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Zbl 1193.65152
Younesian, Davood; Askari, Hassan; Saadatnia, Zia; Kalamiyazdi, Mohammad
Frequency analysis of strongly nonlinear generalized Duffing oscillators using he's frequency-amplitude formulation and He's energy balance method. (English)
[J] Comput. Math. Appl. 59, No. 9, 3222-3228 (2010). ISSN 0898-1221

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 $$x''+x+\alpha_3 x^3+\alpha_5 x^5+\alpha_7 x^7+\dots+\alpha_n x^n=0.$$ 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 2000:: *65L99 Numerical methods for ODE
34C15 Nonlinear oscillations of solutions of ODE

Keywords: nonlinear vibration; generalized Duffing equation; he's frequency-amplitude formulation; He's energy balance method

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