# FEM - eigenvectors from eigensystem

1. Oct 27, 2016

### skrat

1. The problem statement, all variables and given/known data
It's not really a homework so I will try to be as clear as possible. Hopefully, somebody will understand me and be able to help.

I used Euler-Bernoulli theory to analyze the dynamics of a free-free beam (for the problem it is not important to understand what it is). If one discreticizes a beam into $n=5$ equal parts (as in the picture)

than each NODE has two variables (the deflection and the angle of rotation). Declaring a vector $\vec d$ of variables simplifies the writing a bit $$\vec d=(W_1,\Phi_1,W_2,\Phi_2,W_3,\Phi_3,W_4,\Phi_4,W_5,\Phi_5,W_6,\Phi_6)^\intercal.$$
Now let's assume we have the global stiffness ($\ K$) and mass ($\ M$) matrices. The equation we have to solve than is $$\ M \ddot{ \vec d}+\ K\vec d=\vec 0$$

Assuming the system will respond harmonically (in that case $\ddot{ \vec d}=-\omega_0^2\vec d$) the equation of motion rewrites into $$(K-\omega_0^2\ M)\vec d=\vec 0$$ or even better (assuming $\ M$ is reversible $$(\ M^{-1}\ K-\omega_0^2)\vec d=0$$

Now this is an eigenproblem now. Eigenvalues are frequencies of corresponding modes (eigenvectors).
2. Relevant equations

3. The attempt at a solution
Ok...

Solving that eigensystem is really not a problem, so let's assume I have the eigenvalues and eigenvectors.

Now let's say I would like to visualize the displacement (so every odd component of a vector $\vec d$). How do I do that? Would it make sense to simply multiply an eigenvector by vector $(1,0,1,0,1,0,1,0,1,0,1,0)$ to simply ignore the angles or is that completely wrong?

2. Nov 1, 2016

### Staff: Admin

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

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