Dynamic stability and vibrations of thin-walled structures considering heredity properties of the material

Abstract

Dynamic stability and vibrations of thin-walled structures, taking into account the heredity properties of the material are considered in the paper. Mathematical models were constructed in a two-dimensional statement, using the Karman theory of plane plate strains and A.A. Ilyushin aerodynamic theory. When realizing the physico-mechanical properties of the material of the object, the systems of integro-differential equations (IDE) in partial derivatives with the corresponding initial and boundary conditions are taken as mathematical models of the problems under consideration. The obtained nonlinear partial IDEs using the Bubnov-Galerkin method under considered boundary conditions are reduced to solving systems of nonlinear ordinary IDEs with constant or variable coefficients with respect to the time function. The integration of equations obtained using the polynomial approximation of the deflections was carried out by a numerical method based on the use of quadrature formulas. Based on this method, an algorithm for the numerical solution of the problem was developed suitable for all viscoelastic and elastic elements of thin-walled structures of plate type.