Modern Control Engineering

Example Problems and Solutions 89

Solution.In Figure 3–23(a) we assume that displacements xi, xo,andyare measured from their

respective steady-state positions. Then the equations of motion for the mechanical system shown

in Figure 3–23(a) are

By taking the Laplace transforms of these two equations, assuming zero initial conditions, we have

If we eliminate Y(s)from the last two equations, then we obtain

or

Hence the transfer function Xo(s)/Xi(s)can be obtained as

For the electrical system shown in Figure 3–23(b), the transfer function Eo(s)/Ei(s)is found to be

=

AR 1 C 1 s+ 1 BAR 2 C 2 s+ 1 B

AR 1 C 1 s+ 1 BAR 2 C 2 s+ 1 B+R 2 C 1 s

Eo(s)

Ei(s)

=

R 1 +

1

C 1 s

1

A 1 R 2 B+C 2 s

+R 1 +

1

C 1 s

Xo(s)

Xi(s)

=

a

b 1

k 1

s+ 1 ba

b 2

k 2

s+ 1 b

a

b 1

k 1

s+ 1 ba

b 2

k 2

s+ 1 b+

b 2

k 1

s

Ab 1 s+k 1 BXi(s)= ab 1 s+k 1 +b 2 s-b 2 s

b 2 s

b 2 s+k 2

bXo(s)

b 1 CsXi(s)-sXo(s)D+k 1 CXi(s)-Xo(s)D=b 2 sXo(s)-b 2 s

b 2 sXo(s)

b 2 s+k 2

b 2 CsXo(s)-sY(s)D=k 2 Y(s)

b 1 CsXi(s)-sXo(s)D+k 1 CXi(s)-Xo(s)D=b 2 CsXo(s)-sY(s)D

b 2 Ax#o-y#B=k 2 y

b 1 Ax#i-x#oB+k 1 Axi-xoB=b 2 Ax#o-y#B

(a) (b)

xi

xo

k 2 y

k 1

b 2

b 1

R 2

R 1

ei eo

C 2

C 1

Figure 3–23
(a) Mechanical
system;
(b) analogous
electrical system.