Analysis on the motion of nonlinear vibration with fractional order and time variable mass

Abstract

In this paper, an analytical method on the motion of nonlinear oscillator with fractional-order restoring force and time variable mass is developed. The approximate solution for the periodic motion with the form of the trigonometric function form can describe the amplitude and phase of motion. The analytical solution is compared with the numerical solutions, which illustrate the accuracy and validity of theoretical results. This paper adopts an analytical procedure to investigate the dynamics of mechanical systems of the van der Pol type. First, the approximate solution can be considered as the small perturbed version of the nearly exact solution of the equation with constant parameters (ɛ=0). A procedure for an approximate solution of trigonometric function with time variable period function is introduced. Second, the obtained approximate analytic solutions are applied for the van der Pol oscillator with a restoring force with a fractional order and mass variables. It is shown the approximate solutions discussed here inherits the advantages of accuracy, mathematically simple and easy-to-program. This approach could be generalized to more extensive mechanical systems with slowing varying parameter and fractional order of nonlinearity.