What is the correct way to compute physical displacement in FEM modal analysis?

[dblanky]

Hi all,

I have an FEM model that I am doing a modal analysis of. I wanted to check that how I am computing the physical displacement is the correct way, as I've read a lot of about normalising modes, participation factors, effective masses, etc. and I'm not 100% sure on it.

I've got the various mode shapes and frequencies but now I want to compute how much actual displacement I get if I apply a harmonic force to it. However, I don't want to compute this using the FEM tool as I want to compute a displacement in another tool.

So if I have made my FEM model using units m/kg/N/s and computed a set a modal mass normalised modes, {\phi_i}, whose respective amplitudes are q_i. The physical displacement of my model would then be:

$$U(x,y,z) = \sum_i q_i \phi_i(x,y,z)$$

Does this have units of meters as my modes are mass normalised or do I need another scaling factor here?

The actual mode amplitudes I compute with:

$$q_i = F_i / (-w^2 + j\,C\,w - w_i^2)$$

Here w is the frequency I'm looking at, C some damping constant, w_i the resonance frequency of the mode. The actual force I apply in my other tool is some factor of the generalised modal force, F_i. Which is just a projection of the forces I choose to apply in units of Newtons into the various modes.

Does the above sound correct?

If so, do FEM models typically output using modal mass normalisation just because the calculations of displacements are straight forward like this?

Thanks for any help!

 

[Randy Beikmann]

Meirovitch, in Fundamentals of Vibrations, has a good treatment of this. He normalizes each mode phi by scaling, so that phi(transpose)*M*phi=1 for all mode shapes phi. Although the modes are thus scaled to "unit modal mass," they still are in the same units as before (translations in meters, any rotations in radians). The coefficient q for each mode is a dimensionless, time-varying, multiplier of the mode shape in question.

I also think your last equation has a sign error. The last term in the denominator should be positive, shouldn't it?