The free oscillations of an oscillator with a power-nonlinear elasticity characteristic under the action of linear viscous resistance are considered. Using the energy balance method, which is widespread in mechanics, the calculation of the amplitudes of free damped oscillations is reduced to calculating the roots of an algebraic equation, which has an exact analytical solution only with linear elasticity of the oscillator. In the case of an arbitrary positive indicator of nonlinear elasticity, a numerical solution of the equation is required. For this, the Newton's iterative method was used in the work, which has fast convergence of iterations at an arbitrary initial approximation. According to the results of the analysis of the coefficients of the equation established, that in the case of a rigid characteristic of elasticity, when the nonlinearity is greater than unity, the oscillations are reduced to a finite number of decaying ranges, that is, they are limited in time, and in the case of a soft characteristic of elasticity, when the nonlinearity is less than unity, they continue to infinity, as linear dissipative oscillator. The research is given by the method of energy balance and numerical integration of the differential equation of oscillations on a computer. The work of the force of viscous resistance is calculated approximately using periodic Ateb functions that accurately describe free undamped oscillations in the absence of resistance. As a result, approximate iterative dependences are obtained for calculating the amplitudes of the ranges that decay during movement. The numerical results obtained using approximate formulas and numerical computer integration of the nonlinear Cauchy problem are compared. Their satisfactory agreement was noted. A satisfactory agreement was noted between the results for both hard and soft elastic characteristics, which confirmed the adequacy of approximate analytical solutions to the dynamics problem. The main advantage of the described approximate calculation method is that there is no need to build an analytical solution to the nonlinear differential equation of motion of the oscillator, which is a rather complicated mathematical problem. Furthermore, it made it possible to establish conditions under which the oscillator with a viscous and dry friction resistance have similar oscillation properties.
Study of the Apsidal Precession of the Physical Symmetrical Pendulum
