FEM modal analysis computing displacements

Question

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 normalizing 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 normalized 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 normalized or do I need another scaling factor here?

The actual mode amplitudes I compute with: $$ q_i = \frac{F_i}{-\omega^2 + j\ C\ \omega - \omega_i^2} $$ Here $\omega$ is the frequency I'm looking at, $C$ some damping constant, $\omega_i$ the resonance frequency of the mode. The actual force I apply in my other tool is some factor of the generalized 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 normalization just because the calculations of displacements are straight forward like this?

Answer 1

As modal analysis is performed in the frequency domain, your modes and amplitudes are also in the frequency domain.The modal amplitudes ($q_i$) are scaled by the same amount as the applied load. If you want to know the displacement at each mode, then just multiply $q_i$ by whatever the magnitude of the load (for the mode) is scaled by. The maximum amplitude at that frequency is $q_i$ (e.g. the largest displacement).

If you want to know the displacements with respect to time, then you would need to perform an inverse Fourier transform on the amplitude and phase at each frequency, giving you the displacement with respect to time at all spatial locations.

To answer your question about the units and normalized mass, perform basic dimensional analysis to see if the right hand side of your equation gives you units of mass.

Left hand side: $U$: units of $m$

Right hand side: $q$: $N/(kg/s^2)$,

as ($C$ generally has units of $kg/s$ and $\omega$ has units of $1/s$)

$\phi$: dimensionless (as normalized modal mass means no dimensions)

As $1$ $N$ $=$ $1$ $kg*m/s^2$

$q$ has units of: $m$

therefore to answer your question, the units work out, and $U$ is in $m$.