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多稳态动力系统中随机共振的研究进展

靳艳飞 许鹏飞 李永歌 马晋忠 许勇

靳艳飞, 许鹏飞, 李永歌, 马晋忠, 许勇. 多稳态动力系统中随机共振的研究进展. 力学进展, 2023, 53(2): 357-394 doi: 10.6052/1000-0992-22-047
引用本文: 靳艳飞, 许鹏飞, 李永歌, 马晋忠, 许勇. 多稳态动力系统中随机共振的研究进展. 力学进展, 2023, 53(2): 357-394 doi: 10.6052/1000-0992-22-047
Jin Y F, Xu P F, Li Y G, Ma J Z, Xu Y. Stochastic resonance of multi-stable dynamical systems: A review. Advances in Mechanics, 2023, 53(2): 357-394 doi: 10.6052/1000-0992-22-047
Citation: Jin Y F, Xu P F, Li Y G, Ma J Z, Xu Y. Stochastic resonance of multi-stable dynamical systems: A review. Advances in Mechanics, 2023, 53(2): 357-394 doi: 10.6052/1000-0992-22-047

多稳态动力系统中随机共振的研究进展

doi: 10.6052/1000-0992-22-047
基金项目: 国家自然科学基金资助项目(12072025, 12202253, 12272296, 12120101002), 重庆市自然科学基金项目(cstc2021jcyj-msxmX0738), 广东省基础与应用基础研究基金用基础研究基金(2023A1515012329).
详细信息
    作者简介:

    靳艳飞, 北京理工大学宇航学院教授, 博士生导师, 全国徐芝纶力学优秀教师. 主要从事非线性随机动力学与控制、随机振动能量采集、不确定性多体系统动力学等方面的研究, 已发表学术论文80余篇, 出版专著1部. 主持国家自然科学基金面上项目, 参与完成国家自然科学基金重点项目等, 以第一完成人获中国振动工程学会科学技术二等奖. 现任中国振动工程学会副秘书长等, 担任《Journal of Vibration Testing and System Dynamics》编委、《应用力学学报》青年编委等

    许勇, 西北工业大学教授, 博士生导师, 国家杰出青年科学基金获得者, 德国洪堡学者. 从事飞行器结构与强度、随机与非线性动力学、振动与控制等研究, 发表学术论文100余篇. 主持国家自然科学基金重点国际合作研究项目和国家重点研发计划课题等项目, 以第一完成人分别获陕西省科学技术一等奖、教育部自然科学二等奖和陕西省教学成果特等奖. 现任陕西省振动工程学会理事长、中国振动工程学会理事等

    通讯作者:

    hsux3@nwpu.edu.cn

  • 中图分类号: O324

Stochastic resonance of multi-stable dynamical systems: A review

More Information
  • 摘要: 非线性随机动力学是力学、数学、工程等多个领域关注的热点, 在航空航天、机械工程、生物生态等领域有广泛的应用. 多稳态动力系统作为其最重要的研究对象, 在随机扰动下具有丰富的动力学行为, 如随机分岔、随机共振等, 尤其是随机共振, 已经被应用于机械故障诊断、微弱信号检测和振动能量俘获等工程实际问题中. 本文主要综述了多稳态动力系统中的随机共振理论、方法及工程应用. 首先, 通过几类典型的非线性随机动力学系统, 介绍了随机共振的经典理论和度量指标; 其次, 重点阐述了多稳态动力学系统, 尤其是三稳态和周期势系统, 在各类噪声激励下的随机共振现象, 分析了其诱发机理、演化规律和研究方法; 最后, 介绍了多稳态动力系统中随机共振的几类应用实例, 并进一步给出了随机共振当前面临的难题和未来的发展趋势等开放性问题.

     

  • 图  1  双稳态势函数示意图

    图  2  对称双稳态系统(1)的随机共振示意图(Gammaitoni et al. 1998)

    图  3  不同噪声强度下模型(7)的动力学响应, 其中$a = 1.05$, $\varepsilon = 0.01$, $(x(0),y(0)) = (0,0)$ (Pikovsky & Kurths 1997)

    图  4  势函数(11)随刚度系数的变化情况($ b = - 0.2 $): (a) $ c = 0.52 $, (b) $ {R_T} $$ {R_M} $分别代表三稳态和单稳态区域

    图  5  (a)周期势函数($ a = 1 $), (b)离散的多稳态过程(靳艳飞和许鹏飞 2021)

    图  6  耦合势函数(15)的平衡点在$ x - y $相平面上的分布, 其中$ s $, $ u $$ o $分别表示稳定结点、鞍点和不稳定结点(Xu & Jin 2018)

    图  7  最大李雅普诺夫指数作为耦合强度$ c $的函数曲线, 其中 $ {{{\varepsilon _y}} \mathord{\left/ {\vphantom {{{\varepsilon _y}} {{\varepsilon _x}}}} \right. } {{\varepsilon _x}}} = 1 $, $ {{{\omega _y}} \mathord{\left/ {\vphantom {{{\omega _y}} {{\omega _x}}}} \right. } {{\omega _x}}} = {1 \mathord{\left/ {\vphantom {1 {12}}} \right. } {12}} $ (Xu & Jin 2018)

    图  8  不同热噪声强度下系统平均输入能量相对初始位置出现的相外状态和相内状态(Liu & Jin 2013)

    图  9  Lévy噪声的概率密度函数随不同参数$\alpha $$ \beta $的变化情况

    图  10  功率谱放大因子作为噪声强度的函数随不同Lévy噪声参数$ \alpha $$ \beta $的变换情况(Liu & Kang 2018)

    图  11  多值噪声的时间历程图. (a)三值噪声($ D = 0.5 $, $ \tau = 0.42 $), (b)二值噪声($ D = 1 $, $ \tau = 0.5 $)

    图  12  非高斯$ q $噪声的时间历程图($ D = 0.2 $, $ \tau = 0.1 $, $ q = 1.05 $)

    图  13  三稳态势函数在一个信号周期$ T $内的变化

    图  14  三稳态系统跃迁示意图(Nicolis 2012)

    图  15  三稳态随机共振悬臂梁结构示意图(Lu et al. 2013)

    图  16  系统在粗糙三势阱中的跃迁示意图(Li et al. 2016)

    图  17  信噪比($ {\rm{SNR}} $)作为阻尼系数$ {\gamma _0} $的函数随五次非线性刚度系数$ {\kappa _5} $的变化曲线(Xu et al. 2019)

    图  18  记忆时间$ {\tau _c} $对噪声诱导共振的影响. (a)特征相关时间和(b)功率谱放大因子(Xu & Jin 2020)

    图  19  美国凯斯西储大学(CWRU)的滚动轴承故障实验平台

    图  20  轴承内圈故障信号检测. (a)和(b)原始信号和频谱, (c)和(d)随机共振条件下输出信号和频谱(Xu & Jin 2020)

    图  21  海洋环境下目标探测. (a)原始船舶辐射声信号频谱, (b) 基于匹配随机共振检测理论输出的频谱

    图  22  旋转环境下三稳态能量采集装置示意图. (a)安装在车轮上的三稳态能量收集装置, (b)三稳态压电能量采集器(THE)原理图, (c)与THE相连的标准整流电路(Zhang et al. 2022)

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  • 收稿日期:  2022-11-28
  • 录用日期:  2023-02-27
  • 网络出版日期:  2023-03-11
  • 刊出日期:  2023-06-25

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