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 摘 要本文利用微分值積法(Differential Quadrature Method, DQM)配合漢米爾頓原理，推導高速旋轉條件下，具非均勻斷面預扭渦輪葉片的運動方程式，為驗證微分值積法用於分析此類非均勻樑結構的適用性，文中就具均勻等向性(Isotropic)材料及具正交性(Orthotropic)強化碳纖維材料，兩種材質之非均勻預扭葉片或葉片組進行模式推導與參數分析。在正交性(Orthotropic)強化碳纖維葉片模式中，亦分別考慮了纖維編織方向與Kelvin-Voigt線性阻尼效應。為分析微分值積法中選點對計算結果精度之影響，文中亦就均勻分佈取樣點與依Chebyshev-Gauss-Lobatto函數分佈取樣點所得結果進行比對，由數值結果顯示使用Chebyshev-Gauss-Lobatto取樣點得到的結果比均勻分配的取樣點所得到的結果好。文中分別採用四種不同類型的非均勻樑問題，驗證微分值積法對不同問題之適用性與計算精度。其分別為：單一非均勻斷面預扭渦輪葉片之自然頻率分析，渦輪葉片組系統之振動分析，具正交性材料非均勻斷面旋轉預扭葉片之振動分析及不同形狀靜電式致動器(Electrostatic Actuator)之非線性殘振分析。本文中除針對旋轉速度、預扭角度、纖維角度、內阻尼及外阻尼對旋轉預扭樑動態行為的影響作探討，亦對系統的動態響應加以分析討論。在靜電式致動器分析結果，顯示不同形狀的電極(Electrode)及懸臂樑(Actuator)對靜電式致動器之吸入不穩定(Pull-in)及殘振均有明顯的影響。文中研究的結果與文獻的數值解與實驗結果做比較均甚吻合。不同分析例之數值結果，顯示出微分值積法用於分析此類問題具有頗佳之計算效率與準確性。
 AbstractThe dynamic models for different linear or nonlinear beam structures are proposed in this dissertation. The proposed mathematical model for a turbo-disk, which is valid for whatever isotropic or orthotropic turbo-blades with or without shrouds, accounts for the geometric pretwist and taper angles, and considers coupling effect among bending and torsion effect as well. The Kelvin-Voigt internal and external damping effects have been included in the formulation. The effect of fiber orientation on the natural frequencies of a fiber-reinforced orthotropic turbo-blade has also been investigated. The eigenvalue problems of a single pre-twisted taper-blade or a shrouded turbo-blade group are formulated by employing the differential quadrature method (DQM). The DQM is used to convert the partial differential equations of a tapered pre-twisted beam system into a discrete eigenvalue problem. The Chebyshev-Gauss- Lobatto sample point equation is used to select the sample points in these analyses. The effect of the number of sample points on the accuracy of the calculated natural frequencies has also been studied. The integrity and computational efficiency of the DQM in this problem will be demonstrated through a number of case studies. The effects of design parameters, i.e. Kelvin-Voigt internal and linear external damping coefficients, the fiber orientation, and the rotation speed on the dynamic behavior for a pretwisted turbo-blade are investigated.The dynamic response of a nonlinear electrode actuator used in the MEMS has also been formulated and analyzed by employing the proposed DQM algorithm. The transitional responses of the derived nonlinear systems are calculated by using the Wilson– method. Results indicated the curve shape of the electrode and the cantilever actuator may affect the pull-in behavior and the residual vibration of the electrostatic actuators significantly. Numerical results demonstrated the validity and the efficiency of the DQM in solving different type beam problems.