MECH 642 - Advanced Dynamics

Variational methods. Hamilton's principle and equations of motion of engineering systems. Lagrangian formulations for discrete systems. Methods of discretizing continuous systems. Rigid body dynamics. Dynamic behaviour of linear and nonlinear systems. Response of engineering systems to deterministic inputs by classical methods. Stability of linear and nonlinear systems.

Instructor: Prof. James Richard Forbes

Course Overview

MECH 642 introduces advanced kinematics and dynamics in three dimensions. Students will learn how to derive the equations of motion of systems composed of particles and rigid bodies using the Newton-Euler approach, Lagrange’s equation, Hamilton’s (extended) principle, and the Gibbs-Appell equations.

Topics include:

  • Three-dimensional kinematics
  • Direction cosine matrices (DCMs)
  • Euler angles and quaternions
  • Transport theorem and Poisson’s equation
  • Newtonian and Eulerian dynamics
  • Rigid-body dynamics
  • Energy methods
  • Holonomic and nonholonomic constraints
  • Virtual work and D’Alembert’s principle
  • Lagrangian dynamics
  • Calculus of variations
  • Hamilton’s principle
  • Stability theory

Prerequisite and Corequisite Courses

There are no official prerequisite or corequisite courses. However, students are expected to be comfortable with:

  • Calculus
  • Differential equations
  • Linear algebra
  • Classical mechanics
  • Python programming

Python Resources

Assignments require computational work in Python.

Useful resources:

Learning Outcomes

By the end of the course, students will be able to:

  • Describe the motion of particles and rigid bodies in three dimensions.
  • Construct and manipulate direction cosine matrices, Euler angles, and quaternion parameterizations.
  • Apply Newtonian, Eulerian, and Lagrangian methods to derive equations of motion.
  • Analyze systems subject to holonomic and nonholonomic constraints.
  • Use variational methods and Hamilton’s principle to formulate dynamic models.
  • Assess stability using Lyapunov methods and LaSalle’s invariant set theorem.
  • Apply computational tools to solve advanced dynamics problems.

Textbooks

There is no required textbook. However, lectures are based on the following references.

  • G. M. T. D’Eleutario and G. R. Heppler, Newton’s Second Law And All That. (In preparation) Cambridge University Press, 2011.
  • D. S. Bernstein, Geometry, Kinematics, Statics, and Dynamics. (In preparation) Princeton University Press, 2013.
  • P. C. Hughes, Spacecraft Attitude Dynamics, 2nd ed. Mineola, NY: Dover, 2004.
  • F. L. Markley and J. L. Crassidis, Fundamentals of Spacecraft Attitude Determination and Control. New York, NY: Springer, 2014.
  • A. H. J. de Ruiter, C. J. Damaren, and J. R. Forbes, Spacecraft Dynamics and Control: An Introduction. West Sussex, UK: John Wiley & Sons, Ltd., 2013.

Additional Resources

  • N. J. Kasdin and D. A. Paley, Engineering Dynamics: A Comprehensive Introduction. Princeton, NJ: Princeton University Press, 2011.
  • D. T. Greenwood, Advanced Dynamics. Cambridge University Press, 2003.
  • A. V. Rao, Dynamics of Particles and Rigid Bodies: A Systematic Approach. New York, NY: Cambridge University Press, 2006.
  • L. Meirovitch, Methods of Analytical Dynamics. Toronto, ON: McGraw-Hill, 1970.
  • H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, 2nd ed. Reston, VA: AIAA, 2009.

Schedule

Week Date Topic Materials
1 Sep/Jan Kinematics I

Physical vectors, reference frames, and direction cosine matrices (DCMs).
2 Sep/Jan Kinematics II

The Transport Theorem.
3 Sep/Jan Attitude Representations

Axis-angle parameters, Euler angles, and quaternions.
4 Sep/Jan Angular Velocity Kinematics

Poisson’s equation.
5 Oct/Feb Newtonian Dynamics

Single-particle and multi-particle system dynamics.
6 Oct/Feb Rigid-Body Dynamics

Newtonian-Eulerian formulation of rigid-body motion.
7 Oct/Feb System Energy

Kinetic energy, potential energy, and energy methods.
8 Oct/Feb Constraints and Virtual Work

Holonomic and nonholonomic constraints, virtual work, and D’Alembert’s principle.
9 Nov/Mar Lagrangian Dynamics

Lagrangian formulation for particle and rigid-body systems.
10 Nov/Mar Constrained Dynamics

Lagrange multipliers and the null-space method.
11 Nov/Mar Calculus of Variations

The brachistochrone problem and variational calculus.
12 Nov/Mar Hamilton's Principle

Hamilton’s principle, Hamilton’s extended principle, and discretization methods.
13 Dec/Apr Stability Theory

Lyapunov stability, LaSalle’s invariant set theorem, and applications.