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A dynamic nonlinear model with contact and buckling for an elastic plate :the time spectral analysis

Stavroulakis Georgios, Aliki D. Muradova

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Summary

A dynamic generalized nonlinear model for an elastic plate involving contact and buckling phenomena, is considered. The model consists of two coupled nonlinear partial differential equations of hyperbolic type. The equations describe vibrations of the plate, subjected to compressive and tensile (stretching) loading forces together or separately. The forces are applied at the edges of the plate. The structure is unilaterally supported by the upper and lower elastic foundations. We study the case when the foundations are modeled in terms of a nonlinear elastic Winkler-type and shear Pasternak-type. The studied system is a generalization of the von Ka ́rma ́n-Winkler plate model on a dynamic case, which takes into account compressive and tensile loadings and nonlinearity of the foundations. The physically important initial and boundary conditions are considered for the system of the equations. The initial-boundary value problems are solved numerically, using the time spectral method for spatial discretization and the Newmark-beta iterative scheme for the time discretization. First, we expand the solution into partial sums of double Fourier’s series and use Galerkin’s projections for a variation formulation of the problem. The global basis is composed of combinations of the trigonometric functions. After the spatial discretization, the obtained system of nonlinear ordinary differential equations is solved by employing the time-stepping iterative scheme, based on the Newmark-beta formulas. The techniques are tested for several values of the physical constants of the foundations.

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