Analytical Dynamics – Lagrangian and 3D Rigid Body Dynamics
Dr. Shane Ross, Virginia Tech, Aerospace and Ocean Engineering
Education: PhD & BS, Physics, Caltech
Description
The purpose of this course is to ensure that the student has a firm grasp of the
fundamentals of analytical dynamics and 3D rigid body motion. The course will extend
the student’s knowledge of dynamics into areas not covered or covered only briefly at
the undergraduate level, e.g., Lagrange’s equations, D’Alembert’s principle, quasivelocities, phase plane analysis. In summary, we will discuss ways of (1) describing
the motion for some classes of mechanical systems in terms of configuration variables,
(2) writing down the equations of motion for those systems, usually ODEs for the
configuration variables, and (3) methods for finding and classifying the possible
motions (i.e., solving the equations of motion) and analyzing the characteristics of the
motion (e.g., using phase plane analysis).
Texts
The class will largely be based on the instructor’s notes. There is one required
textbook, Engineering Dynamics by Kasdin & Paley [2011]. Kasdin & Paley offer a
good and careful introduction to Newtonian and then Lagrangian dynamics, easing the
student into analytical dynamics, through many worked problems, including the use of
Matlab. Another useful text is Advanced Dynamics by Greenwood [2003], which
covers several methods of deriving equations of motion, and has material unique to
this course, e.g., it describes nonholonomic constraints, Coulomb friction, and phase
portrait analysis. Other useful texts are Analytical Mechanics of Space Systems by
Schaub & Junkins [2009], Applied Dynamics by Moon [2008], and Principles of
Dynamics by Greenwood, 2nd edition [1988]. You may find the non-required texts
valuable, for the additional descriptions provided and for the worked examples.
Topics to be Covered (tentative)
Kinematics of a single particle: reference frames, vector differentiation formula
Dynamics of a single particle
Kinematics and dynamics of systems of particles
Newton-Euler methods as well as other methods, such as D’Alembert’s method of
virtual work/power, and Lagrange’s equations
Categories of forces: conservative, non-conservative, constraint
Solving the differential equations of motion in special cases, e.g., oscillatory motion
Constraints: holonomic (position-based) and non-holonomic (velocity-based) and the
use of Lagrange multipliers
Constants of motion: quantities conserved during motion, e.g., energy, momentum
Linear oscillations in a single degree of freedom
Visualization of a system’s motion
Phase portraits: geometric interpretation of mechanical behavior in phase space
Solving for motion computationally
Routh’s equations: when there are ‘ignorable’ coordinates
Rigid body kinematics in 3D (rotation matrix and Euler angles)
Rigid body dynamics using both Euler’s rigid-body equations and Lagrange’s eqs
Small oscillations (a.k.a. free vibrations, normal modes, rigid-body modes)
Necessary Background
In terms of prerequisites, you should have previous experience with kinematics and
dynamics (e.g., using Newton’s equations in an undergrad course), calculus and
differential equations, and computational methods (e.g., using Matlab to solve ordinary
differential equations and to visualize the solutions). And to be honest, to get the most
out of this course, you should like math and dynamics.