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(mech300g)[2007](s)final~PPSpider^sol_10519.pdf
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MECH300G SPR07
Solution to MECH300G Final Examination
Spring 2007

29 May 2007
Student Name:______________

Student ID:_________________
Student Signature:____________
Problem 1 (15 pts)
Please answer the following questions briefly.
1.1 What is Leakage problem in the vibration measurement? How can this problem be solved?
Ans: When doing Fourier Transform on the measured data with finite length of time history and with the periodicity assumption, the component at one frequency will leak into the vicinity of another components as shown in the following. Windowing (Hanning window, Exponential window, etc) can solve this problem.

1.2 What is the advantage and disadvantage of passive and active vibration control?
Ans:Advantage Passive simple cheaper Active more effective wide bandwidth
Disadvantage narrow bandwidth complicated

expensive, need longer development time
1.3 What kind of vibration control was used in IFF Antenna of F-100 jet fighter? Why do we have to consider the temperature effect on the vibration control efficiency? Ans: (1) Viscoelastic damper (VPCO-15080 polyvinyl chloride copolymer), (2) the firing of 4 cannons can easily
increase the temperature of the viscoelastic damper. This may greatly decrease the damping ratio of some viscoelastic material.

1.4 What kind of vibration control was used in the worlds tallest buildingC Taipei 101?
MECH300G SPR07
Ans: Vibration absorber (Tuned Mass Damper): 2 spherical TMDs (4.5 tons 2)

1.5 What is the advantage of the random signal analysis for vibration testing?
Ans: The main advantage is that it spends much less time to obtain the vibration testing result (frequency response function).
Problem 2 Vibration under arbitrary force condition (20pt)
Find the time response of the compacting machine using the convolution integral, Eq.(4.33) of the textbook:
1 ..w (t . )
2
n
x(t) = F( )e sin(d (t . ))d , where d =n
1.. =

k / m
n
md
when it is subjected to the force shown in the following figure. Assume there is no damping in the system.

F(t)
1
0 t 1
Ans:
x(t) = 1 t sin[n (t .1. )]d
0
mn
.1 t

= sin[ (t .1. )]d (t .1. )
n
mn 0 n
1

.1
= 2 [. cos[d (t .1. )]] = 12 [1. cos(n (t .1))]
mn
mn
0
Problem 3 Transverse vibration of beam (20 pt)
Estimate the fundamental frequency of a cantilever beam whose cross-sectional area and moment of inertia vary as
xx
A(x) =A0 and I (x) = I0
ll where x is measured from the free end.
MECH300G SPR07
x dW . 2 xd 2W 2
Let W (x) = (1. )2. = (1. ), =
l dxl l dx2 l 2 2 22
l dW 2 EI04 2EI02 l 2 A0 x 4 A0l
2V = EI () dx = x()dx = , 2T = AW (x) dx = x(1. ) dx =
max 2 43
0 dx lll max 0 ll 30
1/2 1/2
. EI0 .. EI0 .
2EI (30) 60EI
20 0
= 3 = 4 =
60.. 4 .. = 7