FEM - eigenvectors from eigensystem

[skrat]

Homework Statement

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).

Homework Equations

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?

 

[mighty2000]

Hello,

Thank you for sharing your problem with us. It seems like you are on the right track with solving the eigenproblem to obtain the frequencies and corresponding modes. To visualize the displacement, you can use the eigenvectors to plot the deflection of each node as a function of time. This will give you a representation of the displacement over time for each mode.

Alternatively, you can also use the eigenvectors to plot the displacement for each node at a specific time point. This will give you a snapshot of the displacement at a particular moment in time for each mode.

Multiplying the eigenvectors by a vector to ignore the angles may not give you an accurate representation of the displacement. It would be best to plot the full eigenvector to get a complete understanding of the displacement at each node.

I hope this helps. Let me know if you need any further clarification. Good luck with your analysis!