In this paper the results of a study of the dynamic characteristics of thin-walled composite curved viscoelastic pipes under the influence of internal pulsating pressures is presented. The relationship between stress and strain is described by the Aviary equations. Based on the principle of calculus of variations, the equations of the dynamics of curvilinear shells are obtained. The obtained integral and integro-differential equations are solved using the finite element method, the Mueller and Gauss methods. The results of calculations of the natural frequencies of the shell and the curved rod are compared. As a result of the calculation, it was found that, shell vibrations occur at higher frequencies than rod vibrations (manifestation of a boundary effect). With this difference, the natural frequency, which has no analogue in the rod model decreases. This is due to the fact that, in this case the transverse vibrations of the section itself become significant, the rod. Some mechanical effects were discovered when taking into account the rheological properties of the pipe material.

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.

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.

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.

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.