Torsional vibrations of a rotating viskoelastic rod

Abstract

A homogeneous and isotropic round viscoelastic rod rotating around its axis of symmetry with a constant angular velocity is considered in a cylindrical coordinate system. It is believed that the behavior of the rod is described within the framework of the linear theory of viscoelasticity, where the relationship between stresses and deformations is given in the form of relations Boltzmann-Volterra. In this case, the condition of its reversibility is imposed on the kernel of the integral operator. The equations of motion of the rod concerning non-zero stress components are written, taking into account the centrifugal force caused by the rotation. It is assumed that torsional vibrations are caused by given stress on its surface. A general equation of torsional vibrations of such a rod is obtained, which is an integral-differential equation of infinitely high order for the main part of the torsional displacement. Limiting the general equations to the zero and first approximations, the equations of the second and fourth orders are obtained, which, in the case of the absence of rotation, exactly coincide with the known equations of other authors. The resulting refined equation of the fourth order in its structure considers the angular velocity of rotation, the deformation of the transverse shear, and the inertia of rotation. Based on the derived oscillation equations, a particular problem is solved to study the influence of rotation and viscoelastic properties of the material on the stress-strain state of the rod, according to the results of which graphs of the dependencies of elastic and viscoelastic changes on time at points of three different sections of the rod are constructed. A comparative analysis of the results obtained with the results of other authors is performed.