應用Mindlin板理論與高階剪切形變理論解析固體耦合的振動特性

Theoretical Analysis of Vibration Characteristics for Solids Based on Mindlin and High Order Shear Deformation Theories.

吳亦莊(Yi-Chuang Wu)

指導教授 ： 馬劍清

國立臺灣大學/工學院/機械工程學研究所/博士(2018年)

https://doi.org/10.6342/NTU201800457

摘要

本文主要建構Mindlin板理論與高階剪切形變理論，且將其應用於分析全自由矩形板彎曲與伸展為主導的動態特性；首先將Mindlin板理論的解析解求出，接著由理論計算之結果與有限元素數值模擬相互比對以確認位移假設之正確性；為了驗證理論是否可以應用於解決工程實務問題，文中將Mindlin理論應用於分析壓電材料的共振頻率與模態振形，並進一步的分析物體振動時產生的應力場與電場，接著提出電極設計來達到最佳激振壓電材料的方法；最後利用處理Mindlin理論的概念推導高階剪切形變理論，期望利用更高階的形變假設深入且詳細的解析固體三維耦合的動態問題。
理論分析上，首先由Mindlin位移假設建構厚板的本構方程式，進而導出三維運動方程式和對應的邊界條件；並利用疊加法求得不同尺寸下厚板的共振頻率與三維位移模態圖，將分析結果與有限元素法相互比較以驗證理論之正確性。由Mindlin板理論計算之結果可知其克服以往Kirchhoff薄板裡論受限於厚度的影響；且有別於其他厚板分析理論侷限於較簡單的邊界條件，利用疊加法可解析複雜的邊界條件問題；接著由分析等向性材料的方式，利用Mindlin板理論解析PZT壓電厚板的動態特性，以便深入了解壓電厚板的三維振動特性與力電耦合之關係。
因Mindlin理論於分析高頻振動會出現較大的誤差，因此利用高階剪切形變假設(HOD theory)與漢米爾頓原理(Hamilton's principle)推導厚板運動方程式和力學表示式；接著以變分法(variation method)求導其平衡方程式與邊界條件，最後利用疊加法求得自由邊界厚板振動的共振頻率和模態振形；並將計算所得的頻率和振形與有限元素三維分析的結果相互比對以驗證高階位移形變理論來分析三維問題之正確性。顯示高階剪切形變理論克服了以往Mindlin板理論分析厚板產生過度修正共振頻率的情況，並且疊加法亦可應用於高階形變理論解析複雜邊界條件問題。
最後利用不同的實驗設計以驗證理論對物體振動特性解析的正確性；實驗方法主要是對鋁板與壓電試片進行激振，並以AF-ESPI記錄共振頻率與模態振形，以驗證理論的正確性並探討最佳的電極設計。厚板的實驗則是利用壓電薄膜(PVDF)和光纖光柵做為感測器以便量測物體的暫態波傳訊號，並利用快速傅立葉轉換(Fast Fourier Transform, FFT)將其轉換為頻域訊號以得出三維結構體的共振頻率，並以理論結果討論固體三維的動態特性。

關鍵字

Mindlin板理論 ； 高階位移假設 ； 漢米爾頓原理 ； 變分法 ； 疊加法 ； 有限元素法 ； 厚板 ； 壓電厚板 ； 電子斑點干涉術

並列摘要

In this dissertation, the Mindlin plate theory and high-order shear deformation plate theory are presented and applied to analyzing the flexural and extensional dominated dynamic characteristics for completely free rectangular plates. First, the analytic solution of Mindlin plate theory will be derived. Next, the theoretical analysis and numerical calculation results are compared. In order to solve the vibration problem for engineering application, the Mindlin plate theory is utilized to calculate the reasonant frequencies and mode shapes of piezoelectric plate. The mechanical fields and electrical fields are also presented to design the effective electrodes for excitation. Finally, the high-order shear deformation theory is presented to solve the three-dimensional vibration problem, and the dynamic characteristic of soild is discussed in detail.
First, the Mindlin plate theory is applied to analyze the resonant vibration of a isotropic thick plate. The resonant frequencies and mode shapes of a rectangular thick plate with completely free boundary conditions are analyzed. Three displacements of the flexural mode and extensional mode are presented base on the superposition method. This solution provides the result for the coupling of out-of-plane and in-plane vibrations with the dominated motion of flexural or extensional motion. The solution obtained from this superposition method has excellent convergence for numerical calculation. Furthermore, this method can be easily applied to construct the results for different boundary conditions. From the same way, Mindlin plate theory can be also applied to solving the dynamic characteristic of piezoelectric thick plate. Base on this investigation, the three-dimensional dynamic characteristics and the relation between mechanical fields and electrical fields of piezoelectric material are presented.
The high-order shear deformation theory is derived to have better prediction than Mindlin plate theory for high reasonant frequencies. The high-order displacement assumption and Hamilton’s principle have been used to construct the equation of motions and the boundary conditions. Utilizing the superposition method and Levy solution, the displacement functions can be derived, and the resonant frequencies and corresponding mode shapes could also be obtained. To verify the accuracy of theoretical solution, the resonant frequencies and the corresponding mode shapes are compared with that obtained by FEM calculation. The result of high-order shear deformation theory and FEM are consistent with great accuracy for three-dimensional solids.
Some experimental results are used to verify the accuracy of Mindlin plate theory and high-order shear deformation theory. First, the full-field optical technique, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) is utilized to get the resonant frequencies and corresponding mode shapes to comfirm the accuracy of the results by Mindlin plate theory. To excite the thick plate resonant frequency, we use free fall steel ball to impact the thick plate to generate the transient time response. By using Fast Fourier Transform of the time response, the resonant frequencies can be obtained. From the comparison result of theoretical analysis and experimental measurement, the analytical results obtained in this study can be used to determine the mode shapes and resonant frequencies of the thick plate with good accuracy.

並列關鍵字

Mindlin plate theory ； high-order shear deformation theory ； Hamilton’s principle ； variations ； superposition method ； finite element method ； AF-ESPI ； thick plate ； piezoelectric thick plate

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