本研究以共旋轉有限元素法及非線性梁理論的一致性二階線性化推導挫屈梁的平 衡方程式及運動方程式。將挫屈梁的運動方程式用泰勒級數在挫屈梁的初始變形位置展 開，取到一次項，求得挫屈梁線性振動的運動方程式。本研究探討了探討挫屈梁的初始 變形與自然振動，及不同拱起高度之挫屈梁的受側力的非線性行為。
本研究的數值結果發現挫屈梁初始變形之解析解的適用範圍與挫屈梁細長比
的大小有關，細長比愈大，則解析解的適用範圍愈大。
本研究的數值結果發現，當挫屈梁拱起高度與梁之厚度比(/h)小於 50 時，挫屈梁 前兩個無因次自然振動頻率與梁之細長比幾乎無關，這與文獻的結果一致；但當挫屈梁 之/h大於10 時其第三及第四個無因次自然振動頻率與/h及梁之細長比都有很大的 關係。這與文獻上的結論不同，文獻上認為挫屈梁之第奇數個無因次自然頻率僅與挫屈 梁之/h有關，與梁之細長比無關，第偶數個無因次自然頻率為一常數，與梁之/h、 細長比都無關。
本研究的數值結果發現，當挫屈梁拱起高度與梁之厚度比(/h)小於 10 時，不同細 長比的挫屈梁臨界點對應的無因次負載與高度都很接近，所以不同細長比的挫屈梁可以 共用同一摺線圖；但當挫屈梁之/h大於60 時，不同細長比的挫屈梁不可以共用同一 摺線圖。
參 考 文 獻
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1971.
[30] 黃智傑, “旋轉三維 Timoshenko 梁之振動分析” 國立交通大學機械工程學系, 碩士 論文, 台灣, 新竹, 2001
圖1 直梁一端受軸向壓縮及側向挫屈示意圖
) (a
) (b
C X
2X
2X
1X
1L T
A C B
A B
M B
M A
P
b
h
14
0.00 0.0
b
ah v C
P
b
a
L
b1B
2bB
1bS
2aS
1L
a1L
a2S
2bL
b2
圖2 挫屈梁受側向集中力 P 的位移-負荷曲線圖
圖3 元素座標系統及總體座標系統關係圖
圖4 梁的變形圖
2
O
x
1X
1X
oY
oX
2
1
2
e1
x
21 P
Q y
x1
x2
s
) (x v )
(x xp
1
x
2x
1y x
Q
16
0 5 10 15 20 25 30
0 50 100 150 200
h
Analy.
0.0 0.5 1.0
0 20 40
圖5 挫屈梁的
h
曲線
0.00 0.05 0.10 0.15 0.20 1.00
1.01 1.02 1.03 1.04 1.05
0.5 10
( 10
3)
L T P P CR
圖6 挫屈梁的無因次端點軸向反力
P / P
CR-中點側向位移 / L
T曲線18
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0
10 20 30 40 50 60 70 80
i/ ( EI / mL
T4)
1/2 h
Present
Present
[29] Exp
[29]
[29]
[8]
[8]
圖7 挫屈梁的無因次自然振動頻率
i /(EI
/mL
T4)12-無因次側向位移
/h
的曲線 (
/h
3,i =1,2)
0 1 2 3 4 5 6 7 8
0 100 200 300 400 500 600
[8]
Present
i/ ( EI / mL
T4)
1/2 h
i = 1 i = 2 i = 3
i = 4 i = 5 i = 6 i = 7
圖8 挫屈梁的無因次自然振動頻率
i /(EI
/mL
T4)12-無因次側向位移
/h
的曲線 (
/h
8,i =1,7)
20
0 10 20 30 40 50
0 50 100 150 200 250 300
i / ( EI / AL 4 ) 1/ 2
h
i = 3
i = 4
i = 1
i = 2
圖9 挫屈梁的無因次自然振動頻率
i /(EI
/mL
4T)12-無因次側向位移/h的曲線
圖10 挫屈梁結構中點
C 受側向集中力 P 的結構圖
b
h X
2X
1L T
A C B
C X
2X
1A B
2
/ 2
/
) (a
) (b
v
CP ,
u
C22
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -0.1
0.0 0.1 0.2
B 2.2 u L 2.2 u
L 1.7 d B 2.2 d
L 2.2 d L 3 d B 3 d L 3 u
L 1.7 u 2 PL / EI T O
=3
=2.2
=1.7 B 3 u
- v C / h
圖11 挫屈梁受側向集中力
P 的
h v EI
PL
T2
C 曲線圖(
1.7,2.2,3)
0 20 40 60 80 0
2 4 6 8 10
0.5 10
C / h
CR
CR L
圖12 不同細長比
挫屈梁臨界點在 h
C 平面的摺線圖(
80)
24
0 20 40 60 80
0 1 2 3
P B CR P L CR
0.5 10
PL 2 / E I (1 0 4 )
圖13 不同細長比
挫屈梁臨界點在
EI
PL
T2平面的摺線圖(
80)
0 1 2 3 0
50 100 150 200
Analy.
CR L
CR
1 3 10
C / h
圖14 不同細長比
挫屈梁臨界點在
h
C 平面的摺線圖(
3 10
4)
26
0 1 2 3
0 1 2 3 4 5 6
1 3 10
PL T 2 / E I (1 0 5 )
P L CR
P B CR
圖15 不同細長比
挫屈梁臨界點在 EI
PL
T2平面的摺線圖(
3 10
4)
國科會補助專題研究計畫成果報告自評表
請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價值
（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）、是否適合在 學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。
1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估
達成目標
□ 未達成目標（請說明，以 100 字為限）
□ 實驗失敗
□ 因故實驗中斷
□ 其他原因 說明：
本研究內容與原計畫完全相符，並達成預期目標。本研究的數值結果發現挫
屈梁初始變形之解析解的適用範圍與挫屈梁細長比
的大小有關，細長比愈大，則解析解的適用範圍愈大
本研究的數值結果發現挫屈梁之/h大於 10 時其第三及第四個無因次自然振動 頻率與/h及梁之細長比都有很大的關係。糾正了文獻上認為挫屈梁之第奇數 個無因次自然頻率僅與/h有關，第偶數個無因次自然頻率為一常數，與/h、 梁之細長比都無關的錯誤說法。本人的發現在非淺挫屈梁的研究及應用上應有 相當的貢獻。
2. 研究成果在學術期刊發表或申請專利等情形：
論文：□已發表 □未發表之文稿撰寫中 □無 專利：□已獲得 □申請中 □無
技轉：□已技轉 □洽談中 □無 其他：（以 100 字為限）
28
3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價值
（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以 500 字 為限）
本研究的數值結果發現挫屈梁初始變形之解析解的適用範圍與挫屈梁細長比
的大小有關，細長比愈大，則解析解的適用範圍愈大
本研究的數值結果發現挫屈梁之/h大於 10 時其第三及第四個無因次自然振動 頻率與/h及梁之細長比都有很大的關係。糾正了文獻上認為挫屈梁之第奇數 個無因次自然頻率僅與/h有關，第偶數個無因次自然頻率為一常數，與/h、 梁之細長比都無關的錯誤說法。本人的發現在非淺挫屈梁的研究及應用上應有 相當的貢獻。本研究的結果影可以發表在學術期刊，供設計挫屈梁時參考。
計畫編號 100-2221-E-009-098-
計畫名稱 受側向負荷之挫屈梁的靜態及動態行為研究
出國人員姓名 服務機關及職稱
蕭國模
國立交通大學機械系 教授 會議時間地點
London, U.K., 4 - 6 July, 2012
會議名稱 The 2012 International Conference of Mechanical Engineering (ICME'12) 發表論文題目 Free Vibration Analysis of Rotating Euler Beam by Finite Element Method
一 參加會議的經過
本人承國科會補助才能參加此會議，在此深表感謝。
本會議為2012年國際機械工程會議（The 2012 International Conference of Mechanical Engineering, London, United Kingdom），於今年7月4日至7月6日在英國倫敦帝國理工學院(Imperial College)之 the Wolfson Education Centre at the Hammersmith Campus舉行。本會議是由International Association of Engineers (IAENG)主辦之the World Congress on Engineering 2012(WCE 2012)中的一個子會議。
本會議為一國際性的學術研討會，其主要的目的是要讓在機械工程中不同領域的研究人員聚 在一起，交換機械工程力學之最新發展的資訊及想法。本研討會在機械工程的研究上有很高的學 術地位。
WCE 2012為一個大型的國際會議，共有1183篇論文投稿，但只有接受646篇，接受率約55%。
本會議並邀請振動學界的大師做專題演講，如 Prof. Len Gelman, Cranfield University, UK, Prof.
Alexander M. Korsunsky, Trinity College, University of Oxford, U.K, University of Ontario Institute of Technology, Canada Bale Viswanadha Reddy 教授。
本會議論文的主題分為
Aerodynamics Fluid Dynamics Compressible Flows Computational Mechanics Biomechanics
Automotive Engineering Heat and Mass Transfer Nanomaterial Engineering Plasticity Mechanics Fracture
Geomechanics
Multibody Dynamics Nonlinear Dynamics Structural Dynamics Vibrations
Acoustics Noise Control
Material Engineering Transport Phenomena Manufacturing Process Mechatronics
本人與研究生在會議中發表論文一篇，名稱為「Free Vibration Analysis of Rotating Euler Beam by Finite Element Method」，本人的論文探討了旋轉梁之側向及軸向振動的耦合現象，本人的研究提 出一個共旋轉有限元素法求旋轉梁的穩態變形及自然頻率，也對耦合振動做了進一步的研究，探 討了在不同細長比之旋轉梁的穩態變形及自然振動頻率及振態。
二 與會心得與建議
本會議雖然很大，但一切都安排的井井有條，招待上相當用心。
三 攜回資料
本會議論文發表在IAENG journal 的 Engineering Letters, 20:3, pp253-258, ISSN: 1816-0948 (online version); 1816-093X (print version), Subject Category: Computer science and engineering, Published by: International Association of Engineers
極具參考價值。
Abstract—In this paper a corotational finite element formulation is employed to derive the equations of motion for a three dimensional rotating Euler beam with constant angular velocity. The steady state deformation and natural frequency of the infinitesimal free vibration measured from the position of the corresponding steady state deformation are investigated for rotating Euler beams with different setting angle. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the position of steady state deformation. Numerical examples are studied to demonstrate the accuracy of the proposed method and to investigate the effects of the steady twist deformation on the natural frequency of rotating beams with different setting angle.
Index Terms—Rotating beam, Corotational formulation, Setting angle, Natural frequency.
I. INTRODUCTION
Rotating beams are often used as a simple model for propellers, turbine blades, and satellite booms. Rotating beam differs from a non-rotating beam in having additional centrifugal force and Coriolis effects on its dynamics. It is well known that the spinning elastic bodies sustains a steady state deformations induced by constant rotation [1]. For doubly symmetric rotating beams with setting angle other than 0 and 90 , that steady state deformations include axial deformation and twist deformation. The bending vibration, torsional vibration, and axial vibration of rotating beams are coupled due to the Coriolis effects [2] and the steady state deformation [3]. However, to the authors’ knowledge, the steady state deformation and its effects on the bending, torsional, and axial vibration of doubly symmetric rotating beams with setting angle other than 0and 90 are not reported in the literature. The objective of this paper is to derive the equations of motion for a rotating doubly symmetric Euler beam with constant angular velocity using a corotational finite element formulation. The steady state deformation and natural frequency of the infinitesimal free
Manuscript received March 5, 2012. This work was supported in part by the National Science Council, Taiwan under Grant NSC96-2221-E009-176.
Kuo Mo Hsiao is with Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan (phone:
886-3-5712121-55107; fax: 886-3-5720634; e-mail:
Wen Yi Lin is with Department of Mechanical Engineering, De Lin Institute of Technology, 1 Alley 380, Ching Yun Road, Tucheng, Taiwan (e-mail: [email protected]).
Fumio Fujii is with Department of Mathematical and Design Engineering,
vibration measured from the position of the corresponding steady state deformation are investigated using numerical examples.
II. FINITE ELEMENT FORMULATION
A. Description of Problem
Consider a doubly symmetric uniform Euler beam of length LT rigidly mounted with a setting angle on the periphery of rigid hub with radius R rotating about its axis fixed in space at a constant angular velocity as shown in Fig. 1. The axis of the rotating hub is perpendicular to the beam axis. The beam sustains a steady state axial and torsional deformation induced by constant rotation. In this study, large displacement and rotation with small strain are considered in the steady state deformation. The vibration of
Consider a doubly symmetric uniform Euler beam of length LT rigidly mounted with a setting angle on the periphery of rigid hub with radius R rotating about its axis fixed in space at a constant angular velocity as shown in Fig. 1. The axis of the rotating hub is perpendicular to the beam axis. The beam sustains a steady state axial and torsional deformation induced by constant rotation. In this study, large displacement and rotation with small strain are considered in the steady state deformation. The vibration of