基于Caputo导数的分数阶非线性振动系统响应计算

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导数和Grunwald-Letnikov导数间的关系，推导计算了Caputo导数的一般数值迭代格式。本文方法不要求状态方程中各分数阶导数阶数相等，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，比较Caputo和GL两种算法所得结果，说明了用GL算法求解存在的问题。

Calculation of the response of nonlinear vibration system based on the Caputo fractional derivative

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 derived based on the relationship between Caputo derivative and Riemann-Liouville derivative and Grunwald-Letnikov derivative.This method does not requires the orders of the various fractional order derivatives in the state equations to be equal,which not only can reduce the restriction on the orders of the fractional derivatives in existing algorithms,but also can be extended to multi-degree-of-freedom systems.Then,we select some examples which have the analytical solutions to verify the correctness of the algorithm presented in this paper.Finally,a fractional order Duffing oscillator with multiple basins of attraction is taken as an example,and then comparison of the results obtained by the Caputo and GL algorithms respectively shows the existing problems to solve the Caputo derivative with GL algorithm.