單晶石英加速規之自然頻率理論分析

Abstract

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.