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1 Jounal home page: Non-lnea dynamc of oto stato system wth non-lnea beang cleaance Comptes Rendus Mecanque, Volume 33, Issue 9, Septembe 4, Pages Jean-Jacques Snou and Fabce houveez NON-LINEAR DYNAMIC OF ROOR-SAOR SYSEM WIH NON-LINEAR BEARING CLEARANCE DYNAMIQUE NON-LINEAIRE D'UN ENSEMBLE ROOR-SAOR COMPORAN UN ROULEMEN NON-LINEAIRE AVEC JEU Jean-Jacques Snou* and Fabce houveez Laboatoe de bologe et Dynamque des Systèmes UMR CNRS 553, Ecole Centale de Lyon, 36 avenue Guy de Collongue, 6934 Ecully, Fance. ABSRAC he study deals wth a oto-stato contact nducng vbaton n otatng machney. A numecal oto-stato system ncludng a nonlnea beang wth Hetz contact and cleaance s consdeed. o detemne the non-lnea esponses of ths system, nonlnea dynamc equatons can be ntegated numecally. But ths pocedue s both tme consumng and costly to pefom. he am of ths pape s to apply the Altenate Fequency/me Method and the path followng contnuaton n ode to obtan the non-lnea esponses to ths poblem. Next, obts of oto and stato esponses at vaous speeds ae pefomed. Keywods: dynamc systems, oto dynamcs, nonlnea analyss, beang cleaances, contact. RESUME Une étude potant su la dynamque non-lnéae d un système dans les machnes tounantes est pésentée. Nous consdéons un système oto-stato compotant un oulement non-lnéae avec jeu et contact de Hetz. Afn de détemne la éponse non-lnéae de ce système, les équatons dynamques non-lnéaes peuvent ête ntégées numéquement. Cependant, cette pocédue est coûteuse en teme de temps de calcul et de essouces. Le but de ce pape est de popose l applcaton d une méthode de balance hamonque pou détemne la éponse non-lnéae du système. Ans, les obtes du oto et du stato sont obtenus pou dfféentes vtesses de otaton. Mots-clés: dynamque des systèmes, dynamque des otos, analyse non-lnéae, oulement avec jeux, contact.. INRODUCION he motvaton of ths study comes fom vbaton poblems nduced by oto-stato contact n tubo-machney. In fact, vaous types of non-lnea phenomena and effects appea such as otostato contact and cleaance beang [-]. Dung the ecent yeas, the undestandng of the dynamc behavou of systems wth non-lnea phenomena have been developed n ode to pedct dangeous o favouable condtons and to explot the whole capablty of stuctues though system usng n non-lnea ange. In geneal, tme-hstoy esponse solutons of the full set of non-lnea equatons can detemne the vbaton ampltudes but ae both tme consumng and costly when paametc desgn studes ae needed. Due to the fact that such non-lnea systems occu n many dscplnes of engneeng, consdeable wok has been devoted to development of methods fo the

2 appoxmaton of fequency esponses. One of the most popula method s the Altenate Fequency/me doman (AF) method [3], based on the balance of hamonc components. In ths study, a oto/stato system wth beang, ncludng Hetz contact and cleaance s fstly pesented. Secondly, the effcency of both AF method and path followng contnuaton s demonstated n ode to obtan the non-lnea behavou of a oto-stato system wth beang, ncludng Hetz contact and cleaance; ths method allows to save tme n compason wth a classcal Runge-Kutta ntegaton, by tansfomng non-lnea dffeental equatons nto a set of non-lnea algebac equatons n tems of Foue coeffcents.. ANALYICAL MODEL.. Nonlnea contact In ths model, the Hetz theoy s consdeed n ode to evaluate contact between the balls and the aces [4]. As llustated n Fgue, each ball can be located by ts angula poston θ k. hen, th the adal non-lnea contact foce geneated on the k ball can be defned as follows: 3 F ( ) ( ) adal = KH δ f δ (contact); Fadal ( ) = othewse (no contact) () whee δ and ae the adal cleaances value and the elatve adal dstance between the nne th and the oute aces of the k beang. can be expessed by consdeng hozontal and vetcal th dsplacement of the nne and oute aces of the k beang. One has = cos ( θ k )( xoute xnne ) + sn ( θ k )( youte ynne ). he effectve stffness K H s the combned stffness off a ball to nne ace and oute ace contacts and s defned by [4]: 3/ 3/ ( ) K = K + K () H o he ball-beang model unde consdeaton n ths study has equ-spaced balls ollng on the sufaces of the nne and oute aces. When the oute ng s fxed and the shaft otates, the angle θ k changes wth tme. hen, each ball s located by ts angula poston θ = ω k c. t+ π ( k ) N. hen the pecessonal angula velocty c c = + whee ω c, ω, R, R, N ae the otatonal speed of beang, the otatonal speed of oto, the oute damete of nne ng, the damete of balls, and the numbe of balls, espectvely. Next, the global beang th eacton can be obtaned by summng all the ndvdual contact expessons of each k beang. he total estong foce components n x and y dectons ae N ω of the balls s gven by ω Rω ( R R ) cos( θ ) sn ( θ ) (3) F = F F = F contact / x adal k contact / y adal k k= k=.. Roto-beang-stato model he oto-beang-stato system unde study has the oute ace of the ball beang fxed to a gd suppot and the nne ace fxed gdly to the shaft. A constant vetcal adal foce acts on the beang due to gavty. he exctaton s due to an of unbalance foce whch ntoduces a otatonal fequency. he beang s composed wth 6 balls and s modeled as explaned pevously, by consdeng the non-lneaty due to the Hetz contact wth cleaance. he complete oto-beangstato behavou can be epesented wth the followng equatons: N ( ω ) ( ω ) mx + cx + kx = F mx + cx + kx = meω cos t F my cy ky F mg my cy ky me t F mg s s s s s s contact / x e contact / x s s + s s + s s = contact / y s + + = e ω sn contact / y hs non-lnea system can be also wtten as follows Mx + Cx + Kx = f + f (5) (4)

3 x = xs ys x y. M, C and K ae the mass, the dampng and the stffness matces. and f nclude non-lnea tems, gavty and unbalance, espectvely. whee { } 3. NON-LINEAR MEHOD Both the hamonc balance method and the contnuaton schemes ae well-known numecal tools to study non-lnea dynamcs poblems. Howeve, the AF method seems aely used n engneeng applcatons, and moe patculaly n system wth cleaance and hetz contact. he geneal dea s to epesent each tme hstoy esponse by ts fequency content n ode to obtan a set of equatons ncludng balancng tems wth the same fequency components, and to stat an teatve appoach to obtan oots of these equatons [3]. In ths study, the AF method s used to fnd the esponse solutons of non-lnea oto-beang-stato equatons. 3.. Altenate fequency/tme doman method he non-lnea system (5) can be wtten n the followng way Mx + Cx + Kx + f x, ωτ, f x, ωτ, = g x, ωτ, = (6) ( ) ( ) ( ) whee M, C and K ae the mass, dampng and stffness matces. f s the vecto contanng non-lnea expessons due to the non-lnea contact. Settng x= x + x, x = x + x and x= x + x, the dsplacements x and x ae epesented wth tuncated Foue sees m hamoncs: m = m = x= X + [ X cos( ωt) + X sn( ωt) ], = + [ cos( ωt) + sn( ωt) ] f x X X X (7,8) n whch X, X and X, X, X and X ae the Foue coeffcents of x and x, espectvely. he numbe m of hamonc coeffcents s selected n ode to only take nto account the sgnfcant m + 4 lnea algebac equatons ae obtaned: hamoncs expected n the soluton. ( ) ( ) ω AX + F F + A + J X + Q = (9) n whch A and J ae the Jacoban matces assocated wth the lnea and non-lnea pats of (6). hey ae gven by A= dag ( KB B j Bm ) wth -ω M jωc B j =, -ω jc -( ω j) M+K and ( Γ ). f J = I.( Γ I ). x F and Q epesent the Foue coeffcents of f, and the Foue coeffcents of the devatve of g wth espect to ω, espectvely. F epesents the Foue coeffcents vecto of the non-lnea functon f. X and X contan the Foue coeffcents and Foue ncements of x and x, espectvely. F s dffcult to dectly detemne fom the Foue coeffcents fo many non-lnea elements. Howeve F can be calculated by usng an teatve pocess [3]: DF DF X x( t) f ( t) F whee DF defnes the Dscete Foue ansfom. he DF fom tme to fequency doman s gven by 3

4 ( ) ( ) (( ) π ( )) ( ) ( )( ) π ( ) m + fo = Γ j = m+ cos j m+ fo =,4,...,m m+ sn ( j m+ ) fo =,3,...,m+ he eo vecto R and the assocated convegence ae gven by fo j =,,...,m+ () R = AX+ F F () m m R ( R j R j) and δ = + ( j + j) δ = + + j= X X X (,3) j= 3.. Path contnuaton Usually, the system behavo s of nteest ove a ange of values fo at least one paamete (n ths study, the consdeed paamete s the speed of shaft otaton ω ). In ode to save tme and to obtan moe easly the soluton of the system by consdeng vaatons of paamete values, the path followng technque [3] can be used. In ths study, estmaton of the neghbong pont s obtaned by usng the Lagangan polynomal extapolaton method wth fou ponts. So, fou ponts on the soluton banch ae obtaned a po n ode to begn the extapolaton scheme. Any pont on the soluton banch s epesented at ( X, ω ), X and ω beng the Foue coeffcents and the fequency paamete, espectvely. he ac length between two consecutve ponts ( X +, ω+ ) and ( X, ω ) s gven by ( ) ( ) ( ) δs+ = X+ X X+ X + ω+ ω fo =, and (4) hen, the ac length paametes ae gven by S = ; S = δ s ; S = S + δs ; S = S + δs ; S = S + s (5) and by usng the Lagangan extapolaton scheme, the followng estmated pont at the dstance s can be defned by 3 3 S3 S j X [ X 4ω4] =. fo =,,...,3 S j S j ω (6) = = j 4. APPLICAION he AF method s appled to the oto-beang-stato system defned pevously. he value paametes ae gven n able. Fgue llustates the fequency esponse of ths system obtaned by usng the AF method wth the path followng contnuaton. he esonance peak s obseved nea 5.5 Hz. We can see that at fequences between -9 Hz, unbalance and gavty foces ae of the same ode ampltude, so that the oto and stato esponses ae complex, as llustated n Fgue 3 and n Fgue 4(a). In ode to obtan the non-lnea esponses fo the fequency ange -9 Hz, computatons ae pefomed by usng vaous powe hamoncs: wth 7 o moe fequency components, thee s no vsble dffeence between the obts obtaned wth Runge-Kutta pocess and AF method. When educng the numbe of hamoncs futhe to sx, only the AF method found a totally dffeent soluton. hs emphasses the poblem of the AF method: t s theefoe a method that n geneal can only be used f some a po knowledge about the system s avalable. he calculaton by usng the AF method wth 6, 8 and hamoncs components needs about, and 4 CPU seconds, espectvely. he calculaton by usng the 4 th ode Runge-Kutta pocess needs about 8 CPU seconds. 4

5 At fequences between 3-8 Hz, oto and stato ae always n contact and obts ae ccula and the fst fequency components ae suffcent ( m = ), as llustated n Fgue 3 and n Fgue 4(b). At fequences between - Hz, the same behavou can be obseved, and oto-stato ae always n contact due to the gavty effect. So, Fgue 5 shows the contact evoluton fo each ball of the beang whle nceasng the otaton speed. At fequences between -9Hz, the oto-stato contact s a complex phenomenon wth a successon of contact and no-contact peods. At fequences between 5-8Hz, oto and stato ae always n contact. As explaned pevously, an nteestng pont s the contact s evoluton dung the tanst phase aound -9Hz. As llustated n Fgue 5(b-e), complex non-lnea behavous ae obtaned. 5. SUMMARY AND CONCLUSION he Altenate Fequency /me doman method and the followng path contnuaton wee befly descbed. hey seem nteestng when tme hstoy esponse solutons of the full non-lnea equatons ae both tme consumng and costly. Moeove, extensve paametc desgn studes can be done n ode to appecate the effect of specfc paamete vaaton on the esponse of nonlnea systems. hs method was appled to a oto-beang-stato system wth nonlnea ball beang ncludng hetzan contact and adal cleaance. Complex obts and evolutons of the local contact between the balls and the aceways wee obtaned. 6. REFERENCES. F.F. Ehch, Handbook of otodynamcs, Macgaw-hll, 99.. J.M. Vance, Rotodynamcs of tubomachney, john Wley & Sons, S. Naayanan, and P. Seka, A Fequency Doman Based Numec-Analytcal Method fo Nonlnea Dynamcal Systems, Jounal of Sound and Vbaton, Vol, No. 3 (998) Has, Rollng Beang Analyss, John Wley and Sons, New Yok, 989. Item Unts Value ξ, ξ s Dampng ato fo the oto and the stato -. me e Unbalance magntude kg.m 5.e-3 δ Cleaance m.e-5 K Radal beang stffness N/m.e+ H s ω, ω Natual fequency of the stato and the stato ad/s 5; 5 g Gavty m/s 9.8 able : Numecal model of physcal paametes Valeus numéques des paamètes physques 5

6 th Fgue : Descpton of the beang (a) locaton of the k ball (b) ollng beang ème Descpton du oulement (a) localsaton de la k blle (b) oulement à blles Fgue : Ampltudes of vbatons vesus the otatonal fequency Ampltudes des vbatons pa appot à la vtesse de otaton 6

7 (a) Fequency=3. Hz (b) Fequency=. Hz (c) Fequency=3.4 Hz (d) Fequency=5. Hz (e) Fequency=9. Hz (f) Fequency=5.5 Hz Fgue 3: Obts of the oto and the stato at dffeent fequences (contnuous lne: oto, dashed lne: stato) Obtes du oto et du stato pou dfféentes féquences (lgnes contnues= oto, lgnes en pontllés= stato) 7

8 (a) (b) Fgue 4: X,Y Dsplacements of the oto and stato (a) fequency= 3.4Hz (b) fequency=47.8hz X,Y - Déplacements du oto et stato (a) féquence= 3.4Hz (b) féquence=47.8hz 8

9 detecton detecton 3/ 3/ Foce (N) / me Foce (N) / me (a) Fequency=3. Hz (b) Fequency=. Hz detecton detecton 3/ 3/ Foce (N) / me Foce (N) / me (c) Fequency=3.4 Hz (d) Fequency=5. Hz detecton detecton 3/ 3/ Foce (N) / me foce (N) / me (e) Fequency=9. Hz (f) Fequency=5.5 Hz Fgue 5: Evoluton of the contact and assocated contact foce fo each ball (black zone: contact; whte zone: non-contact) Evoluton du contact et de la foce de contact assocée pou chaque blle (zone noe: contact; zone blanche: pas de contact) 9