https://scholars.lib.ntu.edu.tw/handle/123456789/81628
標題: | 軸向力作用下雙端固定石英振盪器的自然頻率分析 Natural-Frequency Analysis of Axially-Loaded Double-Ended Quartz Resonator |
作者: | 張育瑋 Chang, Yu-Wei |
關鍵字: | 石英振盪器;預力;自然頻率;尤拉樑;提摩盛科樑;漢米爾頓定理;Quartz resonator;Prestressed force;Natural frequency;Euler beam;Timoshenko beam;Hamilton’s Principle | 公開日期: | 2010 | 摘要: | 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,並且討論預力作用下對於石英振盪器自然頻率的影響大小。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,故為本文研究重點。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單石英樑的自由振動行為,分別使用尤拉樑(Euler beam)理論和提摩盛科樑(Timoshenko beam)理論來模擬單樑的變形,利用漢米爾頓定理(Hamilton’s Principle)建立單樑模型的統馭方程式和邊界條件,並且利用Mathematica數學軟體輔助複雜的數學計算來數值解求解頻率相關的特徵方程式,計算出各模態的共振頻率且討論預力大小對於自然頻率變化的影響。對於整組石英振盪器異相(anti-phase mode)振盪之自然頻率的分析方法如同單石英樑,分別對質量塊與中間雙樑假設變形模型,且將中間樑視為尤拉樑,根據幾何邊界條件得知質量塊與中間樑接合處其位移、斜率和力矩必需相等,求得石英振盪器異相(anti-phase mode)振盪時各模態之自然頻率,並且了解預力與自然頻率的變化關係。 The main purpose of this thesis is to analyze the double-ended tuning fork resonator which is made of the (ZYw)+2° quartz, and to discuss the effect of the prestressed force on the natural frequencies of the quartz resonator. 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, my research focuses on the anti-phase mode. Before performing the analysis of the whole quartz resonator, we first analyze the free vibration of the subsystem such at the free vibration of the pair of slender beams modeled by Euler beam theory as well as the free vibration of the proof mass modeled by Timoshenko beam theory. We adopt the Hamilton’s Principle to establish the governing equations and boundary conditions of the above-mentioned beam models, and we use the “Mathematica” software to solve numerically the frequency-related characteristic equation, thus we can obtain the natural frequencies and corresponding mode shapes. Next we analyze the natural frequencies of the whole quartz resonator. We solve simultaneously the governing equation of the proof mass and the governing equation of the slender middle beam as well as the boundary conditions and the interface conditions such as the continuity conditions of the displacement, slope, and moment at interface. Then, we can obtain the natural frequencies of each mode of the double-ended tuning fork type quartz oscillator, and understand the relationship between the prestressed force and the natural frequencies variation. |
URI: | http://ntur.lib.ntu.edu.tw//handle/246246/249988 |
顯示於: | 應用力學研究所 |
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ntu-99-R97543025-1.pdf | 23.53 kB | Adobe PDF | 檢視/開啟 |
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