What do you want to do with it? That would really help. If you are just interested in capturing shear deformations (and not so interested in the non-neglect of rotary inertia of differential elements), then just see the virtual forces part of any classical structural analysis book.
Sorry to be vague. What I am after is the time response of a beam after given an input. I have the 2 PDEs, but am having trouble solving for them in finite form (been a while since my last PDE class).
Sorry, no help from here then. My vibrations book has the solution for beams, but it neglects shear deformations and rotary inertia of the differential element. I assume you've taken those into account and your PDEs somehow can't be solved using the typical separation of variables route. What PDEs are you coming up with? Perhaps if you can post them, I can find a solution in one of my books. I have Timoshenko's book and a math book that is often very helpful for problems like this.
Do you need help finding the modes (eigenvectors) and frequencies (eigenvalues)?
Or how to use that info to find time-domain response for a given input disturbance?
Somehow, I was given a link to the following paper, which gives the solutions for Timoshenko beams in all of the classical end-fixity conditions:
DYNAMICS OF TRANSVERSELY VIBRATING BEAMS USING FOUR ENGINEERING THEORIES, SEON M. HAN, HAYM BENAROYA AND TIMOTHY WEI, Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey,
Journal of Sound and Vibration (1999) 225(5), 935-988
Article No. jsvi.1999.2257, available online at
