Contact Analysis on Large Negative Clearance Four-point Contact Ball Bearing

Transcription

1 Avilble online t rocedi ngineering The Second SR Conference on ngineering Modelling nd Siultion CMS 0 Contct Anlysis on Lrge Negtive Clernce Four-point Contct Bll Bering You Hui-yun, Zhu Chun-xi,Li Wu-xing b, * Deprtent of Mechnicl ngineering, Luoyng Institute of Science nd Technology, Luoyng,470,Chin b Luoyng Building Mteril Mchine Fctory, Luoyng,4700,Chin Abstrct Bsed on the theory of Hertz Contct, this pper presents contct nlysis ethod for lrge negtive clernce four-point contct bll bering. Tke turntble bering -645K for exple, norl contct stresses between steel bll nd inner nd outer ring rcewy re clculted under different negtive clernce when bering idling. The results show tht contct lods of four contct points of the bering increse with the bsolute vlue of negtive clernce, tiny chnges of the negtive clernce hve gret effect on contct stresses. 0 ublished by lsevier Ltd. Selection Open ccess under CC BY-NC-ND license. Keywords: Hertz theory; four-point contct bll bering; negtive clernce; contct stress. Introduction The subject investigted of this pper is kind of lrge four-point contct bll bering, it is kind of turntble bering too. These berings re widely used in crne, wind genertors, rdr s well s hevy chinery such s excvtors. The bering we studied is used on rdr. According to specil working condition of the rdr, it requires certin strting friction torque. Therefore, negtive clernce bering ust be used. In other words, interference fit is required between steel bll nd inner nd outer ring rcewy. At this tie, contct re between steel bll nd rcewys will produce contct stress. Contct stress hs n iportnt ipct on bering friction torque, contct ftigue nd wer, which lrgely deterines bering life. Clcultion of contct stress is the bsis of nlysis of rolling bering. At present, * Corresponding uthor. Tel.: il ddress: lwxgh@6.co ublished by lsevier Ltd. doi:0.06/j.proeng Open ccess under CC BY-NC-ND license.

2 You Hui-yun et l. / rocedi ngineering doestic nd foreign theoreticl nlysis of lrge negtive clernce four-point contct bll bering is blnk. We use Hertz elstic contct theory to nlyze negtive clernce four-point contct bll bering nd nlysis ethod of contct stress of the bering under no lod is given. Contct stresses, shpes nd sizes of contct res of turntble bering -645K As is shown in Figure re clculted. All these cn provide theoreticl bsis nd iportnt dt for bering design.. Bsic eqution of elstic contct Fig. Turntble bering -645K lstic contct is to study two or ore elstoers under externl lod resulting fro locl stresses nd defortion. The sizes of contct res nd the gnitude of contct stresses re relted to initil gps, frictions nd gnitude of lods of the contct res when two objects contct in the boundry. The sizes of contct res chnge constntly with lods in the process of lod. Tht is, the boundry conditions re constntly chnging. They cn not restore nd re not reversible. The uncertinty nd irreversibility kes it no longer liner reltionship between contct stress nd externl lod. The sttes of stresses re relted to the order of lod[]. Therefore, contct proble is kind of highly nonliner behviour, it hs been lwys one of the difficulties in nonliner probles. The contct between steel bll nd inner nd outer ring rcewy of bering is the se proble nd it belongs to typicl stte nonliner proble. lstic contct hs the following two bsic equtions. Blnce eqution x, y dxdy Ac In this eqution, represents lod, represents contct stress nd Ac represents contct re. Coptibility of defortion eqution x', y' dx' dy' z x, / x x' y y' y Ac ' In this eqution, Z represents initil clernce of two contct objects, represents the ount of elstic defortion, nd respectively represent young's odulus of the two objects, nd respectively represent two elstic oisson's rtio. Solving these two equtions is the bsis of elstic contct proble.. Hertz elstic contct theory Hertz contct theory is bsic nd clssicl theory of contct defortion nd stress clcultion of the elstoer. Hertz theory is founded on the following ssuptions []: Mteril of objects in contct with ech other should be hoogeneous nd isotropic. Lod ust be perpendiculr to contct surfce. The contct surfce is copletely sooth, regrdless of friction between the object nd the surfce. Contct object only hs elstic defortion. The defortion obeys Hooke's lw nd cnnot exceed the elstic liit. 4 The size of contct surfce is sll in coprison with rdius of curvture of the contct surfce.

3 76 You Hui-yun et l. / rocedi ngineering Hertz theory discusses the sttus of two objects with curvture in contct with ech other when externl force cts As is shown in Figure. Two elstoers seprtely hve different rdius of curvture on two in plnes. When the vlue of lod is zero they contct with ech other t one point. The contct re chnges into n ellipse when the lod increses grdully. As contct re is prt of the elstoer nd fr less thn rdius of the elstoer, so we cn use sei-infinite plne to nlyze locl defortion. Curvture of the curve which psses through the contct point nd contins the intersection of coordinte surfce of norl line nd curved fce chnges long with the coordinte plne when two objects with rbitrry curved fce coe into contct. Mxiu nd iniu curvtures of which re clled principl curvtures. rincipl curvtures re positive or negtive. The convexity is positive while the concvity is negtive. lne where the principl curvture exists is clled principl plne. rincipl curvture function nd curvture re clculted s follows. F 4 5 In these two foruls is principl curvture of the contct object, it is reciprocl of the rdius R, R, R, R respectively. Concve curvture of the bering rcewy groove is negtive vlue. Assuing tht young's odulus of the objects re nd, oisson's rtios re nd. We cn clculte sizes of sei-jor xis nd sei-inor xis of the contct ellipse s follows: * 6 * b b 7 In the centre of contct re, reltive displceents of the two objects re respective w nd w, the ount of elstic defortion cn be clculted s follows: w w 8 * 8 e K 9 Mxiu contct stress is clculted s follows: x b 0 In bove foruls * nd b* re coefficient of sei-jor xis nd sei-inor xis of the contct ellipse which re relted to the in curvture function f [], ke / * is Hertz coefficient, ke is coplete elliptic integrl of the first kind. Fig. Model of Hertz contct theory

4 You Hui-yun et l. / rocedi ngineering The steel blls of turntble bering -645K re de of GCr5, the inner nd outer ring re de of 4CrMo.These two kinds of terils re siilr in chrcteristic, therefore the forul of contct clcultion cn be siplified, the sizes of contct res cn be clculted s follows: * 0.06 * b b 0.06 Here follows the forul of clculte xiu Hertz contct stress nd the ount of elstic defortion. x b 4 K e.79 0 * 4 Fig. Model of contct of Turntble bering -645K t zero clernce 4 Use Hertz theory to solve contct proble of turntble bering -645K Turntble bering -645K is negtive clernce four-point contct bll bering. The sttus of steel bll contct with the inner nd outer rcewy t zero clernce is shown in figure. Due to the negtive clernce, elstic defortion between steel bll nd rcewys hs occurred when bering idling. Contct res re ellipses fter the defortion. Now the ount of defortion between steel bll nd rcewys is known. Therefore, it needs to be derived reltionship between size of the contct res nd the contct stress nd the ount of elstic defortion. When clernce of the bering is zero, the steel bll dieter d=5.875, the inner ring rcewy rdius of curvture r i =8.4 nd dieter of flute D i =405. Se preters of the outer ring rcewy re r o =8.57 nd D o =49.. This tie, the steel bll nd the inner nd outer ring rcewy respectively hve two points of contct. Contct ngles of the steel bll nd the inner nd outer ring re s follows: g o rcsin 45 r d =i o 5 For convenience, we ssue tht by chnging bll dieter to chnge bering clernce while the reining diensions rein unchnged.the bering clernce is negtive when incresing the bll dieter. Contct res between the steel bll nd inner nd outer ring rcewy re four ellipses. Becuse inner nd outer ring rcewys re syetricl geoetric shpes, two contct ellipses of the inner ring rcewy re exctly the se. So does the outer ring rcewy. When the steel bll dieter d = 5.9, norl totl ount elstic defortion between the steel bll nd the rcewys is s follows: i o d d Rdil negtive clernce is s follows: Do Di d ho hi

5 78 You Hui-yun et l. / rocedi ngineering h d sin d cos d r r r =i o 8 Lod of the steel bll of turntble bering -645K cn be obtined by forul 4 * 0.79 K e 4 i cos i = o cos o 0 The su of curvture is =0.7 when steel bll contct with inner ringthe function of principl curvture F=-0.90, then *=.b*=0.459ke/*= The su of curvture is =0.98 when steel bll contct with outer ringthe function of principl curvture F=-0.869then *=.68b*=0.496Ke/*=0.70. We cn clculte sizes of the inner nd outer of contct res nd xiu contct stresses by forul 6, 9, 0 nd,,. Siilrly, we cn follow se ethod to clculte the se preters with different negtive clernces, the results re listed in Tble. Tble Contct sizes nd xiu contct stresses of different negtive clernces 9 steel bll rdius Totl ount of norl elstic defortion Rdil negtive clernce Steel bll contct with the inner ring jor xis rdius inor xis rdius xiu contct stresses M Steel bll contct with the outer ring jor xis rdius inor xis rdius xiu contct stresses M Conclusion This pper obtins contct nlysis ethod for lrge negtive clernce four-point contct bll bering bsed on Hertz contct theory, tke turntble berings -645K for exple, norl contct stresses of the steel bll nd the inner nd outer ring rcewy re clculted under different negtive clernces when the bering is idling. The results show tht the contct lods of four contct points of bering increse with the bsolute vlue of negtive clernces, sll chnges of the negtive clernce hve gret ipct on the contct stresses. References []Li Run-fng, Gong Jin-xi, Contct robles in Nuericl Method nd Appliction of Mchine Design, 99 []Wn Chng-sen, Anlysis of Rolling Berings, 987 [] Hrris TA, Rolling Bering Anlysis, John Wiley nd Sons, 984. [4] G.LundbergA.lgren, Dynic Cpcity of Rolling Berings, Act olytech, 96 94