1. Newton’s 2nd Law: Translational Motion
Newton’s 2nd law governs the relation between acceleration and force
Acceleration is proportional to force, and inversely proportional to mass
F=ma
where,
F = the vector sum of all forces applied to each body in a system, newton (N)
a = the vector acceleration of each body w.r.t. an inertial reference frame (m/sec2)
m = mass of the body (kg)

2. Newton’s 2nd Law: Rotational Motion
Newton’s 2nd law governs the relation between angular acceleration and moment (torque)
Angular acceleration is proportional to moment, and inversely proportional to moment of inertia
M=I
where,
M = the sum of all external moments about the center of mass of a body in a system, (N-m)
 = the angular acceleration of the body w.r.t. an inertial reference frame (rad/sec2)
I = body’s moment of inertia about its center of mass (kg-m2) I  M

3. Mass Spring Dashpot System
Applying Newton’s 2nd law,
Taking the Laplace transform
Transfer function 𝐺 𝑆 = 𝑌(𝑠) 𝐹(𝑠) = 1 𝑚 𝑠 2 +𝑏𝑠+𝑘 f
𝑚 𝑦 =−𝑏 𝑦 −𝑘𝑦+𝑓
𝑚 𝑠 2 +𝑏𝑠+𝑘 𝑌 𝑠 =𝐹(𝑠)

4. MATLAB Simulation Mass Spring Dashpot System
Transfer function 𝐺 𝑆 = 𝑌(𝑠) 𝐹(𝑠) = 1 𝑚 𝑠 2 +𝑏𝑠+𝑘
m=1, k=1
Case study
b=1 (underdamped <1)
b=2 (critically damped =1)
b=3 (over damped >1) f
num = 1
den = [1 b 1]
sys = tf(num, den)
step(sys)

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

6. Cruise Control Model Example 2.1 Applying Newton’s 2nd law
Since v= 𝑥 , 𝑣 = 𝑥
𝑣 + 𝑏 𝑚 𝑣= 𝑢 𝑚
- Transfer function
𝑚 𝑥 =−𝑏 𝑥 +𝑢
𝑥 + 𝑏 𝑚 𝑥 = 𝑢 𝑚
Free-body diagram

7. Cruise Control Model MATLAB Simulation Transfer function num = 1/m
den = [1 b/m]
Sys = tf(num*u, den)
Step(sys)
Parameter values: u=500, m=1000kg, b=50Ns/m

8. A Two-Mass System: Automobile Suspension
Two masses: Car (m2) and Tire (m1)
Problem
Write equations of motion for the automobile and wheel motion
Find the transfer function 𝑌(𝑠) 𝑅(𝑠)

9. A Two-Mass System: Automobile Suspension
Free body diagram of each body

10. Rotational Motion: Pendulum
Example
Derive equation of motion
Nonlinear equation
Linear approximation
Find transfer function

11. Moment
Moment
𝑀=𝑟×𝐹
𝑀= 𝑥 𝑦 𝑧 𝑟 𝑥 𝑟 𝑦 𝑟 𝑧 𝐹 𝑥 𝐹 𝑦 𝐹 𝑧

12. Pendulum Applying Newton’s 2nd law for rotational motion, M=I
Equation of motion
Linear equation of motion
Is it reasonable assumption?

13. SIMULINK of Pendulum
Linear model
Nonlinear model

14. SIMULINK Mass Spring Dashpot System
Dynamic equation of motion
Draw the block diagram
𝑚 𝑦 =−𝑏 𝑦 −𝑘𝑦+𝑓 f

15. SIMULINK Mass Spring Dashpot System
Dynamic equation of motion
m=1, k=1
Case study
b=1 (underdamped <1)
b=2 (critically damped =1)
b=3 (over damped >1)
𝑚 𝑦 =−𝑏 𝑦 −𝑘𝑦+𝑓 f 𝑦 𝑦

16. Rotational Motion: Satellite Attitude Control Model
Example
Derive the equation of motion
Find transfer function

17. Rotational Motion: Satellite Attitude Control Model
Example
Applying Newton’s 2nd law for rotational motion, M=I
Find transfer function

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

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

20. 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

21. Combined Motion: Rotational and Translational Motion
For a small angle
Free body diagram

22. Flexible Read/Write for a Disk Drive
Examples
Find the equations of motion
Find transfer function

23. Flexible Read/Write for a Disk Drive
Equations of motion
Transfer function