Design of Elastic Couplings with Metallic Flexible Membranes

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1 Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) Desgn of Elas Couplngs wh eall Flexble embanes Dobe Danel, Smon Ionel Depamen of Despve Geomey and Engneeng Gaphs Unvesy POLITEHNICA of Buhaes Splaul Independene, no., Seo 6, Buhaes OANIA ddobed@yahoo.om, onel_smon@yahoo.om Absa: - The loaon of he flexble membane ouplng s spefed whn meall nemeday elemens ouplngs aea, dependng on vaous paamees: naue of omponens, apably o handle nsallaon msalgnmens and he magnude of oque ansmsson. The man aspes ae pened egadng he mehanal ngh alulus and vefaon of he man omponens of he elas ouplng (nne and oue ves, fxng bols and membanes spokes). Calulaon shemes a vaous loads ae sugged o deemne he ss sae n he aea of flle oo of he membane spokes. Compuaon s pefomed o deemne he numbe of membanes fom he elas poke. Ths alulus s ud o a dealed fuhe analyss of he ss and san sae hough he fne elemen mehod, analyze ha wll be he subje of ohe papes. Key-Wods: - Elas ouplng, eall flexble membane, ehanal ngh analyss Inoduon In he mahne-buldng ndusy, poy s n he applaon of he nf eah esuls n he poduon of omponens, uns and enmbles wh hgh values of duably and sably. In ode o ansm he oque beween wo shafs of a knemaal han, unde he ondon of ompensang some mpoan msalgnmens of he jonng shafs, s neessay fo omanan ouplng Podues o handle some mul-ea equemens egadng he sze, safey n exploaon (by guaaneeng a supeo mehanal ngh), behavo dung shoks and vbaons, onsuve smply and eduon of exeuon oss []. Though he funons ha ompensae oaxal eos n he adal, axal and angula deons, and also dumpng osonal osllaons, elas ouplngs deemne mehanal sym mpovemen n effaousness, as well as a smplfaon n he ansmsson manenane. Due o mulple poblems ad by lnk ouplngs n fulfllng he pefomane ea, esuled ha would be neessay and adequae o pefom some omplex eah on flexble nemeday elemens of he elas ouplng suue, whh would lead o funonng opmzaon on a long peod, wh favoable esuls n wha onens nead elably of he sym fomed of he dvng and dven mahney, as well as he ouplng lf. Due o he fa ha he pefomanes of elas ouplngs ae dependng on he geomey of he flexble elemens, he oppouny esuled of flexble elemens behavou sudy fom he pon of vew of mehanal ngh eon, of buklng sably eon, as well as he san and ss sae deemnaon. The eah deons ae also neessay beau hee sn enough nfomaon n hs feld s leaue fom whee an adequae alulaon model ould esul []. The menoned efeenes ove a geneal aea elaed o ouplngs and o he ypes of flexble ouplngs, n he ondons n whh he dedaed papes ae lmed n numbe. On Sengh Cea fo Couplngs wh Spoke Fom embanes The noduon of an elas ouplng n a mehanal ansmsson mus be so aed ou o espond n a lage measue o he equemens mpod by he funonng and susanably of he ansmsson. The analyss s pefomed fo he ouplng vaan n Fg., whh hghlghs he elemens nvolved n he alulaon. In he dsk ype membane he oque s ansmed fom he nne damee o he oue damee by shea loadng of he membane, and lmed flexbly an be obaned by usng vey hn shee [], [5]. By lampng paks of membane ogehe, s possble o ombne an adequae level of oque ansmsson wh a easonable angula and axal msalgnmen apably. The oque s ansmed ISSN: ISBN:

2 Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) hough dvng bols and fxng ves of elas elemens n wo membanes pokes []. l n Cs F, () D D s s whee n s he mean oque, C s s he ve fao defned n he sandads of dffeen ypes of ouplngs (he s esmaed fo an no unfom modeae wokng egme). s s D Ds D D Fg.. Non spae ouplng To alulae he mehanal ngh of he membane he followng elemens ae onsdeed as known: - ouplng s nomnal oque, ; - asmbly damee of oue ves and bols, D ; - asmbly damee of nne ves, D s ; - numbe of oue ves,, and nne ves, s ; - damee of oue and nne ves, d and d s ; - hkness of membanes poke, l ; - damees, maxmum and mnmum, of fon aeas, D and D.. The Asmbly ves Calulus The membanes ae asmbled gdly by ves n he poke a he nne and oue damees. The damees and he numbe of oue ves (plaed on he oue le wh damee D ) ae equal o he nne ves (plaed on he nne le wh damee D s ): d ds and s. The lowes lenghs of ona beween eah vaan ve (oue and nne) wh onneed omponens ae he same wh he hkness poke membanes, l. Theefoe, he ss ae al fo nne ves... The Cona Sess The alulaon s made on he hypohess ha he dsbuon pessue s unfom dsbued on a m-ylndal sufae wh d s damee and l long ona wh he poke membanes [6], [7]. The angenal foe on nne ve s gven by (Fg.): Fg.. Fonal loadng sheme of nne ve The ona pessue s alulaed beween membane poke and ve: p F pa 0, 8. () ds l... The shea ss I s onsdeed ha shea ss s unfom dsbued n al on. In he umsanes he shea ss of ve s: τ F, f τaf 0. π d 7 s () whee K, 5 o ake no aoun of no unfomy of ss n on.. The Calulus of Asmbly Bols beween embanes Poke and m-couplngs The wo m ouplngs, mouned on shafs of a knemaal han, ae asmbled gdly on membanes poke by ws, whh an be fed wh o whou oleanes. If he asmbly s made wh oleanes, he oque s ansmed by fon beween m ouplngs and nemeday dsks, he ompessng ISSN: ISBN:

3 Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) foe beng eaed by ghenng of ws, a fng. The pessue p, unfom dsbued on ona sufae beween m ouplngs and dsks, a D damees an be expesd as: p D and 0 π ( D D ) () z F whee F 0 s he peload foe, z he numbe of bols, D he maxmum damee and D s he mnmum damee of fon aea. Subsung Eq.(), he fon oque beomes: f z F µ π µ z F 0 ( D D ) D D 0 D D D / D / π d dα 0 µ z F 0 m (5) whee D D m s he fon aveage adus. D D The fonal adus an be epened by he expesson D + D m. In he a when he ouplng s nomnal oque s ansmed by fon: f ( D + D ) µ z F0, (6) he peload foe F 0 ang on he w s gven by []: F 0 µ z, ( D + D ) (7) whee µ s he Coulomb fon oeffen, D and D -mnmum and maxmum damees of fon. aeas, -numbe of pas of fon sufaes ( ).. The Tensle Sess The ensle ss s gven by he expesson: F 0 π d, a (8) whee a s he admssble ensle ss and d s he nne damee of w... The Toson ss The oson ss n he od of he w s gven by: τ W p 6 π d (9) whee d d ( d s he majo damee of he head) and F0 d s he wed oque. Beau he od of w s subjeed a ompound ss (ensle and oson), he equvalen ss s alulaed by he hd heoy of ngh [7]: eh + τ,. (0). The Smplfed Calulus of adal Spokes of he embane The ss sae of he membane spoke s deemned by [], []: - oque ansmsson ausng shea, bendng and ensle ss; - he msalgnmen (angula and axal dsplaemens) ha aus bendng and ensle ss; - membane enfugal loadng. The membane enfugal loadng s nonmpoan a low and medum oaonal speed ud (unde pm)... The Shea Sess The shea ss, n al on plaed a mnmum damee, s gven by: whee d m τ f τ af dm l b ns α, () D l s he mnmum damee of flle oo of he membane spokes (Fg.), l - he hkness of membanes poke; b - wdh of spoke; n s - numbe of spokes; l α - he lengh of spoke, dependng on he poson of he angen o flleed pofle. The alulaon sheme shown n Fg., deemnes he al on by means of lnes of angeny o he oo of spoke a α angles gven he spoke symmey axs. Ths hypohess s jusfed by he ss sae of he meall membane obaned by a Fne Elemen Analyss (FEA Fg.,b) whh shows ha he aea ISSN: ISBN:

Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) F e Inead ss n ohe pon han a oo of spoke l α α Gosmea h max eh m τ b α Seunea peuloasã a) Fg.

4 Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) F e Inead ss n ohe pon han a oo of spoke l α α Gosmea h max eh m τ b α Seunea peuloasã a) Fg.. The ss sae sheme a he oo of spoke [], [], [5] a) The ss sae on al on; b) The ss sae of he meall membane obaned by a Fne Elemen Analyss (FEA) b) of maxmum equvalen ss s plaed on he whole lengh of flle [], [8]. The lengh l measued fom he pon of applaon of foe F e unl he on deemned by veal lnes of angeny a he pofle of he aahmen (Fg.,a) s deemned by he expesson: D l ( b + ) os0 () Dependng of he angle α, he poson and dmensons of he sheang on ae gven by: l α l + sn α; b α b+ ( - os α) () Subsung Eq. () no Eq. () we ge he shea ss: τ f [ D ( l + snα )] l [ b + ( osα )] τaf 0, 7. ns ().. Calulaon of Bendng Sess Fo he alulaon of bendng ss, s onsdeed he membane spoke as a fxed beam, loaded by he foe F e foud on s op. Consdeng he faned on as eangula, he bendng ngh ondon s wen as: F e l α l b α a 6, (5) whee l s he hkness of he membane poke and l α s he lengh of a spoke.. Calulaon of Sess due o he oaonal oon The flexble membane has unfom oaon movemen wh angula veloy ω, onsan hkness h and a enal hole wh a adus D /, oue adus beng D/ (Fg.). Due o enfugal foe n he dsk ae podued ss ha ae onsdeed unfom dsbued on he dsk hkness. The alulaon was made n aodane wh he ngh of maeals heoy [6], [7]. The pnpal ss ae gven by: + µ γ + 8 g ω, (6) + µ γ + µ ω + +. (7) 8 g + µ The ss and ae dsbued along he adus afe a hypebol law [7]. ISSN: ISBN:

5 D D Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) df D ω. D d dϕ dϕ ( + d ) ( +d) dϕ df d d The adus of he le along he adal nomal ss have maxmum value s aheved by he anellaon of fs-ode devave of he ss. I follows as: vaable ( ) Fg.. Sess due o he oaonal moon on he membane elemen 0, (8) The equvalen ss aud by enfugal loadng s alulaed by he elaonshp: eh m max + max max max. () whh, by subsuon, gves he maxmum value of adal ss [7]: + µ γ ω 8 g max ( ). (9) The dagams of he dsbuon of pnpal ss n he dsk of onsan hkness ae shown n Fg.5. D max max The adal dsplaemen u of one pon of he membane s gven by: u ( µ ). (0) E Fo he wo oulnes (nne and oue) he dsplaemens has values [6]: u ; E () u. E 0 mn Fg.5. Vaaon dagams of he pnpal ss.5 Calulaon of Equvalen Sess To alulae he equvalen ss, one of he ngh heoes [7] s appled: eh ( + ehm ) + τ f a. () Dependng on he sze of angle α, whh s defned as usual up o he symmey axs of he spoke, he vaaon dagam of he equvalen ss was dawn fo wo analyzed as: whou enfugal loadng and wh enfugal loadng. The enfugal loadng s aken no aoun usng a oaon speed of 500 pm (Fg.6). The dagam shows ha boh he equvalen ss posss a maxmum pon o an angle α of abou 9. The esul s n aod wh he ss sae obaned by FEA, whh shows ha he aeas of maxmum equvalen ss ae plaed on he flle oo of he membane spokes [], [], [5]. Calulaon of he Numbe of embane s poke A membane hkness, h, s deemned by he ea of maeal (shees of on plae, wh hkness of 0. mm, membane manufaued fom OLC 65 A, wh hgh values of ulmae shea ngh and ulmae ensle ngh and onsn elasy modulus) []. τ f z D b ( D l) n l 6 s. () ISSN: ISBN:

6 Poeedngs of he s Inenaonal Confeene on anufaung Engneeng, Qualy and Poduon Syms (Volume I) Equvalen ss eh [N/mm ] Numeal alulaon fo he daa on he ouplng analyzed n he pape shows ha hs ao has he value 0.. Wh hs value, Eq. () s onveed o: Fg.6. Vaaon of he equvalen ss gven by () on flle oo of he spokes (α angle s shown n Fg.,a) eh ( + ehm ) + τ f ( + ehm) + ( 0 ), γ, expesson n whh appea he load fao γ. The fnal elaonshp beomes: (5 γ Fe l eh γ a l b (6) 6 Fom hs expesson esuls he hkness of he membane poke: l 6γ F b e l and he numbe of membanes: n l /h. a (7) Conluson. The sze of he elas ouplng wh flexble meall membanes s hon dependng on he oque value. n Equv. ss (whou enfugal loadng) Equv. ss (wh enfugal loadng) α angle of he angen o flleed pofle [ ]. Dung he wok, neal loadngs opeae on he ouplng (whh appea boh whn he un-ady and he ady opeaon of he ansmsson), shok and vbaon loads as well as loads due o foed defomaon of flexble membanes (due o shafs msalgnmens).. eall membanes ae addonally loaded on hng by enfugal foes, hng beng mpoan o hgh pepheals speeds.. The esuls of mehanal ngh eon (shea foe, foe loadng ws, enfugal loadng nfluene) ae heks of ouplng on ho equess, whh gve a easonable asssmen of a ouplng s ably o fae boh oque and enfugal loadng, o a equed span of lfe of he mehanal ansmsson. 5. Toque ansmsson o enfugal foes whn he ouplng do no au foes o be exeed on he onneed mahnes. The symmeal aangemen of he ouplng elemens ensues he equalzaon of foes whn he ouplng lf efeenes: [] Dobe, D., eahes on mul-cea Opmzaon of Elas Couplngs wh eall Flexble embanes, PhD Dsaon, Unvesy Polehna of Buhaes, 00 [] Dobe, D., Smon, I., aeals Seleon fo Impovng he Pefomane of Flexble embanes fom he Suue of Elas Couplngs, ealuga Inenaonal, No., 009, pp. -5. [] Dobe, D., Smon, I., ul-ea Analyss of Elas Couplngs wh eall Flexble embanes, Annals of Daaam fo 008 & Poeedngs of he 9 h Inenaonal Daaam Symposum, 008, pp [] Dobe, G., Dobe, D., ă,.f., Soahan, S., On San and Sess Sae of he eall embanes a Flexble Couplng, Inenaonal Confeene Powe Tansmssons 0, Vana, vol., 00, pp [5] Dobe, G., Dobe, D., ă,.f., Influene of he embane Geomey on embane Sess and San Sae a he Flexble Elas Couplngs, 9 h Inenaonal Senf Confeene,.T.. 00, Aa Tehna Napoenss, Appled ahemas and ehans 7, vol., 00, pp. -0. [6] Buzdugan, G., Blumenfeld,., Calulul de eznńă al pelo de maşn, Edua Tehnă, 979 [7] oanu, D.., eznńa maealelo, Edua Tehnă, 980. [8] Soohan, S., Consannesu, I.N.,. Paa modelă ş analze u elemene fne, Edua Polehna Pess, 00. ISSN: ISBN:

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma