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| Calculation of the response of nonlinear vibration system based on the Caputo fractional derivative |
| Received:May 18, 2017 Revised:July 08, 2017 |
| View Full Text View/Add Comment Download reader |
| DOI：10.7511/jslx20170518002 |
| KeyWord:fractional derivative algorithm numerical iterative scheme derivative order initial values |
| Author | Institution |
| 李亚杰 |
天津大学 机械工程学院 力学系, 天津 ;天津大学 天津市非线性动力学与混沌控制重点实验室, 天津 |
| 吴志强 |
天津大学 机械工程学院 力学系, 天津 ;天津大学 天津市非线性动力学与混沌控制重点实验室, 天津 |
| 章国齐 |
天津大学 机械工程学院 力学系, 天津 ;天津大学 天津市非线性动力学与混沌控制重点实验室, 天津 |
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| 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 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. |
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