本文選題：圓板 + Mindlin板理論　； 參考：《寧波大學(xué)》2014年碩士論文

【摘要】：在工程應(yīng)用領(lǐng)域,彈性圓板是極其常見的構(gòu)件。對(duì)于它的振動(dòng)研究,一般基于經(jīng)典的薄板理論。但是當(dāng)我們分析中厚板的振動(dòng),以及研究彈性板的厚度模態(tài)如剪切振動(dòng)和它的高階泛音模態(tài)時(shí),經(jīng)典的彈性板理論將不再適用。對(duì)于此類問題的分析,需要用到Mindlin或者Lee高階板理論。這些所謂高階板理論在矩形板和直角坐標(biāo)系的情形下,已經(jīng)有了一個(gè)完整的分析步驟。本文將遵循直角坐標(biāo)的步驟,對(duì)彈性圓板在極坐標(biāo)系下的Mindlin高階板方程進(jìn)行系統(tǒng)推導(dǎo),并對(duì)這些方程進(jìn)行必要的截?cái)�、修正和�?jiǎn)化。首先,本文分別對(duì)直角坐標(biāo)系和柱坐標(biāo)系下無(wú)限大板的高頻振動(dòng)進(jìn)行了分析,得到了無(wú)限大板精確的色散關(guān)系。然后,從柱坐標(biāo)系下三維彈性力學(xué)的基本方程出發(fā),將彈性體的三個(gè)位移展開成厚度坐標(biāo)的冪級(jí)數(shù),然后通過變分原理,消去厚度坐標(biāo),得到了圓板的Mindlin高階板方程。對(duì)高階板方程進(jìn)行截?cái)�、修正和�?jiǎn)化,也可以得到Mindlin的一階圓板方程。沿襲Mindlin對(duì)直角坐標(biāo)系下一階板方程的退化方法,極坐標(biāo)系下的一階板方程也能夠成功退化到經(jīng)典板方程。同時(shí),通過一階板方程的色散關(guān)系與精確的色散關(guān)系的比較,以此來(lái)驗(yàn)證所得到的一階板方程可以用于圓形板的厚度剪切振動(dòng)分析。最后,利用Mindlin一階板理論,分析了彈性圓板的自由振動(dòng),分別求解得到了彈性圓板在軸對(duì)稱和非軸對(duì)稱振動(dòng)時(shí)的頻譜關(guān)系和振動(dòng)模態(tài)波形,并與Mindlin利用坐標(biāo)變換得到的頻譜圖進(jìn)行了對(duì)比,發(fā)現(xiàn)兩者的結(jié)果是完全一致的。本文主要推導(dǎo)了極坐標(biāo)系下彈性圓板的Mindlin高階板方程,分析了各向同性圓板的厚度振動(dòng),這是研究圓板高頻厚度振動(dòng)的第一步。在以后的工作中,我們將借助分析各向同性圓板高頻振動(dòng)時(shí)積累的經(jīng)驗(yàn)和方法,建立各向異性圓板的高階板方程,繼而求得頻率和厚度模態(tài)解,為圓形石英晶體諧振器的設(shè)計(jì)和分析提供方法和理論依據(jù)。
[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.
【學(xué)位授予單位】：寧波大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：TB123
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本文編號(hào)：2008086