HARVESTING PERFORMANCE AND DYNAMIC RESPONSES OF THE BISTABLE HARVESTER WITH A NONLINEAR RESONANT CIRCUIT
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摘要: 双稳态俘能器可实现宽频和高效的俘能效果. 目前的研究主要在双稳态结构中接入单一电阻电路进行俘能. 本文将非线性RLC (电阻−电感−电容)谐振电路引入到三弹簧式双稳态结构中, 构建两自由度非线性系统, 以实现俘能特性的提升. 设计永磁体与线圈的构型, 获得了非线性机电耦合系数. 推导并得到了两自由度非线性俘能器的控制方程. 利用谐波平衡法推导得到了系统的电流与位移的频率响应关系. 基于雅可比矩阵对解的稳定性进行了判别. 将解析解与数值解进行了对比验证. 结果表明, 在双稳态俘能器中引入非线性二阶谐振电路不仅有利于低频俘能, 还可进一步提升俘能响应, 拓宽俘能带宽. 相同的电路参数下, 与线性电路相比非线性电路可通过电流的倍频现象实现结构更低频率的能量俘获. 减小谐振电路与双稳态结构共振频率之比, 增加基础激励幅值, 减小静平衡点之间的距离均可提升俘能器的俘能效果. 通过调控谐振电路与双稳态共振频率之比和基础激励幅值等参数, 可实现系统单倍周期响应、多倍周期响应及混沌响应之间的切换.Abstract: The bistable harvester can achieve wide band and high-efficiency energy harvesting performance under low frequency and low excitation levels. Previous studies mainly use a simple resistor circuit to capture the energy in the bistable structures. This paper proposes a two-degree-of-freedom (DOF) nonlinear system formed by coupling a three-spring bistable structure with a nonlinear RLC (resistance-inductance-capacitance) resonant circuit for energy harvesting enhancement. The nonlinear electromagnetic coupling coefficient between the circuit and structure is obtained by the special configuration between permanents and coils. The governing equation of the two DOF nonlinear systems is acquired. The analytical responses of the current and displacement are derived by the harmonic balance method, whose stability is judged by the Jacobin matrix. The analytical solution is compared with the numerical solution. Results demonstrate that introducing a nonlinear two-order resonant circuit into the bistable energy harvester can further improve the harvesting responses and broaden the energy bandwidth. With the same circuit parameters, the nonlinear resonant circuit can achieve lower frequency energy harvesting performance through frequency doubling of the current compared with the traditional linear circuit. One can enhance the energy harvester performance by decreasing the resonant ratio between the resonant circuit and bistable structure, increasing the excitation amplitude, and decreasing the distance between two static equilibrium points. The system can realize the switching of single-period response, multi-period response, and chaotic responses by adjusting the resonant ratio between circuit and bistable structure, and excitation amplitude.
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表 1 线圈和永磁体的参数配置
Table 1. Parameters of coils and permanent magnets (PM)
Name Value residual magnetic flux density Br/T 1.19 inner radius of PM Rin/mm 9 outer radius of PM Rout/mm 14 height of PM Ht/mm 10 distance between PMs H/mm 3 average radius of coil Ra/mm 16.5 number of coil turn 350 + 350 resistance of coil Re/Ω 24.9 inductance of coil Le/mH 2.01 表 2 仿真分析中用到的固定参数数值
Table 2. Parameters values used in simulation
Parameters name Value m/kg 0.5 k0/(N·m) 1.4061 × 103 kh/(N·m) 2 × 103 k1/(N·m) −2100 k3/(N·m) 2 × 107 c/(N·s·m−1) 2.6515 c1/(N·A−1·m−1) 1.0674 × 103 c3/ (N·A−1·m−3) 9.2731 × 106 C/F 1/(8π2) -
[1] Wang J, Hu G, Zhao L, et al. Perspectives in flow-induced vibration energy harvesting. Applied Physics Letters, 2021, 119: 100502 doi: 10.1063/5.0063488 [2] Yau C, Kwok T , Lei C, et al. Energy Harvesting in Internet of Things. Singapore: Springer, 2018: 35-79 [3] 钱有华, 陈娅昵. 双稳态压电俘能器的簇发振荡与俘能效率分析. 力学学报, 2022, 54(11): 3157-3168 (Qian Youhua, Chen Yani. Bursting oscillations and energy harvesting efficiency analysis of bistable piezoelectric energy harvester. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 3157-3168 (in Chinese) doi: 10.6052/0459-1879-22-298 [4] 赵林川, 邹鸿翔, 刘丰瑞等. 压电与摩擦电复合型旋转能量采集动力学协同调控机制研究. 力学学报, 2021, 53(11): 2961-2971 (Zhao Linchuan, Zou Hongxiang, Liu Fengrui, et al. Hybrid piezoelectric-triboelectric rotational energy harvester using dynamic coordinated modulation mechanism. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(11): 2961-2971 (in Chinese) doi: 10.6052/0459-1879-21-410 [5] Yan B, Yu N, Zhang L, et al. Scavenging vibrational energy with a novel bistable electromagnetic energy garvester. Smart Materials and Structures, 2020, 29: 025022 doi: 10.1088/1361-665X/ab62e1 [6] 张伟, 刘爽, 毛佳佳等. 磁耦合式双稳态宽频压电俘能器的设计和俘能特性. 力学学报, 2022, 54(4): 1102-1112 (Zhang Wei, Liu Shuang, Mao Jiajia, et al. Design and energy capture characteristics of magnetically coupled bistable wide band piezoelectric energy harvester. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(4): 1102-1112 (in Chinese) doi: 10.6052/0459-1879-21-676 [7] Tao K, Chen Z, Yi H, et al. Hierarchical honeycomb-structured electret/triboelectric nanogenerator for biomechanical and morphing wing energy harvesting. Nano-Micro Letters, 2021, 13(1): 123 doi: 10.1007/s40820-021-00644-0 [8] Tan D, Zhou J, Wang K, et al. Bow-type bistable triboelectric nanogenerator for harvesting energy from low-frequency vibration. Nano Energy, 2022, 92: 106746 doi: 10.1016/j.nanoen.2021.106746 [9] Yang Z, Zhou S, Zu J, et al. High-performance piezoelectric energy harvesters and their applications. Joule, 2018, 2(4): 642-697 doi: 10.1016/j.joule.2018.03.011 [10] Yildirim T, Ghayesh M, Li W, et al. A review on performance enhancement techniques for ambient vibration energy harvesters. Renewable and Sustainable Energy Reviews, 2017, 71: 435-449 doi: 10.1016/j.rser.2016.12.073 [11] Li P, Liu Y, Wang Y, et al. Low-frequency and wideband vibration energy harvester with flexible frame and interdigital structure. AIP Advances, 2015, 5(4): 047151 doi: 10.1063/1.4919711 [12] Liu D, Al-Haik M, Zakaria M, et al. Piezoelectric energy harvesting using L-Shaped structures. Journal of Intelligent Material Systems and Structures, 2017, 29(6): 1206-1215 [13] Zhou S, Hobeck JD, Cao J, et al. Analytical and experimental investigation of flexible longitudinal zigzag structures for enhanced multi-directional energy harvesting. Smart Materials and Structures, 2017, 26(3): 035008 doi: 10.1088/1361-665X/26/3/035008 [14] Zhou S, Cao J, Inman D, et al. Impact-induced high-energy orbits of nonlinear energy harvesters. Applied Physics Letters, 2015, 106(9): 093901 doi: 10.1063/1.4913606 [15] Kong X, Li H, Wu C. Dynamics of 1-Dof and 2-Dof energy sink with geometrically nonlinear damping: application to vibration suppression. Nonlinear Dynamics, 2017, 91: 733-754 [16] Zou H, Zhang W, Wei K, et al. A compressive-mode wideband vibration energy harvester using a combination of bistable and flextensional mechanisms. Journal of Applied Mechanics, 2016, 83(12): 121005 doi: 10.1115/1.4034563 [17] 张旭辉, 陈路阳, 陈孝玉等. 线形−拱形组合梁式三稳态压电俘能器动力学特性研究. 力学学报, 2021, 53(11): 2996-3006 (Zhang Xuhui, Chen Luyang, Chen Xiaoyu, et al. Research on dynamics characteristics of linear-arch composed beam tri-stable piezoelectric energy harvester. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(11): 2996-3006 (in Chinese) doi: 10.6052/0459-1879-21-392 [18] Huguet T, Badel A, Lallart M. Exploiting bistable oscillator subharmonics for magnified broadband vibration energy harvesting. Applied Physics Letters, 2017, 111(17): 173905 doi: 10.1063/1.5001267 [19] Yu N, Ma H, Wu C, et al. Modeling and experimental investigation of a novel bistable two-degree-of-freedom electromagnetic energy harvester. Mechanical Systems and Signal Processing, 2021, 156: 107608 doi: 10.1016/j.ymssp.2021.107608 [20] 任博林. 基于双稳态发电系统的非线性吸振器动力学特性研究. [硕士论文]. 西安: 西安理工大学, 2017Ren Bolin. Research on the dynamic characteristics of nonlinear vibration absorber based on bistable generation system. [Master Thesis]. Xi’an: Xi’an University of Technology, 2017 (in Chineses)) [21] Zhou S, Cao J, Inman D, et al. Broadband tristable energy harvester: modeling and experiment verification. Applied Energy, 2014, 133: 33-39 doi: 10.1016/j.apenergy.2014.07.077 [22] Huang D, Zhou S, Litak G. Theoretical analysis of multi-stable energy harvesters with high-order stiffness terms. Communications in Nonlinear Science and Numerical Simulation, 2019, 69: 270-286 doi: 10.1016/j.cnsns.2018.09.025 [23] Kim P, Seok J. A multi-stable energy harvester: dynamic modeling and bifurcation analysis. Journal of Sound and Vibration, 2014, 333(21): 5525-5547 doi: 10.1016/j.jsv.2014.05.054 [24] Pellegrini S, Tolou N, Schenk M, et al. Bistable vibration energy harvesters: a review. Journal of Intelligent Material Systems and Structures, 2012, 24(11): 1303-1312 [25] Wu Y, Ji H, Qiu J, et al. An internal resonance based frequency up-converting energy garvester. Journal of Intelligent Material Systems and Structures, 2018, 29(13): 2766-2781 doi: 10.1177/1045389X18778370 [26] Daqaq M. Transduction of a bistable inductive generator driven by white and exponentially correlated gaussian noise. Journal of Sound and Vibration, 2011, 330(11): 2554-2564 doi: 10.1016/j.jsv.2010.12.005 [27] Cottone F, Vocca H, Gammaitoni L. Nonlinear energy garvesting. Physical Review Letters, 2009, 102(8): 080601 doi: 10.1103/PhysRevLett.102.080601 [28] Elliott S, Zilletti M. Scaling of electromagnetic transducers for shunt damping and energy harvesting. Journal of Sound and Vibration, 2014, 333(8): 2185-2195 doi: 10.1016/j.jsv.2013.11.036 [29] Carrella A, Brennan M, Waters T, et al. Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness. International Journal of Mechanical Sciences, 2012, 55(1): 22-29 doi: 10.1016/j.ijmecsci.2011.11.012 [30] Yan B, Luo Y, Zhang X. Structural multimode vibration absorbing with electromagnetic shunt damping. Journal of Vibration and Control, 2014, 22(6): 1604-1617 [31] Yan B, Zhang X, Niu H. Design and test of a novel isolator with negative resistance electromagnetic shunt damping. Smart Materials and Structures, 2012, 21(3): 035003 doi: 10.1088/0964-1726/21/3/035003 [32] Ma H, Yan B. Nonlinear damping and mass effects of electromagnetic shunt damping for enhanced nonlinear vibration isolation. Mechanical Systems and Signal Processing, 2021, 146: 107010 doi: 10.1016/j.ymssp.2020.107010 [33] 陈予恕. 非线性振动. 北京: 高等教育出版社, 2002Chen Yushu. Nonlinear Vibrations. Beijing: Higher Education Press, 2002 (in Chinese)) -