Refined equations of torsional vibrations of a three-layer cylindrical viscoelastic shell

Abstract

The paper presents approximate equations of torsional vibrations of viscoelastic cylindrical layered shells and round rods arising from the general equations previously obtained by the authors. In this case, viscoelastic operators are taken in the form of Boltzmann -Volterra relations. The case of the zero approximation in infinite sums included in the structure of the oscillation equations is considered. Approximate equations of vibration of a three-layer viscoelastic shell with a thin middle layer, a three-layer viscoelastic shell with constant Poisson coefficients of layers, and an elastic three-layer shell are also derived. In addition, in the case of one of the bearing layers, the oscillation equation of the two-layer shell follows from the obtained vibration equations. In the absence of both bearing layers of the shell, the equations of torsional vibrations of a circular cylindrical homogeneous shell follow from them. For ease of use in solving problems of engineering practice, the equations obtained are given in dimensionless coordinates. For comparative analysis, approximate equations of a homogeneous circular cylindrical viscoelastic shell are also obtained as a special case and compared with the known results of other authors. Equating the inner radius of the shell to zero, the equations of the classical theory of rod vibrations are obtained, as well as refined equations of the Timoshenko type, which has terms in its structure that take into account the inertia of rotation and the deformation of transverse shear.