- #1
psuaero
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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([tex]M^{-1}[/tex]*K + [tex] \omega[/tex]^2)=0
shape functions?
H1=1-[tex]\frac{3x^{2}}{l^{2}}[/tex]+[tex]\frac{2x^{3}}{l^{3}}[/tex]
H2=x-[tex]\frac{2x}{l}[/tex]+[tex]\frac{x^{3}}{l^{2}}[/tex]
H3=[tex]\frac{3x^{2}}{l^{2}}[/tex]-[tex]\frac{2x^{3}}{l^{3}}[/tex]
H4=-[tex]\frac{x^{2}}{l}[/tex]+[tex]\frac{x^{3}}{l^{2}}[/tex]
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.