# AME 40623/60623 – Analytical Dynamics

### Objectives

This course covers fundamental principles and analytical methods for deriving and analyzing the dynamics of mechanical systems. Example applications of these fundamentals span machine design, robot analysis, and spacecraft control. The primary learning objective is for students to be able to model multibody mechanical systems with complex constraints and derive their equations of motion. A secondary objective of the course is for students to be able to apply state-of-the-art symbolic algebra packages (e.g., the Matlab symbolic math toolbox) to automate otherwise tedious and error-prone by-hand derivations. Following this course, students should be able to read and evaluate academic literature that emphasizes the dynamic modeling of mechanisms/spacecraft, numerical methods for dynamics, and other associated topics.

### Prerequisites

Newtonian dynamics in 2D, multivariable calculus (partial vs. total derivative), introductory linear algebra (including eigenvalue decomposition), differential equations, basic Matlab programming.

### Topics

Particle dynamics, moving reference frames, systems of particles, variational calculus, variational principles of mechanics (via D’Alembert, Lagrange, Hamilton, Gauss[not covered in 2019], & Jourdain), holonomic and nonholonomic constraints, dynamics of rigid bodies in 3D.

### Detailed Course Material

A zip file with all homework and lectures can be found here: Link

#### Lectures:

1. Introduction to Analytical Dynamics
2. Impulse and Momentum
3. Energy, Work, Equilibrium
4. Rotations
5. Angular Velocities and Angular Acceleration
6. Moving Reference Frames – Examples
7. Moving References Frames – MATLAB
8. Exam Review
9. No Lecture (Exam 1)
10. Systems of Particles
11. Fundamentals of Analytical and Virtual Work
12. Introduction to Variational Calculus
13. D’Alembert’s Principle
14. Lagrange’s Equations
15. Structure of Kinetic Energy
16. Structure and Vibrations
17. Routhian Reduction
18. Exam 2 Review
19. No Lecture (Exam 2)
20. Hamiltonian Dynamics
21. Rotational Inertia Matrix
22. Eulers Equation
23. D’Alembert’s Principle with 3D Bodies
24. No Lecture (Travelling)
25. Rigid Body Dynamics Review
26. Holonomic vs Nonholonomic Constraints
27. Jourdain’s Principle & Kane’s Method
28. Review

#### Assignments:

1. Newtonian Particle Dynamics
2. Reference Frames and Rotations
3. Planar Rigid Body Dynamics Review & Virtual Work
4. Generalized Forces, D’Alembert’s Principle, & Lagrangian Dynamics
5. Structure of the Equations of Motion, Modal Analysis of Vibrations
6. Routhian Reduction and Hamiltonian Dynamics
7. Rigid-Body Dynamics in 3D and Euler’s Equation
8. D’Alembert’s Principle with 3D Bodies
9. Kane’s Method (…and a variational approach to deriving Euler’s Equation)