三稳态能量收集系统的同宿分岔及混沌动力学分析

李海涛; 丁虎; 陈立群; 秦卫阳

doi:10.21656/1000-0887.410164

三稳态能量收集系统的同宿分岔及混沌动力学分析

doi: 10.21656/1000-0887.410164

¹中北大学理学院力学系, 太原 030051;²上海大学上海市应用数学和力学研究所, 上海 200072;³西北工业大学工程力学系, 西安 710072

基金项目: 国家自然科学基金青年科学基金(11902294)；中国博士后基金(2018M640373)；山西省应用研究基础计划(201801D221037)；山西省高等学校科技创新项目(2019L520)

详细信息

作者简介:
李海涛（1985—），男，副教授，博士(通讯作者. E-mail: lihaitao5884@163.com).

中图分类号: O322
计量
- 文章访问数: 1061
- HTML全文浏览量: 228
- PDF下载量: 300
- 被引次数: 0
出版历程
- 收稿日期: 2020-06-08
- 刊出日期: 2020-12-01

Homoclinic Bifurcations and Chaos Thresholds of Tristable Piezoelectric Vibration Energy Harvesting Systems

¹Department of Engineering Mechanics, School of Science, North University of China, Taiyuan 030051, P.R.China;²Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China;³Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Science Fund for Young Scholars of China(11902294)；China Postdoctoral Science Foundation(2018M640373)

摘要

摘要: 通过考虑动力系统平衡点的变化，构建了三稳态能量收集装置，分析了系统的同宿分岔和混沌等非线性动力学行为，全面研究了势能函数形状对压电能量收集系统响应的影响规律.建立了三稳态能量收集系统的集中参数模型，基于Padé逼近方法得到了同宿轨道解析形式的表达式.根据 Melnikov理论发展了能量收集系统同宿分岔以及混沌动力学的定性研究方法，得到了发生同宿分岔的阈值曲线.利用分岔图、最大Lyapunov指数和相平面图等数值方法验证解析结果，当激励幅值超过Melnikov临界阈值时，系统由阱内运动演变为大幅阱间振动.结果表明，调整对称的稳定平衡位置至非对称情形将导致三稳态能量收集系统非线性动力学行为的变化，不仅使系统在低激励强度下实现大幅阱间跳跃，还抑制了混沌响应产生，相关结果为实现优化能量输出效率提供了一定的理论参考.
- 三稳态 /
- 能量收集 /
- Melnikov函数 /
- 同宿分岔
Abstract: Nonlinear dynamic performances such as homoclinic bifurcation and chaos were investigated for tristable vibration energy harvesting systems. The analytical expression of the symmetric and asymmetric homoclinic solution was obtained through the Padé approximation, which was consistent with the numerical solution. According to the Melnikov theory, the qualitative method of studying the homoclinic bifurcation of the energy harvesting system with a triple well was developed, and the necessary condition for the occurrence of homoclinic bifurcation was obtained. Numerical simulations yielded bifurcation diagrams and maximum Lyapunov exponents that demonstrated the inter-well responses predicted with the Melnikov method. Compared with the system with symmetric potential energy, the system with asymmetric potential energy has a lower threshold of homoclinic bifurcation. For a low excitation level, the system with asymmetric potential energy witnesses inter-well chaos, while the response of the system with symmetric potential energy still keeps trapped in a single well. The change of symmetry of the system potential energy function improves the output voltage due to the increase in the probability of generating a large periodical inter-well oscillation response. The research on the homoclinic bifurcation of nonlinear energy harvesting systems with symmetric and asymmetric triple potential wells provides an effective tool for the parametric design of high-performance energy harvesters.
- tristability /
- energy harvesting /
- Melnikov function /
- homoclinic bifurcation

HTML全文

参考文献(21)

[1]	PRIYA S, INMAN D J. Energy Harvesting Technologies [M]. New York: Springer, 2009.
[2]	董彦辰, 张业伟, 陈立群. 惯容器非线性减振与能量采集一体化模型动力学分析[J]. 应用数学和力学, 2019,40(9): 968-979.(DONG Yanchen, ZHANG Yewei, CHEN Liqun. Dynamic analysis of the nonlinear vibration absorber-energy harvester integration model with inerter[J]. Applied Mathematics and Mechanics, 2019,40(9): 968-979.(in Chinese))
[3]	HARNE R L, WANG K W. A review of the recent research on vibration energy harvesting via bistable systems[J]. Smart Materials and Structures,2013,22(2): 023001.
[4]	张小静, 刘丽兰, 任博林, 等. 势阱深度对双稳态电磁发电系统发电性能的影响研究[J]. 应用数学和力学, 2017,38(6): 622-632.(ZHANG Xiaojing, LIU Lilan, REN Bolin, et al. Influence of potential well depth on power generation performance of bistable electromagnetic energy harvesting systems[J]. Applied Mathematics and Mechanics,2017,38(6): 622-632.(in Chinese))
[5]	吴子英, 牛峰琦, 刘蕊, 等. 有色噪声激励下双稳态电磁式振动能量捕获器动力学特性研究[J]. 应用数学和力学, 2017,38(5): 570-580.(WU Ziying, NIU Fengqi, LIU Rui, et al. Dynamic characteristics of bistable electromagnetic vibration energy harvesters under colored noise excitation[J]. Applied Mathematics and Mechanics, 2017,38(5): 570-580.(in Chinese))
[6]	刘蕊, 吴子英, 叶文腾. 附加非线性振子的双稳态电磁式振动能量捕获器动力学特性研究[J]. 应用数学和力学, 2017,38(4): 432-446.(LIU Rui, WU Ziying, YE Wenteng. Dynamics research of bistable electromagnetic energy harvesters with auxiliary nonlinear oscillators[J]. Applied Mathematics and Mechanics, 2017,38(4): 432-446.(in Chinese))
[7]	MASANA R, DAQAQ M F. Energy harvesting in the super-harmonic frequency region of a twin-well oscillator[J]. Journal of Applied Physics,2012,111(4): 044501.
[8]	LI H T, QIN W Y, LAN C B, et al. Dynamics and coherence resonance of tri-stable energy harvesting system[J]. Smart Materials and Structures,2016,25(1): 015001.
[9]	LI H T, QIN W Y, ZU J, et al. Modeling and experimental validation of a buckled compressive-mode piezoelectric energy harvester[J]. Nonlinear Dynamics,2018,92(4): 1761-1780.
[10]	ZHOU S X, CAO J Y, ERTURK A, et al. Enhanced broadband piezoelectric energy harvesting using rotatable magnets[J]. Applied Physics Letters,2013,102(17): 173901.
[11]	周生喜, 曹军义, ERTURK A, 等. 压电磁耦合振动能量俘获系统的非线性模型研究[J]. 西安交通大学学报, 2014,48(1): 106-111.(ZHOU Shengxi, CAO Junyi, ERTURK A, et al. Nonlinear model for piezoelectric energy harvester with magnetic coupling[J]. Journal of Xi’an Jiaotong University,2014,48(1): 106-111.(in Chinese)
[12]	ZHOU S X, ZUO L. Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting[J]. Communications in Nonlinear Science and Numerical Simulation,2018,61: 271-284.
[13]	CHEN L Q, LI K. Equilibriums and their stabilities of the snap-through mechanism[J]. Archive of Applied Mechanics,2016,86(3): 403-410.
[14]	WANG G, ZHAO Z, LIAO W H, et al. Characteristics of a tri-stable piezoelectric vibration energy harvester by considering geometric nonlinearity and gravitation effects[J]. Mechanical Systems and Signal Processing,2020,138: 106571.
[15]	李海涛, 秦卫阳. 双稳态压电能量获取系统的分岔混沌阈值[J]. 应用数学和力学, 2014,35(6): 652-662.(LI Haitao, QIN Weiyang. Bifurcation and chaos thresholds of bistable piezoelectric vibration energy harvesting systems[J]. Applied Mathematics and Mechanics,2014,35(6): 652-662.(in Chinese))
[16]	SUN S, CAO S Q. Analysis of chaos behaviors of a bistable piezoelectric cantilever power generation system by the second-order Melnikov function[J]. Acta Mechanica Sinica,2017,33(1): 200-207.
[17]	KITIO KWUIMY C A, NATARAJ C, LITAK G. Melnikov’s criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation[J]. Chaos: an Interdisciplinary Journal of Nonlinear Science,2011,21(4): 043113.
[18]	OUMB TKAM G T, KITIO KWUIMY C A, WOAFO P. Analysis of tristable energy harvesting system having fractional order viscoelastic material[J]. Chaos: an Interdisciplinary Journal of Nonlinear Science,2015,25(1): 013112.
[19]	TIAN R, ZHOU Y, ZHANG B, et al. Chaotic threshold for a class of impulsive differential system[J]. Nonlinear Dynamics, 2016,83(4): 2229-2240.
[20]	LI H T, ZU J, YANG Y F, et al. Investigation of snap-through and homoclinic bifurcation of a magnet-induced buckled energy harvester by the Melnikov method[J]. Chaos: an Interdisciplinary Journal of Nonlinear Science,2016,26(12): 123109.
[21]	FENG J, ZHANG Q, WANG W. Chaos of several typical asymmetric systems[J]. Chaos, Solitons and Fractals,2012,45(7): 950-958.