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標題: | 雙端固定壓電樑的振動理論分析 Free-Vibration Analysis of Piezoelectric-Quartz Double-Ended Tuning Fork |
作者: | Chun-Tse Chang 張鈞策 |
指導教授: | 周傳心(Chan-Shin Chou) |
共同指導教授: | 張家歐(Chia-Ou Chang) |
關鍵字: | 石英振盪器,壓電效應,自然頻率(共振頻率), quartz resonators,piezoelectricity,natural frequency(resonant frequency), |
出版年 : | 2010 |
學位: | 碩士 |
摘要: | 本文主要目的為分析”雙端固定音叉式石英(ZYw+2°)振盪器”的壓電效應對自然頻率之影響。此振盪器主要是以一對音叉(雙樑)再加上兩端的質量塊所構成,對表面所佈的電極輸入電壓訊號驅動之,其振動的行為共分為同向振盪(in-phase mode)與異相振盪(anti-phase mode)兩種情況,而後者才是本文欲探討的重點。
在此採用的分析方式為純理論計算並以數學套裝軟體Mathematica輔助。首先,先忽略質量塊的效應將中間音叉視為一簡單的單樑模型並以尤拉樑(Euler beam)假設出位移做計算而電位則是以級數展開之觀點做假設,以上皆是以二維空間變數的觀點來探討,再使用漢米爾頓定理(Hamilton’s principle)推到出運動統御方程式與邊界條件進而求出特徵解,最後再使用邊界條件計算出共振頻率。而整組振盪器的分析只要再加入兩邊質量塊的假設模型,以先前方式個別推出統馭方程式與邊界條件;最後再將質量塊與音叉的統御方程式聯立並以設定好的幾何條件以及音叉與質量塊交界的連續性質來當作邊界條件可用來求得自然頻率。 最後我們可以再依據計算好的自然頻率做進一步的討論,可以推導出音叉樑在每個模態之下,其振形、內部電場以及應變的分佈情形,再根據這些物理量於音叉內部分佈的情況與實際振盪器的電極配置做相關的討論。 The purpose of this paper is to study the effect of piezoelectricity on the natural frequencies of DETF-type(double-ended tuning fork) resonator. which is made of ZYw+2°-cut quartz. This resonator is a tuning fork which consist of a pair of slender Euler beams and two proof masses at two ends of tuning fork. The tuning fork is driven by inputting alternative voltage into the electrodes coated on the surface of tuning fork. There are two different types of vibrational mode of tuning fork for the same order mode shape. That is “in-phase mode” and ”anti-phase mode”. We focus on the analysis of anti-phase mode. The approach of this research is theoretical analysis, and using software ”Mathematica” to help us to calculate some detail compute. First of all, we neglect the effect of proof mass then construct a beam model to simulate tuning fork. Here we use “Euler beam” theory and “power series” method to describe the displacement and electric potential. Second, using Hamilton’s principle to derive the governing equation and boundary conditions of this problem. Finally, we can get characteristic solution and use boundary conditions to find out the resonant frequencies. As to the natural-frequency evaluation of the entire DEFT resonator, we need, in advance, to establish the proper displacement model of the proof mass and derive its equations of motion and natural boundary conditions. Then we solve simultaneously the equations of motion of the tuning-fork beams and two proof masses together with their geometric and natural boundary conditions, and the continuity equations of displacement, slopes, moment at the interfaces, to get the natural frequencies of the flexible vibrations. Based on these obtained natural frequency of each mode, we can find out the corresponding mode shape, the electric field, and the strain field of the quartz tuning-fork beams. From the distributions and the relationships of the strain and electric fields, we can design the layout of the electrodes for efficiently driving the resonator into vibration at a particular natural frequency. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/47587 |
全文授權: | 有償授權 |
顯示於系所單位: | 應用力學研究所 |
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