An analytical approximate technique for a class of strongly non-linear oscillators

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

“…To determine the second-order approximate solution it is necessary to substitute (23) into (10). Then secular terms are eliminated and parameter α 2 can be calculated.…”

Linearized Harmonic Balancing Approach for Accurate Solutions to the Dynamically Shifted Oscillator

“…In Tables 3-8 we compare A linearized harmonic balance method proposed by Wu et al [37] was used to obtain two approximate frequencies for the dynamically shifted oscillator. We can conclude that the approximate frequencies obtained are valid for the complete range of oscillation amplitude, including the limiting cases of amplitude approaching zero and infinity.…”

“…To improve this method, Wu et al [37] presented an approach obtained by combining Newton's method with the harmonic balance method. This method is established by successfully linearizing the governing equation and, subsequently, appropriately imposing the harmonic balance method in order to obtain linear algebraic equations instead of non-linear algebraic equations.…”

“…The most commons and most widely studied methods of all approximation methods for nonlinear differential equations are perturbation methods [1]. Some of other techniques include variational and variational iteration methods [4][5][6][7][8][9][10][11][12], exp-function [13,14], homotopy perturbation [15][16][17][18][19][20][21][22], equivalent linearization [23,24], standard and modified Lindstedt-Poincaré [25][26][27][28][29], artificial parameter [30,31], parameter expanding [32][33][34], harmonic balance methods [1,[35][36][37][38][39][40], etc. Surveys of the literature with numerous references and useful bibliography and a review of these methods can be found in detail in [28] and [41].…”

“…To applied the harmonic balance method to the dynamically shifted oscillator [2,42,43], we use the analytical approach developed Wu et al [37] which incorporates salient features of both Newton's method and the harmonic balance method. Wu et al's approach is established by successfully linearizing the governing equation and, subsequently, appropriately imposing the harmonic balance method in order to obtain linear algebraic equations, which can be easily solved.…”