Structural Vibrations
Big Trouble!
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Earthquake-induced vibrational failure:
Kobe, Japan 1995
http://www.strimoo.com/video/17063414/Kobe-Earthquake-Metacafe.html
Hanshin Expressway
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Safeguards: Base Isolation Keep Structure
intact by providing soft supports
Rubber pads being installed under a
building
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With and without
Other Suspension Systems (same idea)
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Disk Drive
Energy Dissipation: Passive Damping
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Active Viscous
Damping:
Rheomagnetic fluid
Active Mass Damping
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Typical General Vibration Response
Displ
Period, T
y(t)
Peak to Peak
Amplitude A
time
Cycle: One back and forth motion
Period: T Time to complete one cycle
Amplitude: X Max departure from equilbrium=2A
Frequency: f  1 (Hertz, or cycles per sec),
T
Angular Frequency   2 f  2 (radians per sec)
T
Common Features of Vibrations
As systems vibrate, they typically pass
through their equilibrium state.
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Displacement typically follows a sinusoidal
variation with time
x(t )  X 0 sin t   
x(t)
sx(t)
k, d
m
t
Parameters in the typical response
x(t )  X 0 sin t   
Angular Frequency:  (radians per unit time)
Frequency: f   (cycles per unit time, eg Hertz=cycles/sec),
2
1
2
Period: T    (Time to complete one cycle)
f
Amplitude: X o (Maximum displacement from equilibrium)
Phase:  sin 1  xo X o  (Describes ratio of initial displ x(0) to the amp X 0 )
x( t )
sx(t)
Xo
xo
k, d
t
m
T
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Example: Pendulum
Period: T  2 gl seconds
Frequencies f  1 g Hertz,
2 l
  g radians per sec
l
t
T
b
g
x(t )  X 0 sin t  
Example: Spring-Mass
Period: T  2 m seconds
k
k Hertz,
Frequencies f  1 m
2
k radians per sec
 m
d
x(t)
s
T
k, d
t
m
x(t )  s(t )  d  X 0 sin t   
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Frequency and Period are System Dependent
s
k, d
m
T  2 m seconds
k
T  2 gl seconds
k Hertz,
f 1 m
2
k radians per sec
 m
f  1 g Hertz,
2 l
  g radians per sec
l
Amplitude and Phase depend on Initial
Conditions: x(0)=xo and v(0)=vo
x( t )
X0
t
xo
x(t )  X 0 sin t   
Amplitude: Maximum displacement from
static equilib. position
X 0  xo2  vo2 /  2
Phase: Describes the ratio of the initial
displacement x0 to the amplitude of vibration
X0
  sin1  Xxo  (radians)

o 
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