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三稳系统的动态响应及随机共振

赖志慧 冷永刚

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三稳系统的动态响应及随机共振

赖志慧, 冷永刚

Dynamic response and stochastic resonance of a tri-stable system

Lai Zhi-Hui, Leng Yong-Gang
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  • 以平衡点参数p, q构造出一类对称三稳势函数, 进而提出微弱信号和噪声共同驱动的三稳系统模型. 深入研究并总结参数p, q对势垒高度ΔU1, ΔU2及两势垒高度差的影响. 从定常输入的角度提出了系统稳态解曲线的概念, 并进一步研究低频谐波信号输入时系统的输出动态响应. 引入噪声, 三稳系统在合适的参数条件下实现随机共振, 从稳态解曲线的角度分析了噪声诱导的三稳系统随机共振机理. 最后研究了阻尼比k和平衡点参数p, q对系统随机共振的影响.
    Stochastic resonance (SR) describes a nonlinear phenomenon in nature, of which the essential ingredients are a nonlinear system, a weak signal, and a source of noise. Using the nonlinear system, the signal-to-noise ratio (SNR) of the output signal of the system will peak at a certain value of noise intensity under a synergistic action of input signal and noise. Besides the traditional Langevin equation, the new SR models such as monostable oscillators, chaotic systems, time-delay systems and bistable Duffing systems, can also produce SR phenomena. In this paper, a normalized symmetrical tri-stable potential function is constructed by using equilibrium parameters p and q, and a tri-stable system model simultaneously driven by weak signal and noise is further proposed. The tri-stable system model can be understood through a cantilever beam structure with three magnets, and deduced from the Brownian motion equation. We study in-depth and summarize the influences of parameters p and q on the potential barrier heights ΔU1, ΔU2 and their difference value. By analyzing the steady-state solution of the tri-stable system under invariable input, the concept of system steady-state solution curve (SSS curve) is proposed, and is used to further study the system dynamic response under low-frequency harmonic signal input. In these situations, the system response can be obtained by combining the steady-state solutions of the system following time t under a group of tempolabile inputs. Moreover, with the noise injection, the tri-stable system can realize SR under appropriate parameter condition, which can be demonstrated by the output amplitude curve and also the output SNR curve of the system against noise intensity. The mechanism of noise-induced SR of tri-stable system can be analyzed from the perspective of SSS curve. Finally, we further study the influence of tri-stable SR against system parameters. The value of damping ratio k affects the value of damping force acting on the Brownian particle, thus the tri-stable system needs noise with larger intensity to produce SR under a larger k. The values of equilibrium parameters p and q both affect the shape of the SSS curve, a larger p or a smaller q may result in larger-intensity noise for the system to produce SR.
    • 基金项目: 国家自然科学基金(批准号: 51275336)和天津市应用基础与前沿技术研究计划(批准号: 15JCZDJC32200)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51275336) and the Tianjin Research Program of Application Foundation and Advanced Technology, China (Grant No. 15JCZDJC32200).
    [1]

    Benzi R, Sutera A, Vulpiana 1981 Physica A 14 453

    [2]

    Bensi R, Parisi G, Srutera A 1982 Tellus 34 11

    [3]

    Nicolis C 1982 Tellus 34 1

    [4]

    Fauve S, Heslot F 1983 Phys. Lett. A 97 5

    [5]

    McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626

    [6]

    Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rew. Mod. Phys. 70 223

    [7]

    Fan J, Zhao W L, Zhang M L, Tan R H, Wang W Q 2014 Acta Phys. Sin. 63 110506 (in Chinese) [范剑, 赵文礼, 张明路, 檀润华, 王万强 2014 物理学报 63 110506]

    [8]

    Li Y B, Zhang B L, Liu Z X, Zhang Z Y 2014 Acta Phys. Sin. 63 160504 (in Chinese) [李一博, 张博林, 刘自鑫, 张震宇 2014 物理学报 63 160504]

    [9]

    Qin Y, Tao Y, He Y, Tang B P 2014 J. Sound Vib. 333 7386

    [10]

    Wang J, He Q B 2015 IEEE Trans. Instrum. Meas. 64 564

    [11]

    Stocks N G, Stein N D, McClintock P V E 1993 J. Phys. A 26 385

    [12]

    Gomes I, Mirasso C R, Toral R, Calvo O 2003 Physica A 327 115

    [13]

    Masoller C 2002 Phys. Rev. Lett. 88 034102

    [14]

    Gammaitoni L, Marchesoni F, Menichella-Saetta E, Santucci S 1989 Phys. Rev. Lett. 62 349

    [15]

    Lai Z H, Leng Y G, Fan S B 2013 Acta Phys. Sin. 62 070503 (in Chinese) [赖志慧, 冷永刚, 范胜波 2013 物理学报 62 070503]

    [16]

    Lu S L, He Q B, Zhang H B, Zhang S B, Kong F R 2013 Rev. Sci. Instrum. 84 026110

    [17]

    Li J M, Chen X F, He Z J 2013 J. Sound Vib. 332 5999

    [18]

    Zhang H Q, Xu Y, Xu W, Li X C 2012 Chaos 22 043130

    [19]

    Arathi S, Rajasekar S 2011 Phys. Scr. 84 065011

  • [1]

    Benzi R, Sutera A, Vulpiana 1981 Physica A 14 453

    [2]

    Bensi R, Parisi G, Srutera A 1982 Tellus 34 11

    [3]

    Nicolis C 1982 Tellus 34 1

    [4]

    Fauve S, Heslot F 1983 Phys. Lett. A 97 5

    [5]

    McNamara B, Wiesenfeld K, Roy R 1988 Phys. Rev. Lett. 60 2626

    [6]

    Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rew. Mod. Phys. 70 223

    [7]

    Fan J, Zhao W L, Zhang M L, Tan R H, Wang W Q 2014 Acta Phys. Sin. 63 110506 (in Chinese) [范剑, 赵文礼, 张明路, 檀润华, 王万强 2014 物理学报 63 110506]

    [8]

    Li Y B, Zhang B L, Liu Z X, Zhang Z Y 2014 Acta Phys. Sin. 63 160504 (in Chinese) [李一博, 张博林, 刘自鑫, 张震宇 2014 物理学报 63 160504]

    [9]

    Qin Y, Tao Y, He Y, Tang B P 2014 J. Sound Vib. 333 7386

    [10]

    Wang J, He Q B 2015 IEEE Trans. Instrum. Meas. 64 564

    [11]

    Stocks N G, Stein N D, McClintock P V E 1993 J. Phys. A 26 385

    [12]

    Gomes I, Mirasso C R, Toral R, Calvo O 2003 Physica A 327 115

    [13]

    Masoller C 2002 Phys. Rev. Lett. 88 034102

    [14]

    Gammaitoni L, Marchesoni F, Menichella-Saetta E, Santucci S 1989 Phys. Rev. Lett. 62 349

    [15]

    Lai Z H, Leng Y G, Fan S B 2013 Acta Phys. Sin. 62 070503 (in Chinese) [赖志慧, 冷永刚, 范胜波 2013 物理学报 62 070503]

    [16]

    Lu S L, He Q B, Zhang H B, Zhang S B, Kong F R 2013 Rev. Sci. Instrum. 84 026110

    [17]

    Li J M, Chen X F, He Z J 2013 J. Sound Vib. 332 5999

    [18]

    Zhang H Q, Xu Y, Xu W, Li X C 2012 Chaos 22 043130

    [19]

    Arathi S, Rajasekar S 2011 Phys. Scr. 84 065011

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出版历程
  • 收稿日期:  2015-04-22
  • 修回日期:  2015-05-18
  • 刊出日期:  2015-10-05

三稳系统的动态响应及随机共振

  • 1. 南昌大学机电工程学院, 南昌 330031;
  • 2. 天津大学机械工程学院, 天津 300072
    基金项目: 国家自然科学基金(批准号: 51275336)和天津市应用基础与前沿技术研究计划(批准号: 15JCZDJC32200)资助的课题.

摘要: 以平衡点参数p, q构造出一类对称三稳势函数, 进而提出微弱信号和噪声共同驱动的三稳系统模型. 深入研究并总结参数p, q对势垒高度ΔU1, ΔU2及两势垒高度差的影响. 从定常输入的角度提出了系统稳态解曲线的概念, 并进一步研究低频谐波信号输入时系统的输出动态响应. 引入噪声, 三稳系统在合适的参数条件下实现随机共振, 从稳态解曲线的角度分析了噪声诱导的三稳系统随机共振机理. 最后研究了阻尼比k和平衡点参数p, q对系统随机共振的影响.

English Abstract

参考文献 (19)

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