臺灣博碩士論文加值系統

本文主要目的在探討旋轉梁的支承端之剛度及自由端加掛配重其分別對自然振動頻率的影響，還有探討旋轉梁挫屈的臨界轉速。本文中考慮了Euler梁及Timoshenko梁。本文利用d*Alembert原理及虛功原理推導旋轉梁振動的統御方程式。本文中在分析旋轉梁之自然頻率時考慮了轉動慣量及科氏力的影響。本文利用非線性梁理論的一致線性化以得到所有慣性效應及變形間的耦合效應。本文中將旋轉梁均分成多個元素，每個元素的統御方程式都用一個級數來解，並由旋轉梁兩端點及相鄰兩元素在共同節點上的邊界條件可求得自然頻率。
本文最後以數值例題分析了不同轉速下，不同設定角及不同細長比的旋轉梁在不同邊界條件下之自然頻率並分析了旋轉梁挫屈的臨界轉速，同時並探討了科氏力、轉動慣量對自然頻率的影響。

The objective of this paper is to investigate the effects of stiffness of elastically restrained root, tip mass on the natural frequency of vibration of rotating beam, and investigate the critical rotation speed for buckling of rotating off-axis beam. Here, Euler beam and Timoshenko beam are considered. The governing equations for linear vibration of rotating Euler beam and Timoshenko beam are derived by the d*Alembert principle and the virtual work principle. The effect of Coriolis force and rotary inertia on the natural frequency of the rotating beam is investigated. In order to capture all inertia effect and coupling between extensional and flexural deformation, the consistent linearization of the fully geometrically non-linear beam theory is used. A method based on the power series solution is proposed to solve the natural frequency. The rotating beam is divided into serveral segments. The governing equations of each segment are solved by a power series. From the conditions of the rotating beam at two end nodes and common node between two adjacent segments the natural frequency can be calculated.
Numerical examples are studied for the nature frequency of rotating beam with different angular velocity, setting angle, and slenderness ratio, and for the critical rotation speed for buckling. The effect of Coriolis force and rotary inertia on the natural frequency of the rotating beam is investigated by numerical examples.

中文摘要..........................................I
英文摘要.........................................II
目錄.............................................IV
表目錄...........................................VI
圖目錄...........................................XI
符號說明.........................................XV
第一章 緒論......................................1
1.1 研究的動機與目的..........................1
第二章 理論推導..................................4
2.1 問題描述..................................4
2.2 座標系統..................................4
2.3 Euler梁的變形描述.......................5
2.3.1 梁之位移與應變..........................5
2.3.2 梁之速度及加速度........................8
2.4 旋轉梁運動的統御方程式...................10
第三章 旋轉梁的穩態與振動.......................15
3.1 旋轉梁的穩態...........................15
3.2 旋轉梁的振動方程式及其解...............17
3.3 旋轉梁結構的節點位移與節點力...........28
3.4 旋轉梁結構的邊界條件...................30
3.5 自然頻率及振動模態.....................34
第四章 數值計算方法與程序.......................37
4.1 旋轉梁之自然頻率與振動模態...............37
4.2 旋轉梁挫屈時之臨界轉速...................38
第五章 數值例題.................................40
5.1 具端點質量的旋轉梁.......................41
5.2 端點彈性支承剛度對振動頻率的影響.........42
5.3 旋轉梁之臨界轉速.........................43
第六章 結論與展望...............................44
參考文獻.........................................46
附表.............................................50
附圖.............................................94
附錄A 旋轉Timoshenko梁之自然振動................147

1. A. Leissa, "Vibrational Aspects of Rotating Turbomachinery Blades," ASME Applied Mecanics Reviews, Vol.34, 1981, pp. 629-635.
2. V. Ramamurti and P. Balasubramanian, "Analysis of Turbomachinery Blades - A Review," The Shock and Vibration Digest, Vol. 16, 1984, pp. 13-28.
3. M. J. Schilhansl, "Bending Frequency of a Rotating Cantilever Beam," ASME Journal of Applied Mechanics, Vol. 25, 1958, pp. 28-30.
4. R. M. Krupka and A. M. Baumanis, "Bending-Benging Mode of a Rotating Tapered-Twisted Turbomachine Blade Including Rotatory Inertia and Shear Deformation," ASME Journal of Engineerung for Industry, Vol. 91, No.4, 1969, pp. 1017-1024.
5. T. Yokoyama, "Free Vibration Characteristics of Rotating Timoshenko Beam," International Journal of Mechanical Science, Vol. 30, No. 10, 1988, pp. 743-755.
6. S. Y. Lee and Y. H. Kuo, "Bending Frequency of a Rotating Beam with an Elastically Restrained Root," ASME Journal of Applied Mechanics, Vol. 58, 1991, pp. 209-214.
7. S. Y. Lee and S. M. Lin, "Bending Vibration of Rotating Nonuniform Timoshenko Beams with an Elastically Restrained Root," ASME Journal of Applied Mechanics, Vol. 61, 1994, pp. 949-955.
8. S. Y. Lee and Y. H. Kuo, "Bending Vibrations of a Rotating Non-Uniform Beam with an Elastically Restrained Root," Journal of Sound and Vibration, Vol. 154. No. 3, 1992, pp. 441-451.
9. S. Y. Lee and Y. H. Kuo, "Bending Frequency of A Rotating Timoshenko Beams with General Elastically Restrained Root," Journal of Sound and Vibration, Vol. 105, No. 2, 1993, pp. 243-250.
10. S. V. Hoa, "Vibration of A Rotating Beam with Tip Mass," Journal of Sound and Vibration, Vol. 67, 1979, pp.369-381.
11. R. B. Bhat, "Transverse Vibrations of A Rotating Uniform Cantilever Beam With Tip Mass as Predicted by Using Beam Characteristic Orthogonal Polynomials in the Rayleigh-Ritz Method, " Journal of Sound and Vibration, Vol. 105, No. 2, 1986, pp. 199-210.
12. Y. A. Khulief, "Vibration Frequencies of a Rotating Tapered Beam with End Mass," Journal of Sound and Vibration* Vol. 134. No. 1, 1989, pp. 87-97.
13. H. P. Lee, "Vibration on an Inclined Rotating Cantilever Beam With Tip Mass," Journal of Vibration and Acoustics, Vol. 115, 1993, pp. 241-245.
14. J. C. Bruch, Jr. and T. P. Mitchell, "Vibrational of A Mass-Loaded Clamped-Free Timoshenko Beam," Journal of Sound and Vibration, Vol. 114. No. 2, 1988, pp. 341-345.
15. Y. A. Khulief and L. Yi , "Lead-Lag Vibrational Frequencies of A Rotating Beam with End Mass,"Computers & Structures ,Vol. 29, No. 6, 1988, pp. 1075-1085.
16. K. H. Low , "Vibration Analysis of A Tip-Loaded Beam Attached To A Rotating Joint," Computers & Structures, Vol. 52. No. 5, 1994, pp. 955-968.
17. D. C. Kammer and A. L. Schlack, Jr. , "Critical Spin Rate of Rotating Beams by Liapunov's Direct Method," Journal of Vibration, Acoustics, Stress, and Reliability in Design , Vol. 108, 1986, pp. 389-393.
18. C. D. Eick and M. P. Mignolet, "Vibration and Buckling of Flexible Rotating Beams, " AIAA Journal, Vol. 33, No. 3,1995, pp528-538.
19. C. H. J. Fox and J. S. Burdess, "The Natural Frequencies Of A Thin Rotating Cantilever With Offset Root, " Journal of Sound and Vibration, Vol. 65, No. 2, 1979, pp. 151-158.
20. F. Baur and W. Eidel, "Vibration of A Rotating Uniform Beam, PartII : Orientation Perpendicular To The Axis of Rotation," Journal of Sound and Vibration, Vol. 122. No. 2, 1988, pp. 357-375.
21. K. B. Subrahmanyam and K. R. V. Kaza, " Vibration and buckling of Rotating, Pretwisted, Preconed Beams Including Coriolis Effectsd," Trnsactions of the ASME, Vol. 108, 1986, pp. 140-149.
22. 洪船島, "旋轉梁結構之振動分析 ," 國立交通大學機械工程研究所碩士論文, 臺灣, 新竹, 1997.
23. 周志芳, "旋轉梁之自由振動的級數解法 ," 國立交通大學機械工程研究所碩士論文, 臺灣, 新竹, 1998.
24. J. C. Simo and L. Vu-Quac, "The Role of Non-Linear Theories in Transient Dynamic Analysis of Flexible Structures," Journal of Sound and Vibration, Vol. 119, No. 3, 1987, pp. 487-508. T. J. Chung, Continuous Mechanics, Prentice-Hall, Inc., Englewood Cliff, New Jersey, 1988.
25. T. J. Chung, Continuous Mechanics, Prentice-Hall, Inc., Englewood Cliff, New Jersey, 1988.
26. A. P. Boresi, R. J. Schmidt, and O. M. Sidebottom, Advanced Mechanics of Materials, John Wiley & Sons, Inc. New York, 1985. 27. L. Meirovitch, "A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems," AIAA Journal, Vol. 12, No. 10, 1974, pp. 1337-1342.
28. 楊榮哲, "梁元素之一致性共旋轉有限元素推導法及其應用," 國立交通大學機械工程研究所博士論文, 臺灣, 新竹, 1995.
29. W. F. White, R. G. Kvaternik and K. R. V. Kaza, "Buckling of Rotating Beams ," International Journal of Mechanical Science, Vol. 21, 1979, pp. 739-745.
30. W. D. Lakin and A. Nachman, "Unstable Vibrations and Buckling of Rotating Flexible rods," Quarterly of applied Mathematics, Vol. 35, 1978, pp. 479-493.