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The need for mesh refinement in a structural analysis module such as Static or Transient is quite clear; if we want the overall results such as displacement and stresses to be more accurate within the elements, then by refining the mesh size we should be able to achieve this. This happens because the mesh refinement improves the quality of the mesh elements and hence the shape functions (of each element) also becomes more accurate and closer to its ideal form, which in turn improves the accuracy of the results within the elements.

But for modal analysis, I couldn't understand the reason that why would inserting more mesh elements within my model would change the natural frequency results? I mean the purpose of modal analysis is not to acquire nodal displacements, neither obtain any elemental strain or stress results; displacements are random and strain/stress results don't exist since there is no force application at all. So why modal fequencies have dependence on the mesh size?

Below shows an example I just did to further elaborate my point.

enter image description here

I mean this is just one body which has a certain mass, and this body has an overall only 6 DOFs, with stiffness in all of these DOFs. Why can't I just idealize this body with a single mass 'm', and with six springs with certain stiffnesses in all 6 directions? Why do I need to associate a mass and stiffness with each separate element and then conduct a modal analysis in FEA?

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I mean the purpose of modal analysis is not to acquire nodal displacements. Displacements are random.

I disagree with both of those statements.

  • Part of the reason for performing a modal analysis is to get the mode shape. The mode shape is shown by the displacement result.
  • The displacements are not random. If they were random, the mode shape would not appear to be smooth. It would be random. The maximum magnitude is scaled to some value. (Perhaps this is what you meant by random.)

So why modal frequencies have dependence on the mesh size?

The frequency is dependent on the stiffness. The stiffness of an element is dependent on the size of the element. This is especially true when the model is thin and bending. More elements through the thickness result is a better bending stiffness calculation.

The frequency is also related to the mass distribution. Depending on how the mathematics are handled, smaller elements may give a better approximation of the mass distribution which changes the frequency.

Why can't I just idealize this body with a single mass 'm'

As implied by Solar Mike's comment, the body does not vibrate as a single point mass. It has surface area, and the 2D mode shape (bending in some modes, twisting in other modes) cannot be represented by a single point.

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  • $\begingroup$ I mean you talk about mass distribution here, but that is the concept which falls under FEA's roof. If I just directly assume that FEA is the only possible method to obtain the natural freqs for my system, then yes, the better the mass distribution is modelled and approximated, the better the results. But the question here was why do I actually need FEA in the first place to obtain natural freqs and mode shapes for my system? $\endgroup$ Mar 14 at 19:03
  • $\begingroup$ I guess you already addressed this fundamental question of mine that to propely obtain the mode shape for my system (like bending, torsion, etc), I need to run a FEA over my system. Are there any analytical solutions to obtain natural freqs and Mode shapes of 2D or 3D basic structures in the literature? The only way to obtain them that we learnt in School is to essentially idealize the bodies as concentrated masses and move forward with it. $\endgroup$ Mar 14 at 19:06
  • $\begingroup$ The mod deleted my comment - but its spirit lives on in your answer :) at least you understood my point. $\endgroup$
    – Solar Mike
    Mar 14 at 21:46
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I am not knowledgeable to answer this question but to lead you to read this article, which says "In order to obtain greater number of eigenmodes (and natural frequencies as well) finer mesh of finite elements must be used. This must be fulfilled for both beam and plane elements.

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    $\begingroup$ So kind of like Nyquist...but spatially. $\endgroup$
    – DKNguyen
    Mar 14 at 20:33

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