Parametric oscillations of a viscous-elastic orthotropic shell of variable thickness

A solution to the problem of parametric oscillations of a viscous-elastic orthotropic shallow shell of variable thickness is presented. Dynamic loading acts along one side of the shell in the form of a periodic load. Unlike linear problems, the nonlinear problem under consideration could not be solved by applying analytical methods; therefore, approximate methods were used. The mathematical model of the problem is built within the Kirchhoff-Love theory. In this case, tangential inertial forces and geometric non-linearity are taken into account. Deflection and displacements approximation is performed using the Galerkin method in higher order approximations, which allows reducing the problem solution to a system of nonlinear integro-differential equations (IDE) with variable coefficients. The weakly singular Koltunov-Rzhanitsyn kernel with three rheological parameters is used as the relaxation kernel; it describes the viscous-elastic properties of the shallow shell. A numerical method based on the use of quadrature formulas is used to obtain a resolving system of equations for the problem. To obtain numerical results, a computer software was compiled in the Delphi environment for a computational algorithm of the problem solution. The effects of viscous-elastic, orthotropic, nonlinear properties of the shell material, thickness variability, and other physical, mechanical, and geometrical parameters on the dynamic strength of a shallow shell are studied.