Dynamics and Vibrations - Michigan State University
Transcript of Dynamics and Vibrations - Michigan State University
Student Code Number:_____________
Ph.D. Qualifying Exam
Dynamics and Vibrations
January, 2010
T. Pence and B. F. Feeny
Directions: Work all four problems. Note that the problems are EVENLY WEIGHTED.
You may use two books and two pages of notes for reference.
3. A trailer is pulled down a wavy road at a speed v. The road profile is approximated as sinusoidal with a wavelength of L. Modeling the tires as rigid and the trailer as a single degree of freedom mass spring system with base excitation, schematically depicted below, where the deflection of the base is
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y(t) =Y sin(ωt), the absolute vertical deflection of the mass is x, and the deflection of the mass relative to the road surface is z = x – y. The damper is designed for
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ζ = 12
.
(a) Write ω in terms of v and L. (b) Design the suspension spring k such that at a frequency
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ω or higher, the trailer
deflection x has magnitude
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XY
<12
. Use
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ζ =12
for this calculation.
(c) The driver finds that the driving on these roads, the shocks often need replacement. Write an expression for the magnitude
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Fd of the steady-state dynamic force in the
dashpot as a function of r
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r =ωωn
, where
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ωn is the undamped natural frequency of the
trailer.
(d) Plot the nondimensional force magnitude, i.e.
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FdcYωn
, and show the features of the plot
as
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r→ 0, r = 1, and
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r→∞.
4. Some tests are performed on the three-mass structure depicted below. The masses are known, but the stiffnesses are unknown. The equations of motion of the system with the displacement coordinates shown is
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M ˙ ̇ x + C ˙ x + K x = f , where
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x =
x1x2x3
"
#
$ $ $
%
&
' ' ' and
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f =
f1(t)f2(t)f3(t)
"
#
$ $ $
%
&
' ' '
are the displacement vector and applied force vector, and where
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M ,
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C , and
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K are the mass, damping and stiffness matrices. Damping is modeled as “proportional” (or “Rayleigh”), and in the form
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C =αM +βK , where
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α is unknown and
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β = 0 . Information on the mode shapes and corresponding undamped modal frequencies (in rad/sec) is partially known as
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u1 =
a2.44952
"
#
$ $ $
%
&
' ' ' ,
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u2 = γ
−101
$
%
& & &
'
(
) ) ) ,
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u3 =
b−2.44955
2
#
$
% % %
&
'
( ( ( ,
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ω12 = 6.2020,
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ω22 =16,
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ω32 = 25.7980
The mode shapes are normalized (units are kg-1/2), except that γ is an unknown normalization constant, and vector elements a and b are unknown. Also, the stiffness matrix K is unknown, but it is known that
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M =
2m 0 00 m 00 0 m
"
#
$ $ $
%
&
' ' '
, where m = 1/12 kg.
(a) Determine the unknown element a of the first modal vector. (b) Determine normalization constant γ for mode 2. (c) The second mode is excited into a free vibration. The response of one of the coordinates is measured and plotted on the next page. Estimate the modal damping of this mode, i.e.
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ζ 2 , and then obtain the unknown proportional damping coefficient α. (Recall that undamped modal frequencies squared are given above, and
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β = 0 .) You may make simplified calculations by assuming
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ζ 2 is small. (But do not assume
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ζ 2 = 0!) (d) Now an impulse is applied such that
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f = (0 F0δ(t) 0)T in where F0 = 2 Ns, when the system starts at rest (zero initial conditions). Find the response
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x 3(t) of the third mass for the undamped case (for simplicity).