REFINED TRANSFORMATION MODEL OF DEFORMATION OF A ROD-STRIP FIXED ON A DOUBLE-SIDED SUPPORT ELEMENT THROUGH ELASTIC INTERLAYERS. 2. ANALYNICAL SOLUTION. NUMERICAL EXPERIMENTS

A rod-strip consisting of clamped and free sections along its length is considered. The clamped section of finite length is considered to be connected to a rigid and fixed support element through elastic layers. To represent the displacements of the rod and layers, approximations are adopted according to the S.P. Timoshenko's shear model taking into account their transverse compression. Based on the motion equations of the clamped and free sections of the rod, formulated in displacements, an exact analytical solution is found for the problem of free and forced harmonic vibrations. It is shown that within the framework of the approximations used, the solution found is divided into equations corresponding to mutually independent longitudinal-transverse and flexural-shear vibrations. From it the latter type of oscillations occurring in a comparatively low frequency range is of greatest interest. Numerical experiments were performed to determine the natural modes and frequencies, as well as the dynamic response to resonant vibrations of a rod clamped to a two-sided support element through elastic layers and made of a unidirectional fibrous composite based on ELUR-P unidirectional carbon fiber and XT-118 epoxy. A significant transformation of transverse shear stresses in the rod during the transition through the boundary from the free section of the rod to the clamped one is shown. The pronounced localization of transverse compression stresses in the region of the fixed section located near the noted boundary was revealed. An analysis of the structure of dimensionless roots of the characteristic equation corresponding to the rotation angles of the cross sections and deflections of the clamped section of the rod is performed. It is noted that when analyzing resonant vibratios of the rod, these roots are complex, which is due to the introduction of complex elastic moduli to take into account the damping properties of the material.