應用Mindlin板理論解析壓電雙晶圓板耦合問題之動態 分析與能量擷取的實驗量測

Abstract

本文先以Mindlin厚板理論及薄板理論解析壓電單層圓板的振動模態，將理論得到的共振頻、模態振形及面外、面內模態分量的比例關係與有限元素法的分析結果作比較。並以不同的厚度及邊界條件比較Mindlin板理論及薄板理論的差異， Mindlin板理論有較佳的分析結果，可作為一個厚板理論來處理較厚的圓板問題。 接著以Mindlin板理論及薄板理論解析壓電雙層圓板於自由邊界及固定邊界的的振動模態，並與有限元素軟體的分析及實驗的量測結果作比較，可得知壓電雙層圓板於不同邊界條件的振動特性。 本文亦配合能量擷取系統研究壓電材料的正壓電效應，以振動器激振試片引發電壓輸出並使發光二極體發亮。整體而言兩種壓電雙層圓板的實驗結果與理論及有限元素軟體的分析結果相當一致，亦可驗證正、逆壓電效應之間的對應關係。

This paper presents the free vibration analysis of piezoelectric thick circular plate based on Mindlin plate theory and thin plate theory. The analytical solutions are derived and validated by comparing the resonant frequencies, corresponding mode shapes and the ratio between in-plane and out-of-plane vibration modes of the piezoelectric circular plates with those obtained by finite element analysis. To investigate the difference between Mindlin plate theory and thin plate theory, we adopt different radius-thickness ratio of the plate under free and clamped boundary conditions. Mindlin plate theory provides better results than thin plate theory with different radius-thickness ratio. This is the reasonwhy Mindlin plate theory is also called thick plate theory. This study investigates the transverse and planar vibration characteristics of two-layered piezoceramic bimorph under free and clamped boundary conditions by theoretical analysis mentioned above, finite element numerical calculation and experimental measurements. It is noted that the resonant frequencies and mode shapes of the series-type and parallel-type disk bimorph present different dynamic characteristics under different boundary conditions. With piezoceramic materials and energy harvesting system, it is possible to harvest power by vibrating structures. An electromechanical shaker is used to excite the specimens of piezoceramic bimorph to generate the electric voltage and to light up the LED. Good agreements of dynamic characteristics determined by theoretical analysis, experimental measurements and numerical calculation are presented for two types of disk bimorphs. Finally, we can conclude the relationship between the inverse piezoelectric effect and the direct piezoelectric effect.