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Unformatted text preview: Mechanical Vibrations Exam 1 Student Name: ﬂ So i n 1. [5 pts] The response of a single degree Reéponse Amplitude of freedom mass, spring, damper system
is recorded over a range of frequencies.
This test is performed for three different
dampers and the measured amplitudes .
for each damper are shown in Figure ,  7 
1. If the damping ratios are (1 = 0.02, 1
C2 = 0.04, C3 = 0.08, label the ampli—
tude curve that corresponds to the each
damping ratio. Derivation is not re
quired. IX(Q) (m/N) 5 10 15 20 25 so
frequency (Hertz) Figure 1: Response amplitude from three different vibration
test; each test used a different damper. 2. {5 pts] Finish the following sentence: F i c r (2r 1‘9 S
Any periodic function can be expressed as a 0" r '
Write one form of this expression. j((9: f (w CHI/2
“71%)
T  JH
c ‘: L M
" r/‘Wé 3. [15 pts] The position of a point mass, shown as point B in Figure 2, is written with respect
to the origin (A) of a rotating coordinate system. The vector describing the position Within the
rotating coordinate frame is FB/A = 31527; + 103', where t is the time in seconds. The (cc, 3/, 2) frame rotates at Q = 10% + 53'. Find the velocity of the mass, Fem as measured in the fixed frame at t = 1
second. U Figure 2: Fixed coordinate system (U, V, W) and a rotating coordinate system (as, y, z). (b: rpL rm
g: 7‘; + d M 1&4?” +JH§
\J » dt' 91* K” A 4. [15 pts] Estimate the natural frequency (nun), damping ratio (g), and the mass (m) of the
single degree of freedom transfer function, H (0), shown in Figure 3. The value of H ((2 = 0) x
2.2517 X 10—4. x104 Transfer Function Real Hm) (m/N) 5 1o 15 20 25 30
frequency (Hertz) 4'01sz 1o :5 20 25 so
frequency (Hertz) rug—J
r , m
Figure 3: SDOF Transfer function 2 j, G wit
u _ r 2 , 7’? Xp— Dem rm rm “309/” efﬁ— "’f
.. APU
(D H(fL — eorpof : ___1_r1\____________.
ﬂ .
I ' Wf Wat—ﬁlflc'anjL 415w! ‘
(9/ K 3 HM”): 1 1.1517100” 5. [30 pts] Develop the equation of motion for the system shown in Figure 4, Where f(t) = Aces Qt. Use either 213%: = % (ml/E) or Lagranges method. Solve for the transient and steady—state displacement of the system. (771 = 1 Kg, 01 = 0.5 Ns/meter, c2 = 1.5 Ns/meter, k1 = 3 N/meter,
k2 = 6 N/meter). The initial conditions are :50 = 0 and v0 = 1 meter/s. x(t) gt— FR b
Ft! ”.7 {[4 )
& e—~— F“
(—d ¥ Figure 4: Single degree of freedom system
F“ (2
" ’f’fanriﬁ'fl'oﬂéi Moria/1 F,“ T F. K/ {‘72 ; {/7 X A‘f: I‘ \_ .I (\‘t .I _ ,‘t‘
n : A e 2 "KM/4% / Xk We f SUB W007 ©2+2fﬁlnd+ wn2)ACI‘=C)
é—= /\ _ >{wn ream/{Ll if? " _ 1""13 7W”!
4 '1—4ii2‘r7: m) : A.)r+)\1B+iﬂ'D= l/f’l f) /\1(ﬂ{Q'B'D)1\1B+iﬂD:I «"3: Limo + XD 6. [30 pts] Derive the equation of motion and transfer function for the system shown
in Figure 5. Assume a motor is used to apply an external moment, Ma, about point 0 (Where Ma 2 Acos Qt (Nm)). Use the following symbols: k3 is a torsional spring (N/radian), mass of
object 1 is m1, mass of object 2 is mg, the moment of inertial about each members center of gravity I91 and I32. Include gravity and assume small angles: sin9 m 9. in W Figure 5: Rotating system with torsional spring, kg, and external moment M,1 about point 0. A {hKeﬁg'lme CM,+M1)j(3)/+ng 9 I0 5 l  d] +rum FB 0/ e _: m, [0/2 4* M2,?
(“1+ m2} ...
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 Spring '08
 CHEN

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