A free oscillations of a system with one degree of freedom, caused either by the initial deviation from the stable equilibrium position or by the initial velocity provided by the oscillator in this position was considered. Analytical solutions of the nonlinear Cauchy problem for a second-order differential equation were constructed. The solutions are expressed in terms of Jacobi's periodic elliptic functions relating to occultation of special functions. Compact equals are derived for calculating the displacements of the oscillator and the oscillation periods for various methods of motion perturbation and for various variants of the elastic characteristic. The restrictions on the initial excitations for an oscillator with a soft elastic characteristic are determined, when its free oscillations are possible. The existence of a solution of the nonlinear dynamics problem in elementary functions is established. The behavior of an oscillator with a soft characteristic of elasticity under conditions of its freezing are studied. It is shown that from the derived equals, as special cases, the results known in the theory of linear oscillators, as well as oscillators with a purely quadratic nonlinearity, without a linear component, follow when the solution of the problem can be expressed in terms of Ateb-functions. The aim of the work was to derive new calculation equals for determining the displacements of a mechanical system with one degree of freedom under conditions of free oscillations, in the absence of friction. To achieve this objective, the representation of the second integral of the differential equation of motion due to the incomplete elliptic integral of the first kind were used. Using the known tables of the indicated integral, examples of calculations are given in which the probability of the derived equals is confirmed. According to the results of the study, it is also established that in the case of a quadratic elasticity characteristic of the linear component, the motion of the oscillator is described by the periodic elliptic Jacobi function, both in providing it with an initial deviation from the stable equilibrium position, and giving it the initial velocity in this position. In the case of a soft elasticity characteristic, free oscillations are possible only with certain restrictions on the initial perturbations of the system.

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