理論解析與實驗量測壓電平板的面外振動及特性探討

Theoretical Analysis and Experimental Measurement of Out-Plane Vibration and Characteristics for Piezoelectric Rectangular Plate

吳亦莊(Yi-Zhuang Wu)

指導教授 ： 馬劍清

國立臺灣大學/工學院/機械工程學研究所/碩士(2009年)

https://doi.org/10.6342/NTU.2009.01202

摘要

本文主要討論等向性平板的面外振動並計算PZT壓電平板的本構方程式且將之等效為等向性平板；接著利用解析等向性平板的方式應用於壓電材料上。首先我們利用力學平衡的方式來推導等向性平板的統御方程式和力學條件，接著以疊加法的方式求出面外的振動頻率以及面外位移之特性；於邊界條件上的探討，我們討論自由邊界與單邊固定之懸臂板兩種不同的邊界條件，並且以有限元素法和電子斑點干涉術量測來驗證理論的準確性；接著我們利用變分法找出壓電平板的本構方程式，並利用漢米爾頓原理求出統御方程式和力學、電學的方程式，接著可將壓電平板等效為等向性平板的形式並求出共振頻率與面外位移方程式；又可利用位移函數導出力電條件的函數表示式，並比對有限元素法與電子斑點干涉術與雷射都普勒振盪器之結果以驗證理論之正確與否，最後由Maxwell方程式我們可以推導出電場生成之等效磁通量場，並觀察其分佈特性。本文不僅以理論分析共振之頻率和位移模態，並且由力電關係討論壓電模態的激振趨勢；如此可進一步了解壓電材料的動態特性與力電耦合之關係。

關鍵字

等向性平板 ； 疊加法 ； 壓電平板 ； 變分法 ； 漢米爾頓原理 ； 有限元素法 ； 共振頻率 ； 電子斑點干涉術 ； 雷射都普勒振盪器

並列摘要

In this thesis first we will analyze the response frequencies and mode shapes of isotropic thin plate for out-plate displacement and then we could use the method to find out the dynamic characteristics of piezoelectric rectangular plate. So we use the equilibrium of mechanics to derive the governing equation of isotropic plate and mechanical conditions. The analytical solutions for out-plane displacement field with boundary conditions and response frequencies are obtained by superposition method. In this thesis the issues we consider the are completely free plate and cantilever plate.
The analytical solution are compared with two methods to confirm the validity. One is the numerical computation from the finite element method (FEM), another is amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI) technique. Next we use the variation method to obtain the constitutive equations of piezoelectric plate. To find out the governing equation and the formulas of mechanics and electricity, Hamilton’s principle is the method we usually use. Then we could find out the governing equation and function of mechanics and electricity. The PZT plate is the transverse isotropic material, so it can be equivalent to isotropic plate. Then we could use the solution of isotropic plate to solve the out-plate displacement field and response frequencies for PZT plate. After getting the out-plane displacement field formula, we could obtain the mechanical and electric equations and comparing the result by FEM method, AF-ESPI and laser Doppler vibrometer (LDV) to confirm the validity. Finally we use the Maxwell equation to find out the equivalent magnetic flux, and observe the magnetic quantity distribution. In this thesis, we not only discuss the response frequencies and out-plane displacement, but use the relationship between mechanics and electricity to obtain the tendency of modal excitation. Then we could get more information of the relation between mechanical and electric coupling.

並列關鍵字

sotropic rectangular plate ； superposition method ； piezoelectric rectangular plate ； variation method ； Hamilton’s principle ； finite element method ； response frequencies ； AF-ESPI ； LDV

參考文獻

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