1. Lecture 3: Dynamic Models Spring: stores energy Damper: dissipates energy

2. Newton’s 2 nd Law: Translational Motion

3. Newton’s 2nd Law: Rotational Motion I

4. Moment of Inertia I It is a measure of an object’s resistance to changes to its rotation. Equivalent to mass of an object. It should be specified with respect to a chosen axis of rotation.

5. Moment of Inertia I Moment of inertia becomes smaller when mass is concentrated on the axis of rotation

6. Moment of Inertia I L m L m L m Lumped mass Distributed mass Rotation in the middle of bar

7. Moment of Inertia I L m L m L m Lumped mass Distributed mass Rotation in the middle of bar

8. Spring Model Two springs in parallel Two springs in series

9. Spring Model Two springs in parallel Two springs in series

10. Spring and Damper Model mm ? ?

11. Mass Spring Damper System

13. MATLAB Simulation: Mass Spring Dashpot System Matlab code: num = 1 den = [1 b 1] sys = tf(num, den) step(sys)

14. MATLAB Simulation: Mass Spring Dashpot System Matlab code: num = 1 den = [1 b 1] sys = tf(num, den) step(sys)

15. MATLAB Simulink: Mass Spring Dashpot System

16. Mass Spring Damper System Automobile suspension system Problem: Find the transfer function

17. Mass Spring Damper System Automobile suspension system The equation of motion for the system Taking the Laplace transform Transfer function

18. Cruise Control Model Example 2.1 ─Write the equations of motion ─Find the transfer function o Input: force u o Output: velocity Cruise control model Free-body diagram

19. Cruise Control Model Free-body diagram

20. Cruise Control Model Free-body diagram Matlab code: num = 1/m den = [1 b/m] sys = tf(num, den) step(sys)

21. Combined Motion: Rotational and Translational Motion Inverted pendulum mounted cart −Input: force u −Output:  Derive equations of motion Unstable system

22. Combined Motion: Rotational and Translational Motion Position of the center of gravity of the pendulum rod Rotational motion of pendulum Free body diagram

23. Combined Motion: Rotational and Translational Motion Horizontal motion of the center of pendulum Vertical motion of the center of gravity of pendulum Horizontal motion of cart Free body diagram

24. Combined Motion: Rotational and Translational Motion Free body diagram

25. Week 2, Lecture 3: Reading and Practice Reading for week 2: -Franklin Textbook Chapter 2, Dynamic Models: -2.1: Dynamics of Mechanical Systems -2.2: Models of Electric Circuits -Modern Control Engineering by K. Ogata -Chapter 3 Mathematical Modeling of Mechanical Systems and Electrical Systems