Natural Frequency Analysis of Axially-Loaded on the Double-Ended Tuning Fork Piezoelectric-Quartz Resonator

本文主要分析(ZYw)+2°雙端固定音叉式石英振盪器之共振頻率，分析預力作用與壓電效應對於共振頻率的影響，以獲得更接近真實情況與應用的共振頻率變化。雙端固定音叉式石英振盪器可由兩端的質量塊以及中間音叉雙樑所組成，其振動模態可分為同相 (in-phase mode) 振盪和異相(anti-phase mode)振盪兩種，而雙端固定音叉式石英振盪器理想的振盪模態為異相(anti-phase mode)振盪，故本文針對異相振盪行為進行分析，在做石英振盪器解析分析之前，先討論不考慮質量塊效應下石英單樑的自由振動行為，使用尤拉樑(Euler beam)理論來模擬單樑的變形，利用漢米頓原理(Hamilton’s Principle)建立單樑模型的統御方程式和邊界條件，並利用Mathematica數學軟體輔助複雜的數學計算，以數值解求解頻率相關的特徵方程式，並且計算出各模態的共振頻率。 兩端質量塊依據音叉樑異相振盪對質量塊所造成的力矩來建立翹曲形變的模型；耦合部分，假設其結構為彈性體，分別討論各自結構自由振動行為，由上述漢米頓原理所推導出運動統御方程式及邊界條件，根據單樑與質量塊接合處的幾何邊界條件，將其進行耦合，以獲得整體石英振盪器各模態之自然頻率，並討論預力與自然頻率變化關係。The thesis mainly studied about a Z-cut2° double-ended tuning fork quartz resonator, analyzing the impact of the pre-stressed force and piezoelectric effect on a natural frequency, for the sake of approaching the real situation and application of resonance frequency change. The double-ended tuning fork type quartz resonator is composed of a pair of slender Euler beams and two proof masses located at the two ends of the resonator. There are two vibration modes of the tuning fork for the same order mode shape that is in-phase mode and anti-phase mode. The thesis is mainly focus on the analysis of the anti-phase mode, Before performing the analysis of the whole quartz resonator. First, it doesn''t consider the free vibration behavior of the mass effect on quartz single beam. Simulate single beam deformation by Euler beam theory, using the Hamilton''s principle building the governing equation and the boundary condition of the single beam model, and to use the “Mathematica” software to solve the frequency-related characteristic equation numerically, and to calculate the natural frequencies of mode sharps. For two ends of the proof masses building the warping model which is according to the moment that caused by the anti-phase mode from the tuning fork to the proof masses； Assuming the coupling structure as an elastic body, discussing the free vibration to each structure, By Hamilton''s principle getting the governing equation and the boundary condition. For coupling, to base on the single beam and proof mass of geometric boundary conditions at interface, obtaining the natural frequencies of each mode for the double ended tuning fork type quartz oscillator, and to acquire the relationship between the pre-stressed force and the natural frequencies variation