Natural-Frequency Analysis of Axially-Loaded Double-Ended Quartz Resonator
Date Issued
2010
Date
2010
Author(s)
Chang, Yu-Wei
Abstract
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.
Subjects
Quartz resonator
Prestressed force
Natural frequency
Euler beam
Timoshenko beam
Hamilton’s Principle
Type
thesis
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