Md. Abdur Razzak

A simple harmonic balance method for solving strongly nonlinear oscillators

Abstract In this paper, a simple harmonic balance method (HBM) is proposed to obtain higher-order approximate periodic solutions of strongly nonlinear oscillator systems having a rational and an irrational force. With the proposed procedure, the approximate frequencies and the corresponding periodic solutions can be easily determined. It gives high accuracy for both small and large amplitudes of oscillations and better result than those obtained by other existing results. The main advantage of the present method is that its simplicity and the second-order approximate solutions almost coincide with the corresponding numerical solutions (considered to be exact). The method is illustrated by examples. The present method is very effective and convenient method for solving strongly nonlinear oscillator systems arising in nonlinear science and engineering.

An analytical technique for solving quadratic nonlinear oscillator

Purpose The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the approximate solution of quadratic nonlinear oscillator (QNO). Design/methodology/approach This oscillator (QNO) is used as a mathematical model of the human eardrum oscillation. Findings It has been shown that the results by the present technique are very close to the numerical solution. Originality/value The results obtained in this paper are compared with those obtained by Hu (harmonic balance method) and Khan et al. The result shows that the method is more accurate and effective than harmonic balance as well as improved energy balance methods.

An analytical approximate technique to investigate a finite extensibility nonlinear oscillator

Abstract In this paper, an analytical approximate technique based on modified harmonic balance method is presented to study about the dynamics of a finite extensibility nonlinear oscillator described by Febbo (2011) and Beléndez et al. (2012). Generally, a second-order approximation is only considered in this paper. In the proposed method, the approximate period of oscillations and the corresponding periodic solutions are determined, which are valid for both range of amplitudes 0 < A ≤ 0.9 and 0.9 < A < 1 of oscillation. The approximate periods obtained in this paper are compared with numerical result (considered to be exact) and other existing results. Firstly, the results are obtained for the amplitude 0 < A ≤ 0.9 and show that the present method gives high accuracy than other existing results. Moreover, the results are also obtained for the rest of the amplitude 0.9 < A < 1. The relative error measure in this paper is 0.03% for A = 0.9 while the relative errors obtained by Febbo (2011) and Beléndez et al. (2012), were 3.53% and 0.60% respectively. On the other hand, Belendez et al. (2012) did not obtain approximate period for 0.9 < A < 1. In this article, the approximate periods have been determined in the range of value 0.9 < A < 1 and they have provided better results than the existing result of Febbo (2011).