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The calculation of the response of nonlinear vibration system based on the caputo fractional derivative

DOI：

 作者 单位 E-mail 李亚杰 天津大学 yajieli06@163.com 吴志强 天津大学 章国齐 天津大学

摘 要：本文研究了含分数阶Caputo导数的非线性振动系统响应的数值计算方法。首先，由Caputo分数阶导数算子的叠加关系，得到了含分数阶导数项非线性振动系统状态方程的标准形式。其次，基于Caputo导数与Riemann-Liouville导数、Grunwald-Letnikov导数间的关系推导计算Caputo导数的一般数值迭代格式。此方法不要求状态方程中各分数阶导数阶数是相等的，弱化了已有算法中对分数阶导数阶数的限制，并可推广到多自由度的情形。随后，选择若干有解析解的算例验证了本文所得计算方法的正确性。最后，以多吸引子共存的分数阶Duffing振子系统为例，通过对Caputo、GL两种算法所得结果进行比较，说明了用GL算法求解存在的问题。

Abstract：In this paper, the method of calculating the response of a nonlinear vibration system with fractional order Caputo derivative is studied. Firstly, by the superposition relation of the fractional operator of Caputo derivative, we obtain the standard form of the state equation of nonlinear vibration system with fractional derivative. Secondly, the general numerical iterative scheme of Caputo derivative is deduced based on the relationship between Caputo derivative and Riemann-Liouville derivative and Grunwald-Letnikov derivative. This method does not demand the orders of the fractional order derivative of the state equations to be equal, which not only can weaken the restriction of the orders of the fractional derivative in existing algorithms, but also can be extended to the case of multi-degree of freedom. Then, we select some examples which have the analytical solutions to verify the correctness of the algorithm obtained in this paper. Finally, a fractional order Duffing oscillator system with multi basins of attraction is taken as an example, and then by comparing the results obtained by the Caputo and GL algorithms respectively, it shows the existing problems to solve the Caputo derivative with GL algorithm.
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