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.