In the work, the problems of proper and forced oscillations of dissipative mechanical systems, consisting of rigid and deformable bodies are solved. To quantify the dissipative properties of the system, two values are proposed: the minimum resonance frequency of natural oscillations and the maximum resonant amplitude. In the study of the problem of dissipative inhomogeneous mechanical systems, a nonmonotonic dependence of the damping coefficients on the parameters of the system was observed. The concepts are derived Global damping factor, which characterizes the Damping properties of the dissipative mechanical system as a whole.

The vibrations of deformed bodies interacting with an elastic medium are considered. The problem reduces to finding those values of complex Eigen frequencies for which the system of equations of motion and the radiation conditions have a nonzero solution to the class of infinitely differentiable functions. It is shown that the problem has a discrete spectrum located on the lower complex plane and the symmetric spectrum is an imaginary axis.

A mathematical model and a technique for assessing the efficiency of the dissipative ability of structurally inhomogeneous mechanical systems consisting of multilayer cylinders bonded to a thin viscoelastic shell of finite length have been developed. A detailed analysis of the known works devoted to this problem is given. A model, methodology, and algorithm for studying the natural and forced oscillations of a system to assess the damping ability of structurally inhomogeneous elastic and viscoelastic mechanical systems, taking into account the influence of the geometric and physico-mechanical parameters of the shell and cylinderhave been developed. In solving the problems considered, the method of divided variables, the method of the theory of potential functions, the Mueller method, the Gauss method and the orthogonal sweep method were used. The complex eigenfrequencies, amplitudes of forced oscillations are determined, and the largest dephasing abilities of the considered structurally inhomogeneous systems are estimated. It has been revealed that, the effect of the greatest damping ability in structurally heterogeneous systems is manifested when the real parts of complex natural frequencies come closer due to the interaction of close natural forms with each other.

The propagation of natural waves in a cylindrical shell (elastic or viscoelastic) that is in contact with a viscous liquid is considered. The problem reduces to solving spectral problems with a complex incoming parameter. The system of ordinary differential equations is solved numerically, using the method of orthogonal rotation of Godunov with a combination of the Muller method. The dissipative processes in the mechanical system are investigated. A mechanical effect is obtained that describes the intensive flow of mechanical energy.

In this article an analysis of well-known works related to wave propagation and dispersion dependences in a cylindrical waveguide are presented. A mathematical model, methodology and algorithm for solving the problem of wave propagation in a cylindrical waveguide, having a sector cut are developed. The obtained equations are solved by the orthogonal sweep method in combination with the Mueller and Gauss methods. The dispersion relation for a viscoelastic cylindrical waveguide, having a sector cut in cross section at an arbitrary angle is obtained. Based on the obtained results, it was found that, there are no waves in the elastic cylinder of a sector section with real parts of the phase velocity. It has been established that, in the case of a wedge-shaped viscoelastic cylindrical panel, for each mode, there are limiting wave propagation velocities and they change with a change in the radius of curvature. The spectral sets of normal waves with an increase in the angular parameter of the sector cut-out, corresponding to lower non-zero frequencies of wave locking, slowly decrease.