Determination of Approximate Periods of Duffing-harmonic Oscillator
Keywords:Approximate periods, harmonic balance method, Duffing-harmonic oscillator, Power series solutions, Perturbation Method
We introduced an analytical technique based on harmonic balance method (HBM) to determine approximate periods of a nonlinear Duffing-harmonic oscillator. Generally, a set of nonlinear algebraic equations are appeared when HBM is formulated. Investing analytically of such kinds of algebraic equations are a tremendously difficult task and cumbersome. In the present study, the offered technique gives desired results and to avoid numerical complexity. It is remarkable important that a third-order approximate period gives excellent agreement compared with numerical solution. The method is mainly illustrated by strongly nonlinear Duffing-harmonic oscillator but it is also useful for many other nonlinear oscillating systems arising in nonlinear sciences and engineering.
AH Nayfeh. Perturbation Methods. John Wiley & Sons, New York, 1973.
N Elmas and H Boyaci. A new perturbation technique in solution of nonlinear differential equations by using variable transformation. Appl. Math. Comput. 2014; 227, 422-7.
AK Azad, MA Hosen and MS Rahman. A perturbation technique to compute initial amplitude and phase for the Krylov-Bogoliubov-Mitropolskii method. Tamkang J. Math. 2012; 43, 563-75.
A Belendez, A Hernandz, T Belendez, E Fernandez and C Neipp. Application of He’s homotopy perturbation method to the Duffing harmonic oscillator. Int. J. Nonlinear Sci. Numer. Simulat. 2007; 8, 89-8.
S Nourazar and A Mirzabeigy. Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method. Sci. Iran. 2013; 20, 364-8.
AE Ebaid. Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method. Comm. Nonlinear Sci. Numer. Simulat. 2010; 15, 1921-7.
JH He. Max-Min approach to nonlinear oscillator. Int. J. Nonlinear Sci. Numer. Simulat. 2008; 9, 207-10.
DD Ganji and M Azimi. Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems. Appl. Math. Phys. 2012; 74, 131-40.
MR Akbari, DD Ganji, A Majidian and AR Ahmadi. Solving nonlinear differential equations of Vanderpol Rayleigh and Duffing by AGM. Front. Mech. Eng. 2014; 9, 177-90.
M Azimi and A Azimi. Application of parameter expansion method and variational iteration method to strongly nonlinear oscillator. Trends Appl. Sci. Res. 2012; 7, 514-22.
SQ Wang and JH He. Nonlinear oscillator with discontinuity by parameter-expansion method. Chaos Soliton. Fract. 2008; 35, 688-91.
M Baghani, M Fattahi and A Amjadian. Application of the variational iteration method for nonlinear free vibration of conservative oscillators. Sci. Iran. 2012; 19, 513-8.
SS Ganji, DD Ganji, MG Sfahani and S Karimpour. Application of AFF and HPM to the systems of strongly nonlinear oscillation. Curr. Appl. Phys. 2010; 10, 1317-25.
BMI Haque, MS Alam and MM Rahman. Modified solutions of some oscillators by iteration procedure. J. Egyptian Math. Soc. 2013; 21, 142-7.
H Hu. Solutions of the Duffing-harmonic oscillator by an iteration procedure. J. Sound Vib. 2006; 298, 446-52.
T Ozis and A Yildirim. Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method. Comput. Math. Appl. 2007; 54, 1184-7.
N Jamshidi and DD Ganji. Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Curr. Appl. Phys. 2010; 10, 484-6.
S Durmaz and OK Metin. High-order energy balance method to nonlinear oscillators. J. Appl. Math. 2012; 2012, Article ID 518684.
DD Ganji, M Gorji, S Soleimani and M Esmaeilpour. Solution of nonlinear cubic-quintic duffing oscillators using He’s energy balance method. J. Zhej. Univ. Sci. A 2009; 10, 1263-8.
M Daeichin, MA Ahmadpoor, H Askari and A Yildirim. Rational energy balance method to nonlinear oscillators with cubic term. Asian-European J. Math. 2013; 6, Article ID 1350019.
SB Yamgoue, JR Bogning, AK Jiotsa and TC Kofane. Rational harmonic balance-based approximate solutions to nonlinear single-degree-of-freedom oscillator equations. Phys. Scr. 2010; 81, Article ID 035003.
M Xiao, WX Zheng and J Cao. Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method. Math. Comput. Simulat. 2013; 89, 1-12.
AYT Leung and G Zhongjin. Residue harmonic balance approach to limit cycles of non-linear jerk equations. Int. J. Nonlinear Mech. 2011; 46, 898-906.
AYT Leung, HX Yang and ZJ Guo. The residue harmonic balance for fractional order van der Pol like oscillators. J. Sound Vib. 2012; 331, 1115-26.
CA Borges, L Cesari and DA Sanchez. Functional analysis and the method of harmonic balance. Q. Appl. Math. 1975; 32, 457-64.
NA Bobylev, YM Burman and SK Korovin. Approximation Procedures in Nonlinear Oscillation Theory. Walter deGruyter, Berlin, 1994.
RE Mickens. Oscillation in Planar Dynamic Systems. World Scientific, Singapore, 1996.
RE Mickens. A generalization of the method of harmonic balance. J. Sound Vib. 1986; 111, 515-8.
RE Mickens. Mathematical and numerical study of Duffing-harmonic oscillator. J. Sound Vib. 2001; 244, 563-7.
A Belendez, E Gimeno, ML Alvarez, S Gallego, M Ortuno and DI Mendez. A novel rational harmonic balance approach for periodic solutions of conservative nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simulat. 2009; 10, 13-26.
S Karkar, B Cochelin and C Vergez. A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems. J. Sound Vib. 2014; 333, 2554-67.
CW Lim and BS Wu. A new analytical approach to the Duffing-harmonic oscillator. Phys. Lett. A 2003; 311, 365-73.
BS Wu, WP Sun and CW Lim. An analytical approximate technique for a class of strongly nonlinear oscillators. Int. J. Nonlinear Mech. 2006; 41, 766-74.
MS Alam, ME Haque and MB Hossian. A new analytical technique to find periodic solutions of nonlinear systems. Int. J. Nonlinear Mech. 2007; 42, 1035-45.
H Hu and JH Tang. Solution of a Duffing-harmonic oscillator by the method of harmonic balance. J. Sound Vib. 2006; 294, 637-9.
ZK Peng, G Meng, ZQ Lang, WM Zhang and FL Chu. Study of the effects of cubic nonlinear damping on vibration isolations using Harmonic Balance Method. Int. J. Nonlinear Mech. 2012; 47, 1073-80.
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