正交各向異性懸臂板的振動特性與暫態波傳之理論分析、數值計算與實驗量測

Theoretical Analysis, Numerical Calculation and Experimental Measurement on Vibration Characteristics and Transient Wave Propagation of Orthotropic Cantilever Plate

陳竑均(Hung-Chun Chen)

指導教授 ： 馬劍清

國立臺灣大學/工學院/機械工程學研究所/碩士(2021年)

https://doi.org/10.6342/NTU202104173

摘要

本文探討正交各向異性矩形懸臂板的動態特性，結合理論解析、有限元素數值模擬和實驗量測，分析薄板的穩態自由振動特性，以及承受動態外力加載而產生的暫態波傳行為。
理論解析首先說明正交各向異性材料的本構方程式，並探討薄板理論假設，將結構振動問題簡化為二維平面應力問題，接著以力學平衡法推導薄板的統御方程式與邊界條件，並以疊加法求解懸臂板的邊界值問題，獲得薄板面外共振頻率與對應的模態形狀，以及推導面內應變場。最後將求解之共振頻率、模態形狀與有限元素模擬的結果進行比較以驗證理論計算的準確性。
碳纖複材懸臂板材料常數的反算使用Simplex反算法，以穩態振動分析為基礎，搭配ESPI振動實驗的量測結果，反算碳纖複材懸臂板的材料常數。將反算獲得之材料常數代入理論解析和有限元素模擬分別得到共振頻率與模態形狀之正算解，與ESPI振動實驗量測結果進行比較，確認材料常數反算結果的正確性，以建立一套完整正確而快速的材料參數量測方法。
暫態波傳分析運用模態展開法的概念，以模態形狀和時間函數建構薄板的暫態位移，此解析解適用於分析薄板承受任何形式動態外力所產生的暫態行為。本文亦提出有限元素-模態展開法，以有限元素分析的模態形狀作為暫態位移的基底函數，代入暫態位移解析解以求得矩形薄板的暫態位移。最後將波源歷時輸入理論解析、有限元素模擬和有限元素-模態展開法做數值計算，並比較三種方法所得之時間域與頻率域分析結果。

關鍵字

正交各向異性平板 ； 疊加法 ； 模態展開法 ； 振動 ； 暫態波傳 ； 電子斑點干涉術 ； 材料常數反算

並列摘要

This study investigates the dynamic characteristics of an orthotropic rectangular thin plate by theoretical analysis, finite element simulation, and experimental measurements. The dynamic characteristics involve the vibration properties and the transient behaviors of the thin plate due to the externally applied loading.
The constitutive equation of the orthotropic material is first introduced, followed by the assumptions of the classical thin plate, establishing the vibrational problem of the thin structure to a two-dimensional plane-stress condition. The governing equation and boundary conditions are derived by means of force equilibrium, with the boundary-value problem of the cantilever plate solved by the superposition method. The resonant frequencies and the corresponding mode shapes are then obtained, as well as the in-plane strain fields, and compared with the results obtained from the finite element simulation.
The Simplex method is used to inversely determine the material constants of the carbon reinforced composite plate, based on the results of the steady-state vibration analysis by using the experimental data measured by the ESPI. With the inversely estimated material constants and using the theoretical analysis and finite element simulation, the validation of the inverse estimation of the material constants is completed by comparison of the results of theoretical analysis, finite element simulation, and the original data measured by the ESPI experiment.
Theoretical derivation of the transient behavior for an orthotropic cantilever plate subjected to impact loading indicates that transient displacement is a product of the time and space functions (mode shapes). The orthogonality of the mode shape functions is used to construct the time function. The analytical solution thus established for the transient displacement adapts to general form of dynamic loading. The finite element-normal mode expansion method (FENMEM) is introduced, which is the method using the mode shapes obtained from FEM to obtain the transient displacement. The results obtaind from theoretical analysis, FEM, and FENMEM are thus compared with good agreement.

並列關鍵字

orthotropic plate ； method of superposition ； normal mode expansion ； vibration ； transient wave propagation ； ESPI ； inverse estimation of material constants

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