Post-Impact Vibrations of a Square Nonlinear Dispative Oscillator

Abstract

An oscillator damped by viscous linear resistance, due to the instantaneous increase in its mass after impact, can become a dissipative oscillatory system under the action of dry or positional friction. In the article describes the oscillations of a dissipative oscillator with an asymmetric quadratically nonlinear elastic characteristic and dry Coulomb friction, arising as a result of an inelastic vertical impact of a rigid body on it. In the article, the Cox model is used, which does not take into account local deformations of solid bodies subjected to impact. The paper establishes the dependences on the impact velocity and the values of other parameters at which the effect of asymmetry of the elastic characteristic of the system may appear or may not appear. The conditions are derived when the dynamic effect of asymmetry of the power characteristic is manifested in the system. It consists in the fact that the maximum displacement of the oscillator (oscillation range) in the direction of the shock pulse is less than the opposite extreme displacement (range) after the shock oscillations. The existence of such a critical value of the shock impulse is established, the excess of which leads to the loss of motion stability. The second integral of the oscillation equation describes the movement of the oscillator in time, expressed in terms of Jacobi elliptic functions. An approximate formula for their calculation is proposed. Formulas are also derived to determine the time to reach extreme deviations of the system from the equilibrium position. This time is expressed in terms of elliptic integrals of the first kind, which refer to the tabulated functions. Examples of calculations are considered, where, in addition to using the derived formulas, numerical computer integration of the original nonlinear differential equation of motion is carried out. A comparison of the results obtained for the displacement values of a quadratically nonlinear oscillator with dry friction expressed in terms of Jacobi elliptic functions and obtained by numerical integration is carried out. Good consistency of the calculation results in two ways confirmed the adequacy of the obtained analytical solutions of the nonlinear Cauchy problem.