The influence of the nonlinear component of viscous resistance on the duration of free oscillations of the oscillator

Abstract

The free oscillations of a linear elastic oscillator with a power nonlinear viscous resistance are considered. The expression of the resistance force in the equation of motion consists of two terms. The first term is proportional to the speed of movement, and the second to the degree of speed. The study was conducted by the method of energy balance. Two variants of the specified method are implemented. The first is associated with the preparation and solution of the differential equation of the envelope of the graph of the oscillatory process. In the second version of the method, the calculation of the sequence of decreasing amplitudes of the ranges is reduced to a recurrence relation, which, for an arbitrary positive non-linearity index, has to be solved numerically. Newton's iterative method is involved. Cases of nonlinearity are established when the recurrence relation has closed analytical solutions and they are constructed. It is proved that when the non-linearity index is greater than zero, but less than unity, the free oscillations of the oscillator are limited in time and are reduced to a finite sequence of ranges, that is, an oscillator with viscous resistance has the same property as an oscillator with dry friction. Since the calculation formulas were obtained without solving the nonlinear differential equation of motion of the oscillator, the numerical results they bring to are compared with the results of numerical computer integration of the Cauchy problem. Their satisfactory consistency was obtained and it was found that the use of recurrence relations gives higher accuracy than using the expression for the envelope of the graph of free oscillations. It is shown that from the derived formulas, as a special case, the previously obtained relationships for calculating the amplitudes of free oscillations of the oscillator under the combined action of the forces of dry and linear viscous friction follow. The study is accompanied by examples of calculations and a comparative analysis of the results.