A class of wavelet-based Rayleigh-Euler beam element for analyzing rotating shafts

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1 Shock and Vibration 18 (11) DOI 1.333/SAV-1-55 IOS Prss A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts Jiawi Xiang a,b,, Zhansi Jiang a and Xufng Chn b a School of Mchantronic Enginring, Guilin Univrsity of Elctronic Tchnology, Guilin, 5414, P.R. China b Stat Ky Laboratory for Manufacturing Systms Enginring, Xi an Jiaotong Univrsity, Xi an, 7149, P.R. China Rcivd 6 Jun 9 Accptd 8 Octobr 9 Abstract. A class of wavlt-basd Rayligh-Eulr rotating bam lmnt using B-splin wavlts on th intrval (BSWI) is dvlopd to analyz rotor-baring systm. Th ffcts of translational and rotary inrtia, torsion momnt, axial displacmnt, cross-coupld stiffnss and damping cofficints of barings, hystric and viscous intrnal damping, gyroscopic momnts and bnding dformation of th systm ar includd in th computational modl. In ordr to gt a gnralizd formulation of wavltbasd lmnt, ach boundary nod is collocatd six dgrs of frdom (DOFs): thr translations and thr rotations; whras, ach innr nod has only thr translations. Typical numrical xampls ar prsntd to show th accuracy and fficincy of th prsntd mthod. Kywords: Finit lmnt mthod, wavlt-basd lmnt, shafts, intrnal damping, dynamic analysis 1. Introduction A rcnt trnd in modrn rotating machinry dsign has bn th volution of highr rotating spds and lowr wight of rotating componnts, such as shaft, turbin blads, aircraft propllr blads and drill bits, tc. This trnd has bn accompanid by a mor accurat numrical analysis mthod to prdict dynamic bhaviors of rotating structurs. Rcntly, thr wr a numbr of studis rlating to this fild in th past dcads as indicatd in th book by Ehrich [1] and in th survy papr by Mng [] and Huang t al. [3]. Numrous rsarchrs focusd thir intrst on th fild of analyzing th dynamic charactristics of rotating structurs, which ar concrnd with dtrmining critical spds, whirl spds (natural frquncis), instability thrsholds and unbalanc rspons. High prcision numrical approximations hav bn dvlopd to analyz th dynamic bhavior of rotating structurs by using finit lmnt mthod. A rotating shaft lmnt using Timoshnko bam thory was proposd by Nlson [4] and Grnhill t al. [5]. Th computational modl for rotor-baring systms to comput natural whirl spds and instability thrsholds wr invstigatd by Zorzi and Nlson [6], Ku [7], Kalita and Kakoty [8]. Thos works showd that th us of finit lmnts for th modling of rotor-baring systms maks it possibl to formulat incrasingly complicatd problms and to yild highly accurat and succssful rsults. Howvr, th abov mntiond litraturs mployd th convntional finit lmnt mthod. In ordr to gain accurat rsults, numrous lmnts ar ndd. In ordr to promot th analysis prcision and fficincy, Hashmi t al. dvlopd a nw dynamic finit lmnt (DFE) formulation for th vibration analysis of spinning bams [9]. Wang t al. proposd a novl bam finit lmnt having two nods and 16 dgrs of frdom to analyz fr vibration charactrs of Corrsponding author. wxw867@163.com. ISSN 17-96/11/$ IOS Prss and th authors. All rights rsrvd

2 448 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts stppd shafts [1]. Banrj and Su dvlopd a dynamic stiffnss matrix for fr vibration analysis of spinning bams [11,1]. Unlik convntional finit lmnt mthod, th dsirabl advantags of wavlt-basd lmnts ar various basis functions for structural analysis [13,14]. By mans of two-scal rlations of wavlts, th scal adoptd can b changd frly according to rquirmnts to improv analysis accuracy. Xiang t al. constructd som classs of 1D and D BSWI lmnts for structural analysis with high prformanc [15 18]. In addition, Han t al. constructd som splin wavlt lmnts for analyzing structural mchanics problms undr th thory fram of splin lmnts [19]. Xiang t al. also prsntd th BSWI lmnts to dtct cracks in bam and shaft with high prcisions [,1]. Howvr, th wavlt-basd rotating bam lmnt that considrs intrnal damping was not constructd in th publishd litraturs. Although th ffcts of various factors on th dynamic charactristics of rotor-baring systms hav bn invstigatd via th improvd finit lmnt modl, a common fatur of th publishd works is that thos finit lmnts blong to convntional finit lmnt mthod. Th objct of th prsnt articl is to dvlop a class of high prformanc of BSWI Rayligh-Eulr bam lmnts to analyz spinning structurs in nginring, spcially complx rotor-baring systms. In th prsnt finit lmnt modl, ach lmnt has a crtain nods according to th lvl of wavlt basis. Each boundary nod has six dgrs of frdom (DOFs): thr translations and thr rotations, whras ach innr nod has only thr translations. Th outlin of this papr is as follows. In Sctions, a class of novl BSWI rotating Rayligh-Eulr shaft lmnts is constructd. Th lmnts includ th ffcts of translational and rotary inrtia and th gyroscopic momnts, th combind ffcts of bnding dformations and th intrnal viscous and hystrtic damping, cross-coupld stiffnss and damping cofficints of barings, In Sction 3, som numrical studis ar mad by compard with th othr prviously publishd works to invstigat th prsnt wavlt-basd shaft lmnt.. BSWI bam lmnt formulation.1. Th construction of BSWI rotating Rayligh-Eulr bam lmnts Classical approachs to wavlt construction dal with multi-rsolution analysis (MRA) on th whol ral spac R and th corrsponding wavlts ar oftn dfind on th whol squar intgrabl ral spac L(R ). Somtims numrical instability phnomnon will b occurrd whn this kind of wavlts is applid to numrical simulation of partial diffrntial quations (PDEs) []. To ovrcom this limitation, Chui and Quak constructd BSWI functions, and prsntd a dcomposition and rconstruction algorithm [3]. Th scaling functions φ j m,k (ξ) for ordr m at th scal j ar simply dnotd as BSWI m j scaling functions Φ mj, i.. } Φ mj = {φ j m, m+1 (ξ)φj m, m+ (ξ)...φjm, j 1 (ξ). (1) Th xplicit xprssion of ach trm φ j m,k (ξ) for ordr m at th scal j is shown in [17]. Th slndr shaft is modld by a Rayligh-Eulr bam considring th ffcts of th cross-sction inrtia, torsion momnt and axial displacmnt, th lmnt potntial nrgy U can b writtn as U = 1 l EI z ( d w dx ) dx+ 1 l EI y ( d v dx ) dx+ 1 l GJ x ( dθx dx ) dx+ 1 l EA ( ) du dx,() dx whr E is th Young s modulus, I z and I y ar th momnt of inrtia, J x is th polar momnt of inrtia. w(x, t) and v(x, t) ar th transvrs displacmnt, u(x, t) is th axial displacmnt, θ x (x, t) is th rotation of torsion, l is th lmnt lngth, A is th cross-sction ara, G is th shar modulus. Th lmnt kintic nrgy T of th Rayligh-Eulr bam allowing for th rotatory inrtia ffct, including th translational and rotational forms, is givn by T = 1 [ l ( w ) ( ) ] v ρa + dx + 1 l ( ) θz ρi Z dx + 1 l ( ) θy ρi y dx + ΩJ xρ l [ θ z ( θy ) θ y ( θz ) ] dx + 1 l ρi x ( θx ) dx + 1 l ρa ( ) u dx (3)

3 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts 449 y l z θ x1 v1 w1 w w n-1 w n+1 θ z 1 θ z n+ 1 u1 1 n θ y1 u u n-1 v v n-1 θ x n+1 v n+1 n+1 θ y n+1 u n+1 x Fig. 1. Th layout of lmnt nods and th corrsponding DOFs for wavlt-basd lmnt. whr ρ is th dnsity, Ω is th rotational spd(rad/s), θ z (x, t) and θ y (x, t) ar th rotation of th bam sction du to bnding and can b givn by θ z = dw dx = 1 dw l dξ θ y = dv dx = 1 dv l dξ. In ordr to satisfy th displacmnt and slop compatibility among nighboring lmnts, th lmnt boundary nods should includ th transvrs displacmnts and slops [16]. In th prsnt study, th transvrs displacmnts ar intrpolatd by BSWI4 j scaling functionsφ 4j, th axial displacmnt is intrpolatd by BSWI j scaling functions Φ j, and th rotation of torsion is intrpolatd by convntional linar intrpolation. Thrfor, lmnt dgrs of frdom (DOFs) on ach boundary nod in physical spac includ thr displacmnts and thr slops, i.. u i, v i, w i, θ xi, θ yi, θ zi (i = 1, n + 1). Whil on ach innr nod, it only hav thr displacmnts, i.. u i, v i, w i (i =,3,, n). In this study, th bam lmnt is dividd for solving domain Ω into n = j (j is th scal of BSWI) sgmnts, th nod numbr is n + 1, th lmnt dgrs of frdom (DOFs) = 3 j + 9. Th layout of lmnt nods is shown in Fig. 1. Th lmnt is abbrviatd to BSWI m j Rayligh-Eulr rotating bam lmnt. Th lmnt physical DOFs can b rprsntd by δ = {u 1 v 1 w 1 θ x1 θ y1 θ z1 u v w u n v n w n u n+1 v n+1 w n+1 θ xn+1 θ yn+1 θ zn+1 } T, (5) whr θ z1 = 1 dw 1 l, θ dξ y 1 = 1 dv 1 l, θ dξ z n+1 = 1 dw n+1 l and θ dξ yn+1 = 1 dv n+1 l dξ ndpoint. Th unknown fild functions v(ξ, t) and w(ξ, t) ar intrpolatd by Φ 4j as { w(ξ, t) = Φ4j T b w v(ξ, t) = Φ 4j T, b v whr transformation matrix T b is givn by [16], as [ ] T T b = Φ T 4j(ξ 1 ) 1 dφ T 4j (ξ 1) Φ T l dξ 4j(ξ )...Φ T 4j(ξ n )Φ T 4j(ξ n+1 ) 1 dφ T 4j (ξ n+1) l dξ (4) dnot rotation on ach lmnt 1 (6), (7) and physical DOFs vctor w and v ar givn by { w = {w 1 θ z1 w w 3 w n w n+1 θ zn+1} T v = {v 1 θ y1 v v 3 v n v n+1 θ yn+1 } T. (8) Th unknown fild function u(ξ, t) is intrpolatd by Φ j as u(ξ, t) = Φ j T a u, whr transformation matrix T a is givn by [16], as (9)

4 45 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts T a = ([ΦT j (ξ 1)Φ T j (ξ )...Φ T j (ξ n)φ T j (ξ n+1)] T ) 1, (1) and physical DOFs vctor u is givn by u = {u 1 u u n u n+1 } T. (11) Th unknown fild function θ x (x, t) is intrpolatd by convntional linar bass as θ x (x, t) = Nθ x, (1) whr N = {N 1 N }, (13) whr {N1 = 1 ξ, N = ξ and physical DOFs vctor θx is givn by θx = { } T θ x1 θ xn+1. Substitution Eqs (6), (9) and (11) into Eqs () and (3), rspctivly, w obtain (14) (15) U = 1 (w ) T K by (w ) + 1 (v ) T K bz (v ) + 1 (θ x )T K tx (θ x ) + 1 (u ) T K ax (u ), (16) T = ( w ( v ) T M by ) T M rz ( ) θ T x M tx ( w ( v ( θ x ) + 1 ) + 1 ) + 1 ( v ( w ( ) v T ) M bz + 1 ( ) w T ) M ry ( ) w T G (v ) 1 ( ) v T G (w ), (17) ( ( ) u u ) T M ax whr th lmnt bnding stiffnss matrics K by and K bz, torsion stiffnss matrix K tx and axial stiffnss matrix K ax ar K by = EI z l 3 (T b ) T Γ, T b, K bz = EI y l 3 (T b )T Γ, T b, K tx = GJ x l A 1,1, K ax = EA l (T a) T Γ 1,1 1 T a, th lmnt translational mass matrics M by, M bz and M ax ar (18) (19) () (1) M by = M bz = ρal (T b) T Γ, T b, () M ax = ρal (T a )T Γ, 1 T a, th lmnt rotatory inrtia mass matrics M ry and M rz ar M ry = ρi z l (T b) T Γ 1,1 T b, (3) (4)

5 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts 451 M rz = ρi y l (T b )T Γ 1,1 T b, th lmnt torsion mass matrix M tx is M tx = ρj xl A,, and th lmnt gyroscopic matrix G is whr G = ΩJ xρ l (T b) T Γ 1,1 T b, Γ, = Γ 1,1 = Γ, = d Φ T 4j dξ dφ T 4j dξ d Φ 4j dξ dξ, dφ 4j dξ dξ, Φ T 4jΦ 4j dξ, Γ 1,1 dφ T j dφ j 1 = dξ dξ dξ, Γ, 1 = A 1,1 = Φ T jφ j dξ, dn T dξ dn dξ dξ, (5) (6) (7) (8a) (8b) (8c) (8d) (8) (8f) A, = N T Ndξ. Applying Hamilton s principl, th lmnt fr vibration quation can b obtaind as w M by + M ry v G M bz + M rz M tx + G M u ax θx + K by K bz K ax K tx w v u θ x = w v u θ x Rwrit Eq. (9) according to th layout of lmnt physical DOF as mntiond on Eq. (5), w hav (8g). (9) M δ δ + G + K δ =, (3) whr M, G and K ar th lmnt mass matrix, lmnt gyroscopic matrix and lmnt stiffnss matrix that rarrang from Eq. (9) according to Eq. (5) by lmntary transformation.

6 45 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts.. Incorporation of intrnal damping As all ral matrials hav th capability of dissipating mchanical nrgy, it is important that, for high-spd opration, th intrnal damping is ncssary to b considrd in th finit lmnt modl of rotor-baring systms. Zorzi and Nlson [6] considrd th combind ffcts of both viscous and hystrtic intrnal damping in thir finit lmnt formulation of th rotor-baring systm. Ku xtndd th intrnal damping modl to whirl spd and stability of Rayligh-Timoshnko shafts [7]. Grnhill t al. also mployd th intrnal damping modl to construct conical bam lmnt for rotor dynamic analysis [4]. Using η V and η H to dnot th viscous damping cofficint and th hystrtic loss factor of th shaft matrial, according to th formr studis, vibration quation can b xprssd as M δ + (η V K c1 + G ) δ + (η ak + η b K c)δ = F, (31) whr η a = 1 + η H 1 + η H η b = η H, + Ωη 1 + η V H and matrics K c1 and K c ar { K c1 = R 1 K K c = R K, (3) (33) whr th transformation matrix R 1 is dfind by R 1 = R 1 1 R1 R 1... R 1 R 1 1 (3 j +9) (3 j +9), (34) whr 1 R 1 1 = 1 and R 1 = 1, (35) th transformation matrix R is dfind by R 1 R R R =... R R 1 whr (3 j +9) (3 j +9), (36)

7 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts R 1 = 1 and R = 1, (37) and F is th vctor of xciting forcs..3. Barings Th classic linarizd modl with ight spring and damping cofficints is mployd for th modling of baring in th prsnt work. Th forcs at ach baring ar assumd to oby th govrning quations of th following form C b δ b + Kb δ b = Fb, whr F b is th vctor of th baring forcs, δ b is th vctor of baring DOFs, th baring damping matrix Cb and stiffnss matrix K b ar [ ] C b Cww C = wv [ C vw C vv ], K b Kww K = wv K vw C vv whr C ij and K ij ar th baring damping and stiffnss cofficints..4. Discs Th discrt disc, which is thin and symmtric about th axis of rotation, has th following form of govrning quations M d δ d + Gd δ d = Fd, whr F d is th 6 1 vctor of th xciting forcs, δd is th 6 1 vctor of rigid disc DOFs, th disc gyroscopic matrix G d and mass matrix M d ar dfind by G d = Jx d, (41) Jx d J d M d = Jx d, (4) J d Jx d whr th rigid disc diamtrical and polar mass momnts of inrtia ar J d = m ( ) d rd 4 + h d 3 Jx d = m drd, whr h d is th disc thicknss, r d is th disc radius and m d is th disc mass. (38) (39) (4) (43)

8 454 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts b C K b d C b b K L Fig.. Rotor-baring systms..5. Systm quation of motion Th quations of motion of th complt systm can b obtaind by assmbling lmnt vibration quation, that is M δ + G δ + Kδ = F, whr M, G, K and F ar systm mass matrix, gyroscopic and damping matrix, stiffnss matrix and xciting forc, rspctivly. For th analysis of natural whirl spds and instability thrsholds of th rotor-baring systm, th forc trm can b omittd. Th right hand of systm quations of th motion thn is st to zros. Nglcting th xciting forc, Eq. (44) is writtn in th first ordr stat vctor form as whr E q + Fq =, q = E = F = [ δ δ ], [ M M G [ ] M. K ], Th associatd ignvalu problm for Eq. (45) is sought from an assumd solution form as (44) (45) (46) (47) (48) q = q λt, (49) Substituting Eq. (49) into Eq. (46), th global fr vibration frquncy quations ar givn by Eλ + F =, (5) whr λ = σ + i = σ + i πf is th complx ignvalu. (rad/s) is th natural whirl spd, f (Hz) is th modal frquncy of structural dynamic systms. σ rprsnts th instability thrshold whn σ >. Th paramtr of logarithmic dcrmnt δ is dfind as δ = πσ, whr δ rprsnts th instability thrshold whn δ <. (51)

9 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts 455 Tabl 1 Whirl spds in rad/s of a uniform shaft with isotropic undampd flxibl barings at a spin spd of 4 rpm η H =. η V =. s Mod Prsnt Rf. [7] Rf. [8] Rf. [5] Prsnt Rf. [7] Rf. [8] Rf. [5] 1F B F B F B F B Numrical studis Tabl Logarithmic dcrmnt δ of a uniform shaft with isotropic undampd barings at a spin spd of 4 rpm (suppos η H =.) Mod Prsnt Rf. [6] Rf. [7] Rf. [5] 1F B F B F B F B Exampl 1 A stl shaft having diamtr.116 m and lngth 1.7 m supportd by two idntical isotropic barings at both th nds is modld by on BSWI4 3 rotating Rayligh-Eulr bam lmnt with 33DOFs (Computd by = 33). Th matrial proprtis of shaft ar: Young s modulus E = Pa, matrial dnsity ρ = 7833 kg/m and Poisson s ratio µ =.3. Suppos th hystrtic damping η H =. and viscous damping η V =. s, whras th isotropic barings having stiffnss cofficints K wv = K vw = and K ww = K vv = N/m. Th prsnt rsults ar found to b in good agrmnt with thos obtaind by Zorzi and Nlson [6], Ku [7], Kalita and Kakoty [8] and ÖzgÜvn and Özkan [5] as shown in Tabl 1. Th whirl spd maps of this xampl ar prsntd in Fig. 3(a) and (b), rspctivly. Th lttrs F and B rfr to th forward and backward prcssional mods. It has bn obsrvd that for th shaft matrial with hystrtic damping η H =., critical spds for th first thr forward mods ar found to b 4976 rpm ( = Ω), 1476 rpm ( = Ω) and 1558 rpm ( = 3Ω), rspctivly, whras th first thr backward natural whirl spds ar 4969 rpm, 143 rpm and 198 rpm, rspctivly. Whn th viscous damping η V =. s, th critical spds for th first and scond forward mods ar 4989 rpm and 1488 rpm, rspctivly, whras ths ar agr with 496 rpm and 15 rpm by Ku [7] and 5 rpm and 178 rpm by Kalita [8]. Th third forward mod is 17 rpm and th first thr backward natural whirl spds ar 4985 rpm, 1466 rpm and 1184 rpm, rspctivly. To dmonstrat th accuracy of wavlt-basd finit lmnt modl that considrs th ffct of baring damping, a comparison btwn th prsnt solutions of logarithmic dcrmnts δ and thos which ar obtaind by using othr finit lmnt modls, is summarizd in Tabl. Suppos th shaft rotating with a spcific spd Ω = 4 rpm for η H =.. From Tabl, w can s that th prsnt solutions ar in clos agrmnt with th publishd rsults, whras 7 traditional lmnts with 48 DOFs wr usd in th publishd litraturs. Exampl As a scond xampl, th rotating shaft studid in th first xampl with th sam gomtry and matrial constants is usd, but is supportd at th two nds by flxibl dampd barings. Suppos th rotating spd Ω = 4 rad/s. Th stiffnss cofficints of th barings ark ww = K vv = N/m, K wv = K vw = N/m and th damping cofficints ar C ww = C vv = Ns/m and C wv = C vw =. Only on BSWI rotating Rayligh-Eulr shaft lmnt with 33DOFs is usd to modl th rotating shaft. Th natural frquncis and logarithmic dcrmnts δ of th prsnt lmnt and traditional finit lmnt mthod with 7 DOFs ar

10 456 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts 3F 3B Natural whirl frquncy / rad/s =3Ω = Ω = Ω F B 1F 1B Spin spd Ω/rpm (a) 4 x 1 5 3F 3B Natural whirl frquncy / rad/s =3Ω = Ω = Ω F B 1F 1B Spin spd Ω/rpm (b) 4 x 1 Fig. 3. Whirl spd maps of a rotor-baring systm supportd on undampd isotropic baring. (a) Th shaft matrial with hystrtic damping η H =., and (b) viscous damping η V =. s.

11 J.W. Xiang t al. / A class of wavlt-basd Rayligh-Eulr bam lmnt for analyzing rotating shafts 457 Tabl 3 Th comparison of dampd frquncis of th rotor systm of Exampl Mod Prsnt Mohiuddin [6] (rad/s) δ (rad/s) δ 1F B F B F B F B st torsional mod F B st axial mod F B tabulatd in Tabl 3. Th comparison shows that th prsnt lmnt produc modal charactristics ar in good agr with thos of traditional finit lmnt mthod computd by Mohiuddin, and Khulif [6]. In ordr to gain th sam analytical prcision, th prsnt mthod nds only a half of solving DOFs of traditional finit lmnt mthod. Th first torsional and axial modal frquncy ar rad/s and rad/s, rspctivly, and th corrsponding logarithmic dcrmnts δ is clos to th instability thrshold. This numrical xampl conform th high prcision of wavlt-bas finit lmnt modl. In viw of th abov, th modl can b applid to modl rotor-baring systms, which considr intrnal damping, baring stiffnss, gyroscopic momnts, axial dformation and torsion momnt, tc. 4. Conclusions Th objctiv in this papr is to prsnt a novl class of BSWI Rayligh-Eulr rotating bam lmnts to analyz rotor-baring systm. In this papr, th concpts of th wavlt finit lmnt mthod and th traditional linar intrpolation to formulat BSWI rotating bam lmnt ar usd. Th complx notation and formulation mployd hav provd to b asily managabl and computationally fficint. Good agrmnts of numrical xampls ar obtaind btwn th wavlt-basd lmnt and th othr publishd litraturs. Th proposd wavlt-basd lmnts ar suitabl to dal with high prformanc computation for rotor-baring systm. Advantags of th prsnt mthod ovr th convntional finit lmnt mthod ar th lowr solving DOFs and various wavlt-basd lmnts undr diffrnt wavlt scals. Acknowldgmnts Authors ar gratfully acknowldging th financial support by th projcts of National Natural Scinc Foundation of China (Nos. 5858, ), Youth Scinc Foundation of GuangXi Provinc of China (No. 838), Opn Foundation of th Stat Ky Laboratory of Structural Analysis for Industrial Equipmnt (No. GZ815), GuangXi Ky Laboratory of Manufacturing Systm & Advanc Manufacturing Tchnology (No Z). W also gratfully thank two anonymous rviwrs for thir suggstions. Rfrncs [1] F.F. Ehrich, Handbook of rotordynamics, McGraw-Hill Inc, Nw York, 199. [] G. Mng, Rtrospct and prospct to th rsarch on rotordynamics, Chins Journal of Vibration Enginring 15(1) (), 1 9. (In Chins)

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