SOME ASPECTS OF RUBBING MECHANISM IN ROTORDYNAMICS

Transcription

1 AITC-AIT 6 Internatonal Conference on Trbology - September 6, Parma, Italy SOME ASPECTS OF RUBBING MECHANISM IN ROTORDYNAMICS N. BACHSCHMID, P. PENNACCHI Poltecnco d Mlano, Dpartmento d Meccanca Va La Masa, 34, 156 Mlano, Italy ncolo.bachschmd@polm.t, paolo.pennacch@polm.t, ABSTRACT In recent tmes, the occurrence of full annular rub or partal rub condtons n turbomachnery has become frequent due to the reduced clearances used to mprove the effcency n new desgn machnes. In the past, the authors have developed a rather sophstcated model to smulate and reproduce the phenomenon of spral vbratons, whch s generated by the nteracton of the vbratng shaft n the full annular rub condton wth the thermal bow of the shaft, nduced by the heatng due to the frcton forces. In the present paper other aspects are analysed: rub of short duraton, whch can occur for nstance when the shaft s passng a crtcal speed durng the coast down transent, s not able to nduce consstent thermal bows and the assocated spral vbratons, but should also be dentfed by sutable measurements. From lterature t s known that these phenomena can be dentfed by means of vbraton measurements on the stator of the machne as well as by the assocated emtted nose. The frcton forces affect also the rotatng speed of the shaft: the accurate measurement of the speed and of ts harmonc components could also allow to dentfy a rubbng condton. These aspects are nvestgated by means of the MODAROT test-rg whch had been modfed to allow rub condtons to occur. A model of the test-rg and of the partal arc rub condton has also been set, n order to verfy the effects of the rub on the vbratons, nose and rotatng speed. The behavour of the model s analysed n the tme doman, due to the non-lnearty of the phenomenon. KEYWORDS Rotor-to-stator rub, rotor dynamcs, contacts, frcton. 1 INTRODUCTION Rubbng phenomena can occur frequently n rotatng machnery, especally when clearances are tght and problems n algnment condtons between rotatng part and statonary parts of the

2 machne have shown. Ths can occur n the start up procedures of a new machne or of an older machne after an overhaul. The problem can also occur due to a thermal bow of the shaft, or to a thermal dstorton of the statonary casng. Rubs can be of two types: type 1 when the contact occurs n one pont or n a short arc of the statonary part and type when the contact occurs on a long arc (partal arc rub) or on the complete crcumference (full annular rub) of the statonary part. Type 1 occurs when the msalgnment (the statc offset of the rotatng shaft) s hgher then the vbraton ampltude n correspondence of the secton where the rub occurs, then the contact occurs n one pont only of the statonary part and the contact pont s sldng on the surface of the shaft. The contact may be contnuous or ntermttent (also wth rebounds) dependng on stffness and dampng of both rotatng and statonary part, and on the sze and shape of unbounded orbt as mposed by unbalance and bow of the shaft. If the contact s contnuous, the heat ntroduced n the shaft due to frcton has a polar symmetrcal dstrbuton and does not produce any addtonal bow. The heat ntroduced n the statonary part through the hot spot generally on a sealng rng mounted on the casng of the machne has only lttle effect on the development of the rub condtons. Many of the physcal phenomena whch occur durng rotor to stator rub have been descrbed frstly n a lterature survey of Muszynska [1]. If the contact s contnuous and a full annular rub s generated, the frcton force may excte a very dangerous dry frcton reverse whrl n smaller machnes. If the contact s occasonal and assocated to rebounds, also chaotc moton can result n lght shafts. Durng the contact, the stffness felt by the rubbng shaft s hgher than the flexural stffness of the shaft: durng one rotaton the shaft experences a non-lnear stffness and has consequently the typcal behavour of a vbratng system wth non-lnear stffness. All these phenomena have never been measured on bg szed ndustral machnes lke steam turbnes and generators, whch are the man obects of the present analyss on rubbng rotors. When the contact s not contnuous, then the frcton generated heat s ntroduced only durng a lmted angle of rotaton, durng whch the contact pont s sldng on a lmted arc of the shaft crcumference. The part of the orbt where the whrlng shaft gets n contact wth the statonary part s defned by the radal deflecton: where radal deflecton exceeds the remanng clearance, calculated by consderng the statc offset of the shaft due to algnment condtons, there contact occurs. The same contact condtons are repeated each revoluton. Sometmes, when the rub s heavy enough, the heat due to frcton ntroduced through the ntermttent contact pont along the same arc n the shaft, provdes unsymmetrcal heatng of the shaft whch gets a thermal bow. In ths case also the vbratons (measured n general n correspondence of the bearngs of the machne) change and the vbraton vector can experence a spral development (spral vbratons) wth tme. The spral can be wth ncreasng ampltude (unstable spral) or wth decreasng ampltude (stable spral), n ths last case the phenomenon wll become generally perodcal. The shaft starts rubbng, the thermal bow generated by the heat deflects the shaft n such a way that the shaft looses the contact. When heat s agan dstrbuted unformly the process wll repeat. Type of rub occurs when the unbounded orbt of the shaft exceeds the clearance durng a consstent part of the shaft revoluton (partal arc rub), or durng the complete revoluton (full annular rub). Durng the revoluton of the shaft always the same pont or short arc (hot spot) s n contact wth the statonary part and sldes along the arc or the complete crcumference of the statonary part or casng of the machne. All the heat developed by frcton s ntroduced n the rotatng shaft through the same pont (hot spot), and the shaft gets a thermal bow. In ths stuaton the development of spral vbratons s rather common. Spral vbratons have been descrbed frstly n [] and later roughly modelled frstly by Kellenberger [3]. Ths phenomenon

3 has then been extensvely analysed by means of sutable models both for the thermal behavour and for the consequent vbraton behavour n [4][5]. In recent years when the requrement of hgher effcency n turbo-machnery has reduced the clearances between rotatng shafts, mpellers and blade rows and statonary parts, the analyss of rubbng rotors has become a matter of great nterest. From a dagnostc pont of vew t s necessary to recognze rubs from few measurements. For rubs of type spral vbratons consttute a strong symptom of full annular or partal arc rub, but sometmes also spral vbratons were attrbuted to other causes (whrlng moton n ol-flm bearngs). Rubs of type 1 are more dffcult to be recognzed, because apparently they affect only lghtly the vbratons of the casng ntroducng some hgh frequency nose, whch could be detected by accelerometers or also by acoustc measurements [6]. A theoretcal and expermental research has been started n order to analyse the dynamcal behavour of a rubbng rotor wth a rub of type 1. A test rg wth a shaft on two ol-flm bearngs has been modfed to allow dfferent measurements n rubbng condtons. Frstly a theoretcal nvestgaton has been started for analysng all possble symptoms of a rotatng shaft, whch s brought n contact wth an obstacle, whch s also a vbratng system, equpped wth a partal arc seal rng. In the present paper the man results of ths nvestgaton are presented. For the smulatons the model of the test rg has been used, the rotor s represented by a f.e.m. wth 5 degrees of freedom per node (to nclude also torsonal vbratons), the obstacle s a 6 d.o.f. system wth a sprng representng the seal rng, the normal contact force s calculated consderng the (lnear) stffness of the seal rng sprng and ts deflecton, when the shaft orbt exceeds the clearance. The tangental frcton force s calculated consderng the frcton coeffcent tmes the normal contact force. The frcton force exctes both flexural and torsonal vbratons. Orbts n bearngs and n correspondence of the seal rng are calculated wth and wthout rub ntegratng the equatons n the tme doman. Also vbratons of the obstacle, contact force and torsonal vbratons are calculated each tme step. DESCRIPTION OF THE MODEL In ths secton the complete model of the system s descrbed, even some smplfcaton n regards to the supportng structure wll be actually ntroduced n the followng. The rotor s modelled by means of fnte beam elements, takng also nto account the shear and the secondary effect of rotatory nerta. Due to the fact that not only the lateral response but also the torsonal vbraton wll be consdered, 5 d.o.f.s (two translatonal and three rotatonal) are consdered per each node. Axal vbratons wll be neglected. The generalzed dsplacement ( ) vector x r of the rotor -th node s: { x y ϑ ϑ ϑ } T x = (1) x y z Two subsequent nodes, the -th and the +1-th, defne the -th element of the machne as shown n fgure 1 along wth the reference system on the rotor. If the rotor has n r nodes, thus nr 1 elements, the vector x of the generalzed dsplacements of all the rotor nodes s ( ) composed by all the ordered vectors x r as shown n eq. (): { x1 y1 ϑ ϑ ϑ x y ϑ ϑ ϑ r r n n n } x y z n n x y z r r r x = () T

4 Dfferent methods can be used to model the foundaton. For the sake of brevty, only pedestals,.e. lumped d.o.f.s systems, wll be consdered. A dscusson about other methods s reported n [7]. In a smlar manner to the rotor, also the d.o.f.s of the foundaton, horzontal and vertcal dsplacements, whch are connected by the n b bearngs to the rotor, can be ordered n a vector: { x y x } T n yn x = (3) ( f ) ( f ) ( f ) ( f ) ( f ) 1 1 b b Fgure 1. Reference systems of a general rotor element. Fnally, the housng of the seal strp and ts housng wll be consdered as the rgd body P (see Fgure ) that has 6 d.o.f.s: x = { x y z ϑ ϑ ϑ } P P P P xp yp zp T The complete vector of the generalzed dsplacements of the model s therefore: ( f ) { } T P x= x x x (5) Usng Lagrange s method and neglectng the effect of external forces and of the rub, the d.o.f.s of the rotor, of the foundaton and of the housng can be consdered separately: d T T U δw + = dt x x x δ x δw M x + ( C +Ω G ) x + K x = δ x (4) (6) d T T U δw δw + = + + = dt x x x δx M x C x K x δx ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) ( f ) (7) d T T U δw δw + = [ MP] xp + [ CP] x P + [ KP] xp = dt x P xp xp δ xp δ x P (8)

5 Fgure. General layout of the test-rg. In regards to the rotor, the mass matrx [ M ], whch takes also nto account the secondary ( r ) ( r ) effect of the rotatory nerta, the nternal dampng matrx [ C ], the stffness matrx [ K ], whch takes also nto account the shear effect, and the gyroscopc matrx ], all of order (4n 4 n ), can be defned by means of standard Lagrange s methods, beam elements and r r ( f ) ( f ) ( f ) lumped dsks as shown e.g. n [8][9]. Whlst the structure of [ M ], [ C ] and [ K ] s not relevant at ths stage and depends on how the pedestals are mplemented, the matrces of the housng are all smply dagonal matrces: [ M ] = dag( m, m, m, I, I, IP ) P P P P Px Py z [ G ( r ) (9) [ C ] = dag( c, c, c, c, c, c ) (1) P Px Py Pz tpx tpy tpz [ K ] = dag( k, k, k, kt, kt, kt ) (11) P Px Py Pz Px Py Pz In order to consder the r.h.s of eqs. (6) and (7), the test-rg rotor s supported by n b ol-flm bearngs that realze a couplng between the rotor and the supportng structure. Even f the exact calculaton of the forces exchanged between the ournal and the bearng case, due to the

6 ol-flm, requres approprate methods, the dscusson of whch s far from the scope of present paper, a wdely accepted smplfcaton n rotor-dynamcs smulaton (see agan [8]) s the modellng of the ol-flm force feld by means of lnearzed stffness and dampng coeffcents functon of the rotatng speed. Therefore, the expresson of the lnearzed forces of the ol-flm of the -th bearng on the rotor ournal located n the -th node, due to the rotor d.o.f. dsplacements only, s: F ( br) x x k ( Ω) k ( Ω) r ( Ω) r ( Ω) ( ) k ( ) k ( ) y r Ω Ω r ( Ω) r ( Ω) ( Ω ) = ϑ x = ϑy ϑz ( b) ( b) ( b) ( b) xx xy xx xy ( b) ( b) y ( b) ( b) yx yy ( r ) ϑ yx yy x ϑ y ϑz ( b) ( b) ( ) ( ) = K Ω x C Ω x whle that of the forces on the supportng structure, correspondng to the -th bearng and due to the foundaton d.o.f. only, s: F k ( Ω) k ( Ω) x r ( Ω) r ( Ω) x Ω = = ( b) ( f ) ( b) ( f ) = K ( Ω) x ( Ω) C x ( b) ( b) ( f ) ( b) ( b) ( f ) ( bf ) xx xy xx xy ( ) ( b) ( b) ( f ) ( b) ( b) ( f ) kyx ( Ω) k ( ) ( ) ( ) yy Ω y ryx Ω r yy Ω y The actual bearng force between the rotor and the foundaton s gven by the dfference between eq. (1) and eq. (13). Ths way the couplng effect of the ol-flm forces s taken nto account by the relatve dsplacements of the nodes of the rotor and of the foundaton n correspondence of the bearngs and the fully assembled system of equaton s bult up. ( rr ) ( rf ) ( fr) Ths requres the defnton of the stffness couplng matrces [ K ], [ K ], [ K ], ( ff ) ( rr) ( rf ) ( fr) ( ff ) [ K ] and the correspondng dampng matrces [ C ], [ C ], [ C ], [ C ], whch are sparse and respectvely of order (5nr 5 nr), (5nr nb), (nb 5 nr) and ( nb nb). The structure for the stffness matrces s: K b ( K ) = Ω (14) ( rr) ( ) [ ] dag ( ) (1) (13) ( b) ( b) ( fr ) kxx ( Ω) k ( ) xy Ω [ K ] = ( b) ( b) kyx ( Ω) k ( ) yy Ω (15) k ( Ω) k ( Ω) ( ) ( Ω) = ( b) ( b) xx xy ( b) ( b) k ( ) yx Ω k rf yy [ K ] (16)

7 K b = ( K Ω ) (17) ( ff ) ( ) [ ] dag ( ) Dampng matrces have smlar structure, whle the dependence of Ω s omtted hereafter for the sake of brevty. Ths way, the fully assembled system of equatons, wthout exctaton, results: wth: [ Mx ] + [ Cx ] + [ Kx ] = (18) [ M ] M = M [ P ] M ( f ) [ ] [ ] (19) ( rr) ( rf ) [ C ] +Ω [ G ] + [ C ] [ C ] C = C C + C ( fr) ( f ) ( ff ) [ ] [ ] [ ] [ ] [ C ] P () ( rr) ( rf ) [ K ] + [ K ] [ K ] ( fr) ( f ) ( ff ) [ K] = [ K ] [ K ] + [ K ] [ KP ] Due to the layout of the test-rg n whch the drecton of the contact s manly horzontal, the rub between the rotor and the seal strp occurs when the relatve horzontal vbraton between the node correspondng to the rub secton and the housng exceeds the seal clearance δ. If ths condton s verfed, a force system s exchanged between the rotor and the stator, composed by: a normal component determned by the elastc reacton proportonal to the penetraton of the rotor n the sealng; a frcton tangental reacton determned by Coloumb s law; the torques caused by the aforementoned forces on the rotor and on the housng. In other terms: ( y t y t ) f ( ) ( ) > δ : 19 P d F housng node 19 N = k y t y t m19 c () t = μn N μn μn N μna + Nb, c ( () () δ ) 19 P The frcton torque actng on the rotor has the effect to decelerate the rotor. Anyhow, the actual test-rg s drven by an electrc motor that can hold a constant average rotatng speed. Snce the am of the paper s not to accurately model the motor controller, ts effect on the rotor s smply modelled by means of a motor torque actng on the frst rotor node that balances the effect of the frcton torque, when the contact happens: T (1) ()

8 ( y t y t ) f () () > δ : 19 P dm 19 M( t) = μn node 1 T The remanng external forcng systems actng on the rotor are the weght W and the resdual unavodable unbalance dstrbuton whch s modelled by means of a lumped external force n correspondence of the second nerta dsk (node 15): Funb () t = meω cos( Ω t+ ϕunb ) meω sn( Ω t+ ϕunb ) node 15 By consderng all the exctatons, the fully assembled system of equatons s non-lnear due to the rub effect and results: [ ] [ ] [ ] () t () t () t () t Mx + Cx + Kx= F + M + F + W= F + W (5) c unb The non-lnear system of equatons n eq. (7) s ntegrated n the tme doman usng the Newmark s mplct method, n whch the forcng vector F() t s recalculated at each tme step n order to evaluate f rub happens. The algorthm s the followng: 1. Startng from a sutable set of ntal condtons x and x for t =, whch actually correspond to a pont on the steady state orbt wthout rub for the rotor nodes and null dsplacements and veloctes for the housng, the force vector F() s calculated, usng eqs. ()-(4).. The ntal acceleraton vector s calculated by: ( ) x = [ M] F() + W [ C] x [ K] x (6) 1 3. Startng from the frst tme step, n a general tme step -th, the new force vector F( t ) s calculated and the generalzed dsplacements, acceleratons and veloctes are equal to: T (3) (4) 1 b F( t) + W+ [ M] + [ C] x aδt aδt 1 1 b 1 b x = [ M] + [ C] + [ K] + [ M] 1 [ C] x + aδt aδt aδt a 1 1 b + a [ M] Δt 1 [ C] x a a (7) x = ( x x 1 x 1Δt) a x 1 (8) aδt a b ( ) b 1 b 1 x (9) aδt a a x = x x 1 + x 1+Δt 1

9 The constants a and b of the Newmark s method are assumed respectvely equal to.5 and.5; ths s equvalent to the trapezum rule and assures that the mplct ntegraton s uncondtonally stable, wthout addng numercal spurous dampng (hgh values of b ). In fact t can be proven that Newmark s method s uncondtonally stable f b.5 and a.5(.5 + b). 3 SIMULATION RESULTS Some smulatons have been made usng the proposed method and mplementng the model of the test-rg. Snce n the actual layout the supportng structure s rgd n the speed range of analyss, the foundaton d.o.f.s are neglected n the calculaton. Three dfferent cases have been smulated n order to evaluate the dfferences n the dynamcal behavour when the test-rg s operatng below, close or over the frst lateral crtcal speed. From the Bode dagram of the system response wthout rub (Fgure 3) t s possble to see that, due to the system ansotropy, horzontal and vertcal crtcal speeds are actually splt. Anyhow t wll be decded to consder as crtcal speed that of 19 rpm, due to the hgh dynamcal amplfcaton, even only n vertcal drecton. Proect name Strscamento Re vson 1. Date :44: Prepared Checked Approved 3.5 x Frequency response n node 19 (Bode plot) X: 19 Y:.3379 x y Module [m] RAFT Rotatng speed [rpm] 18 9 Phase [degrees] -9 RAFT Rotatng speed [rpm] Unbalance - node: 15 - module:. [kgm] - phase: [ ] Fgure 3. Bode dagram n the rubbng secton due to the consdered unbalance, w/o rub. Also the torsonal egenfrequences have been evaluated before the smulaton. The three frst ones resulted 13.48, and Hz. All the smulatons have been made by usng a tme step of 1e 4 s and consderng the duraton of 1 s. The frcton coeffcent μ has been consdered equal to.3.

10 The damped natural frequences of the obstacle (when not n contact wth shaft) have been set equal to 5.41, 43.1, 51.41, 7.73 (twce) and 74. Hz. The rotor s loaded by ts own weght and by an unbalance of 1e 4 kg m whch s set to brng the shaft n contact wth the seal. 3.1 Dynamcal behavour below the crtcal speed (7 rpm) At low speeds, n rubbng condtons, the sze of the shaft orbts n the bearngs (Fgure 4 and Fgure 5) and n the rubbng node (Fgure 6) are reduced wth respect to the non rubbng condton, and present some rregularty n ts shape n the angular postons where the rub occurs Rotor orbt - Node 4 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Rotor orbt - Node 9 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Fgure 4. Rotor orbts, bearng #1, rotatng speed 7 rpm. Fgure 5. Rotor orbts, bearng #, rotatng speed 7 rpm. Also some random nose s superposed to the steady state mean orbt. The spectra of the vertcal and horzontal shaft vbratons also show some hgher harmonc components (X, 3X and so on) and some nose n a hgher frequency range (6-1 Hz), as can be seen n Fgure 7 for bearng locaton. Hgher harmoncs and nose on the orbts are due to rubbng and are absent when contact does not exst Rotor orbt - Node 19 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Fgure 6. Rotor orbts, rubbng node, rotatng speed 7 rpm. Ampltude Ampltude 1 Node 9 - Vbraton spectrum Horzontal Vertcal X: 11.7 Y:.4174 Horzontal Vertcal Fgure 7. Shaft vbraton spectra, bearng #, rotatng speed 7 rpm (low frequency range). The normal contact force shown n Fgure 8 s ntermttent: durng the theoretcal rubbng arc, n whch contact can occur, contact occurs randomly between and 4 tmes, and also ts

11 duraton vares randomly between two extreme values. Surprsngly the maxmum contact force s almost constant. The spectrum of the contact force (Fgure 9) shows a rather rch content n the hgher frequency range (15-5 Hz). The ntermttent normal contact force, and consequently also the tangental frcton force, excte both the vbratons of the obstacle and the torsonal vbraton of the shaft. The spectrum of the horzontal component of the vbraton of the obstacle s shown n Fgure 1, whle the RMS value s.1415 μm. On real mechnes, accelerometers could be used to measure casng vbratons, therefore the RMS values of the acceleraton would be much hgher due to the hgher frequency content of the vbraton. These values could certanly be measured. 1 Contact force 1 Spectrum of the contact force Force [N] Tme [s] Ampltude [N] Ampltude [N] X: 11.7 Y: Fgure 8. Close up of the normal contact force, rotatng speed 7 rpm. Fgure 9. Normal contact force spectrum, rotatng speed 7 rpm (low frequency range). Several harmoncs of the rotatng speed frequency are excted but also some nose appears n the frequency range of the natural frequences of the obstacle. Ampltude 1 D.o.f Spectrum 1-5 Ampltude [μrad] 1 Node 19 - Torsonal vbraton spectrum Torsonal Ampltude Ampltude [μrad] X: 13.3 Y:.566 X: Y:.3711 X: 113 Y:.1776 Torsonal Fgure 1. Spectrum of the horzontal dsplacement of the obstacle, rotatng speed 7 rpm (hgh frequency range). Fgure 11. Torsonal vbraton spectrum, rubbng node, rotatng speed 7 rpm. The torsonal vbraton spectrum (Fgure 11) shows a strong exctaton of the frst 3 natural torsonal frequences and the RMS value s 1.77 μrad. Several symptoms seem to be avalable for detectng a rubbng condton: a) orbt dstorton n the bearngs whch could be measured by the standard proxmty probes of ndustral turbomachnery;

12 b) the change n vbratons of the statonary casng of the machne, although rather small could be detected by accelerometers placed on the casng; c) torsonal vbratons, whch normally are not measured n ndustral machnes, could represent an addtonal symptom. All these results need to be valdated by expermental results obtaned on a laboratory test-rg or better on a real machne. Snce the flexural natural frequences of the shaft on ts bearngs and supports are representatve of a full sze steam turbne, from dynamcal pont of vew the systems are smlar The possblty that these results could be transferred to bg szed ndustral machnes depends then on the scalng factor of contact forces and nerta. Regardng the shaft orbt dstorton n correspondence of the bearngs, the senstvty to rubbng contact force can be roughly estmated consderng the rato of contact force to statc bearng load: ths rato s around.5 for the model of the test rg. Smlar values could be sutable for real full sze machnes. Regardng the casng vbratons not only a scalng factor must be consdered, but the dynamcal behavour of the complete casng should be taken nto account, also n the hgh frequency range. Regardng the torsonal vbratons, the rotary nerta of full sze machnes s much hgher, the torsonal crtcal speed are at lower frequences, and ts exctaton depends also strongly on the locaton where rub s appled. Therefore t s dffcult to predct the behavour of full sze machnes. 3. Dynamcal behavour over the crtcal speed (15 rpm) The sze of the shaft orbts n the bearngs n rubbng condtons (Fgure 1 and Fgure 13) are somewhat bgger wth respect to the non-rubbng condton Rotor orbt - Node 4 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Rotor orbt - Node 9 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Fgure 1. Rotor orbts, bearng #1, rotatng speed 15 rpm. Fgure 13. Rotor orbts, bearng #, rotatng speed 15 rpm. Dstorton of shape s also remarkable. The contact force tme hstory (Fgure 14) s smlar to the stuaton below the crtcal speed, ts maxmum value s lower (ths depends on the actual nterference condtons whch are not controlled), ts frequency content (Fgure 15) s more rch n hgher frequency components.

13 8 Contact force 1 Spectrum of the contact force Force [N] Tme [s] Ampltude [N] Ampltude [N] Fgure 14. Close up of the normal contact force, rotatng speed 15 rpm. Fgure 15. Normal contact force spectrum, rotatng speed 15 rpm (low frequency range). The exctaton of the vbratons of the obstacle (Fgure 16) and of the torsonal vbratons (Fgure 17) s stronger, wth a respectve RMS value of.783 μm and.491 μrad. Ampltude 1 D.o.f Spectrum 1-5 Ampltude [μrad] 1 5 Node 19 - Torsonal Vbraton spectrum 1 Torsonal Ampltude Ampltude [μrad] X: 13.3 Y:.7471 X: 54.8 Y:.4984 X: 115 Y:.663 Torsonal Fgure 16. Spectrum of the horzontal dsplacement of the obstacle, rotatng speed 15 rpm (hgh frequency range) Fgure 17. Torsonal vbraton spectrum, rubbng node, rotatng speed 15 rpm. 3.3 Dynamcal behavour close to the crtcal speed (19 rpm) Approachng the crtcal speed the behavour changes from one revoluton to the other and a quas steady state stuaton s reached only after about 3 seconds (see e.g. the contact force tme hstory of Fgure ). The orbts of the shaft n the bearngs (Fgure 18 and Fgure 19) are hghly enlarged and twsted, but are smooth wth small rregulartes. Ths could be attrbuted to the resonance effects whch flters hgher harmonc components. The contact force (Fgure ) appears 1X and s contnuous durng rubbng (not ntermttent), therefore the vbratons of the obstacle (Fgure 1) and the torsonal vbratons (Fgure ) are weakly excted n the hgher frequency range. Ther RMS s respectvely μm and μrad.

14 -4 Rotor orbt - Node Rotor orbt - Node 9 Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Orbts w/ rub Last orbt w/ rub Steady state orbt w/o rub Fgure 18. Rotor orbts, bearng #1, rotatng speed 19 rpm Fgure 19. Rotor orbts, bearng #, rotatng speed 19 rpm. 7 Contact force 1 5 D.o.f Spectrum Force [N] Ampltude Tme [s] Ampltude Fgure. Close up of the normal contact force, rotatng speed 19 rpm. Fgure 1. Spectrum of the horzontal dsplacement of the obstacle, rotatng speed 19 rpm (low frequency range). 1 5 Node 19 - Torsonal vbraton spectrum Torsonal Ampltude Torsonal Ampltude 1.5 X: 13.4 Y:.87 X: 545 Y: Fgure. Torsonal vbraton spectrum, rubbng node, rotatng speed 19 rpm.

15 4 CONCLUSION The smulaton of the dynamcal behavour of the rubbng rotor shows that several dfferent symptoms could be avalable to detect a rubbng condton, also f the rub s lght. References [1] Muszynska, A., Rotor to statonary element rub related vbraton phenomena n rotatng machnery lterature survey, Shock and Vbraton Dgest, 3-11 (1989). [] Kroon, R.P., Wllams, W.A., Spral vbratons n rotatng machnery, 5 th Int. Conf. of Appled Mechancs, Wley N.Y., 71 (1937). [3] Kellenberger, W., Spral vbratons due to the seal rngs n turbogenerators: thermally nduced nteracton between rotor and stator, Journal of Mechancal Desgn,, (198). [4] Bachschmd, N., Pennacch, P. and Venn, P., Spral Vbratons n Rotors Due to a Rub, Proc. of IMechE-7 th Int. Conf. on Vbratons n Rotatng Machnery, 1-14 September, Unversty of Nottngham, UK, (). [5] Bachschmd, N., Pennacch, P. and Vana, A., Rotor-to-stator rub causng spral vbratons: modellng and valdaton on expermental data of real rotatng machne, IMechE paper C63/8/4, Proc. of 8 th Internatonal Conference on Vbratons n Rotatng Machnery, 7-9 September 4, Swansea, Wales, (4). [6] Stegemann, D., Remche, W., Beermann, H. and Suedmersen, U., Analyss of short duraton rubbng processes n steam turbnes, VGB Kraftwerkstechnk, 73, n.1, (1993). [7] Pennacch, P., Bachschmd, N., Vana, A., Zanetta, G.A. and Gregor, L., Use of Modal Representaton for the Supportng Structure n Model Based Fault Identfcaton of Large Rotatng Machnery: Part 1 Theoretcal Remarks, Mechancal Systems and Sgnal Processng,, No. 3, (6). [8] Lalanne, M. and Ferrars, G., Rotordynamcs Predcton n Engneerng, John Wley & Sons Inc, Chchester, England, (1998). [9] Chlds, D., Turbomachnery Rotordynamcs, John Wley & Sons Inc, Chchester, England, (1993).