Name ___________________________
Section _____________________
ES205
Examination III
May 14, 1999
Problem
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2
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3
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Total
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Name
ES205 Examination III
30 pts
May 14, 1999
Problem 1
Problem 1
1.1 Frequency response plots for a physical system were determined experimentally and are
shown below. Assuming this system was forced with f(t) = 10cos(0.2t), what would the
steady state response be? (3pts)
Bode Phase Plot for System X
Bode Magnitude Plot for System X
20
40
10
20
0
-10
Magnitude (dB)
Phase (degrees)
0
-20
-40
-60
-20
-30
-40
-80
-50
-100
-60
0.2
0.4
0.6 0.8 1
2
4
6
8
0.2
0.4
0.6
0.8 1
2
4
6
8
Frequency Rad/Sec
Frequency Rad/Sec
1.2 Draw a block diagram for the system described by the equation below: (3pts)
h& + 2 h = 10
1.3 Determine the steady state response, yss, for the system shown below using the transfer
function approach. (3pts)
5sin3t
5s
s + 3s + 18
2
1.4 The response of a second order system to a step, f(t) = 10u(t), is shown below. The system
shown may be described as follows: (2pts)
a) undamped
b) underdamped
c) critically damped
d) overdamped
e) none of these
Response of 2nd Order System to a Step
1.8
1.6
1.4
Reponse
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Time Sec
6
7
8
9
10
1.5 If the system shown in problem 1.4 is excited with a step, f(t) = 6u(t), what is the steady
state response of the system? (2pts)
1.6 A mechanical system which has a natural frequency ωn and a damping ratio ζ is subjected to a
forced excitation frequency ω. At steady state, the system will vibrate at (2pts)
a) ω n
b) ω d
c) ω n 1 − ζ 2
d) ω
e) none of the above
1.7-1.10
Given a system that is described by the following differential equation:
&x& + 2 x& + 4x = f ( t ) where f(t) is a unit step, determine the following values:
1.7 the maximum % overshoot (3pts)
1.8 the time to the first peak, tp. (3pts)
1.9 the 2% settling time (3pts)
1.10 If the input to the system is 4u(t), where u(t) is the unit step, what is the steady state
response of the system? (3pts)
1.11
What transfer function would yield the following frequency response plot? (3pts)
Magnitude, dB
Phase (degrees)
40
0
20
-90
0.1
1
10
Frequency (Rad/s)
-180
0.1
1
10
100
Frequency (Rad/s)
Name
ES205 Examination III
30 pts
May 14, 1999
Problem 2
Sketch the straight line asymptotic frequency response plots for the following transfer function.
Use the semilog paper given below for this purpose. Show all work.
25.3s
(s + 10)(s 2 + 4.2s + 0.8)
70
60
50
40
30
20
10
0
0.01
0.1
1
10
100
10
100
Fr e q u e n c y
70
60
50
40
30
20
10
0
0.01
0.1
1
Fr e q u e n c y ( H z )
Name
ES205 Examination III
40 pts
May 14, 1999
Problem 3
You have been hired to determine the dynamic characteristics of the tail rotor of a helicopter. You have
only limited equipment so you perform two tests:
1) A static deflection test where you apply a known force and measure the displacement,
2) A free response test where you give the rotor an initial displacement and measure the
displacement of the end of the rotor as a function of time.
The force-deflection curve and the free response curve are shown below. For convenience, the
displacement is shown along the x-axis for the force-deflection curve and the least squares curve fit is
shown on the figure. Assume for parts a), b) and c) the root of the tail is fixed so that the equation of
motion for a single degree of freedom model of the tail is given by
keq, c
SEP
m eq &x& + cx& + k eq x = f (t )
meq
x
Determine:
a) The spring constant, damping and equivalent mass of the tail of the helicopter (20 pts) Note: If you
are unable to answer part a), assume values for the remainder of the problem.
b) Determine the steady state response of the tail assuming that the force applied to the tip of the tail is
f(t)=50sin100t N when it is operating. (10 pts)
c) At approximately what forcing frequency would you expect the tail to have a maximum displacement. (5 pts)
c) Determine the steady state response assuming the root of the tail is not fixed, but rather has a
prescribed displacement of y(t) = 2 sin100t mm. Use the spring constant, damping and equivalent
mass found in part a). (5 pts)
keq, c
y
SEP
meq
Free Response for tail rotor
300
2.5
y = 100.4x
250
2
200
150
100
50
0
0
1
2
Displacement (mm)
3
Displacement (mm)
Force (N)
x
1.5
1
0.5
0
-0.5 0
0.1
0.2
-1
-1.5
-2
Time (s)
Continue your work on the next page if necessary
0.3
0.4