One of the characteristic features of the development of the hereditary theory is the wide possibilities for describing the dynamic processes of deformation of various materials. However, due to the lack of an adequate mathematical apparatus, the implementation of these possibilities is in many cases difficult, especially when studying nonlinear dynamic processes. In recent years, the power of computing has increased interest in nonlinear problems. Under these conditions, it is important to create and develop such effective methods of solution that could be applied to the widest possible class of problems. In this work, a mathematical model of the problem of the dynamics of thin-walled structures is constructed taking into account the hereditary properties of the material. Using the Bubnov-Galerkin method under various boundary conditions, the problem under consideration is reduced to solving systems of integro-differential equations. The analysis of the influence of various properties of the construction material on the amplitude-frequency characteristics is carried out.

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In present paper, the problem of the vibration of a viscoelastic dam-plate of a hydro-technical structure is investigated, based on the Kirchhoff-Love hypothesis in the geometrically nonlinear statement. This problem is reduced to a system of nonlinear ordinary integro-differential equations by using the Bubnov-Galerkin method. The resulting system with a weakly-singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The behavior of the viscoelastic dam-plate of hydro-technical structure is studied for the wide ranges of physical, mechanical, and geometrical material parameters.