Designing a Vibration Absorber for a Two Degree of Freedom System

[renta]

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

How do I do a two degree of freedom system (the machine and absorber together)? I have to calculate the natural frequencies and write the equations of motion in matirx form, and find the K and M matrices, form M^(-1)*K, and for designed k2 and c2, find the eigenvalues of M^(-1)*K. I really don't have much experience with this area, can someone please help me?

 

[renta]

please help me. I should also calculate k2 so that the natural frequency of the absorber is equal to the excitation frequency. I'm so confused with vibrations.

 

[renta]

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 ]
(sorry about the periods, but this board messes up my format in the matrix)

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?

 

[renta]

please help!

 

[Gokul43201]

It's not clear what you have and what needs to be designed. Also, when you say 60% reduction in amplitude, you need to specify at what frequency. Any isolator will give you attenuation only at frequencies above the natural frequency - in this case there are 2 - and the transmissibility is a function of frequency.

Do you have a real machine and a real isolator, or is this just a theoretical problem ? What does the isolator consist of ?

My understanding so far is that you've got m1, m2, k1, c1, and you need to design k2, c2. But if you have an isolator, that fixes k2, c2 and m2. But you seem to have m2. Where does that come from ?

Or you probably don't have an isolator, and want to design one... I'm not sure.

Anyway, your matrix equations are correct so far. To get the 2 natural frequencies, solve :

det|[K] - [M]w^2| = 0

This gives you w1 and w2 in terms of m2, k2 and c2. For a good isolator design, you want w1 and w2 to be as small as possible - preferably small compared to the typical driving frequency w0. So you pick m2, k2 and c2 accordingly, ie. to minimize the w's.

Before I go on, I'd like you to answer the questions I've asked 'cause this may be completely along the wrong track.

 

[Gokul43201]

Errata : I think you want det|[K] - [M]w^2 + iw[C]| = 0

 

[Gokul43201]

Actually, I'm more confused than I let on. If you have just one mass (the machine - the isolator is relatively light) and one isolator, why are you doing 2 dof analysis ? What are m2, c2 and k2 ? I guess need to know what your system really is like.