How to plot Mode shapes of a structure using its frequencies

[Patkin]

Dear friends,

I am a beginner in the mechanical vibration field. I have the EOM ( Equation Of Motion ) of my problem in the standard form ( Mx_tt+Cx_t+Kx=0).Then, in the next stage, I uesd State-space form ( AX_t+BX=0 ) of my EOM to calculate the eigenvalues of the system in order to extract the natural frequencies and damping parameters ( Landa=c+iw : c= damping parameter and w= frequency ). Now I want to plot the mode shapes of my system. I want to put my frequency in a equation and then it plots the corresponding mode shapes of that frequency. I don't know how to do that.

Could you please help me with the problem ?!

Thank you in advance.

 

[FeX32]

Hi Patkin,

I assume that your Mxtt+Cxt+Kx=0 is a lumped model?
What kind of system is it?

You need to model your system as distributed one to obtain the mode shapes.
For example a beam:

http://iitg.vlab.co.in/?sub=62&brch=175&sim=1080&cnt=1

 

[electricpete]

Even lumped models have mode "shapes"... they are just discrete values... one per modeled node, rather than continuous functions of the spatial variables.

When you solve the system ( Mxtt+Cxt+Kx=0), you are solving an eigenvalue problem. The eigenvalues are the modal frequencies, the eigenvectors are the associated modeshapes.

=====================================
(2B)+(2B)' ?

 

[electricpete]

Even lumped models have mode "shapes"... they are just discrete values... one per modeled node, rather than continuous functions of the spatial variables.

should have been

Even lumped models have mode "shapes"... they are just defined at discrete spatial values... one per modeled node, rather than being defined over continuous ranges functions of the spatial variables.

=====================================
(2B)+(2B)' ?

 

[Patkin]

My structure is a continuous simply supported beam. I have problem with calculating its mode shapes using eigenvectors !! I mean I don't know how to use eigenvectors or eigenvalues to calculate mode shapes of a n-DOF ( n Degree Of Freedom ) system!! ( Here, my problem is a 6 DOF system! )

Does anyone know how to do that ??

 

[FeX32]

I agree electricpete. I just assumed the OP was talking about a distributed system because he wanted to plot the mode shapes.
I'm not sure how useful plotting a lumped mode shape of say eg. [1 -1 1 1] would be. Although useful in other respects for sure.

Patkin, as electricpete says, the mode shapes are the eigenvectors themselves.
For a continous system, you need to arrive at the general solution by usually assuming the solution is of the form A*e^(sig*x).
Your mode shapes are then simply the discritised solutions for n=1,2,3....etc.

 

[GregLocock]

I suggest you post your eigenvectors.

The main problem is that none of the standard animation packages understand mode shapes. I think there is a mode shape display in abravibe, a matlab toolbox.

Basically the task is to join the spatial coordinates of your nodes together, typically with straight lines, and then move the nodes with a sinusoid of the correct relative phase and magnitude.

Obviously if you only have normal miodes the phase is just +/- pi/2, and a static display is easy.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[Patkin]

Thank you all for your great help.

I want to know how it is possible to plot the mode shapes ( a static display of a mode shape ) using Complex Eigen vectors ?!

Assume that these are my Eigenvavlues :

Landa1= 5+2i
Landa2=5-2i

and corresponding Eigenvectors are :

V1=[1 1-2i]^T
V2=[1 1+2i]^T

 

[GregLocock]

A complex frequency is an odd thing to get

Anyway, not easy to plot those mode shapes statically as phase is important. You'd probably be best to use some sort of phasor diagram

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[FeX32]

I suspect that lambda is the eigenvalues of an underdamped system. ie.. lam1=-zeta*omega-omega*sqrt(1-zeta^2)*i etc.

Forget about plotting the eigenvectors. There is no use to it unless the system is distributed.