On the Distribution of Free Waves on the Surface of a Viscoelastic Cylindrical Cavity

Abstract 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.