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。

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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

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

°

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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

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

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

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(

= 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

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

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

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

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

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

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

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

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

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

r = 1

，

α

=10^D) k ε₀ (10⁻³) ε_b (10⁻³)

=90^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

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

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

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

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

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，

=90^D) k ε₀ (10⁻⁵) ε_b (10⁻³)

T tip

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

O

A ^X¹

X3 Ω X2

O A X₁

X3 ^X₂

β Ω

R

X

O A

α

β

R α L

A

O X

X

圖三旋轉傾斜梁的上視圖

O

β β

X

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

X x

₁

X

o 2

θ

1

2 o x

θ

Ω

X

O A R

α

A

O X

X

L

R α

圖六具傾斜角的旋轉梁結構( β = 0

)

θ

Q

P y

v θ

s

u x + o

x

P

x

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

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

， 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

， 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

， 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

， 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

， 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

， 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

， 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

， 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

， 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^D^θ

及 f

_b^D^θ

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

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

} 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

= α + (B.4)

sin α

2 2

R L

EA q k

= − (B.5)

E

k L

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

R

L

(a) (b)

B X

X

O

q

L

)

( x q

x

B X

X

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

R L x L x

L k AE

F

T T

= − + −

= 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 59-155)

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

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

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

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

參考文獻

[1] A. Leissa, “Vibrational Aspects of Rotating Turbomachinery Blades”, ASME Applied Mechanics Reviews, Vol. 34, 1981, pp. 629-635.

[2] V. Ramamurti, P. Balasubramanian, “Analysis of Turbomachinery Blades - A Review,” The Shock, Vibration Digest, Vol. 16, 1984, pp. 13-28.

[3] M. J. Schilhansl, “Bending Frequency of a Rotating Cantilever Beam”, ASME Journal of Applied Mechanics, Vol. 25, 1958, pp. 28-30.

[4] S. Y. Lee, Y. H. Kuo, “Bending frequency of a rotating beam with an elastically restrained root”, ASME Journal of Applied Mechanics, Vol. 58, 1991, pp. 209-214.

[5] T. Yokoyama, “Free Vibration Characteristics of Rotating Timoshenko Beam”, International Journal of Mechanical Science, Vol. 30, No. 10, 1988, pp. 743-755.

[6] S. Y. Lee, S. M. Lin, “Bending Vibration of Rotating nonuniform Timoshenko Beams with an Elastically Restrained Root”, ASME Journal of Applied Mechanics, Vol. 61, 1994, pp. 949-955.

[7] C. D. Eick, M. P. Mignolet, “Vibration and buckling of flexible Rotating Beams”, AIAA Journal, Vol. 33, No. 3, 1995, pp. 526-538.

[8] J. C. Simo, L. V. Quoc, “The role of non-linear theories in transient dynamic analysis of flexible structures”, Journal of Sound and Vibration, Vol. 119, 1987, pp. 487-508.

[9] S. C. Lin, K. M. Hsiao, “Vibration analysis of a rotating Timoshenko beam”, Journal of Sound and Vibration, Vol. 240(2), 2001, pp. 303-322.

[10] 洪船島, 旋轉梁結構之振動分析, 國立交通大學機械工程研究所碩士

論文, 臺灣, 新竹, 1997.

[11] 周志芳, 旋轉梁之自由振動的級數解法, 國立交通大學機械工程研究 所碩士論文, 臺灣, 新竹, 1998.

[12] 黃志傑, 旋轉三維 Timoshenko 梁之振動分析, 國立交通大學機械工程 研究所碩士論文, 臺灣, 新竹, 2001.

[13] 許哲嘉, 旋轉傾斜梁之動態分析, 國立成功大學機械工程學系博士論 文, 臺灣, 臺南, 2006.

[14] S. Y. Lee, J. J. Sheu, “Free Vibrations of a Rotating Inclined beam”, ASME Journal of Applied Mechanics, Vol. 74, 2007, pp. 406-414.

[15] S. Y. Lee, J. J. Sheu, “Free Vibrations of an Extensible Rotating Inclined Timoshenko beam”, Journal of Sound and Vibration, Vol. 304, 2007, pp.

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.

η = 10

η = 20

η = 50

η = 100

η = 500

η = 1000

r = 1

η = 70

r = 1

ε

α

α

α

α

α

α

α

α

α

r = 1

α

r = 1

α

r = 1

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

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

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、

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

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。

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

[11] 周志芳, 旋轉梁之自由振動的級數解法, 國立交通大學機械工程研究所碩士論文, 臺灣, 新竹, 1998.

[12] 黃志傑, 旋轉三維 Timoshenko 梁之振動分析, 國立交通大學機械工程研究所碩士論文, 臺灣, 新竹, 2001.

[13] 許哲嘉, 旋轉傾斜梁之動態分析, 國立成功大學機械工程學系博士論文, 臺灣, 臺南, 2006.

[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.

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 )

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 )

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 )

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 )

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 )

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 )

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 )

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 )

圖一無傾斜角的旋轉梁結構

圖二具傾斜角的旋轉梁結構

圖三旋轉傾斜梁的上視圖

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