CHAPTER 3 KINEMATICS OF CYLINDRICAL ROLLER BEARING

Transcription

1 34 CHAPTER 3 KINEMATICS OF CYLINDRICAL ROLLER BEARING 3. CAGE AND ROLLER SPEED Th latv mtns f th ag spaat, th lls, an th as f ylnal llng lmnt bangs a mptant t unstan th pfman. Th latv vlts n a ball bang a smwhat m mplx than ths n ll bangs, th latt bng analgus t th spalz as f a z f ntat angl, f that asn th ball bang s us as an xampl h t vlp appxmat xpssns f latv vlts. Whn a ball bang pats at hgh sps, th ntfugal f atng n th ball ats a vgn f th nn an ut a ntat angls, n t mantan f qulbum n th ball (Stvn Lay an Hsash Kawamua 998). Th quatns f ball an spaat angula vlty f all mbnatns f nn an ut a tatn w vlp by Jns (959). Ra ntl assums that pu llng us at th ntllng a, wth all f th ball spn ung at th th a ntat. Th ntatn f th ball tatnal axs s thn asly tmn f bang gmty. Ra ntl pbably us nly n y bangs y flm lubat bangs wh Culmb ftn ntns xst n th ball-a ntat llpss, th mmnt-sstng spn wll always b gat at n f th a ntats. Pu llng wll u at th a ntat wth th hgh magntu mmnt-sstng spn. Ths s usually th nn a at lw sps an th ut a at hgh sps.

2 35 In l lubat bangs n whh hyynam flms xst, llng wth spn us at bth ball-a ntats. Thf ps ball mtns an nly b tmn thugh us f a mput analyss. It an b appxmat that th stuatn wth a asnabl g f auay, hwv, by assumng that th ball llng axs s nmal t th ln awn thugh th nts f th tw ball-a ntats as shwn n Fgu 3.. Inn a Rllng lmnt Out a Fgu 3. Angula vlty f bang lmnts Th angula vlty f th ag an th ll a gvn n th fllwng quatns. Th angula vlty f th ag an b btan by th avag f lna vlts f nn an ut as gvn n th quatn (3.) (Ansn 970). Lkly th angula vlty f th ll an b btan by th ffn f lna vlty at nn an ut as as gvn n th

3 36 quatn (3.). Th lna vlty f nn a an ut a a btan fm th angula vlty f nn an ut as sptvly as gvn n th quatns (3.3) an (3.4). By substtutng th lna vlts n th quatn (3.), th xpssn f angula vlty f ag s v an gvn n th quatn (3.9). Smlaly th lna vlts f nn an ut as a substtut n th quatn (3.) an th quatn f angula vlty f th ll s v an gvn n th quatn (3.). Th quatns (3.) t (3.4) a us t v th latnshp btwn angula vlty f ag an ll. ( v ' v ') / / ( v ' v ') / / (3.) (3.) v ' (3.3) v ' (3.4) (3.5) (3.6) F fx ut a, th angula vlty f ut a must b qual t z ( 0 ), (3.7)

4 37 s R (3.8) ( s) (3.9) (3.0) ( s) s( s) (3.) ( s ) (3.) ( s) ( s ) s( s) (3.3) ( s) s (3.4) Th pylal valus f angula vlty f th ag an ll f pu llng ntn an b btan f vaus shaft sps usng th knmat quatn (3.9) an (3.) sptvly. F xpmntal pupss, th quatns t alulat th pntag f ag an ll slp a smplf an v n tms f knmats an masu angula vlts as gvn n quatn (3.5) an (3.6) sptvly. % f Cag slp =, masu - x00%, kn (3.5) % f Rll slp =, masu - x00%, kn (3.6)

5 38 3. DEFORMATION OF ROLLER BEARING Mst llng lmnt bang applatns nvlv stay stat tatn f th th nn ut a bth. Hwv, th sps f tatn a usually nt s gat as t aus ll ntfugal fs gysp mmnts f sgnfant magntus. In analyzng th la stbutn n th llng lmnts, t s usually satsfaty t gn ths ffts n mst applatns. In ths stn th la fltn latnshp f ll bang s gvn, alng wth aal la stbutn f statally la ll bang. Th fmatn f sngl ll an ts supptng as, ngltng any ynam ffts, has bn xamn n tal by Dwsn an Hggnsn (963). In that pap t s shwn that fltn s pptnal t (La) 0.9, an that a lna latn gvs a g wkng appxmatn. Stbk alulat th stbutn f la n a gmtally pft ll bang, wh th fltn f ah lmnt s pptnal t (La) /3. Th fs an aal mpnnts f splamnt n a ll bang a shwn n Fgu 3.. In analyzng th fltn f th whl bang, t s assum that th splamnt f ah llng lmnt s lna. If t s assum that th bang assmbly an all ts mpnnts a gmtally pft, th fltns f llng lmnts a gvn n th quatns (3.7) an (3.8). Th latnshp btwn bang aal la an spf la n ll a gvn n th quatn (3.0). 0 s (3.7) 0 s (3.8)

6 39 Q Q Q Q Q Q Q Fgu 3. Cmpnnts f f an splamnt f ll bang 360 Z (3.9) Q Q Q s Q s... Q sn (3.0) 0 n Q0 k 0 (3.) Q k (3.) Q k (3.3) Q Q 0 s s... s n (3.4)

7 40 It shws that n a gmtally pft bang th mst havly la ll as a la appxmatly qual t 4Q/Z. It has bn shwn lswh (Dwsn an Hggnsn 963) that th ttal fltn f a sngl ll an ts as n a alst bang aangmnt s gvn t a hgh g f auay by th quatn (3.5). Q ' K ' (3.5) 7.5 Q ' K ' (3.6) A g wkng appxmatn t th pvus quatn s th lna latn gvn n quatn (3.6) whh nats that, wthn th appxmatns nvlv, th fltn s npnnt f th ll aus. Th fltn f a mplt an gmtally pft bang wll b gvn by Q ', th la p unt lngth n th mst havly la ll whh s 4Q/Z. Hn th xpssn an b mf an gvn n th quatn (3.7). 70 QK ' Z (3.7) 3.3 CONTACT WIDTH AND PRESSURE Th la a by th bang s tansmtt thugh th llng lmnts fm n a t th th. Th magntu f th la a by an nvual llng lmnt pns n th ntnal gmty f th bang an th latn f th llng lmnt at any nstant. Th fmatn wthn th ntat s a funtn f, amng th thngs, th lastty paamt an th llpt ntgals f th fst an sn kns. Smplf xpssns that allw quk alulatns f th stsss an fmatns t b ma asly

8 4 fm knwlg f th appl la, th matal ppts, an th gmty f th ntatng lmnts a psnt n ths stn. F ths stuatn th ntat llps suss f ll s f nfnt lngth n th tansvs tn. Ths typ f ntat s xmplf by a yln la aganst a plat, a gv, anth paalll yln by a ll la aganst an nn ut ng. Th Htz thy f ntat f tw paalll last ylns s us t v th ntat pssu stbutn, fmatn an sm ntat wth f th tw mnsnal ln ntat. In ths stuatns th sm ntat wth s gvn by th quatn (3.8). Th Htz ntat pssu s v an gvn n th quatn (3.9) (Jhnsn 985). Th maxmum Htz ntat pssu an b btan by usng th quatn (3.30). Smlaly th ll fmatn s v an gvn n th quatn (3.34). Th ns n th quatns (3.3) an (3.33) f t by an sptvly. b 4 Q '( R) E ' / (3.8) P Q ' b y b / (3.9) P max Q ' b (3.30) Q ' Q l ' (3.3) ( R) R R (3.3) E ' E E (3.33)

9 4 Q l E ln E ' ( R) Q ' ' 4( ') ' (3.34) 3.4 LUBRICATION REGIME Th gnal slutn psnt f th Rynls quatn vs th fllwng lubatn gms; Is-vsus hyynam (IHD) Classal hyynam lubatn n whh lubant vssty s npnnt f ntat pssu. Pz-vsus hyynam (PHD) lubatn, n whh lubant vssty s a funtn f pssu n th ntat. Elasthyynam (EHD) lubatn, n whh bth th nas f vssty wth pssu an th fmatns f th llng mpnnt sufas a ns. Makh t al (979) stablsh a paamt gvn n th quatn (3.35), all C hn, f fx valu f R ths fat was us t fn th lubatn gm. F alulatn f th lubant flm thknsss n llng lmnt away ntats, nly th PHD an EHD gms n t b ns. F alulatns assat wth th ag-llng lmnt ntats, pbably nsatn f th hyynam gm s suffnt. C lg.5x0 0 6 G 5000 W U 3 (3.35)

10 43 Tabl 3. Lubatn gm an ts paamt lmts Paamt Lmts C < - Lubatn Rgm IHD - < C < PHD C > EHD Chaatsts Lw ntat pssu, n sgnfant sufa fmatn. N sgnfant sufa fmatn, lubant vssty nass wth pssu. Sufa fmatn an lubant vssty nas wth pssu. 3.5 MATERIAL OF ROLLER BEARING Inn a, ut a an lls f g qualty, llng bangs a nmally manufatu fm stls that hav th apablty f bng han t a hgh g, at last n th sufa. In unvsal us by th ll bang nusty s AISI 500, a stl matly h n hmum an asly han thughut (thugh-han) th mass f mst bang mpnnts t 6-65 Rkwll C sal hanss (My C.A. 008). As p th stana f Aman In an Stl Insttut (AISI), th mpstn f th bang matal s gvn as C-0.98%, Mn-0.5%, S-0.5%, C-.3%. Cag matals, as mpa t matals f lls, an ngs, a gnally qu t b latvly sft. Thy must als pssss g stngth t wght at; thf, matals as wly vs n physal ppts as ml stl, bass, bnz, alumnum, plyam (nyln), plyttafluthyln (Tfln PTFE), fbglass, an plasts fll wth abn fbs fn us as ag matal.