Internal Loading Distribution in Statically Loaded Ball Bearings Subjected to a Centric Thrust Load: Numerical Aspects

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Numecl spects the tetve pcedue e dscussed. Numecl exmples esults 8 gul-ctct bll beg hve bee cmped wth thse m the ltetue.

1 Vl:4, N:3, Itel Ldg Dstbut Sttclly Lded Bll Begs Subjected t Cetc Thust Ld: Numecl Aspects Má C. Rcc Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 Abstct A kw tetve cmputtl pcedue s used tel ml bll lds clcult sttclly lded sgle-w, gul-ctct bll begs, subjected t kw thust ld, whch s ppled the e g t the gemetc beg cete le. Numecl spects the tetve pcedue e dscussed. Numecl exmples esults 8 gul-ctct bll beg hve bee cmped wth thse m the ltetue. Twety gues e peseted shwg the gemetcl etues, the behv the cvegece vbles d the llwg pmetes s ucts the thust ld: ml bll lds, ctct gle, dstce betwee cuvtue cetes, d ml bll d xl delects betwee the cewys. Keywds Bll, Beg, Sttc, Ld, Itetve, Numecl, Methd. I. INTRODUCTION ALL d lle begs, geeclly clled llg B begs, e cmmly used mche elemets. They e emplyed t pemt ty mts, but, shts smple cmmecl devces such s bcycles, lle sktes, d electc mts. They e ls used cmplex egeeg mechsms such s ct gs tubes, llg mls, detl dlls, gyscpes, d pwe tsmsss. The stddzed ms bll lle begs pemt ty mt betwee tw mche elemets d lwys clude cmplemet bll lles tht mt the sht d usully stty supptg stuctue, equetly clled husg, dlly xlly spced-pt eltshp. Usully, beg my be bted s ut, whch cludes tw steel gs ech whch hs hdeed cewy whch hdeed blls lles ll. The blls lles, ls clled llg elemets, e usully held gully spced eltshp cge, ls clled sept ete. Thee e my deet kds llg begs. Ths wk s cceed wth sgle-w gul-ctct bll begs (Fg. ) tht e desged t suppt cmbed dl M. C. Rcc s wth the Bzl Isttute Spce Resech, Sã Jsé ds Cmps, 7- Bzl (e-ml: mcescc@ul.cm.b). M. C. Rcc thks the cl suppt pvded the Bzl Isttute Spce Resech (INPE), the Bzl Scetc d Techlgcl Develpmet Cucl (CNPq), d The Stte Sã Pul Resech (FAPESP). d thust lds hevy thust lds depedg the ctct gle mgtude. The begs hvg lge ctct gle c suppt heve thust lds. Fg. shws begs hvg smll d lge ctct gles. The begs geelly hve gve cuvtue d the ge 5-53% the bll dmete. The ctct gle des t usully exceed 4. Fg. Agul-ctct bll beg Ths wk s devted t study the tel ldg dstbut sttclly lded bll begs. Sevel eseches hve studed the subject s, exmple, Stbeck [], Sjväll [], Jes [3] d Rumbge [4]. The methds develped them t clculte dstbut ld mg the blls d lles llg begs c be used mst beg pplcts becuse ttl speeds e usully slw t mdete. Ude these speed cdts, the eects llg elemet cetugl ces d gyscpc mmets e eglgble. At hgh speeds tt these bdy ces becme sgct, tedg t lte ctct gles d clece. Thus, they c ect the sttc ld dstbut t get extes. Hs [5] descbed methds tel ldg dstbut sttclly lded begs ddessg pue dl; pue thust (cetc d eccetc lds); cmbed dl d thust ld, whch uses dl d thust tegls tduced Sjväll; d bll begs ude cmbed dl, thust, d mmet ld, tlly due t Jes. Thee e my wks descbg the pmetes vt mdels ude sttc lds but ew demstte such vts pctce, eve ude smple sttc ldgs. The uth beleves tht the lck pctcl exmples s mly due t the heet dcultes the umecl pcedues tht, geel, del wth the eslut sevel -le Itetl Schlly d Scetc Resech & Ivt 4(3) 3 schl.wset.g/37-689/84

2 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 lgebc equts, whch must t be slved smulteusly. I ttempt t cve ths gp studes e beg develped pllel [6]-[]. Ptcully, ths wk kw tetve cmputtl pcedue (see [5], p. 45) s used t bt tel ml bll lds sttclly lded sgle-w, gul-ctct bll begs, subjected t kw thust ld, whch s ppled the e g t the gemetc beg cete le. Althugh the methd s well kw, spects the umecl pcedue d the behv the vbles ude cvegece hve t bee sucetly expled the ltetue. S, umecl spects the tetve pcedue e dscussed d umecl exmples esults 8 gul-ctct bll beg hve bee cmped wth thse m the ltetue. Twety gues e peseted shwg the gemetcl etues, the behv cvegece vbles d the llwg pmetes s ucts the extel thust ld: ml bll lds, ctct gle, dstce betwee cuvtue cetes, d ml bll d xl delects. II. SYMBOLS Semmj xs the pjected ctct, m A Dstce betwee cewy gve cuvtue cetes t ulded pst, m b Semm xs the pjected ctct, m B, Ttl cuvtue d Rcewy dmete, m d Beg ute dmete, m d b Beg e dmete, m d e Beg ptch dmete, m D Bll dmete, m E Mdulus elstcty, N/m E Eectve elstc mdulus, N/m E Ellptc tegl secd kd, s Rcewy gve dus D; shck ct F Appled ld, N k /b K Ld-delect ct, N/m 3/ K Ellptc tegl st kd P d Dmetl clece, m P e Fee edply, m Q Bll-cewy ml ld, N Rcewy gve cuvtue dus; slds cuvtue dus, m s Dstce betwee lc e d ute cewy gve cuvtue cetes, m R Cuvtue dus, m Z Numbe llg elemets β, β, β Ctct gle, d, β Fee ctct gle, d, γ D cs β / d e Γ Cuvtue deece δ Delect ctct demt, m Δψ Agul spcg betwee llg elemets, d, υ Pss s t φ ψ Subscpts: b x,y j Auxly gle Azmuth gle, d, Rees t sld xl dect Rees t sld b Rees t cdte system Rees t e cewy Rees t llg elemet pst Rees t dect clle wth ml ld Rees t ute cewy III. GEOMETRY OF BALL BEARINGS I ths sect, the pcpl gemetcl eltshps ulded bll beg e summzed. The dl css sect sgle-w bll beg shw Fg. depcts the dmetl clece d vus dmetes. The ptch dmete, d e, s the me the e- d ute-ce dmetes, d d d, espectvely, d s gve e ( d d ) d. () Fg. Rdl css sect sgle-w bll beg The dmetl clece, P d, c be wtte s P d d D. () d Rce cmty s mesue the gemetcl cmty the ce d the bll ple pssg thugh the beg xs (ls med cete le tt xs), whch s le pssg thugh the cete the beg pepedcul t ts ple d tsvese t the ce. Fg. 3 depcts css sect bll beg shwg ce cmty, expessed s / D. (3) Itetl Schlly d Scetc Resech & Ivt 4(3) 33 schl.wset.g/37-689/84

3 Vl:4, N:3, A Pd / csβ. (5) A Fm (5), the dmetl clece, P d, c be wtte s d ( cs β ) P A. (6) Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 Fg. 3 Css sect bll d ute ce shwg ce cmty Rdl begs hve sme xl ply sce they e geelly desged t hve dmetl clece, s shw Fg. 4(). Fg. 4(b) shws dl beg wth ctct due t the xl sht the e d ute gs whe mesuble ce s ppled. The dl dstce betwee the cuvtue cetes the tw ces e the sme the Fgs. 4() d (b). Detg quttes eeed t the e d ute ces subscpts d, espectvely, ths dl dstce vlue c be expessed s A P d /, whee A D s the cuvtue cetes dstce the shted pst gve Fg. 4(b). Usg (3) we c wte A s A BD, (4) whee B s kw s the ttl cmty t d s mesue the cmbed cmty bth the ute d e ces t the bll. The ctct gle, β, s deed s the gle mde le, whch psses thugh the cuvtue cetes bth the ute d e cewys d tht les ple pssg thugh the beg tt xs, wth ple pepedcul t the beg xs tt. The ee-ctct gle, β, (Fg. 4(b)) s the ctct gle whe the le ls psses thugh the pts ctct the bll d bth cewys d mesuble ce s ppled. Fm Fg. 4(b), the expess the ee-ctct gle c be wtte s Fee edply, P e, s the mxmum xl mvemet the e ce wth espect t the ute whe bth ces e cxlly ceteed d mesuble ce s ppled. Fee edply depeds ttl cuvtue d ctct gle, s shw Fg. 4(b), d c be wtte s P As β. (7) e Csdeg the gemety tw ctctg slds (ellpsds d b) bll beg we c ve t the tw quttes sme mptce the lyss ctct stesses d demts: The cuvtue sum, /R, d cuvtue deece, Γ, whch e deed s whee, R R x R y Γ R, R x R y R x R y, x y bx, wth x, bx, y d, beg the d cuvtue the bll-ce ctct. A css sect bll beg petg t ctct gle β s shw Fg. 5. Equvlet d cuvtue bth e- d ute-ce ctcts, d ml t, the dect llg c be clculted m ths gue. Csdeg x the dect the mt d y the tsvese dect the d cuvtue the bll-e-ce ctct e x y D /, de D cs β bx, cs β D. () (b) The d cuvtue the bll-ute-ce ctct e Fg. 4 Css sect dl bll beg shwg bll-ce ctct due t xl sht e d ute gs. () Itl pst. (b) Shted pst bx D /, x y de D cs β, cs β Itetl Schlly d Scetc Resech & Ivt 4(3) 34 schl.wset.g/37-689/84

4 Vl:4, N:3, D. Fm [5], k c be wtte tems the cuvtue deece, Γ, d the ellptcl tegls the st d secd kd, K d Ε, s whee E( Γ) ( Γ) K J ( k), E / π / K s ϕ dϕ, k / π / E s ϕ dϕ. k Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 Let The R Fg. 5 Css sect bll beg D cs β γ. d e D γ bx, γ x bx y γ, (8) 4 D γ γ γ Γ R γ, (9) x bx y 4 γ the bll-e-ce ctct, d R x bx D γ bx, γ y 4 D Γ R x bx y 4 the bll-ute-ce ctct. IV. CONTACT STRESS AND DEFORMATIONS γ, () γ γ γ, () γ γ Whe tw elstc slds e bught tgethe ude ld, ctct e develps, the shpe d sze whch deped the ppled ld, the elstc ppetes the mtels, d the cuvtues the suces. F tw ellpsds ctct the shpe the ctct e s ellptcl, wth beg the semmj xs the y dect (tsvese dect) d b beg the sem-m xs the x dect (dect mt). The ellptcl eccetcty pmete, k, s deed s k /b. A e-pt tet methd, whch hs bee used successully the pst [3], s used hee, whee k J(k ). Whe the ellptcty pmete, k, the ellptc tegls the st d secd kds, K d Ε, espectvely, the ml ppled ld, Q, Pss s t, ν, d the mdulus elstcty, E, the ctctg slds e kw, we c wte the sem-mj d -m xes the ctct ellpse d the mxmum demt t the cete the ctct, m the lyss Hetz [4], s whee / 3 6k E QR, () πe 6EQR b πke / 3, (3) / 3 9 Q δ K, (4) ER πke. E υ υb E V. STATIC LOAD DISTRIBUTION UNDER CENTRIC THRUST LOAD Methds t clculte dstbut ld mg the blls d lles llg begs sttclly lded c be ud vus ppes, [5]. The methds hve bee lmted t, t mst, thee degees eedm ldg d demd the slut smulteus le system lgebc equts hghe degees eedm. Slut such equts geelly ecessttes the use dgtl cmpute. I cet cses, hweve exmple, pplcts wth pue dl, pue thust dl d thust ldg wth ml clece the smpled methds wll pbbly pvde sucetly ccute clcultl esults. E b Itetl Schlly d Scetc Resech & Ivt 4(3) 35 schl.wset.g/37-689/84

5 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 Hvg deed smple lytcl expess the demt tems ld the pevus sect, t s pssble t csde hw the beg ld s dstbuted mg the llg elemets. Mst llg-elemet beg pplcts vlve stedy-stte tt ethe the e ute ce bth; hweve, the speeds tt e usully t s get s t cuse bll lle cetugl ces gyscpc mmets sgct mgtudes. I lyzg the ldg dstbut the llg elemets, t s usully stscty t ge these eects mst pplcts. I ths sect the ld delect eltshps bll begs e gve, lg wth specc ld dstbut csstg cetc thust ld sttclly lded llg elemets. A. Ld-Delect Reltshps Bll Begs Fm (4) t c be see tht gve bll-cewy ctct (pt ldg) whee 3 / Q Kδ, (5) ER K π ke. 3 9K The ttl ml ppch betwee tw cewys ude ld septed llg elemet s the sum the ppches betwee the llg elemet d ech cewy. Hece δ δ δ. Theee, d K / K / 3 / K 3 / K δ / 3 3 / Q. (6) B. Bll Begs ude Cetc Thust Ld Let bll beg wth umbe blls, Z, symmetclly dstbuted but ptch ccle ccdg t Fg. 6, t be subjected t cetc thust ld. The, eltve xl dsplcemet, δ, betwee the e d ute g cewys my be expected. Fg. 7 shws the psts bll cete d cewy gve cuvtue cetes t y gul pst ψ, bee d te ldg, whees the cuvtue cetes the cewy gves e xed wth espect t the cespdg cewy. Fm Fg. 7 A Pd / β cs (7) A δ d δ A δ sβ Asβ. (8) ( ) Fm (5) d (7), the ttl ml ppch betwee tw cewys t y gul pst ψ, te the thust ld hs bee ppled, c be wtte s cs β δ A. (9) cs β Fg. 6 Bll gul psts the dl ple tht s pepedcul t the beg s xs tt, Δψ π/z, ψ j π/z(j) Itl pst, e cewy gve cuvtue cete β β A Fg. 7 Psts bll cete d cewy gve cuvtue cetes t gul pst ψ, wth d wthut ppled ld Fm Fg. 7 d (9) t c be detemed tht s, the dstce betwee the cuvtue cetes the e d ute g cewy gves t y llg elemet pst ψ, s gve cs β s A δ A. () cs β Fm (8) d () yelds δ Fl pst, e cewy gve cuvtue cete s A δ A P d / Oute cewy gve cuvtue cete, xed Itetl Schlly d Scetc Resech & Ivt 4(3) 36 schl.wset.g/37-689/84

6 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 Fm (6) d (9) yelds ( β β ) s δ A. () cs β 3 / 3 / cs β Q K A. () cs β I the extel thust ld, F, s ppled t the beg s xs tt the, sttc equlbum t exst Substtut () t (3) yelds F QZsβ. (3) 3 / cs 3 / β F s cs ZA K β. (4) β Equt (4) s le equt wth ukw β. Sce K s uct l ctct gle, β, the equt must be slved tetvely t yeld exct slut β. Tkg K s cstt (4) my be slved umeclly the Newt-Rphs methd. The equt t be stsed tetvely s well kw [5] F ZA β β cs β csβ cs β 3 / K 3 / 3 / cs β sβ cs β 3 cs β t β cs β / cs β. (5) Equt (5) s stsed whe β β s essetlly ze. F ech ew vlue β, ew vlue K must be bted, utl thee s mesuble deece the K vlue. Ths c be cheved thugh ute lp whee the gl s t d the deece β β vsh, whee β, s well s β, s uxly vble. VI. NUMERICAL RESULTS The Newt-Rhps methd ws chse t slve the le equt (4). Chse the llg beg, s put must be gve the gemetc pmetes: d, d, D, Z, d, ccdce wth the Fgs. d 4, d the elstc ppetes E, E b, ν d ν b. Next, the llwg pmetes must be bted:,, B, A, E, d e, P d d β. The teest hee s t bseve the behv gulctct bll beg ude kw thust ld, whch s t be ppled sttclly t the gemetc beg cetele. Let F ges m ze up t the lst vld vlue Newts. Itlly the vlues β, β d β wee dpted s beg equl β. The, ech ew vlue F gg m ze, d β s β, whee s s the shck ct. Whle the ute lp deece β β s gete th mml e, d β β d clculte the vlues: /R, /R, Γ, Γ, k, k, K, K, E, E, K, K d K ccdg t pevus sects. D β s β. I the deece β β s lesse th the mml e, ew thust ld vlue s cqued d the pcedue s epeted up t the lst vld thust ld vlue, whe the pgm eds. F ech tet the ute lp ew vlue β s bted the e lp. The ew β vlue s cmped wth the ld β d the deece β β s gete th mml e ew tet the e lp ccus. I the deece β β s lesse th the mml e, the e lp eds. T shw pplct the they develped ths wk umecl exmple s peseted hee. It ws chse the 8 gul-ctct bll beg, whch ws ls used Hs [5]. Thus, the esults geeted hee c be cmped t cet degee wth the Hs esults. The put dt ths llg beg wee the llwg: Ie cewy dmete, d.79 m Oute cewy dmete, d.4773 m Bll dmete, D.3 m Bll umbe, Z 6 Ie gve dus,.63 m Oute gve dus,.63 m Mdulus elstcty bth blls d ces, E.75 N/m Pss s t bth blls d ces, υ.3 The emg pmetes hs bee clculted, yeldg Ie ce cmty, Oute ce cmty, Ttl cmty t, B Itl cuvtue cetes dstce, A.3 m Eectve elstc mdulus, E N/m Beg ptch dmete, d e.56 m Dmetl clece, P d.48 m Fee-ctct gle, β The tl estmtes wee the llwg: Ctct gle, β β β β. Sce t s the qulttve behv sluts tht s the teest, the esults e peseted hee gphcl m. The Fg. 8 shws the ml bll ld, Q, s uct the extel thust ld, F. F 7,8 N extel thust ld Hs ud the mgtude,676 N ll blls (p. 49). Ths wk ud the mgtude, N ll blls the sme extel thust ld. Assumg cect the esults ths wk, ths mes tht Hs mde e but.34% the ml bll ld detemt. The Fg. 9 shws the ctct gle, β, s uct the extel thust ld, F. Whle Hs hs bee ud ctct gle mgtude 4.6 ll blls d 7,8 N extel thust ld (p. 49), ths wk ud the mgtude ll blls the sme extel thust ld. Ths epesets e but.44% the ctct gle detemt. Itetl Schlly d Scetc Resech & Ivt 4(3) 37 schl.wset.g/37-689/84

7 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 The Fg. shws the eltve xl dsplcemet betwee e d ute g cewys, δ, s uct the extel thust ld, F. Whle Hs hs bee ud xl dsplcemet mgtude.386 mm (p. 49), ths wk ud the mgtude mm the sme extel thust ld. Ths epesets e but 7.9% the eltve xl dsplcemet detemt. Nml Bll Ld, Q [N] Cetc Thust Ld - 8 Agul-ctct Bll Beg Rcc,68 N Hs(),676 N Thust ld, F [N] x 4 Fg. 8 Nml bll ld, Q, s uct the thust ld, F. Ctct gle, β [ ] Reltve xl dsplcemet betwee e d ute g cewys, δ [m] Cetc Thust Ld - 8 Agul-ctct Bll Beg β Hs () 4.6 Rcc Thust ld, F x 4 [N] Fg. 9 Ctct gle, β, s uct the thust ld, F x -5 Cetc Thust Ld - 8 Agul-ctct Bll Beg Hs ().386 mm Rcc.36 mm Thust ld F [N] x 4 Fg. Axl delect, δ, s uct the extel thust ld, F The Fgs. d shw the dstce betwee cuvtue cetes, s, d the ttl bll delect, δ, s ucts the extel thust ld, F, espectvely. The ttl ml bll delect c be bted summg the mxmum ml elstc cmpesss the e d ute ces, δ d δ, subtctg A m s, ce δ s A. The Fgs. 3 d 4 shw the behv the ctct gle β d the ute lp uxly vble β dug the ute lp umecl pcedue. The shck ct dpted ws. d evey level, shw detl, epesets cstt vlue the extel thust ld. The pcedue demded 84 ute lp tets t cve the ge m ze t, N the extel thust ld, wth steps N. The Fgs. 5 d 6 shw the behv the deece betwee the ute lp uxly vble β d the ctct gle β. Dstce betwee the cuvtue cetes, s [m] x Cetc Thust Ld - 8 Agul-ctct Bll Beg Thust ld, F x 4 [N] Fg. Dstce betwee cuvtue cetes, s, s uct the extel thust ld, F Ttl bll delect, δ [m] x Cetc Thust Ld - 8 Agul-ctct Bll Beg Thust ld, F [N] x 4 Fg. Ttl bll delect, δ, s uct the extel thust ld, F β, β'' [d].75 s Cetc Thust Ld - 8 Agul-ctct Bll Beg Oute lp tets Fg. 3 Cvegece pcedue the ctct gle β d the ute lp uxly vble β A Itetl Schlly d Scetc Resech & Ivt 4(3) 38 schl.wset.g/37-689/84

8 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 β, β'' [d].77 s β Cetc Thust Ld - 8 Agul-ctct Bll Beg β'' Oute lp tets Fg. 4 Cvegece pcedue the ctct gle β d the ute lp uxly vble β (detl) β''-β [d] 6 s x -4 Cetc Thust Ld - 8 Agul-ctct Bll Beg Oute lp tets Fg. 5 Cvegece pcedue the deece betwee the ute lp uxly vble β d the ctct gle β β''-β [d] 6 s x -4 Cetc Thust Ld - 8 Agul-ctct Bll Beg Oute lp tets Fg. 6 Cvegece pcedue the deece betwee the ute lp uxly vble β d the ctct gle β (detl) The Fgs. 7 d 8 shw the behv the ctct gle β d the e lp uxly vble β dug the e lp umecl pcedue. The shck ct dpted ws. d evey level, shw detl, epesets cstt vlue the extel thust ld. The pcedue demded 3,879 e lp tets t cve the ge m ze t, N the extel thust ld, wth steps N. The Fgs. 9 d shw the behv the deece betwee the e lp uxly vble β d the ctct gle β. VII. CONCLUSION A kw tetve cmputtl pcedue ws used t tel ml bll lds clcult sttclly lded sgle-w, gul-ctct bll begs, subjected t kw thust ld whch s ppled the e g t the gemetc beg cete le. Aspects the umecl pcedue d the behv the cvegece vbles wee dscussed. Results 8 gul-ctct bll beg wee cmped wth ltetue dt. Pecse pplcts, s exmple, spce pplcts, eque pecse detemt the sttc ldg. Mdels vlble ltetue e ppxmte d te e t cmptble wth the desed degee ccucy. Ths wk c be exteded t deteme the ldg hgh-speed begs whee cetugl d gyscpc ces d t be dscded. The esults ths wk c be used the ccute detemt the ct tque the bll begs, ude y petg cdt tempetue d speed. β, β' [d].75 s Cetc Thust Ld - 8 Agul-ctct Bll Beg Ie lp tets Fg. 7 Cvegece pcedue the ctct gle β d the e lp uxly vble β β, β' [d].7 s β Cetc Thust Ld - 8 Agul-ctct Bll Beg β' Ie lp tets Fg. 8 Cvegece pcedue the ctct gle β d the e lp uxly vble β (detl) REFERENCES [] Stbeck, R. Bll Begs Vus Lds, Ts. ASME 9, 4-463, 97. [] Sjväll, H. The Ld Dstbut wth Bll d Rlle Begs ude Gve Extel Rdl d Axl Ld, Teksk Tdskt, Mek., h.9, 933. [3] Jes, A. Alyss Stesses d Delects, New Deptue Egeeg Dt, Bstl, C., 946. [4] Rumbge, J. Thust Begs wth Eccetc Lds, Mch. Des., Feb. 5, 96. Itetl Schlly d Scetc Resech & Ivt 4(3) 39 schl.wset.g/37-689/84

9 Vl:4, N:3, Itetl Scece Idex, Mechcl d Mechtcs Egeeg Vl:4, N:3, wset.g/publct/84 [5] Hs, T. Rllg Beg Alyss, 4 th ed., Jh Wley & Ss Ic., New Yk,. [6] Rcc, M. C. Bll begs subjected t vble eccetc thust ld, DINCON 9 Pceedgs the 8 th Bzl Ceece Dymcs, Ctl d Applcts, My, 8-, Buu, Bzl, 9. ISBN: [7] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs, ICCCM9 st Itetl Ceece Cmputtl Ctct Mechcs, Pgm d Abstcts, p. -, Sept. 6-8, Lecce, Itly, 9. [8] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs subjected t cmbed dl d thust ld, cludg the eects tempetue d t, Pceedgs Wld Acdemy Scece, Egeeg d Techlgy, Vlume 57, Septembe 9, WCSET 9, Amstedm, Sept. 3-5, 9. ISSN: [9] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs subjected t cmbed dl d thust ld, 6th ICCSM Pceedgs the 6th Itetl Cgess Ct Scety Mechcs, Sept. 3 t Oct., Dubvk, Ct, 9. ISBN [] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs subjected t cmbed dl, thust, d mmet ld, Pceedgs the 6th Itetl Astutcl Cgess, Octbe, -6, Deje, Suth Ke, 9. ISSN [] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs subjected t eccetc thust ld, Submtted t Mthemtcl Pblems Egeeg, 9. [] Rcc, M. C. Itel ldg dstbut sttclly lded bll begs subjected t cmbed dl, thust, d mmet ld, cludg the eects tempetue d t, t be peseted t th P- Amec Cgess Appled Mechcs, Juy, 4-, Fz d Iguçu, Bzl,. [3] Hmck, B. J. d Ades, W. J. Ached-Oute-Rce Bll-Beg Csdeg Cetugl Fces. NASA TN D-6765, 97. [4] Hetz, H. O the Ctct Rgd Elstc Slds d Hdess, Mscelleus Ppes, McMll, Ld , 896. [5] Hmck, B. J. d Ades, W. J. Rllg-Elemet Begs. NASA RP 5, 983. β'-β [d].5 x -3 s Cetc Thust Ld - 8 Agul-ctct Bll Beg Ie lp tets Fg. 9 Cvegece pcedue the deece betwee the e lp uxly vble β d the ctct gle β β'-β [d] x -3 s Cetc Thust Ld - 8 Agul-ctct Bll Beg Ie lp tets Fg. Cvegece pcedue the deece betwee the e lp uxly vble β d the ctct gle β (detl) Itetl Schlly d Scetc Resech & Ivt 4(3) 33 schl.wset.g/37-689/84

Elastic-Plastic Transition of Transversely. Isotropic Thin Rotating Disc

Elastic-Plastic Transition of Transversely. Isotropic Thin Rotating Disc otempoy Egeeg Sceces, Vol., 9, o. 9, 4-44 Elstc-Plstc sto o svesely Isotopc h ottg Dsc Sjeev Shm d Moj Sh Deptmet o Mthemtcs JII Uvesty, -, Secto 6 Nod-7, UP, Id sjt@edml.com, moj_sh7@edml.com stct Elstc-plstc