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Frequency analysis of strongly nonlinear generalized Duffing oscillators using he’s frequency-amplitude formulation and He’s energy balance method. (English) Zbl 1193.65152

Summary: He’s Frequency-Amplitude Formulation (HFAF) and He’s Energy Balance Method (HEBM) are employed to solve the generalized Duffing equation in the form of

${x}^{\text{'}\text{'}}+x+{\alpha }_{3}{x}^{3}+{\alpha }_{5}{x}^{5}+{\alpha }_{7}{x}^{7}+\cdots +{\alpha }_{n}{x}^{n}=0·$

For any arbitrary power of $n$, a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HFAF, HEBM and exact integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results.

##### MSC:
 65L99 Numerical methods for ODE 34C15 Nonlinear oscillations, coupled oscillators (ODE)
##### References:
 [1] Geng, Lei; Cai, Xu-Chu: He’s frequency formulation for nonlinear oscillators, European journal of physics 28, 923-931 (2007) · Zbl 1162.70019 · doi:10.1088/0143-0807/28/5/016 [2] Fan, J.: He’s frequency–amplitude formulation for the Duffing harmonic oscillator, Computers and mathematics with applications 58, 2473-2476 (2009) · Zbl 1189.65163 · doi:10.1016/j.camwa.2009.03.049 [3] He, J. H.: Comment on ’he’s frequency formulation for nonlinear oscillators’, European journal of physics 29, L19-L22 (2008) [4] Zhao, Ling: He’s frequency–amplitude formulation for nonlinear oscillators with an irrational force, Computers and mathematics with applications 58, 2477-2479 (2009) · Zbl 1189.65185 · doi:10.1016/j.camwa.2009.03.041 [5] Zhang, Hui-Li: Application of he’s amplitude–frequency formulation to a nonlinear oscillator with discontinuity, Computers and mathematics with applications 58, 2197-2198 (2009) · Zbl 1189.65181 · doi:10.1016/j.camwa.2009.03.018 [6] Cai, Xu-Chu; Wu, Wen-Ying: He’s frequency formulation for the relativistic harmonic oscillator, Computers and mathematics with applications 58, 2358-2359 (2009) · Zbl 1189.65162 · doi:10.1016/j.camwa.2009.03.024 [7] Zhang, Ya-Nan; Xu, Fei; Deng, Ling-Ling: Exact solution for nonlinear Schrödinger equation by he’s frequency formulation, Computers and mathematics with applications 58, 2449-2451 (2009) · Zbl 1189.81064 · doi:10.1016/j.camwa.2009.03.015 [8] He, J. H.: Iteration perturbation method for strongly nonlinear oscillations, Journal of vibration and control 7, 631 (2001) · Zbl 1015.70019 · doi:10.1177/107754630100700501 [9] Özis, Turgut; Yıldırım, Ahmet: Generating the periodic solutions for forcing van der Pol oscillators by the iteration perturbation method, Nonlinear analysis. Real world applications 10, 1984-1989 (2009) · Zbl 1163.34355 · doi:10.1016/j.nonrwa.2008.03.005 [10] He, Ji-Huan: The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied mathematics and computation 151, 287-292 (2004) · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2 [11] He, Ji-Huan: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999) [12] He, Ji-Huan: Homotopy perturbation method: a new nonlinear analytical technique, Applied mathematics and computation 135, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5 [13] He, Ji-Huan: A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International journal of non-linear mechanics 35, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7 [14] He, Ji-Huan: An elementary introduction to the homotopy perturbation method, Computers and mathematics with applications 57, 410-412 (2009) · Zbl 1165.65374 · doi:10.1016/j.camwa.2008.06.003 [15] Zhang, Hui-Li: Periodic solutions for some strongly nonlinear oscillations by he’s energy balance method, Computers and mathematics with applications 58, 2480-2485 (2009) · Zbl 1189.65182 · doi:10.1016/j.camwa.2009.03.068 [16] Mehdipour, I.; Ganji, D. D.; Mozaffari, M.: Application of the energy balance method to nonlinear vibrating equations, Current applied physics 10, 104-112 (2010) [17] Xu, Lan: He’s parameter-expanding methods for strongly nonlinear oscillators, Journal of computational and applied mathematics 207, 148-154 (2007) · Zbl 1120.65084 · doi:10.1016/j.cam.2006.07.020 [18] Darvishi, M. T.; Karami, A.; Shin, Byeong-Chun: Application of he’s parameter-expansion method for oscillators with smooth odd nonlinearities, Physics letters A 372, 5381-5384 (2008) · Zbl 1223.70065 · doi:10.1016/j.physleta.2008.06.058 [19] Tao, Zhao-Ling: Frequency–amplitude relationship of nonlinear oscillators by he’s parameter-expanding method, Chaos, solitons and fractals 41, 642-645 (2009) · Zbl 1198.65155 · doi:10.1016/j.chaos.2008.02.036 [20] Zeng, De-Qiang: Nonlinear oscillator with discontinuity by the MAX–MIN approach, Chaos, solitons and fractals 42, 2885-2889 (2009) · Zbl 1198.65159 · doi:10.1016/j.chaos.2009.04.029 [21] Liu, J. -F.: He’s variational approach for nonlinear oscillators with high nonlinearity, Computers and mathematics with applications 58, 2423-2426 (2009) · Zbl 1189.65167 · doi:10.1016/j.camwa.2009.03.074 [22] Younesian, D.; Esmailzadeh, E.; Sedaghati, R.: Existence of periodic solutions for the generalized form of Mathieu equation, Nonlinear dynamics 39, 335-348 (2005) · Zbl 1097.70017 · doi:10.1007/s11071-005-4338-y [23] Younesian, D.; Esmailzadeh, E.; Sedaghati, R.: Asymptotic solutions and stability analysis for generalized non-homogeneous Mathieu equation, Communications in nonlinear science and numerical simulation 12, 58-71 (2007) · Zbl 1111.34036 · doi:10.1016/j.cnsns.2006.01.005 [24] Nayfeh, A. H.; Mook, D. T.: Nonlinear oscillations, (1979) · Zbl 0418.70001 [25] He, J. H.: Solution of nonlinear equations by an ancient chinese algorithm, Applied mathematics and computation 151, No. 1, 293-297 (2004) · Zbl 1049.65039 · doi:10.1016/S0096-3003(03)00348-5 [26] He, Ji-Huan: Preliminary report on the energy balance for nonlinear oscillations, Mechanics research communications 29, 107-111 (2002) · Zbl 1048.70011 · doi:10.1016/S0093-6413(02)00237-9 [27] He, J. -H.; Wu, G. -C.; Austin, F.: The variational iteration method which should be followed, Nonlinear science letters A 1, 1-30 (2010) [28] He, Ji-Huan: An improved amplitude–frequency formulation for nonlinear oscillators, International journal of nonlinear sciences and numerical simulation 9, No. 2, 211-212 (2008)