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This work is devoted to the study of the propagation and dispersion of natural waves in oil-gas wells. A detailed analysis of well-known works devoted to this problem is given. In this work, a mathematical formulation and a methodology for studying the propagation, dispersion and attenuation of tube and Lamb waves in a well filled with liquid have been developed. To solve the problem and assess the damping properties of tube and Lamb waves in a well filled with liquid, the following methods were used: separation of variables, the theory of potential functions, an orthogonal sweep and central difference schemes. The complex roots (phase velocities) of the dispersion equation are determined by the methods of Mueller and Gauss. A number of new mechanical effects have been identified that have practical significance: the interference and dispersion of Lamb waves depends on the parameters of the well; the presence of a sliding contact between the pipe and the medium leads to the appearance of pipe and Lamb waves, and taking the viscoelastic properties of the pipe into account leads to a damping effect; with a system of zero frequency, despite the contact conditions between the elements of the system, both L and T waves have the same speed, and with an increase in the frequency of oscillations, the difference in the phase velocities of these waves increases.

In this paper, the integro-differential equations of natural oscillations of a viscoelastic ribbed truncated conical shell are obtained based on the Lagrange variational equation. The general research methodology is based on the variational principles of mechanics and variational methods. Geometrically nonlinear mathematical models of the deformation of ribbed conical shells are obtained, considering such factors as the discrete introduction of edges. Based on the finite element method, a method for solving and an algorithm for the equations of natural oscillations of a viscoelastic ribbed truncated conical shell with articulated and freely supported edges is developed. The problem is reduced to solving homogeneous algebraic equations with complex coefficients of large order. For a solution to exist, the main determinant of a system of algebraic equations must be zero. From this condition, we obtain a frequency equation with complex output parameters. The study of natural vibrations of viscoelastic panels of truncated conical shells is carried out, and some characteristic features are revealed. The complex roots of the frequency equation are determined by the Muller method. At each iteration of the Muller method, the Gauss method is used with the main element selection. As the number of edges increases, the real and imaginary parts of the eigenfrequencies increase, respectively.

The aim of the work is to develop algorithms and a set of programs for studying the dynamic characteristics of viscoelastic thin plates on a deformable base on which it is installed with several dynamic dampers. The theory of thin plates is used to obtain the equation of motion for the plate. The relationship between the efforts and the stirred plate obeys in the hereditary Boltzmann Voltaire integral. With this, a system of integro-differential equations is obtained which is solved by the method of complex amplitudes. As a result, a transcendental algebraic equation was obtained to determine the resonance frequencies, which is solved numerically by the Muller method. To determine the displacement of the point of the plate with periodic oscillations of the base of the plate, a linear inhomogeneous algebraic equation was obtained, which is solved by the Gauss method. The amplitude - frequency response of the midpoint of the plate is constructed with and without regard to the viscosity of the deformed element. The dependence of the stiffness of a deformed element on the frequency of external action is obtained to ensure optimal damping of vibrational vibrations of the plate.