Mathematical Modeling of Elastically Deformed States of Thin Isotropic Plates Using Chebyshev Polynomials

In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.