準確性分析

為探討本文方法的準確性，本節中分析了文獻[14]的例題並與其結果比 較。本節中首先考慮了無因次轉軸半徑 r = 1 、細長比 η = 70 、無因次轉速 k

= 5/70 的旋轉梁，表七為本文的結果及文獻[14]的結果。由表七可以發現除 了傾斜角 α = 0 外，最大膜應變 ε

₀

、最大撓曲應變 ε

_b

都遠大於本文的允許 最大應變 10

⁻²

，所以表七的結果僅供比較用。由表七可以發現當傾斜角 α = 0 時，側向位移的穩態解為零，本文的結果及文獻[14]的結果相當接 近，但因文獻[14]沒有考慮梁的轉動慣量項，故仍有些微的不同；當傾斜角

α 逐漸增加時，穩態側向位移逐漸增加，本文及文獻[14]的無因次振動頻率

之差異逐漸增加，這可能是因文獻[14]沒有考慮梁的側向位移的穩態解 造成的。

K

1

本節中還分析了無因次轉軸半徑 r = 1 、不同細長比 η 、不同無因次轉速 k 之旋轉梁的穩態變形，並與附錄 B 的線性解及文獻[16]的結果比較，由表 八可以發現當 η = 100 時，線性解與本文的非線性解的差異不太大，但當

= 500

η 、1000 時，線性解比本文的非線性解大很多，這應是離心力引起的

軸向拉力會增加梁的側向剛度，且其影響隨細長比 η 增加而增加。當 時，文獻[16]的級數解與本文的非線性解的相當接近，但隨著 側向位移的穩態解增大兩者的差異逐漸增大，這應是因文獻[16]沒有考慮離 心力引起的側向力是側向位移的函數，且文獻[16]在推導其統御方程式時，

僅取到變形二次項，所以側向位移太大時，其結果應不準確。由表八可以 發現對相同的傾斜角

2 . 0 /

_T

<

tip

L V

α ，當 k η 有相同的值時，本文的 之值非常接近，

文獻[16]的 有相同的值，由文獻[16]可以發現當 有相同的值

時，旋轉傾斜梁在不同的無因次轉速下之無因次穩態側向位移 有相同

的無因次統馭方程式，故其 有相同的值。

T tip

L V /

T tip

L

V / η

²

k

²

r sin α

L

T

V/

T tip

L V / 4.3 個案分析

本節中將探討具不同無因次轉軸半徑 r、細長比 η 及傾斜角 α _之旋轉傾 斜 Euler 梁在不同無因次轉速 k 的穩態解、無因次振動頻率 K 及振態。

當無因次轉速 k = 0 時，無因次振動頻率 K 僅與細長比 η 有關且其軸向 和側向振態不互相耦合，表九為 k = 0 時不同細長比 η 之旋轉梁的的前七個 無因次振動頻率 (i = 1~7)，表九中(a)表示該振動頻率對應的振態為軸向 振態，由表九可以發現當

K

i

η = 38-40 時，第一個軸向振態對應的振動頻率和 第三個側向振態對應的振動頻率相當接近，且第二個軸向振態對應的振動 頻率和第五個側向振態對應的振動頻率相當接近， η = 75-80 時，第一個軸 向振態對應的振動頻率和第四個側向振態對應的振動頻率相當接近。

當 r = 0 、 k ≠ 0 時，如前所述，因科氏力不為零，故其軸向和側向振態 互相耦合，表十為 r = 0 、 k ≠ 0 時，不同細長比 η 之旋轉梁在不同無因次轉 速 k 的前七個無因次振動頻率 (i = 1~7)，表十中(a)表示該振動頻率對應 的振態在 k = 0 時為軸向振態，當

K

i

≠ 0

r 、 k ≠ 0 、 α = 0 時，由文獻[16]或

(2.5.17)、(2.5.28)、(2.6.33)式可知旋轉梁的穩態解僅有軸向變形，但由文獻

[16]或(2.7.17)式可知因科氏力不為零，故其軸向和側向振態互相耦合。

表十一到表三十七為無因次轉軸半徑 r = 0.5、1、1.5，傾斜角 α = 、 、

、 、 、 、 、 、 ，細長比

0 5 10 15 30 45 60 75 90 η = 10、20、50、100、500、

1000 之旋轉梁在不同無因次轉速 k 的最大膜應變 ε

₀

、撓曲應變 ε

_b

、無因次

端點側向位移 及前七個無因次振動頻率 (i = 1~7)。由表中可以發現

當傾斜角

T tip

L

V / K

_i

α = 0 、 ，細長比 5 η = 10、20 時，旋轉傾斜梁最大的無因次轉速 k 主要由最大膜應變 ε

₀

決定，當傾斜角 α ≠ 0 、細長比 η > 20 時，旋轉傾斜 梁最大的無因次轉速 k 主要由最大撓曲應變 ε

_b

決定；本文中考慮的最大無 因次轉速隨著傾斜角 α 和細長比 η 的增加而減少，當傾斜角 α = 0 、 、

、 時，本文中最大無因次轉速取 k = 0.06、0.03、0.02、0.01，當傾斜 角

5 10 15

≥ 30

α 、細長比 η ≤ 100 時，本文中最大無因次轉速取 k = 0.01，當傾斜角

≥ 30

α 、細長比 η ≥ 100 時，本文中最大無因次轉速 k 隨著傾斜角 α 和細長

比 η 的增加而減少。由表十一到表三十七中可以發現無因次轉速 k 對無因次

振動頻率 K

_i

(i = 1~7)的影響隨著 i 的增加而減少，但隨著細長比 η 及無因次 轉軸半徑 r 的增加而增加；側向振動頻率 K

_i

隨著無因次轉速 k 增加而增加，

且 i 愈小、細長比 η 愈大、無因次轉軸半徑 r 愈大時，其增加率愈明顯。在 細長比 η ≤ 100 ，相同無因次轉速 k 下，傾斜角 α 對側向振動頻率 似乎影 響不是很大。

K

i

圖八至圖十六為細長比 η = 80、100、1000、傾斜角 α = 5 、 、 時，

旋轉傾斜梁在不同無因次轉速

30 90 k 的穩態變形、穩態軸向及側向位移，圖中的 穩態變形在 X、Y 方向是依相同的尺度繪製，故為旋轉梁真正的變形圖。由 圖八及圖十一可以發現當傾斜角 α = 5 、 η = 80、100 時，側向位移很小，

因離心力的效應，軸向位移是正的；其餘的情況，因側向位移較大，其效 應大於離心力的效應，故軸向位移是負的。

本章中將旋轉梁的振態分成 X

₁

和 X

₂

的分量， U

_R

+ iU

_I

= U e

^iφu

及

i v

I

R

iV V e

V + =

^φ

，其中 U = ( U

_R²

+ U

_I²

)

^1/2

、 V = ( V

_R²

+ V

_I²

)

^1/2

為振態的大小，

R I

u

U

= U φ

tan 、

R I

v

V

= V φ

tan ， φ

_u

、 φ

_v

為相角(Phase angle)。本章中的振動模態 圖中取 − π ≤ φ

_u

≤ π 、 − π ≤ φ

_v

≤ π ， U = U sign (sin φ

_u

) 、 V = V sign (sin φ

_v

) ， 其中 sign ( x ) = 1 若 x ≥ 0 ， sign ( x ) = − 1 若 x < 0 。

圖十七至圖二十八為無因次轉軸半徑 r = 1、細長比 η = 50、100、1000、

傾斜角 α = 0 、 、 、 時，在不同無因次轉速 k 下，對應於前六個振 動頻率的振動模態及相角。因為在無因次轉速 k=0 時，

5 30 90

= 0

=

_v

u

φ

φ ，且在傾

斜角 α = 0 、無因次轉速 k ≠ 0 時， φ

_u

= 0 、

2

φ

_v

= π ，所以在其振動模態圖中

中沒有繪出相角。由圖十七至圖二十八可以發現當無因次轉速 k = 0 時，其 軸向和側向振態不互相耦合。

由圖十七到二十四可以發現當細長比 η = 50 及 100 時，其第四個振態及

第五振態分別為其第一個軸向振態，當 k ≠ 0 時，可以看出該振態有側向分

量，其側向分量的大小隨著無因次轉速 k、傾斜角 α 和細長比 η 增加而增加，

但其軸向分量的形狀和大小與 k = 0 時幾乎一樣；由圖十七到二十四亦可以 發現其餘的振態皆為側向振態，當 k ≠ 0 時，可以看出這些振態有軸向分量，

其軸向分量的大小隨著無因次轉速 k、傾斜角 α 和細長比 η 增加而增加，但 其側向分量的形狀和大小與 k = 0 時幾乎一樣。

由圖二十五到二十八可以發現當細長比 η = 1000 時，所有的振態皆為側

向振態，當 k ≠ 0 時，可以看出這些振態有軸向分量，其軸向分量的大小隨

著無因次轉速 k、傾斜角 α 增加而增加，且其側向分量的形狀和大小與 k = 0 時有相當的改變。

當傾斜角 α ≠ 0 時，側向穩態位移隨傾斜角 α 、無因次轉速 k 及細長比

η 增加而變大，所以以穩態解為平衡點的振動都是軸向與側向耦合的振動，

且其耦合的程度隨傾斜角 α 、無因次轉速 k 及細長比 η 增加而增加。當傾斜

角 α = 0 時，旋轉梁的穩態解僅有軸向變形，但因科氏力不為零，故其軸向

和側向振態互相耦合，由圖十七，細長比 η = 50 的第四個振態及圖二十一，

細長比 η = 100 的第五振態可以發現振態耦合的現象，文獻[22]中提到當兩 個振動頻率愈接近，其對應振動模態的耦合愈明顯，由表九可以發現在無 因次轉速 k = 0 時，細長比 η = 50 的第三及第四個無因次振動頻率為 1.21530 及 1.57086，細長比 η = 100 的第四及第五個無因次振動頻率為 1.20047 及 1.57086，這兩組振動頻率分別為 η = 50 和 100 之前六個無因次振動頻率中 最接近的無因次振動頻率，所以其耦合的現象最明顯。

4.4 旋轉梁之特徵值曲線轉向(Eigenvalue curve veering)分析

由上節的例題可以發現旋轉傾斜梁的無因次振動頻率會隨著無因次轉 速 k 改變，且兩個振動頻率愈接近，其對應振動模態的耦合愈明顯，但上 節的例題因振動頻率不夠接近，並未發現特徵值曲線轉向(Eigenvalue curve veering)及振態交換[23,24]的現象，本節中將考慮不同細長比 η 的旋轉梁，

探討旋轉梁之特徵值曲線轉向及振態交換的現象。

表三十八到表四十五為無因次轉軸半徑 r = 1，細長比 η = 38、38.5、39、

40、77、78、79、80，傾斜角 α = 0 、 、 、 、 、 、 、 、 之旋轉梁在不同無因次轉速 k 的前六個無因次振動頻率 (i = 1 - 6)，圖 二十九為無因次振動頻率-無因次轉速曲線，圖三十到四十五為細長比

5 10 15 30 45 60 75

90 K

_i

η = 38、39、40、77，傾斜角 α = 0 、 、 、 時，在不同無因次轉速 k 下，對應於前六個振動頻率的振動模態。由上述的表及圖中可以發現當傾 斜角

5 30 90

= 0

α ，細長比 η = 38.5、39、77、78、79、80 時，有無因次振動頻率 曲線轉向及振態交換的現象，圖三十四中，無因次轉速 k = 0、0.004，振態 3 和 4 對應的無因次振動頻率分別為 和 ，但無因次轉速 k = 0.005、

0.006，振態 3 和 4 對應的無因次振動頻率分別為 和 ；圖四十二中，

K

3

K

₄

K

4

K

₃

無因次轉速 k = 0、0.002，振態 4 和 5 對應的無因次振動頻率分別為 和

，但無因次轉速 k = 0.003、0.004，振態 4 和 5 對應的無因次振動頻率分 別為 和 。除了上述情況外，圖三十到四十五中，振態 i(i = 1-6)對應的

無因次自然頻率為 。由表九可以發現在無因次轉速 k = 0 時，細長比

K

4

K

5

K

5

K

₄

K

i

η = 38、39、40 的旋轉梁除了第三及第四個無因次振動頻率外，第六及第七

個無因次振動頻率亦相當接近，所以由圖三十到四十一中可以發現除了第

三及第四個振態外，第六振態亦有明顯的振態耦合現象。

第五章 結論與展望

本研究採用共旋轉有限元素法探討設定角為 之等速旋轉的傾斜尤拉

梁的穩態變形及以該穩態變形為平衡點的微小振動之自然振動頻率。本文 中僅考慮梁的軸向位移及單一個側向位移和旋轉二維運動，本研究採用 d'Alembert 原理、虛功原理、幾何非線性梁理論的一致線性化，在梁元素當 前之元素座標上推導節點慣性力和節點變形力，將元素的節點力轉到總體

座標後組合成系統的非線性運動方程式。當設定角為 時，旋轉梁的軸向

和側向穩態變形都不為零，本研究將旋轉梁的運動方程式的時間函數去掉 求得系統穩態平衡方程式，再用基於牛頓法的增量迭代法求出軸向位移及 側向位移的穩態解，將運動方程式在穩態平衡位置用泰勒級數展開，取到 一次項，求得旋轉傾斜梁的振動方程式，假設自然振動頻率存在，可獲得 一組代數齊次方程式，該組齊次方程式係數形成之矩陣的行列式為零時，

即可求得旋轉梁以穩態解為平衡點的自然振動的頻率及其對應的振態。

° 0

° 0

由本研究的數值例題可以得到以下的結論：

1. 旋轉梁的側向穩態位移隨傾斜角 α 、無因次轉速 k 及細長比 η 增加而 變大，且以穩態解為平衡點的振動都是軸向與側向耦合的振動，且 其耦合的程度隨傾斜角 α 、無因次轉速 k 及細長比 η 增加而增加。

2. 旋轉梁的最大的無因次轉速 k 在傾斜角 α = 0 、 ，細長比 5 η = 10、

20 時，主要由穩態解之最大膜應變決定，當傾斜角 ≠ α 0 、細長比 η > 20 時，主要由穩態解最大撓曲應變決定。

3. 由離心力引起的軸向拉力會增加梁的側向剛度，且其影響隨細長比

增加而增加。

4. 無因次轉速 k 對第 i 個無因次振動頻率 的影響隨著 i 的增加而減 少，但隨著梁的細長比

K

i

η 及無因次轉軸半徑 r 的增加而增加；無因次 側向振動頻率隨著無因次轉速 k 增加而增加，且 i 愈小、細長比 η 愈 大、無因次轉軸半徑 r 愈大時，其增加率愈明顯。

5. 梁的無因次振動頻率會隨著無因次轉速 k 改變，且兩個振動頻率愈

接近，其對應振動模態的耦合愈明顯。在傾斜角為 0 ° 且細長比

η = 38-40、77-80 時，可發現當兩個振動頻率很接近時會發生特徵值 曲線轉向(Eigenvalue curve veering)及振態交換的現象。

本研究僅探討設定角為 時，旋轉傾斜尤拉梁穩態變形及振動，其運

動是二維的運動，以後的研究應考慮設定角不為 時的旋轉傾斜尤拉梁的

穩態變形及振動，其運動是三維的運動。

° 0

°

0

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606-624.

[16] 顏宏儒, 旋轉傾斜梁之穩態及自由振動分析, 國立交通大學機械工程 研究所碩士論文, 臺灣, 新竹, 2008.

[17] T. J. Chung, Continuous Mechanics, Prentice-Hall, Inc., Englewood Cliff, New Jersey, 1988.

[18] J. M. Gere, Mechanics of Materials, fifth edition, 2000.

[19] A. J. Sommese, C. W. Wampler, The Numerical solution of systems of polynomials arising in engineering and science, 2005.

[20] K. J. Bathe, Finite element procedure, Prentice-Hall, New Jersey, 1996.

[21] S. S. Rao, Engineering optimization: theory and practice, third edition, 2000.

[22] P. Marugabandhu, J. H. Griffin, “A Reduced-Order Model for Evaluating

the Effect of Rotational Speed on the Natural Frequencies and Mode Shapes

of Blades”, ASME Journal of Engineering for Gas Turbines and Power, Vol.

125, 2003, pp. 772-776.

[23] X. Chen, A. Kareem, “Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects”, Journal of Engineering Mechanics, Vol. 129, No. 2, 2003, pp. 146-159.

[24] J. L. du Boisa, S. Adhikarib, Nick A. J. Lieven, “Eigenvalue curve veering

in stressed structures: An experimental study”, Journal of Sound and

Vibration, Vol. 322, 2009, pp. 1117-1124.

表一 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 10

^，r =1.5)

α 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) ^{tip T} L V /

) 10

( ⁻³ K ¹ K₂(a) K ₃ K ₄ K₅(a) K ₆ K ₇

5 D 10 0.03 1.72672 1.00662 1.44842 .346300 1.57297 1.91534 4.64752 4.75786 7.82258 8.05791 20 1.76145 1.00806 1.44846 .346288 1.57176 1.91526 4.64593 4.72569 7.81539 7.90662 40 1.77854 1.00843 1.44847 .346286 1.57146 1.91524 4.64560 4.71775 7.81356 7.87006 50 1.78193 1.00848 1.44847 .346285 1.57142 1.91524 4.64556 4.71681 7.81324 7.86581 80 1.78701 1.00853 1.44847 .346285 1.57138 1.91524 4.64552 4.71578 7.81285 7.86126 100 1.78870 1.00854 1.44847 .346285 1.57137 1.91524 4.64551 4.71554 7.81275 7.86022 30 D 10 0.01 .173261 .647778 .936098 .343934 1.57247 1.91385 4.64995 4.75630 7.82680 8.05717 20 .176628 .648601 .936101 .343933 1.57126 1.91380 4.64903 4.72357 7.82103 7.90479 40 .178279 .648807 .936101 .343932 1.57096 1.91380 4.64894 4.71542 7.82046 7.86705 50 .178607 .648832 .936101 .343932 1.57092 1.91380 4.64893 4.71445 7.82039 7.86255 80 .179097 .648859 .936101 .343932 1.57088 1.91380 4.64893 4.71339 7.82032 7.85769 100 .179260 .648865 .936101 .343932 1.57087 1.91380 4.64892 4.71314 7.82030 7.85657 75 D 10 0.01 .0867650 1.25238 1.81023 .343750 1.57247 1.91378 4.65000 4.75630 7.82699 8.05717

20 .0878387 1.25396 1.81024 .343749 1.57126 1.91374 4.64909 4.72357 7.82123 7.90479 40 .0883433 1.25435 1.81024 .343749 1.57096 1.91373 4.64900 4.71542 7.82066 7.86705 50 .0884416 1.25440 1.81024 .343749 1.57092 1.91373 4.64899 4.71445 7.82060 7.86255 80 .0885874 1.25445 1.81024 .343749 1.57088 1.91373 4.64899 4.71339 7.82052 7.85769 100 .0886356 1.25446 1.81024 .343749 1.57087 1.91373 4.64898 4.71314 7.82050 7.85657 90 D 10 0.01 .0498854 1.29698 1.87490 .343672 1.57247 1.91375 4.65002 4.75630 7.82707 8.05717 20 .0499869 1.29861 1.87490 .343671 1.57126 1.91370 4.64911 4.72357 7.82131 7.90479 40 .0500053 1.29902 1.87490 .343671 1.57096 1.91370 4.64902 4.71542 7.82074 7.86705 50 .0500064 1.29907 1.87490 .343671 1.57092 1.91370 4.64902 4.71445 7.82068 7.86255 80 .0500064 1.29912 1.87490 .343671 1.57088 1.91370 4.64901 4.71339 7.82061 7.85769 100 .0500059 1.29913 1.87490 .343671 1.57087 1.91370 4.64901 4.71314 7.82059 7.85657

表二 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 20

，r=1.5)

α 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) V^tip /L^T ) 10

( ⁻³ K ¹ K ₂ K₃ (a) K ₄ K₅(a) K ₆ K ₇

5 D 10 0.03 1.72680 1.93655 5.48934 .180882 1.06562 1.57335 2.82949 4.75639 5.19801 8.00220 20 1.76150 1.94011 5.48989 .180860 1.06554 1.57214 2.82878 4.72359 5.19374 7.90467 40 1.77856 1.94104 5.49003 .180855 1.06552 1.57184 2.82872 4.71541 5.19341 7.86675 50 1.78195 1.94115 5.49004 .180854 1.06552 1.57180 2.82871 4.71442 5.19339 7.86221 80 1.78702 1.94128 5.49006 .180853 1.06552 1.57176 2.82871 4.71336 5.19337 7.85729 100 1.78871 1.94131 5.49007 .180853 1.06552 1.57175 2.82871 4.71312 5.19337 7.85615 30 D 10 0.01 .173298 1.29046 3.72389 .175392 1.06018 1.57252 2.82541 4.75613 5.19573 8.00213

20 .176647 1.29216 3.72393 .175390 1.06014 1.57131 2.82480 4.72333 5.19170 7.90455 40 .178289 1.29258 3.72394 .175389 1.06014 1.57100 2.82476 4.71514 5.19143 7.86661 50 .178615 1.29263 3.72394 .175389 1.06014 1.57097 2.82476 4.71416 5.19142 7.86207 80 .179102 1.29269 3.72395 .175389 1.06014 1.57093 2.82476 4.71310 5.19141 7.85715 100 .179264 1.29270 3.72395 .175389 1.06014 1.57092 2.82476 4.71285 5.19141 7.85601 75 D 10 0.01 .0869040 2.50173 7.22633 .174999 1.05987 1.57253 2.82520 4.75612 5.19562 8.00207

20 .0879121 2.50493 7.22636 .174998 1.05984 1.57132 2.82460 4.72332 5.19160 7.90451 40 .0883807 2.50573 7.22637 .174997 1.05983 1.57102 2.82456 4.71513 5.19134 7.86659 50 .0884715 2.50583 7.22637 .174997 1.05983 1.57098 2.82456 4.71415 5.19132 7.86205 80 .0886060 2.50593 7.22638 .174997 1.05983 1.57094 2.82456 4.71309 5.19132 7.85713 100 .0886504 2.50596 7.22638 .174997 1.05983 1.57093 2.82456 4.71284 5.19132 7.85599 90 D 10 0.01 .0500346 2.59388 7.49567 .174830 1.05974 1.57253 2.82511 4.75612 5.19557 8.00206

20 .0500657 2.59716 7.49569 .174829 1.05971 1.57132 2.82451 4.72331 5.19156 7.90451 40 .0500454 2.59798 7.49570 .174829 1.05971 1.57102 2.82447 4.71513 5.19129 7.86659 50 .0500385 2.59808 7.49570 .174829 1.05971 1.57098 2.82447 4.71415 5.19128 7.86205 80 .0500263 2.59819 7.49570 .174829 1.05971 1.57094 2.82447 4.71309 5.19127 7.85712 100 .0500217 2.59821 7.49570 .174829 1.05971 1.57093 2.82447 4.71284 5.19127 7.85599

表三 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 50

，r=1.5) 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) V_tip /L_T K ₁ K ₂ K ₃ K₄(a) K ₅ K ₆ K₇ (a) α

5 D 10 0.03 1.72718 3.89539 .0250785 .0848761 .455824 1.23339 1.57392 2.37221 3.85294 4.75649 20 1.76166 3.91291 .0250895 .0848304 .455686 1.23296 1.57270 2.36997 3.84395 4.72369 40 1.77860 3.91756 .0250922 .0848189 .455654 1.23291 1.57239 2.36977 3.84326 4.71551 50 1.78196 3.91814 .0250925 .0848175 .455650 1.23290 1.57236 2.36976 3.84323 4.71453 80 1.78699 3.91878 .0250929 .0848161 .455646 1.23290 1.57232 2.36975 3.84320 4.71346 100 1.78866 3.91893 .0250929 .0848157 .455645 1.23290 1.57231 2.36975 3.84320 4.71322 30 D 10 0.01 .173543 3.14014 .0224082 .0718399 .439698 1.21709 1.57305 2.35587 3.83741 4.75622 20 .176773 3.14519 .0224097 .0718343 .439671 1.21680 1.57183 2.35382 3.82874 4.72341 40 .178350 3.14647 .0224101 .0718329 .439667 1.21678 1.57153 2.35368 3.82814 4.71523 50 .178662 3.14662 .0224101 .0718327 .439667 1.21677 1.57149 2.35367 3.82812 4.71425 80 .179129 3.14679 .0224101 .0718325 .439667 1.21677 1.57145 2.35367 3.82810 4.71318 100 .179284 3.14683 .0224102 .0718325 .439666 1.21677 1.57145 2.35367 3.82810 4.71294 75 D 10 0.01 .0878436 6.19826 .0444794 .0709001 .438758 1.21517 1.57464 2.35531 3.83649 4.75645

20 .0883942 6.20706 .0444809 .0708973 .438738 1.21488 1.57342 2.35326 3.82783 4.72365 40 .0886110 6.20927 .0444812 .0708966 .438736 1.21486 1.57311 2.35312 3.82724 4.71547 50 .0886496 6.20953 .0444812 .0708966 .438735 1.21486 1.57307 2.35311 3.82722 4.71449 80 .0887045 6.20982 .0444813 .0708965 .438735 1.21486 1.57303 2.35311 3.82720 4.71343 100 .0887220 6.20989 .0444813 .0708964 .438735 1.21486 1.57303 2.35311 3.82720 4.71318 90 D 10 0.01 .0510541 6.47860 .0466112 .0704770 .438377 1.21466 1.57485 2.35498 3.83613 4.75648 20 .0505878 6.48726 .0466120 .0704755 .438359 1.21438 1.57363 2.35294 3.82748 4.72368 40 .0502942 6.48943 .0466122 .0704751 .438357 1.21436 1.57332 2.35280 3.82689 4.71550 50 .0502306 6.48969 .0466122 .0704751 .438357 1.21436 1.57328 2.35279 3.82687 4.71452 80 .0501320 6.48997 .0466122 .0704750 .438357 1.21436 1.57325 2.35279 3.82685 4.71346 100 .0500983 6.49004 .0466122 .0704750 .438357 1.21436 1.57324 2.35279 3.82685 4.71321

表四 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 100

，r=1.5) 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) V_tip /L_T K ₁ K ₂ K ₃ K ₄ K₅ (a) K ₆ K ₇

α

5 D 10 0.03 1.72775 5.09357 .0511913 .0591219 .254778 .652135 1.24001 1.57551 2.02262 2.99338 20 1.76185 5.15588 .0512287 .0590608 .254541 .651746 1.23871 1.57422 2.01772 2.97847 40 1.77853 5.17329 .0512379 .0590456 .254483 .651675 1.23857 1.57390 2.01730 2.97732 50 1.78184 5.17548 .0512390 .0590437 .254476 .651668 1.23856 1.57386 2.01728 2.97726 80 1.78678 5.17791 .0512402 .0590418 .254469 .651659 1.23855 1.57382 2.01726 2.97721 100 1.78842 5.17847 .0512404 .0590413 .254467 .651657 1.23855 1.57381 2.01725 2.97721 30 D 10 0.01 .174250 5.74549 .0788668 .0383279 .223645 .617615 1.20301 1.57860 1.98709 2.95692 20 .177102 5.76051 .0788849 .0383174 .223613 .617446 1.20193 1.57731 1.98247 2.94234 40 .178466 5.76440 .0788893 .0383147 .223606 .617431 1.20185 1.57700 1.98215 2.94131 50 .178734 5.76488 .0788899 .0383144 .223606 .617430 1.20185 1.57696 1.98213 2.94126 80 .179132 5.76540 .0788905 .0383141 .223605 .617428 1.20185 1.57692 1.98212 2.94124 100 .179264 5.76552 .0788906 .0383140 .223605 .617428 1.20185 1.57691 1.98212 2.94123 75 D 10 0.01 .0907471 11.9290 .166216 .0368604 .221453 .612094 1.19273 1.59872 1.99177 2.95541

20 .0897082 11.9519 .166237 .0368544 .221434 .611941 1.19168 1.59719 1.98718 2.94085 40 .0890411 11.9577 .166242 .0368529 .221430 .611929 1.19161 1.59684 1.98684 2.93981 50 .0888954 11.9584 .166242 .0368527 .221430 .611928 1.19160 1.59680 1.98682 2.93976 80 .0886691 11.9591 .166243 .0368525 .221430 .611928 1.19160 1.59676 1.98680 2.93974 100 .0885916 11.9593 .166243 .0368524 .221429 .611928 1.19160 1.59675 1.98680 2.93973 90 D 10 0.01 .0543132 12.7940 .179785 .0361111 .220583 .610488 1.19029 1.60306 1.99241 2.95467

20 .0520411 12.8146 .179797 .0361073 .220569 .610339 1.18923 1.60148 1.98784 2.94012 40 .0507494 12.8197 .179800 .0361063 .220567 .610328 1.18916 1.60113 1.98749 2.93907 50 .0504783 12.8203 .179801 .0361062 .220567 .610327 1.18916 1.60109 1.98747 2.93903 80 .0500633 12.8210 .179801 .0361061 .220566 .610327 1.18916 1.60105 1.98745 2.93900 100 .0499228 12.8211 .179801 .0361060 .220566 .610327 1.18916 1.60104 1.98745 2.93899

表五 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 500

，r=1.5) 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) V_tip /L_T K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

α

5 D 10 0.03 1.73038 5.04954 .0803070 .0464280 .134390 .247496 .393200 .571561 .784736 1.03544 20 1.76394 5.76426 .0804024 .0463541 .133696 .246343 .391776 .569491 .780885 1.02742 40 1.77967 6.04540 .0804161 .0463376 .133519 .246063 .391472 .569153 .780442 1.02671 50 1.78268 6.08649 .0804174 .0463356 .133498 .246030 .391438 .569118 .780406 1.02667 80 1.78709 6.13455 .0804186 .0463336 .133475 .245995 .391401 .569083 .780372 1.02663 100 1.78852 6.14641 .0804189 .0463331 .133470 .245986 .391392 .569075 .780365 1.02663 30 D 10 0.008 .114268 8.46079 .352547 .0144174 .0553632 .133507 .250244 .406843 .604350 .843603

20 .114306 8.60956 .352775 .0144002 .0553013 .133485 .250083 .405945 .601303 .835644 40 .113982 8.65147 .352830 .0143958 .0552855 .133486 .250089 .405911 .601126 .835109 50 .113885 8.65675 .352837 .0143953 .0552836 .133486 .250091 .405913 .601122 .835091 80 .113717 8.66258 .352844 .0143947 .0552815 .133486 .250094 .405915 .601122 .835083 100 .113655 8.66395 .352845 .0143946 .0552810 .133487 .250094 .405916 .601123 .835083 75 D 10 0.003 .00848906 4.97759 .325332 .00799673 .0446289 .121364 .238962 .396654 .595333 .835863 20 .00817969 4.99116 .325406 .00799381 .0446229 .121333 .238737 .395692 .592299 .828112 40 .00799694 4.99460 .325425 .00799307 .0446215 .121331 .238720 .395623 .592080 .827535 50 .00795804 4.99501 .325427 .00799298 .0446213 .121330 .238719 .395619 .592070 .827510 80 .00789818 4.99546 .325429 .00799288 .0446211 .121330 .238719 .395617 .592064 .827495 100 .00787782 4.99556 .325430 .00799286 .0446211 .121330 .238719 .395617 .592063 .827493 90 D 10 0.003 .00526392 5.50358 .363581 .00774943 .0441721 .120239 .237680 .395230 .593820 .834230 20 .00479652 5.51497 .363632 .00774726 .0441682 .120208 .237453 .394262 .590779 .826470 40 .00453290 5.51777 .363645 .00774671 .0441673 .120206 .237435 .394191 .590556 .825889 50 .00447769 5.51810 .363646 .00774664 .0441672 .120206 .237434 .394187 .590546 .825863 80 .00439327 5.51846 .363648 .00774657 .0441671 .120206 .237433 .394185 .590539 .825847 100 .00436471 5.51854 .363648 .00774655 .0441671 .120205 .237433 .394184 .590538 .825845

表六 不同傾斜角與不同轉速的旋轉梁之振動頻率的收斂分析(

η = 1000

，r=1.5) 元素

數目 k ε₀ (10⁻³) ε_b (10⁻³) Vtip /LT K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

α

5 D 10 0.03 1.73113 3.88613 .0835235 .0454694 .127442 .217642 .323061 .442957 .577154 .726538 20 1.76526 5.17029 .0837847 .0453581 .126503 .215621 .320468 .439874 .573192 .720772 40 1.78106 5.89061 .0838230 .0453354 .126247 .215078 .319816 .439172 .572419 .719852 50 1.78397 6.01548 .0838255 .0453331 .126216 .215013 .319740 .439096 .572342 .719773 80 1.78809 6.17213 .0838275 .0453309 .126183 .214944 .319660 .439015 .572263 .719697 100 1.78938 6.21315 .0838279 .0453304 .126175 .214927 .319641 .438996 .572245 .719680 30 D 10 0.008 .117176 8.73688 .429697 .0129066 .0405573 .0836364 .143457 .221474 .319570 .439088

20 .116206 9.19910 .429937 .0128898 .0404478 .0836092 .143553 .221286 .318124 .434692 40 .114745 9.34364 .429982 .0128852 .0404185 .0836045 .143617 .221403 .318213 .434607 50 .114344 9.36265 .429987 .0128846 .0404148 .0836039 .143625 .221421 .318236 .434627 80 .113663 9.38393 .429993 .0128840 .0404109 .0836032 .143634 .221442 .318264 .434657 100 .113413 9.38899 .429994 .0128839 .0404101 .0836030 .143637 .221447 .318271 .434665 75 D 10 0.003 .00938191 7.02594 .663462 .00571542 .0238973 .0587160 .115623 .193319 .292133 .412286

20 .00834191 7.07182 .663647 .00570942 .0238954 .0587390 .115534 .192822 .290543 .408259 40 .00771222 7.08294 .663692 .00570781 .0238942 .0587461 .115530 .192785 .290420 .407943 50 .00757673 7.08424 .663697 .00570762 .0238940 .0587470 .115531 .192783 .290414 .407928 80 .00736725 7.08564 .663703 .00570740 .0238939 .0587479 .115531 .192781 .290408 .407916 100 .00729575 7.08595 .663705 .00570735 .0238938 .0587481 .115531 .192781 .290408 .407914 90 D 10 0.003 .00632598 8.11019 .747141 .00561366 .0232167 .0566047 .113316 .190888 .289635 .409720

20 .00489502 8.14495 .747230 .00560767 .0232188 .0566244 .113214 .190376 .288040 .405709 40 .00405954 8.15225 .747251 .00560603 .0232182 .0566290 .113204 .190328 .287902 .405375 50 .00388231 8.15303 .747254 .00560584 .0232181 .0566295 .113203 .190324 .287893 .405356 80 .00360996 8.15384 .747256 .00560562 .0232180 .0566301 .113202 .190321 .287886 .405341 100 .00351746 8.15402 .747257 .00560557 .0232180 .0566302 .113202 .190320 .287884 .405339

表七 不同傾斜角的旋轉梁之振動頻率的準確性分析(

70

= 5

k ，

r = 1

，

η = 70

) α ε₀ ε_b V_tip/L_T K ₁ K [14] ₁ K ₂ K [14] ₂ 0 D .00761579 0 0 .105427 .104966 .410792 .417489 10 D .00753381 .0220374 .119890 .105377 .104674 .410400 .416567 20 D .00728963 .0438841 .237606 .105225 .102756 .409219 .414031 30 D .00688882 .0653510 .351025 .104971 .100040 .407246 .410294 40 D .00634057 .0862526 .458121 .104612 .0960628 .404475 .404936 50 D .00565758 .106408 .557013 .104146 .0912700 .400900 .398364 60 D .00485594 .125641 .646008 .103568 .0850186 .396518 .389997 70 D .00395486 .143786 .723643 .102875 .0771729 .391331 .381219 80 D .00297641 .160683 .788722 .102058 .0683600 .385349 .371329 90 D .00194513 .176180 .840342 .101109 .0571429 .378595 .361034

表八 不同傾斜角與不同細長比的旋轉梁之端點位移的準確性分析(

r = 1

)

α ^η k ε₀ (10⁻⁴) ε₀^L(10⁻⁴)

ε

_b ε_b^L Vtip /LT V_tip^L /L_T V_tip /L_T[16]

5 D 500 0.02 5.94466 5.98478 .00335361 .0150958 .0734169 1.08945 .073612 1000 0.01 1.48600 1.49619 .00167595 .00754791 .0733957 1.08945 .073612 1000 0.03 13.3890 13.4658 .00519998 .0679312 .0825422 9.80502 .082773 10 D 500 0.01 1.47373 1.48481 .00305801 .00751919 .119162 .542651 .12020

1000 0.005 .368421 .371202 .00152880 .00375959 .119151 .542651 .12020 15 D 500 0.004 .232945 .234548 .00133657 .00179315 .0825672 .129410 .082973

1000 0.01 1.45421 1.46593 .00501583 .0224144 .218450 3.23524 .224477 30 D 100 0.01 1.35791 1.36603 .00399154 .00433013 .0553226 .0625000 .055441

500 0.008 .862862 .874256 .00691072 .0138564 .310627 1.00000 .331620 1000 0.004 .215709 .218564 .00345508 .00692820 .310607 1.00000 .331620 45 D 100 0.01 1.20113 1.20711 .00571738 .00612372 .0794779 .0883883 .079854 1000 0.003 .106851 .108640 .00349175 .00551135 .367905 .795495 .406945 60 D 100 0.008 .637717 .640000 .00464576 .00480000 .0656798 .0692820 .065896

500 0.006 .349676 .360000 .00904678 .0135000 .473591 .974279 .568146 1000 0.003 .0874165 .0900000 .00452322 .00675000 .473573 .974279 .568146 75 D 100 0.008 .485134 .485644 .00525596 .00535370 .0745883 .0772741 .074912 1000 0.003 .0645893 .0682937 .00544045 .00752865 .565891 1.08667 .756726 90 D 100 0.008 .321229 .320000 .00552865 .00554256 .0788019 .0800000 .079189 100 0.01 .502076 .500000 .00860877 .00866025 .121597 .125000 .123034 1000 0.003 .0398360 .0450000 .00621305 .00779423 .642909 1.12500 .988891

0 k = 表九 旋轉傾斜梁在不同細長比下的振動頻率( )

η K ₁ K₂ K₃ K₄ K ₅ K ₆ K₇

10 .343681 1.57086(a) 1.91364 4.64936 4.71413(a) 7.82132 7.86206(a) 20 .174787 1.05953 1.57086(a) 2.82431 4.71413(a) 5.19120 7.86206(a) 38 .0923781 .573449 1.57086(a) 1.58182 3.03502 4.71413(a) 4.88776 38.5 .0911822 .566163 1.56231 1.57086(a) 2.99909 4.71413(a) 4.83278 39 .0900168 .559057 1.54325 1.57086(a) 2.96396 4.71413(a) 4.77893 40 .0877730 .545362 1.50647 1.57086(a) 2.89600 4.67451 4.71413^(a) 50 .0702550 .437859 1.21530 1.57086(a) 2.35176 3.82646 4.71413(a) 75 .0468608 .292950 .817035 1.57086(a) 1.59193 2.61236 3.86790 76 .0462448 .289117 .806427 1.57086(a) 1.57150 2.57930 3.81980 77 .0456446 .285383 .796090 1.55157 1.57086(a) 2.54705 3.77285 78 .0450599 .281744 .786013 1.53214 1.57086(a) 2.51558 3.72700 79 .0444900 .278196 .776186 1.51318 1.57086(a) 2.48487 3.68223 80 .0439342 .274736 .766600 1.49468 1.57086(a) 2.45488 3.63849 100 .0351520 .219989 .614602 1.20047 1.57086(a) 1.97619 2.93707 500 .00703197 .0440661 .123375 .241735 .399539 .596720 .833235 1000 .00351601 .0220341 .0616949 .120893 .199838 .298509 .416903

表十 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r= ) 0

k ε₀ (10 )⁻⁴ ε_b ^tip

T

V

η L K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.01 .499954 0 0 .343670 1.57092^(a) 1.91370 4.64902 4.71445^(a) 7.82068 7.86255(a) 0.02 2.00007 0 0 .343639 1.57111^(a) 1.91389 4.64802 4.71536^(a) 7.81886 7.86394(a) 0.04 8.00427 0 0 .343513 1.57185^(a) 1.91465 4.64426 4.71880^(a) 7.81245 7.86861(a) 0.06 18.0246 0 0 .343306 1.57310^(a) 1.91591 4.63867 4.72385^(a) 7.80359 7.87457(a) 20 0.01 .499954 0 0 .174823 1.05972 1.57096^(a) 2.82446 4.71417(a) 5.19127 7.86208(a) 0.02 2.00007 0 0 .174930 1.06027 1.57127^(a) 2.82489 4.71427(a) 5.19148 7.86213(a) 0.04 8.00427 0 0 .175354 1.06248 1.57251^(a) 2.82663 4.71467(a) 5.19234 7.86236(a) 0.06 18.0246 0 0 .176054 1.06615 1.57458^(a) 2.82952 4.71533(a) 5.19377 7.86273(a) 50 0.01 .499954 0 0 .0703844 .438455 1.21592 1.57096^(a) 2.35238 3.82704 4.71417(a) 0.02 2.00007 0 0 .0707689 .440240 1.21780 1.57125^(a) 2.35425 3.82879 4.71426(a) 0.04 8.00427 0 0 .0722524 .447313 1.22528 1.57240^(a) 2.36171 3.83581 4.71466(a) 0.06 18.0246 0 0 .0745530 .458872 1.23765 1.57433^(a) 2.37409 3.84748 4.71531(a) 100 0.01 .499954 0 0 .0354205 .221216 .615938 1.20187 1.57096^(a) 1.97760 2.93847 0.02 2.00007 0 0 .0361954 .224860 .619929 1.20605 1.57124^(a) 1.98183 2.94267 0.04 8.00427 0 0 .0389181 .238890 .635620 1.22261 1.57239^(a) 1.99865 2.95942 0.06 18.0246 0 0 .0425305 .260607 .660876 1.24968 1.57431^(a) 2.02636 2.98715 500 0.01 .499954 0 0 .00814757 .0498972 .130004 .248856 .406927 .604274 .840899

0.02 2.00007 0 0 .0100978 .0642386 .147927 .268932 .428225 .626321 .863435 0.04 8.00427 0 0 .0135687 .102725 .202962 .335905 .503267 .706764 .947569 0.06 18.0246 0 0 .0164590 .145081 .268690 .421555 .605093 .821022 1.07120 1000 0.01 .499954 0 0 .00504927 .0321223 .0739765 .134505 .214203 .313344 .432052

0.02 2.00007 0 0 .00677821 .0513635 .101497 .168006 .251762 .353634 .474228 0.04 8.00427 0 0 .00942143 .0942683 .169116 .257543 .360229 .477630 .610650 0.06 18.0246 0 0 .0116029 .138373 .240912 .355544 .483657 .624839 .779391

表十一 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=0^D) k ε₀ (10⁻³) ε_b

T tip

L V

1 2 4

K K K ₃ K K ₅ K ₆ K ₇

η

10 0.01 .0994970 0 0 .343771 1.57092(a) 1.91374 4.64898 4.71445^(a) 7.82057 7.86255(a) 0.02 .398032 0 0 .344042 1.57111(a) 1.91404 4.64789 4.71536^(a) 7.81843 7.86392(a) 0.04 1.59285 0 0 .345120 1.57186(a) 1.91526 4.64377 4.71877^(a) 7.81085 7.86841(a) 0.06 3.58661 0 0 .346903 1.57311(a) 1.91730 4.63762 4.72373^(a) 7.80021 7.87390(a) 20 0.01 .0994970 0 0 .175041 1.05988 1.57096^(a) 2.82457 4.71417(a) 5.19132 7.86208(a) 0.02 .398032 0 0 .175798 1.06092 1.57127^(a) 2.82535 4.71427(a) 5.19171 7.86213(a) 0.04 1.59285 0 0 .178785 1.06505 1.57252^(a) 2.82847 4.71466(a) 5.19326 7.86236(a) 0.06 3.58661 0 0 .183629 1.07192 1.57459^(a) 2.83366 4.71533(a) 5.19584 7.86273(a) 50 0.01 .0994970 0 0 .0709373 .438931 1.21639 1.57096^(a) 2.35284 3.82747 4.71417(a) 0.02 .398032 0 0 .0729406 .442131 1.21966 1.57125^(a) 2.35608 3.83052 4.71426(a) 0.04 1.59285 0 0 .0803894 .454706 1.23266 1.57241^(a) 2.36900 3.84268 4.71466(a) 0.06 3.58661 0 0 .0912769 .474929 1.25401 1.57434^(a) 2.39038 3.86289 4.71531(a) 100 0.01 .0994970 0 0 .0365097 .222183 .616926 1.20289 1.57096^(a) 1.97864 2.93950

0.02 .398032 0 0 .0402768 .228637 .623843 1.21012 1.57124^(a) 1.98596 2.94678 0.04 1.59285 0 0 .0524164 .252784 .650685 1.23857 1.57240^(a) 2.01495 2.97574 0.06 3.58661 0 0 .0674213 .288455 .692841 1.28440 1.57432^(a) 2.06229 3.02335 500 0.01 .0994970 0 0 .0119675 .0540290 .134663 .253929 .412249 .609756 .846491

0.02 .398032 0 0 .0200813 .0762383 .163340 .286920 .447870 .647045 .884889 0.04 1.59285 0 0 .0370596 .130652 .243280 .387911 .564823 .775613 1.02188 0.06 3.58661 0 0 .0541445 .188305 .332907 .507593 .711566 .945280 1.21033 1000 0.01 .0994970 0 0 .0100424 .0381228 .0816862 .143506 .224040 .323729 .442813

0.02 .398032 0 0 .0185394 .0653273 .121659 .194022 .282569 .388114 .511474 0.04 1.59285 0 0 .0356716 .123413 .212746 .316745 .434993 .567018 .713250 0.06 3.58661 0 0 .0528045 .182500 .307134 .445787 .599005 .765364 .944405

表十二 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=5^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.005 .0248264 .00943349 .0136160 .343703 1.57088(a) 1.91366 4.64926 4.71421^(a) 7.82113 7.86218(a) 0.01 .0993087 .0377248 .0544395 .343771 1.57092(a) 1.91374 4.64898 4.71445^(a) 7.82057 7.86255^(a) 0.02 .397280 .150753 .217366 .344040 1.57111(a) 1.91404 4.64789 4.71536^(a) 7.81843 7.86392^(a) 0.03 .894049 .338649 .487608 .344488 1.57142(a) 1.91455 4.64613 4.71683^(a) 7.81511 7.86595(a) 20 0.005 .0248264 .0188599 .0544314 .174851 1.05962 1.57089(a) 2.82438 4.71414(a) 5.19123 7.86206^(a) 0.01 .0993088 .0753371 .217236 .175040 1.05988 1.57096^(a) 2.82457 4.71417(a) 5.19132 7.86208^(a) 0.02 .397280 .299724 .861205 .175794 1.06091 1.57127^(a) 2.82535 4.71427(a) 5.19171 7.86213^(a) 0.03 .894050 .668432 1.90939 .177043 1.06264 1.57179^(a) 2.82665 4.71443(a) 5.19235 7.86222^(a) 50 0.005 .0248265 .0470270 .338773 .0704258 .438127 1.21557 1.57089(a) 2.35203 3.82671 4.71414(a) 0.01 .0993089 .186410 1.33534 .0709353 .438929 1.21639 1.57096(a) 2.35284 3.82747 4.71417(a) 0.02 .397281 .720242 5.04737 .0729336 .442122 1.21964 1.57127(a) 2.35608 3.83051 4.71427(a) 0.03 .894048 1.53748 10.4047 .0761333 .447393 1.22500 1.57184(a) 2.36148 3.83557 4.71444(a) 100 0.005 .0248265 .0931927 1.33514 .0354959 .220538 .615182 1.20108 1.57089^(a) 1.97680 2.93767

0.01 .0993091 .359940 5.04454 .0365061 .222179 .616918 1.20288 1.57098^(a) 1.97864 2.93949 0.02 .397272 1.27852 16.5298 .0402677 .228619 .623782 1.20999 1.57150^(a) 1.98602 2.94677 0.03 .893967 2.48198 28.6429 .0457772 .238965 .635024 1.22164 1.57246^(a) 1.99833 2.95888 500 0.005 .0248252 .371275 22.7407 .00857670 .0467568 .126287 .244824 .402727 .599970 .836523

0.01 .0992814 1.03467 46.1594 .0119649 .0540176 .134605 .253821 .412091 .609550 .846218 0.02 .397089 2.47033 64.4901 .0200785 .0762298 .163282 .286748 .447537 .646508 .884061 0.03 .893645 3.93617 71.4377 .0285341 .102679 .201130 .333403 .500339 .703443 .943719 1000 0.005 .0248196 .517285 46.1559 .00598284 .0270110 .0673132 .126944 .206128 .304950 .423444 0.01 .0992609 1.23470 64.4758 .0100409 .0381185 .0816572 .143422 .223882 .323491 .442492 0.02 .397143 2.69658 75.0800 .0185380 .0653250 .121647 .193975 .282446 .387868 .511055 0.03 .893990 4.13175 78.9386 .0270995 .0941368 .166441 .253819 .355873 .472835 .605502

表十三 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=10^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.005 .0246855 .0187952 .0271285 .343703 1.57088(a) 1.91366 4.64926 4.71421^(a) 7.82113 7.86218(a) 0.01 .0987451 .0751628 .108465 .343769 1.57092(a) 1.91374 4.64899 4.71445^(a) 7.82057 7.86255(a) 0.015 .222187 .169049 .243866 .343880 1.57100(a) 1.91386 4.64853 4.71483^(a) 7.81967 7.86314(a) 0.02 .395025 .300364 .433088 .344035 1.57111(a) 1.91404 4.64789 4.71536^(a) 7.81843 7.86392(a) 20 0.005 .0246855 .0375765 .108449 .174850 1.05962 1.57089(a) 2.82438 4.71414(a) 5.19123 7.86206(a) 0.01 .0987452 .150104 .432832 .175037 1.05988 1.57096^(a) 2.82457 4.71417(a) 5.19132 7.86208(a) 0.015 .222188 .336981 .970271 .175349 1.06031 1.57109^(a) 2.82489 4.71421(a) 5.19148 7.86210(a) 0.02 .395027 .597219 1.71605 .175785 1.06091 1.57128^(a) 2.82535 4.71427(a) 5.19171 7.86213(a) 50 0.005 .0246856 .0936995 .674997 .0704242 .438125 1.21557 1.57089(a) 2.35203 3.82671 4.71414(a) 0.01 .0987458 .371451 2.66097 .0709292 .438923 1.21638 1.57097(a) 2.35284 3.82746 4.71417(a) 0.015 .222190 .823659 5.84641 .0717626 .440249 1.21772 1.57111(a) 2.35418 3.82872 4.71421(a) 0.02 .395032 1.43571 10.0625 .0729123 .442098 1.21956 1.57135(a) 2.35607 3.83048 4.71428(a) 100 0.005 .0246857 .185701 2.66058 .0354928 .220535 .615179 1.20107 1.57089(a) 1.97680 2.93767

0.01 .0987467 .717494 10.0569 .0364955 .222166 .616894 1.20284 1.57106^(a) 1.97865 2.93948 0.015 .222186 1.53142 20.7296 .0381045 .224855 .619711 1.20569 1.57149(a) 1.98178 2.94251 0.02 .394995 2.55085 32.9871 .0402404 .228564 .623602 1.20957 1.57228^(a) 1.98623 2.94675 500 0.005 .0246805 .741032 45.3966 .00857017 .0467402 .126231 .244745 .402627 .599852 .836378

0.01 .0986356 2.06691 92.1628 .0119570 .0539832 .134432 .253499 .411625 .608941 .845417 0.015 .221809 3.48592 115.715 .0159239 .0642325 .147134 .267622 .426439 .624048 .860520 0.02 .394262 4.93625 128.676 .0200699 .0762036 .163106 .286233 .446545 .644923 .881650 1000 0.005 .0246582 1.03336 92.1559 .00597887 .0269938 .0672270 .126785 .205904 .304674 .423120

0.01 .0985537 2.46722 128.648 .0100366 .0381054 .0815699 .143170 .223412 .322786 .441549 0.015 .221780 3.92973 142.420 .0142671 .0513339 .100529 .166561 .249851 .351204 .471215

0.02 .394474 5.38896 149.716 .0185334 .0653175 .121609 .193831 .282074 .387128 .509801

表十四 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=15^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.004 .156492 .0179294 .0258795 .343695 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82120 7.86214(a) 0.006 .352109 .0403387 .0582220 .343712 1.57088(a) 1.91367 4.64922 4.71425^(a) 7.82105 7.86224(a) 0.008 .625978 .0717070 .103489 .343736 1.57090(a) 1.91370 4.64912 4.71433^(a) 7.82084 7.86237(a) 0.01 .978104 .112030 .161667 .343768 1.57092(a) 1.91374 4.64899 4.71445^(a) 7.82058 7.86255(a) 20 0.004 .156492 .0358505 .103479 .174827 1.05959 1.57088(a) 2.82435 4.71414(a) 5.19122 7.86206(a) 0.006 .352109 .0806353 .232692 .174876 1.05966 1.57090^(a) 2.82441 4.71415(a) 5.19124 7.86207(a) 0.008 .625979 .143281 .413335 .174945 1.05975 1.57093^(a) 2.82448 4.71415(a) 5.19128 7.86207(a) 0.01 .978107 .223735 .645157 .175033 1.05987 1.57096^(a) 2.82457 4.71417(a) 5.19132 7.86208(a) 50 0.004 .156492 .0894812 .645051 .0703617 .438028 1.21547 1.57088(a) 2.35193 3.82662 4.71414(a) 0.006 .352111 .200857 1.44579 .0704948 .438239 1.21569 1.57090(a) 2.35214 3.82682 4.71415(a) 0.008 .625985 .355903 2.55653 .0706808 .438535 1.21598 1.57093(a) 2.35244 3.82710 4.71415(a) 0.01 .978121 .553765 3.96727 .0709192 .438914 1.21636 1.57097(a) 2.35283 3.82746 4.71417(a) 100 0.004 .156493 .177937 2.55629 .0353672 .220336 .614968 1.20085 1.57088^(a) 1.97658 2.93745

0.006 .352117 .396613 5.66435 .0356346 .220768 .615422 1.20132 1.57093^(a) 1.97707 2.93793 0.008 .625999 .696076 9.86035 .0360058 .221372 .616052 1.20196 1.57102^(a) 1.97776 2.93860 0.01 .978140 1.07029 15.0050 .0364778 .222145 .616853 1.20276 1.57118^(a) 1.97866 2.93947 500 0.004 .156464 .761716 49.2689 .00804423 .0457777 .125164 .243606 .401446 .598645 .835150

0.006 .351812 1.48039 85.4447 .00914593 .0478334 .127322 .245838 .403692 .600876 .837319 0.008 .624910 2.27095 115.249 .0104759 .0505844 .130323 .248950 .406813 .603957 .840287 0.01 .975640 3.09429 137.855 .0119437 .0539259 .134145 .252969 .410863 .607954 .844133 1000 0.004 .156224 1.13540 115.243 .00523820 .0252941 .0651716 .124510 .203500 .302189 .420578

0.006 .350988 1.96781 154.772 .00674778 .0288771 .0693729 .128986 .208055 .306744 .425095 0.008 .623450 2.82562 177.669 .00836615 .0332373 .0749036 .135153 .214484 .313247 .431579 0.01 .973789 3.69537 192.213 .0100294 .0380837 .0814241 .142751 .222639 .321644 .440042

表十五 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=30^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.004 .148580 .0346372 .0499958 .343693 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82120 7.86214(a) 0.006 .334307 .0779295 .112479 .343708 1.57088(a) 1.91367 4.64922 4.71425^(a) 7.82105 7.86224(a) 0.008 .594330 .138531 .199932 .343730 1.57090(a) 1.91370 4.64912 4.71433^(a) 7.82085 7.86237(a) 0.01 .928654 .216433 .312334 .343757 1.57092(a) 1.91373 4.64899 4.71445^(a) 7.82059 7.86255(a) 20 0.004 .148580 .0692600 .199914 .174823 1.05959 1.57088(a) 2.82435 4.71414(a) 5.19121 7.86206(a) 0.006 .334308 .155786 .449566 .174868 1.05965 1.57090^(a) 2.82440 4.71415(a) 5.19124 7.86207(a) 0.008 .594334 .276832 .798628 .174931 1.05974 1.57093^(a) 2.82447 4.71415(a) 5.19127 7.86207(a) 0.01 .928663 .432305 1.24665 .175012 1.05986 1.57096^(a) 2.82456 4.71417(a) 5.19132 7.86208(a) 50 0.004 .148581 .172899 1.24646 .0703529 .438020 1.21546 1.57088^(a) 2.35192 3.82661 4.71414(a) 0.006 .334316 .388196 2.79463 .0704751 .438222 1.21566 1.57090(a) 2.35213 3.82680 4.71415(a) 0.008 .594357 .688087 4.94376 .0706461 .438503 1.21594 1.57095(a) 2.35242 3.82707 4.71416(a) 0.01 .928716 1.07108 7.67594 .0708659 .438864 1.21629 1.57102(a) 2.35279 3.82741 4.71418(a) 100 0.004 .148586 .344016 4.94331 .0353499 .220320 .614949 1.20083 1.57090^(a) 1.97657 2.93743

0.006 .334337 .767509 10.9666 .0355969 .220732 .615371 1.20124 1.57102^(a) 1.97705 2.93789 0.008 .594407 1.34866 19.1194 .0359418 .221305 .615941 1.20178 1.57129^(a) 1.97777 2.93854 0.01 .928788 2.07668 29.1467 .0363833 .222037 .616645 1.20239 1.57180^(a) 1.97874 2.93937 500 0.002 .0371501 .415273 29.1417 .00727916 .0444794 .123797 .242174 .399984 .597168 .833678

0.004 .148469 1.49004 96.5785 .00799487 .0456820 .124887 .243241 .401001 .598129 .834529 0.006 .333126 2.91366 168.308 .00908053 .0476379 .126599 .244787 .402350 .599279 .835362 0.008 .590060 4.48524 227.218 .0104055 .0503120 .129142 .247046 .404274 .600853 .836444 1000 0.002 .0371172 .745007 96.5767 .00399752 .0228423 .0624517 .121651 .200581 .299243 .417623

0.004 .147512 2.24248 227.205 .00520298 .0251579 .0645832 .123571 .202276 .300760 .418964 0.006 .329805 3.89809 304.452 .00671105 .0287201 .0685214 .127308 .205649 .303769 .421631 0.008 .584146 5.60339 348.562 .00832811 .0330971 .0740247 .133053 .211145 .308859 .426295

表十六 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=45^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.004 .135993 .0489849 .0707057 .343690 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82120 7.86214(a) 0.006 .305987 .110212 .159074 .343703 1.57088(a) 1.91367 4.64923 4.71425^(a) 7.82106 7.86224(a) 0.008 .543985 .195921 .282763 .343720 1.57090(a) 1.91369 4.64912 4.71433^(a) 7.82086 7.86237(a) 0.01 .849990 .306103 .441747 .343742 1.57092(a) 1.91373 4.64899 4.71445^(a) 7.82061 7.86255(a) 20 0.004 .135994 .0979532 .282739 .174818 1.05958 1.57088(a) 2.82435 4.71414(a) 5.19121 7.86206(a) 0.006 .305990 .220339 .635872 .174856 1.05964 1.57090^(a) 2.82440 4.71415(a) 5.19124 7.86207(a) 0.008 .543992 .391575 1.12972 .174909 1.05973 1.57093^(a) 2.82446 4.71415(a) 5.19127 7.86207(a) 0.01 .850008 .611558 1.76373 .174977 1.05983 1.57097^(a) 2.82454 4.71417(a) 5.19131 7.86208(a) 50 0.004 .135997 .244592 1.76347 .0703388 .438008 1.21545 1.57088^(a) 2.35191 3.82660 4.71414(a) 0.006 .306005 .549378 3.95577 .0704437 .438194 1.21563 1.57091^(a) 2.35210 3.82677 4.71415(a) 0.008 .544039 .974308 7.00265 .0705909 .438454 1.21588 1.57098^(a) 2.35238 3.82702 4.71416(a) 0.01 .850114 1.51763 10.8821 .0707807 .438786 1.21617 1.57108^(a) 2.35273 3.82734 4.71419(a) 100 0.004 .136007 .487119 7.00207 .0353222 .220295 .614919 1.20079 1.57093^(a) 1.97655 2.93741

0.006 .306049 1.08839 15.5633 .0355363 .220674 .615292 1.20113 1.57114^(a) 1.97703 2.93784 0.008 .544140 1.91629 27.2008 .0358385 .221199 .615774 1.20151 1.57166(a) 1.97777 2.93844 0.01 .850256 2.95766 41.5870 .0362301 .221866 .616333 1.20185 1.57266^(a) 1.97883 2.93923 500 0.003 .0764842 1.27694 86.8781 .00752478 .0449027 .124084 .242418 .400176 .597310 .833733

0.004 .135750 2.15140 139.954 .00791215 .0455302 .124466 .242694 .400341 .597372 .833626 0.005 .211495 3.15828 194.485 .00839937 .0463332 .124905 .242933 .400383 .597235 .833224 0.006 .303365 4.25202 246.054 .00896911 .0473213 .125475 .243204 .400376 .596978 .832624 1000 0.003 .0758403 2.12594 246.045 .00448470 .0236623 .0627492 .121649 .200324 .298812 .417025

0.004 .133598 3.29272 332.757 .00514289 .0249340 .0636582 .122163 .200505 .298748 .416744 0.005 .206917 4.50987 397.106 .00587487 .0265538 .0651281 .123182 .201115 .299037 .416766 0.006 .295795 5.75361 444.206 .00664893 .0284572 .0671547 .124804 .202295 .299848 .417276

表十七 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=60^D)

εb (10⁻³)

T tip

L

V (10⁻³) K₇

η k ε₀ (10⁻⁵) K1 K2 K₃ K4 K₅ K₆

10 0.004 1.19590 .0599949 .0865980 .343687 1.57087(a) 1.91365 4.64930 4.71418(a) 7.82121 7.86214(a) 0.006 2.69081 .134985 .194833 .343695 1.57088(a) 1.91367 4.64923 4.71425(a) 7.82107 7.86224(a) 0.008 4.78373 .239966 .346338 .343706 1.57090(a) 1.91369 4.64913 4.71433(a) 7.82088 7.86237(a) 0.01 7.47472 .374931 .541089 .343721 1.57092(a) 1.91372 4.64900 4.71445(a) 7.82063 7.86255(a) 20 0.004 1.19591 .119975 .346312 .174811 1.05958 1.57088(a) 2.82435 4.71414(a) 5.19121 7.86206(a) 0.006 2.69084 .269898 .778925 .174840 1.05963 1.57090^(a) 2.82439 4.71415(a) 5.19123 7.86207(a) 0.008 4.78384 .479701 1.38407 .174880 1.05970 1.57093^(a) 2.82444 4.71415(a) 5.19126 7.86207(a) 0.01 7.47498 .749299 2.16122 .174932 1.05980 1.57097^(a) 2.82452 4.71417(a) 5.19130 7.86207(a) 50 0.004 1.19596 .299686 2.16094 .0703204 .437992 1.21543 1.57088^(a) 2.35190 3.82658 4.71414(a) 0.006 2.69108 .673466 4.85053 .0704027 .438158 1.21559 1.57092^(a) 2.35207 3.82674 4.71415(a) 0.008 4.78454 1.19522 8.59432 .0705185 .438390 1.21580 1.57100^(a) 2.35232 3.82696 4.71417(a) 0.01 7.47658 1.86339 13.3707 .0706689 .438685 1.21603 1.57115^(a) 2.35265 3.82724 4.71420(a) 100 0.004 1.19611 .597569 8.59368 .0352859 .220262 .614882 1.20075 1.57095^(a) 1.97652 2.93737

0.006 2.69173 1.33781 19.1488 .0354566 .220599 .615196 1.20100 1.57126^(a) 1.97699 2.93776 0.008 4.78606 2.36162 33.5787 .0357013 .221062 .615575 1.20121 1.57204^(a) 1.97774 2.93831 0.01 7.47867 3.65658 51.5416 .0360246 .221647 .615965 1.20124 1.57358^(a) 1.97888 2.93903 500 0.001 .0747628 .186314 13.3687 .00707423 .0441520 .123463 .241827 .399632 .596813 .833328

0.002 .299134 .731223 51.5340 .00720735 .0444011 .123669 .242024 .399815 .596982 .833467 0.003 .672458 1.59370 108.905 .00744485 .0447970 .123862 .242147 .399860 .596957 .833321 0.004 1.19176 2.71183 177.375 .00779555 .0453339 .123957 .242049 .399573 .596503 .832608 1000 0.001 .0747834 .365610 51.5338 .00360371 .0222017 .0618417 .121039 .199979 .298648 .417038

0.002 .297937 1.35589 177.372 .00389785 .0226682 .0619880 .121061 .199888 .298486 .416803 0.003 .661312 2.72290 316.194 .00440406 .0234508 .0620462 .120707 .199203 .297583 .415696 0.004 1.15365 4.25712 429.053 .00505566 .0246273 .0624752 .120479 .198480 .296520 .414351

表十八 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=75^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.004 1.00489 .0669167 .0965896 .343683 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82121 7.86214(a) 0.006 2.26102 .150562 .217318 .343686 1.57088(a) 1.91366 4.64923 4.71425^(a) 7.82108 7.86224(a) 0.008 4.01966 .267665 .386323 .343691 1.57090(a) 1.91368 4.64913 4.71433^(a) 7.82089 7.86237(a) 0.01 6.28085 .418223 .603587 .343696 1.57092(a) 1.91371 4.64901 4.71445^(a) 7.82066 7.86255(a) 20 0.004 1.00490 .133825 .386298 .174802 1.05957 1.57088^(a) 2.82434 4.71414(a) 5.19121 7.86206(a) 0.006 2.26107 .301082 .868966 .174821 1.05962 1.57090^(a) 2.82438 4.71415(a) 5.19123 7.86207(a) 0.008 4.01980 .535195 1.54432 .174847 1.05968 1.57093^(a) 2.82443 4.71415(a) 5.19125 7.86207(a) 0.01 6.28119 .836120 2.41197 .174880 1.05976 1.57097^(a) 2.82449 4.71416(a) 5.19128 7.86207(a) 50 0.004 1.00496 .334416 2.41170 .0702990 .437974 1.21541 1.57088^(a) 2.35188 3.82657 4.71414(a) 0.006 2.26136 .751960 5.41752 .0703547 .438117 1.21555 1.57093(a) 2.35203 3.82670 4.71415(a) 0.008 4.02067 1.33562 9.60911 .0704337 .438315 1.21571 1.57102(a) 2.35225 3.82689 4.71417(a) 0.01 6.28317 2.08447 14.9696 .0705373 .438568 1.21588 1.57120(a) 2.35254 3.82713 4.71420(a) 100 0.004 1.00515 .667774 9.60849 .0352433 .220224 .614840 1.20070 1.57097^(a) 1.97648 2.93733

0.006 2.26217 1.49846 21.4738 .0353624 .220512 .615093 1.20087 1.57136^(a) 1.97692 2.93767 0.008 4.02256 2.65356 37.8065 .0355376 .220906 .615368 1.20093 1.57234^(a) 1.97767 2.93816 0.01 6.28571 4.12454 58.3140 .0357766 .221398 .615595 1.20068 1.57431^(a) 1.97884 2.93879 500 0.001 .0628291 .208424 14.9676 .00706102 .0441400 .123450 .241812 .399616 .596797 .833311

0.002 .251417 .824818 58.3065 .00715769 .0443511 .123597 .241943 .399725 .596885 .833360 0.0025 .392696 1.27764 89.1996 .00723839 .0445011 .123643 .241954 .399698 .596823 .833239 0.003 .564857 1.81927 125.027 .00734494 .0446775 .123639 .241883 .399558 .596622 .832939 1000 0.001 .0628543 .412407 58.3062 .00357888 .0221767 .0618061 .120999 .199935 .298601 .416990

0.002 .249711 1.56838 206.724 .00382250 .0225558 .0617268 .120742 .199523 .298094 .416383 0.0025 .386233 2.35163 293.354 .00403219 .0228344 .0615122 .120295 .198890 .297343 .415521 0.003 .548173 3.22129 376.281 .00429559 .0231971 .0612738 .119726 .198072 .296367 .414405

表十九 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=0.5，

α

=90^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻³) K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇ η

10 0.004 .799902 .0692784 .0999992 .343679 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82122 7.86214^(a) 0.006 1.79980 .155879 .224996 .343677 1.57088(a) 1.91366 4.64923 4.71425^(a) 7.82109 7.86224^(a) 0.008 3.19971 .277126 .399988 .343674 1.57090(a) 1.91368 4.64914 4.71433^(a) 7.82091 7.86237^(a) 0.01 4.99966 .433023 .624970 .343671 1.57092(a) 1.91370 4.64902 4.71445^(a) 7.82068 7.86255^(a) 20 0.004 .799911 .138557 .399966 .174793 1.05956 1.57088^(a) 2.82434 4.71414(a) 5.19121 7.86206^(a) 0.006 1.79985 .311758 .899828 .174800 1.05960 1.57090^(a) 2.82437 4.71415(a) 5.19122 7.86207^(a) 0.008 3.19986 .554251 1.59945 .174810 1.05965 1.57093^(a) 2.82441 4.71415(a) 5.19124 7.86207^(a) 0.01 5.00002 .866044 2.49866 .174824 1.05972 1.57097^(a) 2.82446 4.71416(a) 5.19127 7.86207^(a) 50 0.004 .799975 .346391 2.49843 .0702760 .437954 1.21539 1.57088^(a) 2.35186 3.82655 4.71414(a) 0.006 1.80016 .779385 5.61696 .0703030 .438072 1.21550 1.57093(a) 2.35199 3.82666 4.71415(a) 0.008 3.20079 1.38557 9.97430 .0703421 .438236 1.21563 1.57103(a) 2.35218 3.82682 4.71417(a) 0.01 5.00216 2.16490 15.5613 .0703945 .438444 1.21575 1.57122(a) 2.35243 3.82702 4.71421(a) 100 0.004 .800182 .692755 9.97376 .0351974 .220183 .614798 1.20065 1.57098^(a) 1.97644 2.93729

0.006 1.80104 1.55847 22.3628 .0352597 .220421 .614994 1.20076 1.57139^(a) 1.97684 2.93758 0.008 3.20282 2.76949 39.5477 .0353571 .220743 .615180 1.20070 1.57248^(a) 1.97753 2.93799 0.01 5.00481 4.32362 61.3395 .0354991 .221139 .615273 1.20027 1.57467(a) 1.97867 2.93853 500 0.001 .0500194 .216470 15.5596 .00704669 .0441272 .123436 .241798 .399601 .596782 .833295

0.002 .200183 .864647 61.3328 .00710214 .0442990 .123534 .241874 .399651 .596807 .833277 0.0025 .312610 1.34817 94.6072 .00715405 .0444180 .123531 .241829 .399561 .596678 .833081 0.003 .449395 1.93425 133.848 .00722824 .0445546 .123452 .241670 .399322 .596369 .832658 1000 0.001 .0500457 .432322 61.3326 .00355111 .0221506 .0617743 .120964 .199898 .298563 .416950

0.002 .198042 1.69560 225.723 .00373046 .0224392 .0614996 .120477 .199228 .297782 .416054 0.0025 .304202 2.58480 326.051 .00391126 .0226422 .0610578 .119757 .198288 .296707 .414852 0.003 .426938 3.59170 423.658 .00415519 .0229167 .0605189 .118825 .197068 .295312 .413304

表二十 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=0^D) k ε₀ (10⁻³) ε_b

T tip

L

η V K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.01 .148998 0 0 .343872 1.57092(a) 1.91378 4.64895 4.71445^(a) 7.82046 7.86255(a) 0.02 .596058 0 0 .344444 1.57111(a) 1.91420 4.64776 4.71536^(a) 7.81800 7.86390(a) 0.04 2.38527 0 0 .346719 1.57186(a) 1.91588 4.64328 4.71873^(a) 7.80924 7.86822(a) 0.06 5.37076 0 0 .350462 1.57312(a) 1.91869 4.63657 4.72360^(a) 7.79677 7.87329(a) 20 0.01 .148998 0 0 .175258 1.06004 1.57096^(a) 2.82469 4.71417(a) 5.19138 7.86208(a) 0.02 .596058 0 0 .176661 1.06156 1.57127^(a) 2.82581 4.71427(a) 5.19194 7.86213(a) 0.04 2.38527 0 0 .182150 1.06762 1.57252^(a) 2.83030 4.71467(a) 5.19418 7.86236(a) 0.06 5.37076 0 0 .190897 1.07766 1.57460^(a) 2.83779 4.71533(a) 5.19792 7.86273(a) 50 0.01 .148998 0 0 .0714858 .439405 1.21686 1.57096^(a) 2.35330 3.82790 4.71417(a) 0.02 .596058 0 0 .0750486 .444012 1.22152 1.57125^(a) 2.35791 3.83224 4.71426(a) 0.04 2.38527 0 0 .0877636 .461975 1.23999 1.57241^(a) 2.37627 3.84954 4.71466(a) 0.06 5.37076 0 0 .105339 .490429 1.27012 1.57435^(a) 2.40655 3.87823 4.71531(a) 100 0.01 .148998 0 0 .0375668 .223145 .617913 1.20392 1.57096^(a) 1.97967 2.94053

0.02 .596058 0 0 .0439754 .232350 .627728 1.21418 1.57124^(a) 1.99008 2.95089 0.04 2.38527 0 0 .0630513 .265905 .665347 1.25429 1.57240^(a) 2.03110 2.99195 0.06 5.37076 0 0 .0852550 .313660 .723130 1.31800 1.57433^(a) 2.09747 3.05905 500 0.01 .148998 0 0 .0148226 .0578458 .139137 .258881 .417488 .615179 .852038

0.02 .596058 0 0 .0265111 .0864577 .177158 .303589 .466489 .666975 .905720 0.04 2.38527 0 0 .0505357 .153227 .276760 .432184 .618642 .837278 1.08978 0.06 5.37076 0 0 .0746277 .222828 .385008 .578158 .800313 1.05084 1.33083 1000 0.01 .148998 0 0 .0132579 .0432330 .0885977 .151847 .233362 .333715 .453260

0.02 .596058 0 0 .0252836 .0766152 .138402 .216170 .309504 .418994 .545501 0.04 2.38527 0 0 .0494470 .146591 .247884 .364608 .495920 .640731 .799060 0.06 5.37076 0 0 .0735850 .217470 .360269 .518160 .691561 .878499 1.07803

表二十一 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=5^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.005 .0371543 .0188650 .0272283 .343728 1.57088(a) 1.91367 4.64926 4.71421^(a) 7.82110 7.86218(a) 0.01 .148622 .0754183 .108819 .343871 1.57092(a) 1.91378 4.64895 4.71445^(a) 7.82046 7.86255(a) 0.02 .594553 .301007 .433772 .344441 1.57111(a) 1.91420 4.64776 4.71536^(a) 7.81800 7.86390(a) 0.03 1.33799 .674794 .970406 .345388 1.57142(a) 1.91490 4.64584 4.71682^(a) 7.81418 7.86588(a) 20 0.005 .0371543 .0377019 .108796 .174905 1.05966 1.57089(a) 2.82441 4.71414(a) 5.19124 7.86206(a) 0.01 .148622 .150388 .433403 .175256 1.06004 1.57096^(a) 2.82469 4.71417(a) 5.19138 7.86208(a) 0.02 .594555 .594973 1.70572 .176655 1.06156 1.57128^(a) 2.82581 4.71427(a) 5.19194 7.86213(a) 0.03 1.33800 1.31504 3.73772 .178959 1.06408 1.57180^(a) 2.82768 4.71443(a) 5.19287 7.86222(a) 50 0.005 .0371544 .0937658 .674872 .0705638 .438245 1.21569 1.57089(a) 2.35214 3.82682 4.71414(a) 0.01 .148623 .368372 2.62960 .0714819 .439402 1.21685 1.57096^(a) 2.35330 3.82790 4.71417(a) 0.02 .594560 1.37849 9.53161 .0750365 .443994 1.22144 1.57134^(a) 2.35792 3.83222 4.71428(a) 0.03 1.33800 2.81927 18.5488 .0805930 .451541 1.22898 1.57208^(a) 2.36562 3.83940 4.71448(a) 100 0.005 .0371547 .184151 2.62905 .0357700 .220780 .615428 1.20133 1.57089(a) 1.97706 2.93793

0.01 .148624 .688752 9.52424 .0375607 .223136 .617890 1.20387 1.57105^(a) 1.97969 2.94052 0.02 .594541 2.24028 27.6708 .0439644 .232313 .627575 1.21380 1.57195^(a) 1.99029 2.95088 0.03 1.33786 4.02653 42.8179 .0528363 .246845 .643385 1.23009 1.57328^(a) 2.00786 2.96808 500 0.005 .0371523 .623972 35.8503 .00964371 .0478904 .127486 .246092 .404032 .601297 .837856

0.01 .148592 1.53057 59.7156 .0148209 .0578323 .139055 .258706 .417215 .614805 .851525 0.02 .594466 3.35362 73.4169 .0265094 .0864514 .177104 .303407 .466095 .666282 .904574 0.03 1.33806 5.16695 77.9682 .0384949 .119171 .224942 .363905 .536149 .743176 .986233 1000 0.005 .0371469 .765180 59.7101 .00741092 .0289187 .0695390 .129389 .208697 .307592 .426132

0.01 .148600 1.67595 73.3957 .0132571 .0432299 .0885713 .151759 .233177 .333411 .452825 0.02 .594715 3.47188 80.2091 .0252828 .0766140 .138394 .216134 .309404 .418777 .545103 0.03 1.33890 5.19998 82.5422 .0373627 .111397 .192494 .289119 .400294 .525712 .665915

表二十二 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=10^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.005 .0368725 .0375866 .0542495 .343728 1.57088(a) 1.91367 4.64926 4.71421^(a) 7.82110 7.86218(a) 0.01 .147495 .150264 .216812 .343869 1.57092(a) 1.91378 4.64895 4.71445^(a) 7.82047 7.86255(a) 0.015 .331879 .337786 .487132 .344103 1.57100(a) 1.91395 4.64845 4.71483^(a) 7.81942 7.86313(a) 0.02 .590045 .599746 .864286 .344432 1.57111(a) 1.91419 4.64776 4.71536^(a) 7.81801 7.86390(a) 20 0.005 .0368725 .0751176 .216766 .174904 1.05966 1.57089(a) 2.82441 4.71414(a) 5.19124 7.86206(a) 0.01 .147495 .299644 .863556 .175251 1.06004 1.57096^(a) 2.82468 4.71417(a) 5.19138 7.86208(a) 0.015 .331881 .671134 1.92998 .175830 1.06066 1.57109^(a) 2.82515 4.71421(a) 5.19161 7.86210(a) 0.02 .590052 1.18562 3.39920 .176636 1.06154 1.57128^(a) 2.82580 4.71426(a) 5.19193 7.86213(a) 50 0.005 .0368727 .186831 1.34473 .0705607 .438242 1.21568 1.57089(a) 2.35214 3.82681 4.71414(a) 0.01 .147498 .734133 5.24094 .0714704 .439390 1.21682 1.57099^(a) 2.35329 3.82789 4.71417(a) 0.015 .331891 1.60542 11.3083 .0729618 .441293 1.21868 1.57121^(a) 2.35522 3.82967 4.71423(a) 0.02 .590072 2.74889 19.0110 .0750004 .443941 1.22121 1.57163^(a) 2.35793 3.83216 4.71432(a) 100 0.005 .0368734 .366998 5.23986 .0357642 .220774 .615419 1.20132 1.57091^(a) 1.97706 2.93792

0.01 .147501 1.37346 18.9963 .0375426 .223109 .617820 1.20373 1.57131^(a) 1.97974 2.94050 0.015 .331883 2.80853 36.9624 .0403322 .226941 .621727 1.20753 1.57238^(a) 1.98436 2.94481 0.02 .589994 4.47238 55.2464 .0439313 .232203 .627119 1.21266 1.57405(a) 1.99092 2.95085 500 0.005 .0368639 1.24607 71.5843 .00963699 .0478600 .127360 .245903 .403787 .601000 .837484

0.01 .147373 3.05801 119.162 .0148156 .0577917 .138807 .258185 .416409 .613704 .850035 0.015 .331594 4.87936 137.386 .0205807 .0712323 .156015 .277757 .437207 .635045 .871422 0.02 .589691 6.70119 146.399 .0265043 .0864319 .176943 .302860 .464916 .664231 .901243 1000 0.005 .0368421 1.52880 119.151 .00740825 .0288984 .0694156 .129133 .208312 .307095 .425532

0.01 .147405 3.34889 146.356 .0132546 .0432202 .0884912 .151491 .232621 .332510 .451545 0.015 .331945 5.15678 155.338 .0192519 .0595841 .112452 .181853 .267806 .371091 .492432

0.02 .590688 6.93832 159.879 .0252802 .0766099 .138369 .216026 .309101 .418120 .543898

表二十三 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=15^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .232993 .0358565 .0517545 .343710 1.57087(a) 1.91366 4.64929 4.71418^(a) 7.82118 7.86214(a) 0.006 .524239 .0806657 .116422 .343747 1.57088(a) 1.91369 4.64921 4.71425^(a) 7.82101 7.86224(a) 0.008 .931991 .143377 .206907 .343799 1.57090(a) 1.91373 4.64910 4.71433^(a) 7.82077 7.86237(a) 0.01 1.45626 .223969 .323161 .343865 1.57092(a) 1.91377 4.64895 4.71445^(a) 7.82047 7.86255(a) 20 0.004 .232994 .0716798 .206879 .174860 1.05961 1.57088(a) 2.82437 4.71414(a) 5.19122 7.86206(a) 0.006 .524241 .161163 .464982 .174952 1.05971 1.57090^(a) 2.82445 4.71415(a) 5.19126 7.86207(a) 0.008 .931995 .286224 .825404 .175079 1.05985 1.57093^(a) 2.82455 4.71415(a) 5.19131 7.86207(a) 0.01 1.45627 .446646 1.28723 .175243 1.06003 1.57096^(a) 2.82468 4.71417(a) 5.19138 7.86208(a) 50 0.004 .232995 .178621 1.28693 .0704475 .438102 1.21554 1.57088^(a) 2.35200 3.82668 4.71414(a) 0.006 .524249 .399999 2.87568 .0706875 .438404 1.21584 1.57090^(a) 2.35231 3.82696 4.71415(a) 0.008 .932020 .706460 5.06355 .0710224 .438828 1.21626 1.57095^(a) 2.35273 3.82736 4.71416(a) 0.01 1.45632 1.09469 7.81586 .0714511 .439370 1.21678 1.57102^(a) 2.35328 3.82787 4.71418(a) 100 0.004 .233001 .353188 5.06288 .0355386 .220486 .615117 1.20100 1.57090^(a) 1.97674 2.93761

0.006 .524272 .780114 11.0871 .0360170 .221104 .615750 1.20164 1.57102(a) 1.97745 2.93829 0.008 .932074 1.35273 18.9983 .0366779 .221965 .616617 1.20248 1.57128^(a) 1.97848 2.93924 0.01 1.45640 2.05010 28.3642 .0375124 .223064 .617706 1.20350 1.57175^(a) 1.97983 2.94047 500 0.004 .232945 1.33657 82.5672 .00878114 .0464921 .125811 .244238 .402050 .599215 .835658

0.006 .523761 2.40353 127.695 .0105614 .0493800 .128805 .247270 .405036 .602116 .838383 0.008 .930567 3.49014 158.100 .0126123 .0531886 .133036 .251633 .409363 .606322 .842347 0.01 1.45349 4.57922 178.071 .0148067 .0577239 .138395 .257330 .415102 .611946 .847700 1000 0.004 .232637 1.74492 158.089 .00630646 .0265964 .0665293 .125858 .204796 .303429 .421755

0.006 .523165 2.83438 191.617 .00854092 .0314085 .0723732 .132190 .211269 .309887 .428128 0.008 .930220 3.92558 208.459 .0108760 .0370658 .0798239 .140759 .220369 .319173 .437418 0.01 1.45421 5.01583 218.450 .0132503 .0432039 .0883572 .151045 .231702 .331041 .449500

表二十四 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=30^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .217169 .0692704 .0999838 .343707 1.57087(a) 1.91366 4.64929 4.71418^(a) 7.82119 7.86214(a) 0.006 .488635 .155839 .224918 .343740 1.57088(a) 1.91368 4.64921 4.71425^(a) 7.82102 7.86224(a) 0.008 .868696 .276998 .399741 .343786 1.57090(a) 1.91372 4.64910 4.71433^(a) 7.82079 7.86237(a) 0.01 1.35736 .432711 .624368 .343845 1.57092(a) 1.91377 4.64896 4.71445^(a) 7.82049 7.86255(a) 20 0.004 .217170 .138483 .399691 .174853 1.05961 1.57088^(a) 2.82437 4.71414(a) 5.19122 7.86206(a) 0.006 .488640 .311386 .898436 .174936 1.05970 1.57090^(a) 2.82444 4.71415(a) 5.19126 7.86207(a) 0.008 .868711 .553076 1.59506 .175052 1.05983 1.57093^(a) 2.82453 4.71415(a) 5.19130 7.86207(a) 0.01 1.35740 .863181 2.48796 .175200 1.06000 1.57097^(a) 2.82466 4.71416(a) 5.19137 7.86207(a) 50 0.004 .217177 .345204 2.48741 .0704300 .438086 1.21552 1.57088^(a) 2.35199 3.82667 4.71414(a) 0.006 .488671 .773414 5.56161 .0706487 .438369 1.21579 1.57093^(a) 2.35228 3.82693 4.71415(a) 0.008 .868804 1.36686 9.80109 .0709550 .438764 1.21615 1.57103^(a) 2.35269 3.82730 4.71417(a) 0.01 1.35761 2.11970 15.1438 .0713488 .439268 1.21657 1.57120^(a) 2.35322 3.82777 4.71420(a) 100 0.004 .217197 .683351 9.79987 .0355048 .220453 .615073 1.20094 1.57097^(a) 1.97673 2.93758

0.006 .488757 1.51201 21.5075 .0359457 .221027 .615619 1.20142 1.57136^(a) 1.97747 2.93822 0.008 .869005 2.62751 36.9515 .0365619 .221821 .616314 1.20192 1.57227^(a) 1.97863 2.93913 0.01 1.35791 3.99154 55.3226 .0373502 .222830 .617116 1.20232 1.57395(a) 1.98028 2.94031 500 0.002 .0543143 .798138 55.3097 .00747272 .0446394 .123905 .242269 .400063 .597231 .833714

0.004 .216965 2.62907 162.583 .00872059 .0462982 .125115 .243280 .400855 .597815 .833958 0.006 .486672 4.75037 251.546 .0105026 .0490506 .127347 .245018 .402093 .598573 .834055 0.008 .862862 6.91073 310.627 .0125602 .0528088 .131069 .248188 .404621 .600444 .835114 1000 0.002 .0542410 1.31451 162.579 .00436038 .0231505 .0625672 .121676 .200527 .299137 .417464

0.004 .215709 3.45508 310.607 .00628041 .0264065 .0655495 .124158 .202506 .300707 .418660 0.006 .484207 5.61912 374.732 .00851749 .0312451 .0712882 .129769 .207592 .305203 .422605 0.008 .861283 7.78577 406.544 .0108529 .0369462 .0788909 .138171 .215887 .312999 .429813

表二十五 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=45^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .191997 .0979652 .0141402 .343702 1.57087(a) 1.91366 4.64929 4.71418^(a) 7.82119 7.86214(a) 0.006 .431998 .220400 .0318102 .343728 1.57088(a) 1.91368 4.64922 4.71425^(a) 7.82103 7.86224(a) 0.008 .768008 .391767 .0565383 .343765 1.57090(a) 1.91371 4.64911 4.71433^(a) 7.82081 7.86237(a) 0.01 1.20004 .612026 .0883145 .343813 1.57092(a) 1.91376 4.64897 4.71445^(a) 7.82053 7.86255(a) 20 0.004 .191999 .195864 .0565319 .174842 1.05960 1.57088^(a) 2.82436 4.71414(a) 5.19122 7.86206(a) 0.006 .432007 .440463 .127094 .174911 1.05968 1.57090^(a) 2.82442 4.71414(a) 5.19125 7.86206(a) 0.008 .768038 .782471 .225689 .175008 1.05980 1.57093^(a) 2.82451 4.71415(a) 5.19129 7.86207(a) 0.01 1.20011 1.22146 .352126 .175132 1.05994 1.57097^(a) 2.82462 4.71416(a) 5.19135 7.86207(a) 50 0.004 .192011 .488498 .352056 .0704020 .438062 1.21550 1.57089^(a) 2.35197 3.82665 4.71414(a) 0.006 .432069 1.09530 .787936 .0705867 .438313 1.21572 1.57096^(a) 2.35223 3.82688 4.71416(a) 0.008 .768224 1.93774 1.39041 .0708469 .438663 1.21598 1.57113^(a) 2.35261 3.82720 4.71419(a) 0.01 1.20053 3.00891 2.15186 .0711843 .439107 1.21625 1.57145^(a) 2.35312 3.82761 4.71424(a) 100 0.004 .192052 .968774 1.39025 .0354506 .220402 .615005 1.20085 1.57107^(a) 1.97670 2.93753

0.006 .432241 2.14959 3.06198 .0358306 .220906 .615421 1.20110 1.57183^(a) 1.97749 2.93811 0.008 .768627 3.74874 5.28369 .0363729 .221596 .615862 1.20110 1.57367^(a) 1.97883 2.93894 0.01 1.20113 5.71738 7.94779 .0370830 .222461 .616235 1.20062 1.57710^(a) 1.98091 2.94004 500 0.003 .108006 2.37654 15.7344 .00792274 .0451555 .123910 .242068 .399657 .596644 .832837

0.004 .191502 3.83283 23.7537 .00861721 .0459857 .124039 .241840 .399096 .595793 .831570 0.005 .298065 5.39008 30.8957 .00945862 .0470968 .124360 .241633 .398426 .594730 .830002 0.006 .427412 6.98381 36.7920 .0104019 .0485093 .125076 .241706 .397955 .593799 .828543 1000 0.003 .106851 3.49175 36.7905 .00520115 .0242565 .0625541 .120920 .199178 .297385 .415351

0.004 .188592 5.09603 45.2284 .00623616 .0260909 .0640141 .121694 .199390 .297176 .414802 0.005 .293352 6.70054 50.5786 .00733889 .0283735 .0664014 .123470 .200564 .297868 .415117 0.006 .421568 8.30520 54.1259 .00847821 .0309696 .0695305 .126204 .202707 .299497 .416342

表二十六 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=60^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 1.59191 .119986 .0173188 .343695 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82120 7.86214(a) 0.006 3.58186 .269950 .0389627 .343713 1.57088(a) 1.91367 4.64922 4.71425^(a) 7.82105 7.86224(a) 0.008 6.36788 .479868 .0692552 .343739 1.57090(a) 1.91370 4.64912 4.71433^(a) 7.82084 7.86237(a) 0.01 9.95007 .749705 .108188 .343771 1.57092(a) 1.91374 4.64898 4.71445^(a) 7.82057 7.86255(a) 20 0.004 1.59194 .239914 .0692486 .174828 1.05959 1.57088^(a) 2.82436 4.71414(a) 5.19122 7.86206(a) 0.006 3.58200 .539609 .155715 .174879 1.05966 1.57090^(a) 2.82441 4.71414(a) 5.19124 7.86206(a) 0.008 6.36832 .958813 .276593 .174950 1.05975 1.57093^(a) 2.82448 4.71415(a) 5.19128 7.86207(a) 0.01 9.95114 1.49716 .431705 .175043 1.05988 1.57097^(a) 2.82458 4.71416(a) 5.19133 7.86207(a) 50 0.004 1.59213 .598775 .431633 .0703655 .438030 1.21546 1.57090^(a) 2.35194 3.82662 4.71414(a) 0.006 3.58293 1.34391 .967280 .0705054 .438241 1.21562 1.57100^(a) 2.35217 3.82681 4.71416(a) 0.008 6.37111 2.38084 1.70989 .0707047 .438532 1.21579 1.57124^(a) 2.35251 3.82708 4.71420(a) 0.01 9.95750 3.70333 2.65209 .0709666 .438899 1.21588 1.57171^(a) 2.35298 3.82741 4.71428(a) 100 0.004 1.59275 1.19032 1.70972 .0353793 .220335 .614922 1.20074 1.57117^(a) 1.97666 2.93746

0.006 3.58552 2.65115 3.78356 .0356773 .220751 .615186 1.20073 1.57232^(a) 1.97748 2.93796 0.008 6.37717 4.64576 6.56798 .0361176 .221308 .615331 1.20019 1.57517^(a) 1.97899 2.93869 0.01 9.96627 7.12453 9.94533 .0367164 .221990 .615201 1.19873 1.58053^(a) 1.98153 2.93967 500 0.001 .0995708 .370269 2.65157 .00710412 .0441743 .123473 .241833 .399633 .596811 .833319

0.002 .398630 1.42467 9.94347 .00734584 .0444712 .123552 .241842 .399566 .596676 .833067 0.003 .895261 3.00402 20.0010 .00779944 .0449239 .123279 .241278 .398722 .595591 .831610 0.004 1.58240 4.90168 30.4970 .00846717 .0455709 .122699 .240117 .397048 .593498 .828946 1000 0.001 .0996573 .712333 9.94341 .00367296 .0222367 .0617842 .120950 .199862 .298514 .416886

0.002 .395596 2.45079 30.4964 .00423367 .0227869 .0613626 .120111 .198677 .297113 .415275 0.003 .874165 4.52322 47.3573 .00512761 .0238860 .0611240 .118999 .196928 .294949 .412763 0.004 1.52872 6.63384 57.8448 .00617231 .0256517 .0620640 .118883 .196085 .293610 .411048

表二十七 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=75^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 1.20988 .133831 .0193175 .343687 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82121 7.86214(a) 0.006 2.72229 .301112 .0434614 .343696 1.57088(a) 1.91367 4.64923 4.71425^(a) 7.82107 7.86224(a) 0.008 4.83976 .535292 .0772574 .343708 1.57090(a) 1.91369 4.64913 4.71433^(a) 7.82087 7.86237(a) 0.01 7.56239 .836356 .120700 .343723 1.57092(a) 1.91372 4.64900 4.71445^(a) 7.82063 7.86255(a) 20 0.004 1.20992 .267629 .0772516 .174811 1.05958 1.57088^(a) 2.82435 4.71414(a) 5.19121 7.86206(a) 0.006 2.72246 .602055 .173753 .174841 1.05963 1.57090^(a) 2.82439 4.71414(a) 5.19123 7.86206(a) 0.008 4.84031 1.07005 .308735 .174883 1.05970 1.57093^(a) 2.82445 4.71415(a) 5.19126 7.86206(a) 0.01 7.56372 1.67140 .482078 .174938 1.05980 1.57097^(a) 2.82452 4.71416(a) 5.19130 7.86206(a) 50 0.004 1.21015 .668484 .482015 .0703228 .437993 1.21542 1.57090^(a) 2.35190 3.82658 4.71414(a) 0.006 2.72363 1.50213 1.08182 .0704101 .438157 1.21553 1.57102(a) 2.35209 3.82674 4.71416(a) 0.008 4.84380 2.66550 1.91635 .0705370 .438382 1.21559 1.57132(a) 2.35238 3.82694 4.71421(a) 0.01 7.57166 4.15461 2.98011 .0707082 .438661 1.21552 1.57191(a) 2.35279 3.82719 4.71431(a) 100 0.004 1.21092 1.33266 1.91621 .0352951 .220258 .614832 1.20063 1.57124^(a) 1.97660 2.93738

0.006 2.72687 2.98161 4.26491 .0354939 .220575 .614945 1.20039 1.57270^(a) 1.97740 2.93779 0.008 4.85134 5.25596 7.45883 .0358065 .220983 .614804 1.19935 1.57639^(a) 1.97903 2.93839 0.01 7.58219 8.11670 11.3912 .0362605 .221459 .614181 1.19700 1.58348^(a) 1.98195 2.93922 500 0.001 .0757132 .415405 2.97964 .00707818 .0441499 .123443 .241800 .399598 .596774 .833279

0.002 .303269 1.62313 11.3894 .00725456 .0443646 .123361 .241618 .399312 .596396 .832748 0.0025 .473408 2.48620 17.1154 .00741641 .0445017 .123093 .241199 .398746 .595706 .831861 0.003 .679771 3.48824 23.4115 .00763826 .0446579 .122624 .240485 .397803 .594575 .830452 1000 0.001 .0758172 .811561 11.3894 .00362732 .0221834 .0616888 .120839 .199738 .298382 .416746

0.002 .298516 2.89324 36.2410 .00413243 .0225406 .0606359 .119228 .197673 .296043 .414143 0.0025 .457932 4.14845 47.5851 .00454687 .0228868 .0599396 .118028 .196109 .294259 .412171 0.003 .645894 5.44045 56.5891 .00502734 .0234253 .0595231 .117027 .194724 .292633 .410357

表二十八 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(

r = 1

，

α

=90^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .799911 .138557 .0199998 .343679 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82122 7.86214^(a) 0.006 1.79985 .311759 .0449992 .343677 1.57088(a) 1.91366 4.64923 4.71425^(a) 7.82109 7.86224^(a) 0.008 3.19986 .554252 .0799975 .343674 1.57090(a) 1.91368 4.64914 4.71433^(a) 7.82091 7.86237^(a) 0.01 5.00003 .866046 .124994 .343670 1.57092(a) 1.91370 4.64902 4.71445^(a) 7.82068 7.86255^(a) 20 0.004 .799949 .277114 .0799932 .174793 1.05956 1.57088^(a) 2.82434 4.71414(a) 5.19121 7.86206^(a) 0.006 1.80004 .623516 .179965 .174801 1.05960 1.57090^(a) 2.82437 4.71414(a) 5.19122 7.86206^(a) 0.008 3.20045 1.10850 .319889 .174811 1.05965 1.57093^(a) 2.82441 4.71415(a) 5.19124 7.86206^(a) 0.01 5.00146 1.73208 .499725 .174826 1.05971 1.57097^(a) 2.82446 4.71416(a) 5.19127 7.86206^(a) 50 0.004 .800204 .692777 .499678 .0702768 .437953 1.21538 1.57090^(a) 2.35186 3.82655 4.71414(a) 0.006 1.80129 1.55871 1.12331 .0703070 .438068 1.21543 1.57103(a) 2.35201 3.82666 4.71417(a) 0.008 3.20420 2.77081 1.99439 .0703545 .438223 1.21541 1.57135(a) 2.35223 3.82680 4.71422(a) 0.01 5.01000 4.32854 3.11046 .0704248 .438412 1.21523 1.57200(a) 2.35257 3.82696 4.71432(a) 100 0.004 .801032 1.38534 1.99428 .0352036 .220177 .614745 1.20053 1.57127 1.97653 2.93730

0.006 1.80479 3.11508 4.46722 .0352910 .220389 .614728 1.20013 1.57288 1.97726 2.93760 0.008 3.21229 5.52865 7.88019 .0354544 .220646 .614359 1.19875 1.57704 1.97887 2.93806 0.01 5.02076 8.60877 12.1597 .0357312 .220916 .613344 1.19577 1.58523 1.98193 2.93871 500 0.001 .0500975 .432814 3.11011 .00704972 .0441242 .123414 .241769 .399566 .596740 .833244 0.002 .200817 1.72159 12.1583 .00714860 .0442553 .123200 .241437 .399113 .596183 .832514 0.0025 .313142 2.66832 18.5342 .00726241 .0443213 .122774 .240833 .398338 .595264 .831370 0.003 .448275 3.79060 25.7176 .00743838 .0443815 .122046 .239816 .397053 .593764 .829553 1000 0.001 .0502041 .860795 12.1582 .00357433 .0221288 .0616084 .120749 .199640 .298279 .416639

0.002 .194123 3.21654 40.7207 .00400094 .0222722 .0599363 .118425 .196788 .295121 .413183 0.0025 .290777 4.68882 54.0274 .00440751 .0224722 .0587409 .116668 .194632 .292733 .410601 0.003 .398359 6.21305 64.2909 .00489484 .0228875 .0578803 .115176 .192739 .290593 .408272

表二十九 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=0^D) k ε₀ (10⁻³) ε_b

T tip

L

η V K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.01 .198500 0 0 .343973 1.57092(a) 1.91382 4.64892 4.71445^(a) 7.82035 7.86255(a) 0.02 .794084 0 0 .344846 1.57111(a) 1.91435 4.64763 4.71536^(a) 7.81756 7.86388(a) 0.04 3.17769 0 0 .348310 1.57186(a) 1.91650 4.64279 4.71870^(a) 7.80763 7.86804(a) 0.06 7.15492 0 0 .353985 1.57313(a) 1.92007 4.63552 4.72348^(a) 7.79328 7.87272(a) 20 0.01 .198500 0 0 .175475 1.06020 1.57096^(a) 2.82480 4.71417(a) 5.19144 7.86208(a) 0.02 .794084 0 0 .177520 1.06221 1.57127^(a) 2.82627 4.71427(a) 5.19217 7.86213(a) 0.04 3.17769 0 0 .185453 1.07019 1.57252^(a) 2.83214 4.71467(a) 5.19509 7.86236(a) 0.06 7.15492 0 0 .197894 1.08337 1.57461^(a) 2.84192 4.71534(a) 5.20000 7.86273(a) 50 0.01 .198500 0 0 .0720301 .439880 1.21732 1.57096^(a) 2.35376 3.82833 4.71417^(a) 0.02 .794084 0 0 .0770980 .445886 1.22338 1.57125^(a) 2.35974 3.83396 4.71426^(a) 0.04 3.17769 0 0 .0945541 .469124 1.24727 1.57241^(a) 2.38351 3.85639 4.71466^(a) 0.06 7.15492 0 0 .117702 .505425 1.28599 1.57436^(a) 2.42258 3.89351 4.71531^(a) 100 0.01 .198500 0 0 .0385945 .224102 .618898 1.20494 1.57096^(a) 1.98070 2.94156

0.02 .794084 0 0 .0473812 .236000 .631585 1.21823 1.57124^(a) 1.99418 2.95499 0.04 3.17769 0 0 .0721063 .278366 .679633 1.26977 1.57240^(a) 2.04709 3.00805 0.06 7.15492 0 0 .0999008 .336839 .751962 1.35058 1.57434^(a) 2.13195 3.09426 500 0.01 .198500 0 0 .0172021 .0614083 .143445 .263719 .422648 .620545 .857542

0.02 .794084 0 0 .0316424 .0955023 .189772 .319172 .484211 .686184 .925973 0.04 3.17769 0 0 .0610727 .172703 .305993 .471303 .666924 0.893468 1.15254 0.06 7.15492 0 0 .0905376 .252467 .430045 .639373 .877838 1.14394 1.43827 1000 0.01 .198500 0 0 .0158242 .0477558 .0949072 .159645 .242235 .343339 .463416

0.02 .794084 0 0 .0305566 .0863532 .153022 .235739 .333668 .447131 .576950 0.04 3.17769 0 0 .0601094 .166457 .278177 .405893 .548602 .704779 .874116 0.06 7.15492 0 0 .0896171 .247409 .406028 .580433 .771135 .975828 1.19320

表三十 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=5^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.005 .0494824 .0282946 .0408368 .343753 1.57088^(a) 1.91368 4.64925 4.71421^(a) 7.82108 7.86218(a) 0.01 .197935 .113080 .163138 .343971 1.57092(a) 1.91382 4.64892 4.71445^(a) 7.82035 7.86255(a) 0.02 .791827 .450766 .649227 .344841 1.57111(a) 1.91435 4.64763 4.71536^(a) 7.81757 7.86389(a) 0.03 1.78193 1.00848 1.44847 .346285 1.57142(a) 1.91524 4.64556 4.71681^(a) 7.81324 7.86581(a) 20 0.005 .0494824 .0565259 .163093 .174959 1.05970 1.57089(a) 2.82444 4.71414(a) 5.19126 7.86206(a) 0.01 .197936 .225155 .648508 .175473 1.06020 1.57096(a) 2.82480 4.71417(a) 5.19144 7.86208(a) 0.02 .791831 .885872 2.53403 .177511 1.06220 1.57128(a) 2.82627 4.71426(a) 5.19217 7.86213(a) 0.03 1.78195 1.94115 5.49004 .180854 1.06552 1.57180(a) 2.82871 4.71442(a) 5.19339 7.86221(a) 50 0.005 .0494825 .140220 1.00833 .0707017 .438364 1.21580 1.57089(a) 2.35226 3.82692 4.71414(a) 0.01 .197937 .546082 3.88463 .0720245 .439874 1.21731 1.57097^(a) 2.35376 3.82833 4.71417(a) 0.02 .791843 1.98428 13.5413 .0770825 .445857 1.22323 1.57144^(a) 2.35976 3.83392 4.71429(a) 0.03 1.78196 3.91814 25.0925 .0848175 .455650 1.23290 1.57236^(a) 2.36976 3.84323 4.71453(a) 100 0.005 .0494829 .272974 3.88359 .0360421 .221021 .615673 1.20158 1.57090(a) 1.97732 2.93819

0.01 .197939 .991225 13.5281 .0385867 .224088 .618855 1.20485 1.57114^(a) 1.98075 2.94155 0.02 .791821 3.01351 35.6750 .0473706 .235948 .631344 1.21758 1.57239^(a) 1.99455 2.95498 0.03 1.78184 5.17548 51.2390 .0590437 .254476 .651667 1.23856 1.57386^(a) 2.01728 2.97726 500 0.005 .0494807 .817670 44.3501 .0106024 .0489976 .128676 .247355 .405335 .602621 .839187

0.01 .197920 1.88557 66.1259 .0172010 .0613954 .143356 .263512 .422306 .620056 .856854 0.02 .791934 3.99912 76.9430 .0316414 .0954978 .189728 .319005 .483819 .685449 .924684 0.03 1.78268 6.08649 80.4174 .0463356 .133498 .246030 .391438 .569118 .780407 1.02667 1000 0.005 .0494786 .942627 66.1187 .00860106 .0307005 .0716908 .131795 .211248 .310231 .428825

0.01 .197962 1.99825 76.9150 .0158237 .0477536 .0948853 .159564 .242052 .343020 .462936 0.02 .792371 4.05799 82.0756 .0305562 .0863526 .153016 .235713 .333588 .446949 .576597 0.03 1.78397 6.01549 83.8255 .0453331 .126216 .215013 .319740 .439096 .572342 .719773

表三十一 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=10^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.005 .0490596 .0563741 .0813632 .343753 1.57088(a) 1.91368 4.64925 4.71421^(a) 7.82108 7.86218(a) 0.01 .196245 .225304 .325040 .343968 1.57092(a) 1.91381 4.64892 4.71445^(a) 7.82036 7.86255(a) 0.015 .441571 .506213 .729800 .344327 1.57100(a) 1.91404 4.64838 4.71483^(a) 7.81918 7.86312(a) 0.02 .785067 .898152 1.29361 .344828 1.57111(a) 1.91435 4.64764 4.71536^(a) 7.81758 7.86389(a) 20 0.005 .0490597 .112623 .324951 .174957 1.05970 1.57089(a) 2.82444 4.71414(a) 5.19125 7.86206(a) 0.01 .196246 .448625 1.29219 .175465 1.06020 1.57096^(a) 2.82480 4.71417(a) 5.19144 7.86208(a) 0.015 .441576 1.00250 2.87930 .176309 1.06102 1.57110^(a) 2.82540 4.71421(a) 5.19174 7.86210(a) 0.02 .785081 1.76544 5.05037 .177484 1.06217 1.57128^(a) 2.82626 4.71426(a) 5.19216 7.86212(a) 50 0.005 .0490602 .279401 2.00925 .0706971 .438359 1.21580 1.57089(a) 2.35226 3.82692 4.71414(a) 0.01 .196251 1.08843 7.74345 .0720079 .439855 1.21726 1.57102(a) 2.35375 3.82831 4.71418(a) 0.015 .441598 2.34914 16.4217 .0741434 .442333 1.21961 1.57136(a) 2.35626 3.83061 4.71425(a) 0.02 .785127 3.95809 27.0172 .0770360 .445772 1.22277 1.57201(a) 2.35980 3.83382 4.71438(a) 100 0.005 .0490615 .544079 7.74139 .0360338 .221012 .615657 1.20155 1.57094 1.97732 2.93818

0.01 .196260 1.97723 26.9912 .0385634 .224046 .618727 1.20457 1.57167 1.98087 2.94153 0.015 .441595 3.90304 49.9992 .0424474 .229008 .623708 1.20927 1.57342 1.98700 2.94712 0.02 .785042 6.01783 71.2332 .0473385 .235792 .630623 1.21568 1.57579 1.99567 2.95497 500 0.005 .0490526 1.63324 88.5459 .0105966 .0489579 .128494 .247069 .404953 .602151 .838591 0.01 .196181 3.76742 131.904 .0171975 .0613566 .143089 .262894 .421296 .618625 .854865 0.015 .441578 5.88407 146.430 .0243492 .0775561 .164368 .287590 .447903 .646167 .882631 0.02 .785486 7.99105 153.391 .0316382 .0954838 .189594 .318503 .482645 .683263 .920925 1000 0.005 .0490439 1.88339 131.889 .00859928 .0306811 .0715578 .131491 .210766 .309586 .428027

0.01 .196348 3.99293 153.335 .0158221 .0477466 .0948190 .159319 .241501 .342069 .461519 0.015 .442307 6.07260 160.161 .0231753 .0667503 .123010 .195661 .284370 .389811 .512762 0.02 .787232 8.10983 163.575 .0305546 .0863505 .152998 .235631 .333346 .446397 .575528

表三十二 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=15^D) k ε₀ (10⁻³) ε_b (10⁻³)

T tip

L

V (10⁻³)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .0309495 .0537813 .0776251 .343726 1.57087(a) 1.91367 4.64929 4.71418^(a) 7.82116 7.86214(a) 0.006 .0696370 .120981 .174598 .343782 1.57088(a) 1.91370 4.64920 4.71425^(a) 7.82097 7.86224(a) 0.008 .123800 .215010 .310253 .343861 1.57090(a) 1.91375 4.64908 4.71433^(a) 7.82070 7.86237(a) 0.01 .193441 .335818 .484481 .343962 1.57092(a) 1.91381 4.64892 4.71445^(a) 7.82036 7.86255(a) 20 0.004 .0309496 .107488 .310199 .174894 1.05964 1.57088(a) 2.82439 4.71414(a) 5.19123 7.86206(a) 0.006 .0696373 .241584 .696872 .175027 1.05977 1.57090^(a) 2.82449 4.71415(a) 5.19128 7.86207(a) 0.008 .123801 .428829 1.23621 .175214 1.05995 1.57093^(a) 2.82462 4.71415(a) 5.19135 7.86207(a) 0.01 .193443 .668737 1.92625 .175453 1.06019 1.57097^(a) 2.82479 4.71417(a) 5.19143 7.86208(a) 50 0.004 .0309500 .267421 1.92566 .0705333 .438175 1.21561 1.57088(a) 2.35207 3.82675 4.71414(a) 0.006 .0696391 .597455 4.28994 .0708799 .438569 1.21600 1.57092(a) 2.35247 3.82711 4.71415(a) 0.008 .123807 1.05181 7.52242 .0713630 .439120 1.21652 1.57098(a) 2.35302 3.82762 4.71416(a) 0.01 .193456 1.62331 11.5507 .0719802 .439825 1.21718 1.57110(a) 2.35374 3.82828 4.71419(a) 100 0.004 .0309512 .525825 7.52113 .0357094 .220635 .615265 1.20115 1.57093(a) 1.97691 2.93777

0.006 .0696442 1.15128 16.2826 .0363965 .221438 .616069 1.20193 1.57116^(a) 1.97785 2.93865 0.008 .123819 1.97390 27.4868 .0373406 .222553 .617158 1.20294 1.57167^(a) 1.97923 2.93989 0.01 .193475 2.95281 40.3241 .0385245 .223976 .618517 1.20413 1.57254^(a) 1.98108 2.94148 500 0.004 .0309468 1.79896 106.459 .00946359 .0472001 .126459 .244862 .402637 .599760 .836128

0.006 .0695981 3.08931 152.668 .0118082 .0508952 .130344 .248780 .406467 .603445 .839546 0.008 .123703 4.36918 180.195 .0144315 .0556872 .135817 .254481 .412137 .608946 .844717 0.01 .193296 5.64182 196.988 .0171916 .0612918 .142645 .261880 .419662 .616350 .851783 1000 0.004 .0309252 2.18435 180.181 .00721610 .0278460 .0679212 .127287 .206199 .304787 .423059

0.006 .0695996 3.45494 207.897 .0100128 .0337441 .0753004 .135456 .214650 .313264 .431442 0.008 .123827 4.72050 221.085 .0128992 .0405083 .0844469 .146246 .226329 .325330 .443605 0.01 .193667 5.98046 228.768 .0158194 .0477350 .0947077 .158910 .240586 .340509 .459239

表三十三 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=30^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .285759 .103900 .0149964 .343721 1.57087(a) 1.91366 4.64929 4.71418^(a) 7.82117 7.86214(a) 0.006 .642965 .233728 .0337319 .343771 1.57088(a) 1.91370 4.64920 4.71425^(a) 7.82099 7.86224(a) 0.008 1.14307 .415401 .0599427 .343842 1.57090(a) 1.91374 4.64908 4.71433^(a) 7.82072 7.86237(a) 0.01 1.78607 .648832 .0936101 .343932 1.57092(a) 1.91380 4.64893 4.71445^(a) 7.82039 7.86255(a) 20 0.004 .285761 .207670 .0599330 .174884 1.05963 1.57088^(a) 2.82438 4.71414(a) 5.19123 7.86206(a) 0.006 .642976 .466801 .134661 .175004 1.05975 1.57090^(a) 2.82447 4.71414(a) 5.19127 7.86206(a) 0.008 1.14310 .828735 .238931 .175173 1.05992 1.57093^(a) 2.82460 4.71415(a) 5.19134 7.86207(a) 0.01 1.78615 1.29263 .372394 .175389 1.06014 1.57097^(a) 2.82476 4.71416(a) 5.19142 7.86207(a) 50 0.004 .285776 .516920 .372289 .0705071 .438152 1.21558 1.57089^(a) 2.35205 3.82673 4.71414(a) 0.006 .643045 1.15569 .830125 .0708225 .438516 1.21591 1.57097(a) 2.35243 3.82706 4.71416(a) 0.008 1.14331 2.03651 1.45737 .0712644 .439022 1.21632 1.57115(a) 2.35296 3.82753 4.71419(a) 0.01 1.78662 3.14662 2.24101 .0718327 .439667 1.21677 1.57149(a) 2.35367 3.82812 4.71425(a) 100 0.004 .285821 1.01811 1.45714 .0356599 .220585 .615190 1.20103 1.57109 1.97691 2.93772

0.006 .643236 2.23459 3.16418 .0362954 .221318 .615832 1.20151 1.57189 1.97795 2.93855 0.008 1.14376 3.84233 5.35981 .0371829 .222328 .616599 1.20183 1.57374 1.97965 2.93973 0.01 1.78734 5.76488 7.88899 .0383144 .223606 .617430 1.20185 1.57696 1.98213 2.94126 500 0.002 .0714904 1.15266 7.88670 .00766574 .0447961 .123986 .242326 .400093 .597237 .833677 0.004 .285616 3.54846 20.9805 .00940468 .0469252 .125366 .243308 .400674 .597445 .833320 0.006 .641279 6.11430 30.0040 .0117623 .0504960 .128400 .245619 .402257 .598330 .833313 0.008 1.13885 8.65675 35.2837 .0143953 .0552836 .133486 .250091 .405913 .601122 .835091 1000 0.002 .0714036 1.77418 20.9799 .00470244 .0234641 .0626937 .121696 .200456 .299002 .417269

0.004 .284703 4.32794 35.2810 .00719798 .0276442 .0667603 .125121 .203192 .301152 .418895 0.006 .641188 6.85030 40.5251 .00999761 .0336000 .0742047 .132746 .210299 .307548 .424599 0.008 1.14344 9.36264 42.9987 .0128846 .0404149 .0836039 .143625 .221421 .318236 .434627

表三十四 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=45^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .248001 .146941 .0212090 .343713 1.57087^(a) 1.91366 4.64929 4.71418(a) 7.82118 7.86214(a) 0.006 .558010 .330565 .0477086 .343754 1.57088(a) 1.91369 4.64921 4.71425(a) 7.82100 7.86224(a) 0.008 .992037 .587540 .0847861 .343811 1.57090(a) 1.91373 4.64909 4.71433(a) 7.82076 7.86237(a) 0.01 1.55010 .917769 .132420 .343884 1.57092^(a) 1.91378 4.64895 4.71445(a) 7.82045 7.86255(a) 20 0.004 .248005 .293732 .0847741 .174867 1.05962 1.57088^(a) 2.82438 4.71414(a) 5.19123 7.86206(a) 0.006 .558031 .660372 .190520 .174967 1.05972 1.57090(a) 2.82445 4.71414(a) 5.19127 7.86206(a) 0.008 .992104 1.17269 .338153 .175107 1.05987 1.57093(a) 2.82456 4.71415(a) 5.19132 7.86206(a) 0.01 1.55026 1.82971 .527260 .175287 1.06006 1.57097(a) 2.82471 4.71416(a) 5.19139 7.86206(a) 50 0.004 .248034 .731720 .527129 .0704654 .438115 1.21554 1.57091^(a) 2.35202 3.82669 4.71414(a) 0.006 .558170 1.63777 1.17709 .0707307 .438431 1.21578 1.57104(a) 2.35236 3.82698 4.71417(a) 0.008 .992520 2.89042 2.07049 .0711059 .438867 1.21602 1.57138(a) 2.35286 3.82738 4.71422(a) 0.01 1.55120 4.47432 3.19124 .0715945 .439418 1.21618 1.57205(a) 2.35355 3.82787 4.71433(a) 100 0.004 .248125 1.44503 2.07020 .0355805 .220507 .615076 1.20086 1.57130(a) 1.97689 2.93765

0.006 .558553 3.18433 4.51787 .0361312 .221131 .615479 1.20089 1.57292(a) 1.97807 2.93839 0.008 .993424 5.50164 7.69684 .0369233 .221973 .615765 1.20024 1.57670(a) 1.98022 2.93946 0.01 1.55261 8.29579 11.3931 .0379644 .223022 .615795 1.19858 1.58334(a) 1.98367 2.94090 500 0.003 .139610 3.33394 21.4531 .00832692 .0454066 .123645 .241572 .398947 .595752 .831679

0.004 .247562 5.19859 30.7060 .00930348 .0464778 .123670 .240997 .397844 .594204 .829546 0.005 .385611 7.10727 38.1086 .0104433 .0479552 .124184 .240736 .396936 .592763 .827492 0.006 .553756 9.01112 43.7125 .0116840 .0498366 .125387 .241080 .396574 .591815 .825942 1000 0.003 .138435 4.50532 43.7104 .00584218 .0249203 .0627121 .120621 .198530 .296494 .414262

0.004 .245339 6.39048 51.0738 .00716734 .0273072 .0649396 .122034 .199241 .296668 .414017 0.005 .383287 8.26164 55.4241 .00855346 .0301769 .0682729 .124779 .201274 .298110 .415002 0.006 .553007 10.1272 58.2072 .00997229 .0333560 .0724036 .128708 .204555 .300774 .417182

表三十五 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=60^D) k ε₀ (10⁻⁴) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .198793 .179973 .0259771 .343703 1.57087(a) 1.91366 4.64929 4.71418^(a) 7.82119 7.86214(a) 0.006 .447293 .404895 .0584381 .343731 1.57088(a) 1.91368 4.64922 4.71425^(a) 7.82103 7.86224(a) 0.008 .795210 .719706 .103864 .343771 1.57090(a) 1.91372 4.64911 4.71433^(a) 7.82080 7.86237(a) 0.01 1.24256 1.12432 .162236 .343822 1.57092(a) 1.91376 4.64897 4.71445^(a) 7.82052 7.86255(a) 20 0.004 .198799 .359815 .103852 .174845 1.05960 1.57088^(a) 2.82437 4.71414(a) 5.19122 7.86206(a) 0.006 .447324 .809133 .233468 .174918 1.05969 1.57090^(a) 2.82443 4.71414(a) 5.19125 7.86206(a) 0.008 .795309 1.43733 .414557 .175021 1.05980 1.57093^(a) 2.82452 4.71415(a) 5.19130 7.86206(a) 0.01 1.24280 2.24357 .646744 .175154 1.05995 1.57098^(a) 2.82464 4.71416(a) 5.19136 7.86205(a) 50 0.004 .198842 .897266 .646613 .0704108 .438067 1.21548 1.57092^(a) 2.35198 3.82665 4.71415(a) 0.006 .447534 2.01131 1.44665 .0706099 .438321 1.21562 1.57112^(a) 2.35227 3.82688 4.71418(a) 0.008 .795936 3.55683 2.55122 .0708963 .438668 1.21568 1.57162^(a) 2.35272 3.82719 4.71426(a) 0.01 1.24423 5.51964 3.94454 .0712769 .439099 1.21550 1.57262^(a) 2.35337 3.82756 4.71442(a) 100 0.004 .198980 1.77823 2.55092 .0354753 .220405 .614938 1.20067 1.57153^(a) 1.97685 2.93755

0.006 .448113 3.93982 5.60471 .0359105 .220891 .615063 1.20019 1.57400^(a) 1.97815 2.93818 0.008 .797297 6.85215 9.62555 .0365677 .221521 .614785 1.19846 1.57993^(a) 1.98079 2.93911 0.01 1.24628 10.4061 14.3658 .0374757 .222275 .613860 1.19495 1.59043^(a) 1.98539 2.94039 500 0.001 .0124417 .551847 3.94359 .00713528 .0441953 .123472 .241825 .399619 .596790 .833288

0.002 .0498478 2.08080 14.3625 .00749782 .0445298 .123336 .241523 .399149 .596174 .832430 0.003 .111919 4.25027 27.4813 .00818433 .0450453 .122528 .240164 .397282 .593888 .829537 0.004 .197757 6.69837 39.5492 .00915509 .0458747 .121541 .238260 .394619 .590640 .825581 1000 0.001 .0124619 1.04039 14.3624 .00374895 .0222661 .0616767 .120794 .199664 .298290 .416636

0.002 .0494384 3.34911 39.5483 .00457764 .0229389 .0607865 .119196 .197503 .295781 .413801 0.003 .109668 5.85810 55.9504 .00578511 .0244655 .0608303 .118068 .195568 .293314 .410915 0.004 .193409 8.33265 64.7602 .00712348 .0268352 .0626337 .118602 .195239 .292390 .409558

表三十六 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=75^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 1.41488 .200743 .0289755 .343692 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82120 7.86214(a) 0.006 3.18358 .451651 .0651887 .343705 1.57088(a) 1.91367 4.64923 4.71425^(a) 7.82106 7.86224(a) 0.008 5.65995 .802882 .115875 .343724 1.57090(a) 1.91370 4.64912 4.71433^(a) 7.82086 7.86237(a) 0.01 8.84416 1.25440 .181024 .343749 1.57092(a) 1.91373 4.64899 4.71445^(a) 7.82060 7.86255(a) 20 0.004 1.41496 .401411 .115865 .174820 1.05958 1.57088^(a) 2.82435 4.71414(a) 5.19121 7.86206(a) 0.006 3.18398 .902921 .260568 .174862 1.05964 1.57090^(a) 2.82440 4.71414(a) 5.19124 7.86206(a) 0.008 5.66119 1.60456 .462908 .174921 1.05973 1.57093^(a) 2.82447 4.71415(a) 5.19128 7.86206(a) 0.01 8.84715 2.50583 .722637 .174997 1.05983 1.57098^(a) 2.82456 4.71415(a) 5.19132 7.86205(a) 50 0.004 1.41550 1.00220 .722528 .0703470 .438011 1.21542 1.57093^(a) 2.35193 3.82660 4.71415(a) 0.006 3.18659 2.25048 1.62014 .0704678 .438195 1.21546 1.57118(a) 2.35216 3.82677 4.71419(a) 0.008 5.66902 3.98939 2.86597 .0706477 .438440 1.21534 1.57181(a) 2.35254 3.82698 4.71429(a) 0.01 8.86496 6.20953 4.44812 .0708965 .438735 1.21486 1.57307(a) 2.35311 3.82722 4.71449(a) 100 0.004 1.41722 1.99454 2.86571 .0353506 .220289 .614793 1.20047 1.57170^(a) 1.97677 2.93743

0.006 3.19387 4.44822 6.34880 .0356431 .220619 .614646 1.19956 1.57488^(a) 1.97813 2.93792 0.008 5.68599 7.80121 11.0163 .0361261 .221012 .613820 1.19683 1.58269^(a) 1.98115 2.93867 0.01 8.88954 11.9584 16.6242 .0368527 .221430 .611928 1.19161 1.59680(a) 1.98682 2.93976 500 0.001 .0886451 .620857 4.44731 .00709709 .0441582 .123422 .241770 .399559 .596726 .833217

0.002 .355552 2.39131 16.6211 .00737310 .0443605 .122979 .241098 .398660 .595634 .831812 0.0025 .554825 3.61686 24.4522 .00763597 .0444769 .122296 .240118 .397400 .594146 .829984 0.003 .795804 4.99501 32.5427 .00799298 .0446213 .121330 .238719 .395619 .592070 .827510 1000 0.001 .0888877 1.19565 16.6210 .00368659 .0221814 .0614983 .120583 .199426 .298036 .416365

0.002 .348595 3.99716 47.2665 .00447598 .0225720 .0596109 .117788 .195925 .294117 .412061 0.0025 .534911 5.54461 58.4738 .00506404 .0230877 .0588892 .116401 .194095 .292013 .409741 0.003 .757671 7.08424 66.3697 .00570761 .0238941 .0587470 .115531 .192783 .290414 .407928

表三十七 旋轉傾斜梁在不同細長比與不同轉速下的振動頻率(r=1.5，

α

=90^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

L

V (10⁻²)

η K ₁ K ₂ K ₃ K ₄ K ₅ K ₆ K ₇

10 0.004 .799927 .207835 .0299998 .343679 1.57087(a) 1.91365 4.64930 4.71418^(a) 7.82122 7.86214(a) 0.006 1.79993 .467638 .0674988 .343677 1.57088(a) 1.91366 4.64923 4.71425^(a) 7.82109 7.86224(a) 0.008 3.20011 .831377 .119996 .343674 1.57090(a) 1.91368 4.64914 4.71433^(a) 7.82091 7.86237(a) 0.01 5.00064 1.29907 .187490 .343671 1.57092(a) 1.91370 4.64902 4.71445^(a) 7.82068 7.86255(a) 20 0.004 .800011 .415670 .119990 .174793 1.05956 1.57088^(a) 2.82434 4.71414(a) 5.19121 7.86206(a) 0.006 1.80035 .935273 .269947 .174801 1.05960 1.57090^(a) 2.82437 4.71414(a) 5.19122 7.86206(a) 0.008 3.20144 1.66274 .479828 .174813 1.05965 1.57093^(a) 2.82441 4.71415(a) 5.19125 7.86206(a) 0.01 5.00385 2.59808 .749570 .174829 1.05971 1.57098^(a) 2.82447 4.71415(a) 5.19128 7.86205(a) 50 0.004 .800586 1.03915 .749498 .0702781 .437952 1.21536 1.57094^(a) 2.35187 3.82654 4.71415(a) 0.006 1.80316 2.33792 1.68475 .0703135 .438061 1.21532 1.57120(a) 2.35204 3.82664 4.71419(a) 0.008 3.20987 4.15539 2.99040 .0703752 .438200 1.21505 1.57188(a) 2.35233 3.82676 4.71430(a) 0.01 5.02306 6.48969 4.66122 .0704751 .438357 1.21436 1.57328(a) 2.35279 3.82687 4.71452(a) 100 0.004 .802448 2.07760 2.99023 .0352140 .220167 .614656 1.20032 1.57177^(a) 1.97667 2.93730

0.006 1.81104 4.66797 6.68756 .0353429 .220337 .614290 1.19909 1.57532^(a) 1.97797 2.93764 0.008 3.22810 8.26754 11.7482 .0356146 .220490 .613028 1.19568 1.58431^(a) 1.98111 2.93818 0.01 5.04783 12.8203 17.9801 .0361062 .220567 .610327 1.18916 1.60109^(a) 1.98747 2.93903 500 0.001 .0502277 .648908 4.66066 .00705477 .0441193 .123377 .241721 .399507 .596672 .833160

0.002 .201893 2.56381 17.9777 .00722366 .0441871 .122676 .240754 .398276 .595220 .831351 0.0025 .314261 3.93783 26.9195 .00743089 .0441828 .121667 .239394 .396589 .593272 .829019 0.003 .447767 5.51810 36.3646 .00774664 .0441672 .120206 .237434 .394187 .590546 .825864 1000 0.001 .0504732 1.28190 17.9776 .00361187 .0220947 .0613474 .120412 .199237 .297838 .416158

0.002 .191177 4.51428 53.6436 .00434099 .0221568 .0584298 .116472 .194500 .292648 .410551 0.0025 .283431 6.34013 66.3355 .00494287 .0225120 .0571632 .114502 .192068 .289934 .407619 0.003 .388229 8.15303 74.7254 .00560584 .0232181 .0566295 .113203 .190324 .287893 .405356

表三十八 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 38 )

α k K ₁ K ₂ K₃ (a) K ₄ K ₅ K₆ (a) 0^D 0 .0923781 .573449 1.57086 1.58182 3.03502 4.71413

0.01 .0933104 .574596 1.57093 1.58297 3.03606 4.71416 0.02 .0960486 .578022 1.57116 1.58637 3.03918 4.71424 0.03 .100433 .583687 1.57158 1.59197 3.04436 4.71438 0.04 .106241 .591528 1.57221 1.59975 3.05162 4.71457 0.05 .113232 .601458 1.57305 1.60967 3.06092 4.71483 0.06 .121175 .613377 1.57410 1.62169 3.07226 4.71514 5 D 0.01 .0933074 .574593 1.57087 1.58302 3.03606 4.71416 0.02 .0960381 .578009 1.57056 1.58697 3.03917 4.71423 0.025 .0980332 .580557 1.57023 1.59003 3.04150 4.71427 0.03 .100414 .583656 1.56983 1.59376 3.04435 4.71432 10D 0.005 .0926090 .573734 1.57086 1.58212 3.03528 4.71414 0.01 .0932982 .574585 1.57071 1.58318 3.03605 4.71416 0.015 .0944361 .575999 1.57013 1.58528 3.03734 4.71417 0.02 .0960067 .577971 1.56900 1.58854 3.03914 4.71418 15D 0.004 .0925234 .573629 1.57086 1.58202 3.03518 4.71414 0.006 .0927048 .573854 1.57082 1.58230 3.03539 4.71414 0.008 .0929582 .574168 1.57070 1.58275 3.03567 4.71415 0.01 .0932831 .574571 1.57046 1.58343 3.03604 4.71415 30D 0.004 0.0925101 .573618 1.57082 1.58205 3.03517 4.71414 0.006 .0926750 .573828 1.57063 1.58246 3.03536 4.71414 0.008 .0929059 .574121 1.57016 1.58326 3.03563 4.71414 0.01 .0932029 .574496 1.56930 1.58454 3.03598 4.71413 45D 0.004 .0924058 .573487 1.57086 1.58187 3.03505 4.71413 0.006 .0924888 .573600 1.57077 1.58208 3.03516 4.71414 0.008 .0926275 .573787 1.57038 1.58268 3.03533 4.71413 0.01 .0928224 .574047 1.56948 1.58389 3.03557 4.71412 60D 0.004 .0924610 .573576 1.57071 1.58211 3.03514 4.71413 0.006 .0925653 .573733 1.57014 1.58287 3.03528 4.71413 0.008 .0927130 .573951 1.56884 1.58445 3.03549 4.71411 0.01 .0929054 .574227 1.56671 1.58695 3.03576 4.71405 75D 0.004 .0924285 .573549 1.57068 1.58213 3.03511 4.71413 0.006 .0924927 .573671 1.56997 1.58299 3.03523 4.71413 0.008 .0925846 .573840 1.56839 1.58481 3.03540 4.71410 0.01 .0927063 .574053 1.56586 1.58767 3.03561 4.71403 90D 0.004 .0923937 .573519 1.57066 1.58212 3.03509 4.71413 0.006 .0924144 .573605 1.56990 1.58300 3.03517 4.71412 0.008 .0924457 .573722 1.56821 1.58489 3.03529 4.71409 0.01 .0924900 .573868 1.56551 1.58787 3.03545 4.71402

表三十九 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 38 . 5 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K₆ (a)

0^D 0 .0911822 .566163 1.56231 1.57086^(a) 2.99909 4.71413 0.01 .0921270 .567326 1.56340 1.57100(a) 3.00015 4.71416 0.02 .0949006 .570802 1.56655 1.57153(a) 3.00333 4.71423 0.03 .0993372 .576549 1.57034(a) 1.57387 3.00863 4.71436 0.04 .105208 .584500 1.57179(a) 1.58095 3.01603 4.71454 0.05 .112264 .594566 1.57282(a) 1.59085 3.02552 4.71478 0.06 .120272 .606642 1.57394(a) 1.60299 3.03709 4.71509 5 D 0.01 .0921240 .567323 1.56330 1.57110 3.00015 4.71416

0.02 .0948900 .570789 1.56500 1.57308 3.00332 4.71421 0.025 .0969096 .573374 1.56537 1.57549 3.00570 4.71423 0.03 .0993184 .576517 1.56544 1.57881 3.00861 4.71426 10D 0.005 .0914162 .566451 1.56255 1.57092 2.99935 4.71414 0.01 .0921147 .567315 1.56303 1.57136 3.00014 4.71415 0.015 .0932676 .568750 1.56304 1.57289 3.00146 4.71416 0.02 .0948584 .570751 1.56224 1.57586 3.00330 4.71413 15D 0.004 .0913295 .566345 1.56246 1.57090 2.99925 4.71414 0.006 .0915133 .566573 1.56259 1.57101 2.99946 4.71414 0.008 .0917701 .566891 1.56267 1.57127 2.99975 4.71414 0.01 .0920994 .567300 1.56262 1.57176 3.00013 4.71414 30D 0.004 .0913159 .566333 1.56239 1.57096 2.99924 4.71414 0.006 .0914831 .566547 1.56230 1.57128 2.99944 4.71413 0.008 .0917172 .566844 1.56186 1.57205 2.99971 4.71412 0.01 .0920182 .567225 1.56095 1.57339 3.00007 4.71410 45D 0.004 .0912943 .566315 1.56230 1.57103 2.99923 4.71413 0.006 .0914350 .566505 1.56193 1.57162 2.99940 4.71413 0.008 .0916326 .566769 1.56091 1.57294 2.99965 4.71410 0.01 .0918881 .567106 1.55919 1.57508 2.99997 4.71404 60D 0.004 .0912662 .566291 1.56221 1.57110 2.99921 4.71413

0.006 .0913720 .566450 1.56156 1.57194 2.99936 4.71412 0.008 .0915218 .566671 1.56006 1.57372 2.99957 4.71407 0.01 .0917170 .566952 1.55769 1.57648 2.99984 4.71398 75D 0.004 .0912333 .566263 1.56213 1.57115 2.99918 4.71413

0.006 .0912984 .566388 1.56129 1.57216 2.99930 4.71411 0.008 .0913917 .566559 1.55945 1.57424 2.99947 4.71406 0.01 .0915154 .566775 1.55665 1.57739 2.99969 4.71393 90D 0.004 .0911980 .566234 1.56209 1.57117 2.99916 4.71413

0.006 .0912191 .566321 1.56115 1.57223 2.99924 4.71411 0.008 .0912510 .566439 1.55917 1.57441 2.99937 4.71405 0.01 .0912963 .566587 1.55617 1.57770 2.99953 4.71391

表四十 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 39 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K₆(a)

0^D 0 .0900168 .559057 1.54325 1.57086^(a) 2.96396 4.71413 0.01 .0909741 .560237 1.54439 1.57097(a) 2.96504 4.71415 0.02 .0937830 .563763 1.54780 1.57130(a) 2.96829 4.71421 0.03 .0982717 .569591 1.55343 1.57189(a) 2.97369 4.71431 0.04 .104204 .577652 1.56112 1.57287(a) 2.98124 4.71447 0.05 .111325 .587853 1.56961 1.57547(a) 2.99092 4.71469 0.06 .119397 .600085 1.57330(a) 1.58523 3.00272 4.71498 5 D 0.01 .0909710 .560234 1.54436 1.57100 2.96504 4.71415 0.02 .0937723 .563750 1.54735 1.57176 2.96828 4.71416 0.025 .0958163 .566372 1.54928 1.57264 2.97071 4.71415 0.03 .0982529 .569559 1.55131 1.57405 2.97368 4.71411 10D 0.005 .0902539 .559349 1.54353 1.57090 2.96423 4.71414 0.01 .0909617 .560225 1.54427 1.57109 2.96503 4.71414 0.015 .0921296 .561681 1.54524 1.57168 2.96637 4.71411 0.02 .0937404 .563711 1.54610 1.57302 2.96826 4.71401 15D 0.004 .0901660 .559242 1.54342 1.57088 2.96413 4.71414 0.006 .0903523 .559473 1.54362 1.57093 2.96434 4.71414 0.008 .0906125 .559796 1.54386 1.57104 2.96464 4.71413 0.01 .0909462 .560211 1.54412 1.57123 2.96502 4.71412 30D 0.004 .0901523 .559230 1.54339 1.57090 2.96412 4.71413 0.006 .0903217 .559446 1.54351 1.57103 2.96432 4.71412 0.008 .0905589 .559748 1.54354 1.57132 2.96460 4.71409 0.01 .0908640 .560134 1.54340 1.57191 2.96496 4.71403 45D 0.004 .0901304 .559211 1.54335 1.57093 2.96410 4.71413 0.006 .0902730 .559404 1.54335 1.57115 2.96428 4.71411 0.008 .0904734 .559672 1.54311 1.57171 2.96453 4.71404 0.01 .0907324 .560013 1.54244 1.57279 2.96486 4.71391 60D 0.004 .0901019 .559187 1.54331 1.57095 2.96408 4.71413 0.006 .0902092 .559349 1.54318 1.57127 2.96423 4.71409 0.008 .0903611 .559573 1.54266 1.57208 2.96445 4.71399 0.01 .0905592 .559857 1.54149 1.57364 2.96473 4.71378 75D 0.004 .0900686 .559159 1.54326 1.57097 2.96405 4.71412 0.006 .0901347 .559285 1.54303 1.57136 2.96418 4.71408 0.008 .0902294 .559459 1.54229 1.57235 2.96435 4.71395 0.01 .0903551 .559677 1.54075 1.57424 2.96458 4.71368 90D 0.004 .0900328 .559129 1.54323 1.57098 2.96403 4.71412 0.006 .0900543 .559217 1.54293 1.57140 2.96412 4.71407 0.008 .0900868 .559338 1.54208 1.57245 2.96424 4.71394 0.01 .0901331 .559487 1.54036 1.57447 2.96441 4.71364

表四十一 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 40 )

α k K ₁ K ₂ K ₃ K₄(a) K ₅ K₆(a) 0^D 0 .0877730 .545362 1.50647 1.57086 2.89600 4.67451

0.01 .0887554 .546576 1.50765 1.57096 2.89713 4.67548 0.02 .0916345 .550201 1.51120 1.57127 2.90050 4.67838

0.03 .0962265 .556192 1.51709 1.57178 2.90612 4.68319 0.04 .102280 .564471 1.52529 1.57251 2.91396 4.68982 0.05 .109529 .574941 1.53572 1.57348 2.92402 4.69803 0.06 .117725 .587482 1.54828 1.57477 2.93628 4.70664 5 D 0.01 .0887522 .546573 1.50764 1.57098 2.89712 4.67547

0.02 .0916237 .550188 1.51100 1.57147 2.90049 4.67826 0.025 .0937162 .552883 1.51341 1.57197 2.90302 4.68026

0.03 .0962077 .556158 1.51623 1.57269 2.90611 4.68262 10D 0.005 .0880164 .545663 1.50676 1.57089 2.89628 4.67475 0.01 .0887426 .546564 1.50759 1.57102 2.89712 4.67544 0.015 .0899404 .548061 1.50886 1.57136 2.89851 4.67652 0.02 .0915913 .550147 1.51041 1.57208 2.90047 4.67789 15D 0.004 .0879261 .545552 1.50665 1.57088 2.89618 4.67466 0.006 .0881173 .545790 1.50687 1.57091 2.89640 4.67484 0.008 .0883844 .546122 1.50716 1.57098 2.89671 4.67509 0.01 .0887268 .546549 1.50752 1.57109 2.89711 4.67539 30D 0.004 .0879121 .545540 1.50663 1.57089 2.89617 4.67464 0.006 .0880860 .545762 1.50681 1.57096 2.89638 4.67480 0.008 .0883295 .546073 1.50699 1.57112 2.89667 4.67498 0.01 .0886427 .546470 1.50713 1.57143 2.89704 4.67515 45D 0.004 .0878897 .545521 1.50660 1.57090 2.89615 4.67462 0.006 .0880360 .545719 1.50671 1.57102 2.89634 4.67473 0.008 .0882418 .545994 1.50674 1.57131 2.89660 4.67481 0.01 .0885079 .546345 1.50660 1.57189 2.89694 4.67481 60D 0.004 .0878604 .545496 1.50657 1.57091 2.89613 4.67460 0.006 .0879707 .545662 1.50660 1.57108 2.89629 4.67465 0.008 .0881268 .545892 1.50647 1.57151 2.89652 4.67463 0.01 .0883305 .546184 1.50603 1.57236 2.89681 4.67444 75D 0.004 .0878263 .545467 1.50653 1.57092 2.89610 4.67457 0.006 .0878942 .545596 1.50649 1.57113 2.89623 4.67458 0.008 .0879918 .545775 1.50623 1.57165 2.89641 4.67446 0.01 .0881214 .545999 1.50554 1.57271 2.89665 4.67411 90D 0.004 .0877895 .545436 1.50650 1.57093 2.89608 4.67454 0.006 .0878117 .545526 1.50641 1.57115 2.89617 4.67451 0.008 .0878456 .545650 1.50606 1.57171 2.89630 4.67434 0.01 .0878938 .545804 1.50523 1.57284 2.89648 4.67389

表四十二 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 77 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K ₆

0^D 0 .0456446 .285383 .796090 1.55157 1.57086(a) 2.54705 0.01 .0475274 .287806 .798607 1.55415 1.57097(a) 2.54962 0.02 .0527487 .294954 .806106 1.56182 1.57132(a) 2.55733 0.03 .0603897 .306488 .818433 1.57127(a) 1.57514 2.57012 0.04 .0695879 .321922 .835345 1.57226(a) 1.59254 2.58791 0.05 .0797415 .340707 .856536 1.57316(a) 1.61497 2.61060 0.06 .0904762 .362297 .881655 1.57424(a) 1.64193 2.63804 5 D 0.01 .0475221 .287800 .798593 1.55379 1.57135 2.54962

0.02 .0527362 .294925 .805992 1.55723 1.57616 2.55735 0.025 .0563121 .300156 .811477 1.55835 1.58123 2.56313

0.03 .0603752 .306428 .818132 1.55934 1.58777 2.57019 10D 0.005 .0461168 .285985 .796712 1.55212 1.57098 2.54769 0.01 .0475062 .287780 .798548 1.55281 1.57240 2.54961 0.015 .0497366 .290742 .801540 1.55205 1.57682 2.55284 0.02 .0526986 .294837 .805649 1.55006 1.58410 2.55739 15D 0.004 .0459424 .285764 .796483 1.55189 1.57097 2.54745 0.006 .0463123 .286238 .796967 1.55207 1.57134 2.54796 0.008 .0468261 .286900 .797634 1.55197 1.57225 2.54868 0.01 .0474797 .287748 .798475 1.55139 1.57392 2.54960 30D 0.004 .0459158 .285739 .796452 1.55165 1.57121 2.54743 0.006 .0462548 .286181 .796881 1.55100 1.57246 2.54792 0.008 .0467292 .286795 .797440 1.54916 1.57526 2.54861 0.01 .0473383 .287577 .798100 1.54610 1.57975 2.54952 45D 0.004 .0458733 .285700 .796405 1.55132 1.57153 2.54740 0.006 .0461624 .286091 .796750 1.54966 1.57385 2.54784 0.008 .0465726 .286630 .797151 1.54607 1.57863 2.54850 0.01 .0471080 .287309 .797542 1.54082 1.58577 2.54939 60D 0.004 .0458176 .285650 .796346 1.55098 1.57184 2.54735 0.006 .0460404 .285976 .796592 1.54841 1.57513 2.54775 0.008 .0463639 .286418 .796809 1.54338 1.58155 2.54835 0.01 .0467976 .286966 .796890 1.53643 1.59090 2.54919 75D 0.004 .0457522 .285592 .796281 1.55072 1.57207 2.54729 0.006 .0458960 .285842 .796426 1.54749 1.57602 2.54762 0.008 .0461138 .286176 .796464 1.54148 1.58356 2.54815 0.01 .0464205 .286578 .796249 1.53332 1.59449 2.54891 90D 0.004 .0456815 .285530 .796216 1.55058 1.57215 2.54723 0.006 .0457381 .285702 .796271 1.54705 1.57636 2.54749 0.008 .0458366 .285923 .796162 1.54056 1.58439 2.54791 0.01 .0459954 .286176 .795716 1.53175 1.59611 2.54856

表四十三 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 78 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K ₆

0^D 0 .0450599 .281744 .786013 1.53214 1.57086^(a) 2.51558 0.01 .0469663 .284199 .788565 1.53476 1.57096(a) 2.51820 0.02 .0522436 .291438 .796166 1.54258 1.57127(a) 2.52602 0.03 .0599473 .303111 .808656 1.55549 1.57182(a) 2.53901 0.04 .0692007 .318714 .825783 1.57125(a) 1.57470 2.55707 0.05 .0793981 .337682 .847230 1.57309(a) 1.59649 2.58010 0.06 .0901664 .359459 .872635 1.57421(a) 1.62377 2.60794 5 D 0.01 .0469609 .284193 .788550 1.53458 1.57116 2.51819

0.02 .0522311 .291409 .796050 1.54038 1.57374 2.52604 0.025 .0558388 .296704 .801610 1.54380 1.57659 2.53192 0.03 .0599330 .303050 .808353 1.54733 1.58068 2.53909 10D 0.005 .0455382 .282354 .786643 1.53274 1.57094 2.51623 0.01 .0469449 .284173 .788504 1.53405 1.57176 2.51819 0.015 .0492011 .287173 .791537 1.53504 1.57449 2.52147 0.02 .0521936 .291319 .795702 1.53521 1.57969 2.52609 15D 0.004 .0453615 .282130 .786411 1.53251 1.57093 2.51599 0.006 .0457362 .282610 .786902 1.53285 1.57114 2.51651 0.008 .0462565 .283281 .787578 1.53314 1.57168 2.51724 0.01 .0469182 .284140 .788430 1.53323 1.57270 2.51818 30D 0.004 .0453346 .282105 .786380 1.53237 1.57106 2.51597 0.006 .0456780 .282553 .786814 1.53223 1.57181 2.51647 0.008 .0461587 .283175 .787380 1.53140 1.57364 2.51717 0.01 .0467756 .283966 .788046 1.52955 1.57694 2.51811 45D 0.004 .0452916 .282065 .786331 1.53217 1.57125 2.51593 0.006 .0455846 .282462 .786680 1.53139 1.57270 2.51639 0.008 .0460005 .283007 .787084 1.52918 1.57613 2.51707 0.01 .0465433 .283694 .787476 1.52523 1.58203 2.51798 60D 0.004 .0452352 .282014 .786271 1.53196 1.57144 2.51589 0.006 .0454612 .282344 .786519 1.53055 1.57358 2.51629 0.008 .0457895 .282791 .786735 1.52706 1.57850 2.51691 0.01 .0462300 .283345 .786810 1.52133 1.58670 2.51779 75D 0.004 .0451690 .281955 .786205 1.53178 1.57158 2.51583 0.006 .0453150 .282209 .786350 1.52988 1.57422 2.51617 0.008 .0455366 .282546 .786383 1.52546 1.58023 2.51672 0.01 .0458490 .282951 .786155 1.51844 1.59012 2.51752 90D 0.004 .0450974 .281892 .786139 1.53167 1.57163 2.51577 0.006 .0451551 .282067 .786193 1.52953 1.57448 2.51603 0.008 .0452560 .282290 .786076 1.52463 1.58097 2.51648 0.01 .0454190 .282544 .785612 1.51691 1.59171 2.51717

表四十四 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 79 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K ₆

0^D 0 .0444900 .278196 .776186 1.51318 1.57086^(a) 2.48487 0.01 .0464199 .280683 .778773 1.51584 1.57096(a) 2.48752 0.02 .0517527 .288014 .786476 1.52379 1.57126(a) 2.49547 0.03 .0595184 .299824 .799129 1.53692 1.57177(a) 2.50865 0.04 .0688257 .315595 .816471 1.55502 1.57255(a) 2.52699 0.05 .0790658 .334745 .838172 1.57256(a) 1.57894 2.55035 0.06 .0898666 .356704 .863859 1.57415(a) 1.60610 2.57859 5 D 0.01 .0464145 .280677 .778757 1.51571 1.57111 2.48752

0.02 .0517403 .287984 .786358 1.52233 1.57298 2.49549 0.025 .0553795 .293343 .791992 1.52673 1.57495 2.50146 0.03 .0595043 .299763 .798825 1.53171 1.57768 2.50875 10D 0.005 .0449744 .278814 .776825 1.51381 1.57093 2.48553 0.01 .0463983 .280656 .778711 1.51534 1.57155 2.48752 0.015 .0486801 .283695 .781783 1.51706 1.57362 2.49085 0.02 .0517030 .287893 .786005 1.51848 1.57763 2.49555 15D 0.004 .0447954 .278587 .776589 1.51357 1.57091 2.48529 0.006 .0451749 .279074 .777087 1.51397 1.57108 2.48581 0.008 .0457017 .279753 .777771 1.51440 1.57149 2.48655 0.01 .0463715 .280623 .778635 1.51475 1.57227 2.48751 30D 0.004 .0447682 .278562 .776557 1.51346 1.57102 2.48526 0.006 .0451161 .279015 .776997 1.51351 1.57159 2.48577 0.008 .0456029 .279645 .777569 1.51311 1.57301 2.48649 0.01 .0462277 .280447 .778243 1.51193 1.57567 2.48744 45D 0.004 .0447247 .278522 .776509 1.51331 1.57116 2.48523

0.006 .0450216 .278923 .776861 1.51288 1.57228 2.48570 0.008 .0454431 .279475 .777268 1.51138 1.57503 2.48639 0.01 .0459935 .280170 .777660 1.50837 1.58004 2.48733 60D 0.004 .0446676 .278470 .776447 1.51314 1.57130 2.48518 0.006 .0448967 .278804 .776697 1.51222 1.57298 2.48560 0.008 .0452299 .279256 .776911 1.50965 1.57703 2.48624 0.01 .0456773 .279817 .776980 1.50496 1.58426 2.48715 75D 0.004 .0446006 .278410 .776380 1.51300 1.57141 2.48512 0.006 .0447488 .278667 .776525 1.51169 1.57350 2.48547 0.008 .0449742 .279007 .776553 1.50828 1.57853 2.48604 0.01 .0452925 .279416 .776311 1.50232 1.58745 2.48688 90D 0.004 .0445280 .278346 .776314 1.51290 1.57145 2.48506 0.006 .0445869 .278522 .776365 1.51137 1.57371 2.48533 0.008 .0446902 .278748 .776240 1.50753 1.57920 2.48580 0.01 .0448576 .279003 .775757 1.50088 1.58898 2.48653

表四十五 旋轉梁在不同傾斜角與不同轉速下的振動頻率( r = 1 ， η = 80 )

α k K ₁ K ₂ K ₃ K ₄ K ₅ K ₆

0^D 0 .0439342 .274736 .766600 1.49468 1.57086(a) 2.45488 0.01 .0458877 .277256 .769222 1.49738 1.57096^(a) 2.45758 0.02 .0512757 .284677 .777027 1.50544 1.57125^(a) 2.46564 0.03 .0591023 .296624 .789842 1.51876 1.57175^(a) 2.47902 0.04 .0684624 .312561 .807398 1.53718 1.57247^(a) 2.49763 0.05 .0787439 .331891 .829353 1.56032 1.57356^(a) 2.52132 0.06 .0895762 .354031 .855320 1.57395(a) 1.58903 2.54996 5 D 0.01 .0458823 .277249 .769206 1.49728 1.57108 2.45758 0.02 .0512633 .284647 .776907 1.50433 1.57264 2.46567 0.025 .0549336 .290070 .782615 1.50923 1.57419 2.47173 0.03 .0590884 .296564 .789537 1.51496 1.57627 2.47913 10D 0.005 .0444248 .275362 .767247 1.49532 1.57092 2.45555 0.01 .0458660 .277228 .769158 1.49698 1.57146 2.45757 0.015 .0481732 .280306 .772271 1.49909 1.57319 2.46096 0.02 .0512261 .284555 .776549 1.50126 1.57654 2.46574 15D 0.004 .0442436 .275132 .767009 1.49508 1.57091 2.45531 0.006 .0446278 .275626 .767513 1.49552 1.57105 2.45584 0.008 .0451611 .276314 .768206 1.49602 1.57140 2.45659 0.01 .0458389 .277194 .769081 1.49650 1.57206 2.45756 30D 0.004 .0442160 .275107 .766977 1.49499 1.57100 2.45529 0.006 .0445684 .275566 .767421 1.49514 1.57148 2.45580 0.008 .0450614 .276204 .768000 1.49496 1.57269 2.45653 0.01 .0456941 .277015 .768681 1.49418 1.57499 2.45751 45D 0.004 .0441720 .275066 .766927 1.49486 1.57112 2.45525

0.006 .0444728 .275472 .767283 1.49461 1.57207 2.45573 0.008 .0449000 .276031 .767692 1.49351 1.57445 2.45643 0.01 .0454579 .276734 .768085 1.49113 1.57889 2.45741 60D 0.004 .0441142 .275014 .766865 1.49472 1.57124 2.45520 0.006 .0443466 .275351 .767116 1.49406 1.57267 2.45563 0.008 .0446846 .275809 .767329 1.49203 1.57622 2.45629 0.01 .0451389 .276376 .767391 1.48810 1.58277 2.45724 75D 0.004 .0440464 .274953 .766797 1.49459 1.57133 2.45514 0.006 .0441969 .275213 .766941 1.49359 1.57312 2.45550 0.008 .0444261 .275557 .766963 1.49082 1.57757 2.45609 0.01 .0447503 .275970 .766707 1.48569 1.58578 2.45698 90D 0.004 .0439729 .274888 .766729 1.49450 1.57136 2.45508 0.006 .0440330 .275066 .766778 1.49330 1.57331 2.45536 0.008 .0441387 .275294 .766645 1.49013 1.57818 2.45585 0.01 .0443105 .275551 .766141 1.48431 1.58725 2.45663

圖一 無傾斜角的旋轉梁結構

圖二 具傾斜角的旋轉梁結構

R

O

A ^X1

X3 Ω X2

β

Ω

LT

O A X₁

X3 ^X₂

β

Ω

β Ω

R

X

1

X

2

X

3

O A

α

β

R α L

T

A

O X

¹

X

2

圖三 旋轉傾斜梁的上視圖

O

β β

X

3

X

2

Ω

圖四 旋轉傾斜梁的側視圖

圖五 元素座標及總體座標關係圖

O Y

o

X x

₁

X

1

X

o 2

θ

1

θ

²

1

2

o x

2

θ

e

Ω

X

1

X

3

X

2

O A R

α

A

O X

1

X

2

L

T

R α

圖六 具傾斜角的旋轉梁結構( β = 0

^D

)

θ

Q

P y

v θ

s

u x + o

x

2

x

1

P

x

2

x

3

z y

圖七 梁的變形圖

0.0 0.2 0.4 0.6 0.8 1.0

)

)

)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

圖十九 不同轉速下的第一至第六振動模態( α = 30

^D

， r = 1 ， η = 50 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

Modal deflection U,V Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection

圖二十三 不同轉速下的第一至第六振動模態( α = 30

^D

， r = 1 ， η = 100 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

x/L

0 5

Mode

Phase angle FU,FV

0.0 0.5 1.0 Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

Modal deflection U,V

圖二十七 不同轉速下的第一至第六振動模態( α = 30

^D

， r = 1 ， η = 1000 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

x/L

0 5

Mode

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

0 2 4 6

圖三十一 不同轉速下的第一至第六振動模態( α = 5

^D

， r = 1 ， η = 38 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

Phase angle FU,FV

0.0 0.5 1.0

圖三十三 不同轉速下的第一至第六振動模態( α = 90

^D

， r = 1 ， η = 38 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

圖三十五 不同轉速下的第一至第六振動模態( α = 5

^D

， r = 1 ， η = 39 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

0 3

Mode

tion U,VcModal defle

0.0 0.5 1.0

U V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

圖三十九 不同轉速下的第一至第六振動模態( α = 5

^D

， r = 1 ， η = 40 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

V

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L

0 5

Mode

VModal deflection U,

U V

Phase angle FU,FV

Modal deflection U,V

圖四十三 不同轉速下的第一至第六振動模態( α = 5

^D

， r = 1 ， η = 77 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

0.0 0.5 1.0

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

圖四十五 不同轉速下的第一至第六振動模態( α = 90

^D

， r = 1 ， η = 77 )

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

Modal deflection U,V

Phase angle FU,FV

x/L 6

Mode

U,V

附錄A f

_a^Dθ

及 f

_b^Dθ

的推導

附錄 B 旋轉傾斜梁的線性穩態解

若不考慮旋轉傾斜梁穩態變形對慣性力及梁之側向剛度的影響，則可 用 d’Alembert 原理及剛體動力學求得慣性分布力，再用材料力學可以求得 旋轉傾斜梁的線性穩態解。

令圖 B1 中 為轉軸 r A 到旋轉梁上任一點 P 的位置向量， 及 為作用 在

q

1

q

₂

P 點 X

₁

及 X

₂

方向的慣性分布力，則作用在 P 點之一小段梁 dx 的慣性分布 力可以表示成

{ q

_x,

q

_y

} dx = Ω × ( Ω × r ) ρ Adx (B.1) Ω = { 0, 0, Ω } (B.2) r = { R cos α + x , − R sin α , 0 } (B.3) 其中 ρ 、 A 為梁的密度及斷面積， Ω 為轉速。

將(B.2)、(B.3)式代入(B.1)式整理可得

( cos )

2 2

x R

L EA q k

T

x

= α + (B.4)

sin α

2 2

R L

EA q k

T

y

= − (B.5)

E

k L

^T

2 2 2

= ρ Ω

(3.3)

其中 k 為無因次化轉速。由(B.4)及(B.5)式可知旋轉傾斜梁相當於受到如圖

B2 所示的分布力。

圖 B1

P

A O

B

Ω

r α

x

dx q

₁

dx

dx q

₂

X

1

X

2

R

L

T

(a) (b)

B X

¹

X

2

O

q

2

L

T

)

1

( x q

x

B X

¹

X

2

O

圖 B2

由(B.5)式及圖 B2(a)可得旋轉傾斜梁在距離固定端 x 處的膜應變 (membrane strain) ε

₀

( x ) 可表示成

)]

2 ( cos 1 ) (

[

^{2 2}

2 2

0

R L x L x

L k AE

F

T T

T

x

= − + −

= α

ε (B.6)

由(B.6)式可知當 x = 0 時，旋轉傾斜梁有最大的膜應變 ε

)

= 2 ，由(B.5)式可得梁之任一斷面的最大撓曲應變(flexural strain)為

[18]

在文檔中 以有限元素法分析旋轉傾斜尤拉梁的穩態變形與自由振動 (頁 53-0)