Modern Control Engineering

98 Chapter 3 / Mathematical Modeling of Mechanical Systems and Electrical Systems

y 1

y 2

u 1

m 2

b 1

u 2

k 2

k 1

m 1

Figure 3–32Mechanical system.

B–3–3.Obtain a state-space representation of the mechan-

ical system shown in Figure 3–32, where u 1 andu 2 are the

inputs and y 1 andy 2 are the outputs.

B–3–4.Consider the spring-loaded pendulum system shown

in Figure 3–33. Assume that the spring force acting on the

pendulum is zero when the pendulum is vertical, or u=0.

Assume also that the friction involved is negligible and the

angle of oscillation uis small. Obtain a mathematical model

of the system.

kk

a



mg

u

Figure 3–33Spring-loaded pendulum system.

B–3–5.Referring to Examples 3–5 and 3–6, consider the

inverted-pendulum system shown in Figure 3–34. Assume

that the mass of the inverted pendulum is mand is evenly

distributed along the length of the rod. (The center of

gravity of the pendulum is located at the center of the rod.)

Assuming that uis small, derive mathematical models for

the system in the forms of differential equations, transfer

functions, and state-space equations.

M

y

x

u

G

O





x

y

x u

Figure 3–34Inverted-pendulum system.

B–3–6.Obtain the transfer functions X 1 (s)/U(s)and

X 2 (s)/U(s)of the mechanical system shown in Figure 3–35.

m 1 m 2

k (^1) k 3
u x 1 x 2
b 1
k 2
b 2
Figure 3–35Mechanical system.
B–3–7.Obtain the transfer function Eo(s)/Ei(s)of the elec-
trical circuit shown in Figure 3–36.
R 1
eo
R 2
ei L C
i 1 i 2
Figure 3–36Electrical circuit.
B–3–8.Consider the electrical circuit shown in Figure 3–37.
Obtain the transfer function Eo(s)/Ei(s)by use of the block
diagram approach.
R 1
C 1 eo
R 2
ei C 2
i 1 i 2
Figure 3–37Electrical circuit.
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