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My Natural frequencies seem to shift between a Normal Modes Analysis and a
Frequency Response Analysis. Why?

I did a normal modes analysis to get the natural frequency of a rod like structure. I then
applied a pressure loading which varied as a function of frequency in a frequency
response analysis. Finally I input an acceleration at the attachment point.

The Frequency response of the rod under the pressure loading showed a peak
response at the longitudinal natural frequency. The Frequency response of the rod
under input acceleration showed a different natural frequency. Is this due to damping
perhaps?

Answer

There were in effect two different models being analyzed here. In the case of the normal
modes analysis and the pressure loading the structure it was unrestrained. It is what is
called a free-free model. The pressure does not apply any form of displacement
boundary condition and the structure will have a net motion given by F(t) = M*a(t).

In the case of the input acceleration this is an enforced motion and you are applying a
boundary condition to the structure which will change it's natural frequencies. Imagine
each node has to be constrained to move in your specified input direction. Any relative
movement of the fixed face will be eliminated, preventing the rod end from having lateral
displacements due to poissons ratio and any possible end 'belling' shapes. The flexural
modes will also be higher as you are stiffening the structure with the constraints.

If the base of the rod is actually the attachment point to the rest of the structure, then
pressure loading and use the same degrees of freedom at the base when applying the
base motion (acceleration). All these analyses should then have the same natural
frequencies.

There should not be a frequency shift due to damping at these levels. In fact most FE
solvers ignore the correction term between damped natural frequencies and undamped
natural frequencies. You won't shift the frequencies even if you put in really high
damping.