Modal analysis

[michaelthomas]

Hy,

I try to do a damped modal analysis but I do not understand really the results. I get complex values (and the conjugate complex) and 2 mode shapes: one for the real and one for the imaginary part. How can this interpreted ? And what is the relationship to the conjugate complex eigenvalue ?

Thanks for your help
michael

 

[cbrn]

Hi,
the complete explanation is a bit complicated - and long... - to put here in a message board... I'd suggest to first thoroughly read a vibration analysis book.
Anyway, the complex eigenvectors can be interpreted as this:
if s = p+iq is an eigenvalue (in the complex domain), it means that the displacement can be expressed as x(t) = Xe^((p+iq)*t) = Xe^(pt)e^(iqt). Here, X is the particular solution associated to the initial conditions, e^(pt) is an exponential term associated to the decay of the vector in time (decay if p<0 i.e stability, increment if p>0 i.e. unstability), and e^(iqt) is an oscillating term which gives the frequency of the eigenvibration. If you have a multi-dof system, as with a FE model, the above is valid for every node, so the plots of ANSYS give the real part and the imaginary part of the eigenvalue at every node.

Hope this helps...

Regards

 

[michaelthomas]

Hello cbrn,
thank you for your posting.

>if s = p+iq is an eigenvalue (in the complex domain), it >means that the displacement can be expressed as
>x(t) = Xe^((p+iq)*t) = Xe^(pt)e^(iqt).

That I understand..

>Here, X is the particular solution associated to the >initial conditions,

I thought, X is the eigenvector..?

>e^(pt) is an exponential term associated to the decay of >the vector in time (decay if p<0 i.e stability, increment >if p>0 i.e. unstability), and e^(iqt) is an oscillating >term which gives the frequency of the eigenvibration.

ok...

>If you have a multi-dof system, as with a FE model, the >above is valid for every node, so the plots of ANSYS give >the real part and the imaginary part of the eigenvalue at >every node.

I do not understand...You mean, you have different eigenvalues at every node ??

michael

 

[cbrn]

Hi,
eh eh, I thought that I also might be somewhat confusing...
especially in my last sentence... it was a bit late yesterday, should have drunk one more cup of coffee...
So, X is the modulus part of the eigenvector. In fact, the eigenvector is, in this case, an "eigenphasor" in the complex domain; however, the meaning remains the same, as you say.
There are no different eigenvalues at each node, my sentence was VERY inexact! There are as many eigenvalues as the number of DOFs of the system (so, for a continuum, the eigenvalues are expressable as an eigenfunction), and these are the different data stored in the Ansys' "sets". For each eigenmode, X vary with the location (that's the meaning of the U-plots, normalized either to unity or to total mass). For each eigenmode, Ansys plots X together with p and q, separately.

I hope that now it's clearer and more exact...

Regards