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在受二进制非周期信号和周期方波信号激励的分数阶双稳系统中，研究了非周期振动共振问题，用于微弱非周期信号的检测和增强.当非周期信号脉宽较大时，系统为小参数，通过调节周期方波信号的幅值，能够实现非周期振动共振.当非周期信号脉宽较小时，分别通过变尺度法和二次采样法实现了非周期振动共振.使用变尺度法，得到的大参数等价系统能够匹配任意小的非周期信号脉宽，其中变尺度系数是该方法在使用过程中需要选择的关键参数.使用二次采样法，二次采样后得到的非周期信号具有较大的脉宽，能够匹配原先的小参数系统，其中二次采样频率比是该方法使用过程中的关键参数.这两种方法虽然实现非周期振动共振的物理过程不同，但能够达到相同的效果.系统阶数对振动共振产生影响，随着阶数的增大，发生最佳振动共振时所需要的辅助信号幅值变大，同时系统输出的最佳时间序列与输入非周期信号之间的相似性增强.Aperiodic signal is widely used in different engineering fields.It is important to detect or enhance a weak aperiodic signal.In this work,we investigate the aperiodic vibrational resonance (AVR) in a fractional-order bistable system excited by an aperiodic binary signal and a square waveform signal simultaneously.The weak aperiodic binary signal is the characteristic signal which usually carries the useful information.The square waveform signal is the auxiliary signal which is used to induce the AVR.By tuning the amplitude of the auxiliary signal,the AVR may occur and the aperiodic binary signal is enhanced.The occurrence of the AVR is measured by the cross-correlated coefficient between the input aperiodic binary signal and the output time series.When the cross-correlated coefficient achieves a large enough value, the AVR may occur and the weak aperiodic signal is enhanced excellently by the auxiliary signal.If the aperiodic binary signal has large pulse width and the system has small parameters (usually on the order of 1),the AVR can be realized by tuning the amplitude of the square waveform.If the aperiodic binary signal has small pulse width,the AVR cannot be realized in the system with small parameters directly.For this case,we realize the AVR by the re-scaled method and the twice sampling method separately.By the re-scaled method,through a scale transformation,the equivalent system with large system parameters can match the input characteristic signal with arbitrary small pulse width.When the re-scaled method is used,the scale parameter is a key factor.By the twice sampling method,the reconstructed characteristic signal after the twice sampling has a large pulse width.Then,it can match the original system with small system parameters.When the twice sampling method is used,the ratio of the twice sampling frequency to the first sampling frequency is a key factor.Although these two methods have different physical processes,they can achieve the same goal. The AVR also depends on the fractional-order value closely.Specifically,with the increase of the fractional-order,the resonance region in the cross-correlated coefficient curve turns wider.Moreover,the amplitude of the square waveform signal which induces the optimal AVR to turn larger.Simultaneously,the similarity between the optimal output and the input binary aperiodic signal is enhanced.The method and the results of this paper not only can be used to enhance the weak aperiodic binary signal but also have a certain reference value in processing other kinds of aperiodic signals, such as the linear or nonlinear frequency modulated signal,etc.Furthermore,the results in this paper also present rich dynamical behaviors of a fractional-order system and may provide reference value in the study of fractional-order systems.
[1] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[2] Landa P S, McClintock P V 2000 J. Phys. A: Math. Gen. 33 L433
[3] Collins J J, Chow C C, Capela A C, Imhoff T T 1996 Phys. Rev. E 54 5575
[4] Chizhevsky V N, Giacomelli G 2008 Phys. Rev. E 77 051126
[5] Yang J H, Sanjuán M A F, Liu H G, Litak G, Li X 2016 Commun. Nonlinear Sci. Numer. Simulat. 41 104
[6] Yang J H 2017 Bifurcation and Resonance in Fractional-order Systems (Beijing: Science Press) (in Chinese) [杨建华2017 分数阶系统的分岔与共振 (北京: 科学出版社)]
[7] Monje C A, Chen Y Q, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer) p11
[8] Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[9] Thomsen J J 2003 Vibrations and Stability: Advanced theory, Analysis, and Tools (Berlin: Springer-Verlag) pp287-334
[10] Thomsen J J 2002 J. Sound Vib. 253 807
[11] Balachandran B, Magrab E B 2008 Vibrations (Australia: Cengage Learning) pp210-212
[12] Hu H Y 2004 Foundation of Mechanical Vibration (Harbin: Harbin Institute of Technology Press) p26 (in Chinese) [胡海岩 2004 机械振动基础 (哈尔滨: 哈尔滨工业大学出版社) p26]
[13] Palm W J 2005 System Dynamics (2nd Ed.) (New York: McGraw-Hill Higher Education) p498
[14] Ogata K 2004 System Dynamics (4th Ed.) (New Jersey: Prentice Hall) pp384-388
[15] Liu X, Liu H, Yang J, Litak G., Cheng G, Han S 2017 Mech. Sys. Signal Pr. 96 58
[16] Huang D, Yang J, Zhang J, Liu H 2017 P. I. Mech. Eng. C-J. Mec. doi:0954406217719924
[17] Yang J H, Sanjuán M A F, Liu H G 2017 J. Comput. Nonlin. Dyn. 12 051011
[18] Liu H G, Liu X L, Yang J H, Sanjuán M A F, Cheng G 2017 Nonlinear Dynam. 89 2621
[19] Magin R, Ortigueira M D, Podlubny I, Trujillo J 2011 Signal Process. 91 350
[20] Leng Y G, Wang T Y 2003 Acta Phys. Sin. 52 2432 (in Chinese) [冷永刚, 王太勇 2003 物理学报 52 2432]
[21] Leng Y G, Wang T Y, Guo Y, Xu Y G, Fan S B 2007 Mech. Syst. Signal Pr. 21 138
[22] Li Q, Wang T, Leng Y, Wang W, Wang G 2007 Mech. Syst. Signal Pr. 21 2267
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[1] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[2] Landa P S, McClintock P V 2000 J. Phys. A: Math. Gen. 33 L433
[3] Collins J J, Chow C C, Capela A C, Imhoff T T 1996 Phys. Rev. E 54 5575
[4] Chizhevsky V N, Giacomelli G 2008 Phys. Rev. E 77 051126
[5] Yang J H, Sanjuán M A F, Liu H G, Litak G, Li X 2016 Commun. Nonlinear Sci. Numer. Simulat. 41 104
[6] Yang J H 2017 Bifurcation and Resonance in Fractional-order Systems (Beijing: Science Press) (in Chinese) [杨建华2017 分数阶系统的分岔与共振 (北京: 科学出版社)]
[7] Monje C A, Chen Y Q, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer) p11
[8] Blekhman I I 2000 Vibrational Mechanics (Singapore: World Scientific)
[9] Thomsen J J 2003 Vibrations and Stability: Advanced theory, Analysis, and Tools (Berlin: Springer-Verlag) pp287-334
[10] Thomsen J J 2002 J. Sound Vib. 253 807
[11] Balachandran B, Magrab E B 2008 Vibrations (Australia: Cengage Learning) pp210-212
[12] Hu H Y 2004 Foundation of Mechanical Vibration (Harbin: Harbin Institute of Technology Press) p26 (in Chinese) [胡海岩 2004 机械振动基础 (哈尔滨: 哈尔滨工业大学出版社) p26]
[13] Palm W J 2005 System Dynamics (2nd Ed.) (New York: McGraw-Hill Higher Education) p498
[14] Ogata K 2004 System Dynamics (4th Ed.) (New Jersey: Prentice Hall) pp384-388
[15] Liu X, Liu H, Yang J, Litak G., Cheng G, Han S 2017 Mech. Sys. Signal Pr. 96 58
[16] Huang D, Yang J, Zhang J, Liu H 2017 P. I. Mech. Eng. C-J. Mec. doi:0954406217719924
[17] Yang J H, Sanjuán M A F, Liu H G 2017 J. Comput. Nonlin. Dyn. 12 051011
[18] Liu H G, Liu X L, Yang J H, Sanjuán M A F, Cheng G 2017 Nonlinear Dynam. 89 2621
[19] Magin R, Ortigueira M D, Podlubny I, Trujillo J 2011 Signal Process. 91 350
[20] Leng Y G, Wang T Y 2003 Acta Phys. Sin. 52 2432 (in Chinese) [冷永刚, 王太勇 2003 物理学报 52 2432]
[21] Leng Y G, Wang T Y, Guo Y, Xu Y G, Fan S B 2007 Mech. Syst. Signal Pr. 21 138
[22] Li Q, Wang T, Leng Y, Wang W, Wang G 2007 Mech. Syst. Signal Pr. 21 2267
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