1. A.ZH. SEYTMURATOV - Korkyt Ata Kyzylorda State University, Kyzylorda, Kazakhstan.

2. U.U. UMBETOV - Karaganda Industrial University, Temirtau, Kazakhstan.

This paper presents the results of the study of natural and forced vibrations of flat elements, taking into account the layering of the element material, rheological viscous properties, environmental influences, deformable base, anisotropy, etc. Various formulations of boundary value problems of vibrations of a rectangular flat element are considered, both taking into account viscosity. and taking into account the above factors of a geometric and mechanical nature, which are trance dental frequency equations, which is reduced to algebraic ones, and the influence of both boundary conditions along the edges of a rectangular plate and the parameters of a geometric and mechanical nature on the frequencies of natural oscillations of rectangular flat elements is considered and the previous results are generalized for a rectangular plate, the material of which satisfies the viscoelastic Maxwell model. [One] When studying oscillatory processes in a solid deformable body, it is advisable to take the kernel of viscoelastic operators as regular, since only such operators describe instantaneous elasticity and then viscous flow, which is typical for deformable solids. Integro-differential equations with regular kernels are known to be equivalent to partial differential equations. Depending on the considered particular types of a flat element in the general solutions of a three-dimensional problem, the main unknown functions are chosen: displacements or deformations at points of a fixed plane of a flat element, in particular, in the middle plane of a plate of constant thickness. Displacements and stresses at an arbitrary point of a flat element are expressed in terms of the main unknown functions, which are determined from the boundary conditions on the surfaces of a flat element. The equations obtained for the main unknown functions and are the general equations for the vibration of a plane element, containing the derivatives of functions with respect to coordinates and time of any arbitrarily large order. General solutions are presented as power series over the thickness of a flat element. The general solution refers to an equation of the hyperbolic type, which describes the oscillatory and wave processes in a flat element. Restricting ourselves in the series of the general equation to a finite number of first terms, we obtain approximate equations for the vibration of one or another flat element. The proposed approach makes it possible to rigorously construct approximate theories of the oscillations of flat elements of various types.

Boundary conditions, deformation, flat element, movement, natural vibrations vibration theories.