Dynamic Stability Analysis of Cages in High-Speed Oil-Lubricated Angular Contact Ball Bearings *

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1 Trns. Tinjin Univ. 011, 17: DOI /s Dynmi Stility Anlysis of Cges in High-Speed Oil-Lurited Angulr Contt Bll Berings * LIU Xiuhi( 刘秀海 ) 1,DENG Sier( 邓四二 ),TENG Hongfei( 滕弘飞 ) 1 ( 1. Shool of Mehnil Engineering, Dlin University of Tehnology, Dlin 11604, Chin;. Shool of Mehtronis Engineering, Henn University of Siene nd Tehnology, Luoyng , Chin) Tinjin University nd Springer-Verlg Berlin Heidelerg 011 Astrt:To investigte the ge stility of high-speed oil-lurited ngulr ontt ll erings, dynmi model of ges is developed on the sis of Gupt s nd Meeks work. The model n simulte the ge motion under oil lurition with ll si degrees of freedom. Prtiulrly, the model introdues oil-film dmping nd hysteresis dmping, nd dels with the ollision ontt s imperfet elsti ontt. In ddition, the effets of inner ring rottionl speed, the rtio of poket lerne to guiding lerne nd pplied lod on the ge stility re investigted y simulting the ge motion with the model. The results n provide theoretil sis for the design of ll ering prmeters. Keywords:dynmi nlysis; high-speed ngulr ontt ll ering; ge; stility; simultion In high-speed pplitions suh s jet-engines nd gyros, the instility of rolling ering elements might led to the sudden nd tstrophi filure long efore the ntiipted ftigue life. This leds to the omputtionl modeling nd simultion of ering dynmis to predit the onditions under whih suh instle motion might our. Cge stility is one of the most importnt ftors ffeting the performne nd life of rolling erings. Thus the investigtion of ge instility hs reeived rod ttention. Bll rolling erings re the primry hoie of high speed. Mny reserhers hve developed some dynmi models, nd investigted the influening ftors on ge stility. Wlter [1] initilly developed n nlytil model for ll ering. Gupt [,3] developed ering simultion model ADORE, whih is ple of simulting the motion of eh ering element in ll si degrees of freedom. By simulting, Gupt investigted the orreltion etween ge instility nd fritionl hrteristis of lurint esides the effets of ge lerne, dynmi lod nd ge unlne on the ge instility of solidlurited ll ering. Meeks et l [4,5] proposed dynmi model for the ll ering in whih there re si degrees of freedom for ge. Knnel, Boesiger, Kingsury Tteishi nd other reserhers mde gret del of studies on the dynmis of the ge of ll erings. In dditionl, severl reserhers studied the dynmi hrteristis of the ge of ylindril or tpered roller rolling erings [6,7]. The ge nd lls with si degrees of freedom re onsidered in Gupt s dynmi model, ut oil-film dmping nd hysteresis dmping re not onsidered, nd the ontt is ssumed s the perfet elsti ontt. Although the ontt is onsidered s the non-perfet ontt in Meeks model, ut the motion of ll is onstrined. Moreover, Meeks model pplies to the solidlurited ll ering. In Chin, the studies of the simultion of rolling erings re mostly foused on ylindril roller rolling erings [8-1], nd some mrked hievements hve een mde. By ontrst, the studies of ngulr ontt ll erings re fewer, nd the qusi-dynmi model rther thn three-dimensionl dynmi model is used to nlyze the ngulr ontt ll erings [13,14]. The plne model is minly dopted to nlyze the dynmis of the ges in ll erings [15], while the study of three-dimensionl dynmi model of the ge is few. The ojetive of this pper is to develop dynmi model of the ge whih is pplied to oil-lurited ngulr ontt ll erings. The model n simulte the mo- Aepted dte: *Supported y Ntionl Key Tehnology Reserh nd Development Progrm of Chin during the 11th Five-Yer Pln Period (No. JPPT ) nd Ntionl Nturl Siene Foundtion of Chin (No ). LIU Xiuhi, orn in 1981, mle, dotorte student. Correspondene to TENG Hongfei, E-mil: Tenghf@dlut.edu.n.

2 LIU Xiuhi et l: Dynmi Stility Anlysis of Cges in High-Speed Oil-Lurited Angulr Contt Bll Berings tion of the ge with si degrees of freedom. Prtiulrly, the model introdues oil-film dmping etween lls nd re s well s hysteresis dmping ting in ollision, while it does not introdue the motion onstrint of lls. With the model, the ge stility in n oil-lurited ngulr ontt ll ering will e investigted s funtions of inner ring rottionl speed, the rtio of poket lerne to guiding lerne nd pplied lod. 1 Dynmi nlytil model An ngulr ontt ll ering under oil lurition is investigted in this pper. The ge of the ering is guided y the outer ring whih is fied in spe. It is ssumed tht the inner ring rottes t onstnt speed. Furthermore, the geometries of the ering elements re ssumed to e perfet, the pokets of ge re ylindril, nd ll the mss nd geometri enters re ssumed to e oinident. To estlish the dynmi equtions of the elements, it is required to desrie the geometri nd kineti hrteristis of the elements. Then the intertions etween the elements re omputed. 1.1 System desription Vrious oordinte frmes, inluding n inertil frme, ge-fied oordinte frme, ll-zimuth frme nd lol frme, re defined to desrie the geometri nd kineti hrteristis of the elements nd determine the intertions etween the elements. A vetor trnsformtion from frme to nother frme n e omplished y trnsformtion mtri whih is orreltive with the Crdn ngles etween two frmes [1,3]. 1. Anlysis of fores The intertions etween the ering elements hve to e investigted for the dynmi model of the ge. Aiming t high-speed oil-lurited ngle ontt ll ering, we developed dynmi model in whih oilfilm dmping nd hysteresis dmping were onsidered. So the nlysis of severl fores in this pper is different from tht in Refs.[ 5] Intertions etween the ll nd re The intertions etween the ll nd re onsist of the norml fore, rolling resistne nd tngentil frition fore. The hysteresis dmping nd oil dmping etween the ll nd re re onsidered, nd thus the norml fore F rn is lulted y 3/ F = K δ + ( C + C ) u (1) rn P h o rn The deformtion δ etween the ll nd re n e determined y using Gupt s method [3]. The Hertzin stiffness K P is lulted in Ref.[16]. Considering the effet of hysteresis, hysteresis dmping C h n e otined y [17,18] 3/ 3KPαδ e Ch = where α e is etween 0.08 nd 0.3 s/m for steel or ronze. Oil dmping Co is lulted y [19] k 8 Λ 10 Co = 1ER G W U ( 5 k e ) Λ e γ where 1 10 re onstnts desried in Ref.[19]. The hydrodynmi rolling resistne F R resulting from inlet oil film is lulted y [0] FR =.86ER k G U W () where F R onforms to the reverse diretion of u. The frition fore n e lulted y integrting the produt of norml stress nd trtion oeffiient over the ontt region. To simplify the lultion, the ontt region is divided into severl elementry strips whih re prllel to the minor is of ontt ellipse. The frition fore μ ifrni ting on eh strip is lulted. After summing up the fores, the totl frition fore F rt t the ontt n e otined. F = μ F (3) rt i i rni The frition or trtion oeffiient is determined y lurition regime whih is evluted y the oil film prmeter Λ= h / σ + σ. The lultion of the entrl film thikness h is referred to Refs.[1 3]. The oil-lurited ering is investigted in this pper, so the trtion of the oil film is onsidered. Under the elstohydrodynmi lurition (EHL), the frition oeffiient μ hd is time-vrying s funtion of the slide-roll rtion nd other ftors. It is lulted y the empiril formul [4] : μhd = ( A + Bs)e Cs + D where the oeffiients A, B, C nd D re the funtion of norml lod, lurint-inlet temperture, veloity of two ontt ojets. The detiled proedure is desried in Ref.[4]. Under the oundry lurition, the frition oeffiient μ d is [7] μd = ( s) e Intertions etween the ge nd ll The ge is ssumed to e rigid rotor. When there is interferene etween the ge nd ll, the intertion 1

3 Trnstions of Tinjin University Vol.17 No etween them is ssumed to e Hertzin ontt. Otherwise the intertion is ssumed to e purely hydrodynmi without interferene [3]. The intertion n e determined y the reltive position etween the ll nd ge s shown in Fig.1. With the interferene etween the ge nd ll, the norml fore F N is lulted ording to the dry ontt y 3/ F = K δ + C u (4) N P h rn With no interferene etween the ge nd ll, the norml fore is lulted ording to Brewe s isovisous-rigid lurition model of point ontt: U[0.131rtn( αr / ) ] 18α rr F N = 1+ /(3 αr) h (5) Fig.1 Reltive position of ll nd ge Beuse the slide etween the ge nd ll is lrge, the frition oeffiient etween the ll nd ge is ssumed to e onstnt μ. Thus the tngentil frition fore is FT = μfn (6) 1..3 Intertions etween the ge nd outer ring The hypotheses of the ge-ll intertions re used to define the ge-ring intertions shown in Fig.. It hs to e done first to nlyze whether there is ontt etween the ring nd ge. If there is ontt, the fore is lulted ording to the Hertzin ontt, otherwise the fore is lulted ording to the short journl ering model [5]. Fig. Cge-ring intertions The zimuth of the point with the minimum lerne or mimum interferene in the ge frme is solved y glolly onvergent Newton s method. Then the interferene etween the ge nd ring δ e is otined. In the se of δ e Δ r, the intertions etween the ring nd ge re evluted y the slie tehnique in order to hndle the inline of the ge. The proess inludes the following steps: dividing the ge into severl thin disks; judging the mount of the disks in ontt with the ring; nd then omputing the intertions etween the ring nd disks. Consequently these lol intertions re integrted to otin the totl ring-ge intertion. In this se, the norml fore etween the ge nd the jth disk is lulted y 10/ 9 FrN j = KLδ j + ChurN j (7) The tngentil fore etween the ge nd the jth disk is FrTj = μrfrn j (8) where μ r is Coulom frition oeffiient. If δe < Δr, the intertions etween the ge nd ring n e evluted ording to the short journl ering model. The fores nd moment re otined y [5] 3 ηul ε Fr1 = (9) (1 ε ) 3 πηul ε Fr = (10) 3/ 4 (1 ε ) πηvrrl M r = (11) In the ove three-dimensionl mehnil model of the ge nd ll, the time-vrying trtion fores of lurint re onsidered, menwhile oil-film dmping etween the lls nd re s well s hysteresis dmping ting in ollision re introdued. The model etter orresponds with the tul onditions. 1.3 Dynmi differentil equtions Bsed on Newton s lw of motion, the trnsltionl motion of the ge mss enter n e written in the Crtesin oordintes s m = F my = F y (1) mz = F z The differentil equtions for the rottionl motion of the ge round its mss enter n e desried y the lssil Euler equtions of motion s I 1ω1 ( I I3) ωω3 = M1 I ω ( I3 I1) ω1ω3 = M (13) I 3ω3 ( I1 I) ω1ω = M3

4 LIU Xiuhi et l: Dynmi Stility Anlysis of Cges in High-Speed Oil-Lurited Angulr Contt Bll Berings where M 1, M nd M 3 re the omponents of vrious pplied moments in the ge-fied oordinte frme. The orienttion of the ge is trked y defining the orienttion of the ge-fied frme reltive to the inertil frme through Crdn ngles. The reltionship etween Crdn ngles nd ngulr veloity nd elertion n e found in Refs.[1] nd [3]. In the ylindril system of the inertil frme, Newton s equtions of motion of single ll mss enter is m = F mr mr θ = Fr (14) mr θ + mr θ = F θ In the ll-zimuth frme, Euler equtions of motion of single ll is I ω1 = M1 I ω Iωθ 3 = M (15) I ω3 + Iω θ = M3 In this pper, the inner ring of the ering rottes t onstnt ngulr veloity, nd the overturning moment nd vrile torque re not onsidered, so only three differentil equtions for the trnsltionl motion of the inner ring, whih re the sme s Eq.(1), re tken into ount. 1.4 Solving proedure of the dynmi model The si steps of solving the dynmi model re shown in Fig.3. The initil onditions for the integrtion of the differentil equtions of motion re first determined from the geometri onstrints nd kineti onditions y qusi-stti nlytil model. Then the lol intertions etween the ering elements re identified depending on the reltive position nd reltive motion of the elements, thus the pplied fores nd moments re otined y Eqs.(1) (11). After tht, the totl fores nd moments re determined. Finlly, the differentil equtions re integrted with the fourth-order method whih n vry step size utomtilly. Then the reltime position nd veloity vriles of the ge re otined. 1.5 Verifition nd omprison A lssi ll ering, the ge mss enter orit of whih ws mesured nd simulted y Gupt, ws onsidered to verify the dynmi nlytil model developed in this pper. The prmeters of the ering re listed in T.1, nd more dt re desried in Ref.[3]. The ge mss enter orits otined through the eperiment nd simultion re shown in Fig.4. T.1 Geometries of the ering for verifition Prmeter Vlue Bering inside dimeter/mm 100 Bering outside dimeter/mm 180 Bll dimeter/mm Numer of lls 18 Contt ngle/( ) 5 Cge guide surfe Outer ring Inner re urvture ftor 0.54 Outer re urvture ftor 0.5 Cge-ring guiding dimetril lerne/mm Poket lerne/mm () Orit mesured y Gupt () Orit predited y Gupt Fig.3 Solving proedure of the dynmi model () Orit predited y the model in this pper Fig.4 Cge mss enter orit t r/min 3

5 Trnstions of Tinjin University Vol.17 No From Fig.4, it is seen tht the orit simulted y the model developed in this pper is in resonle greement with the eperimentl orit. The orit is under stedy ondition. The smll differenes in the orit rdii, s Gupt remrked, re proly due to the unertinty of the lerne etween the ge nd ring [3]. The orit simulted y Gupt does not reh stedy stte euse there re limited numer of irles in the proess. It is proved tht the pility of the model developed in this pper to predit the motion of ge is desirle, nd it is fesile nd relile to nlyze the dynmi performne of the ge y the model. 1.6 Instility riterion of the ge Sine instle orits orrespond to errti motion, the trnsltionl speed of the ge mss enter n e used s quntittive riterion for ge stility. For stle opertion, the ge mss enter orit is lmost irulr, nd very smll vritions in speed re oserved, while lrge vritions orrespond to errti opertion [6]. In the following setion, we will disuss the ge stility whih is ffeted y inner ring rottionl speed, the rtio of poket lerne to guiding lerne nd pplied lod. The riterion of the ge instility in Ref.[6] is introdued for the disussion in this pper. The degree of instility of ge is defined s the rtio of the stndrd devition of the ge enter trnsltionl speed to its men vlue, i.e., the speed devition rtio. The smller vlue of the rtio orresponds to more stle ge motion. stte from the initil stte. Therefore, we investigted the dt of simultion from the 5th to the 45th shft revolution. The effets of vrious prmeters on the ge stility re estimted y nlyzing the dt. We normlized the rdil nd il positions of the ge with the guiding lerne etween the ge nd the outer ring in the following setion..1 Effet of inner ring rottionl speed on ge stility Fig.5 nd Fig.6 show the orits nd speed devition rtio of the ge mss enter t vrious inner ring rottionl speeds when the il lod is 100 N. In these ses, ll the ge mss enter orits re lmost irulr nd regulr, whih mens the ge is stle. However, in Fig.6 there is still smll flutution in the ge enter trnsltionl motion, inditing tht there re some ollisions etween the lls nd ge or etween the re nd ge. When the inner ring rottionl speed is higher thn r/min, the speed devition rtio of the ge dereses, nd the stility inreses with inresing inner ring rottionl speed. When the speed is high, the geometri oupling etween the ge nd ll is high, nd thus the ollision redues nd the vrition in the speed of the ge mss enter is smll. Simultion results () n =0 000 r/min () n = r/min The geometries nd mss prmeters of the highspeed ngle ontt ll ering investigted in this pper re shown in T.. The ering is lurited with vition lurint. It tkes period of time to reh stedy T. Bering geometries nd mss prmeters Prmeter Vlue Cge outside dimeter/mm 34.5 Cge inside dimeter/mm 9.15 Cge poket dimeter/mm 6.7 Cge mss/g 15.7 Cge width/mm Cge-ring guiding rdil lerne/mm 0.5 Bering ore/mm 0 Bering outside dimeter/mm 4 Pith dimeter/mm 31 Inner re urvture ftor 0.53 Outer re urvture ftor 0.55 Numer of lls 1 Bll dimeter/mm 6.35 () n= r/min (d) n= r/min Fig.5 Cge mss enter orits t vrious inner ring rottionl speeds Fig.6 Cge instility t vrious inner ring rottionl speeds 4

6 LIU Xiuhi et l: Dynmi Stility Anlysis of Cges in High-Speed Oil-Lurited Angulr Contt Bll Berings. Effet of ge poket lerne nd guiding lerne on ge stility The effet of the lerne rtio on ge stility ws investigted. The lerne rtio is defined s rtio of the ge poket lerne to the guiding lerne etween the ge nd the outer ring. The poket lerne vries while the guiding lerne etween the ge nd the outer ring remins unhnged t the inner ring rottionl speed of r/min when the il lod is 100 N. Fig.7 nd Fig.8 show the orits nd speed devition rtio of the ge mss enter for vrious lerne rtios. The orits re regulr for lerne rtio less thn 1. The speed devition rtio inreses with the inrese of the lerne rtio. If the lerne rtio is more thn 1, the ge mss enter orit eomes irregulr, nd ge instility rises. Therefore, lrge lerne rtio is not suitle for the stle motion of the ge. This is euse there re more offset spes for the lls if the lerne rtio is greter thn 1, nd the rndomness of ollisions etween the ge nd lls inreses. Menwhile, the ge might ollide with the outer ring, nd lrge frition is introdued. Then the ge orit eomes irregulr, nd the speed devition rtio eomes lrge..3 Effet of il lod on ge stility It is ssumed tht the ering opertes only under n il lod F when the inner ring rottionl speed is r/min. For the il lods of 50 N, 100 N, 00 N, 300 N nd 500 N, the ge orits nd speed devition rtios re shown in Fig.9 nd Fig.10. It is seen tht the ge orits re regulr in ll the ses, while ge stility is enhned with the inrese of il lod. The lrge il lod restrits the slide of lls nd mkes the ll operte stly, thus the ollision etween the ll nd ge redues. Therefore, loding n pproprite il fore on n ngulr ontt ll ering is required to keep the ge motion stle. However, the il lod n not e too lrge, in tht se the eessive frition nd wer would tke ple. () F = 50 N () F = 00 N () =0.4 () =1.0 () F = 300 N (d) F = 500 N Fig.9 Cge mss enter orits under vrious il lods () =1. (d) =1.5 Fig.7 Cge mss enter orits for vrious lerne rtios Fig.8 Cge instility for vrious lerne rtios Fig.10 Cge instility under vrious il lods.4 Effet of dmping on ge stility The oeffiient α e is n importnt ftor to define the mgnitude of dmping. The lrgerα e, the more powerful dmping. Fig.11 shows the effet of α e on ge stility. Cge stility eomes etter with the rise of α e, i.e., lrge dmping is enefiil to ge stility. In ddition, the introdution of dmping mkes it quiker to solve the dynmi equtions. 5

7 Trnstions of Tinjin University Vol.17 No Fig.11 Cge instility t vrious vlues of α e 3 Conlusions In this pper, iming t the ge stility of highspeed oil-lurited ngulr ontt ll erings, dynmi nlytil model of the ge is developed. In the model, n empiril formul is employed to lulte the trtion oeffiient of luriting oil. The model lso introdues oil-film dmping etween the lls nd re s well s hysteresis dmping ting in the ollision. The ge motion in n oil-lurited ngulr ontt ll ering ws nlyzed for vrious operting onditions nd geometri prmeters y the model developed in this pper. The effets of the operting onditions nd geometri prmeters on ge stility re s follows. (1) The ge mss enter orit is lmost irulr t high inner ring rottionl speed under n il lod. The ge instility dereses with the inrese of the inner ring rottionl speed. () The ge instility is enhned when inresing the poket lerne. Prtiulrly, when the rtio of the poket lerne to the guiding lerne is more thn 1, the ge instility is enhned shrply, nd the ge mss enter orit eomes irregulr. This trend is similr to wht Gupt showed for solid-lurited ering. Thus, for the ge in n oil-lurited ering, the rtio of the poket lerne to the guiding lerne lso hs lose reltion to the ge stility. (3) The lrge il lod enefits stedy motion of the ge, ut it lso reinfores frition nd heting in the ering. Therefore, it is neessry to pply pproprite il lods to ering design. (4) The lrger the dmping, the more stle the motion of the ge. The si-degree-of-freedom dynmi model n e used in prmeter optimiztion design of ngulr ontt ll erings. Nomenltures rdil guiding lerne; C h hysteresis dmping; C o oil dmping; E' equivlent elsti modulus; F pplied fore; F, F y, F z omponents of pplied fore on the ge; F, F r, F θ omponents of pplied fore on the ll in the ylindril inertil frme; G dimensionless mteril prmeter, G = α E' ; h entrl film thikness; I 1, I, I 3 prinipl moments of inerti of the ge; I prinipl moment of inerti of the ll; L width of the guiding lnd of the ge; l width of the line ontt; k elliptiity rtio; K P Hertzin stiffness of point ontt; K L Hertzin stiffness of line ontt, M pplied moment; M sher torque of oil; m mss of ll; m mss of the ge; R r guided rdius of the ge; 8/9 K = 0.356E'l ; R, R y equivlent urvture rdii long the minor nd mjor es of ontt ellipse; s slide-roll rtio, s = ( u u ) u ; u men veloity; u, u surfe veloities of two mting ellipsoids nd ; u rn norml omponent of reltive veloity; U dimensionless veloity prmeter, U = η 0 u/( E'R ) ; U U = U ; v reltive veloity; W dimensionless lod prmeter, W = w E'R ; w norml lod;, L / y, z position omponents of the ge;, r, θ position omponents of the ll in the ylindril inertil frme; α visosity-pressure oeffiient; α e oeffiient relevnt to the oeffiient of restitution; α r rdius rtio, α = R / R ; γ surfe pttern prmeter; Δ r ritil lerne; δ elsti deformtion; ε eentri rtio; r η 0 dynmi visosity t tmospheri pressure; Λ oil film prmeter; μ frition oeffiient; y μ d frition oeffiient under oundry lurition; μ hd frition oeffiient under EHL; σ, σ surfe roughness of odies nd ; φ T therml orretion ftor of the oil film; ω ngulr veloity. Referenes [1] Wlter C T. The dynmis of ll erings[j]. Journl of Lurition Tehnology, 1971, 93(1):

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