In term of directly implementing the 3rd order polynomial as an assumption to model twist along the length of a an element...how could that assumption be justified in torsion? If the twist rate (the derivative of twist with respect to position along a beam) of a beam is constant, the assumption falls apart. Hence, the Hermite equations become incredibly simple because the slopes are all constant. Is it really this straightforward? I ask because of a discrepancy that I saw in the Rayleigh-Ritz problem concerning the strain energy formulation that he gave us in the homework versus how it is defined in the notes.
I have the stiffness and inertia matrices, but when I go to take the determinant of \(k-\omega^2I\) in order to get the natural frequencies, the computer cannot process it because the answer is so large. Anyone else having this problem? Work is attached.
Just for kicks, I tried applying the Rayleigh solution to the problem as they use physics, but the answer is nonsensical when compared with the Galerkin and Rayleigh-Ritz results I already have (which agree with each other). How are you guys doing with this one?
Well my determinant wasn't fun, but certainly not impossible. I still ended up getting an answer very similar to the other two methods. Not sure if this is the difference, but did you mean the stiffness and MASS matrices? so \( k-\omega^2M\)? or did you actually use Inertia matrices?