弹性圆板的Mindlin高阶板方程及其振动研究

发布时间：2018-06-12 03:25

  本文选题：圆板 + Mindlin板理论　； 参考：《宁波大学》2014年硕士论文

【摘要】：在工程应用领域,弹性圆板是极其常见的构件。对于它的振动研究,一般基于经典的薄板理论。但是当我们分析中厚板的振动,以及研究弹性板的厚度模态如剪切振动和它的高阶泛音模态时,经典的弹性板理论将不再适用。对于此类问题的分析,需要用到Mindlin或者Lee高阶板理论。这些所谓高阶板理论在矩形板和直角坐标系的情形下,已经有了一个完整的分析步骤。本文将遵循直角坐标的步骤,对弹性圆板在极坐标系下的Mindlin高阶板方程进行系统推导,并对这些方程进行必要的截断、修正和简化。首先,本文分别对直角坐标系和柱坐标系下无限大板的高频振动进行了分析,得到了无限大板精确的色散关系。然后,从柱坐标系下三维弹性力学的基本方程出发,将弹性体的三个位移展开成厚度坐标的幂级数,然后通过变分原理,消去厚度坐标,得到了圆板的Mindlin高阶板方程。对高阶板方程进行截断、修正和简化,也可以得到Mindlin的一阶圆板方程。沿袭Mindlin对直角坐标系下一阶板方程的退化方法,极坐标系下的一阶板方程也能够成功退化到经典板方程。同时,通过一阶板方程的色散关系与精确的色散关系的比较,以此来验证所得到的一阶板方程可以用于圆形板的厚度剪切振动分析。最后,利用Mindlin一阶板理论,分析了弹性圆板的自由振动,分别求解得到了弹性圆板在轴对称和非轴对称振动时的频谱关系和振动模态波形,并与Mindlin利用坐标变换得到的频谱图进行了对比,发现两者的结果是完全一致的。本文主要推导了极坐标系下弹性圆板的Mindlin高阶板方程,分析了各向同性圆板的厚度振动,这是研究圆板高频厚度振动的第一步。在以后的工作中,我们将借助分析各向同性圆板高频振动时积累的经验和方法,建立各向异性圆板的高阶板方程,继而求得频率和厚度模态解,为圆形石英晶体谐振器的设计和分析提供方法和理论依据。
[Abstract]:Elastic circular plate is an extremely common component in engineering applications. The vibration research is based on the classical thin plate theory. But when we analyze the vibration of medium thick plate and study the thickness mode of elastic plate such as shear vibration and its high order overtone mode, the classical elastic plate theory will no longer be applicable. Mindlin or Lee's higher order plate theory is needed for the analysis of this kind of problems. In the case of rectangular plates and rectangular coordinate systems, these so-called higher-order plate theories have a complete analysis step. In this paper, the Mindlin higher order plate equations of elastic circular plates in polar coordinate system will be systematically deduced, and these equations will be truncated, modified and simplified. Firstly, the high frequency vibration of infinite plate in rectangular coordinate system and cylindrical coordinate system is analyzed, and the exact dispersion relation of infinite plate is obtained. Then, starting from the basic equations of three-dimensional elastic mechanics in cylindrical coordinate system, the three displacements of elastic body are expanded into power series of thickness coordinate, and then the Mindlin higher order plate equation of circular plate is obtained by eliminating the thickness coordinate by variational principle. Mindlin's first order circular plate equation can also be obtained by truncating, modifying and simplifying the higher order plate equation. Following Mindlin's degenerate method for the first order plate equation in the rectangular coordinate system, the first order plate equation in the polar coordinate system can also be successfully degenerated to the classical plate equation. At the same time, by comparing the dispersion relation of the first order plate equation with the exact dispersion relation, the obtained first order plate equation can be used to analyze the thickness shear vibration of circular plate. Finally, using Mindlin's first-order plate theory, the free vibration of elastic circular plate is analyzed, and the spectrum relation and vibration mode waveform of elastic circular plate under axisymmetric and non-axisymmetric vibration are obtained, respectively. Compared with the spectrum obtained by Mindlin's coordinate transformation, it is found that the two results are consistent with each other. In this paper, the Mindlin high-order plate equation of elastic circular plates in polar coordinate system is derived, and the thickness vibration of isotropic circular plates is analyzed. This is the first step to study the high-frequency thickness vibration of circular plates. In future work, we will establish the higher order plate equations of anisotropic circular plates by means of the accumulated experience and method of analyzing the high frequency vibration of isotropic circular plates, and then obtain the frequency and thickness modal solutions. It provides the method and theoretical basis for the design and analysis of circular quartz crystal resonator.
【学位授予单位】：宁波大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TB123
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本文编号：2008086