# Which eigenvectors should I ignore?

Solving an Euler-Bernoulli equation for a beam using Finite Element Method leads to the following equation $$M \ddot{\vec d}(t)+K \vec d(t)=\vec F$$ where $M$ is system mass matrix, $K$ system stiffness matrix and $\vec d(t)$ a vector of primary node variables. In my case where we discreticise the beam to $n=5$ identical parts (see attached figure) the vector of primary variables can be written as follows $$\vec d (t)=(w_1,\Phi_1,w_2,\Phi_2,w_3,\Phi_3,w_4,\Phi_4,w_5,\Phi_5,w_6,\Phi_6)^\intercal$$ assuming $w_i$ is the vertical displacement of $i$-th node and $\Phi_i$ the angle at $i$-th node.

Note that the $dim(\vec d(t))=12$, therefore both mass and stiffness matrices are 12$\times$12 matrices.

It is also obvious from the attached figure that $\vec F =\vec 0$, since we do not have any external forces or momentums acting on the beam. In that case the dynamic equation can be rewritten to $$-\omega_0^2 M\vec d(t)+K\vec d(t)=\vec 0,$$ where I already assumed that the system will respond harmonically ($\ddot{\vec d}(t)=-\omega_0^2\vec d(t)$). Now what we have here is eigensystem $$(M^{-1}K-\omega_0^2)\vec d(t)=\vec 0$$

THE PROBLEM: Let's assume that we were able to calculate all the eigenvalues and eigenvectors of the system (meaning we were able to get some numerical values). Now what I would like to do is I would like to visualize the displacements of each node - $w_1$, $w_2$, $w_3$, $w_4$, $w_5$ and $w_6$.

But I have 12 eigenvectors each with dimension of 12. Meaning I would have to somehow ignore half of the eigenvectors. But... How do I know which ones?

EXAMPLE: Let's say that one of the eigenvectors is $(1,2,3,4,5,6,7,8,9,10,11,12)$. According to how we defined $\vec d(t)$ I would like to get components $(1,3,5,7,9,11)$. But which eigenvectors do I choose?

The reason you think you need to ignore half the eigenvectors is most likely because your plotting algorithm is ignoring half the displacement data! It is only using the translations, and ignoring the rotations.

If you used the element shape functions to plot the internal displaced shape along the length of each element, every mode shape would look different.

Most Finite Element post processing software doesn't bother to do that, because for accurate results, as a rule of thumb you need about 5 node (grid) points for each "wavelength" of the displaced shape anyway.

So for your model with 5 elements, probably only one eigenvector (the lowest frequency) would be useful for real-life engineering work, and at best it would only be worth plotting the mode shapes for the two or three lowest frequencies.

Generally, the expectation is that you should keep the lowest eigenvalues, as they represent the collective motion of the beam which is well modelled by your discretization. (The higher eigenvalues, on the other hand, model waves with a higher spatial frequency, which you're not sampling tightly enough to get an accurate description.)

However, the real test with any discretization procedure is to check that your observables are numerically converged. Select the first $k$ eigenvalues (going up in frequency), and then repeat your calculation for 14×14, 16×16, 18×18 matrices (i.e. discretizing using smaller length steps), and as high as your code can go. If your observables don't change much, then you can be reasonably confident that you're doing things right.

That's as far as one can go without knowing exactly what you're putting in. However, in your case, there is a separation of your state space into two distinct sectors, and if you're only interested in one sector then you can select for the eigenvalues whose eigenvectors have significant amplitude in that sector.

As an extreme case, suppose that you introduce absolutely no couplings in either $K$ or $M$ between the vertical and torsional displacements. This will mean that $M^{-1}K$ will be the direct sum of two matrices, each with a distinct set of eigenpairs, with the eigenvectors either completely torsional or completely vertical. You then take the eigenvalues that correspond to the vertical displacements.

However, this simple separation is not always possible. In the other extreme case, you can introduce matrices where the coupling between the vertical and torsional displacements is as high as the vertical-vertical and torsional-torsional couplings, and in this case you'll just have completely mixed eigenvectors, so you need to take everything that's converged.

In an intermediate coupling, it may or may not be a good idea to post-select only on "mainly vertical" eigenmodes - that's for you to evaluate given the relative amplitudes of the "mainly vertical" vs the "mainly torsional" modes over the degrees of freedom that you care about.