Talk:Modal analysis using FEM

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Excellent page, nicely done Greglocock 08:45, 27 December 2006 (UTC)

Thanks. I've been contemplating writing about this topic for many years. The existence of Wikipedia and the ability to put equations into html have created the exact circumstances to finally do it. It's good to see someone else want this subject covered. Slffea 17:23, 27 December 2006 (UTC)

Statics vs. Dynamics

[This gets rambling. I hope it sparks conversation or at least serves as notes to myself for future work.]

The page mentions offhand that one eigensystem is for statics while the other (without the acceleration term) is for dynamics. This seems to me to be a subtle distinction that deserves more depth, and that could help people new to modal analysis understand what a mode actually is.

Consider a 2-DoF spring system, the displacement in U=(x,y) of the end of a beam extending one unit in the z direction. For small deflection, you can describe the energy in the beam as a quadradic -- a parabolic surface:

$E={\frac {1}{2}}[U]^{T}[K][U]$

where [K] is symmetric positive definite. If the beam is a cylinder, the energy surface will be circularly symmetric meaning [K] will be a scaled identity matrix. If the beam is a thin strip, thinner in the y direction, then [K] will be diagonal with a smaller (2,2) element than (1,1) element. If the beam were thin in the (1,1) direction, then you would have a [K] matrix with stiffness eigenvectors of (1,1) and (1,-1), with corresponding larger and smaller stiffness eigenvalues.

The distinction that I'm trying to wrap my head around is between these stiffness modes and harmonic modes, in which mass is taken into account. As the text says, the eigenequation

$[M][U]\lambda +[K][U]=[0]\qquad \qquad (1)$

can become

$-[M]^{-1}[K][U]=[U]\lambda \qquad \qquad (2)$

which is a traditional linear-algebra problem. In the simple case above, the mass matrix is just a scaled identity (I think), so (1) and (2) have the same eigenvectors and eigenvalues.

I think this is a deceptively simple case... I think the distinction I'm trying to get at is that for simple cases, [M] will be a scaled identity, but the distinction between dynamic and static mode shapes must come from when this is not the case.

I'm realizing I'm at the edge of my understanding in a few things: (1) Stiffness matrix should really redir to Direct_stiffness_method#System_stiffness_relation or its own page, which should do a better job describe that in more detail. (2) Mass matrix should exist and explain what that matrix really means.

A related thought: Each dynamic eigenmode has a corresponding frequency. This isn't the eigenvalue, lambda, since it has units of inverse time squared. What is the generalized equivalent of

$\omega ={\sqrt {\frac {k}{m}}}$?

Is it

$[\omega ]=[M]^{-1/2}[K]^{1/2}$?

That got out of hand, but at least I have some action items to better understand this stuff. —Ben FrantzDale (talk) 18:56, 20 July 2009 (UTC)

I'm the original author of the article. I'm not signing in as slffea because I now refuse to (significantly) contribute to Wikipedia, but I wanted to adress your questions.

1) "The page mentions offhand that one eigensystem is for statics while the other (without the acceleration term) is for dynamics."

There is only an eigensystem for dynamic problems. The static equation at the bottom of the first section represents the typical static problem encountered in solid mechanics. All time dependent terms, including velocity(associated with the damping) and acceleration are zero for a static problem.

2) "If the beam is a cylinder, the energy surface will be circularly symmetric meaning [K] will be a scaled identity matrix. If the beam is a thin strip, thinner in the y direction, then [K] will be diagonal with a smaller (2,2) element than (1,1) element. "

You may be too focused on beams and trusses, probably because I say you can view the equation of motion as that of a 3 dimensional spring mass system. But even if I were to do modal analysis using brick, shell, quad, etc. elements, you can still view the entire system as a 3 dimensional spring mass system.

3) "In the simple case above, the mass matrix is just a scaled identity (I think), so (1) and (2) have the same eigenvectors and eigenvalues."

The mass is usually NOT "a scaled identity" matrix. It is a banded matrix just like the stiffness. If you can stand it, you can look at my software which has multiple element types, and all of them can do modal analysis. You can go through the sourcecode and see how the mass matrix is assembled, and it is unlikely that the mass is the identity, or even diagonal(although I allow for this option. The results are poor though.)

4) ":$\omega ={\sqrt {\frac {k}{m}}}$?"

You should think of this as:

$\omega ^{2}=\lambda$

I recommend you look at the Català language version of the page here which derives the problem using $\omega ^{2}$.Former contributer (talk) 17:31, 31 July 2009 (UTC)

Good answers. Since discovering this page, I went back to my FEA books and started a page on mass matrix and stiffness matrix and I now do see how a mass matrix could be something other than diagonal (the springs have mass). WRT the eigensystems, I still think more could be said about the distinction between static modes–eigenvectors of the stiffness matrix alone, not considering the mass matrix–and vibration modes.

With regards to $\omega ^{2}=k/m$, I think I have the answer I wanted: Numerical considerations aside, we would like to solve

$[M]^{-1}[K][U]=\omega ^{2}[U]$.

In as much as $[M]^{-1}[K]$ is positive semi-definite, we would still have a real-valued problem with the same solution if we took the square root:

$[M]^{-1/2}[K]^{1/2}[U]=\omega [U]$.

Since that eigenproblem is just an eigendecomposition of $[M]^{-1/2}[K]^{1/2}$, we could write

$[\Omega ]:=[M]^{-1/2}[K]^{1/2}=[D][\omega ][D]^{\top }$.

So I think the interpretation of $[M]^{-1/2}[K]^{1/2}$ is the multidimensional generalization of frequency―frequencies in various directions. Cool. —Ben FrantzDale (talk) 01:31, 1 August 2009 (UTC)

Edit: changed [M]^{-1/1} to [M]^{-1/2}. Oops. —Ben FrantzDale (talk) 14:58, 3 August 2009 (UTC)

1)"WRT the eigensystems, I still think more could be said about the distinction between static modes-eigenvectors of the stiffness matrix, not considering the mass matrixand vibration modes."

I forgot that all square matrices have eigenvectors and eigenvalues. The reason it is rarely considered for the static problem for [K] is that there is really no physical intepretation of these eigenvectors/eigenvalues. Even their usefullness mathemically is limited. You could decompose the stiffness into:

$[Q][\lambda _{K}][Q]^{T}$

where $\lambda _{K}$ is the diagonal matrix of eigenvalues for [K] and [Q] are the eigenvectors. This would allow you to solve the static problem with:

$[U]=[Q][\lambda _{K}]^{-1}[Q]^{T}[F]$

but this would be done only when the system is small. You can see page 286-288 of:

Clough, Ray W. and Joseph Penzien, Dynamics of Structures, 2nd Ed., McGraw-Hill Publishing Company, New York, 1993.

2)$[\Omega ]:=[M]^{-1/1}[K]^{1/2}=[D][\omega ][D]^{\top }$

Do you mean: $[M]^{-1/2}[K]^{1/2}=[D][\omega ][D]^{\top }$?

I think that it is possible to decouple the problem using the matrix of eigenvectors in which case you can then solve for each $\omega$ based on the "k" and "m" which appear in the resulting single degree of freedom system. This is impractical for large systems of course. For your interpretation of "the multidimensional generalization of frequency―frequencies in various directions.", I can't find a reference for that. But you may. Former contributer (talk) 09:43, 2 August 2009 (UTC)

As for the eigenspace of [K], they must have the interpretation of the modes of principal stiffness (for large eigenvalues) and compliance (for small eigenvalues). Using that to solvethe problem

$[U]=[Q][\lambda _{K}]^{-1}[Q]^{T}[F]$.

In particular, you should be able to approximate $[K]^{-1}$ and so [U] for arbitrary [F] by finding the few eigenvalue–eigenvector pairs with the smallest eigenvalue. This you can do quickly (since [K] will generally be spares) using an iterative eigensolver without finding all of the pairs.

Yes, I meant $[M]^{-1/2}$. Fixed. While I don't have references, I'm just thinking that, because it's a matrix with eigenvalues describing frequencies and eigenvectors describing corresponding modes, the entire matrix must have an interpretation of generalized "frequencies in various orthogonal directions". Since eigencecomposition is just an orthogonal rotation of a matrix, you could imagine the case in which $[M]^{-1/2}[K]^{1/2}$ happens to be diagonal, as would be the case of a mass in 2D on two axis-aligned springs. The interpretation then would be that the mass has a natural frequency of $\omega _{x}$ in the x direction and $\omega _{y}$ in the y direction. That is, $[M]^{-1/2}[K]^{1/2}$ is a generalized multidimensional frequency WRT directions. If you change your coordinate system, the springs are no longer axis-aligned, but the interpretation of $[M]^{-1/2}[K]^{1/2}$ shouldn't change, even if it isn't as clear. But I'm rambling and don't have any real sources to back this up, just my intuition, at the moment. —Ben FrantzDale (talk) 14:58, 3 August 2009 (UTC)

I think you have more insight into this than me. I haven't dealt with the "principal stiffness" or used an "iterative eigensolver" to solve a linear system. My interest usually focuses on the implementation, and I don't make enough effort to learn the mathematics behind it or alternative methods unless it's likely I will use them . This is probably why I haven't written any papers. I think it will be up to you to add any of the knowledge above, if you're so inclined.Former contributer (talk) 22:05, 3 August 2009 (UTC)

For reference, Lanczos algorithm is the typical iterative eigenvalue algorithm. I don't know if anyone actually says "principal stiffness" in the literature... I'm just intuiting the meaning of the eigendecomposition of [K]. Good discussion, and thanks for starting the page. —Ben FrantzDale (talk) 12:07, 4 August 2009 (UTC)