Periodic Solution for Strongly Nonlinear Vibration Systems by He’s Energy Balance Method

Abstract

This paper applies He's Energy balance method (EBM) to study periodic solutions of strongly nonlinear systems such as nonlinear vibrations and oscillations. The method is applied to two nonlinear differential equations. Some examples are given to illustrate the effectiveness and convenience of the method. The results are compared with the exact solution and the comparison showed a proper accuracy of this method. The method can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other science.

Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

“…Ganji et al in [34] obtained the same approximation as that in (52). Using a single-term approximate solution ( ) = cos( ) to (36) and the Ritz procedure [40], Tiwari et al [41] obtained an approximate angular frequency as follows:…”

The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

“…Due to the limitation of existing exact solutions, many analytical approaches have been investigated. Many researchers have been working on various analytical methods for solving nonlinear oscillation systems in the last decades, such as Homotopy Perturbation Method (HPM) [1][2][3][4][5], Max-Min Approach (MMA) [5][6][7][8][9][10], Variational Iteration Method (VIM) [2,5], Energy Balance Method (EBM) [5,[11][12][13][14][15][16], Amplitude-Frequency Formulation (AFF) [5-7, 15, 17, 18], Improved Amplitude-Frequency Formulation (IAFF) [5], Parameter Expansion Method (PEM) [5,7,[18][19][20], Homotopy Analysis Method (HAM) [5,21], Modified Homotopy Perturbation Method (MHPM) [5,6], Modified LindstedtPoincare Method [22], Harmonic Balance Method [23,24], and combined Newton's Method with the Harmonic Balance Method [25].…”

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

“…In recent years, many powerful methods are used to find approximate solution as well as the amplitude- frequency relationship to the nonlinear differential equations. Some of these methods are Homotopy Perturbation Method (HPM) (He, 1999;He, 2004a;He, 2004b;He, 2004c;Turgut et al, 2007;Bayat et al, 2012), Max-Min Approach (MMA) (He, 2008;Chen et al, 2011;Dumaz et al, 2011;Yazdi et al, 2012;Bayat et al, 2012), Variational Iteration Method (VIM) (Bayat et al, 2012), Energy Balance Method (EBM) (Ganji et al, 2009;Khah et al, 2010;Younesian et al, 2010;Bayat et al, 2012), Amplitude-Frequency Formulation (AFF) (Chen et al, 2011;Jouyburi et al, 2014;Bayat et al, 2012), Parameter Expansion Method (PEM) (Kayaa et al, 2009;Dumaz et al, 2011;Darvishia et al, 2008;Zhao, 2009;Bayat et al, 2012 ), Homotopy Analysis Method (HAM) (He, 2004c;Bayat et al, 2012, Shahram Shahlaei-Far et al, 2016, Modified Homotopy Perturbation Method (MHPM) (Jouybari et al, 2014), Equivalent linearization Method (ELM) (Krylov et al, 1943;Caughey, 1959;Iyengar, 1988;Anh et al, 1995;Anh et al, 1997;Elishakoff et al, 2009;Anh, 2015) and combining Newton's Method with the Harmonic Balance Method (Lim et al, 2006).…”