應用微分數值積分法於轉子軸承系統之動態分析

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(3) Dynamic Analysis of Rotor-Bearing System by the Differential Quadrature Method .

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(6) . !"#$%&'() [email protected] efficiency, and the great potential of the (Differential DQM for use on rotor-bearing systems. Quadrature Method)

(7) Keywords: rotor-bearing system, differential !"#$%&'() * quadrature method, Timoshenko beam. +,-.(/01234512.

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(10) ; F |}J,qr~*K <= > )Z\ J, Timoshenko ? @A*BCDEF G |};<= * HIJKLM-NOP>QRS k*^#[|*B *TQU3VWX @A u3 ¡¢£ =)YZ[\]*9, *¤¥¦§¨2y. ^_)Y`Z=)YZabK (DQM)#t< c)YZ*de^.\fgh u&'(©ª«)YZ) ijk@67lmno*J,. *¬ Bellman j Casti (1971)$\ -

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(13) qTQRS45·¸>³ y ´¹Bert j Malik (1996)º z { {. »¤¼½¾¿À Timoshenko ?y 67 DQM

(14) y *ÁÂÃ 3Ä Abstract This research is to investigate the ÅÆ,:) Ç#º |} dynamic characteristics of rotor-bearing $%ÈtÈ&'

(15) ) ¶ systems by using the differential quadrature #É<Ê( *s# method (DQM). The DQM is a numerical method that is able to provide accurate ËÌÍ½ÎÏÐ*Æ |} results with small computational effort and is = ÈvIi*ÈÑ suitable for solving initial- and/or boundary-value problems. The purpose of this research is to apply the DQM to analyze 67LMOÒ ÓÔ= the dynamic behaviors of a spinning shaft, )YZ*ÕÖ P×ØÙvÚf which is modeled as a Timoshenko beam. Û*ÜK x 3 y )ÝÞÝ³´ u x 3 Bearings are considered to be linear and modeled as spring-damper sets. Numerical u y *3EºJßià ϕ x 3 ϕ y áo results show the excellent accuracy, âã Timoshenko ?äå* =)Y 1.

(16) f (x ) AOPQR! m bF;aG; H2(Quadrature rule) PQR!T N $ ghi+1jklmnopqrstu F;!b vdwxmy+F;V. Zæç:. ρl 2 ρIl. ∂ 2u x ∂t 2. ∂ ϕx ∂t −. ρl 2 ρIl. 2. 2. +. ∂ ∂ ∂u (κGlϕ x ) − (κG x ) = 0 ∂ζ ∂ζ ∂ζ. + ΩJ pl. ∂ϕ y. Wij(m) 1z{|}~N. ∂t. ∂u ∂ EI ∂ϕx κAG ( )+ (lϕx − x ) = 0 ∂ζ l ∂ξ ∂ζ l. 2. ∂ uy ∂t 2. ∂ 2ϕ y. +. f ( x) = x k −1 , k = 1, 2, L, N OTPQR!BC1f f (x ) !ObF;aVG[Bert 11988] DQM Lpqrse rse1A5P!P QR1z{nP 9~20 TR1$. ∂u y ∂ ∂ (κGlϕ x ) − (κG )=0 ∂ζ ∂ζ ∂ζ. − ΩJ pl. ∂ϕ x ∂t. [ ]. ∂t 2 ∂u y ∂ EI ∂ϕ y κAG − ( )+ (lϕ y − )=0 l ∂ζ l ∂ξ ∂ζ l Ω ρ

(17) E G. bF;!VG Wij(1) /pqrs e ζ !F;$Z[cd:. [ ]. Θ ∂Θ = Wij(1) {Θ j } ∂ζ ζ =ζ i Θ Z[!Jc u , ϕ , M : V DQM '!pqrsekl. I J p

(18) κ A !"#$%&!'()* + ,-.()/0.()1 2345!6789:  u′(ζ 0 , t )  κAG − ϕ (ζ 0 , t )  = l   ∂u (ζ 0 , t )   a Ku (ζ 0 , t ) + C  ∂t   K : C ;<.!=:>?@ /ζ 0 A345!BC10 ζ 0 D #!3412 a = +1 /0 ζ 0 E#! 3412 a = −1. [ ]. !34891VG Wij(1). kl¡¢3489£!¤¥ x1 a34'()¦1A34§!

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(20) ¨ª1f u1 = 012. [ ]. {ui′ } = Wij(1). dxm. {ui } $Z[.  u1′  0 W12  u′  0 W 22  2    M  M = M u′  0 W N −1,2  N −1    u′N  0 WN,2. W1, N  u1  W2, N  u2    O M M  M   L WN −1, N −1 WN −1, N uN −1   L WN, N −1 WN, N  uN . L L. W1, N −1 W2, N −1. «3489¬$®5¯!rHL !F;VG£!°± 0" #±()12"#©. F;G;H!IJKLMNA OPQR!F;G,STPQR!NG UVWXYZ[ \]A x !+^_5 P x1 , x 2 ,..., x N ` N TPQR12N f (x ) A x= xi a m bF;$Z[cd:. d m f ( xi ). u. oª1² [Wij(1) ]M $®5¯!re³° ± ´()!µT3489 67rse1klm¶pqrse !67rseY·Z[(Choi 1 2000) tpqrse$¸¹cd: [ M ]{q&&} + [G ]{q&} + [ K ]{q}. N. ≅ ∑ Wij(m) f ( x j ), i = 1,2,L, N j =1. = [ K b ]{q} + [C b ]{q&}. Wij(m) K m bF;UV@ 5efN 2.

(21) . ½Iw;Ú! ½¾¿À! Ω q! !.1jÝbDQM!;Ú Timoshenko Á1"#Â.()1 ®o"#Â.()! ë ÃÄÅÆcÇ 1 ²[ ÈÉ1 91®oÛ!. 1Ý±¼DQM! Ê 7800 kg/m31@ E = 2.0 × 1011 rHw¼Ù1Õ$³½¾¿! N/m21ËÌÍν = 0.3/ÎÏ 0.05 m1 ¾¿r 1.3 m1 κ = 0.88 o"#Â.()!1½ Z 1 Ð® DQM ¼ 500 !¾¿ß rad/s Ñ13489d!ÒÓ Ô1L 1. DQM ;ÚaÒÓ !"4 ÕÖ×Ø¶ Zu ¶ Han (1992)!ÙÚÙÍÛ. 1$ÜÝ DQM AÞ3489 2. #$.qº%!òó d!;Úßàu!áâ1¶ÙÚÙ 3. ÍÛ DQM !;Ú×Ø¶ÙÚÙ!ã !ãä¬åo 0.1% æçcÇ 1 ! ä .() Timoshenko Á1.!è¶> 4. !F;VG¤¥HwÙyI& DQM o§J3489!'( ? @ ; < K = K xx = K yy = 5 é 107 N/m1 C xx = C yy = 5é102 N-s/m DQM ¶

(22) ßêëìH²aÒÓ ÔÇcÇ 2 Bellman, R. E., and Casti, J., 1971, ²[1íÇ$î"ï!×Øàð “Differential Quadrature and Long-Term Y1² DQM ñ$±â®o34 Integration,” Journal of Mathematical ()a Timoshenko Á Analysis and Application, Vol. 34, pp. æç.!è=!òó1 C xx = 235-238. Bert, C. W., Jang, S. K., and Striz, A. G, C yy = 0 Ñ1è@ K í 0 ôõön÷ 1988, “Two New Approximate Methods for 1012 N/m Ñ1Timoshenko Á! øÔù Analyzing Free Vibration of Structural cÇ 3 ²[ íÇ$î K ôõö Components,” AIAA Journal, Vol. 26, pp. UÑ1ú 1 :ú 2 Tº! øÔ 612-618. ôõÂòó´ûu/K no 103 N/m Bert, C. W., and Malik, M., 1996, Quadrature Method in Ñ1ú 3 Tº! øÔüôõÂ “Differential 10 Computational Mechanics: A Review,” òó/ K no 10 N/m Ñ1! øÔôõýþ1Ñ!3489à ASME Applied Mechanics Review, Vol. 49, No. 1, pp. 1-28. o'()!34 Bert, C. W., Striz, A. G., and Jang, S. K. 1989, “Nonlinear Bending Analysis of Orthotropic Rectangular Plate by the Method ½¾¿, DQM ;Ú. of Differential Quadrature,” Computational !qº»1Ð®!F;VG¤¥ Mechanics, Vol. 5, pp. 217-226. HwÙy3489!1±â;ÚÒ Choi, S.-T., Wu, J.-D., and Chou, Y.-T., Timoshenko Á!qº» æç. 2000, “Dynamic Analysis of a Spinning :ù.qº Timoshenko Beam by the Differential »!òó ,GÀ!×Ø1$Ü Quadrature Method,” AIAA Journal, to Ý DQM ;Ú²a×Ø1 ßàu!á appear. Lin, R. M., Lim, M. K., and Du, H., 1994, â1² p Ñ à

(23) “Deflection of Plates with Nonlinear !qº»f$,5e¼. 3.

(24) Boundary Supports Using Generalized Differential Quadrature,” Computers and. Structures, Vol. 53, No. 4, pp. 993-999. Zu, J. W. Z., and Han, R. P. S., 1992, y “Natural Frequencies and Normal Modes of a c Spinning Timoshenko Beam with General xx Boundary Conditions,” Journal of Applied. Mechanics, Vol. 59, pp. S197-204.. x. sampling points. ........... 1 2. 3. . N−1 N. Ω. c xx. k xx. z. k xx. Ç 1 .[hÇ. Z 1 3489d1aÒÓ Ô (Ŝ = 500 rad/s) Mode 2. Mode 3. (rad/s). (rad/s). (rad/s). B. F. B. F. B. FEM DQM. Whirl Speed (rad/s). Mode 1. F. DQM 732.25 735.81 2869.5 2882.8 6254.0 6280.7 #1 Ref.a 732.25 735.81 2869.5 2882.8 6254.0 6280.7 Error. 0%. 0%. 0%. 0%. 0%. 3000. 0%. 2000. 1000. DQM 1631.0 1626.9 4343.2 4329.4 8163.0 8136.9 #2 Ref.a 1631.0 1626.9 4343.2 4329.4 8157.9 8132.1 Error. 0%. 0%. 0%. 0%. 0.07% 0.06%. DQM. 0. 0. 0. 0. 1658.8 1641.3. #3 Ref.a. 0. 0. 0. 0. 1658.8 1641.3. Error. 0%. 0%. 0%. 0%. DQM. 0. 0. 1147.0 1137.2 3626.2 3602.5. 0. 0. 1147.0 1137.2 3626.2 3602.5. 0%. 0%. #4 Ref.a Error. 0%. 0%. 0%. 0%. 0%. 0%. DQM 263.33 261.63 1618.4 1607.1 4395.5 4370.9 #5 Ref.a 263.33 261.63 1618.4 1607.1 4395.5 4370.9 Error. 0%. 0%. 0%. 0%. 0%. 0%. 0. 0. 2000. 7000. 0%. 0%. 0%. 0%. Natural Frequency (rad/s). a. 0%. 0%. Zhu and Han (1992) B: backward whirl; F: forward whirl #1: hinged-hinged #2: clamped-clamped #3: free-free #4: hinged-free #5: clamped-free #6: clamped-hinged. 6267. Mode 1 Mode 2 Mode 3. DQM 1137.8 1133.8 3587.1 3573.5 7206.4 7179.9 Error. 6000. Ç 2 ;<Ð® DQM ¶ FEM Ñ1 aÒÓ ÔÇ. 6000. #6 Ref.a 1137.8 1133.8 3587.1 3573.5 7206.5 7180.0. 4000. Spin Speed (rad/s). 5000 4000 3000 2000. 2876. 1650. 1000 0 100. 734. 102. 104. 106. 108. 1010. 1012. Spring Stiffness (N/m). Ç 3 .=ù øÔ!òó. 4.

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