Abstract
On the application of mathematical methods for the research of vibration processes in mechanicsArticle represents the study of applied problems of mathematics, whose mathematical modeling leads to boundary problems for equations in partial derivatives.Mathematical methods, applied to these models, enable to obtain exact analytical results.Detailed result is represented for boundary problem of oscillations of thin structures with boundary conditions in general terms.Application of spectral decomposition for sufficiently smooth function, characterizing the membrane deviation from equilibrium state, enables to define exact analytic representation of inflection function for studied problem.To calculate multilayer plates, method of finite elements is applied.