The variations of a system with one degree of freedom in the presence of nonlinear positional friction were considered. The restoring force of the oscillator's elasticity is described by the power function. A nonlinear solution of the Cauchy problem for the case of a power nonlinearity of positional friction were constructed. It is expressed through known periodic Ateb functions, which in recent years have become widespread in the theory of oscillations thanks to the efforts of the Lviv School of Mathematics and Mechanics. Formulas for calculating the displacements of the oscillator in time, as well as the amplitudes and periods of oscillations caused by the initial deviation of the system from the equilibrium position or the initial velocity given to the oscillator in this position were derived. The analytical dependence of the oscillation periods on the amplitude for cases of soft and rigid characteristics of the elastic system was established. It was found that due to the non-linearity of the damping of the system of the amplitudes, it leads to a change in the period of the oscillations of the system during its motion. The period may decrease or increase with oscillation, depending on the value of the indicator of non-linearity in the expression of the restoring force. It is shown, that from the obtained analytical solutions, as separate cases, there were known formulas related to free nonlinear oscillation oscillators without friction or with linear position friction. In order to simplify the using of the obtained solutions, we recommend well-known approximations of the Ateb-functions with the help of elementary functions. Examples of calculations are presented, where comparison of the results obtained with the use of constructed analytical solutions and numerical integration of the equation of oscillation on a computer is carried out. The good consistency of numerical results obtained in different ways is noted.
[1] L. Cveticanin, T. Pogany, “Oscillator with a sum of noninteger-order nonlinearities”, Journal of Applied Mathematics, vol. 2012, pp. 1-20, 2012. doi:10.1155/2012/649050.
[2] V. L. Biderman, Teoriya mehanicheskih kolebaniy [Theory of mechanical oscillations]. Moscow, Russia: Vysshaya shkola Publ., 1980. [in Russian].
[3] N. G. Suryaninov, A. F. Dashchenko, and P. A. Belous, Teoriticheskie osnovyi dinamiki mashin [Theoretical foundations of machine dynamics]. Odessa, Ukraine: OGPU, 2000. [in Russian].
[4] V. P. Olshanskiy, etc., Kolivannya dysypatyvnykh ostsyliatoriv [Fluctuations of dissipative oscillators]. Kharkiv, Ukraine: Mіs'kdruk Publ., 2015. [In Ukrainian].
[5] V. P. Olshanskiy, etc., Dynamika dysypatyvnykh ostsyliatoriv [Dynamics of dissipative oscillators]. Kharkiv, Ukraine: Mіs'kdruk Publ., 2016. [In Ukrainian].
[6] V. P. Olshanskiy, S. V. Olshanskiy, “Nestatsionarnyie kolebaniya mehanicheskoy sistemy lineyno-perenmennoy massy s kombinirovannyim treniem” [Unsteady oscillations of the mechanical system of linear-variable mass with combined friction], Visnik HNTUSG: Problemi nadiynosti mashin ta zasobiv mehanizatsiyi silskogospodarskogo virobnitstva [Bulletin of KhNTUAC: Problems of reliability of machines and means of mechanization of agricultural production], vol. 151, pp. 324-333, 2014. [in Russian].
[7] V. P. Olshanskiy, S. V. Olshanskiy, “Vilni kolivannya ostsilyatora zminnoyi masi z pozitsiynim tertyam” [Free oscillations of the variable mass oscillator with positional friction], Vibratsii v tekhnitsi ta tekhnolohiiakh [Vibrations in technique and technologies], vol. 2 (78), pp. 27-33, 2015. [in Ukrainian].
[8] V. P. Olshanskiy, S. V. Olshanskiy, “Pro ruh ostsilyatora zi stepenevoyu harakteristikoyu pruzhnostI” [On the motion of an oscillator with a power characteristic of elasticity], Vibratsii v tekhnitsi ta tekhnolohiiakh [Vibrations in technique and technologies], vol. 3 (86), pp. 34-40, 2017. [in Ukrainian].
[9] B. I. Sokil, “Pro zastosuvannya Ateb-funktsiy dlya pobudovi rozv’yazkiv deyakih rivnyan, yaki opisuyut neliniyni kolivannya odnovimirnih seredovisch” [On the application of Ateb-functions for construction of solutions of some equations describing nonlinear oscillations of one-dimensional media], Dopovidi Natsionalnoyi akademiy nauk Ukrayini [Reports of the National Academy of Sciences of Ukraine], vol. 1, pp. 55-58, 1997. [in Ukrainian].
[10] V. V. Gritsik, M. A. Nazarkovich, “Matematichni modeli algoritmiv i realizatsiya Ateb-funktsiy” [Mathematical models of algorithms and implementation of Ateb-functions], Dopovidi Natsionalnoyi akademiy nauk Ukrayini [Reports of the National Academy of Sciences of Ukraine], vol. 12, pp. 37-42, 2007. [in Ukrainian].
[11] I. V. Kuzio, T.-N. M. Vankovich, and Ya. A. Zinko, Teoretichna mehanika. Dinamika. Kn.1 [Theoretical mechanics. Dynamics. B.1]. Lviv, Ukraine: Rastr-7 Publ., 2012. [In Ukrainian].
[12] P. Ya. Pukach, Yakisni metodi doslidzhennya neliniynih kolivanih sistem [Qualitative methods for investigating nonlinear oscillate systems]. Lviv, Ukraine: Lviv Polytechnic Publishing House, 2014. [In Ukrainian].
[13] M. Abramovits, I. Stigan, Spravochnik po spetsialnyim funktsiyam (s formulami, grafikami i matematicheskimi tablitsami) [Handbook of special functions (with formulas, graphs and mathematical tables)]. Moscow, Russia: Nauka Publ., 1979. [in Russian].
[14] E. Yanke, F. Elde, and F. Lesh, Spetsialnyie funktsii [Special functions]. Moscow, Russia: Nauka Publ., 1977. [in Russian].
[15] A. P. Prudnikov, Yu. A. Bryikov, O. I. Marichev, Integraly i ryady. Elementarnyie funktsii [Integrals and rows. Elementary functions]. Moscow, Russia: Nauka Publ., 1981. [in Russian].