Department of Mechanical Engineering MAK 333E – SYSTEM DYNAMICS AND CONTROL MIDTERM EXAM 1 (Fall 2014-2015) Instructor: Assoc. Prof Dr. Erdinç ALTUĞ Student Number : Name and Surname :
Date : October 31, 2014 (Exam duration is 90 minutes.) minutes.) Signature : 1
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Sum
QUESTIONS and ANSWERS Problem 1:
[15 points] In a steel rolling mill, the thickness of steel bars is changed. The system measures the thickness of the bars and the gap distance between rolls is adjusted with a motor. Motor is controlled according to difference between the desired thickness and the measured value. a) Determine the input and the output of this system. b) Is this a open-loop or closed-loop system? Why? c) By clearly showing the actuator, controller and sensor, draw the block diagram of the system.
gap distance
Figure 1. Steel Mill
Answer:
1
Problem 2:
[15 points] Solve the following ordinary differential equation using the Laplace Transform
Method. Note that the initial conditions are y (0) = 0, 2
d y dt
2
−
2
dy
dy dt
(0) = 0 and u(t) is the unit-step function.
15 y = 3u (t ) dt −
Answer:
2
Problem 3: [15 points] The block diagram of a multiple-loop feedback control system is shown in the Figure 2. Simplify the block diagram and find C(s)/R(s).
G2 ( s ) U(s)
+ _
Σ
Y(s)
+
G1 ( s )
-
Σ
H ( s ) Figure 2. Block diagram
Answer:
3
G3 (s)
Problem 4:
[25 points] Consider the system shown in Figure 3.
x1
x2
t b1
b2 Figure 3. A mechanical system
a) By drawing free-body-diagrams and showing each force clearly, determine the equations of motion.
y1 = x1 b) Obtain the state-space representation of the system by selecting the states as
y2
=
xɺ1
y3 = x2 y4 = xɺ2
Answer: a)
b)
4
Problem 5:
[15 points] Consider the following control system
G (s) =
1 2s + 1
a) What is the order of this system? Calculate the time constant of this system. What does a time constant mean? Explain. b) Which one of the following systems is faster?
G(s) =
1 2s + 1
G (s) =
2 2s + 2
G (s) =
1 3s + 1
G (s) =
1 2,5 s + 1
c) If you apply a unit-step input to this system, calculate the system response c(t). Draw this response roughly, clearly showing the time constant and the steady-state error.
Answer: st
st
a) 1 order with time constant 2 seconds. Time constant is the time it takes for a 1 order system to reach 63,2% of its final value. b) Second system with time constant 1 seconds. c)
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Problem 6: [15 points] Consider a satellite attitude control system shown in Figure 4a. The output of this system exhibits continued oscillations and is not desirable. This system can be stabilized by use of tachometer feedback, as shown in Figure 4b.
Figure 4
(b)
a) What is the order of these systems? Write closed-loop transfer functions (C(s)/R(s)) of both of the systems.
b) If K/J = 4, and Kh=1, determine the characteristic equations of both systems. What are the poles and zeros of the systems? Plot them on the complex plane. Explain why the tachometer feedback is useful for this system.
