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

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Presentation transcript:

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

Newton’s 2 nd Law: Translational Motion

Newton’s 2nd Law: Rotational Motion I

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

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

Spring Model Two springs in parallel Two springs in series

Spring Model Two springs in parallel Two springs in series

Spring and Damper Model mm ? ?

Mass Spring Damper System

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

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

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

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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

Cruise Control Model Free-body diagram

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

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

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

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

Combined Motion: Rotational and Translational Motion 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