We study the process of nonmonotonic loading of the deformable filler in a cylindrical shell with regard for the Coulomb friction. A numerical-analytic description of the loop of structural damping is obtained by using applied models.In the contemporary industry, a significant role is played by vibrational processes. The operation of all machines and mechanisms without exception is directly connected with the appearance of vibrations. In most cases, vibrations decrease the strength, reliability, and durability of industrial machines, mechanisms, and structures and exert harmful influence on the health of the personnel. Thus, the problem of vibroinsulation proves to be quite urgent both from the viewpoint of engineering and for the labor protection.One of the ways used for the solution of the formulated problem is connected with the application of vibroprotecting devices, such as shock absorbers, dampers, dynamic vibration absorbers, etc. This is why the research and design works and theoretical investigations in the field of development of new means of vibroprotection and methods for their numerical analyses are of crucial importance.Note that new promising vibroinsulators were designed at the Pidstryhach Institute for Applied Problems in Mechanics and Mathematics on the basis of a new type of vibration-protecting devices based on the so-called shell elastic elements [6]. As the main distinctive feature of structures of this type, we can mention the application of thin-wall elements (shells, plates, and bars) as the principal bearing and actuating elements. From the viewpoint of design, these are deformable shell systems with dry friction. A comprehensive survey of works devoted to the investigation of dynamic systems with various laws of friction can be found in [9]. In the mechano-mathematical modeling of the behavior of elastic elements under (generally speaking, nonmonotonic) loading, we get a class of nonlinear nonconservative mixed contact problems of the frictional interaction of thin shells with deformable fillers. The general approaches to the formulation and solution of these problems developed for simplified one-dimensional models are described in [2]. The statements, methods, and solutions of some problems from this class and the results of experimental investigations can be found in [1,3,5,7,8]. Engineering methods for the numerical analysis of shell vibroinsulators were developed on the basis of the asymptotic analysis of the obtained solutions in [6].In the present work, we develop methods for the investigation of the stress-strain state, compliance, and damping ability of "cylindrical shell-deformable filler" systems used to model the main elastic element of vibration-protecting devices.Consider an elastic deformable cylinder (filler) of radius R and length 2a placed into a cylindrical shell with thickness h 0 (Fig. 1). The filler is compressed on the end faces by perfectly rigid pistons subjected to the action of an external load Q nonmonotonic as a function of time.

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