航空发动机高速滚珠轴承振动特性研究.pdf

Chinese Journal of Turbomachinery第65卷，2023年第3期Http:/turbo- Vol.65，2023,No.3Vibration Characteristics of High-Speed Ball Bearing ofVibration Characteristics of High-Speed Ball Bearing ofAero-EnginesAero-Engines*Yun-bing Tang*Jing-xuan ZhangLing-ling ShiXiao-hu Xie（ChangzhouE&ETurbo-PowerCo.,Ltd.)Abstract：Ballbearingiswidelyusedasrotorsupporting.Itisanon-negligiblesourceofvibrationinaero-engines.Thevibrationcharacteristicsofthehighspeedballbearingarestudiedinthispaper.Bearingkinematicstheoryisappliedtodiscussthebasiccharacteristicsofballbearings.Thevibrationfactorscausedbyballbearingsareanalyzed,andthenthevibrationmodelofballbearingsissetup.Theeffectoftheunbalance,theballpassagevibration,theradialclearanceand the surface waviness are considered.It is found that the ball passage vibration is excited by the contact forcebetween the balls and the races.Increasing the numbers of ball can reduce the vibration amplitude.The surfacewaviness which is caused due to the geometric imperfection of ball bearings is considered as one of the importantvibrationsourcesofbearingsystems.Keywords：AerospacePropulsionSystem;BallBearing;BallPassageVibration;SurfaceWavinessDOI:10.16492/j.fjjs.2023.03.0010*Funding：National Natural Science Foundation of China(51775267)*Corresponding Author：Yun-Bing Tang，1IntroductionWith the development of science and technology,theperformance of aero-engines is improved constantly.Theball bearing is widely used in aero-engines.Its vibrationproperties exert important influences on the dynamic charac-teristics of rotor systems,especially at a high speed.There-fore,the research on the vibration characteristics of ball bear-ings is essential in order to meet the development of the newtype aero-engine 1-5.As the rotor support,the ball bearing needs to bear andresist the vibration caused by external force.At the sametime,due to the influence of structure and manufacturingtechnology,ball bearing itself will also cause vibration dur-ing operations.The vibration of the ball bearing mainly in-cludes the vibration caused by structural characteristics andthe vibration caused by manufacturing deviation of parts.The vibration caused by the structural characteristics in-cludes the deformation of the inner and outer rings of thebearing caused by the contact force of the rolling elementand the variable stiffness vibration caused by the radial loadof the bearing.The manufacturing deviation of parts includesball element roughness,roundness,inner and outer ring wavi-ness,etc.Therefore,it is of great significance for the designand use of ball bearings to study the vibration characteristicsof bearings and their systems.The ball passage vibration which is also called varyingcompliance vibration is studied by Sunnersj firstly 6.Thechaotic behavior and bifurcations can be observed in Ehrichand Tiwari s models 7-8.The studies undertaken by Tiwariinvolved the effect of radial internal clearances9.The effectof the surface waviness is considered by Nizami and Gunhee10-12.The vibration caused by ball bearings is discussed inthis paper.The analytical model taking into account the un-balance,the ball passage vibration,the radial clearance andthe surface waviness altogether is set up.Newmark method isusedtosolvetheequationsandtheresultsareobtained13-14.2Equations of MotionThe relative motion between the balls and the races hasimportant influence on the vibration characteristics of ballbearings.Fig.1 shows the motion sketch of a ball bearing.ThereforeFig.1The motion sketch of a ball bearing 59Chinese Journal of TurbomachineryVa=Vi=idi2,Vc=Vo=odo2（1）Where，Vais the ball speed at the pointa,Vcis the ballspeed at the pointc,diis the inner race diameter anddoisthe outer race diameter.When there is no slipping of balls as they roll on the surfaceof races,the speed of the ball(Vb)can be expressed as fol-lows:Vb=Va+Vc2=Vi+Vo2（2）Where，Viis the speed of inner race,Vois the speed of outerrace,iis the angle speed of inner race andois the angelspeed of outer race.Some assumptions are made.The inner race is rigidlyfixed to the rotor.The outer race is fixed rigidly to the sup-port and assumed to be stationary.Thereforei=,o=0（3）Where，is the speed of the bearing.According to Eqs.(2)and(3),the angular velocity ofthe ball about the center of the inner race isb=Vb(di+do)/4=di(di+do)（4）As the ball bearing rotating,the angeljchanges withtime(Fig.1)and is given byj=2Nb(j-1)+bt（j=1,2,Nb）（5）Where，Nbis the number of the ball bearing and t is the time.The radial displacement(uj)at thejth ball is given asuj=xcosj+ysinj（6）Where，xandyare displacements of the center of the innerrace.Considering the internal radial clearance(),the con-tact deformation becomesuj=xcosj+ysinj-（7）3The Vibration Caused by Ball BearingA ball bearing is used as the rotor support.It needs toendure and resist the vibration caused by the exciting force.At the same time,the ball bearing will also excite vibrationby itself,because of the effect of the structure and the fabri-cation technology.The vibration of ball bearings is usuallycaused by the structure characteristics or by the manufactur-ing process.The vibration caused by structure characteristicsis mainly named as ball passage vibration.Surface waviness,roughness and roundness belong to the manufacturing pro-cess deviation.3.1Ball passage vibrationWhile the ball bearing is rotating,applied loads are sup-ported by a few balls restricted to a narrow load region.Fig.2shows the load of ball bearings in different time.Note thatthe stiffness of the bearing varies with the cage position.Asthe position of the balls changing with the applied load mov-ing from the loaded to the unloaded zone,the load distribu-tion on the shaft changes thus producing a relation mo-tion between the inner and outer race.This vibrationwhich is named as ball passage vibration is excited bythe contact force(ball passage force)between the ballsand the races.Ball passage vibration universally occurs in ball bear-ings which bear the radial load.It is generated by the loadand the motion of bearings.The frequency of ball passage vi-bration(fvc)is related with the ball number and the anglespeed of the ball.It can be expressed as follows:fvc=vc2=bNb2（8）3.2Vibration due to surface wavinessSurface waviness,roughness and roundness are all thegeometric imperfection of ball bearings.With the develop-ment of the processing technique,the effects of the rough-ness and the roundness to the vibration are reducing,but theeffect of the surface waviness is still prominent.Althoughball bearings are perfectly manufactured without surfacewaviness,it may be generated by load or operating condi-tions.Therefore,the surface waviness becomes an importantsource of the vibration in ball bearings.The surface waviness of ball bearings can be modeledby the sinusoid function 7.Fig.3 shows the surface wavi-ness of the race with the amplitude of the waviness and thewaviness length.It can be expressed as follows:=psin2L（9）Where，pis the maximum amplitude of waviness andisthe waviness length.Take the inner race surface waviness as an example.The surface waviness of the inner race shows in Fig.4.Whereois the center of the ball bearing,bis the ball center,dmis the pitch diameter.Considering the effect of the initial a-mplitude(T),the inner race surface waviness(i)can beexpressed as follows:i=T+()psin2L（10）Where，Tisreplacedbythedeflectionduetothepreload.The waviness length()is the length of the race cir-cumference divided by the waviness number.It can be ex-pressed as follows:=di/Nw（11）Fig.2The load of ball bearings at different timeFig.3The mathematic model of the surface wavinessVibration Characteristics of High-Speed Ball Bearing of Aero-Engines 60Chinese Journal of Turbomachinery第65卷，2023年第3期Http:/turbo- Vol.65，2023,No.3Where，Nwis the waviness number.L=dij/2（12）Where，the contact anglejisj=2Nb()j-1+()b-it=2Nb()j-1+()b-t（j=1,2,Nb）（13）Considering the inner race waviness(i),the contactdeformation becomesuj=xcosj+ysinj-+i（14）4Calculate ModelThe contact force between the ball and the races is ex-pressedusingtheHertziancontacttheoryasfollows12:Q=Knu3 2（15）Where，Kn=1()1 Ki2 3+()1 Ko2 33 2,Kiis the load-deflectionconstant for the contact of a ball with the inner race andKoisthe load-deflection constant for the contact of a ball with theouter race.According the Eq.(14)and(15),the contact force isQj=Kn()xcosj+ysinj-+i3 2+（16）If the expression in the bracket is greater than zero,thenthe ball at angular locationjis loaded giving rise to a forceQj.If the expression in the bracket is negative or zero,thenthe ball is not in the load zone,and the forceQjis set to zero.The sign“+”signifies the meaning of above.According the Eq.(16),the total force components inthe x and y directions areQxQy=Knj=1Nb(xcosj+ysinj-+i)3 2+cosjsinj（17）Considering the unbalance,the ball passage vibration,the clearance and the surface waviness,the vibration equa-tion of ball bearings isMx+Cx+Knj=1Nb(xcosj+ysinj-+i)3 2+cosj=Fr+Mem2costMy+Cy+Knj=1Nb(xcosj+ysinj-+i)3 2+sinj=Mem2sint（18）Where，Fris the radial force andemis the eccentricity.5Results and DiscussionTake the deep groove ball bearing C204JUT as an exam-ple.The specifications of the ball bearing are following:in-nerracediameterdi=29.1mm,outerracediameterdo=39.96mm,ball diameterDb=5.43mm,groove diameterof inner raceri=1.40mm,groove diameter of outer racero=1.42mm,contact angle=0o,ball numberNb=8and rad-ialclearance=20m.Simultaneity,M=1kg,C=300Ns/m,Fr=5Nandem=0.05mm.The Eq.(18)can be solved by Newmark method to ob-tain the vibration response of the ball bearing.The vibrationbehaviors of the bearing system are analyzed by means of bi-furcation diagram and spectrum diagram.5.1Effect of ball passage vibrationIn order to discuss the effect of ball passage vibration,first of all the unbalance force and the surface waviness areignored(em=0andi=0in Eq.(18).The main factors governing the ball passage vibrationare the rotor speed,the damping,the radial clearance and theradial force.Fig.5(a)shows the displacement bifurcation fig-ure varying the rotor speed.Fig.5(b)shows the displacementbifurcation figure varying the damping.Fig.5(c)shows thedisplacement bifurcation figure varying the internal radialclearance.Fig.5(d)shows the displacement bifurcation fig-ure varying the radial force.It is found that many kinds of pe-riodic(synchronous,sub-and super-synchronous)and non-periodic(quasi-periodic and chaotic)responses are existed inthe system.The non-periodic vibration can be avoided by se-lecting the proper parameters(rotor speed,damping,radialclearance and radial force).For example,when the rotorspeed is between 25005500r/min(in Fig.5(a),the responseis fixed in period 1 which is favorable to the ball bearing.Fig.4The sketch of inner race waviness(a)Varying the rotor speed(b)Varying the damping 61Chinese Journal of Turbomachinery（a）Nb=5（b）Nb=6（c）Nb=9（d）Nb=12Fig.6 shows the frequency distribution at different ballnumber.Tab.1 shows the amplitude and the frequency of ballpassage vibration at different ball number.The results are fol-lowing:1)With the increasing of the ball number,the frequencyof ball passage vibration is increasing and the amplitude ofball passage vibration is reducing.2)Increasing the ball number means increasing the num-ber of balls supporting the bearings,therefore increasing thesystem stiffness and reducing the vibration amplitude.Fig.5The displacement bifurcation figure(c)Varying the clearance(d)Varying the radial forceFig.6The spectrum figure at different ball numberBall number/NbFrequencyfvc/HzAmplitude of vibration/m5141.19362.838e-66169.43231.796e-69254.14857.801e-712338.86461.083e-7Tab.1The amplitude and the frequency at different ball numberWhen the ball passage force and the unbalance force acttogether,the vibration responses will become more complex.Fig.7 shows the frequency response at different rotor speed.Wherefis the frequency of unbalance force.Fig.8 showsthe frequency of the vibration varying the rotor speed.Fig.9shows the amplitude of the vibration varying the rotor speed.Vibration Characteristics of High-Speed Ball Bearing of Aero-Engines 62Chinese Journal of Turbomachinery第65卷，2023年第3期Http:/turbo- Vol.65，2023,No.3The results are following:1)Combination vibration frequencies occur in the re-sponse,except thefvc,2fvc,f,2f.For example,fvc-2f,fvc-fandfvc+fcan been seen in the Fig.7(b).2)With the increasing of rotor speed,the frequency ofthe ball passage vibration and the vibration caused by the un-balance force are increasing(in Fig.8).3)With the increasing of rotor speed,the amplitudecaused by the unbalance force is increasing and the ampli-tude of the ball passage vibration is decreasing(in Fig.9).Fig.7The spectrum figure at different rotor speedFig.9The amplitude varying the rotor speed（a）n=4000r/min（b）n=8000r/min（c）n=10000r/min（d）n=11000r/minFig.8The frequency varying the rotor speed5.2Effect of surface wavinessThe number(Nw),the initial amplitude(HT)and themaximum amplitude(Hp)make up the main parameters ofwaviness.The follows are discussing the effect of those pa-rameters.The unbalance force is ignored in order to makethe analysis easy(em=0in Eq.(18).5.2.1Effect of the waviness numberThe computational parameters:the initial amplitude ofwavinessT=2e-6mm,the maximum amplitude of wavi-nessp=3e-6mmand the rotor speedn=4000r/min.Fig.10 shows the frequency responses at different wavi-ness number.With the increasing of the waviness number,the sub-synchronous and super-synchronous of thefvcfre-quency occur in the vibration response.WhenNw=12in Fig.10(d),broadband noise appears in the spectrum sketch.5.2.2Effect of the initial amplitude of wavinessThe computational parameters:the waviness numberNw=8,the maximum amplitude of wavinessHp=3e-6mmand the rotor speedn=4000r/min.63Chinese Journal of TurbomachineryFig.10The spectrum figure at different waviness number（a）Nw=5（b）Nw=7（c）Nw=9（d）Nw=12Fig.11 shows the frequency response at different initialamplitude of waviness.With the increasing the initial ampli-tude,the sub-synchronous and super-synchronous of thefvcfrequencydisappearinthevibrationresponses.WhenHT=8e-6mmin Fig.11(d),there are onlyfvcand2fvcin thespectrum sketch.（a）HT=1e-6mm（b）HT=2e-6mm（c）HT=4e-6mm（d）HT=8e-6mmFig.11The spectrum figure at different initial amplitude of waviness5.2.3Effect of the maximum amplitude of wavinessThe computational parameters:the waviness numberNw=8,the initial amplitude of wavinessHo=2e-6mmandthe rotor speedn=4000r/min.Fig.12 shows the frequency response at different maxi-mum amplitude of waviness.With the increasing of the maxi-mum amplitude of waviness,sub-synchronous and super-syn-chronous of thefvcfrequency occur in the vibration respons-es.WhenHp=10e-6mmin Fig.12(d),broadband noise ap-pears in the spectrum sketch.Vibration Characteristics of High-Speed Ball Bearing of Aero-Engines 64Chinese Journal of Turbomachinery第65卷，2023年第3期Http:/turbo- Vol.65，2023,No.3Fig.12The spectrum figure at different maximum amplitudes of waviness（a）Hp=1e-6mm（b）Hp=3e-6