CYLINDRICAL SHAPE OF FREE OSCILLATION OF A RECTANGULAR ORTHOTROPIC PLATE LOCATED ON AN INHOMOGENEOUS VISCOELASTIC BASE

Summary: The presented paper is devoted to the solution of cylindrical free oscillation problems of a rectangular plate, taking into account the resistance of the external environment. In the title of the article, accepting a cylindrical oscillation is not chosen randomly and it is a rare issue in the theory of plates and coatings, therefore, the solution of the considered problem is new and relevant. In this paper, the influence of the external environment is considered viscoelastic according to the Voigt model, the plate is assumed to be rectangular, thin-walled, and orthotropic, being heterogeneous along its length (i.e., in the direction of the large side). The conditions for fixing the contours of the plate are chosen in such a way that a heterogeneous boundary condition is obtained, otherwise the shape of the free oscillation cannot be cylindrical. Under the required conditions, the oscillation equation of the plate is formulated, and as a result, the fourth order partial differential equation with a variable coefficient is obtained. The obtained equation is a complicated enough equation modified from the Sophie-Germain equation written for the deflection of the plates according to the condition of the problem, and it is hard to be solved. Such equations do not have an analytical solution, and for now the most rational method for its solution is considered to be Bunov Galerkin's method of orthogonalization and separation of variables. In order to calculate the value of the frequency of the oscillation, the relationship equations were obtained, the calculations were carried out for cases where the characteristic functions change with a linear law, and the material of the plate is homogeneous and heterogeneous. Equations of dependence between dimensionless frequency and characteristic functions and parameters characterizing a heterogeneous viscoelastic base were obtained. An error function was constructed for the obtained equation and the orthogonalization condition was checked with the help of the error function. In order to compare the solution of the problem, the frequency analysis of the plate was carried out using the finite element method, for which the frequency analysis was carried out in 6 frequency cases on plates with different length dimensions. Solidworks software was used here. The results were presented in form of graphs and tables.