石英壓電平板高頻振動分析

The Analysis of High-Order Overtone Vibrations in Quartz Piezoelectric Plates

莊可丞(Ke-Cheng Zhuang)

指導教授 ： 黃自貴

國立虎尾科技大學/工程學院/機械設計工程研究所/碩士(2014年)

https://doi.org/10.6827/NFU.2014.00197

摘要

使用壓電晶體平板理論，可以預測壓電晶體平板之振動資訊，以簡化使用有限元素分析軟體所需計算之範圍，進而簡化實驗範圍，大幅提升實驗效率。現今壓電晶體平板理論中，較為著名的有Mindlin板理論與Lee板理論，其中，Lee使用二維壓電晶體平板理論計算出AT-cut與SC-cut石英平板在一倍頻時之頻散關係，且與三維理論計算出來的數據相當接近，之後接著計算出在一倍頻時之共振響應模態與電容比，其中，對AT-cut方形石英平板之計算結果與另一些學者之實驗數據幾乎吻合，而在當時尚無對SC-cut方形石英平板之共振響應模態以及電容比的實驗數據。
本研究使用Lee的平板理論，並延伸到三倍頻，計算出SC-cut石英平板在x_1與x_3方向之三倍頻頻散曲線圖，並提取一倍頻的頻率範圍與Lee的文獻中計算出來的比對，發現本研究計算出來的頻散曲線圖與Lee文獻圖中三維理論計算出來的幾乎吻合。由頻散曲線圖中能觀察一些模態的截止頻率。接著，我們計算出三倍頻之共振響應模態的關係圖，由共振響應圖中能觀察方形壓電平板在特定尺寸之振動頻率的分布情形。另外，本研究使用延伸到三倍頻之Lee二維壓晶體電平板理論，計算出受外力激振之SC-cut方形石英平板在x1與x3方向之三倍頻之電容比，並將三倍頻之電容比與同樣頻率範圍之共振響模態比對，發現電容比中的激振頻率與共振響應模態中同樣尺寸比的振動模態皆有對應。藉由觀察在三倍頻時之電容比之相對大小，能更確切地得知方形壓電平板在高頻時的每個振動模態之互相干擾情形。
本研究使用有限元素分析軟體Comsol的三維模組，針對AT-cut與IT-cut方形石英平板共兩個尺寸，進行分析其在一倍頻時之電容比，並與使用Lee二維一倍頻壓電晶體平板理論所計算之電容比進行比對，發現使用Comsol三維模組分析之結果與使用Lee二維一倍頻壓電晶體平板理論所計算之結果，其許多共振頻率的電容比變化趨勢相當接近。其中，使用Comsol三維模組針對AT-cut方形石英平板之電容比分析結果，與另一些學者之實驗結果比對，其每個振動模態之共振頻率的電容比趨勢幾乎吻合。

關鍵字

壓電晶體平板理論 ； P.C.Y Lee ； Mindlin ； AT-cut ； SC-cut ； 頻散曲線圖 ； 截止頻率

並列摘要

In the Lee plate theory, 2D theories for piezoelectric plate theory were deduced from 3D theories by expansion in trigonometric function of the thickness coordinate. According to these theories, the first-order dispersion curves for AT-cut and SC-cut rectangular quartz plates can be calculated without using correction coefficients. These dispersion curves approximate those obtained when using 3D theory, which was also used to calculate first-order resonance responses. The first-order resonance responses for AT-cut quartz plates were nearly identical to the experimental results obtained by other researchers. Subsequently, Lee calculated the capacitance ratios of quartz plates vibrating at the first overtone. The calculation results for AT-cut rectangular quartz plates were almost identical to experimental results yielded in other studies. At that time, no data for the resonance responses and capacitance ratios of SC-cut rectangular quartz plates had yet been obtained. In the Lee plate theory, the displacement field and electric potential were expanded to infinity along the thickness coordinate using a trigonometric function. Therefore, the Lee 2D plate theory involves a theory of an infinite system in which a number is calculated and expanded to a limited number of items. Recently, Lee proposed a method for extracting a number at high orders.
This study adopted the Lee plate theory and extended the theory to the third order. Subsequently, the method by which a limited number of items is extracted based on the infinite-system theory was employed to calculate the third-order dispersion curves for SC-cut quartz plates in the x1 and x3 directions. The extracted first-order frequency range was then compared with the results obtained by Lee. The results showed that the dispersion curves obtained in this study were nearly identical to those achieved by Lee using the 3D theory. This study also calculated the third-order resonance frequency-to-size ratios and examined the curve distribution.
This study extends the previous study, retaining the use of Lee’s two-dimensional piezoelectric crystal plate theory, which is applicable to vibration at the third overtone, to calculate the capacitance ratios of an SC-cut rectangular quartz plate that vibrates at the third overtone because of external forces in the x1 and x3 directions. The capacitance ratios of a quartz plate vibrating at the third overtone were also compared with resonance response modes for the same frequency range. The results indicated that the vibration frequencies for various capacitance ratios corresponded with the vibration modes of the same size ratio for various resonance responses. By observing the size of the capacitance ratio of a rectangular piezoelectric plate vibrating at the third overtone, how each vibration mode of the rectangular piezoelectric plate at high frequency interferes with other vibration modes can be clearly understood.

並列關鍵字

Plate theory ； P.C.Y. Lee ； Mindlin ； AT-cut ； SC-cut ； Dispersion curve ； Cut-off frequency

參考文獻

[36]Nicholas Grant Davey, 1998, Resonances of AT-cut Quartz Strips, A thesis from University of Auckland.

[1]Mindlin, R.D., 1951, “Thickness-Shear and Flexural Vibrations of Crystal Plates”, J. Appl. Phys., 22, no. 3, pp. 316-323.

[2]Mindlin, R.D., 1952, “Forced Thickness-Shear and Flexural Vibrations of Piezoelectric Crystal Plates”, J. Appl. Phys., 23, no. 1, pp. 83-88.

[3]Mindlin, R.D., and Deresiewicz, H., 1955, “Thickness-Shear and Flexural Vibrations of Rectangular Crystal Plates”, J. Appl. Phys., 26, no. 12, pp. 1435-1442.

[4]Mindlin, R.D., 1961, “High Frequency Vibrations of Crystal Plates”, Quart. Appl. Math., 19, p.51-61.