Mindlin彈性板理論的修正及其效果

本論文主要用的是修正后的Mindlin高階彈性板理論，是在Mindlin彈性板理論的基礎(chǔ)上進(jìn)行修正后得到的理論,在位移方向增加彎曲模態(tài)分量作了很大的進(jìn)步。其中Mindlin板理論都是分析石英晶體諧振器的基礎(chǔ)。對(duì)于此理論的修正，前人已做過(guò)很多這方面的努力，也得出了有效和精確的結(jié)果。我們首先用Mindlin彈性板理論分析了AT切石英晶片中厚度-剪切和彎曲振動(dòng)模態(tài)之間的色散關(guān)系，再用修正后的Mindlin板理論得到色散關(guān)系，并對(duì)兩者進(jìn)行比較說(shuō)明，修正后的Mindlin板理論得到的結(jié)果明顯要優(yōu)于Mindlin板原理論。用修正后的Mindlin板理論分析了晶片中厚度-剪切、彎曲振動(dòng)和拉伸振動(dòng)模態(tài)耦合的色散關(guān)系圖和頻譜圖，其中的頻譜圖可作為晶片重要參數(shù)選擇的依據(jù)。隨著晶體諧振器向小型化和高性能方向發(fā)展，控制這些模態(tài)的耦合問(wèn)題便成為保證晶片振動(dòng)性能的關(guān)鍵。所以在實(shí)際的生產(chǎn)設(shè)計(jì)過(guò)程中，提出了更高的要求。
關(guān)鍵詞：Mindlin板，修正，石英晶體，諧振器，模態(tài)耦合

This paper mainly uses improved Mindlin higher-order elastic plate theory, which was improved on the basis of Mindlin elastic plate theory . Significant progress has been made increasing flexural mode component representations . Where Mindlin plate equations are the foundation of quartz crystal resonator analysis . For these theories , great efforts have been made to improve the equations and obtain efficient and accurate solutions in past decades .First , we analyze the relation between thickness-shear and flexural vibrations mode in the AT-cut quartz crystal plate by Mindlin elastic plate theory , then obtain the result by the improved Mindlin plate theory , finally compare them and find the differences . The result of improved Mindlin plate theory excelled the Mindlin plate theory . Next we analyze the dispersion relations and frequency versus length to thickness ratio of coupled thickness-shear ,flexural and extensional vibrations in the AT-cut quartz crystal plate by improved Mindlin plate theory , when we determine the optimal important design parameters of quartz crystal plate , above conclusions may be our basis . With the development of smaller and high quality resonators, the control of coupling of modes is the key to the property of the vibration of quartz crystal . So in the resonator design process of a crystal blank ,presents more challenging problems .
Keywords: Mindlin plate, improve , quartz crystal, resonator, coupling mode

第一章 緒 論4
1. 1 畢業(yè)設(shè)計(jì)（論文）的背景、意義..4
1.2 畢業(yè)設(shè)計(jì)（論文）所作的工作..4
第二章 Midnlin板理論介紹..5
2.1 無(wú)窮冪級(jí)數(shù)的二維板方程.5
2.1.1 二維方程中板的位移形式..5
2.1.2 二維方程中板的應(yīng)變應(yīng)力形式5
2.1.3 彈性體的運(yùn)動(dòng)變分方程6
2.2 算例..9
2.2.1 AT切石英晶體的自由振動(dòng).9
2.2.2 厚度-剪切和彎曲振動(dòng)模態(tài)的色散關(guān)系10
第三章 修正后的Mindlin板理論.12
3.1 修正后的Mindlin板理論的二維方程12
3.1.1 修正后的Mindlin板理論中的位移、應(yīng)變和應(yīng)力形式.12
3.1.2 修正后的Mindlin板二維運(yùn)動(dòng)方程.13
3.2 算例14
3.2.1 AT切石英晶體的運(yùn)動(dòng)方程..14
3.2.2 色散關(guān)系.15
3.2.3 結(jié)果比較.16
3.3 厚度-剪切和彎曲耦合振動(dòng)的頻譜關(guān)系..17
第四章 多種模態(tài)耦合振動(dòng)的分析.20
4.1 三種模態(tài)耦合振動(dòng)的色散關(guān)系20
4.2 三種模態(tài)耦合振動(dòng)的頻譜關(guān)系21
4.3 位移關(guān)系.24
第五章 結(jié)論和展望.27
參考文獻(xiàn)...28
致 謝.29
附 錄.30
附錄1：畢業(yè)設(shè)計(jì)說(shuō)明書(shū)、設(shè)計(jì)圖紙等材料刻錄光盤(pán)..30
附錄2：壓電晶體諧振器重要參數(shù)的最優(yōu)設(shè)計(jì)綜述 ……見(jiàn)《附件材料》.30
附錄3：彈性板振動(dòng)的數(shù)學(xué)理論介紹 ……見(jiàn)《附件材料》.30
附錄4：Mindlin高階板理論的修正 ……見(jiàn)《附件材料》..30