Probabilistic approach for the design of an Equal-Leaf Spring

Transcription

1 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août 013 Probablstc approach for th dsgn of an Equal-af Sprng A. DOUKH a, R. BEN SGHAIER b a. Ecol Natonal d Ingénurs d Souss, BP 64 Souss Erradh, 403 (Tuns). b. MS aborator d mécanqu d Souss, Ecol Natonal d Ingénurs d Souss, BP 64 Souss Erradh, 403 (Tuns). Insttut Supérur ds Scncs Applqués t d Tchnolog d Souss, Cté Taffala (Ibn Khaldoun), 4003 Souss, Tuns. Résumé : s ngénurs créatfs dans l doman d constructon ds élémnts d suspnson, chrchnt à optmsr l dmnsonnmnt n srvc ds rssorts à lams durant la concpton. Dans c papr, on utls un approch probablst pour l dmnsonnmnt d un rssort à lams égals n vérfant l crtèr d dmnsonnmnt à un fablté donné c qu nous prmt d ntrodur ds dagramms d dmnsonnmnt consrvaturs. a fablté d dmnsonnmnt st détrmné n utlsant la foncton d prformanc avc smulaton numérqu à travrs la méthod d Mont Carlo. a méthod proposé prmt d obtnr un dmnsonnmnt d un rssort à lams égals plus fabl qu l dmnsonnmnt basé sur l crtèr détrmnst. Abstract : Th cratv ngnrs n th suspnson parts manufacturng fld sk to optmz th dsgn n a srvc of laf sprngs thy want durng ts concpton. In ths papr, w us a probablstc approach to dsgn an Equal-af Sprng (ES) by vrfyng th dsgn crtron for a gvn rlablty that allows ntroducng dsgn consrvator dagrams for svral ES charactrstcs. Th rlablty of th dsgn s dtrmnd usng th prformanc functon wth numrcal Mont Carlo Smulaton (MCS). Ths proposd mthod lads to a mor rlabl dsgn of an ES compard to dtrmnstc dsgn crtron. Mots clfs: rssort à lams, dmnsonnmnt, fablté, méthod d Mont Carlo. 1 Introducton A sprng s a mchancal part that uss lastc proprts of som matrals to absorb mchancal nrgy. It producs a movmnt or xrts an ffort or a coupl ovr a systm. Thy ar varously mployd n motor vhcl suspnson systms, th actual choc mad bng dtrmnd largly by vrsatlty n applcaton and bst conomy of matral n trms of nrgy storag pr unt volum. Th goal of th us of such a sprng s to fulfll a bst suspnson: th stablty of vhcl, th barng of contact of th whl on th ground, th safty of th automotv passngr and transportd matral aganst th mpacts and vbratons that can b found durng th path. W dstngush many typs of sprngs: laf sprng, hlcal sprng and torson sprng. af sprngs ar among sprngs that ar usd for th suspnson n a whld vhcl bcaus of th flxblty of on or many supr-mposd mtallc lavs. Ths flxblty ordrs th oscllaton of sprng du to th vhcl load. Thr ar many typs of laf sprngs [1]: lamnatd laf sprng; parabolc laf sprng; sngl, qual or mult-laf sprng. Th qual-laf sprng (ES) s consttutd of supr-mposd lavs havng th sam dmnson paramtrs. Th modlsaton of ths sprng s bascally carrd out from th modlsaton of a flxbl bam wth unform rctangular scton wth a fxd-on-nd pont (a cantlvrd bam) whch srvs to fnd th dsgn n a srvc crtron for ths sprng [1]. Th dsgn n a srvc crtron s xprssd by a mathmatcal nqualty. Whn t s vrfd, th laf sprng kps ts form and prformanc. Ths crtron dpnds on many paramtrs dfnng th dmnson and th mtallurgy of th laf sprng. Varyng th paramtrs valus, th srvc bhavour of th laf changs; ths paramtrs nflunc th laf flxblty (oscllaton prod). W consdr th paramtrs dfnng th dsgn n a srvc crtron of a laf as random varabls. Thy ar xprssd by a samplng drawn followng a probablstc dstrbuton[3,4]. In ths work, th srvc rlablty s computd by usng Mont Carlo mthod as th rato of th scurd vnts numbr and th total 1

2 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août 013 sz. In ordr to stablz th rlablty rspons of th modllng, th sz numbr should b optmzd. Mathmatcal modllng of an qual-laf sprng In our work, w consdr an ES dfnd by th followng dmnsonal data: lngth l, wdth B, laf thcknss and numbr of qual lavs n. P/ Brdl P/ la l 1 l l Insrtr Fgur 1. Equal-laf sprng A scton part of th ES s fttd n th brdl stoppng th flxur and s thn lmnatd from th lngth calculaton as xprssd n [1]: =l 1+l = l-l a (1) Th load P s unformly dstrbutd on th whol scton of th lavs. Howvr, th frst laf calld mastr laf, whch s drctly rstd on th supports, has to rst on th longtudnal loadng du to th ngn acclraton and brakng, as wll as on th latral loadng. Th mastr laf contans rnforcd xtrmts calld ys whch allow th sprng handlng. 3 Dsgn crtron of an qual-laf sprng In ordr to nsur a good dsgn of an ES, th maxmal strss must b always nfror to th admssbl strss. Ths condton s gvn by th followng nqualty: = 3. P. max. B. n. < () adm s th admssbl strss whch dfns th stat just bfor falur and t s xprssd as follow: adm R = adm (3) s whr s s th safty coffcnt and R s th yld strss of th matral. In ths papr, w consdr th 55Cr3 stl that s th most usful n th laf sprng constructon. Ths matral has th yld strss of 150N/mm at 0.%. W us a safty coffcnt of 5 corrspondng for ths cas of matral and applcaton []. Th appld strss s xprssd as follows: P app = (4) B. th crtron s quvalnt to th strct nqualty blow: 3. app. max = < adm (5). n. Ths dsgn crtron n a srvc may b ntrprtd graphcally on th fgur :

3 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août 013 n=3 ; =11mm =1500mm B=50mm; s=5 R=150N/mm Fgur. Th curv of th maxmal strss as a functon of th appld strss. Th horzontal straght ln rprsnts th admssbl strss and th nclnd straght ln rprsnts th maxmal strss as a functon of th appld strss. Th zon n whch th maxmal strss s strctly lss than th admssbl strss (th pont s stuatd blow th horzontal straght ln) s calld dsgn n a srvc scurty zon. Th ES s thn dsgnd and t s prformant. Whras, th zon n whch th maxmal strss s strctly supror to th admssbl strss s calld th waknss zon and th sprng s dfctv. Th qualty condton s a condton of doubt: w cannot dcd whthr t s th stat of scurty or 3. waknss. Th slop of th oblqu straght ln s.. Th varaton of th slop s dducd from th dsgn varaton of th ES. Whn varyng th valus of and paramtrs, w gt a bam of straght lns wth dffrnt slops dsprsd from th orgn nstad of on nclnd straght ln. Ths s gomtrcally ntrprtd by th fgur 3: n. n=3; R=150N/mm =5:3:0mm; s=5 =1000:100:000mm B=0:10:10mm P=1:49:500N Fgur 3. Curv of th maxmal strss as a functon of appld strss for dffrnt dsgn typs. Hghr valus of a slop man that th lngth valu of th sprng s bg and th thcknss valu s small. A small load appld on th ES producs a bg dflcton and strong oscllatons. Th bhavor of th sprng n a srvc rapdly dcrass wth ncrasng th slop. Othrws, whn th slop s small t mans that th lngth valu s small and th thcknss on s bg. A havy load appld on th ES lad to a small dflcton and th sprng wll b stabl. Thn, th bhavor n a srvc of th ES mprovs. 4 Probablstc approach for dsgn an qual-laf sprng Th am of ths probablstc approach s to lmnat th us of safty coffcnt (s) and to rplac t by dfnng dffrnt rlablty valus wth corrspondng confdnc lvls. 4.1 Rlablty computaton Many xprncs hav bn carrd out on th ES. W vary th dmnsons of th ES small varatons; w vrfy thrfor th dsgn crtron and w tst th bhavour of th lavs sprng. In ths papr, w vrfy th dsgn crtron and thn w comput th rlablty usng th Mont Carlo mthod [4]. Ths 3

4 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août 013 probablstc approach s usd to fnd th dmnsons that carry a way to th bst rlablty: w spak about dsgn crtron wth a gvn rlablty. Thn, w dfn th followng random varabls: R,, wth a coffcnt of varaton (cov) for ach on. Ths varabls can b rprsntd by a vctor. For ach random varabl, w carry out an N random samplng followng a Gaussan normal dstrbuton dfnd by a man valu and a coffcnt of varaton. Ths vctors allow th computaton of th rlablty n a srvc. Th proposd probablstc approach for ths problm of dsgn s basd on th coupl «maxmal strssyld strss» that consst n dtrmnng th probablty that th yld strss of ES wll b supror to th maxmal strss. For a functon of yld strss w hav YldS= R (6) 3. app. and for a functon of maxmal strss w hav Sgmax= max = (7). n. W dfn a prformanc functon G that sparats th dsgn n a srvc fld: a zon whr th sprng fulflls th bst suspnson and th waknss ara by G (x )=YldS(x )-Sgmax(x ), whr x s an lmnt of random vctor {X}, havng [3,4] a jont probablty dstrbuton functon f{x }({X}). G (x )>0 prsnts a bst suspnson of th sprng fld (sprng dsgnd n a srvc) F s for th ES; G (x )<0 prsnts a waknss fld F d for th sam sprng. Th scurty dsgn probablty P S s dtrmnd by: P = 1 { ( ) 0} { }({ }) 1... Pr ( ( ) max( )) Pr ( ( ) 0) S G x > f X dx dx = ob YldS x > Sg x = ob G x > x n R n Whn n s th numbr of paramtrs dfnng th functon G. Ths probablty can b computd by numrcal ntgraton whch rmans most of tm complcatd du to th xstnc of many ntgraton varabls. Consquntly, th Mont Carlo mthod s vry practcal and asy for us. In ths papr, th dsgn probablty P S s approxmatd by numrcal rsoluton usng th Mont Carlo mthod. For a total numbr of smulatons N, w admt that th frquncy of prformanc vnts of th sprng whr G (x )>0 strtch to dsgn probablty P S whn N + : Numbr of prformant vnts ( G( x) > 0) (9) PS = lm N + Total numbr of vnts Ths procdur prsnts th smplcty of th rlablty computng. Th spd of th mthod convrgnc s vry slow as t s proportonal [5] at N. A numbr of smulatons qual to 10 4 s rqurd n ordr to nsur th stmaton [6] of th rlablty whch rprsnts th scurty dsgn probablty P S wth a varaton coffcnt of 10%. Th rlablty rspons bgns to stablz from th numbr N=10 5 whch wll b chosn as a standard sz appld n our work [4]. Th usd random varabls for th followng applcaton ar dfnd by a cov of %. In ordr to xplor th proposd dsgn srvc crtron, w us a graphc rprsntaton (fgur 4) showng th yld strss as a functon of th appld strss whch both ar consdrd as random varabls. (8) N=300; n=3; cov=% =1500mm; =5mm B=50mm; R=150N/mm app=0.075mpa Fgur 4. Surfac of dsprson and cloud of ponts of an ES. 4

5 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août 013 Th straght ln rprsntng th lmt stat of th dsgn crtron ( G(x )=0 ) s transformd nto a bam of straght lns. Instad of th loadng rprsntatv pont dfnd by th appld strss, w gt a cloud of ponts. Th surfac dsprson corrspondng to th crtron and to th loadng dpnds on th fxd sz of th random samplng. Th obtand bam of lns dfns a nw zon calld th uncrtanty zon whch sparats both scurty and waknss zons. 4. Iso probablstc dsgn n a srvc crtron In ordr to obtan an so probablstc dsgn n a srvc crtron for a gvn rlablty, w ncrmnt th strss ampltud loadng and w comput th rlablty n ach ncrmntaton stp untl obtanng th targt on [4]. Th whol random varabls ar usd wth a cov of %. Svral rlablty valus (99,90,75,50,5,10,1%) for th so probablstc dsgn n a srvc crtron ar dtrmnd n ths applcaton. Ths rsults ar llustratd n fgur 5: n=3; cov=% =1500mm; =5mm B=50mm; R=150N/mm Fgur 5. Dsgn dagrams of a gvn rlablty. By applyng th so probablstc dsgn n a srvc approach for dffrnt rlablty valus, w obtan svral confdnc lvl zons nstad of both unqu scurty and waknss zons, charactrzng th classc dtrmnstc approach. Partcularly, th rsult of ths last applcaton s fundamntally basd on th hypothss that so rlablty crtra, n-sghtd by horzontal lns (fgur 5), ar obtand for a spcfc ES ladng to ncrmnt th loadng ampltud accordng to a unqu drcton. In ordr to obtan an so probablstc dsgn crtron vald to many dffrnt typs of ES, many ncrmnt drctons hav bn usd corrspondng ach on to spcfc dsgn charactrstcs. In ach drcton, w ncrmnt th load untl obtanng th dsrabl rlablty valu. W rcord th ponts corrspondng ach on a partcular drcton. Fttng ths rcordd ponts, w obtan th so probablstc dsgn n a srvc crtron curv. An applcaton for 99% rlablty s shown on fgur 6: n=3; cov=% R=150N/mm 99% Rlablty Fgur 6. 99% so rlablty dagram for many ES charactrstcs. 5

6 1 èm Congrès Franças d Mécanqu Bordaux, 6 au 30 août Conclusons Th proposd probablstc approach prmts mprovng th dsgn n a srvc crtron of an ES, n th sns that w ncras th scurty confdnc lvl. Gomtrcally, ths can b ntrprtd as: th probablstc approach allows obtanng a bam of straght lns and a cloud of ponts ladng to gt an uncrtanty zon. Furthrmor, th probablstc dsgn n a srvc crtron for a gvn rlablty allows plottng so probablstc dagrams (consrvatv dagrams). Two applcatons of th so probablstc dsgn crtron ar studd: th frst on s for a spcfc dmnson of ES followng thn a unqu ncrmnt drcton, and th scond on s for svral ES charactrstcs followng so many ncrmnt drctons. Rfrncs [1] M. Duchmn, Tchnqus d l ngénur Rssorts contrants d flxon. Réf. B543. [] Y. Xong "Formular d mécanqu-pècs d constructons" Group Eyrolls, 007, ISBN : [3] Ch. Bouraou R. Bn Sghar R. Fathallah "Effts ds dsprsons ds paramètrs d condton d surfacs sur la fablté ds composants mécanqus grnallés sollctés à la fatgu polycyclqu" 18 èm Congrès Franças d Mécanqu Grnobl, 7-31 Aôut CFM'07 007, Franc. [4] R. Bn Sghar, Ch Bouraou, R. fathallah t al. Probablstc hgh cycl fatgu bhavour prdcton basd on global approach crtra Intrnatonal Journal of Fatgu [5] mar M, Chataunuf- A. & Mttau J.C. Fablté ds structurs: couplag mécano-fablst statqu, Edt. Hrms Pars, 005. [6] Zhao Y.G, Ono T. Momnt for structural rlablty. Structural safty. 3 (001) pg: