1. Lecture 3: Dynamic Models
Dynamics of Mechanical Systems
Newton’s 2nd law
F=ma (translation)
M=I (rotation)
Mass-Spring-Dashpot Model
Mass (m)
Spring (Spring force 𝐹 𝑠 =𝑘𝑥)
Dashpot (Damping force 𝐹 𝑑 =𝑏 𝑥 )
Spring: store energy, dashpot: dissipate energy

2. 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)

3. 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

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 Rotation in the middle of bar Lumped mass
Distributed mass L L m L m m

7. Moment of Inertia I Rotation in the middle of bar Lumped mass
Distributed mass L L m L m m
𝐼= 𝑚𝐿 2
𝐼= 1 3 𝑚𝐿 2
𝐼= 𝑚𝐿 2

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

9. Spring Model Two springs in parallel Two springs in series
When k=k1=k2, keq=2k
When k=k1=k2, keq=0.5k

10. Spring and Dashpot Model? ?

11. Mass Spring Dashpot System
Derive equation of motion
Transfer function 𝐺 𝑆
Input: force f
Output: displacement y f

12. 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)

13. 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)

14. Mass Spring Dashpot System
Automobile suspension system
Problem: Find the transfer function

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

16. 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

17. 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

18. 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