NONLINEAR DYNAMICS CHARACTERISTICS OF TUMBLER-INSPIRED ELECTROMAGNETIC ENERGY HARVESTERS
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摘要: 海洋波浪能作为一种可再生能源, 将其俘获并转化为电能为无线传感器持续供电, 可以推动海洋环境监测的数字化改造升级. 然而, 海浪能的低频与随机性等特征导致其俘获难度大. 不倒翁结构具有不同于传统结构的超低频振动特性, 并且其对低频激励敏感的特点, 可以吸收周围振动能量. 为此, 文章通过引入不倒翁机制与Halbach磁铁阵列, 构建了磁非线性力, 设计一种不倒翁式电磁俘能器, 以实现提高低频波浪能的俘获效果. 首先, 基于拉格朗日方程建立不倒翁式电磁俘能器的理论模型, 并用谐波平衡法推导了不倒翁摆角与输出电压的频率响应关系. 将解析解与数值解进行对比验证. 其次, 探究了激励频率与幅值等参数对系统动力学行为的影响规律. 最后, 研制了不倒翁式电磁俘能器原理样机, 搭建俘能试验平台并进行试验, 验证了理论模型的正确性. 研究表明: 引入磁非线性力使得系统呈现刚度硬化特征, 有利于提升低频俘能效率. 不倒翁式电磁俘能器随激励频率与幅值的变化, 呈现周期、准周期及混沌运动等复杂动力学行为. 低频与大激励条件更容易造成俘能器系统的混沌运动, 有利于提高俘能效果. 本研究为不倒翁式电磁俘能器的设计及在低频波浪能高效俘获的应用提供了理论支持.Abstract: Ocean wave energy, as a prominent renewable source, possesses the potential to be harnessed for the generation of electricity, specifically catering to the power requirements of wireless sensors. It will be the key promoting the digital evolution of the marine environmental monitoring systems. However, the low-frequency and large randomness characteristics restrict the efficient harvesting of ocean waves. The tumbler structure has ultra-low frequency vibration characteristics that are different from traditional structures, and it is sensitive to low-frequency excitation, which can absorb surrounding vibration energy. In this paper, we design a tumbler-inspired electromagnetic energy harvester with a Halbach array, aiming to enhance the performance of low-frequency wave energy harvesting by constructing magnetic nonlinear forces. The theoretical model of the harvester is established according to the Lagrange’s equation. The analytical responses of the tumbler's swing angle and the harvester's voltage are derived by the harmonic balance method. The analytical solution is compared with the numerical solution. Moreover, simulations are conducted to investigate the effect of different excitation conditions such as excitation amplitude and frequency on the dynamic response characteristics of the system. Finally, a prototype of the tumbler-inspired electromagnetic energy harvester was fabricated, and an experimental platform was built. It verifies the correctness of the theoretical model. Both the simulation and experimental results show that the harvester exhibits the hardening stiffness characteristic through introducing magnetic nonlinearity. This characteristic is beneficial to enhance the low-frequency energy harvesting efficiency. The harvester exhibits diverse dynamic behaviors such as periodic motion, quasi-periodic motion, and chaotic motion with the change of the excitation amplitude and frequency. In addition, low frequency and large excitation are more likely to cause chaotic motion, which is beneficial to energy harvesting effect. This study provides a theoretical support for the design and application of the tumbler mechanism in low-frequency ocean wave energy harvesting.
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图 1 不倒翁式电磁俘能器的设计示意图
Figure 1. Schematic diagram of the tumbler-inspired electromagnetic energy harvester
图 2 不倒翁式电磁俘能器的理论模型
Figure 2. Theoretical model of the tumbler-inspired electromagnetic energy harvester
图 3 不倒翁式电磁俘能器摆角与感应电压解析解与数值解对比
Figure 3. Comparison between theoretical solution and numerical solution of the swing angle and the voltage of the tumbler-inspired electromagnetic energy harvester
图 4 摆角响应和电压随激励频率的分岔图
Figure 4. Bifurcation diagram of the swing angle and voltage with respect to the excitation frequency
图 5 激励频率在2.5 Hz时俘能器的摆角和电压响应、相轨迹与庞加莱截面和傅里叶频谱
Figure 5. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation frequency is 2.5 Hz
图 6 激励频率在4 Hz时俘能器摆角和电压响应、相轨迹与庞加莱截面和傅里叶频谱
Figure 6. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation frequency is 4 Hz
图 7 激励频率在10 Hz时俘能器摆角和电压响应、相轨迹与庞加莱截面和傅里叶频谱
Figure 7. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation frequency is 10 Hz
图 8 摆角响应和电压随激励幅值的分岔图
Figure 8. Bifurcation diagrams of the swing angle and voltage with respect to the excitation amplitude
图 9 激励幅值在A = 0.35g时俘能器的摆角和电压响应、相轨迹与庞加莱截面和傅里叶频谱
Figure 9. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation amplitude is 0.35g
图 10 激励幅值在A = 0.5g时俘能器的摆角和电压响应、相轨迹与庞加莱截面和傅里叶频谱
Figure 10. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation amplitude is 0.5g
图 11 激励幅值在A = 0.8g时俘能器摆角和电压响应、相轨迹与庞加莱截面、傅里叶频谱
Figure 11. Time-domain histories, phase orbit, Poincare map and Fourier spectrum of the swing angle and voltage for the excitation amplitude is 0.8g
图 14 不同激励频率时实验电压响应与傅里叶频谱
Figure 14. Experimental time-domain history and Fourier spectrum of voltage under different excitation frequencies
表 1 非线性磁力的刚度系数
Table 1. Stiffness coefficients of nonlinear magnetic force
Parameter k1 k2 k3 Value −0.5181 −11.58 39.05 
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表 2 俘能器的几何和材料参数
Table 2. Geometric and material properties of the harvester
Categories Parameters Values tumbler structure r/mm 30 e/mm 1.9 m/kg 0.4 Jc/(kg·mm2) 269.54 H/mm 26 magnetic structure (a1,b1,c1)/mm 20,10,10 (a2,b2,c2)/mm 10,30,10 (xOA,yOA)/mm −75, −85 $\beta $/(°) 40 Br1/T 1.25 Br2/T 1.25 others c 0.002 L/mH 8.26 Ra/$\Omega $ 20
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