Modern Control Engineering

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Example Problems and Solutions 231

Example Problems and Solutions

A–5–1. In the system of Figure 5–49,x(t)is the input displacement and u(t)is the output angular

displacement. Assume that the masses involved are negligibly small and that all motions are

restricted to be small; therefore, the system can be considered linear. The initial conditions for x

anduare zeros, or x(0–)=0andu(0–)=0. Show that this system is a differentiating element.

Then obtain the response u(t)whenx(t)is a unit-step input.

Solution.The equation for the system is

or

The Laplace transform of this last equation, using zero initial conditions, gives

And so

Thus the system is a differentiating system.

For the unit-step input X(s)=1s, the output becomes

The inverse Laplace transform of gives

u(t)=

1

L

e-(kb)t

Q(s)

Q(s)=

1

L

1

s+(kb)

Q(s)

Q(s)

X(s)

=

1

L

s

s+(kb)

aLs+

k

b

LbQ(s)=sX(s)

Lu

+

k

b

Lu=x#

bAx# -Lu

B=kLu

No friction

x

b

k

u

L

Figure 5–49
Mechanical system.