Dynamical analysis of fractional-order Mathieu equation

Shaofang Wen1 , Yongjun Shen2 , Xianghong Li3 , Shaopu Yang4 , Haijun Xing5

1Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang, China

2, 4, 5Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang, China

3Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China

2Corresponding author

Journal of Vibroengineering, Vol. 17, Issue 5, 2015, p. 2696-2709.
Received 1 February 2015; received in revised form 25 March 2015; accepted 11 April 2015; published 15 August 2015

Copyright © 2015 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

The dynamical characteristics of Mathieu equation with fractional-order derivative is analytically studied by the Lindstedt-Poincare method and the multiple-scale method. The stability boundaries and the corresponding periodic solutions on these boundaries for the constant stiffness δ0=n2 (n = 0, 1, 2, …), are analytically obtained. The effects of the fractional-order parameters on the stability boundaries and the corresponding periodic solutions, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient (ELDC) and the equivalent linear stiffness coefficient (ELSC). The comparisons between the transition curves on the boundaries obtained by the approximate analytical solution and the numerical method verify the correctness and satisfactory precision of the analytical solution. The following analysis is focused on the effects of the fractional parameters on the stability boundaries located in the δ-ε plane. It is found that the increase of the fractional order p could make the ELDC larger and ELSC smaller, which could result into the rightwards and upwards moving of the stability boundaries simultaneously. It could also be concluded the increase of the fractional coefficient K1 would make the ELDC and ELSC larger, which could move the transition curves to the left and upwards at the same time. These results are very helpful to design, analyze or control this kind of system, and could present beneficial reference to the similar fractional-order system.

Keywords: fractional-order derivative, Mathieu equation, Lindstedt-Poincare method, multiple-scale method, stability boundaries.

Acknowledgements

The authors are grateful to the support by National Natural Science Foundation of China (No. 11372198), the Program for New Century Excellent Talents in University (NCET-11-0936), the Cultivation Plan for Innovation Team and Leading Talent in Colleges and Universities of Hebei Province (LJRC018), the Program for Advanced Talent in the Universities of Hebei Province (GCC2014053), and the Program for Advanced Talent in Hebei (A201401001).

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