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Parameter-splitting perturbation method for the improved solutions to strongly nonlinear systems

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Abstract

A procedure named parameter-splitting perturbation method for improving the perturbation solutions to the forced vibrations of strongly nonlinear oscillators is proposed. The idea of the proposed procedure is presented in general first. After that, it is applied to optimize the solutions obtained by the multiple-scales method which is one of well-known perturbation methods. The harmonically forced Duffing oscillator, the harmonically forced oscillator with both nonlinear restoring force and nonlinear inertial force, the harmonically forced purely nonlinear oscillator and harmonically forced two-degree-of-freedom system with cubic nonlinearity are analyzed in various cases to show the advantages of the proposed method. The ratio of nonlinear stiffness coefficient to linear stiffness coefficient is chosen to be larger than one to highlight the validity of the propose method when dealing with strongly nonlinear oscillators. The validity of the proposed procedure is examined by comparing the frequency–response curves obtained by the proposed method, conventional multiple-scales method and numerical continuation method. Moreover, the errors corresponding to the results obtained by multiple-scales method are compared with those obtained by the proposed method to examine the performance of the proposed method. The results show that the proposed method can give much improved solutions in comparison with those obtained by multiple-scales method.

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Acknowledgements

The results presented in this paper were obtained under the supports of the Science and Technology Development Fund of Macau (Grant No. 042/2017/A1) and the Research Committee of University of Macau (Grant No. MYRG2018-00116-FST).

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  1. Department of Civil and Environmental Engineering, University of Macau, Taipa, Macau, China

    Hai-En Du, Guo-Kang Er & Vai Pan Iu

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Du, HE., Er, GK. & Iu, V.P. Parameter-splitting perturbation method for the improved solutions to strongly nonlinear systems. Nonlinear Dyn 96, 1847–1863 (2019). https://doi.org/10.1007/s11071-019-04887-w

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