未來展望

第五章數值例題

6.2 未來展望

對於旋轉傾斜 Timoshenko 梁的穩態變形與自由振動分析上仍有許多問題值得克服與深入探討，未來展望如下

1. 旋轉傾斜 Timoshenko 梁的固定端實際上可能不是剛接在旋轉圓柱上，

未來可以考慮支承的剛度，並探討其對自然頻率的影響，還有旋轉傾斜 Timoshenko 梁的自由端考慮附加質量，並探討其對自然頻率的影響。

2. 本文僅探討設定角為

0 °

及90 時的旋轉傾斜 Timoshenko 梁，當設定角° 為

0 °

時，本文在側向穩態變形與自然振動分析上僅提出理論之推導與可行的方法，未來的研究應可深入探討並考慮其他設定角的旋轉傾斜 Timoshenko 梁之穩態變形與自由振動分析。

3. 本文僅考慮旋轉傾斜 Timoshenko 梁的軸向振動、單一側向振動及二維的旋轉運動，這可能與三維的運動有所差異，未來應考慮三維的軸向振動、雙軸側向振動及扭轉運動的探討。

參考文獻

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國立交通大學機械工程研究所碩

士論文

, 臺灣, 新竹, 1997.

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303-322.

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國立交通大學機械工程研究所碩士論文

, 臺灣, 新竹, 1998.

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241–245.

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表一旋轉傾斜Timoshenko 梁在不同元素數目之自然頻率的收斂分析 (r =0,

α

= 0°,

β

= 0°,

η

=750, k =0.1 )

N 1 2 3 5 10

present [14] present, [14] present, [14] present, [14] present, [14]

K₁N

表二旋轉傾斜Timoshenko 梁在不同元素數目之自然頻率的收斂分析 (r =0,

α

= 0°,

β

= 90°,

η

=750, k =0.1 )

N 1 2 3 5 10

present [14] present, [14] present, [14] present, [14] present, [14]

K₁N

表三不同設定角的旋轉傾斜 Timoshenko 梁之自然頻率的收斂分析 (r =1, α= 0°,

η

=1000, k =0.06)

β

_i _K_i¹ _I¹ _K_i² _I² _K_i^N _I³ _I⁵ _I¹⁰ 0° 1 0.0690970 113 0.0734840 65 0.0734842 51 39 29

2 0.2170014 113 0.2171749 65 0.2171752 51 39 29 3 - - 0.3596817 65 0.3596818 51 39 29 4 - - 0.5173824 66 0.5173823 51 39 29 5 - - 0.6906797 66 0.6906801 51 39 29 6 - - 0.8775444 66 0.8775445 51 39 29 7 - - 1.0769951 66 1.0769950 52 39 29 8 - - 1.2888755 66 1.2888755 52 39 29 9 - - 1.5134188 66 1.5134190 52 40 29 10 - - 1.5742433 66 1.5742433 52 40 29 11 - - 1.7510263 66 1.7510265 52 40 29 12 - - 2.0021639 67 2.0021628 52 40 29 90° 1 0.0910332 113 0.0950297 65 0.0950297 51 39 29 2 0.2297731 113 0.2253830 65 0.2253830 51 39 29 3 0.3441493 113 0.3646995 65 0.3646995 51 39 29 4 0.5128507 112 0.5208861 66 0.5208861 51 39 29 5 0.5679179 112 0.6933106 66 0.6933105 51 39 29 6 0.8132527 111 0.8796176 66 0.8796175 51 39 29 7 1.0816823 111 1.0786836 66 1.0786856 52 39 29 8 1.3070388 111 1.2902893 66 1.2902893 52 39 29 9 1.5377446 111 1.5146235 66 1.5146239 52 40 29 10 1.5593525 110 1.5696500 66 1.5696500 52 40 29 11 - - 1.7520683 66 1.7520682 52 40 29 12 - - 2.0030740 67 2.0030744 52 40 29

K , K_i²：各別為分成1 段及 2 段時第 i 個無因次自然頻率， i = 1~12

K ：代表為分成N 段時第 i 個無因次自然頻率， N=3, 5, 10，i = 1~12 I ：代表為分成 N 段時無因次自然頻率之級數解收斂時所需的項數 N

–：代表無法用雙精度求得無因次自然頻率

表四不同傾斜角的旋轉傾斜 Timoshenko 梁之自然頻率的收斂分析 (r =1.5,

β

= 09 °,

η

=1000, k =0.06, α=0° ,15°)

α

_i _K_i¹ _I¹ _K_i² _I² _K_i^N _I³ _I⁵ _I¹⁰ 0° 1 0.1051062 127 0.1079671 72 0.1079676 56 42 30

2 0.2464092 127 0.2543244 72 0.2543256 56 42 30 3 0.3867831 126 0.4097877 72 0.4097877 56 42 30 4 0.5836241 126 0.5826134 72 0.5826133 56 42 30 5 - - 0.7724185 72 0.7724148 56 42 30 6 - - 0.9765286 72 0.9765289 56 42 30 7 - - 1.1934823 72 1.1934809 56 42 30 8 - - 1.4227884 72 1.4227886 56 42 30 9 - - 1.5696500 72 1.5696500 56 42 31 10 - - 1.6644639 72 1.6644644 56 42 31 11 - - - - 1.9187509 56 42 31 12 - - - - 2.1859970 56 43 31 15° 1 0.1059065 126 0.1067182 71 0.1067184 55 42 30 2 0.2566037 125 0.2515243 71 0.2515240 55 42 30 3 0.3869000 125 0.4054246 71 0.4054247 55 42 30 4 0.5266159 125 0.5766466 71 0.5766468 55 42 30 5 0.7582764 124 0.7647745 71 0.7647739 55 42 30 6 - - 0.9671686 71 0.9671686 55 42 30 7 - - 1.1823859 71 1.1823887 55 42 30 8 - - 1.4099755 71 1.4099755 56 42 30 9 - - 1.5696500 72 1.5696500 56 42 30 10 - - 1.6499585 72 1.6499584 56 42 30 11 - - 1.9025931 72 1.9025926 56 42 30 12 - - 2.1682353 72 2.1682369 56 42 31

K , K_i²：各別為分成1 段及 2 段時第 i 個無因次自然頻率， i = 1~12

K ：代表為分成N 段時第 i 個無因次自然頻率， N=3, 5, 10，i = 1~12 I ：代表為分成 N 段時無因次自然頻率之級數解收斂時所需的項數 N

–：代表無法用雙精度求得無因次自然頻率

表五不同傾斜角的旋轉傾斜 Timoshenko 梁之自然頻率的收斂分析 (r =1.5,

β

= 09 °,

η

=1000, k =0.06, α=30°, 45°)

α

_i _K_i¹ _I¹ _K_i² _I² _K_i^N _I³ _I⁵ _I¹⁰ 30° 1 0.1233033 122 0.1029669 69 0.1029670 54 41 30

2 0.2425943 122 0.2431188 69 0.2431193 54 41 30 3 0.4227925 121 0.3923341 69 0.3923343 54 41 30 4 0.5481693 121 0.5587371 69 0.5587369 54 41 30 5 0.7063993 120 0.7418315 69 0.7418313 54 41 30 6 0.9440046 120 0.9390626 69 0.9390623 54 41 30 7 1.1656773 119 1.1490859 70 1.1490867 54 41 30 8 1.3964557 119 1.3715190 70 1.3715185 54 41 30 9 - - 1.5696500 70 1.5696500 54 41 30 10 - - 1.6064388 70 1.6064388 54 41 30 11 - - 1.8541411 70 1.8541416 55 41 30 12 - - 2.1150136 70 2.1150152 55 42 30 45° 1 0.0920720 115 0.0966937 66 0.0966936 52 39 29 2 0.2276733 115 0.2290956 66 0.2290955 52 39 29 3 0.3644149 114 0.3704853 66 0.3704853 52 39 29 4 0.5567624 114 0.5288158 66 0.5288158 52 40 29 5 - - 0.7034794 66 0.7034794 52 40 29 6 - - 0.8920757 66 0.8920758 52 40 29 7 - - 1.0934354 67 1.0934357 52 40 29 8 - - 1.3072990 67 1.3072990 52 40 29 9 - - 1.5338367 67 1.5338366 52 40 29 10 - - 1.5696500 67 1.5696500 52 40 29 11 - - 1.7734087 67 1.7734092 53 40 29 12 - - 2.0264573 68 2.0264556 53 40 29

K , K_i²：各別為分成1 段及 2 段時第 i 個無因次自然頻率， i = 1~12

K ：代表為分成N 段時第 i 個無因次自然頻率， N=3, 5, 10，i = 1~12 I ：代表為分成 N 段時無因次自然頻率之級數解收斂時所需的項數 N

–：代表無法用雙精度求得無因次自然頻率

表六不同傾斜角的旋轉傾斜 Timoshenko 梁之自然頻率的收斂分析 (r =1.5,

β

= 09 °,

η

=1000, k =0.06, α=60°, 90°)

α

_i _K_i¹ _I¹ _K_i² _I² _K_i^N _I³ _I⁵ _I¹⁰ 60° 1 0.0640434 105 0.0878305 62 0.0878306 49 37 27

2 0.2531377 105 0.2093620 62 0.2093621 49 37 27 3 0.2997511 104 0.3397201 62 0.3397201 49 37 27 4 0.4805814 104 0.4866167 62 0.4866167 49 37 27 5 0.6453988 104 0.6493411 62 0.6493411 49 37 28 6 0.8111740 103 0.8257538 62 0.8257538 49 38 28 7 1.0185970 103 1.0149500 62 1.0149499 49 38 28 8 1.2145896 103 1.2168583 62 1.2168583 49 38 28 9 1.4251300 103 1.4317842 63 1.4317840 49 38 28 10 1.5608151 103 1.5696500 63 1.5696500 50 38 28 11 1.6529672 103 1.6601848 63 1.6601850 50 38 28 12 1.9273824 103 1.9025703 63 1.9025699 50 38 28 90° 1 0.0610550 77 0.0610732 48 0.0610732 38 30 23 2 0.1507153 76 0.1507115 48 0.1507115 38 30 23 3 0.2480073 74 0.2480078 47 0.2480078 38 30 23 4 0.3602173 74 0.3602107 47 0.3602107 38 30 23 5 0.4869539 71 0.4869413 48 0.4869413 38 30 23 6 0.6272713 69 0.6272303 48 0.6272303 38 30 23 7 0.7811530 70 0.7811381 48 0.7811381 39 31 23 8 0.9491835 71 0.9492362 49 0.9492362 39 31 23 9 1.1322287 72 1.1322365 50 1.1322366 40 31 24 10 1.3312623 72 1.3308321 50 1.3308321 41 32 24 11 1.5459719 74 1.5456371 51 1.5456371 41 32 25 12 1.5696500 74 1.5696500 51 1.5696500 41 32 25

K , K_i²：各別為分成1 段及 2 段時第 i 個無因次自然頻率， i = 1~12

K ：代表為分成N 段時第 i 個無因次自然頻率， N=3, 5, 10，i = 1~12 I ：代表為分成 N 段時無因次自然頻率之級數解收斂時所需的項數 N

–：代表無法用雙精度求得無因次自然頻率

表七旋轉傾斜Timoshenko 梁與 Euler 梁在不同細長比的自然頻率 (k =0,

α

= 0°,

β

= 90°, r =1)

η

_Beam K₁ K₂ K₃ K₄ K₅ K₆

10 T 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 T[13] 0.3231 1.4531 1.5708(a) 3.1671 4.7124(a) 4.8228 E[20] 0.34368 1.57080(a) 1.91364 4.64936 4.71239(a) 7.82131

20 T 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) T[13] 0.1718 0.9570 1.5708(a) 2.3376 3.9620 4.7124(a) E[20] 0.17479 1.05953 1.57080(a) 2.82431 4.71239(a) 5.19119

50 T 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 T[13] 0.0701 0.4296 1.1640 1.5708(a) 2.1836

E[20] 0.07026 0.43786 1.21530 1.57080(a) 2.35176 3.82644

100 T 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 E[20] 0.03515 0.21999 0.61460 1.20047 1.57080(a) 1.97618

500 T 0.00703 0.04406 0.12332 0.24152 0.39898 0.59549 E[20] 0.00703 0.04407 0.12338 0.24173 0.39954 0.59671

1000 T 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 E[20] 0.00352 0.02203 0.06169 0.12089 0.19984 0.29851 (a)：表示該振動頻率對應的振態為軸向振態

表八旋轉傾斜Timoshenko 梁與 Euler 梁在不同細長比的自然頻率 (k =0.06,

α

= 0°,

β

= 90°, r =1)

η

_Beam K₁ K₂ K₃ K₄ K₅ K₆

10 T 0.33687 1.46845 1.56965(a) 3.18802 4.71201(a) 4.84684 E[20] 0.35729 1.56965(a) 1.92445 4.65886 4.71201(a) 7.83009

20 T 0.19795 0.98262 1.56965(a) 2.36709 3.99763 4.71201(a) E[20] 0.20094 1.08307 1.56965(a) 2.84703 4.71201(a) 5.21357

50 T 0.12126 0.48764 1.22605 1.56965(a) 2.25114 3.50509 E[20] 0.12153 0.49536 1.27507 1.56965(a) 2.41411 3.88986

100 T 0.10434 0.31879 0.72079 1.29970 1.56965(a) 2.04434 E[20] 0.10442 0.31976 0.72703 1.32243 1.56965(a) 2.10361

500 T 0.09584 0.23058 0.38929 0.58079 0.80197 1.05173 E[20] 0.09585 0.23062 0.38939 0.58102 0.80246 1.05271

1000 T 0.09503 0.22538 0.36470 0.52089 0.69331 0.87962 E[20] 0.09504 0.22540 0.36473 0.52095 0.69342 0.87980 (a)：表示該振動頻率對應的振態為軸向振態

表九旋轉傾斜Timoshenko 梁與 Euler 梁在不同細長比的自然頻率 (k =0.06,

α

= 30°,

β

= 90°, r =1)

η

_Beam K₁ K₂ K₃ K₄ K₅ K₆

10 T 0.33582 1.46724 1.56965(a) 3.18634 4.71201(a) 4.84491 E[20] 0.35625 1.56965(a) 1.92360 4.65810 4.71201(a) 7.82939

20 T 0.19607 0.98066 1.56965(a) 2.36477 3.99480 4.71201(a) E[20] 0.19906 1.08110 1.56965(a) 2.84525 4.71201(a) 5.21180

50 T 0.11815 0.48345 1.22133 1.56965(a) 2.24590 3.49941 E[20] 0.11842 0.49121 1.27052 1.56965(a) 2.40927 3.88488

100 T 0.10077 0.31236 0.71272 1.29046 1.56965(a) 2.03432 E[20] 0.10085 0.31332 0.71900 1.31333 1.56965(a) 2.09387

500 T 0.09207 0.22223 0.37634 0.56310 0.77957 1.02492 E[20] 0.09208 0.22227 0.37643 0.56331 0.78005 1.02589

1000 T 0.09124 0.21695 0.35155 0.50286 0.67019 0.85129 E[20] 0.09125 0.21697 0.35159 0.50292 0.67029 0.85146 (a)：表示該振動頻率對應的振態為軸向振態

表十旋轉傾斜Timoshenko 梁在不同細長比下的自然頻率 (k =0)

η

K₁ K₂ K₃ K₄ K₅ K₆ K₇ K₈

7.8 0.39547 1.57080(a) 1.59016 3.36683 4.66650 4.71239(a) 5.83581 6.53366 7.9 0.39157 1.57080(a) 1.58409 3.35836 4.68507 4.71239(a) 5.84797 6.55088 8.0 0.38773 1.57080(a) 1.57798 3.34978 4.70252 4.71239(a) 5.86127 6.56765 8.05 0.38584 1.57080(a) 1.57492 3.34545 4.71082 4.71239(a) 5.86835 6.57587 8.1 0.38396 1.57080(a) 1.57185 3.34110 4.71239(a) 4.71882 5.87571 6.58396 8.15 0.38210 1.56877 1.57080(a) 3.33673 4.71239(a) 4.72653 5.88336 6.59194

8.2 0.38026 1.56568 1.57080(a) 3.33233 4.71239(a) 4.73394 5.89129 6.59980 8.3 0.37662 1.55950 1.57080(a) 3.32349 4.71239(a) 4.74786 5.90799 6.61516 8.4 0.37303 1.55330 1.57080(a) 3.31458 4.71239(a) 4.76057 5.92579 6.63006 8.5 0.36951 1.54707 1.57080(a) 3.30560 4.71239(a) 4.77207 5.94465 6.64451 10 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 6.28755 6.84520 20 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 5.71642 7.52062 30 0.11599 0.68631 1.57080(a) 1.77800 3.17597 4.71239(a) 4.77098 6.48495 38 0.09193 0.55521 1.47435 1.57080(a) 2.70220 4.15235 4.71239(a) 5.75344 39 0.08960 0.54211 1.44286 1.57080(a) 2.65114 4.08347 4.71239(a) 5.66974 40 0.08739 0.52959 1.41256 1.57080(a) 2.60170 4.01636 4.71239(a) 5.58773 50 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 4.71239(a) 4.85651 100 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 2.80509 3.83656 500 0.00703 0.04406 0.12332 0.24152 0.39898 0.59549 0.83088 1.10492 1000 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.41660 0.55449

表十一旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =0,

α

= 0°,

β

= 0°, k =0, 0.01)

η

K₁ K₂ K₃ K₄ K₅ K₆ K₇ K₈

0 10 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 6.28755 6.84520 20 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 5.71642 7.52062 50 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 4.71239(a) 4.85651 100 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 2.80509 3.83656 500 0.00703 0.04406 0.12332 0.24152 0.39898 0.59549 0.83088 1.10492 1000 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.41660 0.55449 0.01 10 0.32309 1.45317 1.57096(a) 3.16728 4.71241(a) 4.82309 6.28773 6.84539

20 0.17186 0.95720 1.57090(a) 2.33787 3.96235 4.71242(a) 5.71687 7.52114 50 0.07019 0.43018 1.16465 1.57089(a) 2.18436 3.43349 4.71242(a) 4.85738 100 0.03540 0.22015 0.60892 1.17711 1.57089(a) 1.91453 2.80667 3.83817 500 0.00815 0.04989 0.12995 0.24865 0.40639 0.60309 0.83861 1.11275 1000 0.00505 0.03212 0.07397 0.13448 0.21414 0.31320 0.43177 0.56988

(a)表示該振動頻率對應的振態為軸向振態

表十二旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =0,

α

= 0°,

β

= 0°, k 0.03, 0.06) =

η

K₁ K₂ K₃ K₄ K₅ K₆ K₇ K₈

0.03 10 0.32305 1.45374 1.57225(a) 3.16894 4.71255(a) 4.82523 6.28921 6.84692 20 0.17216 0.95913 1.57171(a) 2.34035 3.96542 4.71268(a) 5.72051 7.52528 50 0.07120 0.43509 1.17018 1.57166(a) 2.19042 3.43998 4.71268(a) 4.86429 100 0.03737 0.22981 0.61975 1.18874 1.57166(a) 1.92668 2.81921 3.85107 500 0.01191 0.08258 0.17323 0.29882 0.46079 0.66022 0.89759 1.17305 1000 0.00817 0.07250 0.13429 0.21075 0.30259 0.41066 0.53590 0.67900 0.06 10 0.32292 1.45571 1.57658(a) 3.17451 4.71306(a) 4.83241 6.29417 6.85208

20 0.17314 0.96563 1.57444(a) 2.34872 3.97576 4.71356(a) 5.73277 7.53922 50 0.07432 0.45126 1.18862 1.57426(a) 2.21073 3.46176 4.71354(a) 4.88755 100 0.04243 0.25975 0.65481 1.22706 1.57425(a) 1.96703 2.86108 3.89423 500 0.01629 0.14493 0.26846 0.42125 0.60461 0.82016 1.06961 1.35420 1000 0.01137 0.13821 0.24061 0.35516 0.48321 0.62434 0.77882 0.94733

(a)表示該振動頻率對應的振態為軸向振態

表十三旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =0.5,

α

= 0°,

β

= 0°, k =0, 0.01)

η

K₁ K₂ K₃ K₄ K₅ K₆ K₇ K₈

0 10 0.32309 1.45309 1.57080(a) 3.16707 4.71239 4.82282 6.28755 6.84520 20 0.17182 0.95696 1.57080(a) 2.33755 3.96200 4.71239 5.71642 7.52062 50 0.070061 0.42956 1.16396 1.57080(a) 2.18360 3.43268 4.71239 4.85651 100 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 2.80509 3.83656 500 0.00703 0.04406 0.12332 0.24152 0.39898 0.59550 0.83090 1.10495 1000 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.41660 0.55449 0.01 10 0.32320 1.45329 1.57096(a) 3.16746 4.71241 4.82329 6.28788 6.84554 20 0.17208 0.95741 1.57090(a) 2.33811 3.96265 4.71242 5.71722 7.52154 50 0.07074 0.43067 1.16517 1.57089(a) 2.18492 3.43410 4.71242 4.85802 100 0.03649 0.22112 0.60993 1.17818 1.57089(a) 1.91565 2.80783 3.83937 500 0.01197 0.05402 0.13461 0.25374 0.41172 0.60859 0.84424 1.11846 1000 0.01004 0.03812 0.08167 0.14348 0.22398 0.32359 0.44254 0.58093

(a)表示該振動頻率對應的振態為軸向振態

表十四旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =0.5,

α

= 0°,

β

= 0°, k 0.03, 0.06)=

η

K₁ K₂ K₃ K₄ K₅ K₆ K₇ K₈

0.03 10 0.32407 1.45488 1.57225(a) 3.17052 4.71255(a) 4.82705 6.29052 6.84826 20 0.17415 0.96101 1.57171(a) 2.34255 3.96810 4.71268(a) 5.72366 7.52888 50 0.07597 0.43946 1.17482 1.57166(a) 2.19547 3.44540 4.71268(a) 4.87009 100 0.04576 0.23807 0.62862 1.19828 1.57166(a) 1.93667 2.82958 3.86177 500 0.02852 0.10264 0.20110 0.33339 0.50029 0.70320 0.94307 1.22037 1000 0.02709 0.09409 0.16636 0.25374 0.35582 0.47284 0.60559 0.75489 0.06 10 0.32696 1.46016 1.57667(a) 3.18081 4.71310(a) 4.83963 6.29933 6.85743

20 0.18090 0.97305 1.57445(a) 2.35744 3.98641 4.71356(a) 5.74530 7.55354 50 0.09107 0.46783 1.20671 1.57427(a) 2.23061 3.48321 4.71355(a) 4.91053 100 0.06732 0.28771 0.68740 1.26328 1.57425(a) 2.00568 2.90155 3.93622 500 0.05405 0.18809 0.33258 0.50718 0.71103 0.94446 1.20894 1.50596 1000 0.05271 0.18226 0.30668 0.44520 0.59834 0.76463 0.94360 1.13544

(a)表示該振動頻率對應的振態為軸向振態

表十五旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =1,

α

= 0°,

β

= 0°, k =0, 0.01)

(a)表示該振動頻率對應的振態為軸向振態

k η _K₁ _K₂ K₃ K₄ K₅ K₆ K₇ K₈

0 10 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 6.28755 6.84520 20 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 5.71642 7.52062 50 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 4.71239(a) 4.85651 100 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 2.80509 3.83656 500 0.00703 0.04406 0.12332 0.24152 0.39898 0.59550 0.83090 1.10495 1000 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.41660 0.55449 0.01 10 0.32331 1.45342 1.57096(a) 3.16763 4.71241(a) 4.82349 6.28803 6.84569

20 0.17231 0.95762 1.57090(a) 2.33835 3.96295 4.71242(a) 5.71757 7.52194 50 0.07129 0.43116 1.16569 1.57089(a) 2.18549 3.43470 4.71242(a) 4.85867 100 0.03754 0.22209 0.61094 1.17926 1.57089(a) 1.91677 2.80899 3.84056 500 0.01482 0.05784 0.13909 0.25869 0.41698 0.61404 0.84982 1.12414 1000 0.01326 0.04323 0.08858 0.15182 0.23330 0.33358 0.45300 0.59173

表十六旋轉傾斜Timoshenko 梁在不同轉速與不同細長比的自然頻率 (r =1,

α

= 0°,

β

= 0°, k =0.03, 0.06)

(a)表示該振動頻率對應的振態為軸向振態

k η _K₁ _K₂ K₃ K₄ K₅ K₆ K₇ K₈

0.03 10 0.32508 1.45601 1.57226(a) 3.17210 4.71256(a) 4.82887 6.29182 6.84960 20 0.17613 0.96288 1.57171(a) 2.34474 3.97077 4.71268(a) 5.72680 7.53247 50 0.08044 0.44378 1.17944 1.57166(a) 2.20051 3.45081 4.71268(a) 4.87587 100 0.05282 0.24603 0.63735 1.20772 1.57166(a) 1.94661 2.83990 3.87243 500 0.03848 0.11912 0.22489 0.36387 0.53611 0.74305 0.98592 1.26548 1000 0.03735 0.11133 0.19238 0.28900 0.40019 0.52565 0.66593 0.82178 0.06 10 0.33094 1.46458 1.57678(a) 3.18709 4.71313(a) 4.84682 6.30444 6.86277

20 0.18832 0.98040 1.57446(a) 2.36612 3.99700 4.71356(a) 5.75778 7.56781 50 0.10514 0.48380 1.22446 1.57428(a) 2.25027 3.50449 4.71355(a) 4.93336 100 0.08514 0.31300 0.71822 1.29825 1.57426(a) 2.04341 2.94132 3.97765 500 0.07453 0.22256 0.38459 0.57765 0.79970 1.05000 1.32953 1.57424 1000 0.07348 0.21718 0.35968 0.51738 0.69068 0.87754 1.07700 1.28888

表十七旋轉傾斜Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =0.5,

β

= 90°,

η

=10)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 0.03 0˚ 0.32558 1.45581 1.57051(a) 3.17075 4.71229(a) 4.82704 15˚ 0.32555 1.45577 1.57051(a) 3.17070 4.71229(a) 4.82698 30˚ 0.32545 1.45566 1.57051(a) 3.17054 4.71229(a) 4.82679 45˚ 0.32529 1.45548 1.57051(a) 3.17029 4.71229(a) 4.82650 60˚ 0.32508 1.45524 1.57051(a) 3.16996 4.71229(a) 4.82613 75˚ 0.32483 1.45497 1.57051(a) 3.16958 4.71229(a) 4.82569 90˚ 0.32457 1.45467 1.57051(a) 3.16917 4.71229(a) 4.82521 0.06 0˚ 0.33293 1.46393 1.56965(a) 3.18174 4.71201(a) 4.83963 15˚ 0.33280 1.46377 1.56965(a) 3.18153 4.71201(a) 4.83938 30˚ 0.33240 1.46332 1.56965(a) 3.18090 4.71201(a) 4.83866 45˚ 0.33177 1.46260 1.56965(a) 3.17990 4.71201(a) 4.83751 60˚ 0.33095 1.46166 1.56965(a) 3.17860 4.71201(a) 4.83600 75˚ 0.32998 1.46056 1.56965(a) 3.17708 4.71201(a) 4.83425 90˚ 0.32895 1.45939 1.56965(a) 3.17544 4.71201(a) 4.83237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 6.28755 6.84520 7.85398(a) 8.12603 8.93990 10.29274 0.03 0˚ 6.29054 6.84831 7.85392(a) 8.12957 8.94530 10.29608 15˚ 6.29050 6.84826 7.85392(a) 8.12952 8.94522 10.29604 30˚ 6.29037 6.84813 7.85392(a) 8.12937 8.94499 10.29589 45˚ 6.29016 6.84791 7.85392(a) 8.12912 8.94462 10.29566 60˚ 6.28989 6.84764 7.85392(a) 8.12880 8.94414 10.29536 75˚ 6.28958 6.84731 7.85392(a) 8.12843 8.94358 10.29501 90˚ 6.28924 6.84697 7.85392(a) 8.12803 8.94299 10.29463 0.06 0˚ 6.29944 6.85762 7.85375(a) 8.14020 8.96140 10.30609

15˚ 6.29926 6.85744 7.85375(a) 8.13999 8.96109 10.30589 30˚ 6.29875 6.85690 7.85375(a) 8.13937 8.96017 10.30531 45˚ 6.29793 6.85605 7.85375(a) 8.13840 8.95872 10.30439 60˚ 6.29686 6.85494 7.85375(a) 8.13712 8.95682 10.30319 75˚ 6.29562 6.85365 7.85375(a) 8.13563 8.95460 10.30179 90˚ 6.29428 6.85227 7.85375(a) 8.13404 8.95222 10.30029

(a)：表示該自然頻率對應的振態為軸向振態

表十八旋轉傾斜Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =0.5,

β

= 90°,

η

=20)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 0.03 0˚ 0.17678 0.96157 1.57051(a) 2.34279 3.96826 4.71229(a) 15˚ 0.17671 0.96151 1.57051(a) 2.34272 3.96817 4.71229(a) 30˚ 0.17652 0.96132 1.57051(a) 2.34250 3.96790 4.71229(a) 45˚ 0.17621 0.96102 1.57051(a) 2.34215 3.96747 4.71229(a) 60˚ 0.17580 0.96063 1.57051(a) 2.34169 3.96692 4.71229(a) 75˚ 0.17532 0.96018 1.57051(a) 2.34116 3.96627 4.71229(a) 90˚ 0.17481 0.95970 1.57051(a) 2.34060 3.96558 4.71229(a) 0.06 0˚ 0.19086 0.97528 1.56965(a) 2.35841 3.98703 4.71201(a) 15˚ 0.19061 0.97503 1.56965(a) 2.35812 3.98667 4.71201(a) 30˚ 0.18988 0.97429 1.56965(a) 2.35725 3.98561 4.71201(a) 45˚ 0.18873 0.97312 1.56965(a) 2.35586 3.98392 4.71201(a) 60˚ 0.18720 0.97159 1.56965(a) 2.35406 3.98172 4.71201(a) 75˚ 0.18542 0.96980 1.56965(a) 2.35195 3.97915 4.71201(a) 90˚ 0.18348 0.96787 1.56965(a) 2.34969 3.97639 4.71201(a)

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 5.71642 7.52062 7.85398(a) 9.32700 10.99557 11.01027 0.03 0˚ 5.72377 7.52901 7.85392(a) 9.33625 10.99553 11.01889 15˚ 5.72366 7.52888 7.85392(a) 9.33612 10.99553 11.01876 30˚ 5.72335 7.52852 7.85392(a) 9.33572 10.99553 11.01839 45˚ 5.72285 7.52795 7.85392(a) 9.33509 10.99553 11.01780 60˚ 5.72219 7.52721 7.85392(a) 9.33427 10.99553 11.01704 75˚ 5.72143 7.52634 7.85392(a) 9.33331 10.99553 11.01615 90˚ 5.72062 7.52541 7.85392(a) 9.33228 10.99553 11.01519 0.06 0˚ 5.74573 7.55406 7.85375(a) 9.36388 10.99541 11.04448 15˚ 5.74531 7.55357 7.85375(a) 9.36334 10.99541 11.04398 30˚ 5.74406 7.55215 7.85375(a) 9.36177 10.99541 11.04253 45˚ 5.74207 7.54987 7.85375(a) 9.35926 10.99541 11.04021 60˚ 5.73947 7.54690 7.85375(a) 9.35598 10.99541 11.03719 75˚ 5.73645 7.54345 7.85375(a) 9.35217 10.99541 11.03366 90˚ 5.73320 7.53974 7.85375(a) 9.34808 10.99541 11.02987 (a)：表示該自然頻率對應的振態為軸向振態

表十九旋轉傾斜Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =0.5,

β

= 90°,

η

=50)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 0.03 0˚ 0.08170 0.44051 1.17523 1.57051(a) 2.19569 3.44555 15˚ 0.08156 0.44036 1.17507 1.57051(a) 2.19552 3.44537 30˚ 0.08113 0.43993 1.17461 1.57051(a) 2.19502 3.44483 45˚ 0.08043 0.43924 1.17388 1.57051(a) 2.19421 3.44397 60˚ 0.07953 0.43833 1.17292 1.57051(a) 2.19317 3.44284 75˚ 0.07845 0.43728 1.17180 1.57051(a) 2.19195 3.44154 90˚ 0.07729 0.43615 1.17059 1.57051(a) 2.19064 3.44013 0.06 0˚ 0.10921 0.47179 1.20831 1.56965(a) 2.23149 3.48382

15˚ 0.10878 0.47124 1.20770 1.56965(a) 2.23082 3.48309 30˚ 0.10749 0.46963 1.20591 1.56965(a) 2.22884 3.48096 45˚ 0.10541 0.46704 1.20306 1.56965(a) 2.22569 3.47756 60˚ 0.10264 0.46365 1.19932 1.56965(a) 2.22158 3.47312 75˚ 0.09931 0.45966 1.19495 1.56965(a) 2.21678 3.46794 90˚ 0.09561 0.45535 1.19025 1.56965(a) 2.21161 3.46237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 4.71239(a) 4.85651 6.41102 7.85398(a) 8.06214 9.78423 0.03 0˚ 4.71229(a) 4.87020 6.42556 7.85392(a) 8.07759 9.80064 15˚ 4.71229(a) 4.87000 6.42535 7.85392(a) 8.07737 9.80040 30˚ 4.71229(a) 4.86942 6.42473 7.85392(a) 8.07671 9.79970 45˚ 4.71229(a) 4.86850 6.42375 7.85392(a) 8.07567 9.79859 60˚ 4.71229(a) 4.86730 6.42247 7.85392(a) 8.07430 9.79714 75˚ 4.71229(a) 4.86590 6.42098 7.85392(a) 8.07271 9.79545 90˚ 4.71229(a) 4.86440 6.41938 7.85392(a) 8.07101 9.79364 0.06 0˚ 4.71201(a) 4.91096 6.46894 7.85375(a) 8.12371 9.84963

15˚ 4.71201(a) 4.91018 6.46810 7.85375(a) 8.12282 9.84868 30˚ 4.71201(a) 4.90789 6.46566 7.85375(a) 8.12021 9.84591 45˚ 4.71201(a) 4.90425 6.46176 7.85375(a) 8.11606 9.84148 60˚ 4.71201(a) 4.89949 6.45668 7.85375(a) 8.11064 9.83571 75˚ 4.71201(a) 4.89394 6.45075 7.85375(a) 8.10432 9.82899 90˚ 4.71201(a) 4.88798 6.44439 7.85375(a) 8.09753 9.82177 (a)：表示該自然頻率對應的振態為軸向振態

表二十旋轉傾斜Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =0.5,

β

= 90°,

η

=100)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 0.03 0˚ 0.05474 0.23997 0.62935 1.19866 1.57051(a) 1.93691 15˚ 0.05452 0.23969 0.62905 1.19834 1.57051(a) 1.93658 30˚ 0.05388 0.23889 0.62817 1.19739 1.57051(a) 1.93558 45˚ 0.05284 0.23760 0.62677 1.19588 1.57051(a) 1.93399 60˚ 0.05145 0.23591 0.62494 1.19391 1.57051(a) 1.93192 75˚ 0.04978 0.23392 0.62280 1.19161 1.57051(a) 1.92951 90˚ 0.04793 0.23178 0.62049 1.18913 1.57051(a) 1.92692 0.06 0˚ 0.09030 0.29399 0.69008 1.26477 1.56965(a) 2.00662 15˚ 0.08978 0.29310 0.68900 1.26356 1.56965(a) 2.00532 30˚ 0.08824 0.29049 0.68583 1.25999 1.56965(a) 2.00150 45˚ 0.08574 0.28627 0.68076 1.25430 1.56965(a) 1.99540 60˚ 0.08236 0.28068 0.67408 1.24684 1.56965(a) 1.98742 75˚ 0.07823 0.27401 0.66619 1.23807 1.56965(a) 1.97807 90˚ 0.07353 0.26665 0.65760 1.22858 1.56965(a) 1.96798

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 2.80509 3.83656 4.71239(a) 4.99194 6.25634 7.61587 0.03 0˚ 2.82975 3.86190 4.71229(a) 5.01794 6.28301 7.64322 15˚ 2.82940 3.86153 4.71229(a) 5.01757 6.28262 7.64282 30˚ 2.82836 3.86047 4.71229(a) 5.01647 6.28149 7.64166 45˚ 2.82672 3.85877 4.71229(a) 5.01472 6.27970 7.63982 60˚ 2.82457 3.85655 4.71229(a) 5.01244 6.27736 7.63741 75˚ 2.82207 3.85397 4.71229(a) 5.00979 6.27463 7.63461 90˚ 2.81938 3.85120 4.71229(a) 5.00693 6.27170 7.63159 0.06 0˚ 2.90222 3.93673 4.71201(a) 5.09497 6.36218 7.72455

15˚ 2.90085 3.93531 4.71201(a) 5.09350 6.36067 7.72299 30˚ 2.89684 3.93114 4.71201(a) 5.08919 6.35622 7.71841 45˚ 2.89044 3.92449 4.71201(a) 5.08232 6.34913 7.71112 60˚ 2.88207 3.91581 4.71201(a) 5.07334 6.33989 7.70160 75˚ 2.87229 3.90566 4.71201(a) 5.06287 6.32910 7.69049 90˚ 2.86175 3.89474 4.71201(a) 5.05160 6.31749 7.67855 (a)：表示該自然頻率對應的振態為軸向振態

表二十一旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =0.5,

β

= 90°,

η

=1000)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.03 0˚ 0.04043 0.09876 0.16905 0.25551 0.35709 0.47379 15˚ 0.04015 0.09815 0.16809 0.25421 0.35545 0.47186 30˚ 0.03932 0.09631 0.16525 0.25034 0.35060 0.46612 45˚ 0.03795 0.09332 0.16061 0.24402 0.34268 0.45678 60˚ 0.03608 0.08925 0.15430 0.23545 0.33199 0.44423 75˚ 0.03377 0.08424 0.14654 0.22493 0.31893 0.42897 90˚ 0.03109 0.07846 0.13760 0.21288 0.30407 0.41176 0.06 0˚ 0.07996 0.19194 0.31254 0.44925 0.60136 0.76700 15˚ 0.07940 0.19070 0.31059 0.44657 0.59792 0.76278 30˚ 0.07771 0.18699 0.30479 0.43859 0.58767 0.75024 45˚ 0.07495 0.18092 0.29531 0.42553 0.57088 0.72971 60˚ 0.07119 0.17267 0.28240 0.40774 0.54802 0.70175 75˚ 0.06652 0.16248 0.26646 0.38573 0.51973 0.66720 90˚ 0.06107 0.15071 0.24801 0.36021 0.48694 0.62723

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 0.41660 0.55449 0.71197 0.88902 1.08559 1.30162 0.03 0˚ 0.60633 0.75549 0.92192 1.10613 1.30851 1.52930 15˚ 0.60413 0.75307 0.91931 1.10337 1.30562 1.52630 30˚ 0.59763 0.74591 0.91162 1.09523 1.29709 1.51747 45˚ 0.58708 0.73435 0.89920 1.08211 1.28339 1.50327 60˚ 0.57294 0.71890 0.88267 1.06469 1.26524 1.48451 75˚ 0.55587 0.70033 0.86290 1.04393 1.24367 1.46228 90˚ 0.53674 0.67967 0.84100 1.02104 1.21998 1.43794 0.06 0˚ 0.94552 1.13704 1.34197 1.56087 1.56965(a) 1.79429 15˚ 0.94055 1.13132 1.33555 1.55378 1.56965(a) 1.78658 30˚ 0.92574 1.11433 1.31648 1.53275 1.56965(a) 1.76372 45˚ 0.90152 1.08657 1.28535 1.49846 1.56965(a) 1.72650 60˚ 0.86859 1.04888 1.24317 1.45210 1.56965(a) 1.67627 75˚ 0.82798 1.00251 1.19143 1.39541 1.56965(a) 1.61506 90˚ 0.78114 0.94924 1.13224 1.33083 1.54564 1.56965(a) (a)：表示該自然頻率對應的振態為軸向振態

表二十二旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1,

β

= 90°,

η

=10)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 0.03 0˚ 0.32659 1.45695 1.57051(a) 3.17233 4.71229(a) 4.82886 15˚ 0.32653 1.45687 1.57051(a) 3.17223 4.71229(a) 4.82874 30˚ 0.32632 1.45665 1.57051(a) 3.17191 4.71229(a) 4.82837 45˚ 0.32600 1.45628 1.57051(a) 3.17141 4.71229(a) 4.82779 60˚ 0.32558 1.45581 1.57051(a) 3.17075 4.71229(a) 4.82704 75˚ 0.32509 1.45526 1.57051(a) 3.16999 4.71229(a) 4.82616 90˚ 0.32457 1.45467 1.57051(a) 3.16917 4.71229(a) 4.82521 0.06 0˚ 0.33687 1.46845 1.56965(a) 3.18802 4.71201(a) 4.84684 15˚ 0.33660 1.46814 1.56965(a) 3.18760 4.71201(a) 4.84635 30˚ 0.33582 1.46724 1.56965(a) 3.18634 4.71201(a) 4.84491 45˚ 0.33457 1.46580 1.56965(a) 3.18435 4.71201(a) 4.84262 60˚ 0.33293 1.46393 1.56965(a) 3.18174 4.71201(a) 4.83963 75˚ 0.33102 1.46174 1.56965(a) 3.17871 4.71201(a) 4.83613 90˚ 0.32895 1.45939 1.56965(a) 3.17544 4.71201(a) 4.83237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 6.28755 6.84520 7.85398(a) 8.12603 8.93990 10.29274 0.03 0˚ 6.29185 6.84964 7.85392(a) 8.13112 8.94760 10.29754 15˚ 6.29176 6.84955 7.85392(a) 8.13101 8.94745 10.29744 30˚ 6.29150 6.84929 7.85392(a) 8.13070 8.94698 10.29715 45˚ 6.29108 6.84886 7.85392(a) 8.13021 8.94625 10.29669 60˚ 6.29054 6.84831 7.85392(a) 8.12957 8.94530 10.29608 75˚ 6.28992 6.84766 7.85392(a) 8.12883 8.94418 10.29538 90˚ 6.28924 6.84697 7.85392(a) 8.12803 8.94299 10.29463 0.06 0˚ 6.30455 6.86296 7.85375(a) 8.14635 8.97053 10.31187 15˚ 6.30420 6.86260 7.85375(a) 8.14593 8.96991 10.31148 30˚ 6.30319 6.86153 7.85375(a) 8.14470 8.96809 10.31032 45˚ 6.30156 6.85983 7.85375(a) 8.14275 8.96519 10.30848 60˚ 6.29944 6.85762 7.85375(a) 8.14020 8.96140 10.30609 75˚ 6.29695 6.85504 7.85375(a) 8.13723 8.95698 10.30329 90˚ 6.29428 6.85227 7.85375(a) 8.13404 8.95222 10.30029 (a)：表示該自然頻率對應的振態為軸向振態

表二十三旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1,

β

= 90°,

η

=20)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 0.03 0˚ 0.17873 0.96344 1.57051(a) 2.34498 3.97093 4.71229(a) 15˚ 0.17859 0.96331 1.57051(a) 2.34484 3.97075 4.71229(a) 30˚ 0.17821 0.96294 1.57051(a) 2.34440 3.97021 4.71229(a) 45˚ 0.17759 0.96235 1.57051(a) 2.34370 3.96936 4.71229(a) 60˚ 0.17678 0.96157 1.57051(a) 2.34279 3.96826 4.71229(a) 75˚ 0.17583 0.96067 1.57051(a) 2.34173 3.96697 4.71229(a) 90˚ 0.17481 0.95970 1.57051(a) 2.34060 3.96558 4.71229(a) 0.06 0˚ 0.19795 0.98262 1.56965(a) 2.36709 3.99763 4.71201(a) 15˚ 0.19747 0.98212 1.56965(a) 2.36650 3.99691 4.71201(a) 30˚ 0.19607 0.98066 1.56965(a) 2.36477 3.99480 4.71201(a) 45˚ 0.19383 0.97833 1.56965(a) 2.36202 3.99143 4.71201(a) 60˚ 0.19086 0.97528 1.56965(a) 2.35841 3.98703 4.71201(a) 75˚ 0.18733 0.97172 1.56965(a) 2.35421 3.98190 4.71201(a) 90˚ 0.18348 0.96787 1.56965(a) 2.34969 3.97639 4.71201(a)

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 5.71642 7.52062 7.85398(a) 9.32700 10.99557(a) 11.01027 0.03 0˚ 5.72691 7.53260 7.85392(a) 9.34022 10.99553(a) 11.02257 15˚ 5.72670 7.53236 7.85392(a) 9.33995 10.99553(a) 11.02232 30˚ 5.72607 7.53164 7.85392(a) 9.33915 10.99553(a) 11.02159 45˚ 5.72507 7.53050 7.85392(a) 9.33789 10.99553(a) 11.02041 60˚ 5.72377 7.52901 7.85392(a) 9.33625 10.99553(a) 11.01889 75˚ 5.72225 7.52727 7.85392(a) 9.33434 10.99553(a) 11.01710 90˚ 5.72062 7.52541 7.85392(a) 9.33228 10.99553(a) 11.01519 0.06 0˚ 5.75822 7.56834 7.85375(a) 9.37962 10.99541(a) 11.05896 15˚ 5.75737 7.56737 7.85375(a) 9.37855 10.99541(a) 11.05798 30˚ 5.75488 7.56452 7.85375(a) 9.37541 10.99541(a) 11.05509 45˚ 5.75091 7.55998 7.85375(a) 9.37041 10.99541(a) 11.05049 60˚ 5.74573 7.55406 7.85375(a) 9.36388 10.99541(a) 11.04448 75˚ 5.73970 7.54716 7.85375(a) 9.35626 10.99541(a) 11.03745 90˚ 5.73320 7.53974 7.85375(a) 9.34808 10.99541(a) 11.02987 (a)：表示該自然頻率對應的振態為軸向振態

表二十四旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1,

β

= 90°,

η

=50)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 0.03 0˚ 0.08589 0.44482 1.17985 1.57051(a) 2.20073 3.45096 15˚ 0.08561 0.44453 1.17953 1.57051(a) 2.20039 3.45060 30˚ 0.08479 0.44367 1.17861 1.57051(a) 2.19938 3.44952 45˚ 0.08346 0.44230 1.17715 1.57051(a) 2.19778 3.44780 60˚ 0.08170 0.44051 1.17523 1.57051(a) 2.19569 3.44555 75˚ 0.07960 0.43841 1.17300 1.57051(a) 2.19326 3.44294 90˚ 0.07729 0.43615 1.17059 1.57051(a) 2.19064 3.44013 0.06 0˚ 0.12126 0.48764 1.22605 1.56965(a) 2.25114 3.50509

15˚ 0.12048 0.48658 1.22485 1.56965(a) 2.24981 3.50365 30˚ 0.11815 0.48345 1.22133 1.56965(a) 2.24590 3.49941 45˚ 0.11436 0.47843 1.21570 1.56965(a) 2.23966 3.49265 60˚ 0.10921 0.47179 1.20831 1.56965(a) 2.23149 3.48382 75˚ 0.10288 0.46394 1.19964 1.56965(a) 2.22193 3.47350 90˚ 0.09561 0.45535 1.19025 1.56965(a) 2.21161 3.46237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 4.71239(a) 4.85651 6.41102 7.85398(a) 8.06214 9.78423 0.03 0˚ 4.71229(a) 4.87598 6.43174 7.85392(a) 8.08417 9.80764 15˚ 4.71229(a) 4.87559 6.43131 7.85392(a) 8.08372 9.80716 30˚ 4.71229(a) 4.87443 6.43008 7.85392(a) 8.08241 9.80576 45˚ 4.71229(a) 4.87259 6.42812 7.85392(a) 8.08032 9.80354 60˚ 4.71229(a) 4.87020 6.42556 7.85392(a) 8.07759 9.80064 75˚ 4.71229(a) 4.86740 6.42258 7.85392(a) 8.07442 9.79726 90˚ 4.71229(a) 4.86440 6.41938 7.85392(a) 8.07101 9.79364 0.06 0˚ 4.71201(a) 4.93380 6.49336 7.85375(a) 8.14977 9.87738

15˚ 4.71201(a) 4.93225 6.49170 7.85375(a) 8.14800 9.87549 30˚ 4.71201(a) 4.92769 6.48683 7.85375(a) 8.14280 9.86996 45˚ 4.71201(a) 4.92044 6.47907 7.85375(a) 8.13452 9.86114 60˚ 4.71201(a) 4.91096 6.46894 7.85375(a) 8.12371 9.84963 75˚ 4.71201(a) 4.89989 6.45711 7.85375(a) 8.11110 9.83621 90˚ 4.71201(a) 4.88798 6.44439 7.85375(a) 8.09753 9.82177 (a)：表示該自然頻率對應的振態為軸向振態

表二十五旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1,

β

= 90°,

η

=100)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 0.03 0˚ 0.06077 0.24787 0.63807 1.20810 1.57051(a) 1.94685 15˚ 0.06038 0.24734 0.63748 1.20746 1.57051(a) 1.94617 30˚ 0.05922 0.24578 0.63575 1.20558 1.57051(a) 1.94419 45˚ 0.05732 0.24327 0.63298 1.20259 1.57051(a) 1.94104 60˚ 0.05474 0.23997 0.62935 1.19866 1.57051(a) 1.93691 75˚ 0.05157 0.23605 0.62509 1.19408 1.57051(a) 1.93210 90˚ 0.04793 0.23178 0.62049 1.18913 1.57051(a) 1.92692 0.06 0˚ 0.10434 0.31879 0.72079 1.29970 1.56965(a) 2.04434 15˚ 0.10344 0.31717 0.71875 1.29736 1.56965(a) 2.04179 30˚ 0.10077 0.31236 0.71272 1.29046 1.56965(a) 2.03432 45˚ 0.09637 0.30453 0.70300 1.27938 1.56965(a) 2.02235 60˚ 0.09030 0.29399 0.69008 1.26477 1.56965(a) 2.00662 75˚ 0.08265 0.28116 0.67465 1.24748 1.56965(a) 1.98810 90˚ 0.07353 0.26665 0.65760 1.22858 1.56965(a) 1.96798

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 2.80509 3.83656 4.71239(a) 4.99194 6.25634 7.61587 0.03 0˚ 2.84007 3.87255 4.71229(a) 5.02891 6.29429 7.65482 15˚ 2.83937 3.87183 4.71229(a) 5.02817 6.29353 7.65403 30˚ 2.83731 3.86970 4.71229(a) 5.02598 6.29127 7.65171 45˚ 2.83403 3.86631 4.71229(a) 5.02249 6.28769 7.64803 60˚ 2.82975 3.86190 4.71229(a) 5.01794 6.28301 7.64322 75˚ 2.82475 3.85674 4.71229(a) 5.01264 6.27756 7.63761 90˚ 2.81938 3.85120 4.71229(a) 5.00693 6.27170 7.63159 0.06 0˚ 2.94197 3.97815 4.71201(a) 5.13787 6.40647 7.77020

15˚ 2.93929 3.97535 4.71201(a) 5.13496 6.40346 7.76710 30˚ 2.93139 3.96711 4.71201(a) 5.12642 6.39464 7.75800 45˚ 2.91877 3.95395 4.71201(a) 5.11279 6.38057 7.74350 60˚ 2.90222 3.93673 4.71201(a) 5.09497 6.36218 7.72455 75˚ 2.88279 3.91655 4.71201(a) 5.07411 6.34068 7.70241 90˚ 2.86175 3.89474 4.71201(a) 5.05160 6.31749 7.67855 (a)：表示該自然頻率對應的振態為軸向振態

表二十六旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1,

β

= 90°,

η

=1000)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.03 0˚ 0.04793 0.11531 0.19471 0.29055 0.40132 0.52651 15˚ 0.04745 0.11427 0.19309 0.28833 0.39851 0.52314 30˚ 0.04604 0.11113 0.18823 0.28170 0.39011 0.51309 45˚ 0.04370 0.10595 0.18020 0.27072 0.37624 0.49654 60˚ 0.04043 0.09876 0.16905 0.25551 0.35709 0.47379 75˚ 0.03625 0.08960 0.15485 0.23619 0.33292 0.44531 90˚ 0.03109 0.07846 0.13760 0.21288 0.30407 0.41176 0.06 0˚ 0.09503 0.22538 0.36470 0.52089 0.69331 0.87962 15˚ 0.09408 0.22327 0.36141 0.51637 0.68752 0.87252 30˚ 0.09124 0.21695 0.35155 0.50286 0.67019 0.85129 45˚ 0.08653 0.20648 0.33523 0.48045 0.64142 0.81606 60˚ 0.07996 0.19194 0.31254 0.44925 0.60136 0.76700 75˚ 0.07152 0.17339 0.28353 0.40929 0.55001 0.70418 90˚ 0.06107 0.15071 0.24801 0.36021 0.48694 0.62723

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 0.41660 0.55449 0.71197 0.88902 1.08559 1.30162 0.03 0˚ 0.66661 0.82233 0.99440 1.18342 1.38988 1.57051(a)

15˚ 0.66273 0.81801 0.98969 1.17837 1.38453 1.57051(a) 30˚ 0.65118 0.80515 0.97568 1.16338 1.36870 1.57051(a) 45˚ 0.63223 0.78410 0.95283 1.13899 1.34300 1.56517 60˚ 0.60633 0.75549 0.92192 1.10613 1.30851 1.52930 75˚ 0.57417 0.72023 0.88409 1.06619 1.26680 1.48612 90˚ 0.53674 0.67967 0.84100 1.02104 1.21998 1.43794 0.06 0˚ 1.07869 1.29029 1.51462 1.56965(a) 1.75207 2.00307

15˚ 1.07029 1.28060 1.50369 1.56965(a) 1.73993 1.98978 30˚ 1.04515 1.25164 1.47100 1.56965(a) 1.70366 1.95009 45˚ 1.00348 1.20366 1.41692 1.56965(a) 1.64372 1.88459 60˚ 0.94552 1.13704 1.34197 1.56087 1.56965(a) 1.79429 75˚ 0.87146 1.05216 1.24684 1.45612 1.56965(a) 1.68063 90˚ 0.78114 0.94924 1.13224 1.33083 1.54564 1.56965(a) (a)：表示該自然頻率對應的振態為軸向振態

表二十七旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1.5,

β

= 90°,

η

=10)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.32309 1.45309 1.57080(a) 3.16707 4.71239(a) 4.82282 0.03 0˚ 0.32760 1.45809 1.57051(a) 3.17391 4.71229(a) 4.83068 15˚ 0.32750 1.45797 1.57051(a) 3.17375 4.71229(a) 4.83049 30˚ 0.32720 1.45763 1.57051(a) 3.17328 4.71229(a) 4.82995 45˚ 0.32672 1.45709 1.57051(a) 3.17253 4.71229(a) 4.82908 60˚ 0.32609 1.45638 1.57051(a) 3.17154 4.71229(a) 4.82795 75˚ 0.32536 1.45556 1.57051(a) 3.17040 4.71229(a) 4.82663 90˚ 0.32457 1.45467 1.57051(a) 3.16917 4.71229(a) 4.82521 0.06 0˚ 0.34076 1.47295 1.56965(a) 3.19428 4.71201(a) 4.85403 15˚ 0.34037 1.47249 1.56965(a) 3.19364 4.71201(a) 4.85329 30˚ 0.33920 1.47115 1.56965(a) 3.19177 4.71201(a) 4.85114 45˚ 0.33735 1.46900 1.56965(a) 3.18878 4.71201(a) 4.84772 60˚ 0.33491 1.46619 1.56965(a) 3.18489 4.71201(a) 4.84324 75˚ 0.33205 1.46291 1.56965(a) 3.18034 4.71201(a) 4.83801 90˚ 0.32895 1.45939 1.56965(a) 3.17544 4.71201(a) 4.83237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 6.28755 6.84520 7.85398(a) 8.12603 8.93990 10.29274 0.03 0˚ 6.29315 6.85098 7.85392(a) 8.13266 8.94991 10.29899 15˚ 6.29301 6.85085 7.85392(a) 8.13250 8.94967 10.29884 30˚ 6.29262 6.85045 7.85392(a) 8.13204 8.94898 10.29841 45˚ 6.29200 6.84981 7.85392(a) 8.13130 8.94788 10.29771 60˚ 6.29120 6.84898 7.85392(a) 8.13035 8.94645 10.29681 75˚ 6.29025 6.84801 7.85392(a) 8.12923 8.94478 10.29576 90˚ 6.28924 6.84697 7.85392(a) 8.12803 8.94299 10.29463 0.06 0˚ 6.30962 6.86829 7.85375(a) 8.15249 8.97961 10.31765 15˚ 6.30910 6.86775 7.85375(a) 8.15186 8.97869 10.31706 30˚ 6.30759 6.86615 7.85375(a) 8.15002 8.97597 10.31533 45˚ 6.30517 6.86361 7.85375(a) 8.14709 8.97164 10.31258 60˚ 6.30200 6.86029 7.85375(a) 8.14327 8.96597 10.30898 75˚ 6.29829 6.85642 7.85375(a) 8.13882 8.95935 10.30479 90˚ 6.29428 6.85227 7.85375(a) 8.13404 8.95222 10.30029 (a)：表示該自然頻率對應的振態為軸向振態

表二十八旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1.5,

β

= 90°,

η

=20)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.17182 0.95696 1.57080(a) 2.33755 3.96197 4.71239(a) 0.03 0˚ 0.18065 0.96531 1.57051(a) 2.34718 3.97360 4.71229(a) 15˚ 0.18045 0.96512 1.57051(a) 2.34695 3.97332 4.71229(a) 30˚ 0.17988 0.96456 1.57051(a) 2.34630 3.97253 4.71229(a) 45˚ 0.17896 0.96367 1.57051(a) 2.34525 3.97125 4.71229(a) 60˚ 0.17776 0.96251 1.57051(a) 2.34389 3.96959 4.71229(a) 75˚ 0.17634 0.96115 1.57051(a) 2.34230 3.96766 4.71229(a) 90˚ 0.17481 0.95970 1.57051(a) 2.34060 3.96558 4.71229(a) 0.06 0˚ 0.20479 0.98990 1.56965(a) 2.37573 4.00819 4.71201(a) 15˚ 0.20410 0.98916 1.56965(a) 2.37485 4.00711 4.71201(a) 30˚ 0.20207 0.98698 1.56965(a) 2.37227 4.00395 4.71201(a) 45˚ 0.19879 0.98351 1.56965(a) 2.36814 3.99891 4.71201(a) 60˚ 0.19443 0.97896 1.56965(a) 2.36276 3.99234 4.71201(a) 75˚ 0.18923 0.97363 1.56965(a) 2.35647 3.98466 4.71201(a) 90˚ 0.18348 0.96787 1.56965(a) 2.34969 3.97639 4.71201(a)

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 5.71642 7.52062 7.85398(a) 9.32700 10.99557(a) 11.01027 0.03 0˚ 5.73005 7.53619 7.85392(a) 9.34418 10.99553(a) 11.02625 15˚ 5.72973 7.53583 7.85392(a) 9.34378 10.99553(a) 11.02588 30˚ 5.72879 7.53475 7.85392(a) 9.34259 10.99553(a) 11.02478 45˚ 5.72729 7.53304 7.85392(a) 9.34070 10.99553(a) 11.02302 60˚ 5.72534 7.53080 7.85392(a) 9.33823 10.99553(a) 11.02073 75˚ 5.72306 7.52820 7.85392(a) 9.33536 10.99553(a) 11.01806 90˚ 5.72062 7.52541 7.85392(a) 9.33228 10.99553(a) 11.01519 0.06 0˚ 5.77066 7.58256 7.85375(a) 9.39531 10.99541(a) 11.07331 15˚ 5.76939 7.58111 7.85375(a) 9.39371 10.99541(a) 11.07185 30˚ 5.76566 7.57685 7.85375(a) 9.38901 10.99541(a) 11.06756 45˚ 5.75973 7.57006 7.85375(a) 9.38153 10.99541(a) 11.06071 60˚ 5.75198 7.56121 7.85375(a) 9.37176 10.99541(a) 11.05174 75˚ 5.74294 7.55086 7.85375(a) 9.36035 10.99541(a) 11.04122 90˚ 5.73320 7.53974 7.85375(a) 9.34808 10.99541(a) 11.02987 (a)：表示該自然頻率對應的振態為軸向振態

表二十九旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1.5,

β

= 90°,

η

=50)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.07006 0.42956 1.16396 1.57080(a) 2.18360 3.43268 0.03 0˚ 0.08987 0.44909 1.18444 1.57051(a) 2.20575 3.45636 15˚ 0.08947 0.44866 1.18397 1.57051(a) 2.20524 3.45581 30˚ 0.08829 0.44738 1.18260 1.57051(a) 2.20373 3.45419 45˚ 0.08638 0.44534 1.18041 1.57051(a) 2.20134 3.45162 60˚ 0.08382 0.44267 1.17754 1.57051(a) 2.19821 3.44826 75˚ 0.08074 0.43954 1.17420 1.57051(a) 2.19456 3.44434 90˚ 0.07729 0.43615 1.17059 1.57051(a) 2.19064 3.44013 0.06 0˚ 0.13218 0.50295 1.24348 1.56965(a) 2.27058 3.52620

15˚ 0.13111 0.50141 1.24171 1.56965(a) 2.26860 3.52405 30˚ 0.12791 0.49686 1.23651 1.56965(a) 2.26279 3.51774 45˚ 0.12264 0.48953 1.22818 1.56965(a) 2.25351 3.50766 60˚ 0.11540 0.47979 1.21722 1.56965(a) 2.24134 3.49448 75˚ 0.10633 0.46817 1.20430 1.56965(a) 2.22707 3.47904 90˚ 0.09561 0.45535 1.19025 1.56965(a) 2.21161 3.46237

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 4.71239(a) 4.85651 6.41102 7.85398(a) 8.06214 9.78423 0.03 0˚ 4.71229(a) 4.88176 6.43790 7.85392(a) 8.09074 9.81463 15˚ 4.71229(a) 4.88117 6.43727 7.85392(a) 8.09007 9.81391 30˚ 4.71229(a) 4.87944 6.43542 7.85392(a) 8.08810 9.81182 45˚ 4.71229(a) 4.87668 6.43248 7.85392(a) 8.08497 9.80849 60˚ 4.71229(a) 4.87309 6.42865 7.85392(a) 8.08088 9.80414 75˚ 4.71229(a) 4.86890 6.42418 7.85392(a) 8.07612 9.79908 90˚ 4.71229(a) 4.86440 6.41938 7.85392(a) 8.07101 9.79364 0.06 0˚ 4.71201(a) 4.95649 6.51765 7.85375(a) 8.17570 9.90501

15˚ 4.71201(a) 4.95418 6.51517 7.85375(a) 8.17306 9.90219 30˚ 4.71201(a) 4.94739 6.50790 7.85375(a) 8.16529 9.89392 45˚ 4.71201(a) 4.93656 6.49631 7.85375(a) 8.15292 9.88074 60˚ 4.71201(a) 4.92240 6.48116 7.85375(a) 8.13676 9.86352 75˚ 4.71201(a) 4.90584 6.46346 7.85375(a) 8.11787 9.84341 90˚ 4.71201(a) 4.88798 6.44439 7.85375(a) 8.09753 9.82177 (a)：表示該自然頻率對應的振態為軸向振態

表三十旋轉傾斜Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1.5,

β

= 90°,

η

=100)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.03513 0.21891 0.60755 1.17564 1.57080(a) 1.91300 0.03 0˚ 0.06624 0.25551 0.64665 1.21745 1.57051(a) 1.95672 15˚ 0.06570 0.25474 0.64578 1.21650 1.57051(a) 1.95571 30˚ 0.06410 0.25247 0.64322 1.21371 1.57051(a) 1.95276 45˚ 0.06146 0.24881 0.63912 1.20924 1.57051(a) 1.94805 60˚ 0.05783 0.24395 0.63373 1.20340 1.57051(a) 1.94189 75˚ 0.05329 0.23816 0.62738 1.19654 1.57051(a) 1.93469 90˚ 0.04793 0.23178 0.62049 1.18913 1.57051(a) 1.92692 0.06 0˚ 0.11665 0.34165 0.74998 1.33348 1.56965(a) 2.08119 15˚ 0.11545 0.33939 0.74706 1.33008 1.56965(a) 2.07746 30˚ 0.11187 0.33267 0.73842 1.32004 1.56965(a) 2.06648 45˚ 0.10591 0.32166 0.72441 1.30386 1.56965(a) 2.04885 60˚ 0.09758 0.30666 0.70564 1.28238 1.56965(a) 2.02559 75˚ 0.08684 0.28812 0.68298 1.25679 1.56965(a) 1.99806 90˚ 0.07353 0.26665 0.65760 1.22858 1.56965(a) 1.96798

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 2.80509 3.83656 4.71239(a) 4.99194 6.25634 7.61587 0.03 0˚ 2.85034 3.88317 4.71229(a) 5.03986 6.30555 7.66640 15˚ 2.84929 3.88209 4.71229(a) 5.03874 6.30440 7.66521 30˚ 2.84622 3.87891 4.71229(a) 5.03546 6.30103 7.66175 45˚ 2.84132 3.87384 4.71229(a) 5.03024 6.29566 7.65622 60˚ 2.83491 3.86723 4.71229(a) 5.02343 6.28865 7.64902 75˚ 2.82743 3.85951 4.71229(a) 5.01548 6.28048 7.64062 90˚ 2.81938 3.85120 4.71229(a) 5.00693 6.27170 7.63159 0.06 0˚ 2.98106 4.01903 4.71201(a) 5.18030 6.45036 7.81549

15˚ 2.97709 4.01487 4.71201(a) 5.17599 6.44589 7.81088 30˚ 2.96543 4.00266 4.71201(a) 5.16330 6.43276 7.79733 45˚ 2.94675 3.98314 4.71201(a) 5.14304 6.41181 7.77571 60˚ 2.92218 3.95751 4.71201(a) 5.11648 6.38438 7.74742 75˚ 2.89323 3.92739 4.71201(a) 5.08531 6.35223 7.71430 90˚ 2.86175 3.89474 4.71201(a) 5.05160 6.31749 7.67855 (a)：表示該自然頻率對應的振態為軸向振態

表三十一旋轉傾斜 Timoshenko 梁在不同轉速與不同傾斜角的自然頻率 (r =1.5,

β

= 90°,

η

=1000)

α

K₁ K₂ K₃ K₄ K₅ K₆

0 0˚ 0.00352 0.02203 0.06169 0.12087 0.19977 0.29835 0.03 0˚ 0.05437 0.12967 0.21696 0.32099 0.43997 0.57302 15˚ 0.05375 0.12828 0.21480 0.31804 0.43622 0.56849 30˚ 0.05188 0.12411 0.20834 0.30920 0.42498 0.55494 45˚ 0.04875 0.11715 0.19756 0.29445 0.40626 0.53244 60˚ 0.04434 0.10738 0.18241 0.27374 0.38005 0.50108 75˚ 0.03855 0.09464 0.16265 0.24680 0.34617 0.46089 90˚ 0.03109 0.07846 0.13760 0.21288 0.30407 0.41176 0.06 0˚ 0.10797 0.25433 0.40979 0.58261 0.77241 0.97653 15˚ 0.10672 0.25152 0.40542 0.57665 0.76477 0.96717 30˚ 0.10297 0.24312 0.39233 0.55874 0.74183 0.93906 45˚ 0.09669 0.22910 0.37049 0.52882 0.70348 0.89208 60˚ 0.08783 0.20936 0.33972 0.48662 0.64934 0.82575 75˚ 0.07617 0.18359 0.29949 0.43129 0.57828 0.73876 90˚ 0.06107 0.15071 0.24801 0.36021 0.48694 0.62723

α

K₇ K₈ K₉ K₁₀ K₁₁ K₁₂

0 0˚ 0.41660 0.55449 0.71197 0.88902 1.08559 1.30162 0.03 0˚ 0.72035 0.88259 1.06044 1.25454 1.46541 1.57051(a)

15˚ 0.71510 0.87669 1.05395 1.24751 1.45793 1.57051(a) 30˚ 0.69941 0.85905 1.03458 1.22661 1.43569 1.57051(a) 45˚ 0.67343 0.82996 1.00273 1.19235 1.39933 1.57051(a) 60˚ 0.63742 0.78986 0.95907 1.14564 1.35000 1.57051(a) 75˚ 0.59172 0.73942 0.90465 1.08786 1.28939 1.50949 90˚ 0.53674 0.67967 0.84100 1.02104 1.21998 1.43794 0.06 0˚ 1.19348 1.42279 1.56965(a) 1.66446 1.91875 2.18600

15˚ 1.18239 1.40998 1.56965(a) 1.64996 1.90259 2.16824 30˚ 1.14909 1.37152 1.56965(a) 1.60644 1.85414 2.11502 45˚ 1.09344 1.30730 1.53384 1.56965(a) 1.77341 2.02646 60˚ 1.01495 1.21686 1.43178 1.56965(a) 1.66019 1.90257 75˚ 0.91220 1.09880 1.29906 1.51356 1.56965(a) 1.74288 90˚ 0.78114 0.94924 1.13224 1.33083 1.54564 1.56965(a) (a)：表示該自然頻率對應的振態為軸向振態

表三十二旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 0°，

η =

8.1, 8.15)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.1 0 0.38396 1.57080(a) 1.57185 3.34110 4.71239(a) 4.71882 0.01 0.38413 1.56853 1.57456 3.34161 4.71163 4.72002 0.02 0.38465 1.56624 1.57818 3.34314 4.71017 4.72281 0.03 0.38550 1.56434 1.58229 3.34568 4.70873 4.72644 0.04 0.38668 1.56284 1.58689 3.34924 4.70749 4.73075 0.05 0.38819 1.56171 1.59198 3.35379 4.70648 4.73568 0.06 0.39002 1.56096 1.59759 3.35934 4.70570 4.74123 0.07 0.39216 1.56055 1.60370 3.36587 4.70513 4.74740 0.08 0.39460 1.56048 1.61034 3.37336 4.70475 4.75418 0.09 0.39733 1.56073 1.61750 3.38180 4.70455 4.76157 0.1 0.40033 1.56130 1.62518 3.39118 4.70451 4.76955

8.15 0 0.38210 1.56877 1.57080(a) 3.33673 4.71239(a) 4.72653 0.01 0.38227 1.56694 1.57307 3.33724 4.71203 4.72734 0.02 0.38279 1.56477 1.57657 3.33877 4.71114 4.72957 0.03 0.38365 1.56297 1.58059 3.34132 4.71007 4.73289 0.04 0.38484 1.56155 1.58512 3.34489 4.70902 4.73705 0.05 0.38636 1.56050 1.59015 3.34946 4.70810 4.74197 0.06 0.38821 1.55981 1.59569 3.35502 4.70735 4.74757 0.07 0.39036 1.55948 1.60175 3.36157 4.70677 4.75384 0.08 0.39282 1.55947 1.60834 3.36909 4.70637 4.76075 0.09 0.39557 1.55979 1.61545 3.37756 4.70614 4.76830 0.1 0.39860 1.56041 1.62310 3.38696 4.70606 4.77647 (a)：表示該自然頻率對應的振態為軸向振態

表三十三旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 0°，

η =

8.2, 8.3)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.2 0 0.38026 1.56568 1.57080(a) 3.33233 4.71239(a) 4.73394 0.01 0.38043 1.56465 1.57227 3.33285 4.71216 4.73462 0.02 0.38095 1.56292 1.57535 3.33439 4.71157 4.73658 0.03 0.38181 1.56133 1.57916 3.33694 4.71078 4.73965 0.04 0.38302 1.56006 1.58354 3.34052 4.70994 4.74366 0.05 0.38455 1.55913 1.58846 3.34510 4.70916 4.74851 0.06 0.38641 1.55855 1.59391 3.35068 4.70850 4.75411 0.07 0.38858 1.55830 1.59990 3.35725 4.70797 4.76043 0.08 0.39106 1.55838 1.60642 3.36479 4.70760 4.76743 0.09 0.39383 1.55877 1.61348 3.37329 4.70737 4.77510 0.1 0.39688 1.55947 1.62107 3.38272 4.70729 4.78341

8.3 0 0.37662 1.55950 1.57080(a) 3.32349 4.71239(a) 4.74786 0.01 0.37679 1.55912 1.57162 3.32401 4.71227 4.74845 0.02 0.37732 1.55826 1.57383 3.32555 4.71195 4.75019 0.03 0.37820 1.55730 1.57702 3.32813 4.71148 4.75301 0.04 0.37942 1.55649 1.58096 3.33172 4.71095 4.75682 0.05 0.38098 1.55591 1.58555 3.33633 4.71043 4.76156 0.06 0.38286 1.55562 1.59074 3.34195 4.70995 4.76714 0.07 0.38507 1.55562 1.59650 3.34855 4.70957 4.77351 0.08 0.38759 1.55592 1.60284 3.35614 4.70929 4.78065 0.09 0.39040 1.55651 1.60974 3.36468 4.70913 4.78851 0.1 0.39349 1.55738 1.61720 3.37418 4.70910 4.79706 (a)：表示該自然頻率對應的振態為軸向振態

表三十四旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 0°，

η =

8.4, 8.5)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.4 0 0.37303 1.55330 1.57080(a) 3.31458 4.71239(a) 4.76057 0.01 0.37321 1.55317 1.57137 3.31510 4.71232 4.76113 0.02 0.37375 1.55285 1.57304 3.31665 4.71212 4.76279 0.03 0.37464 1.55249 1.57565 3.31924 4.71182 4.76552 0.04 0.37588 1.55219 1.57909 3.32286 4.71147 4.76926 0.05 0.37746 1.55207 1.58325 3.32749 4.71112 4.77396 0.06 0.37938 1.55216 1.58808 3.33314 4.71081 4.77956 0.07 0.38162 1.55249 1.59354 3.33978 4.71055 4.78601 0.08 0.38417 1.55307 1.59963 3.34741 4.71038 4.79327 0.09 0.38702 1.55391 1.60631 3.35601 4.71031 4.80130 0.1 0.39016 1.55501 1.61359 3.36555 4.71034 4.81008

8.5 0 0.36951 1.54707 1.57080(a) 3.30560 4.71239(a) 4.77207 0.01 0.36970 1.54707 1.57125 3.30612 4.71234 4.77263 0.02 0.37024 1.54709 1.57259 3.30769 4.71221 4.77427 0.03 0.37114 1.54717 1.57477 3.31029 4.71202 4.77698 0.04 0.37240 1.54735 1.57775 3.31393 4.71179 4.78071 0.05 0.37401 1.54768 1.58147 3.31859 4.71156 4.78542 0.06 0.37595 1.54819 1.58591 3.32426 4.71136 4.79108 0.07 0.37822 1.54889 1.59103 3.33094 4.71121 4.79763 0.08 0.38081 1.54982 1.59680 3.33861 4.71113 4.80503 0.09 0.38371 1.55096 1.60322 3.34726 4.71114 4.81325 0.1 0.38689 1.55233 1.61027 3.35686 4.71124 4.82224 (a)：表示該自然頻率對應的振態為軸向振態

表三十五旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 90°，

η =

8.1, 8.15)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.1 0 0.38396 1.57080(a) 1.57185 3.34110 4.71239(a) 4.71882 0.01 0.38428 1.57076(a) 1.57223 3.34164 4.71238(a) 4.71923 0.02 0.38524 1.57067(a) 1.57336 3.34324 4.71235(a) 4.72047 0.03 0.38682 1.57051(a) 1.57526 3.34592 4.71229(a) 4.72252 0.04 0.38903 1.57029(a) 1.57791 3.34966 4.71222(a) 4.72539 0.05 0.39185 1.57000(a) 1.58130 3.35445 4.71212(a) 4.72905 0.06 0.39527 1.56965(a) 1.58544 3.36029 4.71201(a) 4.73349 0.07 0.39927 1.56924(a) 1.59030 3.36717 4.71187(a) 4.73870 0.08 0.40383 1.56876(a) 1.59590 3.37507 4.71171(a) 4.74464 0.09 0.40894 1.56822(a) 1.60220 3.38398 4.71153(a) 4.75130 0.1 0.41457 1.56761(a) 1.60921 3.39388 4.71133(a) 4.75865

8.15 0 0.38210 1.56877 1.57080(a) 3.33673 4.71239(a) 4.72653 0.01 0.38242 1.56915 1.57076(a) 3.33727 4.71238(a) 4.72695 0.02 0.38338 1.57029 1.57067(a) 3.33888 4.71235(a) 4.72821 0.03 0.38498 1.57051(a) 1.57219 3.34156 4.71229(a) 4.73031 0.04 0.38720 1.57029(a) 1.57485 3.34531 4.71222(a) 4.73323 0.05 0.39004 1.57000(a) 1.57826 3.35012 4.71212(a) 4.73696 0.06 0.39347 1.56965(a) 1.58241 3.35598 4.71201(a) 4.74149 0.07 0.39750 1.56924(a) 1.58729 3.36287 4.71187(a) 4.74680 0.08 0.40209 1.56876(a) 1.59290 3.37080 4.71171(a) 4.75286 0.09 0.40722 1.56822(a) 1.59923 3.37973 4.71153(a) 4.75965 0.1 0.41288 1.56761(a) 1.60626 3.38966 4.71133(a) 4.76715 (a)：表示該自然頻率對應的振態為軸向振態

表三十六旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 90°，

η =

8.2, 8.3)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.2 0 0.38026 1.56568 1.57080(a) 3.33233 4.71239(a) 4.73394 0.01 0.38058 1.56607 1.57076(a) 3.33287 4.71238(a) 4.73437 0.02 0.38155 1.56721 1.57067(a) 3.33449 4.71235(a) 4.73565 0.03 0.38315 1.56912 1.57051(a) 3.33718 4.71229(a) 4.73779 0.04 0.38539 1.57029(a) 1.57179 3.34094 4.71222(a) 4.74077 0.05 0.38824 1.57000(a) 1.57520 3.34576 4.71212(a) 4.74457 0.06 0.39169 1.56965(a) 1.57937 3.35164 4.71201(a) 4.74919 0.07 0.39574 1.56924(a) 1.58427 3.35855 4.71187(a) 4.75460 0.08 0.40035 1.56876(a) 1.58990 3.36650 4.71171(a) 4.76078 0.09 0.40551 1.56822(a) 1.59625 3.37546 4.71153(a) 4.76771 0.1 0.41120 1.56761(a) 1.60331 3.38541 4.71133(a) 4.77535

8.3 0 0.37662 1.55950 1.57080(a) 3.32349 4.71239(a) 4.74786 0.01 0.37694 1.55988 1.57076(a) 3.32403 4.71238(a) 4.74830 0.02 0.37792 1.56104 1.57067(a) 3.32566 4.71235(a) 4.74964 0.03 0.37954 1.56296 1.57051(a) 3.32836 4.71229(a) 4.75185 0.04 0.38180 1.56564 1.57029(a) 3.33214 4.71222(a) 4.75495 0.05 0.38469 1.56908 1.57000(a) 3.33699 4.71212(a) 4.75890 0.06 0.38818 1.56965(a) 1.57328 3.34290 4.71201(a) 4.76370 0.07 0.39227 1.56924(a) 1.57821 3.34985 4.71187(a) 4.76932 0.08 0.39693 1.56876(a) 1.58388 3.35784 4.71171(a) 4.77574 0.09 0.40215 1.56822(a) 1.59027 3.36685 4.71153(a) 4.78294 0.1 0.40789 1.56761(a) 1.59738 3.37686 4.71133(a) 4.79088 (a)：表示該自然頻率對應的振態為軸向振態

表三十七旋轉傾斜 Timoshenko 梁在不同細長比與不同轉速的自然頻率 (r =1，

α

= 0°，

β

= 90°，

η =

8.4, 8.5)

η

_k K₁ K₂ K₃ K₄ K₅ K₆

8.4 0 0.37303 1.55330 1.57080(a) 3.31458 4.71239(a) 4.76057 0.01 0.37336 1.55368 1.57076(a) 3.31512 4.71238(a) 4.76103 0.02 0.37435 1.55484 1.57067(a) 3.31676 4.71235(a) 4.76241 0.03 0.37599 1.55678 1.57051(a) 3.31948 4.71229(a) 4.76471 0.04 0.37828 1.55948 1.57029(a) 3.32328 4.71222(a) 4.76792 0.05 0.38120 1.56295 1.57000(a) 3.32815 4.71212(a) 4.77202 0.06 0.38473 1.56717 1.56965(a) 3.33409 4.71201(a) 4.77700 0.07 0.38886 1.56924(a) 1.57214 3.34108 4.71187(a) 4.78283 0.08 0.39357 1.56876(a) 1.57784 3.34911 4.71171(a) 4.78950 0.09 0.39884 1.56822(a) 1.58428 3.35817 4.71153(a) 4.79697 0.1 0.40464 1.56761(a) 1.59143 3.36823 4.71133(a) 4.80522

8.5 0 0.36951 1.54707 1.57080(a) 3.30560 4.71239(a) 4.77207 0.01 0.36985 1.54746 1.57076(a) 3.30615 4.71238(a) 4.77255 0.02 0.37085 1.54863 1.57067(a) 3.30779 4.71235(a) 4.77399 0.03 0.37251 1.55058 1.57051(a) 3.31053 4.71229(a) 4.77637 0.04 0.37482 1.55330 1.57029(a) 3.31435 4.71222(a) 4.77969 0.05 0.37777 1.55679 1.57000(a) 3.31924 4.71212(a) 4.78394 0.06 0.38134 1.56104 1.56965(a) 3.32521 4.71201(a) 4.78909 0.07 0.38552 1.56604 1.56924(a) 3.33224 4.71187(a) 4.79514 0.08 0.39028 1.56876(a) 1.57179 3.34031 4.71171(a) 4.80205 0.09 0.39560 1.56822(a) 1.57826 3.34942 4.71153(a) 4.80979 0.1 0.40146 1.56761(a) 1.58546 3.35954 4.71133(a) 4.81835 (a)：表示該自然頻率對應的振態為軸向振態

圖一無傾斜角的旋轉梁結構三視圖

Ω R

O

A X ¹

X 1

A O

L

X 3

X 2

Ω

β

圖二具傾斜角的旋轉梁架構

X 3

R β

X 1

X 2

Ω

β

O A

α

圖三 (a)Lagwise bending vibration (b)撲翼振動(Flapping vibration) (a)

(b)

Ω

100

圖四旋轉傾斜梁的俯視圖

R O

A

X 1

X 2

L

α

β

圖五旋轉傾斜梁的側視圖

O

X 2

X 3

β

Ω

101

圖六梁的變形圖

X 2

X 3

z

y P Q

X 1

X 2

Q y

v

γ

u x +

ϕ

102

x

N l = L

L

1 2 m m +1 N

x

₃

x

_m₊₁

x

_m₊₂

x

_N ₊₁

... ...

l m 1 ) ( −

F

M

s

1

2 X

圖七作用於梁中任一小段的端點負荷

圖八梁的分段元素

103

10 100 1000

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

η

K

/ K

iE 1

2 3 4 5 6

50 500

i

圖九旋轉傾斜梁在不同細長比下側向振態對應的自然頻率之比值 (k =0,

α

= 0°,

β

= 90°, r =1)

10 100 1000

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

i

K

/ K

iE 1

2 3 4 5 6

20 50

η

500

圖十旋轉傾斜梁在不同細長比下側向振態對應的自然頻率之比值

(k =0.06,

α

= 0°,

β

= 90°, r =1)

104

10 100 1000

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

i

K

/ K

iE 1

2 3 4 5 6

20 50

η

500

圖十一旋轉傾斜梁在不同細長比下側向振態對應的自然頻率之比值 (k =0.06,

α

= 30°,

β

= 90°, r =1)

105

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

Mode 1

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 2

Modal def lection

x/L

圖十二不同轉速下的第一至第六個振動模態 (r

= 1

α

= 0°,

β

= 0°,

η

=10)

k =

k = U V

0 0.03 0.06

k = k = k = U V

0 0.03 0.06

106

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 3

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 4

Modal def lection

x/L

圖十二 (續)

k =

k = U V

0 0.03 0.06

k =

k = U V

0 0.03 0.06

107

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 5

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 6

Modal def lection

x/L

圖十二 (續)

k =

k = U V

0 0.03 0.06

k =

k = U V

0 0.03 0.06

108

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 1

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 2

Modal def lection

x/L

圖十三不同轉速下的第一至第六個振動模態 (r

= 1

α

= 0°,

β

= 0°,

η

=50)

k =

k = U V

0 0.03 0.06

k =

k = U V

0 0.03 0.06

109

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 3

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 4

Modal def lection

x/L

圖十三 (續)

k =

k = U V

0 0.03 0.06

k =

k = U V

0 0.03 0.06

110

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5

1.0 Mode 5

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

Mode 6

圖十三 (續)

k =

k = U V

0 0.03 0.06

k =

k = U V

0 0.03 0.06

111

112

113

114

0.00 0.02 0.04 0.06 0.08 0.10 1.54

115

0.00 0.02 0.04 0.06 0.08 0.10 1.54

116

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.1

Modal deflection η

x/L

Mode 1

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.1

η

Mode 2

Modal def lection

x/L

圖十九不同轉速下的第一至第六個振動模態 (r =1,

α

= 0°,

β

= 0°,

η

=8.1⁾

k =

U V

0 0.03 0.06 0.1

k =

U V

0 0.03 0.06 0.1

117

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.1

η

Mode 3

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.1

η

Mode 4

Modal def lection

x/L

圖十九 (續)

k =

U V

0 0.03 0.06 0.1

118

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

=8.1

η

Mode 5

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

=8.1

η

Mode 6

圖十九 (續)

k =

U V

0 0.03 0.06 0.1

119

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

=8.15 η^v

Mode 1

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

=8.15 η^v

Mode 2

圖二十不同轉速下的第一至第六個振動模態 (r

= 1

α

= 0°,

β

= 0°,

η

=8.15⁾

k =

U V

0 0.03 0.06 0.1

k =

U V

0 0.03 0.06 0.1

120

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

Modal def lection

x/L

=8.15 η^v

Mode 3

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.15 η^v

Mode 4

Modal def lection

x/L

圖二十 (續)

k =

k = U V

0 0.03 0.06 0.1

k =

k = U V

0 0.03 0.06 0.1

121

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.15 η^v

Mode 5

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

=8.15 η^v

Mode 6

Modal def lection

x/L

圖二十 (續)

k =

k = U V

0 0.03 0.06 0.1

122

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.1

η

K

k

mode 3 mode 2

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.15

η

K

k

mode 3 mode 2

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.2

η

K

k

mode 3 mode 2

圖二十一不同細長比的自然頻率與轉速之特徵值曲線

(r =1，

α

= 0°，

β

= 90°，

η

=8.1, 8.15, 8.2)

123

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.3

η

K

k

mode 3 mode 2

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.4

η

K

k

mode 3 mode 2

0.00 0.02 0.04 0.06 0.08 0.10 1.54

1.55 1.56 1.57 1.58 1.59 1.60 1.61

=8.5

η

K

k

mode 3 mode 2

圖二十二不同細長比的自然頻率與轉速之特徵值曲線

(r =1，

α

= 0°，

β

= 90°，

η

=8.3, 8.4, 8.5)

124

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₁ =0.38026 Mode 1

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₂ =1.56568 Mode 2

Modal def lection

x/L

圖二十三轉速為零的第一至第六個振動模態 (r

= 1

α

= 0°,

β

= 90°,

η

=8.2⁾

U V

125

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₃=1.57080 Mode 3

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₄ =3.33233

Modal def lection

x/L

Mode 4

圖二十三 (續)

U V

126

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₅ =4.71239 Mode 5

Modal def lection

x/L

0.0 0.5 1.0

-1.0 -0.5 0.0 0.5 1.0

K₆ =4.73394

Modal def lection

x/L

Mode 6

圖二十三 (續)

U V

127

在文檔中等速旋轉傾斜Timoshenko梁的穩態變形與自由振動分析 (頁 74-145)

第五章 數值例題

6.2 未來展望

0 °

0 °

參考文獻

,

國立交通大學機械工程研究所碩

士論文

國立交通大學機械工程研 究所碩士論文

國立成功大學機械工程學系博士 論文

國立交通大學機械工 程研究所碩士論文

國立交通大學機械工程研究所碩士論文

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第五章數值例題

國立交通大學機械工程研究所碩士論文

國立成功大學機械工程學系博士論文

國立交通大學機械工程研究所碩士論文

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