Clulton Metho of Dynm Lo Beng Cuve of Doubleow Foupont Contt Bll Beng Shohun L College of Mehnl Engneeng Tnn Unvesty of Tehnology n Euton Tnn, Chn bstt On the bss of the stt nlyss of oubleow foupont ontt bll bengs, the eltonshp between the bs te lfe of bengs n the ynm los of bengs wee eue. ong to the efnton of ynm lo beng uve of beng, the ynm beng uve of oubleow foupont ontt bll beng ws plotte. The lulton se n pplton se of ynm beng uves of oubleow foupont ontt bll bengs wee lso gven. Key wos Doubleow FouPont Contt Bll Beng; Dynm Lo; Beng Cuve I. INTODUCTION Wn enegy s enewble len enegy hs been p ttenton ll ove the wol, n the wn powe geneton tehnology evelope ply. Yw bengs n vble pth bengs e the ey omponents of wn tubnes, n the stutue s mostly oubleow foupont ontt bll bengs wth nne ng o oute ng wth teeth. The wn tubnes n the wo e usully subete to ombne los (xl foe, l foe n ovetunng moment) n mpt los, whh eques tht the yw bengs n the vble pth bengs hve suffent yng pty. The ynm beng uve of the beng shows the mxmum ynm lo beng n be une the pemse of the gven lfe, whh s of get sgnfne fo the seleton of the beng n the lo. t pesent, the eseh of beng lo uve mnly fouses on the stt lo beng uve [9]. In, Wng Hong n othes gve the theoetl lulton fomul of beng pty n lfe estmton of mult ow olle slewng beng, n ntoue smplfe wng metho of beng ynm n stt beng pty uve bse on Hetz ontt theoy, LunbegPlmgen ftgue lfe theoy n the spel geomet stutue htests n wong ontons of mult ow olle slewng beng []. Ths metho s genelly use to quly test whethe the lo s vlble, but the lulton esults e not ute enough. bo, Gönz P pesente lulton moel of ynm n stt beng pty of thee ow olle beng n nlyze the stt beng pty of lge oubleow foupont ontt bll beng []. Kn L et l nlyze the beng pty of s olle slewng bengs, n gve the beng uves of bengs une ffeent s ngles Ynshung Wng College of Mehnl Engneeng Tnn Unvesty of Tehnology n Euton Tnn, Chn [,5]. In ths ppe, we eue the mthemtl elton between the bs te lfe n the ynm lo of oubleow foupont ontt bll beng on the bss of stt nlyss, ntoue the wng metho of ynm lo uve, n gve the lulton n pplton se. II. EXCT SOLUTION OF CONTCT FOCE The stutue of the oubleow foupont ontt bll beng s shown n Fgue : Fg. Stutue of oubleow foupont ontt bll beng Bul the beng oonte system s shown n Fgue. x xs long the beng xs eton, s the nne mete eton, eh bll poston ngle φ n be expesse s: = ( )/(z/) (=,,... Z/) whee Z s the numbe of steel blls n ouble ow bengs. o Fg. Coonte system of the beng The ontt ps mnly subete to xl foe symmety fo ontt (uppe), ontt (below), the othe two ontt ps, espetvely lle ontt, ontt. Befoe beng subete to n extenl lo, when te xl lene u nto ount, the ente of uvtue of the nne n oute goove of ny p of steel bll ontts n be obtne by the followng fomul: f fe D W u () x φ y z IJETV6IS969 www.et.og (Ths wo s lense une Cetve Commons ttbuton. Intentonl Lense.)
Whee f s the us oeffent of the nne ewy uvtue; f e s the uvtue us fto of the oute ewy; D w s the mete of the steel bll; s the ontt ngle of ntl poston. When the xl lene u =, the uvtue ente stne of the nne n oute goove n be obtne by the followng fomul: W e D f f () Suppose the oute ng s fxe n the nne e ottes, the oute lo te on the nne ng s shown n Fgue. Whee F s the xl ynm lo, F s the l lo, M s the the ovetunng moment ynm Fg. Extenl pple los on beng lo, m s the pth le mete of the beng. s the ente stne between the two ows of bll bengs of oubleow foupont ontt bll beng. When the oubleow foupont ontt bll beng s subete to n extenl lo, the nne ng s sple, n the ente of uvtue of the goove of ll ps of ontts hs hnge. The ente of uvtue between the nne n oute goove of the ontt ps (=,,,) t the poston ngle φ s φ:.5 sn.5 sn.5 sn.5 sn () Whee, n espetvely e the xl splement, the l splement n the nlnton ngle of the nne ng when the nne ng bes the xl ynm lo F, the l lo F n the ovetunng moment ynm lo M; the us of uvtue of the ewy = / m+(f.5) D w /u ( ) ; Whee m s the mete of the beng pth le; φ s the poston ngle of steel bll. fte the splement of the nne ng, the ontt ngle of ontt ps (=,,,) t poston s: ) sn sn( () The nne ng s n equlbum une the ton of extenl lo n noml ontt lo, n the foes tng on the nne ng e shown n Fgue M F F φ φ φ φ Fg. Foes tng on nne ewy Fg.5 the gm of ewy goove of beng ong to mehnl equlbum equton: sn sn sn sn sn sn sn sn M F F m (5) Whee s the noml ontt lo of ontt ps t poston ; s the ente stne between the two ows of bll bengs of oubleow foupont ontt bll beng. n be get ong to the Hetz ontt theoy: IJETV6IS969 (Ths wo s lense une Cetve Commons ttbuton. Intentonl Lense.) www.et.og
Kn,.5, Whee, K n s the totl lo efomton onstnt of the ollng boy n the nne n oute ngs; s the totl elst ontt efomton between the steel bll n the nne n oute ewy, long the eton of ontt ps, t poston : (6) (7) ong to the gven geomet pmetes of the beng n n ntl vlue of nne ng splement (,, ),, n n be lulte though fomul ~. Put the vlues of, n nto the fomul 7 to obtn, Then, n e lulte by fomul 6 n, espetvely. Put n nto the fomul 5, whle mng F =, F n M fo ontnuous vlues, ong to fomul 5, usng the Newtonphson metho, to obtn the fnl vlue of beng nne ng splement s (,, ) une eh wong ontons (F, M, F ). By fomul 6, the noml ontt lo of eh poston ngle of the beng s obtne. III. CLCULTE THE BSIC TED LIFE OF THE BEING The ewy of oubleow foupont ontt bll bengs s typl peh shpe goove. The steel bll hs fou ontt ponts wth the nne n oute ewy, whh oespon to fou hnnels. Nme the fou hnnels s hnnel,,,, s shown n Fgue 5.. Bs Dynm Lo of Beng. Fo oubleow foupont ontt bll bengs, the te ynm lo of the ngs (e) s: ( e) D f 98. f W m. ( e) ( e). /. Z.6D W In the fomul, the symbol stns fo the nne ng, n the symbol e stns fo the oute ng; λ n η e oeton ftos fo oubleow foupont ontt bll bengs..9 B. Bs Equvlent Dynm Lo of Beng. s the oute ng s fxe n the nne e ottes, the equvlent ollng lo on the ewy of the nne e s: / / (8) (9) e Z The equvlent ollng lo on the oute ewy s:. / ev () Z Whee s the ontt lo of steel bll. C. te Lfe Clulton of Inne ng. The te lfe of eh ewy on the nne ng s: L ( / e ) te lfe of nne ng s: ().9 /9 L L () D. te Lfe Clulton of Oute ng. The te lfe of eh ewy on the oute ng s: L () e ( e / ev ) te lfe of oute ng s: /9 L e ( L e ).9 () The te lfe of oubleow foupont ontt bll bengs L n be obtne by fttng the te lfe of the nne ng n the te lfe of the oute ng: /9 /9 6 L L L (5) e IV. PPLICTION CSE The stutul pmetes n mtel pmetes of etn type oubleow foupont ontt bll beng e shown n Tb : TBLE. Pmetes of Doubleow Foupont Contt Bll Beng pmete vlues Pth mete of bll set m [mm] 5 Bll mete D W [mm].5 The ente stne of ouble ow steel bll [mm] The us of uvtue of nne hnnel [mm]. The us of uvtue of oute hnnel e [mm]. 69 IJETV6IS969 www.et.og (Ths wo s lense une Cetve Commons ttbuton. Intentonl Lense.)
The numbe of blls Z 8 Posson to of bll n feule v. xl ply u [mm]. Elst moulus of bll n beng ngs E [Gp] 7 The beng's stutul pmetes, mtel pmetes n n ntl vlue (,, ) of the splement of the nne ng e substtute nto bove lulton metho to obtn the te lfe of the beng L. The xl ynm lo F n ovetunng moment ynm lo M whh onfom to the L < ε e extte s the ponts on the oonte system. Ths exmple tes ε=. n obtns sees of ponts, s shown n Fgue 6. Fg.6 Foe ombnton poston ponts of oubleow foupont ontt bll bengs Connete bove ponts, the ynm lo beng uves of the oubleow foupont ontt bll bengs e obtne, s shown n Fgue 7. Fg.7 Dynm lo beng gm of ouble ow fou pont ontt bll beng The metho of usng the ynm lo beng uve to etemne whethe the beng selete meets the lfe equement une gven lo e s follows:. Uses pove the xl ynm lo F, ovetunng moment M n te lfe L of the oubleow foupont ontt bll bengs. In ths exmple, the oubleow foupont ontt bll beng hs ottonl spee of./mn, n the eque seve lfe s 75 hous. Tht s to sy L =5. The xl ynm lo n ovetunng moment espetvely e: F =N n M =5N m.. In the ynm lo beng uve, fn the oonte pont oesponng to the gven lo F n M, n use the pont "" to nte. In ths exmple, the oontes of pont "" e ( F, M ), n whh F =N n M =5N m.. Connet the oonte system ogn n pont "", exten bove lne n se the ynm lo beng uve to pont "B". s shown n Fgue 8, fn the oonte vlue of "B" pont. In ths exmple, the oontes of the pont "B" e ( F B, M B ), whee F B=N n M B=8N m.. Clulte lo fto f L. In ths exmple, the lo fto f L=F /F B=/=.66. 5. Clulte the lfe of the selete oubleow foupont ontt bll bengs t gven F n M ontons: L=f L. In ths se, L= f L =5.. 6.Detemne whethe the onton LL s estblshe, f t s estblshe, the esgne bengs n meet the lfe equements une the gven lo F n M, othewse, the esgne bengs n t meet the lfe equements. In ths exmple, L=5.>L =5, so the oubleow foupont ontt bll beng selete n meet the equements of lfe une the gven lo F =N n M=5N m. V. CONCLUSION Ths ppe eue the eltonshp between the bs te lfe n the ynm lo of beng bse on the stt moel of oubleow foupont ontt bll beng. The beng uve of beng ynm lo s wn, xl ynm lo F s the bsss, ovetunng moment ynm lo M s the onte, ong to the efnton of ynm lo beng uve of ollng beng. The pont on the uve n be unestoo s the ynm lo tht the beng n be when the beng lfe s gven vlue. The lulton of ynm yng uve of oubleow foupont ontt bll beng poves bss fo the seleton n pplton of suh bengs. CKNOWLEDGMENT Ths eseh s suppote by the Ntonl Sene Founton of Chn(No. 575) n Tnn Ntul Sene Founton (No.6JCYBJC89). Fg.8 Shemt gm of "" n "B" ponts IJETV6IS969 www.et.og (Ths wo s lense une Cetve Commons ttbuton. Intentonl Lense.) 5
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