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The free oscillations of a system with one degree of freedom are considered under the assumption that the elasticity of a spring is proportional to the cubic root of its deformation. Two forms of the analytical solution of the nonlinear differential equation of motion of the oscillator are obtained. In the first displacement of the oscillator in time is expressed in terms of incomplete elliptic integrals of the first and second kind. In the second form, the solution is expressed in terms of periodic Ateb-functions. The tables of the involved functions are made, which simplify the calculation. Formulas are also derived for calculating the oscillation periods when the oscillator is signaled or the initial deviation from the equilibrium position or the initial velocity (instantaneous pulse) in this position. The dependence of the oscillation period on the parameters of the oscillator and the initial conditions is established. Examples of calculations of oscillations are presented with the use of compiled tables of special functions and using the proposed approximations of the Ateb-functions. Comparison of numerical results obtained by different methods is made.

The reversal of the vertical motion of a spherical particle with variable mass and radius in air flow is described using the analytical solutions of the Cauchy problem. It is shown that a particle with decreasing mass falling against an upward flow slows down to a stop and moves upward. If its mass is increasing, the particle moving with upward flow slows down to a stop and then moves downward Introduction. The classical dynamics of bodies of variable mass was intensively developed to follow the development of rocketry. The especial importance of this problem relegated the other applied problems in this area of mechanics to the background. Besides rocket dynamics, however, there are many processes that involve moving particles of variable mass. Among them are vaporization of dispersed fire-extinguishing substances moving in heated gas, vaporization and burning of moving liquid fuel droplets in engines, flight of burning particles of solid fuels in boiler furnaces, etc. While moving, such bodies decrease in size and weight. Therefore, the ballistic properties of particles with variable parameters are still of interest.Many early works on the mechanics of bodies of variable mass are included in the generalizing publications [6,7,11]. Various models describing the flight of vaporizing droplets in gas with nonlinear drag can be found in [4]. Dynamic problems of the interaction of spherical bodies, including droplets in fluid were solved in [13][14][15][16] and other publications cited in the review [12]. Some peculiar features in the vertical motion of a spherical body with decreasing mass are mentioned in [17]. The extremal properties of the velocity of vertical motion of particles with variable parameters, which are absent when the mass of particles is constant, were studied in [8,9]. Further ballistic analysis showed that a particle with decreasing size and mass moving with an upward gas flow may reverse its direction of motion. A falling particle of decreasing mass may start rising, while a rising particle of increasing mass may start falling. Here we study this effect.The goal of the paper is to quantitatively describe the reversal of the vertical motion of a spherical particle with variable parameters in upward gas flow.1. Reversal of the Falling of a Particle with Decreasing Mass in a Counter Flow. Let the variation in the radius r of a falling spherical body follow the Sreznevsky law (the surface area of the particle proportionally increases with time t):

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