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In physics, the non-linear mode coupling is an important strategy to manipulate the mechanical properties of a vibrational system. Compared with the single-mode nonlinear system, the complex systems with two- or multi-mode nonlinear coupling have garnered considerable attention, among which the analytical solutions to the coupled Duffing equations are widely studied to solve nonlinear coupling. The fact is that the solving of the Duffing coupling equations generally starts with the eigenmodes solution of the linear equations. The trial solution of the coupled equations is the linear superposition of the eigenmodes. Under the secular perturbation theory and similar conditions, the Duffing coupling equation degenerates into two decoupled equations. However, thus far most of the solution methodologies are too complicated to unravel the underlying physical essence clearly. In this paper, first, by applying the representational transformation to the linear terms of the first-order coupled Duffing equations and the secular perturbation theory for the nonlinear terms, a decoupled expression of the first-order Duffing equations is derived, which can be solved more straightforwardly. Subsequently, in order to verify the correctness of the method, we design a coupled tuning fork mechanical vibration system, which consists of two experimental instruments to provide driving force and receive signals, two tuning forks and springs. The amplitude spectra are measured by an experimental instrument of forced vibration and resonance (HZDH4615), which provides a periodic driving signal for the tuning fork. The numerical fitting by software is employed to clarify the mechanism of the spectrum. Theoretically, the obtained fitting parameters can also evaluate some important attributes of the system. Most strikingly, due to the nonlinear coupling the splitting of the resonant peak and the phenomenon of “hysteresis loop” are clearly observed in the experiment. The research shows that the experimental results perfectly match the theoretical results obtained before. The method of solving coupled nonlinear equations in this article provides a solution and improvement of flexible adoption of nonlinear theory. On the other hand, it can be extended to coupled light and electricity systems, offer certain guidance for understanding the dynamic behavior of coupled systems, and will be conductive to the quantitative examination of numerous nonlinear coupling devices.
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Keywords:
- coupled Duffing equations /
- coupled-mode theory /
- representational transformation /
- secular perturbation theory /
- nonlinear effect
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图 1 双模耦合体系的相关参数示意图 (a)非线性振子模型; (b)解耦的CMT
Figure 1. Parameters’ schematic diagram of two-mode coupled system: (a) Nonlinear oscillator model; (b) decoupled CMT.
图 2 (a)音叉耦合实验测量系统; (b)耦合音叉的放大图
Figure 2. (a) Tuning fork coupling experimental measurement system; (b) enlarged view of coupled tuning fork.
图 3 (a)单个音叉振动时的实验振幅谱和拟合谱; (b)两个音叉耦合时的实验振幅谱和拟合谱
Figure 3. (a) Experimental amplitude spectrum and fitting spectrum of single tuning fork vibration; (b) experimental amplitude spectrum and fitting spectrum of two tuning fork coupling.
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[1] Jing H, Özdemir S K, Lü X Y, Zhang J, Yang L, Nori F 2014 Phys.Rev. Lett. 113 053604
Google Scholar
[2] 曹保锋, 李鹏, 李小强, 张雪芹, 宁王师, 梁睿, 李欣, 胡淼, 郑毅 2019 68 080501
Google Scholar
Cao B F, Li P, Li X Q, Zhang X Q, Ning W S, Liang R, Li X, Hu M, Zheng Y 2019 Acta Phys. Sin. 68 080501
Google Scholar
[3] Shu L, Guo L, Wu G C, Chen W 2019 Appl. Therm. Eng. 153 85
Google Scholar
[4] Antonio D, Czaplewski D A, Guest J R, López D, Arroyo S I, Zanette D H 2015 Phys. Rev. Lett. 114 034103
Google Scholar
[5] Cross M C, Zumdieck A, Lifshitz R, Rogers 2004 Phys. Rev. Lett. 93 224101
Google Scholar
[6] Westra H J R, Poot M, van der Zant H S J, Venstra W J 2010 Phys. Rev. Lett. 105 117205
Google Scholar
[7] Peng B, Ozdemir S K, Lei F, Monifi F, Gianfreda M, Long G, Fan S, Nori F, Bender C M, Yang L 2014 Nat. Phys. 10 394
Google Scholar
[8] Zhou X, Chong Y 2016 Opt. Express 24 6916
Google Scholar
[9] Abdollahi S 2017 Ph. D. Dissertation (Edmonton: University of Alberta)
[10] Bernard M, Manzano F R, Pavesi L, Pucker G, Carusotto I, Ghulinyan M 2017 Photonics Res. 5 168
Google Scholar
[11] Assawaworrarit S, Yu X, Fan S 2017 Nature 546 387
Google Scholar
[12] Sarma B, Sarma A K 2018 Sci. Rep. 8 14583
Google Scholar
[13] Yao Z, Ma J, Yao Y, Wang C 2019 Nonlinear Dyn. 96 205
Google Scholar
[14] Ding Z, Qiao K, Ernst N, Kong J, Chen M, Matthews L, Hyde T 2019 New J. Phys. 21 103051
Google Scholar
[15] Zheng Y, Zhou L, Dong Y, Qiu C, Chen X, Guo G, Sun F 2020 Phys. Rev. Lett. 124 223603
Google Scholar
[16] Kuang Y, Zhu M 2019 Appl. Phys. Lett. 114 203903
Google Scholar
[17] Sabarathinam S, Volos C, Thamilmaran K 2017 Nonlinear Dyn. 87 37
Google Scholar
[18] Ramos D, Frank I W, Deotare P B, Bulu I, Loncar M 2014 Appl. Phys. Lett. 105 181121
Google Scholar
[19] 刘海波, 吴德伟, 金伟, 王永庆 2013 62 050501
Google Scholar
Liu H B, Wu D W, Jin W, Wang Y Q 2013 Acta Phys. Sin. 62 050501
Google Scholar
[20] Bernstein A, Rand R H, Meller R 2018 Open Mech. Eng. J. 12 108
Google Scholar
[21] 侯东晓, 赵红旭, 刘彬 2013 62 234501
Google Scholar
Hou D X, Zhao H X, Liu B 2013 Acta Phys. Sin. 62 234501
Google Scholar
[22] Kovacic I, Rand R H, Sah S M 2018 Appl. Mech. Rev. 70 020802
Google Scholar
[23] Daniel D J 2020 Prog. Theor. Exp. Phys. 2020 043A01
Google Scholar
[24] 朱存远, 李朝刚, 方泉, 汪茂胜, 彭雪城, 黄万霞 2020 69 074501
Google Scholar
Zhu C Y, Li C G, Fang Q, Wang M S, Peng X C, Huang W X 2020 Acta Phys. Sin. 69 074501
Google Scholar
[25] Karabalin R B, Cross M C, Roukes M L 2009 Phys. Rev. B 79 165309
Google Scholar
[26] Haus H A 1984 Waves and fields in optoelectronics (New Jersey: Prentice-Hall) pp197−217
[27] Huang W, Lin J, Qiu M, Liu T, He Q, Xiao S, Zhou L 2020 Nanophotonics https://doi.org/10.1515/nanoph-2020-0007
[28] Fan S, Suh W, Joannopoulos J D 2003 J. Opt. Soc. Am. A 20 569
Google Scholar
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