by the following rotary matrices:
and .
The determined transformation matrices between Cartesian coordinate systems have the following characteristics
, i.e. the rotary matrix is an orthogonal matrix.
Generally, we can reach any position of a rigid body by means of a maximum three successive rotations (i.e. by a specification of three rotary axes and three rotary angles). Depending on the order of the rotations, the rotary angles are referred as Briant angle (succession sometimes also vice versa) and as Euler angle (succession
). We obtain the resultant matrix by multiplying the singular matrices, e.g.:
Figure 11.6. Coordinate transformation by rotation around the z-axis.
If, in a coordinate system, the position and the orientation of multiple bodies corresponding position vector to the mass midpoint and the rotary matrix with three angles
Briant and Euler angles), we combine these coordinate into a position vector
are described by the
(e.g.
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(11.5)
Because of the given bearing elements, relations between the body coordinates to each other are defined by algebraic equations:
(11.6)
The position of the MBS can be uniquely described by independent coordinates . In mechanics, these coordinates
are referred to as generalized coordinates (cp. Section [Virtual Displacements]).
This also means that in a general approach, it does not necessarily deal with coordinates in the imagined space.
Generalized coordinates must be
1.
independent,
2.
uniquely define the position of a system,
3.
reconcilable with constraints.
This coordinates are also referred to as minimal coordinates or position variables.
The introduction of minimal coordinate makes it necessary that the position vectors of the system bodies be expressed independent to . The explicit representation of the position vector
is a prerequisite.
In order to completely define the MKS, we must add the velocities degree of freedom leads to two state quantities. The state vectors of a MKS results into:
. Thus, one mechanical
(11.7)
The vector uniquely described the position and the velocity of the multi-body system at each time point .
The space describe by is referred to as state space (cp. Section [State Equations]).
1.5.2. Translation
The position of a rigid body is described by a position vector to its mass midpoint
We obtain the velocity and the acceleration by successive differentiation in respect to time
(11.8)
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(11.9)
(11.10) where is the functional matrix or the Jacobi matrix of translation. Under rheonom conditions, the local
velocity vector additionally appears. The vector is
(11.11) describes the centrifugal, Coriolis and gyroscopic parts of the translation acceleration of the system.
Similar to the translation of mass midpoint, the rigid body can be restricted by constraints. In this case, we obtain:
(11.12)
(11.13) where is the functional matrix or Jacobi matrix of rotation. The vector resumes the centrifugal,
Coriolis and gyroscopic parts of the rotations acceleration. In rheonom constraints, the local velocity vector
furthermore appears.
1.5.3. Kinematical Differentials
MAPLE) can be problematic because the provisional results can be so voluminous that they practically can not be utilised later. That is why we consider an alternative solution approach in which no analytical derivation need to be accomplished.
This shall be exemplified in the following on a translation.
First Derivation
Time derivatives of absolute coordinates of all bodies for arbitrary values of generalized velocities , for given positions, can be specified by means of elementary-kinematic expressions. This is made possible by the incorporation of global kinematics into general mechanism.
Especially pseudo-velocities for special dimensionless pseudo-velocities of generalized coordinates
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(11.14) can be determined, in which the ”unit” vector of the -th element possesses a “ ”, else only zeroes.
Since the real time derivations are linear combinations of the generalized velocities , these are again independent from each other and it yields:
(11.15)
The comparison of equation (11.15) with equation (11.9) finally provides the simple rule:
Second derivation
(11.16)
At given position and velocity of the system the acceleration of all bodies can also be determined for arbitrary values of the generalized acceleration by means of simple kinematic applications. Especially pseudoaccelerations for different generalized accelerations, i.e. where
can be determined. Out of equation
(11.17)
coordinated and the absolute coordinates of the bodies. They are only determinable by elementary-kinematic expressions (particularly laws of Relative Kinematics). Thus, they are referred to as kinematic differentials.
Kinematic Differentials
The time derivatives of the summarized absolute coordinates can now be split into their translator parts , .
The corresponding statements are as follows:
(11.18)
by means of “physical” vectors which are independent of the choice of the coordinate systems used. In contrast to that, in the analytical approach it is only possible to establish these relationships between differentials, by differentiating the specific components when they are all represented in a common coordinate system.
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The transition to a component representation can be arbitrarily “retarded”, i.e. the choice of the coordinate system can be adjusted to the analyzed term, which also leads to a reduction in calculation efforts. This in a very compact form is made possible by the formulation of the equations of motion for general multi-body systems.
Example 11.3: cp. Section [ Kinematics], “[Equations of Motion of a Double Pendulum]”.
2. Principle of Linear and Angular Momentum
Newton’s Equations (principal of linear momentum) of a rigid body is
.
(11.19)
Euler Equations (principle of conservation of angular momentum) in an Inertial System with given fixed body is
inertial tensor
(11.20) and given a vector of the external moments with respect to the mass midpoint is
These equations are identical to the equations in a body fixed coordinate system.
Example 11.4:
(11.21)
Euler Equations of a free body, which is affected by a torque and in which we assume a principal axis system for the inertial tensor, yields
3. Consideration of Constraints and the Principle of d’Alembert
split up into imposed forces and moments and reaction forces and moments
(11.22)
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(11.23)
Thus, the complete Newton-Euler equations yields
(11.24)
(11.26)
(11.27)
(11.25)
the form
D’Alembert’s Principle proves that the virtual work of the reaction forces disappears.
(11.28)
: or with
(11.29)
Since is arbitrary, it yields
(11.30)
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4. Equations of Motion
the left hand side, by the transpose of the Jacobi matrix of the translation or the rotation respectively and, subsequently, by adding up each singular summand:
(11.31)
We therefore obtain a general, non-linear equation of motion for a (holomonic) multi-body system:
(11.32) with the symmetric and positive definite mass matrix , the vector of the Coriolis, centrifugal and gyroscopic forces and the vector of the generalized forces . We obtain the same equation system in the
5. Equations of Motion of a Double Pendulum
As a simple example of the formulation of motion equations of a multi-body system we now derive the nonlinear equations of a double pendulum. The double pendulum consists of two homogenous rods, each with the mass and the length . The rods are simply and frictionless supported at the points and .
Figure 11.7. Double pendulum.
Position vectors of the mass midpoints:
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We obtain the velocities by differentiating the position vectors, which gives
Successive differentiation of (11.34) and (11.35) leads to the accelerations
(11.33)
(11.34)
(11.35)
(11.36)
.
Solution with kinematic differentials
Velocity
Let and . Thus, we obtain in the first column of the Jacobi matrix
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(11.37)
Modelling of Mechanical Systems and let and which gives for the second column of the Jacobi matrix thus,
We therefore obtain for the velocity
(11.38)
(11.39)
(11.40)
(11.41)
If and , we obtain in the first column of the Jacobi matrix, and with
(11.42)
and in the second column of the Jacobi matrix
(11.43) hence,
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Acceleration
Modelling of Mechanical Systems
The applied forces and moments are
From (11.30) and (11.31) we obtain the Jacobian matrices:
and
Thus, we obtain the Newton-Euler Equations of the double pendulum
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(11.44)
(11.45)
(11.46)
(11.47)
(11.48)
(11.49)
Modelling of Mechanical Systems
(11.50)
(11.51)
(11.52)
(11.53)
according to (11.31), the non-linear motion equations of the systems result into
(11.54)
The reaction forces are eliminated in the multiplication process and can therefore be already omitted when we formulate the Newton-Euler Equations.
The method described here is also qualified for the formulation and solution of motion equations on a computer.
Without computing, it is frequently advantageous to leave out the explicit formulation of the Jacobi matrices and to eliminate the reaction forces by algebraic transformation. In this case however, we must completely incorporate the reaction forces into the equations.
6. Linear Equations of Motion
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Additionally to the general non-linear equations, linearized equations of motions play an important role in practice and especially in control theory.
The linearization of mechanical systems is purposefully based in the proximity of an equilibrium position, or more general, about a reference (target) motion. The reference (target) motion can either be based on the system itself or it is imposed by feedback control externally. Characteristic reference (target) motions of a system are the particular solution of its non-linear motion equations:
In the neighborhood of a reference (target) motion, the system shall now experience small disturbances:
(11.55)
.
Thus, it yields
An appropriate separation must be now undertaken for the externally imposed “adjusting” forces
If we insert this into the original non-linear motion equations, we obtain
(11.56)
(11.57)
(11.58) for the quantities occurring there.
By doing so, we obtain an equation of the reference (target) motion and an equation for the deviation from the target motion respectively:
Target motion:
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Disturbing motion:
Modelling of Mechanical Systems
(11.59)
(11.60)
Or, again, if we substitute by
Annotation 11.2:
We also obtain equation (11.59) out of the non-linear motion equations by term-wise linearization.
(11.61)
The matrices and (and every other square matrix) can additionally be split up into a symmetric and a skew symmetric part.
(11.62)
(11.63)
We therefore obtain the classical linear motion equations of mechanics.
The validity of the statements is simple to pursue
(11.64)
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Modelling of Mechanical Systems
(11.65)
Here, the matrices definitely possess the same physical characteristics like the system. If we multiply by from the left hand side, we obtain an equation giving the powers of the system.
(11.66)
The symmetric mass matrix determines the modification of the kinetic energy and, thus, the mass forces.
The damping matrix denotes the damping forces by means of Rayleigh’s function and refers to the gyroscopic forces, which do not affect any change in the energy balance. The matrix determines the potential energy and therefore also the effect of the position forces whereas describes the non-conservative position forces. The system is conservative when
would be constant.
, i.e. without the action of external forces , the total energy
Example 11.5: Linear Equations of motions of a Double Pendulum.
yields
.
If we consider this in the motion equations (11.54), we obtain the linearized motion equations
(11.67)
(11.68)
Mass and stiffness matrix give and
.
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Modelling of Mechanical Systems
7. State Equations
For the following discussion regarding the dynamics of a multi body system, it is convenient to transfer the motion equation of second order into a state equation. By means of substitution we obtain under consideration of the non-linear state equations
(11.69)
In the linear case we obtain
(11.70)
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Chapter 12. Lagrange’s Equations of
Motion of Second Kind
An important alternative to the formulation of motion equation is the method by Langrange (1788). In contrast to the synthetic methods described above (cutting off of single bodies, then applying the principle of conservation of angular momentum and principal of linear momentum and finally eliminating the reaction forces), in this case it deals with an analytical method, which is based on the analysis of energy expressions of the total system.
Kinetic Energy
For the kinetic energy of a rigid body with the mass the inertial tensor the absolute center of gravity velocity and the angular velocity , it yields
(12.1)
The kinetic energy consists of the translational and rotational parts. Because the kinetic energy is independent from the used coordinate system, it is unimportant in which coordinate system the specific energy parts are calculated. On the other hand it is clear that the angular velocities have to be specified in the same system, as those of the inertial tensor.
The kinetic energy of a multi-body system is built by the sum of kinetic energies of the single bodies. We choose the mass midpoint of the single bodies as the reference point. We obtain
(12.2)
Potential Energy
If the work achieved by the applied forces is independent from the distance covered, the forces have, as is generally known, a potential and can thus be determined by differentiation. It yields
(12.3) with the potential energy which is a scalar local function.
The potential energy of a multi-body system results from the sum of the potential energies of the single bodies
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Lagrange’s Equations of Motion of
Second Kind
(12.4)
Annotation 12.1
therefore referred to as conservative. Non-conservative forces modify their mechanical total energy. If we specially deal with forces which eliminate energy, we refer to them as dissipative forces.
Examples of conservative forces are weights.
and spring forces
.
The appropriate potentials are and
.
Potentials can be determined except for one additive constant, i.e. the potential zero-point can be set arbitrarily.
If a multi-body system has only conservative forces, the whole system is referred to as conservative. Therefore, it yields the theorem of conservation of mechanical energy:
(12.5)
The described energy expressions are used for the derivation of the equations of motion. Here, in contrast to the synthetic method the single bodies are not free cut, but the system will be considered as a whole.
First of all, the kinetic energy will be represented subject to the generalized coordinates and, if necessary, subject to time
(12.6)
The generalized forces result from imposed forces and moments
(12.7)
By means of these quantities we obtain Lagrange’s equation of motion of second kind
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Lagrange’s Equations of Motion of
Second Kind
(12.8)
Annotation 12.2:
1.
The number of equations of motion is identical to the number of degrees of freedom of the system.
Furthermore, it is not necessary to introduce the reaction forces. The opposite way around, they cannot be calculated.
2.
In order to specify the motion equations, we have to calculate the partial and total differentiations of the function . One follows the chain rule to perform the total differentiation of with respect to time t.
these can also be (analogous to (12.3)) calculated by formal differentiation of the potential energy
in accordance with the generalized coordinate. We obtain
If we introduce Lagrange’s function , we obtain Lagrange’s motions equation in classical form
(12.9)
(12.10)
If, in addition to conservative forces, non-conservative forces appear, these (and only these) are considered by
the expression (12.7) on the right hand side of (12.8).
Example 12.1: Double Pendulum, [Double pendulum.]
We obtain the following statements for the kinetic energy of the total system
(12.11) and the potential energy results into
We obtain the following expression for the partial derivation
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Lagrange’s Equations of Motion of
Second Kind
motion
In this case we also obtain the motion equations with form (11.34) where
,
,
.
Annotation 12.3:
1.
The motion equations, which result from Lagrange’s equations, are completely identical with the equations
which we obtained by means of d’Alembert’s principle from Newton-Euler Equations. This is true for the general case also. Both methods exclusively differ in respect to their approaches, not in respect to their results.
2.
In contrast to Newton-Euler’s formalism, Lagrange’s equations (in this form) allow for the calculation of reaction forces (position forces). Conversely, the consideration of these forces is not necessary when we set up equations, which accrues considerable advantages in practice.
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Chapter 13. Nonlinear Single Track
Modell (based on [9]
[ Nonlinear single track model (bicycle model), top view ] depicts a nonlinear single track model.
Figure 13.1. Nonlinear single track model (bicycle model), top view
Figure 13.2. Nonlinear single track model, side view
1. Equations of Motion of the chassis
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Nonlinear Single Track Modell
(based on [9]
(13.1)
Where and are the front and rear wheel forces, the gravity force and the aero dynamical force
(13.2) which acts contrary to the direction of the vehicle velocity.
Decomposed in the inertial system it holds
(13.3)
(13.4) thus
(13.5)
The forces F xf
and F xr
correspond to the circumferential force, and represent the lateral force, and form load of the front and rear wheel, respectively.
For the Euler’s Equations with respect to the vehicle’s center of gravity we obtain
(13.6)
Decomposed in the vehicle fixed coordinate system it yields
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Nonlinear Single Track Modell
(based on [9]
(13.7)
(13.8) thus
(13.9) where we do not consider the first component equation, as we neglect the rolling dynamics of our model.
2. Tyre Model
2.1. Stationary Tyre Model
idea of this model is to approximate a nonlinear tyre characteristic that was previously acquired from measurements, by basic mathematical functions (e.g. sine and arctangent). Besides the tyre profile and material, the tyre characteristic is also dependent on the longitudinal slip, slip angle and the wheel load.
For the stationary front tyre force we obtain
(13.10) and accordingly for the rear one
(13.11)
The parameters and do not have any physical meaning as they are only used for the mathematical approximation. and denote the longitudinal slip and the slip angle, is the wheel load.
For the calculation of and we need the velocity of the front and rear axle. They are obtained by the absolute kinematics according to
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Nonlinear Single Track Modell
(based on [9] for the front axle and for the rear axle. Thus we get for the velocity of the front axle, decomposed in the inertial system
(13.12)
(13.13)
(13.14) thus
(13.15) and accordingly for the velocity of the rear axle
(13.16) respectively
Afterwards we need to decompose the wheel velocities to the corresponding wheel frames and we obtain
(13.17)
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Nonlinear Single Track Modell
(based on [9]
(13.18) and
(13.19) s and α are kinematical values and can be calculated according to
(13.20) for the front axle and for the rear axle
(13.21)
The front and rear wheel loads and which are necessary for the stationary tyre model are discussed in the
the Newton’s Equation from the inertial system to the vehicle fixed system, as the decomposition in both frames identical. Furthermore we assume that the axle load is uniformly distributed on the left and right wheel. Thus we obtain for the front wheel load
(13.22) and for the rear one
(13.23)
2.2. Dynamic Tyre Model
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Nonlinear Single Track Modell
(based on [9]
The stationary wheel forces that we discussed in the previous section are established with a certain delay due to the tyre flexibility. This behavior can be realized by a differential equation of first order. For the front wheel we obtain
(13.24) and for the rear wheel accordingly
The values and are constants, where longitudinal one.
(13.25)
, as the lateral flexibility of the tyre is greater than its
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Chapter 14. Dynamic Wheel Rotation
The dynamic rotation of the front and rear wheel are considered by the following differential equation, decomposed in the corresponding front and rear coordinate system, respectively.
Figure 14.1. Dynamic wheel rotation.
According to [Dynamic wheel rotation.] we obtain
where, each for the front and rear wheel,
inertias,
angular accelerations,
driving torques,
braking torques, depending on the direction of the wheel rotation.
1. Driving Torques
The driving torques for the front and rear axle is realized by the rate of the driving torque axle (thus, means rear wheel drive) and the driving torque :
Again the driving torque depends on the engine torque , according to
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(14.1)
of the rear
(14.2)
Dynamic Wheel Rotation
(14.3)
is the differential ratio and travel
the gear ratio depending on the actual gear. is the acceleration pedal
.
is realized by an engine characteristic ([Engine characteristics.]).
Figure 14.2. Engine characteristics.
For the engine speed we assume this simplified interrelationship:
(14.4)
Interim values can be determined by a linear interpolation.
2. Breaking Torques
b
is plotted versus the scaled braking pedal travel p b
.
Figure 14.3. Braking torque characteristics.
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Dynamic Wheel Rotation
The braking torque rate for front and rear axle can be determined, according to the driving torque rate, by the factor for the rear axle:
(14.5)
and are the magnitudes of braking torques. The direction can be modelled by Coulomb’s friction law by hand of a sign-function.
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Chapter 15. The Overall Modell
equations.
1.
Newton’s Law of the chassis decomposed in the inertial system:
(15.1)
1.
Euler’s Law of the chassis decomposed in the vehicle fixed system:
1.
Dynamics of the wheel rotation decomposed in the corresponding wheel fixed coordinate frame:
(15.2)
(15.3)
)
1.
Dynamics of the tyre forces decomposed in the corresponding wheel fixed frame:
(15.4)
(15.5)
These differential equations of first and second order can be transformed to a system of differential equations of first order, i.e. the nonlinear state space model of the form
(15.6)
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The Overall Modell
is the state vector where
is the vector of the input variables, the state variables depend on nonlinearly: where
steering wheel angle , steering angle , is the steering ratio,
acceleration pedal travel in the interval ,
braking pedal travel in the interval ,
actual gear (-1 (reverse gear); 0 (idle running); 1; 2; 3; 4; 5).
Thus the nonlinear state space model yields
(15.7)
(15.8)
1. Simulation Results
.
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The Overall Modell
In the following some simulation results for different inputs of the steering wheel angle jump are shown. A velocity controller provides for the nearly constant velocity of the vehicle model. By the help of the steering wheel angle jump one can estimate the responding behavior of the vehicle. Major vehicle dynamics variables
Figure 15.1. Nonlinear single track model: different steering wheel angle jumps at v =
100km/h.
2. Animations
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The Overall Modell
Motions of vehicles and robots are simulated and are shown by animations:
• Rolling on vehicle: 3erBoschungRO.avi
• Vehicle in cross wind: Crosswind.avi
• Real time simulation of a passing: Echtzeit_Video.avi
• ESP simulation: ESP_Video_verteilt.avi
•
Dynamic simulation of Vehicle lights: Hella_ohne.avi
• Walking robot on slope: Simulation_freegait_DIVXv5.avi
• Walking robot combined by wheels: Simulation_gehen_schreiten_DIVXv5.avi
• Motion of a baker without optimization: Terex_1_PTP_ohne_Optimierung.avi
• Motion of a baker without optimization:
Terex_3_PTP_mit_Optimierung.avi, Terex_4_PTP_mit_Optimierung.avi
Terex_2_PTP_mit_Optimierung.avi,
3. Videos
Motions of vehicles and robots are simulated and are shown by animations:
• Rolling on vehicle 1: RolloverVideoschnitt3.avi
• Rolling on vehicle 2: L148-Vergleich.avi
• ESP video: ESP_Video.avi
• Maneuver with Bosch ESP: Maneuver_Bosch_ESP_320_240.avi
• Vehicle maneuver: Maneuver_presentation.avi
• Bending of a robot foot: Pruefstand_Arbeitsraum_DIVXv5.avi
• Rotation of a robot foot: Pruefstand_Kreis_Joystick_DIVXv5.avi
• Supporting robot foot: Video_FussHuefte_DIVXv5.avi
• Step of a robot foot: Video_Vergleich_DIVXv5.avi
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Chapter 16. Mathematical Basics
1. Matrix Calculations
matrix:
Row vector
Column vector
matrix:
matrix:
Zero matrix:
It is defined that
Unit matrix:
, if is not the zero matrix, so if at least one matrix element .
Diagonal matrix:
Upper diagonal matrix:
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Lower diagonal matrix:
Band matrix:
Mathematical Basics
Cyclic band matrix:
1.1. Matrix Operations
Addition and subtraction of two matrices
Multiplication of a matrix with a scalar number
Scalar product/dot product/inner product (r )
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Mathematical Basics
Two vectors and are orthogonal, if their Scalar produkt is zero:
Cross produkt ( ), only 3D
The cross product is orthogonal to both vectors and :
Multiplication of two matrices
Typically, the matrix product is not commutative:
Basic operations
Multiplication with the unit matrix
Linear system of equations
Given
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Mathematical Basics with or respectively, the solution vector is
Transpose of a matrix
Basic operations
A matrix is symmetric if holds, and skew symmetric if holds.
Every quadratic matrix can be decomposed in a sum of a symmetric and skew symmetric matrix: with and
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Mathematical Basics
Inverse of a matrix (only for square matrices!)
Matrices which have an inverse are called regular or invertible. Otherwise, they are called singular.
Every matrix has at most one inverse:
A quadratic matrix is invertible if and only if holds.
Basic operations (provided that the inverse exists)
With the help of the inverse a system of equations with a quadratic matrix can be solved:
matrices can be easily inverted:
The calculation of the inverse of an elimination algorithm).
Gaussian elimination algorithm
matrix with is expensive (for example using Gaussian
Given:
Wanted:
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Mathematical Basics
1.2. Determinants
Determinants are only defined for square matrices.
The calculation of a determinant of big matrices can be extremly expensive if one considers certain simplified operations.
The calculation of a determinant can be most easily be explained by the following recursive algorithm, the
Laplace’s formula:
The determinant of a matrix is the only element of the matrix:
The determinant of an matrix is obtained by the calculation of determinant of matrices of , e.g. by “developing for the row“
sub
Thereby, one obtains the matrices by deleting the th column in . The signs in the sum alternate, thus, the last sign is for uneven and for even . In order to save calculation time, one of course develops for a row or column which has as many zeros as possible.
Simple calculation of matrices
Basic operations
1.3. Norms
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1.3.1. Norms of vectors
Taxicab norm or Manhattan norm or -norm
Mathematical Basics
Euclidian norm or -norm
General: -norm
Maximum norm oder -Norm
Basic operations
1.3.2. Norms of matrices
For every matrix norm belonging to a vector norm it holds that
Matrix norm of -norm
Spectral norm (matrix norm of Euclidian norm)
Matrix norm of norm
Spectral radius:
The spectral radius norm.
, which is the greatest absolute eigenvalue of the matrix , is smaller than any matrix
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Appendix A. Worksheets
Worksheet 1: Simple simulation model of a percussion hammer[3]
In the following a simple principle model of a percussion hammer is developed and simulated with SIMULINK.
Figure A.1. Percussion hammer [2]
This is a power tool designed for chiseling and breaking concrete, stone and asphalt, for tamping, bushing and compacting in the building trade. A percussion hammer has a built-in hammer mechanism. The striking energy is independent of operator pressure.
Function description:
The drive piston (1) is actuated in a periodic translator motion by the crank mechanism (2). The force transmission from the drive piston to the percussion piston (3) results from the air cushion (4). Due to this excitation the percussion piston performs an oscillating translator motion and axial percussions on an interposed piston, which itself impacts on the tool (In the following the interposed piston and the tool are neglected.).
Analogous model:
System equations:
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Worksheets
The rotation of the drive crank is assumed to be constant (rheonomic constraint). The position of the drive piston is explicitly described by the length s, which is dependent on the geometric parameters r, q as well as the angular velocity of the drive crank:
.
The percussion piston can move independently in x-direction.
1.
Determine the amount of degrees of freedom for this system.
2.
A friction force (Coulomb’s friction) acts between the percussion piston and the wall of the cylinder, arranged by
.
is the cross-sectional area of the cylinder, the dynamic friction coefficient between piston and wall of the cylinder and p
0
the ambient pressure.
1.
A force acts between the drive and percussion piston due to the air cushion, which depends on the distance
between the piston surfaces in a nonlinear manner.
.
1.
Applying the principle of linear momentum we obtain the equations of motion
.
Modelling the impact process
When the percussion piston impinges on the tool respectively the header, a momentum appears which is approximately described by a spring-damper combination with spring constant and damping coefficient :
.
However this force only acts when the percussion piston tries to penetrate into the tool
1.
Consider this circumstance with an appropriate logical block in the block diagram.
Realization of the simulation
.
1.
Develop a block diagram, which can establish a basis for the input in SIMULINK. Initially generate separate block diagrams for the motion of the drive piston, the friction force, the air force as well as the momentum.
Afterwards develop a block diagram of the equations of motion, which imply the forces to be combined in subsystems being dependent on and .
2.
Enter the block diagram in SIMULINK and perform simulations for different initial conditions and parameters. Pore over the influence of the rotational velocity of the crank on the function of the hammer in particular.
3.
An important characteristic for the function of the hammer is the impulsion velocity of the percussion. How does the velocity change if the percussion mass is changed?
4.
Determine the equivalent coefficient of restitution between percussion piston and tool by the help of the impulsion and rebound velocity of the percussion piston.
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Worksheets
Apply the following numerical values for the simulation:
1.
Percussion mass:
2.
Cross-sectional area of the air cushion:
3.
Air cushion height at ambient pressure:
4.
Ambient pressure:
5.
Crank radius:
6.
Length of con rod:
7.
Initial position of the percussion:
8.
Friction coefficient:
9.
Spring constant for impact modelling:
10.
Damping coefficient for impact modelling:
11.
Input speed:
12.
Length of cylinder:
Wroksheet 2: Mathematical Pendulum
Mathematical pendulum
For the given mathematical pendulum (mass , length ) the time-dependent motion is to be analyzed. We assume that the pendulum is pivoted in point and can only perform a planar motion. The acceleration of gravity acts in vertical direction. The motion of the pendulum is influenced by a damping torque which is proportional to the respective rotational velocity: .
Figure A.2. Mathematical pendulum
Modelling the mathematical pendulum
1.
By how many state variables can the equations of motion be definitely described? This equals to how many degree(s) of freedom of mechanics?
2.
Determine the equations of motion. Transform them to the state space.
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Worksheets
3.
Set up the system of the mathematical pendulum in a block diagram.
Solution of the state equations using MATLAB / SIMULINK:
1.
Enter the plotted block diagram in SIMULINK and study the system’s behavior for the following parameter values:
1.
,
, 2.
3.
with the initial conditions
1.
,
2.
,
3.
.
1.
Plot the corresponding diagrams of the rotational angle and rotational velocity, respectively versus time, as well as the phase diagrams. Experiment with different parameters and further initial conditions.
Linearized equations of motion:
1.
Which linearization do we need to apply? Set up the linear state equations in matrix description
2.
0
and f
0
, respectively of the pendulum with the given parameters from above
3.
Confirm the numerical value for the eigenfrequency using MATLAB and perform a simulation of the nonlinear and linear system with suitable initial conditions. Evaluate the error due to linearization.
Modified pendulum:
Now the pendulum is extended by a spring
Figure A.3. Modified pendulum
1.
Set up the nonlinear equations of motion for this system. And transform them to the nonlinear state space.
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Worksheets
2.
What is the system’s new state of equilibrium? Determine further states of equilibrium if they exist
3.
Modify your model adequately and perform further simulations.
Worksheet 3: Longitudinal oscillations of a continuous beam
The longitudinal oscillations of a continuous beam can be described by the help of the partial differential equation
.
It can be analyzed with analytical or numerical methods.
Figure A.4. Continuous beam
By means of a steal beam analyze how far the accuracy of the eigenfrequencies, which are determined by local discretization, depends on the number of discretized elements.
Worksheet 4: Predator-Prey-Models
In a biotope the relationship of animals of the species (=prey) and (=predator) is examined. The relationship of both species is such that mainly feeds on . An example for this may be the relationship between the species fox and hare.
Which time-dependent process results for the populations of the species and , if the population of is x
B
(t) and the population of is ? In the following this task is solved step-by-step.
Dynamics of a population of a species
Strictly speaking the populations of differential calculus, we assume that
1.
When can this assumption be justified?
and
and
can only be whole-numbered. To apply the methods of
are differentiable functions.
2.
Determine the current growth velocity for the population of a breed.
3.
Determine the growth rate
Constant growth rate
(differential equation!) which is defined by the relation .
1.
Determine the simplified linear differential equation for problem if denotes the initial population?
2.
Which three cases do we need to distinguish?
. What is the general solution for this
Due to the limits of the natural resources the exponential increase of populations is practically not possible. For this reason a more realistic process of growth has to be examined.
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Worksheets
Logistic growth rate
Here we assume that the constant value of the growth rate is weakened by a component (factor proportional to the population.
) which is
1.
Set up the equation for this case (which is also known as the logistic growth rate equation do we obtain now?
. Which population
2.
Determine the general solution for this problem. Hint: Apply the method of expansion into partial fractions.
Towards which constant value does this solution converge for .
Dynamics of a population in a predator-prey community
Two species are considered, of which populations are denoted by the indices (predator) and (prey). For the derivation of suitable growth laws the following simplified assumptions are made:
Without predators the prey population is defined by a growth law with constant growth rate.
The (reducing) influence of predators on the prey population is proportional to the predator population (factor
).
1.
What is the growth rate for the prey population?
Without prey the predator population would become extinct with a constant (negative) growth rate .
The refreshment of the growth rate of the predator population is proportional to the prey population (factor ).
1.
What is the growth rate of the predator population?
2.
What system of differential equations do we obtain now? Be the current population of the prey and the current population of the predator.
These equations are denoted as Volterra-Equations (according to Vito Volterra (1860-1940)). Volterra was the one who first set up a mathematical model for the dynamics of populations.
The Volterra-Equations cannot be solved analytically anymore. We have to resort to numerical solution methods. Therefore MATLAB/SIMULINK is applied now.
Solution of the Volterra-Equations with MATLAB/SIMULINK
1.
Set up the block diagram of the system.
2.
Enter the block diagram in SIMULINK and study the system behavior for different parameter values. Choose e.g.
(each with thousand units)
(time units)
1.
Compare with the behavior of isolated populations. The time response of predator and prey populations is periodic for each. Determine empirically the period and show that the predator and prey populations have the same period.
The predator-prey model with logistic growth
Here a more realistic model can also be set up, if we assume logistic growth rates:
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Worksheets
,
With the base equation
.
the logistic Volterra-System yields
and
Change your model accordingly and analyze the system behavior by repeating the simulation with different sets of parameters. Particularly check, if the chronological process of the populations is still periodic.
Worksheet 5: Modelling of a double pendulum with DYMOLA
Figure A.5. a) Double pendulum b) Cut model of pendulum
[ a) Double pendulum b) Cut model of pendulum ]a shows a planar double pendulum. Each pendulum is
represented by a massless link (length l) and a point mass m at the end of the link. The modelling of this system
1.
Determine the geometrical constraint for the subsystem.
2.
Draw the Multiport/Cut-Set-model. What are the across and through variables of the system?
3.
Determine the necessary equations of motion for the pendulum.
4.
Modelling the double pendulum with DYMOLA:
1.
Generate the subsystem model in the DYMOLA environment: a.
Construct the connections (Ports) for input and output signals (File/New/Connector). Hint: through variables are realized by the prefix flow.
b.
Construct the inertial frame for the system. What are the constraints for the across variables of this inertial frame (File/New/Model).
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Worksheets c.
Generate the subsystem of the pendulum according to the equations that you have determined before
.
2.
Test your subsystem by modelling a single pendulum and check the simulation results.
3.
Set up the total double pendulum and compare the new results with those of subtask 2.
4.
Append further pendulum elements and analyze the behavior.
Worksheet 6: Oscillations of a quarter vehicle model
The equations of motion of a quarter vehicle model are with
1.
Calculate the eigenvalues and eigenvectors with MATLAB.
2.
Determine the angular eigenfrequencies of the system.
3.
Determine the eigenoscillations of the system.
4.
Draw the block diagram of the system.
5.
Create the SIMULINK model of the system according to the block diagram.
6.
The system is excited by a sinusoidal oscillation with the eigenfrequency. What phenomenon occurs?
Modelling parameters
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1.
gravitational acceleration
2.
mass of the chassis
3.
mass of the wheel
4.
spring constant of the suspension
5.
damping constant of the suspension
6.
spring constant of the wheel
Worksheet 7: Kinematics of a rear five-link wheel suspension
With the help of MATLAB the kinematics of the following rear five-link wheel suspension is to be modelled
([Rear five-link wheel suspension]).
Figure A.6. Rear five-link wheel suspension
The vertical deflection c z
of the wheel carriage is considered to be the generalized coordinate (degree of freedom).
1.
How many constraint equations are necessary to determine the position and orientation of the wheel carriage?
What are the dependent coordinates of the wheel carriage?
Given are the position vectors to the mounting points reference coordinate frame (cp. MATLAB file ‚main.m’).
and in respect of the vehicle
1.
Determine on the basis of the given mounting points the lengths of the control arms to . Set up the wheel carriage fixed vectors from to the points to in respect of the wheel carriage reference point .
2.
Determine the constraint equations in the form . Hints: Fill the constraint equations into the prepared file ‚FiveLink.m’. The numerical solution of the constraint equation is realized with the help of the
MATLAB function ‚fsolve’. It calls ‚FiveLink.m’ and internally solves the (nonlinear) system of equation
.
3.
Draw the wheel trajectory and the deflection of the suspension depending on the camber and toe angle.
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4.
Animate the wheel suspension with MATLAB.
Worksheets
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Appendix B. Exercises
Exercise 1
Linearize the following expressions about the according working points.
1.
about
2.
about
3.
about
4.
about
5.
Exercise 2
about .
A cylindrical basin (cross sectional area , water level ) with continuous inflow and outlet (cross sectional area of outlet ).
1.
Generate the differential equations for the state variable h.
2.
Set up the block diagram for the system.
3.
Sketch possible solutions of
Exercise 3
Transform the differential equation
for . Which further information is necessary for a definite solution?
to a system of differential equations of order.
Exercise 4
Linearize the system about its states of equilibrium.
Exercise 5
Calculate the eigenvalues and eigenvectors of the matrix
.
Exercise 6
The following matrix is given:
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Exercises
.
1.
Calculate the eigenvalues and the eigenvectors of the matrix.
2.
Confirm by means of this example the orthogonally property of the eigenvectors: prove this circumstance generally for any symmetric n×n-matrix?
3.
Show that with the relation
. How can you is valid, i. e. the similarity transformation defined by T transforms A to diagonal form.
1.
Calculate by the help of subtasks a) to c) the expressions and confirm the relations
and .
1.
Show that different solutions result by element-wise application of the functions sin, cos and on the matrix .
Exercise 7
Determine for the linear time-invariant systems
and
1.
the eigenvalues and eigenvectors,
2.
the solutions for given initial conditions.
Exercise 8
The initial value problem is given.
1.
Calculate the analytical solution of the initial value problem.
2.
Calculate a numerical solution in the interval applying the explicit Euler method.
with the integration step width
3.
Repeat the calculation in b) for the step widths and solutions with each other and also with the analytical solutions.
Exercise 9
by
. Compare the error of the numerical
The differential equation with the initial conditions and is given.
1.
Formulate the problem as a system of equations of order and calculate the analytical solution.
2.
Determine the solution of the initial value problem at the times method. Apply the integration step width .
and using the explicit Euler
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Exercises
Exercise 10
The equation of motion of a spring-mass oscillator is
.
Determine for the values numerical solution in the interval
1.
the Runge-Kutta method of . order,
2.
the explicit Euler method.
with
and the initial values
3.
Solve the subtasks a) and b) for .
Vary for the subtasks the integration step width and study the behavior of the solution.
Exercise 11
The initial value problem
, a is to be solved with the Runge-Kutta method of order numerically. Find a step width such that the local discretization error is less than .
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