88 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems
A–3–3. Obtain a state-space representation of the system shown in Figure 3–22.
Solution.The system equations are
The output variables for this system are y 1 andy 2 .Define state variables as
Then we obtain the following equations:
Hence, the state equation is
and the output equation is
A–3–4. Obtain the transfer function Xo(s)/Xi(s)of the mechanical system shown in Figure 3–23(a). Also
obtain the transfer function Eo(s)/Ei(s)of the electrical system shown in Figure 3–23(b). Show that
these transfer functions of the two systems are of identical form and thus they are analogous systems.
B
y 1
y 2
R= B
1
0
0
0
0
1
0
0
RD
x 1
x 2
x 3
x 4
T
D
x# 1
x# 2
x# 3
x# 4
T = F
0
-
k
m 1
0
k
m 2
1
-
b
m 1
0
0
0
k
m 1
0
k
m 2
0
0
1
0
VD
x 1
x 2
x 3
x 4
T + E
0
0
0
1
m 2
Uu
x# 4 =
1
m 2
C-kAy 2 - y 1 B+uD=
k
m 2
x 1 -
k
m 2
x 3 +
1
m 2
u
x# 3 =x 4
x# 2 =
1
m 1
C-by# 1 - kAy 1 - y 2 BD=-
k
m 1
x 1 -
b
m 1
x 2 +
k
m 1
x 3
x# 1 =x 2
x 4 =y# 2
x 3 =y 2
x 2 =y# 1
x 1 =y 1
m 2 y$ 2 +kAy 2 - y 1 B=u
m 1 y$ 1 +by# 1 +kAy 1 - y 2 B= 0
m 1 m 2
k
y 1
b
u
y 2
Figure 3–22
Mechanical system.
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