Abstract The frequency-temperature behavior has long been analyzed to ensure the smoothness of functioning frequency of a quartz crystal resonator during the temperature variation and maintain its operations in a stable state. It has been proven that vibrations of quartz crystal plates in the thickness-shear mode can be well described by the theory of incremental thermal field in conjunction with the Mindlin plate equations, demonstrating the possibility that vibration frequencies and mode shapes can also be well predicted with the temperature as a targeted variable. By studying the vibrations with temperature changes in a given range, the frequencies and vibration modes can be calculated as a function of temperature, then they can be used in the calculations of both strain and kinetic energies for vibration mode identification or characterization based on energy proportion and changes. The trends of changes of vibration frequency, amplitudes, and energy with temperature variation make it possible for characterization the variations of vibrations and particularly the weakening or enhancement of the dominance of specific vibration modes. With such relations between vibration properties and temperature known, possible analytical tools can be developed to aid the optimal design by avoiding significant mode changes or occurrences of mode conversion.

中文翻译：

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石英晶体板在应变能和动能热场中振动频率和振型变化的研究

摘要 长期以来，人们一直对频率-温度特性进行分析，以确保石英晶体谐振器在温度变化过程中工作频率的平滑性并保持其工作在稳定状态。已经证明，增量热场理论结合 Mindlin 板方程可以很好地描述石英晶体板在厚度-剪切模式下的振动，证明了振动频率和模式形状也可以很好地预测的可能性温度作为目标变量。通过研究在给定范围内随温度变化的振动，频率和振动模式可以计算为温度的函数，然后它们可用于计算应变能和动能，以根据能量比例和变化进行振动模式识别或表征。振动频率、振幅和能量随温度变化的变化趋势使得表征振动的变化，特别是特定振动模式优势的减弱或增强成为可能。已知振动特性和温度之间的这种关系，可以开发可能的分析工具，通过避免显着的模式变化或模式转换的发生来帮助优化设计。温度变化的能量和能量使得表征振动的变化成为可能，特别是特定振动模式优势的减弱或增强。已知振动特性和温度之间的这种关系，可以开发可能的分析工具，通过避免显着的模式变化或模式转换的发生来帮助优化设计。温度变化的能量和能量使得表征振动的变化成为可能，特别是特定振动模式优势的减弱或增强。已知振动特性和温度之间的这种关系，可以开发可能的分析工具，通过避免显着的模式变化或模式转换的发生来帮助优化设计。