КОЛИВАННЯ ІМПУЛЬСНО НАВАНТАЖЕНОГО ОСЦИЛЯТОРА ЗІ СТЕПЕНЕВИМ ПОЗИЦІЙНИМ ТЕРТЯМ

Abstract

Nonstationary oscillations of the oscillator with nonlinear positional friction caused by an instantaneous force pulse are described. The power dependence of the positional friction force on the displacement of the system, which generalizes the known models, is accepted. The corresponding dynamics problems were solved precisely by the method of adding and approximated by the method of energy balance. In the study, using periodic Ateb-functions, an exact analytical solution of the nonlinear differential equation of motion was constructed. Compact formulas for calculating oscillation ranges and half-cycle durations are derived. It is shown that the decrease in the amplitude of oscillations, as well as under the action of the force of linear viscous resistance, follows the law of geometric progression. The denominator of the progression is less than one and depends on the positional friction constants, in particular on the nonlinearity index. Thus, we have not only a decrease in the amplitude of oscillations, but also an increase in the durations of half-cycles, which is characteristic of nonlinear systems with a rigid force characteristic. Approximate displacement calculations use Pade-type approximations for periodic Ateb-functions. The error of these approximations is less than one percent. From the obtained analytical relations, as separate cases, the known dependences covered in the theory of oscillations for linear positional friction follow. It is shown that even in the case of nonlinear positional friction the process of oscillations caused by an instantaneous momentum has many oscillations and is not limited in time. In the case of power positional friction, the oscillation ranges of the pulse-loaded oscillator can be calculated by elementary formulas. The calculation of displacements in time is associated with the use of periodic Ateb-functions, the values of which are not difficult to determine by known asymptotic formulas. Calculations confirm that the obtained approximate formula does not give large errors. In order to verify the adequacy of the obtained analytical solutions, numerical computer integration of the original nonlinear differential equation of motion was performed. The results of the calculation, which lead to analytical and numerical solutions of the Cauchy problem, are well matched.