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 ($\varepsilon =0$). 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.

本文提出了一种具有分数阶恢复力和时变质量的非线性振子运动的解析方法。具有三角函数形式的周期运动的近似解可以描述运动的幅度和相位。将解析解与数值解进行比较，说明理论结果的准确性和有效性。本文采用分析程序来研究 van der Pol 型机械系统的动力学。首先，近似解可以被认为是具有常数参数的方程的近似精确解的小扰动版本（$\varepsilon =0$）。介绍了带时变周期函数的三角函数的近似求解过程。其次，将获得的近似解析解应用于具有分数阶和质量变量的恢复力的 van der Pol 振荡器。这表明这里讨论的近似解继承了准确性、数学上简单和易于编程的优点。这种方法可以推广到更广泛的机械系统，具有缓慢的变化参数和非线性分数阶。