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基于非厄米趋肤效应的离散系统非线性振动控制

NONLINEAR VIBRATION CONTROL OF DISCRETE DYNAMICS SYSTEMS BASED ON NON-HERMITIAN SKIN EFFECT

  • 摘要: 非线性动力系统低频宽带振动控制是动力学与控制领域的难题. 本文针对多自由度非线性弹簧振子系统,通过引入独立可控的单向作用力构建了含趋肤效应的离散非厄米系统,基于摄动法研究了系统的非线性复数域色散关系,揭示了系统传输波非互易特性、非线性振动能量束缚模式以及趋肤效应对非线性振动响应的调控规律. 结果表明,单向作用力引入了对称性破缺的色散关系,使得非线性弹簧振子系统存在非互易波传输行为,其振动能量沿特定方向被束缚,系统振动呈现出趋肤效应特征;非厄米系统设计不仅能实现弱/强非线性弹簧振子系统的低频宽带振动控制,还可以实现系统周期极限环振动、准周期振动和混沌振动等模式的调控和切换. 本文提出的非厄米系统设计方法为非线性动力学系统低频宽带振动控制及振动模式调控提供了参考.

     

    Abstract: Nonlinear dynamical systems exhibit large wavelengths, broad frequency bands, and a significant degree of nonlinearity in their low-frequency broadband vibration response. Scientific challenges such as dimensional constraints, harmonic frequencies, multi-frequency components, and full-band vibration control have been long-standing technical issues that remain unresolved. This paper presents a multi-degree-of-freedom nonlinear spring-mass system, wherein a discrete non-Hermitian system exhibiting a skin effect is created with the application of independently adjustable unidirectional forces. The nonlinear dispersion relationship within the complex domain of the system is examined using perturbation methods. This elucidates nonreciprocal wave transmission properties, energy confinement zones and modes of nonlinear oscillations, as well as the influence of the skin effect on nonlinear vibrational responses. The results show that the unidirectional force breaks the symmetry of the dispersion relation, leading to nonreciprocal wave transmission that propagates in only one direction. In this case, the vibration energy is localized to a specific direction, and the skin effect is apparent. The design of non-Hermitian systems enables low-frequency broadband vibration control in linear spring-mass systems and across weak, intermediate and strong nonlinearities. Additionally, it can be used to regulate and switch vibration modes, including periodic limit cycles, quasi-periodic vibrations, and chaotic vibrations. The non-Hermitian system design methodology presented in this article provides insights for low-frequency broadband vibration control and the modulation of vibration modes in complex nonlinear dynamical systems, offering a possibility for broadband control applications.

     

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