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

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)

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

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.

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

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

Moment of Inertia I Rotation in the middle of bar Lumped mass Distributed mass L L m L m m 𝐼= 1 12 𝑚𝐿 2 𝐼= 1 3 𝑚𝐿 2 𝐼= 𝑚𝐿 2

Spring Model Two springs in parallel Two springs in series

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

Spring and Dashpot Model ? ?

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

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)

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)

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

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

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

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

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