Analytical and numerical solution of an $$\varvec{n}$$-term fractional nonlinear dynamic oscillator

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.