Background In this article, to the development of the theory and methods for calculating vibrations of dissipative mechanical systems consisting of solids, which include both deformable and non-deformable bodies is discussed. Theoretically, the problem in the mathematical aspect, there are not enough developed solution methods and algorithms for dissipative mechanical systems has not yet been posed. The problems of choosing a nucleus and its rheological parameters, their influence on the frequency and damping coefficient systems have not been studied Purpose The goal of the work is to formulate the statement, develop the solution methods and the algorithm for studying the problems of the dynamics of dissipative mechanical systems consisting of thin-walled plates (or shells) with attached masses and point development theory. Methods A method and algorithm for solving problems of eigen and forced vibrations of dissipative mechanical systems consisting of rigid and deformable bodies, based on the methods of Muller, Gauss, Laplace and Runge integral transform, are developed. Results To describe the dissipative properties of the system as a whole, the concept of a global damping coefficient (GDC) is introduced. In the case of a dissipatively homogeneous system of the GDC, it is determined by the imaginary part of the first modulo complex natural frequency. In the role of GDC in the case of a dissipatively inhomogeneous system, are the imaginary parts of both the first and second frequencies. Moreover, the “Change of Roles” occurs with the characteristic value of the stiffness coefficient of the deformable elements; the real parts of the first and second frequencies are closest. At the indicated characteristic value of the deformable elements, the global damping coefficient has a pronounced maximum. A change in the parameter, on which the global damping coefficient so substantially depends, can be achieved by varying physical properties or geometrical dimensions, thereby opening up the promising possibility of effectively controlling the damping properties of dissipative-inhomogeneous mechanical systems. Conclusions The developed solution methods, algorithms and programs allow determining the dissipative properties of a mechanical system depending on various physic-mechanical parameters, geometrical dimensions and boundary conditions. A method for estimating the dissipative properties of the system as a whole (with forced vibrations) depending on the instantaneous values of the deformable elements (shock absorbers) has been developed. The developed method allows to reduce (several times) the amplitudes of displacements and stresses.

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 thin-walled shell and a thick-walled mass (cylinder) in contact with it, made of a different material, are structural elements of many machines, apparatus, and structures. The paper considers forced steady-state vibrations of cylindrical shell structures filled with a layered viscoelastic material. The study aims to determine the damping properties of vibrations of a structurally inhomogeneous cylindrical mechanical system under the influence of harmonic loads. The dynamic stress-strain state of a three-layer cylindrical shell filled with a viscoelastic material under the action of internal time-harmonic pressure is investigated. The oscillatory processes of the filler and the bonded shell satisfy the Lamé equations. At the contact between the shell and the filler, the rigid contact conditions are satisfied. Dependences between stresses and strains for a linear viscoelastic material are presented in the form of the Boltzmann-Voltaire integral. The method of separation of variables, the method of the theory of potential functions (special functions), and the Gauss method are used to solve this problem. Based on the analysis of the numerical results, it was found that the dependence of the resonant amplitude of the shell displacements on the viscous properties of the filler is 12-15%. Analysis of the results obtained shows that the study of vibrations of shells containing fillers according to the rod theory will lead to rather large erroneous results (up to 20%).

This article provides a detailed analysis of the problem of the optimal choice of parameters of dynamic dampers with two degrees of freedom. The choice method is based on calculating the difference between the resonant frequency of the system without absorbers and the nearest resonant frequency of the system with absorbers. The optimal parameters are determined and the effectiveness of many mass-dynamic vibration dampers with various types of viscosity under vibrational impacts is estimated. It has been shown that frequency-independent friction equally affects how viscous friction affects the behavior of the protected structure and damper and its effectiveness. A sufficient condition for damping vibrations in both coordinates is the presence of two dampers for movement, installed at different points. Frequency limits are set to suppress damping simultaneously in the upper and lower surroundings with respect to the resonant frequency of the main system, and also only in the upper neighborhood. It was found that the use of more than two absorbers while maintaining their total mass affects the frequency band in which it is possible to dampen the oscillations of the system.

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.