Forced and parametric vibrations of a composite plate caused by its resonant bending vibrations

For a rod-strip based on the shear model of S.P. Timoshenko of the first order of accuracy, taking into account the transverse shear and compression in the thickness direction, the two-dimensional equations of the plane problem of the theory of elasticity, compiled in a simplified geometrically nonlinear quadratic approximation, are reduced to one-dimensional geometrically nonlinear equations of equilibrium and motion. Under static loading, the derived equations make it possible to reveal known flexural-shear buckling modes under compression conditions and purely transverse-shear buckling modes under flexural conditions. When considering stationary low-frequency dynamic processes of deformation, the derived equations in the linearized approximation are divided into two systems of equations, of which linear equations describe low-frequency flexural-shear vibrations, and linearized equations describe forced and parametric longitudinal-transverse (“breathing”) vibrations caused by flexural-shear vibrations.