可變剛性邊界應用於壓電平板理論之能量擷取系統

Energy Harvesting System of Piezoelectric Plate Theory with Variable Boundary Condition

陳新承(Hsin-Cheng Chen)

指導教授 ： 黃育熙

國立臺灣大學/工學院/機械工程學研究所/碩士(2021年)

https://doi.org/10.6342/NTU202101434

摘要

本論文利用平板疊加法理論解析搭配有限元素法與實驗量測，探討壓電陶瓷雙晶片在彈簧邊界下的面外振動特性，並利用錐形彈簧的非線性特性，改變壓電試片的共振頻率以對應到環境的激振頻率。理論解析首先利用克希荷夫薄板理論，將壓電陶瓷雙晶片的三層結構簡化成單層矩形平板，並計算不同電極連接下的等效參數，得出壓電平板的統御方程式，本文在彈簧邊界中提出三種模型，分別為均勻彈簧邊界與點支撐彈簧邊界以及質量效應下點支撐彈簧邊界，希望能更接近實際平板連接彈簧的情況，由於平板的邊界較為複雜，不容易直接求得解析解，因此使用疊加法將平板拆分為多個較簡易的結構塊，每個結構塊會共同滿足部分邊界以及統御方程式，疊加後再將剩餘的邊界滿足，利用正交函數展開得到壓電平板在彈簧邊界下的共振頻率與位移模態形狀，與有限元素法比較均有良好的對應性，接著利用模態正交特性求得含阻尼之面外位移以及電壓暫態、穩態響應，並利用程式掃頻得到含阻尼之面外位移以及電壓頻率響應。
實驗量測的部分使用錐形彈簧作為壓電平板的支撐邊界，根據錐形彈簧理論得到不同壓縮量對應的彈簧勁度，並以電子斑點干涉術量測不同壓縮量下，逆壓電效應與正壓電效應個別的壓電平板的共振頻率以及面外位移模態，並與理論模型進行比較，雖然不同壓縮條件下部分共振頻率無法與理論完美對應，但隨著壓縮量邊界增加確實可提升共振頻率，最後利用控制器將壓電片的輸出電壓進行回授控制，控制馬達轉動使彈簧的壓縮量增加或減少，來對應環境的激振頻率以產生最大的輸出電壓。

關鍵字

壓電平板 ； 疊加法 ； 有限元素法 ； 彈性支撐 ； 錐形彈簧 ； 能量擷取系統 ； 可變剛性邊界

並列摘要

This research analyzed the out-of-plane vibration characteristics of the piezoelectric bimorph in spring support boundary conditions which based on theoretical analysis with finite element method (FEM) and experiment measurements. Conical compression spring has nonlinear characteristics which can change stiffness according to different compressed length. This research uses conical spring as spring support boundary conditions to change the resonance frequencies of piezoelectric bimorph in order to fit the input frequency from the environment. Theoretical analysis uses Kirchhoff’s thin plate theory to simplify the three layers structure to a single layer plate, and calculates the equivalent material constants under different electrode connections to obtain the governing equation of the piezoelectric plate. Three modals are proposed in the spring boundary, namely uniform spring boundary, point support spring boundary and point support spring boundary with mass effect. Then, applied superposition method to obtain resonance frequencies and mode shapes of the piezoelectric plate. The result shows good correspondence with the finite element method (FEM). Moreover, the modal orthogonality was used to calculate the transient and steady state behavior of out-of-plane displacement and voltage response under sinusoidal input force, and use program to calculate the frequency response.
The experiment measures the vibration of piezoelectric plate under conical spring support boundary. According to the conical spring theory, the spring stiffness corresponding to different compression amounts is obtained and the resonance frequencies and out-of-plane displacement mode shapes of the piezoelectric plate under the inverse piezoelectric effect and the positive piezoelectric effect under different compressed length are measured by electronic speckle pattern interferometry (ESPI). Compared with the theoretical model, most of resonant frequencies correspond well to the theory during different compressed length. The resonant frequency does increase as the compressed length increases. The output voltage of the piezoelectric plate is feedback controlled by microcontroller, i.e. Arduino, which controlled the motor to increase or decrease the compressed length of the spring in order to fit the excitation frequency of the environment to generate the maximum output voltage.

並列關鍵字

piezoelectric plate ； superposition method ； finite element method ； elastic support ； conical spring ； energy harvesting system ； variable stiffness boundary

參考文獻

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[5] S. Y. Wang, “A finite element model for the static and dynamic analysis of a piezoelectric bimorph.”, International Journal of Solids and Structures, 41, pp. 4075-4096, 2004.