Comparison of Energy Balance Period with Exact Period for Arising Nonlinear Oscillator Equations

Abstract

In this paper, He's energy balance method is applied to nonlinear oscillators. The new algorithm offers a promising approach by constructing a Hamiltonian for the nonlinear oscillator. We proved that the energy balance is very effective, convenient and does not require any linearization or small perturbation. In contradicts of the conventional methods, He's Energy Balance method (HEBM) using just one iteration, leads us to high accuracy of solutions. Energy Balance method is very effective, convenient and adequately accurate to both linear and nonlinear problems in physics and engineering.

A simple harmonic balance method for solving strongly nonlinear oscillators

“…The absolute percentage errors of each method has been calculated and presented in all Tables 1 and 2c. The results of these Tables show that the approximate frequencies (concern by this paper) provide better results than those obtained by Mickens (1996), Ganji et al (2009), Zhao (2009, Akbarzade and Farshidianfar (2014), Khan et al (2012a). Thus, the present method provides better result than other existing results.…”

“…Table 1 indicates that the approximate frequencies (concern of this paper) are better than those obtained by Mickens, 2001;Lim and Wu, 2003;Tiwari et al, 2005;Ozis and Yildirim, 2007;Ghadimi and Kaliji, 2013;Khan et al, 2012a. (45)) with the exact frequency x e and other existing frequencies (those frequencies obtained by Mickens, 1996a;Ganji et al, 2009;Zhao, 2009;Akbarzade and Farshidianfar, 2014;Khan et al, 2012a) Next, we have calculated the second-order approximate frequency (by Eq. (45)) of Eq.…”

“…(45)) of Eq. (33) with several amplitudes of oscillation and some different values of k that are compared with numerical result and other existing results (those results obtained by Mickens, 1996;Ganji et al, 2009;Zhao, 2009;Akbarzade and Farshidianfar, 2014;Khan et al, 2012a) which are presented in Tables 2a-c. The absolute percentage errors of each method has been calculated and presented in all Tables 1 and 2c.…”

“…Recently, many authors (Khan a new analytical approximation by variational iterative method (VIM); but the solution contains many harmonic terms. Many analytical techniques (Hosen et al, 2012;Mickens, 1996Mickens, , 2001Lim and Wu, 2003;Tiwari et al, 2005;Ozis and Yildirim, 2007;Ghadimi and Kaliji, 2013;Ganji et al, 2009;Zhao, 2009;Akbarzade and Farshidianfar, 2014;Khan et al, 2013a,b) have been used to solve strongly nonlinear oscillator systems having a rational force (such as Duffingharmonic oscillator: €…”

Power series approach to nonlinear oscillators

“…These methods were also extended to include different fields of mechanics, specially in structural dynamics [20][21][22][23][24] and fluid mechanics. [25][26][27] Some of these methods include the energy balance method, [28][29][30][31][32][33][34][35][36] the Hamiltonian approach and perturbation method 20,37,38 in addition to the variational approach, [39][40][41][42] and many others.…”