Oscillator with Continual Mass Variation

Abstract

In this Chapter, oscillator with continual mass variation is considered. Mass variation is assumed to be a continual and monotone time function. Besides, it is supposed that oscillator is strong nonlinear. Free vibrations of systems with one and two degrees of freedom are investigated. Model of vibration are one or two strong nonlinear ordinary second order differential equations with time variable parameters. If mass variation of the oscillator is slow, parameters of equation are assumed to be functions of ‘slow time’ which is defined as the product of a small parameter and of time. In spite of the fact that mass variation is slow and the equation is with slow variable parameters, usually, it is impossible to obtain analytical solution of the differential equation of the oscillator in the closed form. Various procedures for approximate solving of equations are developed. In this Chapter a procedure for solving strong nonlinear differential equation with time variable parameters is presented. Method is based on the assumption that oscillator with slow mass variation is a perturbed one to the oscillator with constant parameters. It gives the idea that solution of mass variable oscillator is a perturbed version of solution of the oscillator with constant parameter. Solution of equation with constant parameters is assumed in the form of Ateb function (exact solution) or trigonometric function with exact period of vibration (approximate solution). Based on these solutions approximate solution of equations with time variable parameter is introduced, where amplitude and frequency parameters of oscillator are time dependent. To simplify the problem, averaging of equations of motion is introduced. So obtained solutions are very appropriate and acceptable for use. Influence of the reactive force and reactive torque, which exist due to mass variation, are specially investigated. Dynamics of the system with and without reactive force, and with and without influence force is consideration. Theoretical results are applied for truly nonlinear one-degree-of-freedom oscillators, for cubic and fifth order oscillators with variable mass functions, for Van der Pol oscillator with variable mass, for vibration of the rotor, considered as one-mass oscillator with two degrees of freedom, and for a two-mass system with two-degrees of freedom and mass variation.