Approximate method for calculating the ranges of free vibrations of a quasilinear dispipative oscillator

Abstract

A simplified method for the approximate calculation of the ranges of free damped oscillations of an oscillator with weak power nonlinearity in the expression of the elastic force is presented. The proposed method is based on the assumption that the nonlinear term in the expression of the elastic force has little effect on the equation of the envelope of the graph of free vibrations. The envelope equation used here, according to the accepted assumption, corresponds to the first approximation of the asymptotic method of the theory of nonlinear oscillations and was obtained by studying the free oscillations of a dissipative Duffing oscillator in the works of other authors. But here, the dependence of the half-cycle durations on the changing oscillation amplitude, which is characteristic of nonlinear systems, is additionally taken into account. Using the derived formulas, the ranges of oscillations were calculated under the action of various resistance forces, namely, Coulomb dry friction, linear viscous friction and quadratic viscous resistance. Various versions of power nonlinearities that complement the linear component in the expression of the elastic force are considered. In order to clarify the errors of the approximate method, computer numerical integration of the differential equation of motion for various types of resistance was carried out. Based on the results of comparing the numerical results obtained in various ways, restrictions on the nonlinearity of the elastic force were established, in this case, the errors of approximate compact formulas is several percent. The main advantage of the proposed method is the simplicity of implementation and the absence of the need to build an analytical solution of the nonlinear differential equation of motion of the oscillator caused by its initial deviation from the position of static equilibrium. In addition, the proposed approximate calculation method allows you to take into account the effect of various resistance forces, that is, viscous or dry friction.