Download Lesson 7 - CRV Lab

1 Vibration Isolation
208322 Mechanical Vibrations
Lesson 7
We see from the figure that the motion transmitted from the supporting
structure to the system is less than 1 when the ratio  / n is greater than
We present two schemes in vibration isolation. First is when the
system is subjected to motion of its support, as was discussed in Lesson 6.
The displacement transmissibility is given by
2. Therefore, n must be small compared to . This can be done by
using soft spring whose k is small.
1/ 2

X 
k 2  (c ) 2

Y  (k  m 2 )  (c ) 2 
Second is when a mass-spring-damper system produces a harmonic
force F  t   F0 sin t. We are interested in the force transmissibility, a
1/ 2
 1  (2 r ) 2


2 2
2
 (1  r )  (2 r ) 
ratio between amplitude of the force transmitted by the machine to the
supporting structure and amplitude of the disturbing force F  t  . The
,
force transmissibility is derived similarly to that of Lesson 6 and is given by
where r   / n , X is amplitude of system, and Y is amplitude of
support. The plot of this amplitude ratio is given in Lesson 6 and is shown
again below.
1/ 2

FT  1  (2 r ) 2

2 2
2
F0  (1  r )  (2 r ) 
.
We can see that it has the same form as X / Y of the support motion
case. Because
X
F0 / k
1/ 2
2 2
2

  
 

 1       2
 
   n    n  
,
to reduce X without changing the ratio FT / F0 , we need to increase k
and mount the mass
k /  m  M  and hence
m on a large mass M to keep the ratio
 / n constant.
2 Energy Dissipated by Damping
Damping is present in all oscillatory systems. Its effect is to remove
energy from the system, resulting in smaller oscillation. This section we
want to look closely how damping removes energy from the system.
Consider a damping force Fd  cx acting on a system whose
Figure 1: The plot of X / Y versus frequency
ratio r   / n .
steady-state displacement and velocity are given by
1
Copyright
2007 by Withit Chatlatanagulchai
208322 Mechanical Vibrations
Lesson 7
x  X sin t    ,
 F dx
  cx dx
  cx dt
 
 c X 
cos t    dt
Wd 
x   X cos t    .
Therefore,
d
2
Fd  cx
2
 c X cos t   
  c X 2 .
Damping properties of materials are listed in many different ways.
First is specific damping capacity, defined as the energy loss per cycle Wd
 c X  x .
2
divided by the peak potential energy U :
Rearranging terms, we have an equation of an ellipse
2
Wd
.
U
2
 Fd   x 

    1
 c X   X 
Second is loss coefficient, defined as the ratio of damping energy loss per
radian Wd / 2 divided by the peak potential or strain energy U :
whose graph is given in Figure 2.

Fd  kx
Fd
2
0
 c X 1  sin 2 t   
2
2 /
2
Wd
.
2 U
☻ Example 1: [1] Determine the expression for the power developed by a
force F  F0 sin t    acting on a displacement x  X 0 sin t.
x
x
Figure 2: Energy dissipated by viscous
damping.
The energy loss per cycle due to the damping force is computed
from the equation
2
Copyright
2007 by Withit Chatlatanagulchai
208322 Mechanical Vibrations
Lesson 7
Solution
Power is the rate of doing work, which is given by
3 Sharpness of Resonance
Sharpness of resonance is the same as bandwidth already
discussed in Lesson 6.
dx
dt
  X 0 F0  sin t    cos t.
PF
4 Vibration-Measuring Instruments
The basic element of many vibration-measuring instruments is the
seismic unit of Figure 3.
☻ Example 2: [1] A force F  F0 sin t acts on a displacement of
x  X sin t    . Determine the work done per cycle.
Figure 3: A seismic unit.
The seismic unit is mounted on the surface of the system whose vibration
is to be determined. Suppose x  t  is absolute displacement of the seismic
Solution
We know that
m and y  t  is absolute displacement of the system that
vibrates with y  t   Y sin t. Our objective is to determine the
unit mass
W   F x dt.
Substituting F  F0 sin t
displacement, velocity, and acceleration of the system that the seismic unit
is mounted on. We can achieve this by studying the relative motion
z  x  y. We have done this in Lesson 6 where we studied relative
motion of the support motion case. The relative motion is given by
and x  X sin t    into the equation
above gives the work done per cycle of
W 
2 / 
0
 F0 sin t   X cos t     dt
z (t ) 
  F0 X sin  .
3
m 2Y sin(t  1 )
2 1/ 2
(k  m )  (c ) 
2 2
Copyright
 Z sin(t  1 ) ,
2007 by Withit Chatlatanagulchai
208322 Mechanical Vibrations
Lesson 7
where
Z
m 2Y
(k  m 2 )2  (c )2
Y
r2
(1  r 2 )2  (2 r ) 2
,
 c 
 2 r 
 tan 1 
.
2 
2 
 k  m 
1 r 
The ratio Z / Y is shown in Figure 4, and plot of 1 is in Figure 5.
1  tan 1 
Figure 5: Variation of  with
r.
Next, we will discuss two main types of vibration pickup
equipments. First, seismometer is an instrument with low natural
frequency. From Figure 4, when  / n is large, the relative displacement
Z approaches Y regardless of the value of the damping  .
Second, accelerometer is an instrument with high natural
frequency. When  / n  0, we have
Figure 4: Relative Motion plot.
1
(1  r )  (2 r ) 2
2 2
4
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1
2007 by Withit Chatlatanagulchai
208322 Mechanical Vibrations
Lesson 7
y  t   sin t  sin3t.
and then
Z
 2Y acceleration

.
n2
n2
By measuring, we may have
z  t   sin t  1   sin  3t  2  ,
The useful range of the accelerometer can be seen from Figure 6.
Accelerators often have   0.7 to extend their useful ranges to
1 and 2 are phase shifts. The situation is shown in Figure 7
where the reproduced wave z  t  differs greatly from the original wave
y t .
where
0   / n  0.2.
Figure 6: Acceleration error versus frequency.
The relative motion z is usually converted to an electric voltage by
making the seismic mass a magnet moving relative to coils fixed in the
case. Because the voltage generated is proportional to the rate of cutting
of the magnetic field, the output of the instrument will be proportional to
the velocity of the vibrating body. Such instruments are called velometers.
Phase distortion occurs in the reproduction of a complex wave.
Suppose that the system vibrates with two frequencies
Figure 7: Phase shifts.
This can be avoided by either having all phase angles be zero or all
harmonic components be shifted equally. From Figure 5, the first case
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Copyright
2007 by Withit Chatlatanagulchai
208322 Mechanical Vibrations
Lesson 7
  0 for  / n  1; the second case corresponds to
  0.7 for  / n  1. The second case can be expressed by the
corresponds to
equation

Thus, for
 
.
2 n
  0 or   0.7, the phase distortion is practically eliminated.
Lesson 7 Homework Problems
None.
Homework problems are from the required textbook (Mechanical
Vibrations, by Singiresu S. Rao, Prentice Hall, 2004)
References
[1]
Theory of Vibration with Applications, by William T. Thomson and
Marie Dillon Dahleh, Prentice Hall, 1998
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Copyright
2007 by Withit Chatlatanagulchai