Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler–Bernoulli, Timoshenko, and Sheremet’ev–Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation $O(h^2)$ and the Bubnov–Galerkin finite element method. It is shown that for a relative thickness $\lambda\le$ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.