A combination of asymptotic methods in nonlinear mechanics with basic techniques of perturbation theory to study a mathematical model of the nonlinear oscillation system is proposed in the paper. The system under consideration describes the torsional vibrations of an elastic body, where its elastic properties are under the nonlinear law. The relationships presented as the ordinary differential equations are obtained due to the proposed procedure. Therefore, the main parameters of the single-frequency oscillations and the resonance conditions can be determined. There are proposed applications of the obtained results to the optimization problem concerning the processing equipment.

- Cveticanin L. Strong Nonlinear Oscillator – Analytical Solutions. Mathematical Engineering. Springer (2018).
- Cveticanin L. Period of vibration of axially vibrating truly nonlinear rod. Journal of Sound and Vibration.
**374**, 199–210 (2016). - Cveticanin L., Pogany T. Oscillator with a sum of non-integer order non-linearities. Journal of Applied Mathematics.
**2012**, Article ID 649050, 20 pages (2012). - Gendelman O., Vakakis A. F. Transitions from localization to nonlocalization in strongly nonlinear damped oscillators. Chaos, Solitons and Fractals.
**11**(10), 1535–1542 (2000). - Mitropol'skii Yu. A. On construction of asymptotic solution of the perturbed Klein-Gordon equation. Ukr. Math. J.
**47**(9), 1378–1386 (1995). - Mitropol'skii Yu. A., Limarchenko O. S. On asymptotic approximations for slow wave processes in nonlinear dispersive media. Ukr. Math. J.
**50**(3), 408–424 (1998). - Oleynik O. A. Lectures on partial differential equations. Moscow, Binomial (2005), (in Russian).
- Andrianov I. V. Danishevskyi V. V., Ivankov A. O. Asymptotic methods in the theory of vibrations of beams and plates. Dnepropetrovsk, Pridneprovsk State Academy of Civil Engineering and Architecture (2010), (in Russian).
- Mitropolskii Yu. A., Moseenkov B. I. Asymptotic solutions of partial differential equations. Kyiv, Vyshcha Shkola (1976), (in Russian).
- Pukach P. Ya., Kuzio I. V. Resonance phenomena in quasi-zero stiffness vibration isolation systems. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu.
**3**, 62–67 (2015). - Myshkis A. D., Filimonov A. M. Periodic oscillations in nonlinear one-dimensional continuous media. Proceedings of the IX International Conference on nonlinear oscillations. Part 1, 274–276 (1984), (in Russian).
- Pukach P. Ya., Kuzio I. V., Nytrebych Z. M., Ilkiv V. S. Analytical methods for determining the effect of the dynamic process on the nonlinear flexural vibrations and the strength of compressed shaft. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu.
**5**, 69–76 (2017). - Pukach P. Ya., Kuzio I. V., Nytrebych Z. M., Ilkiv V. S. Asymptotic method for investigating resonant regimes of non-linear bending vibrations of elastic shaft. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu.
**1**, 68–73 (2018). - Pukach P. Ya. Investigation of bending vibrations in Voigt-Kelvin bars with regard for nonlinear resistance forces. J. Math. Sci.
**215**(1), 71–78 (2016). - Pukach P. Y. Qualitative Methods for the Investigation of a Mathematical Model of Nonlinear Vibrations of a Conveyer Belt. J. Math. Sci.
**198**(1), 31–38 (2014). - Pukach P. Ya., Kuzio I. V. Nonlinear transverse vibrations of semiinfinite cable with consideration paid to resistance. Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu.
**3**, 82-86 (2013), (in Ukrainian). - Filimonov A. M. Continual and discrete models of bounded one-dimensional media in viscoelasticity. Journal of Applied Mathematics and Mechanics.
**61**(2), 275–285 (1997). - Myshkis A. D., Filimonov A. M. Continuous solutions of hyperbolic systems of quasilinear equations with two independent variables. Nonlinear analysis and nonlinear differential equations. Moscow, Fizmatlit (2003), (in Russian).
- Senik P. M. Inversion of the incomplete beta function. Ukr. Math. J.
**21**(3), 271–278 (1969). - Nazarkevych M. Investigation of Beta- and Ateb-function dependencies. Bulletin of the National University "Lviv Polytechnic".732: Computer Science and Information Technology, 207–216 (2012) (in Ukrainian).
- Nayfeh A. H. Perturbation methods. New York, Wiley-Interscience (1973).
- Maslov V. P. Asymptotic methods and perturbation theory. Moscow, Nauka (1988), (in Russian).
- Sokil B. I. Periodic Ateb–functions in the study of single-frequency solutions of some wave equations. Proceedings of Shevchenko Scientific Society.
**1**, 588-592 (1997), (in Ukrainian). - Pisarenko G. S., Kvitka O. L., Umansky E. S. Resistance of materials. Kyiv, Vyshcha shkola (2004), (in Ukrainian).
- Sokil B., Senyk A., Sokil M., Andrukhiv A., Kovtonyuk M., Gromaszek K., Ziyatbekova G., Turgynbekov Y. Mathematical models of dynamics of friable media and analytical methods of their research. Przeglad Elektrotechniczny.
**95**(4), 74–78 (2019). - Lyashuk O., Vovk Y., Sokil B., Klendii V., Ivasechko R., Dovbush T. Mathematical model of a dynamic process of transporting a bulk material by means of a tube scraping conveyor. Agricultural Engineering International: CIGR Journal. 21 (1), 74–81 (2019).