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The problems have been considered of natural nonlinear vibrations of an absolutely rigid semiball and a semicylinder on a horizontal plane, assuming that there is no energy dissipation, sliding and tipping on foundation. To adjust the damper to a frequency close to the fundamental tone of vibrations, it is necessary to assess the natural frequency of the damper, which is determined under the assumption on smallness of the vibrations amplitude. This paper represents the comparison of the natural frequency of linearized and nonlinear system. The relative error has been estimated of the natural frequency calculation, which is caused by linearization. It is shown that the ratio of the natural frequency of the linearized system to the natural frequency of the nonlinear system does not depend on the mass and radius. This conclusion made it possible to generalize the results of particular computational solutions and to obtain a formula which takes into account the amplitude influence on the natural vibrations frequency and helps to determine the natural frequency for the initial angles to ninety degrees.

The conclusion has been made about the necessity to choose the optimal strategies for management by geotechnical systems, based on the analysis of geological faults, which are the main indicator of the mining and geological conditions that characterize the mineral deposits, as well as on the parameters for the infrastructure development of the underground space. The methodological peculiarity of solving the problems set is the use of game theory with modified criteria of Wald, maximax and Savage, since the manifestation of specific geological faults is probabilistic in nature. When choosing the optimal strategy, the average linear deviations of gains or risks are taken into account.