MATLAB FEM for natural frequencies and mode shapes of a beam

[psuaero]

Homework Statement

Model the above beam with 3 elements(image provided in attachment).
Calculate and list the first 6 natural frequencies
Plot the mode shapes corresponding to each of the natural frequencies

Homework Equations

DET(M^{-1}*K + \omega^2)=0

shape functions?
H1=1-\frac{3x^{2}}{l^{2}}+\frac{2x^{3}}{l^{3}}

H2=x-\frac{2x}{l}+\frac{x^{3}}{l^{2}}

H3=\frac{3x^{2}}{l^{2}}-\frac{2x^{3}}{l^{3}}

H4=-\frac{x^{2}}{l}+\frac{x^{3}}{l^{2}}

The Attempt at a Solution

My code Assembles the reduced global mass matrix(M) and reduced global stiffness matrix(K)
each of which is 6x6. I found the natural frequencies using eigenvectors and eigenvalues:

[v,d]=eig(M^-1*K). where v contains the eigenvectors and d has eigenvalues

My problem is in finding the mode shapes. I'm not sure where to begin, here is my guess:

realizing that the middle element shares a node with both end elements I can reduce the eigenvectors from 6x1 to 4x1. should I multiply the shape functions and mode shapes and sum them to get the equation for the mode shape. for example:

w=H(1)*v(1,1)+H(2)*v(2,1)+H(3)*v(3,1)+H(4)*v(4,1)

I just want to make sure that my mode shapes are correct since I can't find them in the notes. Also when I did the above the mode shape wasn't what I expected. for the first mode i would expect a parabolic shape.

 

[Jonny6001]

Hello, I don't favour the approach you are using to determine the natural frequencies of beams. Perhaps you ae forced to use that method.

My perfered method involves a series of continuity and beam equations that must be satisified during a natural frequency. It is similar to a Holzer method for determining torsional resonances.

I worked on the method for a while so it could be used on a variety of beam constraints, comparison with experimental and other theoretical methods was very good.

I can scan in some of my notebook scribbles when deriving the method if you would like. If you are stuck with using eigenvectors etc, it won't be of any use.

Generally speaking, the maximum number of natural frequencies a beam can have is equal to the degree of freedom for the beam. If you are told to use 3 elements for the beam, it can only have 3 distinct natural frequencies. Sure it will have the same frequencies in the orthagonal plane but that's not very exciting.

Good luck