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系統識別號 U0002-2002200908314800
中文論文名稱 撓性雙旋轉軸之有限元素法
英文論文名稱 Finite Element Method for Dual Flexible Rotary System
校院名稱 淡江大學
系所名稱(中) 航空太空工程學系碩士班
系所名稱(英) Department of Aerospace Engineering
學年度 97
學期 1
出版年 98
研究生中文姓名 洪于棻
研究生英文姓名 Yu-Fen Hung
學號 694370825
學位類別 碩士
語文別 英文
口試日期 2009-01-08
論文頁數 82頁
口試委員 指導教授-田豐
委員-吳登淵
委員-楊智旭
中文關鍵字 雙軸承  有限元素法  擾性轉子系統  Matlab 
英文關鍵字 Twin spool  Dual rotor  Flexible rotary system  Finite element method  Matlab 
學科別分類 學科別應用科學航空太空
中文摘要 這篇論文的主要目的在於使用有限元素來分析撓性雙旋轉軸。這個方法的優點在於計算過程中所用來計算撓性雙旋轉軸振動頻率的過程中,能夠有效增加它的準確度。此外,在建立撓性雙旋轉系統運動方程時,考慮了轉動慣性力矩、迴轉力矩效應與剪應力與應變的效應。
而不用像傳統的傳遞矩陣方法,需要額外一個一個的把特性矩陣加入
當撓性雙旋轉系統複雜性提高、所使用的節點數增加時,仍然保有了精確度。
英文摘要 The purpose of the research is that we hope to use a reduced order finite element method to analyze a dual rotating flexible shafts. The advantage of the proposed method is that the largest size of the matrices involved in computing the critical speed of rotating shaft increase it's accuracy. In addition, the effects of shear deformation, rotary inertia, gyroscopic moments
and shear strain are all taken into account as well.
However, that didn't like the traditional transfer matrix method need another to add the characteristic matrices.
The accuracy of the obtained results from this method is high, which however inevitably demands computer memory intensivelyfor increasing number of nodes.
論文目次 Contents
致謝 . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Chines Abstract . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . iii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction . . . . . . . . . . . . . . . . . . . . .1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . 3
1.3 Research Method . . . . . . . . . . . . . . .. . . . . 4
2 Equation of Motion of the Flexible Dual-Rotor System ....5
3 The Finite Element Method ..............................12
3.1 Shape Function . . . . . . . . . . . . . . . . . . . 13
3.2 System Equation of Motion . . . . . . . . . . . . . . 16
3.3 Dual-Rotor Bearing . . . . . . . . . . . . . . . . . 19
3.4 Solution of Dual-Rotor Bearing System . . . . . . . 23
4 Numerical Examples .....................................25
5 Conclusion .............................................41
A The Equation of Motion................................. 42
A.1 Kinetic energy . . . . . . . . . . . . . . . . . . . 45
A.2 Potential Energy . . . . . . . . . . . . . . . . . . 48
A.2.1 Proof the Strain and Shear Strain Energy .. . . . . 48
A.3 Lagrange Equation . . . . . . . . . . . . ....... . . 57
B Finite Element Matrices . . . . . . . . . . . . . . . . 66
Bibliography . . . . . . . . . . . . . . . . . . . . . .74

List of Tables
4.1 Dimension of the shaft 1 and 2 in Example 4.1 (Unit : MKS) . . . . . . . . 26
4.2 Dimension of the disk shaft in Example 4.1 (Unit : MKS) . . . . . . . . . . 27
4.3 Dimension of the bearing in Example 4.1 (Unit : MKS) . . . . . . . . . . . 27
4.4 The critical speeds (rad/s) for inner-rotor in 4.1 . . . . . . . . . . . . . . . 27
4.5 The critical speeds (rad/s) for outer-rotor in 4.1 . . . . . . . . . . . . . . . 28
4.6 Nodal coordinate for the dual rotor system . . . . . . . . . . . . . . . . . . 34
4.7 Disk data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8 Bearing stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.9 The critical speeds for inner-rotor in [13] . . . . . . . . . . . . . . . . . . . 36
4.10 The critical speeds for outer-rotor in [13] . . . . . . . . . . . . . . . . . . . 37

List of Figures
2.1 Twin Spool System[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Twin Spool System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The Twin Spool coordinate system . . . . . . . . . . . . . . . . . . . . . . 7
2.4 The definitions of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Internal force of each element . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Definition for  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Bearing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Dual-Rotor system model . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Connect node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1 The definitions of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 A three-dimensional picture and section picture in Example 4.1 . . . . . . 26
4.3 The dual rotor system general by matlab . . . . . . . . . . . . . . . . . . . 28
4.4 The Campbell from [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 The Campbell from Matlab calculation (rad/s) . . . . . . . . . . . . . . . . 30
4.6 The innerrotor for the First mode 884Hz . . . . . . . . . . . . . . . . . . . 31
4.7 The outerrotor for the First mode 856Hz . . . . . . . . . . . . . . . . . . . 31
4.8 The innerrotor for the Second mode 1693Hz . . . . . . . . . . . . . . . . . 32
4.9 The outerrotor for the Second mode 1604Hz . . . . . . . . . . . . . . . . . 32
4.10 The innerrotor for the Third mode 2289Hz . . . . . . . . . . . . . . . . . . 33
4.11 The outerrotor for the Third mode 2271Hz . . . . . . . . . . . . . . . . . . 33
4.12 The dual rotor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.13 A three-dimensional picture and section picture in Example 4.2 . . . . . . 35
4.14 The dual rotor system general by matlab . . . . . . . . . . . . . . . . . . . 36
4.15 The Campbell from Matlab calculation (rpm) . . . . . . . . . . . . . . . . 37
4.16 The innerrotor for the First mode 9448Hz . . . . . . . . . . . . . . . . . . 38
4.17 The outerrotor for the First mode 8396Hz . . . . . . . . . . . . . . . . . . 38
4.18 The innerrotor for the Second mode 12730Hz . . . . . . . . . . . . . . . . . 39
4.19 The outerrotor for the Second mode 12270Hz . . . . . . . . . . . . . . . . 39
4.20 The innerrotor for the Third mode 18180Hz . . . . . . . . . . . . . . . . . 40
4.21 The outerrotor for the Third mode 16780Hz . . . . . . . . . . . . . . . . . 40
A.1 The Twin Spool coordinate system . . . . . . . . . . . . . . . . . . . . . . 42
A.2 The definitions of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.3 The shaft size model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.4 A single shaft model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.5 The shaft’s cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.6 Plane rotate to x axial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.7 strain energy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
參考文獻 Bibliography
[1] L. M. Yang, 轉子現場動平衡技術. China: National Defense Industry
Press, 2007.
[2] E. C. Pestel and F. A. Leckie, Matrix Methods in Elastomechanics. McGraw-Hill, 1963.
[3] M. A. Dokainish, “A new approach for the plate vibration:combination of transfer matrix and finite element technique,” Journal of Sound and Vibration, Transactions of the ASME, vol. 67, no. 1, pp. 35–42, 1972.
[4] A. C. Lee, “The analysis of linear rotor-bearing systems : A general transfer matrix method,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 115, pp. 490–497, 1993.
[5] Y. M. Huang and C. M. Wang, “Combined methodology for analysis of rotary systems,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 123, pp. 428–434, 2001.
[6] P. M. A., “A general method of calculating critical speeds of flexible rotors,” Journal of Engineering for Industry, Trans, ASME, pp. 126–132, 1945.74
[7] R. L.Ruhl, “A finite element model for distributed parameter turborotor systems,” Journal of Engineering for Industry, Trans, ASME,pp. 26–132, 1972.
[8] D. H. Hibner, “Dynamic response of viscous-damped muti-shaft jet engines,” Journal of Aircraft, vol. 12, no. 4, pp. 305–312, 1975.
[9] D. A. Glasgow and H. D. Nelson, “Stability analysis of rotor bearing systems using component mode synthesis,” American Society of Mechanical Engineers, pp. 79–DET–63, September 1979.
[10] T. Huang, “The transfer matrix impendence coupling method for the eigen solutions of multi-spool rotor systems,” Model Testing and Analysis, American Society of Mechanical Engineers, vol. 3, pp. 71–76, December 1987.
[11] Z. C. Zeng and Y. Hu, “The dynamics analysis of a multo-shaft-rotorbearing-case system,” Instituted of Mechanical Engineers Conference Publicaion, pp. 607–614, 1988.
[12] G. K. Gupta, K. D. and K. Athre, “Stability analysis of dual rotor system by extended transfer matrix method,” A merican Society of Mechanical Engineers, pp. 87–GT–194, June 1989.
[13] S. H. Tu, “Rotor-bearing analysis for turbomachinery single-and dualrotor systems,” Journal of Propulation and Power, vol. 20, pp. 1096–1104, Nov-Dec 2004.
[14] D. T. Greenwood, Principles of Dynamics. New Jersey: Prentice Hall,1988.
[15] M. Lalanne and G. Ferraris, Rotordynamics Prediction in Engineering. John Wiley Sons, 1997.
[16] J. S. RAO, Rotor Dynamics. India: New Age International Limited, 1983.
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