THE TOLERANCE FIELD EFFECT ON THE ANGULAR CONTACT BALL BEARINGS SYSTEMS RATING LIFE

Transcription

1 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN THE TOLERANCE IELD EECT ON THE ANGULAR CONTACT BALL BEARINGS SYSTEMS RATING LIE Remes Dael, Racocea Cea Techcal Uvest "Gh Asach of Ias, dem@ahoocom ABSTRACT To assue both good damc load capact ad hghe shaft stffess, two ollg beags ae usuall mouted pa The load dstbuto o the cotacts of the two ollg beags depeds o dvdual stffess of the each beag, o the legth sepaato betwee the beags, ad o the chose toleace values I ths wok we peset a model fve degees of feedom, whch could seve to fd the load dstbuto beag aagemets, cosdeg the temedate elemets as gd bodes The assembl stffess was detemed cosdeg the dvdual stffess of each elemet, ad the atg lfe was epessed as a fucto of toleace values of the temedate elemets Ths was ealsed b solvg a o-lea sstem of equatos, cludg the cetfugal effects ad some of the fcto foces KEYWORDS: Agula cotact ball beag, Quas-damc equlbum, Ratg lfe 1Aaltcal Appoach o a ollg beag pa, (,), the dstace peces L1, L ae cosdeed to have the same tal legth ad the shaft s cosdeed as gd The followg co-odate sstems wee cosdeed: a etal sstem OXYZ wth ts og o the mddle legth of the e g's cuvatue cetes; a ollg elemet fame OX 1 Y 1 Z 1 fo each (,) ball To lmt the complet of the aalss, the followg assumptos wee admtted: the beag was mouted o a elastc shaft ad a gd housg; the sufaces cotact have deal shapes; the pa beag sstem wee cosdeed to be gd ecept the local cotact oes Related to the ball co-odate sstem (OX 1 Y 1 Z 1 ), two degees of feedom, epeseted b the taslatos u ad u, have bee cosdeed fo each ball The eteal load vecto {}, appled to the ete aagemet, cotas 5 compoets that ae futhe dvded to each beag of the aagemet: {} {,,, M, M } (1a) {} {,,, M, M } (1b) The statc equlbum povdes easl the sstem of equatos: (L+B) 1 B (L+B) 1 B () M 1 +M M M 1 +M M The dsplacemet vecto {} of the e g has also fve compoets: {} {,,, γ, γ } (3) a dm/ L α0 Y 1 B1 L1 B Z1 O X1 O e Z α,1 α,1 α 0 R O X Y B1 L B B Y 1 g1 Geeal vew Statc Equlbum of the (,) ball elemet To solve the equlbum sstem (3), s ecessa to fd the compoets of {} vecto whch ae fuctos of dstaces O e, ad O espectvel The followg otatos wee toduced: l oe O e Ro-D w /-Sd/4; (4a) l o O R-D w /-Sd/4; (4b) L e l o +l oe (4c) R Z1 O X 1 C 1

2 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN Cosdeg detcal beags the aagemet pa, the D1 ad D values peseted gues ad 3, ae : D1 L1+(B,1 +B, )/ (5a) D L+(B o,1 +B o, )/ (5b) D1 D (5c) Cosdeg futhe the toleaces TB,o, coespodg to B1, B dstaces ad the toleaces TL 1, coespodg to L1 ad L dstaces, the values D1 ad D become: D p L+TB o,1 +TL + TB o, +(B o,1 +B o, )/ (6) D 1p L1+TB,1 +TL 1 + TB,1 +(B,1 +B, )/ (7) The tal cotact agle α 0 ad L e paametes deped also o D p ad D 1p values, so that ew values α 0, ad L e () have to be cosdeed Also, α 1, ad R paametes ae dffeet vesus the tal values The supplemeta aal cleaace toduced b the effectve values fo D p ad D 1p paametes s: a D p -D 1p (8) p p α0p O Dp D D1 α0 1 g The α 0 ad L e whe L+(B o,1 +B o, )/> L1+(B,1 +B, )/ α 0 O D Dp D1 O 1 p α0p Op gue 3 The α 0 ad L e whe L+(B o,1 +B o, )/< L1+(B,1 +B, )/ Assumg a as decso ctea esults: f a>0 fo α 0, acta((l e s(α 0 )+a)/(l e cos(α 0 )); sd1()[(l e cos(α 0, )) +(L e s(α 0, )+a) ] 05 -L e ; L e () lo+loe+sd1(); fo 1 α 0,1 α 0 ; sd1(1):0; L e (1):L e ad α,1 acta{[ D p ]/[[ dm/+lo()cos(α 0, )]]} R[ D p ]/[s(α,1 )]; f a<0 O fo 1 α 0,1 acta((l e s(α 0 )+a)/(l e cos(α 0 )); sd1(1)[(l e cos(α 0,1 )) +(L e s(α 0,1 )+a) ) 05 -L e ; L e (1)lo+loe+sd1(1); fo α 0, α 0; sd1():0; L e ():L e ad α,1 acta{[ D p ]/[[ dm/+lo()cos(α 0, )]]} R[ D p ]/[s(α,1 )]; If the e g s msalged aoud OY ad O aes wth γ ad γ agles espectvel, the the tal agle α,1 become a fucto of α 1 (,): α 1 (,)α,1 +sg()γ cos(ψ(,))+ sg()γ s(ψ(,)) (9) whee: ψ(,) defes the agula posto of the ball elemet the etal sstem; sg() defes "" ow: 1, 1 sg( ) (10) 1, Because the statc load case the e ad oute cotact agles ae equal fo a dvdual ball elemet, but dffeet fo eve ball, the total defomato that acts o the (,) ball ca be wtte as: (,) (, ) + (, ) lo loe (11) whee: (, ) Le( )cos( α0, ) + cos( ψ(, )) + + s( ψ(, )) + R[cos( α (, )) cos( α )] (, ) L ( )s( α e s( α1(, ))] The cotact agle fo the (,) olle elemet s: 0, 1 ) + + R[s( α,1,1 ) (, ) α s(, ) α(, ) αe(, ) acta (, ) (1) The omal load ad cotact agle ae gve b: Q(,) K ech (,) (13a) α (,) α e (,) (13b) The {} dsplacemet vecto esults b solvg the equlbum equato sstem fo the e g Usg the pevous elatos, the equlbum of foces ad momets ae: Q(, )cos( (, ) Q( (, )cos(, ) α ( α (, ))cos(, ))s( ψ( ψ(,, )) )) (15a) (15b)

3 8 M + M + Q(, (, ) ( (,, )s( (, )b (, (, )b THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN α ( ), )) )b (, ) + )b (, (, ) ) + (15c) (15d) (15e) whee: Q(,) epesets the load actg o the (,) ball; (,), (,) epeset the adal foces whch act, ball; b,, (,), epesets the dstace fom the pot of e acewa - ball cotact to the cete of the etal sstem B D b (, ) + (, b(, ) C + (, s( ψ(, )) b(, ) C + (, cos( ψ(, )) ) ) ) + l o + l + l o o w s( αs(, )) Dw cos( αs(, )) Dw cos( αs(, )) (,)(,)(K ech /K ) 1/ (16) The {} compoets epeset the soluto of the Eq (15a-15e) ad t was foud b a Newto-Raphso algothm To solve the equlbum sstem (15) s ecessa to wte the Jacoba mat fo the two beags The gdt mat M s: γ γ γ γ M (17) γ γ M M M M M γ γ M M M M M γ γ whee: [ K (, ) cos( α ( {} [ K ech (, ) {} M ech cos( α ( {} M (, )b (, ) + {} Z α 0, αe (, )b (, ) +, ))s(, ))cos( (, )b (, )b O' ψ(, (18b) ψ( ))], ))] (18c) (, ) (18d) (, ) (18e) O X e g 4 The cete of mass dsplacemet (u, u) fo the (,) ball f Q l oe O lo mo u c α m * αe u Q o α g 5 The foces o (,) ball fo [ K (, ) {} ech s( α (, ))] (18a) 3 Quas-damc effects Due to cetfugal foce, both the load ad the cotact agle ae modfed vesus the statc values

4 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN Cosdeg the estece of the cetfugal foce the fal posto fo the mass cete of the (,) ball s peseted g 4 as fucto of u ad u paametes The ew posto of e pot s foud also wth the Newto Raphso algothm appled ths tme to all balls The loads that act o the (,) ball ae peseted g5 Cosdeg the gudg ball assumpto [1], the equlbum equatos fo the () ball ae: ECA(,)Q (,)s(α (,))-Q o (,)s(α e (,))- [mcos(α (,))-mocos(α e (,))]0 (19a) ECR(,)Q (,)cos(α (,))- Q o (,)cos(α e (,))+[ms(α (,))- mos(α e (,))]+c0 (19b) whee: m(1-λ)mg/dw; moλmg/dw; foµqo; fµq MgDw m10-7 ω c ω w s(β) β - attude agle, [ad] s( α e(, )) β acta ad λ1, fo oute ace cos( αe(, )) + γ gudg s( α (, )) β acta ad λ0, fo e ace cos( α(, )) γ gudg The followg must be cosdeed: fo ad f act lke blockg foces; If m>f the mm-f else m0; If mo>fo the momo-fo else mo0; I these codtos the gdt mat fo (,) elemet ECA(, ) ECA(, ) s: MC(, ) u ECR(, u ) ECA(, ) (0) The followg otatos wee toduced to smplf the MC(,) compoets: dtodto(,) - cotact defomato fo statc load case at oute cotact level; dtdt(,) - cotact defomato fo statc load case at e cotact level; αsαs(,) - cotact agle the statc case; ZO(l oe +dto)cos(αs)+u; (1a) XO(l oe +dto)s(αs)+u; (1b) ZI(l o +dt)cos(αs)-u; (a) XI(l o +dt)s(αs)-u; (b) (ZI +XI ) 05 -l o; o(zo +XO ) 05 -l oe TqK 15 TqoKoo 15 Td1+(XI/ZI) Tdo1+(XO/ZO) om these otatos esults: 15XI ECA ( + lo )ZI Td Tq u 1 XI ZI Td ZI Td 15XO + ECAo o(o+ loe)zo Tdo Tqo u 1 XO Zo Tdo ZO Tdo ECA ECA ECAo (3) u u u 15XI ZI ECA ( + lo )ZI Td Tq 3 XI XI ZI Td ZI Td 15XOZO + ECAo o(o+ loe)zo Td Tqo 3 XO XO ZO Tdo ZO Tdo ECA ECA ECAo (4) 15XI ECR ( + lo ) Td Tq u XI + ZI Td 15 15XO ECRo o( o + loe ) Tdo Tqo u XO ZO Tdo 15 ECR ECR ECRo u u u 15ZI ECR ( + lo ) Tq XI _ 3 15 ZI Td + Td (5) 15ZO + ECRo o( o + loe ) Tdo Tqo XO ZO Tdo ECR ECR ECRo (6)

5 84 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN The gdt mat of the beag The dsplacemets u ad u ae obtaed solvg the eq (0) to (6) The gdt mat (17) has the followg elemets: [ K (,,u,u ) s( α(,,u,u ))] [ K (,,u,u) cos( α(,,u,u))s( ψ(, ))] [ K (,,u,u) cos( α(,,u,u))cos( ψ(, ))] (,,u,u)b (, ) + (,,u,u)b (, ) M {} (,,u,u)b ( ) + (,,u,u)b (, ) M {} whee: (,,u,u) K (,,u,u)cos( α(,,u,u))cos( ψ(, ))) (,,u,u) K (,,u,u)cos( α(,,u,u))s( ψ(, ))) (,,u,u ) K (,,u,u )s( α (,,u,u )) 5 The () pa beags lfe Usg the Ludbeg-Palmgee lfe atg method appled to the (,) pa beags, the basc damc elemet capact, Q c, s defed as the ball load whch wll esult a lfe of a mllo evolutos of the acewa wth 90 pecet pobablt of suvval o a ball wth damete 5mm, basc damc ball capact ca be calculated as: 041 ( 1! γ) ( 1 ± γ) 139 f Dw 18 1/ 3 Qc A λ D 13 w N f 1 (7) dm whee: A98; λ1; The calculato of the e ad oute ace lves ae gve Eq (8), fo the case of the e ace otatg wth espect to the load The ace lves ae calculated pe Eq (9), fo the case whe the e ace s statoa wth espect to the load The combato of the ace lves to gve the beag lfe s show b Eqs (30) ad (31) If the e ace otates wth espect to load, the lves ca be calculated as: Q L ( ) Q whee: c e 4 03 Qcs Ls( ) Q (8) es 4 N 1 0 Qe( ) Z Q(, ) 1 p p ; Q N 1 pe Q(, ) 0 es( ) Z 1 pe (9) The () beag lfe tems of mllos of evolutos ca be calculated as: 1/ e L10 ( ) ( L ( ) + Ls ( )) (30) Results that the pa beags lfe mllo of evolutos s: 1/ e L10 ( L10(1) + L10( )) (31) Beag lfe tem of hou ca be calculated as: 6 L10 10 B10 (3) 60RPM 6 Numecal eamples The followg geometcal values have bee cosdeed: B1B 1 B 10 [mm]; BB o1 B o 10 [mm] LL110 [mm]; N 1 balls / beag; Dw955 [mm]; dm46 [mm]; α o 15 [deg] Ro/Dw05; R/Dw053; L50 [mm] The effect of the a paamete vesus 1 ad evoluto s elated g 6 The eteal foces ae: 10 [N], 10 [N], 00 [N], L50 [mm] To pot out the fluece of the legth toleaces o load dstbuto, the aal dsplacemet s elated g 7 as fucto of the a paamete 1,, [N] , 8000 [pm], [pm], 1000 [pm] 1, 8000 [pm] 1, [pm] 1, 1000 [pm] "a" paamete, [mm]*100 g 6 a paamete vesus aal dsplacemet Aal dsplacemet, [mm]* [N], 10[N], 00 [N], L50 [mm] "a" paamete, [mm]* [pm] [pm] 1000 [pm] g 7 Aal dsplacemet vesus a paamete fo the beag

6 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN The fluece of the legth toleaces o the beag lfe s elated g 8 ad 9, as fucto of the a paamete The eteal codtos ae: 5[N], 0 [N], 500 [N]; RPM5000 [pm], L100 [mm] B10, [hous] B10, [mm] B1 [mm] B30 [mm] B0 [mm] a, [mm]*1000 B10, [hous] g8 Pa beag's lfe vesus a paamete a0 110 a a B, [mm] g9 Pa beag's lfe vesus B paamete 6 Coclusos The paametes "a" ad "B" fluce the compoets of the load vecto {}, ad also the load dstbuto the "" ollg beags' aagemet - om the vewpot of the mamum atg lfe, a optmum dstace betwee the cuvatue cetes of the aces of the two beags has bee foud (g8 ad 9) - I ode to obta a mamum atg lfe fo a tadem mouted beags (g1), the paamete "a" must appoach to eo Notatos Idees, dstaces ad coodate sstems beag de ball umbe the beag e g o oute g pot cotact costat, 15 N the umbe of balls fo the beag m ball weght,[kg] K ech, K,, K e equvalet, e, ad oute, gdt factos dm ptch damete, [mm] Dw ball damete, [mm] mass cete of the (,) ball; O,e cetes of cuvatue L1,L legths of temedate pats TL 1, toleaces of the L1 ad L legths, [mm] B1, B beag wdth, [mm] B,, B o, e ad oute g wdth, [mm] R, B, C, L legths, [mm] D1, D tal dstaces betwee cuvatue cetes D 1p, D p fal dstaces betwee cuvatue cetes R o, oute ad e acewa adus, [mm] TB,o, legth toleaces of the e ad oute gs, OXYZ etal sstem OX 1Y 1Z 1 ollg elemet fame fo the (,) ball u,u dsplacemet of the mass cete,, de to descbe the aes l oe dstace betwee O e ad pots, [mm] l o dstace betwee O ad pots, [mm] L e, L e() dstace betwee O ad O e pots, [mm] Sd, Sd1() dametcal cleaace of the beag [mm] Q(,) omal load, [N] Q,o(,) omal load e ad outhe aceva m, mo, tagetal foces f, fo a aal cleaace, [mm] {} vecto de oces ad momets {}, {} foce vectos {M} momet vecto Mg goscopc momet, aal load alog OX aes;, adal loads alog OY aes, adal load alog OZ aes M, M, M, M eteal momets whch act aoud of the OY ad OZ as espectvel Lfe paamets L 10 beag lfe, m, 90% pob suvval B 10 beag lfe, hous, 90% pob suvval Q c damc capact Q e damc equvalet load of the otatg ace Q es damc equvalet load of the statoa ace RPM evolutos pe mute Geek otatos α 0, tal cotact agle α 0,, α 0p tal cotact agle affected b a paamete α,1 tal agle betwee R ad OZ as α 1 (,) fal agle betwee R ad OZ as α s (,) cotact agle obtaed fom statc equlbum α (,) e cotact agle obtaed fom the statc equlbum α e (,) oute cotact agle obtaed fom the statc equlbum α (,,u,u) oute cotact agle α e (,,u,u) e cotact agle (,,u,u) local cotact defomato at the e g level fo the (,) de (,,u,u) local cotact defomato at the oute g level fo the (,) de γ γdw/dm, dmesoless paamete µ fcto coeffcet λ Joh s coeffcet agula speed of the cage ad ball, ω c,w

7 86 THE ANNALS O UNIVERSITY DUNĂREA DE JOS O GALAŢI ASCICLE VIII, 00, ISSN REERENCES 1THas, Rollg Beag Aalss Joh Wle & Sos, New Yok, Lodo, Sde, 1966 MDGaftau, DN Olau, MC Cocea, De Veluste wege de Rebug Radal-Aal-Kugellage be hohe Dehahle, Weq, 160 (1993), MD Gaftau, Cetu, Sp, Olau D Rulmet, Poectae s tehologe, vol 1, Ed tehca, Bucuest, PK Gupta, Damcs of Rollg-Elemet Beags, Pat III: Ball Beag Aalss, Tasactos of the ASME, vol 101, Jul Rumbage, JH Poplawsk, J, V, Coelatg Computeed Rollg Beag Aalss Techques to the ISO Stadads o Load Ratg Lfe, Tbolog Tasactos, vol37 (1994), 793