I want to obtain the structural response of circular cylindrical shell to moving load. Because the problem is axissymetric, equations of motion are similar to beam equations. When assuming rotary inertia and shear deformation, it looks like Timoshenko beam. The references that solve the problem of beam used the eigenfunctions that are applicable for Bernouli-Euiler beam. These eigenfunctions simply supported boundary conditions applicable for Timushenko beam. In other cases, what is the eigenfunction?
There are a number of papers that give the mode shape and natural frequency equations for Timoshenko beam. One of the earlier ones is by Huang, titled "The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions". It was published in 1961, in Journal of Applied Mechanics Vol 28, pages 579-584. Eigenfunctions for both the end conditions that you asked for has been given there.
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