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Vibration Absorber
Mohammad Tawfik
Vibration Absorber
The first passive damping
technique we will learn!

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Vibration Absorber
Mohammad Tawfik
For a 2-DOF System
• For the shown 2-DOF
system, the equations
of motion may be
written as:
• Where:
fxx  KM 







2
1
f
f
f

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Vibration Absorber
Mohammad Tawfik
For Harmonic Excitation
• We may write the
equation for each of
the excitation
frequency in the form
of:
• Then we may add
both solutions!
 







0
11 tCosf
KM

xx
 






tCosf
KM
22
0

xx

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Vibration Absorber
Mohammad Tawfik
Consider the first force
• We may write the
equation in the form:
• And the solution in
the form:
• Which will give:
 tCosfKM 1
0
1






 xx
 tCos
x
x








2
1
x
  xx 2
2
12
 






 tCos
x
x


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Vibration Absorber
Mohammad Tawfik
The equation of motion becomes
• Get x1() and find out when does it equal
to zero!





































00
0 1
2
1
22
221
2
2
1
2
f
x
x
kk
kkk
m
m



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Vibration Absorber
Mohammad Tawfik
Using the Dynamic Stiffness
Matrix
• Writing down the dynamic stiffness matrix:
Getting the inverse:




















0
1
2
1
22
2
2
2211
2
f
x
x
KmK
KKKm


   




















0
1
2
222
2
211
2
211
2
2
222
2
2
1 f
KKmKKm
KKmK
KKm
x
x




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Vibration Absorber
Mohammad Tawfik
Obtaining the Solution
• Multiply the inverse by the right-hand-side
• For the first degree of freedom:
  
 





 







12
122
2
21
2
21212
4
212
1 1
fK
fKm
KKKmKKmmmx
x 

 
  
0
21
2
21212
4
21
122
2
1 



KKKmKKmmm
fKm
x



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Vibration Absorber
Mohammad Tawfik
Vibration Absorber
• For the first degree of freedom to be
stationary, i.e. x1=0
• The excitation frequency have to satisfy:
• Note that this frequency is equal to the
natural frequency of the auxiliary spring-
mass system alone
2
2
m
K


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Vibration Absorber
Mohammad Tawfik
Vibration absorber

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Vibration Absorber
Mohammad Tawfik
Vibration absorber

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Vibration Absorber
Mohammad Tawfik
Homework #2
• Repeat the example of this lecture using
f2=f3=0 and f1=1 AND f1=f2=0 and f3=1
• Plot the response of each mass for each
of the excitation functions
• Comment on the results in the lights of
your understanding of the concept of
vibration absorber

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Vibration Absorber
Mohammad Tawfik
Homework #2 (cont’d)
• Use modal decomposition
(diagonalization) to obtain the same
results.