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designing a vibration absorber...
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Posted by: misty2 ®

05/07/2004, 12:39:08

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I'm trying to design a vibration absorber that will reduce the vibration of a machine by about 60%. I found the equations of motion of the machine without the absorber to be..

c1x1’+m1x1”+k1x1+k2(x1-x2)+c2(x1’-x2’)=Fosinwt
and the absorber alone to be..
m1x2”+k2(x1-x2)+c2(x2’-x1’)=0

 

Is this right so far?

equations of motion….

c1x1’+m1x1"+k1x1+k2(x1-x2)+c2(x1’-x2’)=Fosinwt

m1x2"+k2(x1-x2)+c2(x2’-x1’)=0

matrix form…

[m1 0] [x1"]+          [c1+c2       -c2][x1’]+       [k1+k2 k1-k2][x1] =[Fosinwt]

[0 m2] [x2"]           [-c2              c2][x2’]           [k2         -k2] [x2]    [ 0         ]

 

finding eigenvalues…

det[(M^-1)K-Ilambda]=0

[((k1+k2)/m1)-lambda             k1-k2] =0

[k2/m2                   (-k2/m2)-lambda]

how do I find k2 so that I can get my eigenvalues?  I was told I have to calculate k2 such that the natural frequency of the absorber is equal to the excitation frequency (omega).  But I'm not really sure how to do this.  Please help me!!







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Re: designing a vibration absorber...
Re: designing a vibration absorber... -- misty2 Post Reply Top of thread Forum
Posted by: jbel ®

05/22/2004, 07:55:34

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First,

I believe you have formulated an incorrect matrix.

I am assuming the following conditions:

k1 = spring from ground to machine

k2 = spring from machine to absorber

c1 = dashpot from ground to machine

c2 = dashpot from machine to absorber

My equations of motion:

m1x1" = -k1x1 -k2(x1 - x2) - c1x1 -c2(x1-x2)

m2x2" = -k2(x2 - x1) - c2(x2-x1)

These are the same as your equations!!!!!!!

Try this matrix form:

[m1 0] [x1"]+    [c1+c2  -c2][x1’]+    [k1+k2 -k2][x1] =[Fosinwt]

[0 m2] [x2"]      [-c2        c2][x2’]       [-k2      k2] [x2]    [ 0         ]

 Review your equations and redo your matricies and you should get the same answer, I think you just transposed something when writing the stiffness matrix.

As far as calculating k2:

Do just as you said. set (omega) = (omega)(nat. abs.) 

noting that  (omega)(nat. abs.) = sqrt(k2/m2)

Then:

(omega)^2 = (k2/m2)

Now, choose m2 to be some value, and if you know (omega) you then know k2!!!!!!

This allows you to choose your absorbing mass and then calculate the corresponding spring stiffness.







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