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Updated numerical integration method for stability calculation of Mathieu equation with various time delays

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An Erratum to this article was published on 16 January 2017

Abstract

The determination of the stability of systems with time delays is of high importance in many industrial and research applications. In this study, we improve the numerical integration method (NIM) by using the Lagrange form interpolating polynomial to approximate the delayed terms and construct a periodic discrete dynamical map for the damped Mathieu equation with time delays. Hence, we can obtain the Floquet transition matrices without matrix multiplication, which can reduce calculation time when the matrix inversed has a small bandwidth according to the computation based on MATLAB. To compare the calculation efficiency and computational accuracy between the updated and original methods, we compare the stability charts calculated by using three NIM methods for a damped Mathieu equation with multiple delays. To further confirm the efficiency of the presented method, we calculate the stability of the Mathieu equation with distributed delays and time-periodic delays, and then we compare the computational accuracy and calculation time with those of the semi-discretization method.

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Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 51305151, 51535004, 51025518).

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Authors and Affiliations

  1. State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xiao Jian Zhang, Cai Hua Xiong & Han Ding

  2. State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Ye Ding

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  1. Xiao Jian Zhang
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  2. Cai Hua Xiong
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  3. Ye Ding
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  4. Han Ding
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Correspondence to Cai Hua Xiong.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s11071-017-3331-6.

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Zhang, X.J., Xiong, C.H., Ding, Y. et al. Updated numerical integration method for stability calculation of Mathieu equation with various time delays. Nonlinear Dyn 87, 2077–2095 (2017). https://doi.org/10.1007/s11071-016-3096-3

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  • DOI: https://doi.org/10.1007/s11071-016-3096-3

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