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.

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