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In this paper, we consider the natural vibrations of inhomogeneous mechanical systems, i.e., cylindrical bodies located in a deformable viscoelastic medium. The theory and methods for studying the natural vibrations of a cylindrical shell in a viscoelastic medium are constructed. The viscoelastic properties of the medium are taken into account using the hereditary Boltzmann-Walter theory. For the statement of the problem, the general equation of the theory of viscoelasticity in the potentials of displacements in a cylindrical coordinate system is used. An algorithm has been developed to solve the tasks posed on a computer using the Bessel, Hankel, and Mueller and Gauss methods. The considered problems were reduced to finding complex natural frequencies for the system of equations of motion of a cylindrical shell in an infinite viscoelastic medium using radiation conditions. It is shown that the problem has a discrete complex spectrum. The eigen frequencies of oscillations of a low-contrast heterogeneity are found. Revealed that the imaginary part of the eigen frequencies is comparable with the real one, which can lead to aperiodic movements of the systems considered.

This article was devoted to the development of methods of the dynamic calculation based on the finite difference method of laminar structures in the framework of the bimoment theory, which takes into account the spatial stress-strain state. Were given the solutions of the problem of transverse vibrations of the plate model of structures.

The results of calculations of the enclosing dam stability of phosphogypsum sludge collectors are presented in the paper. The stability calculations were carried out according to the “RUZO” program; they are based on the method of circular-cylindrical sliding surfaces under the action of basic and seismic loads. The main objective of this work is to determine the minimum safety factors for dam slopes of a given transverse profile under known loads and characteristics of the dam body material. To develop a rational profile of collector dams, 9 options of design models of the structure and its base were considered. The body of the enclosing dams of collectors of considered options consists, as a rule, of two elements: the primary floodwall built of earth materials, and secondary dams constructed according to a different build-up schemes from stored materials. Secondary dams are constructed from stored phosphogypsum by the method of layer-by-layer filling and compaction. The calculations showed that in all considered schemes the minimum safety factors under the action of basic and seismic loads are higher than the standard values. The maximum permissible values of the slopes are established with an allowable margin of safety, thus, the stability of dam slopes in all considered options of the phosphogypsum collector is ensured under basic and special combinations of loads.

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