Forced Bending Vibrations of a Plane Rod Fixed on a Rigid Support Element of Finite Length Under the Action of an External Transverse Force Aplied to Its Free End

Abstract

The problem of forced bending vibrations of a plane rod with a finite-length fastening section under the action of an external transverse force at its free end was solved. The classical Kirchhoff–Love model in the classical geometrically nonlinear approximation was used to describe the deformation process of the free part of the rod. The deformation of its fixed part was described by the Timoshenko refined shear model that takes into account transverse strains. The conditions of kinematic conjugation of the free and fixed parts of the rod were formulated. The equations of motion, the corresponding boundary conditions, and the force conditions of conjugation of the rod parts were obtained using the Hamilton–Ostrogradsky variational principle. An exact analytical solution of the problem of forced vibrations of a rod under the action of a harmonic transverse force at the free end of the unfastened part of the rod was deduced. Numerical experiments were carried out to study the resonant vibrations of rods made of unidirectional fiber composite. The effect of a noticeable increase of the amplitudes of transverse vibrations of the ends of the cantilever parts of the rods studied due to transverse contraction of the fixed section was revealed. Taking into account the transverse contraction caused an almost twofold reduction of the maximum transverse shear stresses in the fixed part of the duralumin rod.