Solution Reduction of a Dynamic Plates Bending Problem to the Sequential Solution of the First Kind Fredholm Integral Equations: Fid-Bau Portal

Solution Reduction of a Dynamic Plates Bending Problem to the Sequential Solution of the First Kind Fredholm Integral Equations

M.Mirzaeva, M. / Tajibaev, G. O. / Abdurazakov, N. N. / Turgunova, L.

Abstract

This work is devoted to the development of an algorithm for solving linear dynamic bending problems by the method of sources and sinks. The problem is considered for a thin isotropic plate of constant thickness under a transverse dynamic load, taking into account arbitrary border conditions on the contour and various edge contours. The vibration problem is considered in the classical formulation. The solution of the boundary value problem of vibrations is reduced to a consistent solution of a system of two-dimensional integral equations. For the solution of the vibration problems, an unbounded plate under the action of a single concentrated force is used. The expressions for the components of the stress–strain state of the plate through the intensities of the sources are obtained, which are convenient for programming in an algorithmic language. The source and sink method is a numeric-analytical approximate method, and the numerical method for solving a system of integral equations is not stable. Therefore, the regularization method is used for a stable numerical implementation of the method. The reliability of the method is shown by comparing the numerical results with the parameters obtained from the analytical solution of the vibration problem of a rectangular support plate under the action of a trigonometric load Fxcospt. In the work, a universal, stable method for solving plate vibration problems is obtained. Performance evaluation of proposed work against existing analytical solution shows that it has comparable results, but more importantly, can be easily implemented in modern programming languages.