Study of parametric oscillations of viscoelastic cylindrical panel of variable thickness

ABSTRACT Introduction. Isotropic viscoelastic cylindrical panels of variable thickness under the effect of a uniformly distributed vibration load applied along one of the parallel sides, resulting in parametric resonance (with certain combinations of eigenfrequencies of vibration and excitation forces) are considered. Materials and methods. It is believed that under the effect of this load, the cylindrical panels undergo the displacements (in particular, deflections) commensurate with their thickness. Based on the classical Kirchhoff-Love hypothesis, a mathematical model of the problem of parametric oscillations of a viscoelastic isotropic cylindrical panel of variable thickness in a geometrically non-linear formulation is constructed. Corresponding nonlinear equations of vibration motion of panels under consideration are derived (in displacements). The technique of the nonlinear problem solution by applying the Bubnov-Galerkin method at polynomial approximation of displacements (and deflection) and a numerical method that uses quadrature formula are proposed. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. Results. Parametric oscillations of viscoelastic cylindrical panels of variable thickness under the effect of an external load are investigated. The effect on the domain of dynamic instability of geometric nonlinearity, viscoelastic properties of material, as well as other physical-mechanical and geometric parameters and factors (initial imperfections of the shape, aspect ratios, thickness, boundary conditions, excitation coefficient, rheological parameters) are taken into account. Conclusions. A mathematical model and method have been developed for estimating parametric oscillations of a viscoelastic cylindrical panel of variable thickness, taking into account geometric nonlinearity under the action of periodic loads. The results obtained are in good agreement with the results and data of other authors. The convergence of the Bubnov-Galerkin method is verified.