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Cleveland State University

MCE441: Intr. Linear Control Systems

Lecture 3: Dynamic Modeling of Engineering Systems

Mechanical Systems

Prof. Richter

Ideal Spring

⊲ Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

Simple example:quarter-carsuspension

More difficultexample

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Viscous Linear Damper

Ideal Spring

⊲Viscous LinearDamper

Laws for mechanicalsystems

Simple example:quarter-carsuspension

More difficultexample

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Laws for mechanical systems

Ideal Spring

Viscous LinearDamper

⊲

Laws formechanicalsystems

Simple example:quarter-carsuspension

More difficultexample

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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 = I

d~w

dt= I~α

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

Simple example: quarter-car suspension

Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

⊲

Simple example:quarter-carsuspension

More difficultexample

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

Solution

Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

⊲

Simple example:quarter-carsuspension

More difficultexample

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More difficult example

Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

Simple example:quarter-carsuspension

⊲More difficultexample

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

Solution

Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

Simple example:quarter-carsuspension

⊲More difficultexample

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Solution

Ideal Spring

Viscous LinearDamper

Laws for mechanicalsystems

Simple example:quarter-carsuspension

⊲More difficultexample

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Verify that the required equation is

(

M +I

r2

)

y + by +

(

kM +k

2

)

y =F

2