壓電平板以磁力變換軸向剛性之流體致振變頻能量擷取系統

Abstract

本論文使用平板疊加法理論與晶格波茲曼法的流場模擬，探討壓電陶瓷雙晶片於單邊固定邊界且受軸向預應力作用下，在渦流激振下的面外振動特性，搭配有限元素法與實驗量測進行驗證，利用平板受軸向預應力作用會改變剛性之特性，變化壓電片的共振頻率以與渦街的振動頻率相互對應。理論解析先以壓電本構方程式結合克希荷夫薄板理論，將壓電雙晶片的三層結構等效成單層矩形平板，再透過力與力矩平衡關係式，得出受軸向預應力作用下的壓電平板之統御方程式，接著使用疊加法將平板拆分成多個結構塊進行求解，每個結構塊會皆會滿足原平板的部分邊界條件與統御方程，將各結構塊的解疊加後再滿足剩下的邊界條件，透過正交函數展開以求得壓電平板在單邊固定邊界以及軸向預應力作用下的共振頻率與模態振型，並與有限元素法模擬結果相互比較，兩者結果有著良好的對應性。 實驗量測部分使用磁力作為壓電平板的軸向預應力來源，根據磁力理論計算磁鐵在不同距離下的磁力大小，搭配雷射都卜勒測振儀來量測受不同大小磁力作用下壓電平板的振動頻率變化，並結合電子斑點干涉術以量測壓電平板的面外模態振型，部分實驗量測結果雖與理論模型無法完美擬合，但隨著作用力的增加確實能夠增減壓電片第一共振頻率的變化。風洞實驗先使用熱線風速計量測不同流速下圓柱繞流模型之渦街現象，並與二維、三維晶格波茲曼法以及計算流體力學結果進行比較，隨著流速增加，實驗量測結果與三維晶格波茲曼法具有良好對應性。接著量測圓柱複合壓電平板模型下，不同流速的渦流現象，搭配示波器同時量測壓電平板的電壓響應，驗證圓柱繞流所產生的渦街能激振壓電平板，並且當激振頻率與壓電平板的第一共振頻率相互匹配時，壓電平板的能量擷取效率會大幅提升且有較高的電壓輸出。最後架設磁力變換軸向剛性機構於風洞中，量測在不同流速且受不同量值軸向力作用下壓電平板的電壓響應，在相同的流速範圍下利用磁力變化壓電平板軸向剛性的電壓響應，相較於未作變頻設計有著更寬的頻寬顯示具有更好的效率。

This research investigated the vibration characteristics of a piezoelectric plate subjected to pre-stress axial load using the superposition theory and applied to the vortex-induced vibration excitation by lattice Boltzmann method for flow field. The study is complemented by theoretical analysis, finite element method (FEM), and experimental measurements. The stiffness effect of pre-stress axial load on a piezoelectric plate is analyzed and leads to change in the resonant frequency of the piezoelectric energy harvester to correspond with the vortex shedding frequency. The theoretical analysis combines the piezoelectric constitutive equations by Kirchhoff's plate theory to equivalently represent the three-layer structure as a single-layer rectangular plate. The governing equation for the piezoelectric plate subjected to pre-stress axial load is obtained through force and moment balance equations. The superposition method is then employed to divide the plate into multiple structural blocks for the solution, where each block satisfies partial boundary conditions and the governing equation of the original plate. The solutions of individual blocks are superposed to satisfy the remaining boundary conditions. The resonance frequencies and mode shapes of the piezoelectric cantilevered plate under a pre-stress axial load are obtained through orthogonal function expansion and show good agreement with the results in FEM. In experimental measurements, magnetic force applied pre-stress axial load to the piezoelectric plate. Various magnetic forces are provided by the change in distance and analyzed by magnetic theory. Under different magnitudes of magnetic force, a laser Doppler vibrometer is employed to measure the variation of frequency spectrum for the piezoelectric plate. Electronic speckle pattern interferometry is also used to measure the resonant frequencies and their corresponding out-of-plane mode shapes of the piezoelectric plate. At least the first mode in the experiment can fit the theoretical model to change the natural frequency by pre-stress axial load, being used to design the variable-frequency function for the fluid-induced vibration energy harvester. The vortex shedding around a cylindrical bluff body is initially measured using a hot wire anemometer at different flow velocities in the wind tunnel experiments. The vortex frequency results in the 2D and 3D lattice Boltzmann method are compared with CFD results, showing better agreement with the 3D lattice Boltzmann method as the flow-rate changes. Subsequently, the vortex-induced vibration of a piezoelectric plate compound with a cylinder is measured, and the generated voltage of the piezoelectric plate is simultaneously recorded using an oscilloscope. The vortex shedding is intended to excite the first resonant frequency of a piezoelectric plate, as well as the vortex frequency, the efficiency in piezoelectric energy harvesting system increases significantly, resulting in higher voltage output. Finally, an axial force provides different axial stiffness for the piezoelectric plate by modulating magnitudes of magnetic force, to measure the highest voltage of the piezoelectric energy harvester in the wind tunnel, under different flow velocities. Compared to previous cases within the same flow velocity, the piezoelectric plate exhibits a wider bandwidth of voltage response.