Dynamical analysis of fractional-order Mathieu equation
Dynamical analysis of fractional-order Mathieu equation
复制标题
分数阶Mathieu方程的动力学分析
作者：
Shaofang Wen;Yongjun Shen;Xianghong Li;Shaopu Yang;Haijun Xing
利用Lindstedt-Poincare方法和多尺度方法研究了具有分数阶导数的Mathieu方程的动力学性质。对于常刚度δ0=n2（n = 0，1，2，.），得到了稳定边界和在这些边界上的周期解.分数阶参数对稳定边界和相应周期解的影响，包括分数阶系数和分数阶数，用等效线性阻尼系数（ELDC）和等效线性刚度系数（ELSC）表征。将近似解析解与数值方法得到的边界上的过渡曲线进行比较，验证了解析解的正确性和令人满意的精度。以下分析集中于分数参数对位于δ-e平面内的稳定性边界的影响。研究发现，分数阶数p的增加会使ELDC增大，ELSC减小，从而导致稳定边界同时向右和向上移动.分数系数K1的增大会使ELDC和ELSC增大，从而使缓和曲线同时向左和向上移动。这些结果对此类系统的设计、分析和控制都有很大的帮助，对类似分数阶系统的设计也有一定的参考价值。
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-e 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.
登录
查看更多内容
具有硬化刚度和分数阶导数的 Duffing 振子的稳态响应
影响因子：
3.2
作者：
Chen, Lincong;Chen, Lincong;Wang, Weihua;Wang, Weihua;Li, Zhongshen;Li, Zhongshen;Zhu, Weiqiu;Zhu, Weiqiu
通讯作者：
Zhu, Weiqiu
DOI：
--
发表时间：
2012
期刊：
Journal of Shanghai University
影响因子：
--
作者：
Lin Chang-pin
通讯作者：
Lin Chang-pin
DOI：
10.1115/detc2010-28068
发表时间：
2010-11
期刊：
--
影响因子：
--
作者：
R. Rand;S. Sah;M. K. Suchorsky
通讯作者：
R. Rand;S. Sah;M. K. Suchorsky
影响因子：
7.8
作者：
Z. Ge;Chang-Xian Yi
通讯作者：
Z. Ge;Chang-Xian Yi
DOI：
--
发表时间：
2013
期刊：
Communication on Applied Mathematics and Computation
影响因子：
--
作者：
Lin Chang-pin
通讯作者：
Lin Chang-pin