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Significance Statement
The proposed time spectral method solves numerically the nonlinear dynamic problem for the von Kármán elastic plate with simply supported, partially clamped and fully clamped boundary conditions. The technique uses the Fourier transform. A solution is expanded in double series where global basis are trigonometrical functions, which nicely reflect the shape of the solution (deflection of the plate). The Fourier coefficients are functions of time.
First the Fourier transform was introduced by the author for a static problem. The present article is one more step of application of this methodology to the dynamic problem, i.e. using Fourier analysis in investigation of vibrations of the plate. The frequency of the vibrations depends on the intensity of the buckling loading factor which in part is connected with bifurcation points of the static problem. A behavior of the solution changes when the buckling loads get over bifurcation points of the static problem.
The article demonstrates important issues regarding convergence order of the time spectral method. The theoretically established results concerning high accuracy order of the method are confirmed by the carried out numerical experiments. The considered mechanical initial boundary value problem is spatially discretized and then the obtained nonlinear system of ordinary differential equations with respect to time is solved by the classical Runge-Kutta method or in case of non-smoothness of the structure by the implicit time-stepping Newmark scheme. Thus, the introduced approach effectively combines analytical tools with numerical technique. The implementation is performed in MatLab environment.
An extension and modification of the suggested methodology is possible for more complex similar problems, in particular, to nonlinear control models of suppression of vibrations of smart plates with the use of fuzzy and neuro-fuzzy logic. The control functions can be expressed in the form of Fourier series.
Journal Reference
Journal of Engineering Mathematics, 2015, Volume 92, Issue 1, pp 83-101.
Aliki D. Muradova
Department of Production Engineering and Management, Institute of Computational Mechanics and Optimization, Technical University of Crete, Kounoupidiana, University Campus, 73100, Chania, Greece.
Abstract
The nonlinear dynamic equations of vibrations of a von Kármán thin rectangular elastic plate are solved by means of a time spectral method. External constant (compressive or stretching) forces, applied to the edges of the plate, cause oscillations of the plate. Once the initial and boundary conditions for the equations are set up, the initial-boundary value problem has a unique solution. The solution is expanded in double trigonometric series with time-dependent coefficients. Galerkin’s projections are applied for spatial discretization. The Fourier coefficients are estimated, and the rate of convergence of the method is obtained. The resulting system of nonlinear ordinary differential equations is solved by a numerical scheme based on the fourth-order Runge–Kutta method. The implicit Newmark-β method is also tested. Numerical examples with various initial conditions are presented.
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