壓電週期結構疊層板之波傳頻譜分析

Abstract

本文主要是要探討壓電週期疊層板波傳現象，所以由本構方程式配合運動方程式和高斯定理以獲得統御方程式，此等方程式在週期結構內具有平移對稱關係，所以位移場和電位場的形式可以布拉克理論表示，進而得到代表週期結構行為的統御方程式。將應力和電位移場以位移和電位場表示，即可利用層與層間的位移、應力、電位和電位移必須連續等條件求得轉換矩陣，再滿足上下表面的邊界條件即能求得出壓電週期疊層板波傳頻譜。 在實驗上使用電擊激振方式以及雷射都卜勒測振儀和頻譜分析儀等設備量測均質壓克力材料、鋁-壓克力和空氣-壓克力週期結構在平面應力行為時的頻帶關係，並與平面波法所求得的平面應力色散曲線之頻帶間隙作比較，可獲得實驗結果與理論值具有一致性。 在數值分析上，從二維週期結構平行於無限厚板之波向量為零問題開始探討壓電效應對橫向極化的影響，並計算出壓電週期結構頻譜結構以及頻帶間隙與填充比關係；再探討平行於無限厚板之波向量非為零的問題上計算出等向性週期結構複合材也具有頻帶間隙現象；並計算兩種不同厚度之等向性週期結構單層板與由三層組成週期疊層板，利用所推導之理論以求得頻譜圖和各場在空間分布的情形，由頻譜圖中發現僅具有局部的頻帶間隙現象。

The dissertation focuses on the phenomenon of wave propagation of piezoelectric periodical composite laminated plate. By using the motion equations and Gauss theorem cooperative with constitutive relations of piezoelectricity, the governing equations can be determined. Because the governing equations have the relationship of translation symmetry, the displacement fields and the electrical potential fields can be represented by Bloch theorem to get the periodical structural governing equations. For gaining the band structure of piezoelectric periodical composite laminated plate, the boundary conditions of the upper and lower surfaces should be satisfied first, where the boundary conditions can be determined with the transfer matrix from the coefficients between layers and layers, such as displacements, stresses, electrical potentials, and electrical displacements, on condition of continuity. These coefficients result from the formal expressions of displacement fields and electrical potential fields, according to the constitutive relations of piezoelectricity. To verify the derived results from above mathematical analyses, the experiment is applied. The experiment is set up with these two instruments, laser Doppler vibrometers and spectrum analyzer, to observe the response of the electric discharge pulse strikes. The observation is for the band relations from the reactions of two periodical structures, the aluminum alloy cylinders in a PMMA background and the air cylinders in a PMMA background. Compared these observed results with the band gaps of these dispersion curves from the plane stress in the plane wave method, our mathematical analyses provided good agreement. Moreover, the mathematical analyses can be applied to more or even infinite layers structures. The probing for piezoelectric effects to transverse polarization on a two-dimensional structure of the infinite thick plate is designed, where the wave-vector component parallel to the structure is zero. The band structure is calculated for this piezoelectric periodical structure. The relationship of band gaps and filling fraction is therefore obtained. A discussion on isotropic periodical structures without the assumption of the wave-vector component parallel to the structure being zero is provided. We rule out these structures with the band gaps also. According to this rule, the formulations for isotropic periodical plates with different thickness are generated. The formula for three-layer structures can be determined from formulas of single-layer structures. Besides, the frequency spectrum diagram can be plotted according to the formula. In the frequency spectrum diagram, there are only local band gaps found.