Torsional vibrations of layered cylindrical viscoelastic shells and rods
Annotatsiya
The paper analyzes various limiting and special cases of equations of unsteady vibrations of circular cylindrical viscoelastic shells and rods, consisting of three layers of different thicknesses and materials. The materials of the layers of the shell and the rod are considered viscoelastic according to the Bolsmann-Volterra model, while the kernels of the integral operators, in the general case, are not subject to restrictions. It is assumed that the integral operators are reversible and the Poisson's coefficient of the layer materials are generally not constant. General equations of torsional vibrations of circular cylindrical layered shells and rods are given and four limiting and special cases arising from them are considered: 1) equations of torsional vibrations of a two-layer viscoelastic shell; 2) general equations of torsional vibrations of a three-layer cylindrical viscoelastic shell with a thin filler; 3) equations of torsional vibrations of a three-layer elastic shell; 4) equations of torsional vibrations of a viscoelastic rod. In addition, along with the equations of vibration for a three-layer shell, an algorithm is presented that makes it possible to unambiguously determine all stress and displacement components in an arbitrary section of an arbitrary layer in spatial coordinates and time using the field of the required functions. Formulas for stresses and displacements, including all special and limiting cases, allow one to correctly formulate boundary conditions for solving applied problems. From the obtained equations of vibration, if we restrict ourselves to the first few terms in the infinite series included in the structure of the equations, we can obtain equations of the classical type and refined ones of the type of S.P.Timoshenko. Moreover, in the formulas for non-zero stresses and displacements are limited by the same approximations as in the equations.