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TRANSCRIPT
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1Lecture 9- Forced Vibration with Damping
MEEM 3700 1
MEEM 3700MEEM 3700Mechanical VibrationsMechanical Vibrations
Mohan D. Rao Chuck Van Karsen
Mechanical Engineering-Engineering MechanicsMichigan Technological University
Copyright 2003
Lecture 9- Forced Vibration with Damping
MEEM 3700 2
Single Degree of Freedom Forced Vibration
mm
k C
X
F(t)
mx
Free Body Diagram
mg
k F(t)
xc& kx
x x x ( )m c k F t+ + =&& &
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2Lecture 9- Forced Vibration with Damping
MEEM 3700 3
mm
kk CC
XX
F(t)F(t)
If If
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3Lecture 9- Forced Vibration with Damping
MEEM 3700 5
SolveSolve (1) (1) and and (2) (2) simultaneously,simultaneously,
[ ]
=
+=
21
50222
mKCtan
)C()mk(FX .
o
( )x( ) cos( ) sin( ) cos( )nt d dt e A t B t X t = + +
+ = + =
( )X cos( ) sin( )
( )X cos( ) ( )X sin( )
K m C X F
C K mo
2
2 0(1)(1)(2)(2)
Lecture 9- Forced Vibration with Damping
MEEM 3700 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
-0.5
0
0.5
1
1.5
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4Lecture 9- Forced Vibration with Damping
MEEM 3700 7
0.52 2 2
12
( ) ( )
tan
oFXk m C
Ck m
= +
=
kmc
mk :Recall n 2==
Lecture 9- Forced Vibration with Damping
MEEM 3700 8
2 22
Xk 1
1 2o
n n
F
= +
t a n
=
2
12
n
n
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5Lecture 9- Forced Vibration with Damping
MEEM 3700 9
Magnification Factor vs. Frequency Ratio
Frequency Ratio (/n)
Mag
nifi
catio
n Fa
ctor
(XK
/Fo) XK
Fo
n n
=
+
1
1 22 2 2
..15
..05
.2
.3
.4
.5
.6
.81.0
5.0
Lecture 9- Forced Vibration with Damping
MEEM 3700 10
Frequency Ratio (/n)
Phase Angle vs. Frequency Ratio
Phas
e A
ngle
()
/2
tan
=
2
12
n
n
.05.05
.15
5.01.0
.2 ..3.4 .5 .6 .8
ta n
=
2
12
n
n