Transcribed Text
Problem 1, Consider the system with two masses shown in the figure below. Each
mass is on a surface with a friction coefficient.
X1 (t)
x2(t)
fv3
f(t)
K1
K3
0000
M1
K2
M2
0000
fv,
fv2
(a. ) Derive the differential equations that describes the motion of the system.
(b. . ) Express the system in state space form with the displacement and velocity of
both masses as the system outputs.
Problem 2 Consider the pitch control system of a space shuttle in the figure
below.
Commanded
Commanded
pitch
pitch
rate
acceleration
Shuttle
Commanded
Controller
dynamics
+
+
pitch
+
+
+
Pitch
K1
K2
G1(s)
G2(s)
Pitch
Pitch
Pitch
error
rate
acceleration
error
error
Accelerometer
32
Rate gyro
5
Inertial measuring unit
1
(a. ) Find the closed loop transfer function relating pitch to commanded pitch.
(b. . ) Find the closed loop transfer function relating pitch rate to commanded
pitch rate.
(c. Find the closed loop transfer function relating pitch acceleration to
commanded pitch acceleration.
Problem 3 Consider the block diagram shown below.
R(s) +
1
+
50
+
C(s)
S
S + 1

2
2
s
(a. ) Draw the signal flow graph, and find the transfer function G(s) = C(s) / R(s)
for the system.
Problem 4 Consider the approximate model of a tensile tester shown in the figure
below.
Current
Load, frame, specimen,
Desired load
Controller
amplifier
load cell
Actual load
F¿(s)
+
E((s)
0.06
15 x106
F(s)
1666.67
s+720
33+8232+4X 106s + 2x 107

E(s)
1
(a.) Model the system in state space.
(b.) Simulate the unit step response using MATLAB. Is the response predominately
first order or second order?
(c.) Plot the controller output ei(t) for the unit step response in (b.) .
Problem 5 Consider the two stage Operational Amplifier circuit shown below.
100 k§2
4 uF
2 F
v1(t)
V
V1(t)
vi(t)
vo(t)
+
500 kQ2 2F
+
400 k§2
4 F


(a. ) Find the transfer function relating the output voltage Vo(t) to the
input
voltage VI(t). .
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