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Improving convergence of incremental harmonic balance method using homotopy analysis method

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Abstract

We have deduced an iteration scheme in the incremental harmonic balance (IHB) method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.

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Authors and Affiliations

  1. State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, 800 Dongchuan Road, 200240, Shanghai, China

    Yanmao Chen

  2. Department of Mechanics, Sun Yat-sen University, 510275, Guangzhou, China

    Jike Liu

Authors
  1. Yanmao Chen
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  2. Jike Liu
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Correspondence to Jike Liu.

Additional information

The project supported by the National Natural Science Foundation of China (10772202), Doctoral Program Foundation of Ministry of Education of China (20050558032), and Guangdong Province Natural Science Foundation (07003680, 05003295).

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Chen, Y., Liu, J. Improving convergence of incremental harmonic balance method using homotopy analysis method. Acta Mech Sin 25, 707–712 (2009). https://doi.org/10.1007/s10409-009-0256-4

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  • DOI: https://doi.org/10.1007/s10409-009-0256-4

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