The dispersion relations and vibration modes of infinite quartz crystal plates at higher frequencies

Abstract

High frequency vibrations of quartz crystal plates have been subjected to extensive studies for applications in the design and analysis of quartz crystal resonators functioning at the fundamental thickness-shear mode and its overtones. One important result needed in studies and product design is the exact dispersion relations of quartz crystal plates to be used for the validation of approximate theories, notably the Mindlin plate theory and Lee plate theory, for the vibration analysis of finite plates which are essential in the proper selection of resonator configurations. Earlier analyses have been focused on the fundamental thickness-shear vibrations, which have been studied with the first-order plate equations. The analysis of quartz crystal resonators vibrating at overtone modes of the fundamental thickness-shear mode requires the employment of higher-order plate equations, which pose a challenge in the validation before them can be used for solutions of frequency and vibration modes. Clearly, in addition to the improvement of plate equations involving successive higher-order displacements through correction factors, validation of these equations with accurate dispersion relations from the three-dimensional analysis of vibrations of infinite plates are essential. The computational procedure for the accurate dispersion relations of infinite plates have been established, but available numerical results have been restricted to lower frequencies due to the confined interests at resonators of the fundamental thickness-shear mode. To extend the known methods and theory for the analysis of quartz crystal resonators, complete dispersion relations in a large frequency range will be important in deriving the approximate equations suitable for the analysis of vibrations at the higher-order overtones. The piezoelectric effect can also be considered in the dispersion relations under the current implementation. The complicated dispersion relations and patterns of mode intersection can also be used in the prediction of suitable overtone modes with better performance. 无限大石英晶体板高频振动的色散关系和振动模态分析王骥 阳丽君 杜建科 315211 浙江省宁波市江北区风华路 818 号宁波大学机械工程与力学学院压电器件技术实验室 由于石英晶体板高频振动在基频及泛音厚度剪切型石英晶体谐振器的设计和分析中发挥的作用，我们对此进行了 深入研究。在研究和产品设计中需要的一个重要结果是石英晶体板准确的色散关系，它常被用来验证用于分析石 英晶体板振动的近似板理论。Mindlin 板理论和 Lee 板理论是最常用的二维高阶板理论，可以用于分析有限板的 振动，是谐振器选择合适尺寸的基本依据。早期的研究关注于利用一阶板方程对基频厚度剪切振动进行分析。泛 音石英晶体谐振器工作在高阶泛音厚度剪切模态，因此需要利用高阶板方程作振动分析，而在此之前应首先对它 进行验证。除了在包含一系列高阶位移分量的板方程中加入修正系数之外，还需比较无限大板利用高阶板方程得 到的色散关系与由三维运动方程作振动分析得到的准确的色散关系。虽然求解无限大板准确的色散关系的步骤早 就为大家熟知，但由于以前只限于分析基频厚度剪切型谐振器，故已有的计算结果都局限在低频区。为了将已知 的方法和理论扩展用于分析泛音石英晶体谐振器，需要计算更大频率范围的准确的色散关系，以便验证二维高阶 板方程可以用来分析板的高阶泛音振动。在有激励电流作用时需要考虑压电效应得到的色散关系。复杂的色散关 系和模态分析可为谐振器选择合适的泛音工作模态提供参考，以期得到更好的工作性能。 关键词：色散关系，板，Mindlin，振动，石英晶体，谐振器 1. 引言 石英晶体谐振器是利用石英晶体压电效应制成的谐 振器件，是晶体振荡器和滤波器的关键元件，广泛用 于产生高频电子信号，实现精确计时、频率标准、制 导和滤波等测量和信号处理系统中必不可少的频率 控制功能。石英晶体谐振器也是用于温度、压力、重 力、 气体等传感器的敏感元件。 由于高端应用的需求， 其发展趋势为高频率、 频率高精度和高稳定、 小尺寸、 片式化、低噪声等。 978-1-4244-4949-1/09/$25.00

Scalar Differential Equation for Slowly-Varying Thickness-Shear Modes in AT-Cut Quartz Resonators With Surface Impedance for Acoustic Wave Sensor Application

“…For small |X|, the solid lines are approximations of the odd TSh branches of the exact dispersion curves [45] with approximately the same intercepts on the frequency axis, the same slope when X = 0, and the same curvature when X = 0. Therefore (20) can be used to describe these waves approximately for small |X| or long waves.…”