臺灣博碩士論文加值系統

本文主要分析(YXl)-88度石英樑以及雙端固定音叉式石英振盪器之共振頻率，首先分析石英樑的共振頻率，利用漢米爾頓定理(Hamilton''s Principle)與變分法建立統馭方程式與邊界條件，再使用分離變數法求得特徵方程式、並利用解析解求得特徵值，求得到共振頻率。對於雙端固定音叉式石英振盪器分成同向(in-phase mode)與異向(anti-phase mode)振盪，同向振盪(in-phase mode)可將質量塊視為提摩盛科樑，中間樑為尤拉樑，而對於異向振盪(anti-phase mode)，對質量塊提出新位移場模型，中間樑視為尤拉樑。同向(in-phase mode)與異向(anti-phase mode)振盪兩者都與石英樑方法一樣，利用漢米爾頓定理(Hamilton’s Principle)與變分法建立統馭方程式與邊界條件，再使用分離變數法求得特徵方程式、並利用解析解求得特徵值，求得到共振頻率，所得到的解析解與實驗結果相當符合。利用相同方法分析單音叉的同向(in-phase mode)與異向(anti-phase mode)共振頻率，並建立32.768KHz的理論尺寸。

The thesis is mainly to investigate the resonance frequencies of the (YXl)-88度 quartz beam and the (YXl)-88度 double-ended tuning fork quartz oscillator. First, the resonance frequencies of the quartz beam is analyzed in step one. Governing equations and boundary conditions are obtained by using Hamilton’s Principle and variational principle of mechanics. By applying the separation of variables method, we can derive the eigenequations. The eigenvalues can be obtained by using the analytic solutions. Thus, we can calculate the resonance frequencies of the quartz beam. The modes of the double-ended tuning fork quartz oscillator can be divided into the in-phase mode and the anti-phase mode. For the case of in-phase mode, the proof masses are simulated by using the assumption of Timoshenko beam, and the single beams are simulated by using the assumption of Euler beam. In anti-phase mode, we develop the assumption of anti-phase mode shapes of proof masses, and the single beams are simulated by using the assumption of Euler beam. The problem-solving processes of the in-phase mode and the anti-phase mode are the same as those of the former. Governing equations and boundary conditions are obtained using Hamilton’s Principle and variational principle of mechanics. By applying the separation of variables method, we can derive the eigenequations. The eigenvalues can be obtained by using the analytic solutions. Thus, we can calculate the resonance frequencies of the quartz tuning fork oscillator. The analytic solutions are closely consistent with the experimenting results. By using the same methods, we can analyze the resonance frequencies of the in-plane mode and the out-of-plane mode of the single ending tuning fork oscillator, and derive the theoretical sizes when the frequency is 32.768KHz.

口試委員會審定書………………………………………………………………………i
誌謝………………………………………………………………………………………ii中文摘要………………………………………………………………………………iii
英文摘要…………………………………………………………………………………iv目錄……………………………………………………………………………………vi
圖目錄…………………………………………………………………………………ix
表目錄…………………………………………………………………………………xv
第一章　　導論………………………………………………………………………1
1.1　前言………………………………………………………………………………1
1.2　石英加速規原理…………………………………………………………………2
1.3　文獻回顧…………………………………………………………………………3
1.4　本文目的與章節摘要…………………………………………………………10
第二章　　石英材料特性……………………………………………………………12
2.1　石英材料………………………………………………………………………12
2.1.1 晶格對稱性……………………………………………………………………12
2.1.2 對稱操作………………………………………………………………………12
2.1.3 三維晶格類型…………………………………………………………………15
2.1.4 石英晶體的外型………………………………………………………………16
2.2　石英材料常數…………………………………………………………………17
2.2.1 應力與應變……………………………………………………………………17
2.2.2 石英晶體的材料常數…………………………………………………………18
2.2.3 石英晶體的切角設計…………………………………………………………20
第三章　　單樑振動………………………………………………………………23
3.1　理想截面之尤拉樑(Euler beam)……………………………………………23
3.1.1 定義座標方向以及長度………………………………………………………23
3.1.2 尤拉樑(Euler beam)之變形假設模型………………………………………24
3.1.3 漢米爾頓定理(Hamilton''s principle)…………………………………………28
3.1.4 尤拉樑(Euler beam)之應變能及動能…………………………………………28
3.1.5 尤拉樑(Euler beam)之統馭方程式及邊界條件……………………………30
3.1.6 外型尺寸與頻率的影響………………………………………………………37
3.2　理想截面之提摩盛科樑(Timoshenko beam)………………………………39
3.2.1 提摩盛科樑(Timoshenko beam)之變形假設模型…………………………39
3.2.2 提摩盛科樑(Timoshenko beam)之應變能及動能…………………………40
3.2.3 提摩盛科樑(Timoshenko beam)之統馭方程式及邊界條件………………42
3.2.4 外型尺寸與頻率的影響………………………………………………………49
3.3　溼蝕刻截面之尤拉樑(Euler beam)……………………………………………52
3.3.1 基本假設………………………………………………………………………52
3.3.2 溼式蝕刻之尤拉樑的統馭方程式及邊界條件………………………………52
3.3.3 外型尺寸與頻率的影響………………………………………………………60
3.4 溼式蝕刻之提摩盛科樑(Timoshenko beam)…………………………………62
3.4.1 基本假設………………………………………………………………………62
3.4.2 溼式蝕刻之提摩盛科樑的統馭方程式及邊界條件…………………………63
3.4.3 外型尺寸與頻率的影響………………………………………………………71
第四章　　雙端固定音叉式石英振盪器之同向振盪模態…………………………75
4.1　同向振盪之質量塊……………………………………………………………75
4.1.1 同向振盪之質量塊變形假設…………………………………………………75
4.1.2 同向振盪之質量塊的統馭方程式與邊界條件………………………………77
4.2 同向振盪之石英振盪器…………………………………………………………82
4.2.1 同向振盪之統馭方程式與邊界條件…………………………………………82
4.2.2 同向振盪之外型尺寸對頻率的影響…………………………………………95
第五章　　雙端固定音叉式石英振盪器之異向振盪模態………………………101
5.1 異向振盪之線性旋轉變形質量塊………………………………………………101
5.1.1 異向振盪之線性旋轉變形質量塊變形假設…………………………………101
5.1.2 異向振盪之線性旋轉變形質量塊統馭方程式及邊界條件…………………104
5.2 異向振盪之線性旋轉變形石英振盪器………………………………………109
5.2.1 異向振盪之線性旋轉變形的統馭方程式與邊界條件………………………109
5.2.2 異向振盪之外型尺寸對頻率的影響…………………………………………116
5.3 異向振盪之非線性旋轉變形質量塊…………………………………………118
5.3.1 異向振盪之非線性旋轉變形質量塊變形假設………………………………118
5.4 異向振盪之非線性旋轉變形石英振盪器………………………………………124
5.4.1 異向振盪之非線性旋轉變形的統馭方程式與邊界條件……………………124
5.4.2 異向振盪之外型尺寸對頻率的影響…………………………………………132
5.5 單音叉式振盪器之異向振盪……………………………………………………137
第六章　　結論……………………………………………………………………140
參考文獻……………………………………………………………………………142
附錄A………………………………………………………………………………146
附錄B………………………………………………………………………………157
附錄C………………………………………………………………………………160
附錄D………………………………………………………………………………162
附錄E…………………………………………………………………………………166
附錄F…………………………………………………………………………………168
附錄G………………………………………………………………………………170
附錄H…………………………………………………………………………………173
附錄I…………………………………………………………………………………174
附錄J………………………………………………………………………………176
附錄K………………………………………………………………………………180
作者簡歷……………………………………………………………………………183

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