VIBRATION ANALYSIS OF A PIEZOELECTRIC CIRCULAR PLATE ENERGY HARVESTER CONSIDERING A PROOF MASS
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摘要: 基于圆板的压电能量采集技术在取代化学电池为低功耗电子器件提供能源方面具有巨大的潜能. 本文通过理论建模和数值仿真研究了考虑附加质量接触面积的压电圆板能量采集器的采集性能. 首先, 基于基尔霍夫薄板理论, 用广义哈密顿原理推导了带附加质量块的压电圆板能量采集器的机电耦合方程, 并用伽辽金法对方程近似离散, 通过离散方程得到电压、功率输出和最优负载阻抗的闭合解. 用有限元仿真对所提出的理论模型进行了验证, 结果表明该理论模型可以成功地预测压电圆板能量采集器输出电压和功率. 最后, 基于闭合解探讨了负载阻抗、附加质量块、压电圆板的内外半径等相关参数对压电圆板能量采集器固有频率、输出电压和功率的影响. 结果表明, 当质量块与复合板的接触半径足够小(本文中接触半径小于板半径的1/14)时, 质量块与复合圆板的接触面积可以忽略; 相较于无孔的压电片, 内径位于2.5 ~ 4 mm范围内的压电片可以提高能量采集器的采集性能; 附加质量、压电片外径和负载阻抗的合理选择既可以降低压电圆板的固有频率, 还可以提高其采集性能.Abstract: Piezoelectric energy harvesting based on circular plates has great potential in replacing chemical battery to provide power for low-power electronic devices. This paper investigates the harvesting performance of a piezoelectric circular plate energy harvester with a proof mass considering its contact area via theoretical modeling and numerical simulations. Firstly, based on the Kirchhoff thin plate theory, the generalized Hamilton principle is applied to derive the electromechanical coupling equations of the piezoelectric circular plate energy harvester considering the proof mass. The equations are approximately discretized using the Galerkin method, and the discretized equations yield closed solutions of voltage, power output, and optimal load resistance. In addition, the correctness of the theoretical model is verified via the finite element simulations, and the results show that the theoretical model can successfully predict the power output of the piezoelectric circular energy harvester. Finally, the closed solutions are used to explore the effects of related parameters, such as the load resistance, the proof mass, the inner and the outer radius of the piezoelectric circular plate, on the natural frequencies, the output voltage and power of the piezoelectric circular energy harvester. The results show that when the contact radius between the proof mass and the piezoelectric composite plate is small enough (here, the contact radius is less than 1/14 of the plate radius), the contact area between the proof mass and the circular plate can be ignored. It is found that the piezoelectric plate with inner diameter between 2.5 mm and 4 mm can improve the harvesting performance of the energy harvester, compared with the piezoelectric plate without hole. As well as that the proper choice of the mass, the load resistance and the outer radius of piezoelectric circular plate can not only reduce the natural frequencies of the piezoelectric circular plate, but also improve its harvesting performance.
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图 1 附加质量块的圆板能量采集器的剖面图
Figure 1. The cutaway view of the circular energy harvester considering proof mass
图 2 不同模态数下输出电压随基础激励频率变化
Figure 2. The variation of the voltage output with base excitation frequency for different number of modes
图 3 压电圆板能量采集器的有限元模型
Figure 3. Finite element model of the piezoelectric circular plate energy harvester
图 4 负载阻抗对输出电压和功率的影响
Figure 4. The effect of the load resistance on the output voltage and power
图 5 附加质量对能量采集器采集性能的影响
Figure 5. The influence of the mass on the harvesting performance of the energy harvester
图 6 压电片外径对能量采集器的固有频率、最大输出电压和功率的影响
Figure 6. The effect of the outer radius of the piezoelectric plate on the natural frequency, maximum output voltage and power of the energy harvester
图 7 压电片内半径对能量采集器采集性能的影响
Figure 7. The effect of the inner radius of the piezoelectric plate on the harvesting performance of energy harvester
表 1 圆板能量采集器的参数
Table 1. The parameters of the circular energy harvester
Parameter Unit Value Eb GPa 90 $ S_{11}^E $ m2/N 1.65 × 10−11 ρb kg/m3 8500 ρp kg/m3 7500 vb — 0.33 vp — 0.2897 d31 m/V −2.74 × 10−10 $\varepsilon _{33}^T$ F/m 3.021 × 10−8 hb mm 0.2 hp mm 0.2 rb mm 28 μ s−1 57.7715 γ s 7.7022 × 10−6 
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表 2 m0 = 0时压电圆板能量采集器的固有频率理论值与仿真值比较
Table 2. Comparison of natural frequency of circular energy harvester with m0 = 0 from theory and simulation
rpi/mm RL = 0 Ω RL = 10 kΩ RL→∞ FEA/Hz model/Hz error/% FEA/Hz model/Hz error/% FEA/Hz model/Hz error/% 0 581.73 580.72 −0.174 656.66 654.05 −0.397 663.88 662.87 −0.152 3 572.56 571.56 −0.175 640.32 638.01 −0.361 647.41 646.75 −0.102 4 566.81 565.82 −0.175 629.94 627.88 −0.328 636.95 636.46 −0.077 5 560.64 559.64 −0.178 618.72 616.81 −0.309 625.64 625.48 −0.026
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表 3 m0 = 74.3 g时压电圆板能量采集器的固有频率理论值与仿真比较
Table 3. Comparison of natural frequency of circular energy harvester with m0 = 74.3 g from theory and simulation
rpi/mm RL = 0 Ω RL→∞ model/Hz FEA (χ = 1/28)/Hz error/% FEA (χ = 1/14)/Hz error/% model/Hz FEA (χ = 1/28)/Hz error/% FEA (χ = 1/14)/Hz error/% 0 109.81 110.23 −0.383 113.6 −3.33 125.61 126.88 −1.003 130.89 −4.04 3 97.89 97.73 0.164 102.02 −4.05 108.72 108.27 0.411 113.96 −4.60 4 92.85 92.03 0.893 96.81 −4.09 101.54 100.42 1.116 106.55 −4.70 5 87.57 86.58 1.141 91.66 −4.46 94.04 93.26 0.835 99.56 −5.87
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