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Cleveland State University

MCE441: Intr. Linear Control Systems

Lecture 3: Dynamic Modeling of Engineering Systems

Mechanical Systems

Prof. RichterMechanical Engineering Department

Ideal Spring

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Viscous Linear Damper

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Laws for mechanical systems

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■ Newton’s law, translational: ~F =

d(m~v)dt

■ Rotational: ~T =d~L

dt

■ Constant mass: ~F = m~a

■ Constant moment of inertia, rotation about principal axis: ~T = Id~w

dt= I~α

■ Each mass and each d.o.f. contribute a second-order ODE

Examples

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1. Basic in-class example: mass-spring-dampers, single and multi-mass.2. Schaum 6.30 (accelerometer)3. Elevator system starting from motor torque.

Simple example: quarter-car suspension

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Find the I/O differential equation (x, y)

m

y

reference equilibrium

level

x

road pro�le

k �

g

More difficult example

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Find the I/O differential equation (F,y)

k�

k

M

F

x

I� r

b

y

Solution

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Verify that the required equation is(

M +I

r2

)

y + by +

(

kM +k

2

)

y =F

2