一类二自由度分数阶参激系统的混沌阈值研究

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振动与冲击 ›› 2025, Vol. 44 ›› Issue (15) : 102-115.

振动理论与交叉研究

一类二自由度分数阶参激系统的混沌阈值研究

魏想1，2，温少芳*2，3，申永军3

作者信息 +

Chaos threshold of a class of 2DOF fractional order parametrically excited systems

WEI Xiang1,2, WEN Shaofang*2,3, SHEN Yongjun3

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文章历史 +

摘要

研究了二自由度分数阶参激系统在谐波激励下的分岔与混沌行为。利用Melnikov方法分析了二自由度分数阶参激系统产生Smale马蹄形意义下分岔与混沌的必要条件，得到了其解析结果。将得到的解析结果与数值迭代算法的结果进行对比，结果表明，两种算法得到的混沌阈值曲线的变化趋势一致，吻合度较高，证实了得到的混沌阈值解析曲线的准确性。利用最大Lyapunov指数图、相图、时程图、频谱图以及Poincare截面图分析了一些典型点的动力学响应特性，得到了系统进入混沌运动状态的途径，详细说明了基于Melnikov方法求得的系统混沌边界条件的合理性。最后分析了系统各参数对混沌阈值曲线的影响，研究表明：增大分数阶系数、参激系数、线性阻尼系数和耦合阻尼系数可以抑制混沌运动的发生，而增大线性刚度系数、非线性刚度系数以及分数阶阶次有时会增加系统混沌发生的可能性，以上结论可为实际应用提供分析参考，对此类系统的设计和控制具有一定指导价值。

Abstract

The bifurcation and chaotic behaviour of a two-degree-of-freedom fractional-order parametrically excited system under harmonic excitation are studied. The Melnikov method is used to analyse the necessary conditions for bifurcation and chaos in the sense of Smale horseshoe in a two-degree-of-freedom fractional-order parametric excited system, and its analytical results are obtained. The obtained analytical results are compared with the results of the numerical iterative algorithm. The results show that the change trend of the chaotic threshold curves obtained by the two algorithms is consistent and the degree of consistency is high, which confirms the accuracy of the obtained chaotic threshold analytical curve. The dynamic response characteristics of some typical points are analyzed using the maximum Lyapunov exponent diagram, phase diagram, time history diagram, frequency diagram and Poincare section diagram, and the path for the system to enter a chaotic motion state is obtained. The rationality of the chaotic boundary conditions of the system calculated based on the Melnikov method is explained in detail. Finally, the influence of various system parameters on the chaos threshold curve is analyzed. The research shows that increasing the fractional-order coefficient, parametric excitation coefficient, linear damping coefficient and coupling damping coefficient can suppress the occurrence of chaos; while increasing the linear stiffness coefficient, nonlinear stiffness coefficient and the fractional-order number sometimes increases the possibility of chaos in the system. The above conclusions can provide analysis references for practical applications and have certain guiding values for the design and control of such systems.

导出引用

魏想1, 2, 温少芳2, 3, 申永军3. 一类二自由度分数阶参激系统的混沌阈值研究[J]. 振动与冲击, 2025, 44(15): 102-115

WEI Xiang1, 2, WEN Shaofang2, 3, SHEN Yongjun3. Chaos threshold of a class of 2DOF fractional order parametrically excited systems[J]. Journal of Vibration and Shock, 2025, 44(15): 102-115

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