Free and Forced Bending-Torsional Oscillations of an Anisotropic Elongated Plate Fixed on a Spherical Hinge

Refined geometrically nonlinear equations of motion are derived for elongated rod-type plates which are made of composite materials. The equations are derived on the basis of the previously proposed relations of the consistent version of the geometrically nonlinear theory of elasticity at small deformations and the classical Bernoulli–Euler model. In the model, the axes of the chosen coordinate system do not coincide with the orthotropy axes of the plate material in a plane stress-strain state. It is shown that for a plate made of cross-ply reinforced composite material the derived equations, compiled even in the geometrically linear approximation, describe coupled bending-torsional oscillations. As an example of their application, numerical solutions of linear problems on free and forced bending-torsional oscillations of an anisotropic elongated plate fixed on a spherical hinge are found. It is assumed that such a supporting element of the plate is located at some small distance from the end cross-section and subjected to kinematic loading by setting the deflection and torsional angle according to the harmonic law of their variation in time with a given frequency. The model under consideration is intended to simulate natural processes and structures in applied engineering problems aimed at developing innovative oscillatory biomimetic propulsion systems.