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An efficient method is developed to solve the free-vibration problems for arbitrarily shaped orthotropic multilayer plates in a refined formulation. The method is based on the R-function and Ritz methods. Sequences of coordinate functions satisfying kinematic boundary conditions are constructed in an analytic form. The method is used to solve the vibration problem for multilayer square and arbitrarily shaped plates. The results obtained for square plates are analyzed. A comparison of these results with those available in the literature demonstrates the efficiency of the method Keywords: free vibrations, orthotropic multilayer plate, refined formulation, Ritz method, thickness variability, square plate Introduction. Multilayer plates and shells are widely used to model composite elements of many modern structures. The design of multilayer elements should take into account the physical characteristics of layers, which may be essentially dissimilar. To this end, refined theories that account for the shear strains of layers are generally used, which drastically complicates the governing equations. There are a great many studies [3, 4, 8-13, etc.] that propose methods for analyzing the stress state and dynamic behavior of composite plates and shells. However, most of them address shallow shells and plates with a circular or rectangular planform. Much fewer publications consider plates or shells with complex planforms and boundary conditions other than hinged support. Therefore, nonclassical methods for design of composite plates of complex geometry are of theoretical and applied interest. In the present paper, we propose a method for studying the free vibrations of multilayer plates. The method is based on a refined Timoshenko theory, the R-function method, and variational methods. The key advantage of our method is the possibility of describing complex geometries and constructing analytic solutions. The R-function method allows us to resolve a great many challenges in the field of vibrations of multilayer plates and shells.

We have developed an effective approach to the solution of problems on geometrically nonlinear vibrations of orthotropic multilayer plates of irregular shapes in a classical statement based on the use of the R-function theory, Ritz variational method and Bubnov-Galerkin method. Using the proposed method, problems of vibrations of both multilayer rectangular plates and plates of complex geometries have been solved. The effect of the domain geometry and boundary conditions on the amplitude-frequency characteristics has been investigated.

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