Using the basic relationships of the hereditary theory of viscoelasticity and asymptotic methods, the problem of natural oscillations of structural-inhomogeneous, multiply connected, axisymmetric shell structures is reduced to an effectively solvable mathematical problem of complex eigenvalues, in which approximate engineering methods are proposed.

Crystals of niobium and its alloys obtained by low-energy implantation of molybdenum and zirconium ions were studied in a multi-functional installation. The energy distribution curves of N (E) photoelectrons before and after heating niobium – molybdenum – zirconium alloys were studied. The contribution of surface zones formed by molybdenum and zirconium atoms to photoelectron emission in the photon energy region of 8–10 eV was analyzed.

The methods and algorithm to assess dynamic characteristics of high-rise structures are given in the paper taking into account the variability of slopes and the structure thickness in the framework of one-dimensional theory of viscoelasticity. The Boltzmann-Volterra hereditary theory was used to describe dissipative processes in the structure material. The reliability of results was verified by comparing the obtained results with the exact solution of a number of test problems. Natural vibrations of high-rise chimney stacks and ventilation pipes of thermoelectric and nuclear power plants have been investigated. It was revealed that the natural frequencies of the considered structures fall into the dangerous range of earthquake frequencies. The obtained frequencies of natural vibrations of real structures are compared with the results of field experiments.

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The methods for solving the problems of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures based on the hereditary theory of viscoelasticity are given in the paper. Using the basic relations of the hereditary theory of viscoelasticity and asymptotic methods, the problem of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures is reduced to the effectively solvable mathematical problem of complex eigenvalues, in which approximate engineering methods are proposed.