The document discusses vibration damping using the energy method. It introduces modeling of a mass-spring system and calculating its kinetic and potential energy. The conservation of energy principle for such systems is explained. The natural frequency can be determined directly from the maximum kinetic and potential energies. Viscous damping is then introduced through a dashpot model. The equation of motion is derived for a damped single degree of freedom spring-mass system. Depending on the damping ratio, the system response can be over-damped, critically damped, or under-damped. Examples of each case are shown. Design of damping for vibration control is demonstrated through an example problem.
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Vibration Damping
Maged Mostafa
Potential and Kinetic Energy
The potential energy of mechanical systems U is often stored in
“springs” (remember that for a spring F=kx)
Uspring F
0
x0
dx kx
0
x0
dx
1
2
kx0
2
The kinetic energy of mechanical systems T is due to the motion
of the “mass” in the system
M
k
x0
Mass Spring
x=0
)(
2
1
)(
2
1 2
txmtmvTtrasn
)(
2
1
tJTrot &
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Vibration Damping
Maged Mostafa
Conservation of Energy
T U constant
or
d
dt
(T U) 0
For a simple, conservative (i.e. no damper), mass spring system
the energy must be conserved:
At two different times t1 and t2 the increase in potential energy
must be equal to a decrease in kinetic energy (or visa-versa).
U1 U2 T2 T1
and
Umax Tmax
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Vibration Damping
Maged Mostafa
Determining the Natural frequency
directly from the energy
Umax
1
2
kA2
Tmax
1
2
m(n A)2
Since these two values must be equal
1
2
kA2
1
2
m(n A)2
k mn
2
n
k
m
Assuming the solution is given by
x(t)= ASin(t+f)
then the maximum potential and kinetic energies can be used to
calculate the natural frequency of the system
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Vibration Damping
Maged Mostafa August 30, 1999
30Mechanical Engineering at Virginia Tech
Solution: Design Problem
00
01
2
0
2
00
2
tan
)()(
1
frequencynaturaldamped,1
)sin()(
xv
x
xxvA
tAetx
n
d
dn
d
nd
d
tn
f
f
Recall the Underdamped response
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Vibration Damping
Maged Mostafa August 30, 1999
32Mechanical Engineering at Virginia Tech
Solution:
sec,/300
sec/16.8,200,3
0 mmv
radkm n
Maximum amplitude, will occur at:
• Smallest Natural Frequency
• Smallest Spring Stiffness
• Largest Mass
• Largest initial velocity
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Vibration Damping
Maged Mostafa August 30, 1999
33Mechanical Engineering at Virginia Tech
Solution:
281.0
025.0)(
1
tansin)(
2
1
2
1
1
tan
1
0
2
1
1
tan
1
0
d
n
d
n
e
v
tx
OR
e
v
tx
n
d
Maximum amplitude, will occur at:
• First Peak