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研究生:徐明宏
研究生(外文):Ming-Hung Hsu
論文名稱:微分值積法於非均勻樑之振動分析
論文名稱(外文):Vibration Analysis of Non-uniform Beams Using the Differential Quadrature Method
指導教授:光灼華
學位類別:博士
校院名稱:國立中山大學
系所名稱:機械與機電工程學系研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:英文
論文頁數:152
中文關鍵詞:非均勻樑微分值積法
外文關鍵詞:Non-uniform Beamsthe Differential Quadrature Method
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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。
Abstract
The 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.
Contents
Acknowledgments i
Abstract ii
Contents v
List of Figures vii
List of Tables ix
Nomenclature x
Chapter 1 Introduction 1
1-1 Literature review 1
1-2 The Differential Quadrature Method 5
Chapter 2 Eigen solutions of pre-twisted tapered beams 10
2-1 Formulation of the eigen value problem 11
2-2 Numerical results 24
Chapter 3 Eigen value problem for the grouped turbo blades 33
3-1 Formulation of the eigen value problem for a shrouded turbo blade group 34
3-2 Numerical results 52
Chapter 4 Dynamic analysis of orthotropic pre-twisted blades 56
4-1 Natural frequencies of a fiber reinforced pretwisted blade 58
4-2 Dynamic behavior of a composite blade 73
4-3 Numerical results 87
Chapter 5 Residual vibrations of electrostatic actuators with different electrode shapes 101
5-1 Static deflections of the electrostatic actuators 103
5-2 Dynamic behavior of actuators 108
5-3 Numerical results and discussions 117
Chapter 6 Conclusions 129
References 133
Appendix A 145


List of Figures
Figure 2-1 Schematic view of a pre-twisted tapered beam 23
Figure 2-2 The lowest and the second natural frequencies of the tapered beam with different pre-twisted angle 32
Figure 3-1 Geometry of the grouped turbo blades 50
Figure 3-2 Geometry of a pre-twisted taper blade 51
Figure 3-3 The lowest natural frequencies ( ) of the rotating grouped turbo blades with different pre-twisted angles 54
Figure 3-4 The calculated natural frequencies of the grouped turbo blades with different shroud stiffness 55
Figure 4-1 Geometry of a tapered pre-twisted orthotropic blade
72
Figure 4-2 Natural frequencies of an orthotropic blade with different fiber orientation angles 94
Figure 4-3 Natural frequencies of an orthotropic blade with different pre-twisted angles 95
Figure 4-4 Natural frequencies and of a pre-twisted orthotropic blade with different rotation speeds 96
Figure 4-5 Natural frequencies and of a pre-twisted orthotropic blade with different inclined angles 97
Figure 4-6 Comparison of tip deflection of the composite blade solved using the Newmark method and the Wilson- method
98
Figure 4-7 Tip deflection with different external damping 99
Figure 4-8 Tip deflection with different internal damping 100
Figure 5-1 Schematic view of a curved electrode actuator 107
Figure 5-2 Variation of the shapes for curved electrode with different values of the polynomial order 121
Figure 5-3 Comparison of the tip deflections with different applied voltages and electrode shapes 122
Figure 5-4 Comparison of the tip responses of the actuator solved using the Newmark method and the Wilson- method
123
Figure 5-5 Tip responses of the actuator with different shapes 124
Figure 5-6 Tip responses of the actuator with different applied voltage
125
Figure 5-7 Tip responses of the actuator with different external damping 126
Figure 5-8 Tip responses of the actuator with different internal damping 127
Figure 5-9 Tip responses of the actuator with different tapered
ratio 128


List of Tables
Table 2-1 Frequency ratio for a flat beam with a pre-twisted angle of (solved with equally spaced sample points) 26
Table 2-2 Frequency ratio for a flat beam with a pre-twisted angle of (solved with equally spaced sample points) 27
Table 2-3 Frequency ratio for a flat beam with a pre-twisted angle of (solved with equally spaced sample points) 28
Table 2-4 Frequency ratio for a flat beam with a pre-twisted angle of (solved with the Chebyshev- Gauss- Lobatto points) 29
Table 2-5 Frequency ratio for a flat beam with a pre-twisted angle of (solved with the Chebyshev- Gauss- Lobatto points) 30
Table 2-6 Frequency ratio for a flat beam with a pre-twisted angle of (solved with the Chebyshev- Gauss- Lobatto points) 31
Table 4-1 Natural frequencies for an orthotropic beam with a fiber orientation angle of 92
Table 4-2 Natural frequencies for an orthotropic beam with a fiber orientation angle of 93


Nomenclature
= cross section area of the blade
= width at the blade root
= width at the blade tip
= external damping coefficient
= external damping coefficient
= internal damping coefficient
= damping matrix
= the initial gap between electrode and actuator at
= weighting coefficient of the order differentiation
= applied voltage
= Young’s modulus
= Young’s modulus in 1-axis
= Young’s modulus in 3-axis
= Young’s modulus in z-axis
= torsion moment
= lift force in the transverse direction of the blade
= force vector
= modulus of rigidity in 1-2 plane
= modulus of rigidity in 1-3 plane
= modulus of rigidity in 2-3 plane
= modulus of rigidity in x-y plane
= modulus of rigidity in x-z plane
= moment of inertia
= moment of inertia
= moment of inertia
= polar moment of inertia
= shroud stiffness
= dimensionless shroud stiffness
= stiffness matrix
= length of the beam
= bending moments
= torsion moments
= mass matrix
= number of sample points
= total number of blades
= polynomial order of curve
= outer diameter of disk
= the number of the blade
= the shape of curved electrode as function of position
= thickness at the blade root
= thickness at the blade tip
= th time
= kinetic energy of the blade
= kinetic energy of the orthotropic blade
= elastic deflections of the blade
= strain energy of the blade
= strain energy of the orthotropic blade
= Poisson’s ratio
= elastic deflections of the blade
= displacement
= displacement vector
= displacement vector
= displacement vector at the boundary
= displacement vector at the interior degrees of freedom
= taper ratio
= taper ratio
= pre-twisted angle
= fiber orientation
= inclined angle of blades
= parameter of the Wilson- direct integration
= twist angle
= density of the beam
= natural frequency
= dimensionless natural frequency
= rotational angular speed
= dimensionless rotational angular speed
= the dielectric constant in the air
= time interval
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