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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图 1 非线性电磁俘能器的构造及对应物理简化模型
Figure 1. Configurations of nonlinear electromagnetic harvesters and corresponding simplified physical models
图 2 “三弹簧”构型双稳态恢复力的精确解与近似解
Figure 2. Exact and approximate bistable restoring force of the “three springs” configuration
图 3 非线性电磁耦合系数的理论解和多项式拟合结果
Figure 3. Analytical and approximate nonlinear electromagnetic coupling coefficient
图 4 理论解与数值解对比
Figure 4. Comparison between theoretical solution and numerical solution
图 5 5 Hz定频激励下相对位移与电流之间的频率关系
Figure 5. Relationship between relative displacement and current under 5 Hz constant frequency excitation
图 6 电阻式俘能器与共振式俘能器性能对比
Figure 6. Energy harvesting comparison between the energy harvesters with a pure resistor and with a resonant circuit
图 7 电路谐振频率对俘能特性的影响
Figure 7. Effect of circuit resoance frequency on energy harvesting performance
图 8 响应位移和电流随频率比的分叉图
Figure 8. Birfication diagram of displacement and current with respect to frequency ratio
图 9 频率比0.25情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 9. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when the frequency ratio is 0.25
图 10 频率比0.5情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 10. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when the frequency ratio is 0.5
10 频率比0.5情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面 (续)
10. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when the frequency ratio is 0.5 (continued)
图 11 频率比0.75情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 11. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when frequency ratio is 0.75
图 12 激励幅值对俘能特性的影响
Figure 12. Effect of excitation amplitude on energy harvesting performance
图 13 响应位移和电流随激励幅值的分叉图
Figure 13. Birfication diagram of displacement and current with respect to excitation amplitude
图 14 激励幅值x0 = 0.75 mm情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 14. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when x0 = 0.75 mm
14 激励幅值x0 = 0.75 mm情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面 (续)
14. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when x0 = 0.75 mm (continued)
图 15 激励幅值x0 = 1 mm情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 15. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when x0 = 1 mm
图 16 激励幅值x0 = 2 mm情况下位移和电流的时域响应、傅里叶频谱、相轨迹及庞加莱截面
Figure 16. Output responses, Fourier frequency responses, phase orbit, and Poincare map of displacement and current when x0 = 2 mm
图 17 静平衡位置和势能井深度对俘能特性的影响
Figure 17. Effect of static equilibrium position and depth of potential energy on energy harvesting performance
17 静平衡位置和势能井深度对俘能特性的影响 (续)
17. Effect of static equilibrium position and depth of potential energy on energy harvesting performance (continued)
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 
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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)
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