Vibration processes play a significant role in the modern industry. In most cases, vibration reduces strength, reliability, and durability of industrial machines, mechanisms, and structures, as well as adversely affects the health of support staff. This study presents a new dry friction shell shock absorber design. The shock absorber contains two spring sections and a friction module with an open shell and an elastic filler; at the same time, the spring sections and the friction module work in parallel. The proposed device demonstrates good damping characteristics, capable of operating under high operating loads, and at the same time has compact transverse dimensions. With a nonmonotonic loading of such a shock absorber, due to the contact interaction of the filler with an open shell, part of the energy that is supplied to the system will be dissipated. In other words, in response to the action of an external nonmonotonic loading, the phenomenon of structural hysteresis occurs in the friction module of the shock absorber. To describe the deformation of the shock absorber, a mechanical and mathematical model of a shell with a cut along the generatrix, which is the main bearing link, has been developed. By means of the technique of quasistatic analysis of structural damping in nonmobile nonconservative shell systems with a deformable filler, the hysteresis loops of the presented shock absorber are analytically described. Using them, according to the known loading history, it is possible to predict the behavior of the considered nonconservative system at any time after the start of the loading process. At each stage of the cyclic loading, the distribution of stresses and relationships between the external loading and the piston displacement was studied. Inequalities are obtained for permissible loads under which the operation of the shock absorber is safe. Such shell shock absorbers are projected to be used in the mining, oil and gas, and construction industries.