Abstract

This work presents a novel method for the analytical and numerical solution of an n-term fractional nonlinear dynamical system. Two simple methods, commonly known to vibration engineers, namely the method of averaging and harmonic balance method, are utilized to obtain the analytical and numerical solution, respectively. The differential equation is derived from a physical problem. The primary resonance of an n-term fractional nonlinear oscillator is studied analytically by the averaging method. Initially, the amplitude–frequency parametric relation is obtained, and then, the effect of the system parameters such as the excitation amplitude, fractional order and nonlinear stiffness coefficients on the dynamics of the system is investigated. Further, the dynamical system is solved numerically using the harmonic balance method. The main advantage of using this approach is that it reduces the solution of differential equation to those of solving a system of algebraic equations, thus greatly simplifying the problem. The results reveal that the proposed methods are very effective and simple. The fractional-order system is defined in Caputo sense. Moreover, only a small number of harmonics are needed to obtain a satisfactory result with reduced computational time.

中文翻译：

---

$$ \ varvec {n} $$ n项分数阶非线性动态振荡器的解析和数值解

摘要

这项工作为n项分数阶非线性动力系统的解析和数值解提供了一种新颖的方法。振动工程师通常已知的两种简单方法，即平均法和谐波平衡法，分别用于获得解析解和数值解。微分方程是从一个物理问题得出的。n的主共振均值法对项分数阶非线性振荡器进行了解析研究。最初，获得了振幅-频率参数关系，然后，研究了诸如激励振幅，分数阶和非线性刚度系数等系统参数对系统动力学的影响。此外，使用谐波平衡法对动力学系统进行数值求解。使用这种方法的主要优点是，它可以将微分方程的解简化为求解代数方程组的解，从而大大简化了问题。结果表明，所提出的方法非常有效且简单。分数阶系统在Caputo的意义上定义。此外，