Homework Help: 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