Optimization of welding parameters for weld penetration in FCAW

Transcription

1 Optmzaton of wldng paramtrs for wld pntraton n FCAW N.B. Mostafa, M.N. Khajav* Mchancal Engnrng Dpartmnt, Shahd Raja Unvrsty, hran, Iran *Corrspondng author. E-mal addrss:mhrdadnour@.org Abstract In ths papr basd on th statstcal tchnqu of cntral compost rotatabl dsgn a modl has bn dvlopd to prdct wld bad pntraton n flu-cord arc wldng (FCAW) of a grad of hgh strngth low alloy (HSLA) stls. h dvlopd modl s abl to prdct th lnar, quadratc and two-factor ntractv ffcts of such wldng procss paramtrs as wldng currnt, arc voltag, nozzlto-plat dstanc, lctrod-to-wor angl, and wldng spd on wld bad pntraton. h dvlopd modl was usd to mamz th wld bad pntraton by mployng th Constrand Optmzaton Mthod.h optmzaton rsults show that wld bad pntraton wll bcom mamum whn th frst four paramtrs wll attan thr mamum valus and th last on namly wldng spd wll ta ts mnmum valu. Kywords: Wldng paramtr; Wld pntraton; Flu-Cord Arc Wldng; Optmzaton; Squntal quadratc Programmng. Introducton Inadquat wld bad dmnsons such as shallow dpth of pntraton may contrbut to falur of a wldd structur snc pntraton dtrmns th strss carryng capacty of a wldd jont [].o avod such occurrncs th nput or wldng procss varabls whch nflunc th wld bad pntraton must thrfor b proprly slctd and optmzd to obtan an accptabl wld bad pntraton and hnc a hgh qualty jont []. hs nvstgaton s basd on a prvous wor [] carrd out by on of th authors n whch th cntral compost rotatabl dsgn tchnqu had bn usd to dvlop a mathmatcal modl corrlatng th fv wldng procss varabls of wldng currnt (I), wldng spd (S), arc voltag (V), nozzl-to-plat dstanc (N), and lctrod to wor angl () to th wld bad pntraton n flu cord arc wldng (FCAW) of a grad of hgh strngth low alloy (HSLA) stls. hs tchnqu was abl to prdct th lnar, quadratc and two factor ntractv ffcts [-7] of procss paramtrs on wld pntraton. Usng ths modl or quaton th Squntal Quadratc Programmng (SQP) whch s an optmzaton mthod for constrand optmzaton problms [8] was usd to mamz wld bad pntraton thus achvng a hgh qualty jont.. Eprmntal dtals.. Matrals A grad of HSLA stl of mm thcnss had bn usd as th plat matral. Flu cord wr of.6 mm damtr (AWS classfcaton E705) wth 00% CO shldng gas had also bn mployd on a sm automatc flu cord arc wldng (FCAW) machn... Eprmntal dsgn h rqurd dsgn matr (tabl ) wth a total of prmntal runs had bn dvlopd for th fv wldng factors slctd. h uppr () and lowr (-) lvls of all th fv varabls as shown n tabl had bn stablshd by tral runs pror to th actual wldng to nsur dposton of an accptabl wld bad. h ntrmdat lvls of -, 0, of all th varabls had bn calculatd by ntrpolaton... Wldng procdur and pntraton masurmnt Bad-on-plat wlds wr dpostd randomly accordng to th lvls of paramtrs n th dsgn matr. Spcmns wr cut off from ths plats and dpth of pntratons was masurd usng a profl projctor at a magnfcaton of 0.. Slcton and dvlopmnt of th modl A modl rlatng th dpth of pntraton to th lvls of on or mor wldng factors s an ndspnsabl ad n th ntrprtaton of rsults from an prmntal dsgn. Wd prnc had shown that quadratc rlatonshps ar usually adquat and so most surfac dsgns n th ltratur ar for fttng a quadratc quaton [, 6]. Basd on ths consdratons th followng scond dgr polynomal rlatng th rspons or pntraton to all th fv wldng varabls was assumd. y = b b b b ()

2 Optmzaton of wldng paramtrs for wld pntraton n FCAW 68 whr y, Rspons varabl or yld,.g. pntraton.,..., 5, Indpndnt wldng factors such as wldng currnt (I), wldng spd (s), tc. b0, b,..., b5, b, b,..., b5, b,..., b55, coffcnts. Rgrsson abl Cntral compost rotatabl dsgn matr for fv factors, =5 Factors S.No. 0 5 = , A dummy varabl, Wldng currnt (I), Wldng spd (S), Arc voltag (V),, 5 Nozzl-to-plat dstanc (N) Elctrod-to-wor angl () abl Factors and thr lvls N o Factors Unt N ot at o n Lvl Wldng Amp I Currnt Wldng Cm/ S spd mn Arc Volt V voltag Nozzl mm N to plat dstanc 5 Elctrod to wor angl dgr Usng a computr program all th rgrsson coffcnts of th modl wr computd for th rspons or dpth of pntraton. abl Sgnfcant coffcnts for pntraton Coffcnt Pntraton b 0.5 b 0. b -0.6 b 0.066* b * b b 0. b --- b --- b b b --- b b 0.6 b b b --- b --- b --- b --- b * Insgnfcant coffcnts h dvlopd modl was tstd for adquacy by applyng th analyss of varanc (ANOVA) and analyss of rgrsson tchnqus. Smlarly to tst th sgnfcanc of rgrsson coffcnts th Studnt s tst was mployd to fnd th nsgnfcant trms for 95% confdnc lvl. Onc th nsgnfcant trms wr dntfd thy wr droppd from th modl wthout sacrfcng th adquacy of th modl. h sgnfcant coffcnts ar shown n tabl. Usng th sgnfcant rgrsson coffcnts th fnal modl for pntraton could b constructd as gvn blow:

3 68 N.B. Mostafa, M.N. Khajav P =.5 0. * I * S 0. * 0. * I * S * S * V 0.6 * V * N * V * N () h modl so dvlopd could b utlzd n prdctng th valu of dpth of pntraton for ach gvn st of wldng varabls by nsrtng th codd valus of th varabls n th abov quaton. In addton to th adquacy tst prformd, th valdty of th rsults was also tstd wth th hlp of scattr dagram Fg.. As s vdnt from ths fgur thr s a farly good corrlaton btwn th obsrvd and prdctd valu of th bad pntraton. functon and th constrants ar lnar functons of th dsgn varabl, th problm s nown as a Lnar Programmng (LP) problm. Quadratc Programmng (QP) concrns th mnmzaton or mamzaton of a quadratc objctv functon that s lnarly constrand. For both th LP and QP problms, rlabl soluton procdurs ar radly avalabl. Mor dffcult to solv s th Nonlnar Programmng (NP) problm n whch th objctv functon and constrants can b nonlnar functons of th dsgn varabls. A soluton of th NP problm gnrally rqurs an tratv procdur to stablsh a drcton of sarch at ach major traton. hs s usually achvd by th soluton of an LP, a QP, or an unconstrand subproblm... Constrand optmzaton Fg.. Scattr dagram showng obsrvd vs. prdctd pntraton.. Dfnton of optmzaton problm An optmzaton or a mathmatcal programmng problm can b statd as follows: Fnd X = n Subjct to constrants G () = 0, =,, m whch mnmzs f() G () <= 0, I = m,, m () whr s th vctor of lngth n dsgn paramtrs, f() s th objctv functon, whch rturns a scalar valu, and th vctor functon G() rturns a vctor of lngth m contanng th valus of th qualty and nqualty constrants valuatd at. An ffcnt and accurat soluton to ths problm dpnds not only on th sz of th problm n trms of th numbr of constrants and dsgn varabls but also on charactrstcs of th objctv functon and constrants. Whn both th objctv In constrand optmzaton, th gnral am s to transform th problm nto an asr subproblm that can thn b solvd and usd as th bass of an tratv procss. A charactrstc of a larg class of arly mthods s th translaton of th constrand problm to a basc unconstrand problm by usng a pnalty functon for constrants that ar nar or byond th constrant boundary. In ths way th constrand problm s solvd usng a squnc of paramtrzd unconstrand optmzatons, whch n th lmt (of th squnc) convrg to th constrand problm. hs mthods ar now consdrd rlatvly nffcnt and hav bn rplacd by mthods that hav focusd on th soluton of th Kuhn-ucr (K) quatons. h K quatons ar ncssary condtons for optmalty for a constrand optmzaton problm. If th problm s a so-calld conv programmng problm, that s, f() and G (), =,,m ar conv functons, thn th K quatons ar both ncssary and suffcnt for a global soluton pont. A functon f() s sad to b conv f for any par of ponts X and X and all, 0 f[ λx ( λ)x] λf (X ) ( λ)f (X) that s, f th sgmnt jonng th two ponts ls ntrly abov or on th graph of f(x). Rfrrng to th dfnton of optmzaton problm (Eq. ) th Kuhn-ucr quatons can b statd as f (X ) λ.g (X ) = 0 λ 0 * m = λ. G (X ) = 0 =,...,m = m,...,m h frst quaton dscrbs a canclng of th gradnts btwn th objctv functon and th actv constrants at th soluton pont. For th gradnts to b cancld, Lagrang multplrs (, =,,m ) ar ncssary to balanc th dvatons n magntud of th objctv functon and constrant gradnts. Bcaus only actv constrants ar ncludd n ths canclng opraton, constrants that ar not actv must not b ncludd n ths opraton and so ar gvn Lagrang multplrs qual to zro. hs s statd mplctly n th last two quatons of Eq.. h soluton of th K quatons forms th bass to many nonlnar programmng algorthms. hs algorthms attmpt to comput th Lagrang multplrs drctly. Constrand quas- Nwton mthods guarant suprlnar convrgnc by accumulatng scond ordr nformaton rgardng th K quatons usng a quas-nwton updatng procdur. hs mthods ar commonly rfrrd to as Squntal Quadratc ()

4 Optmzaton of wldng paramtrs for wld pntraton n FCAW 68 Programmng (SQP) mthods, snc a QP subproblm s solvd at ach major traton (also nown as Itratv Quadratc Programmng, Rcursv Quadratc Programmng, and Constrand Varabl Mtrc mthods)... Squntal Quadratc Programmng (SQP) SQP mthods rprsnt th stat of th art n nonlnar programmng mthods. Schttows [9], for ampl, has mplmntd and tstd a vrson that outprforms vry othr tstd mthod n trms of ffcncy, accuracy, and prcntag of succssful solutons, ovr a larg numbr of tst problms. Basd on th wor of Bggs [0], Han [], and Powll ([,]), th mthod allows you to closly mmc Nwton s mthod for constrand optmzaton just as s don for unconstrand optmzaton. At ach major traton, an appromaton s mad of th Hssan of th Lagrangan functon usng a quas-nwton updatng mthod. hs s thn usd to gnrat a QP subproblm whos soluton s usd to form a sarch drcton for a ln sarch procdur. An ovrvw of SQP s found n Fltchr [], Gll t al. [5], Powll [6], and Schttows [7]. h gnral mthod, howvr, s statd hr. Gvn th dfnton of optmzaton problm (Eq. ) th prncpal da s th formulaton of a QP subproblm basd on a quadratc appromaton of th Lagrangan functon. m L(X, λ ) = f (X) λ.g (X) (5) = Hr Eq. s smplfd by assumng that bound constrants hav bn prssd as nqualty constrants. You obtan th QP subproblm by lnarzng th nonlnar constrants... Quadratc Problm (QP) Subproblm h quadratc problm s as follows: Mnmz H d f (X ) d g g (X ) d (X ) = 0 =..., m g (X ) d g (X ) 0 = m,...,m (6) hs subproblm can b solvd usng any QP algorthm. h soluton s usd to form a nw trat of th followng form X = X α d h stp lngth paramtr s dtrmnd by an approprat ln sarch procdur so that a suffcnt dcras n a mrt functon s obtand. h matr H s a postv dfnt appromaton of th Hssan matr of th Lagrangan functon (Eq. 5). H can b updatd by any of th quas-nwton mthods, although th BFGS mthod appars to b th most popular. A nonlnarly constrand problm can oftn b solvd n lss traton than an unconstrand problm usng SQP. On of th rasons for ths s that, bcaus of lmts on th fasbl ara, th optmzr can ma nformd dcsons rgardng drctons of sarch and stp lngth. d.. SQP mplmntaton h SQP mplmntaton conssts of thr man stags: Updatng of th Hssan matr of th Lagrangan functon Quadratc programmng problm soluton Ln sarch and mrt functon calculaton hs stags ar pland brfly... Updatng th Hssan matr At ach major traton a postv dfnt quas-nwton appromaton of th Hssan of th Lagrangan functon, H, s calculatd usng th BFGS mthod, whr,( =,,m) s an stmat of th Lagrang multplrs. H whr s q = H = X f (X = f ( q q ) q s X n = ) s = H H H s λ * g (X λ * g (X n ) ) Powll [] rcommnds png th Hssan postv dfnt vn though t mght b postv ndfnt at th soluton pont. A postv dfnt Hssan s mantand provdng q s s postv at ach updat and that H s ntalzd wth a postv dfnt matr. Whn q postv, s s not q s modfd on an lmnt-by-lmnt bass so that q s > 0. h gnral am of ths modfcaton s to dstort th lmnts of q, whch contrbut to a postv dfnt updat, as lttl as possbl. hrfor, n th ntal phas of th modfcaton, th most ngatv lmnt of q * s s rpatdly halvd. hs procdur s contnud untl q s s gratr than or qual to -5. If, aftr ths procdur, q s s stll not postv, modfy q by addng a vctor v multpld by a constant scalar w, that s, q = q wv whr v g ( ).g ( = ) g f ( q ).w < 0 and ( q ).(s ) < 0 ( =,..., m) v = 0 othrws. and ncras w systmatcally untl q s bcoms postv. ( ).g ( )

5 68 N.B. Mostafa, M.N. Khajav... Quadratc Programmng problm soluton At ach major traton of th SQP mthod, a QP problm of th followng form s solvd, whr A rfrs to th th row of th m-by-n matr A. Mnmz A d = b A d b q(d) = d Hd c d =,...,m = m,...,m h mthod usd n th Matlab Optmzaton oolbo s an actv st stratgy (also nown as a projcton mthod) smlar to that of Gll t al., dscrbd n [8] and [9]. It has bn modfd for both Lnar Programmng (LP) and Quadratc Programmng (QP) problms. h soluton procdur nvolvs two phass. h frst phas nvolvs th calculaton of a fasbl pont (f on sts). h scond phas nvolvs th gnraton of an tratv squnc of fasbl ponts that convrg to th soluton. In ths mthod an actv st, A s mantand that s an stmat of th actv constrants (.., thos that ar on th constrant boundars) at th soluton pont. Vrtually all QP algorthms ar actv st mthods. hs pont s mphaszd bcaus thr st many dffrnt mthods that ar vry smlar n structur but that ar dscrbd n wdly dffrnt trms.... Ln sarch and mrt functon calculaton h soluton to th QP subproblm producs a vctor d, whch s usd to form a nw trat of th followng form: X = X α d h stp lngth paramtr s dtrmnd n ordr to produc a suffcnt dcras n a mrt functon. h mrt functon usd by Han [] and Powll [] s usd n Matlab Optmzaton toolbo. 5. Wld pntraton optmzaton problm formulaton h objctv functon for wld pntraton whch must b mamzd was drvd n scton of th papr as Eq. h constrants of th wldng paramtrs ar gvn n tabl. Hr w rstat ths quatons. Mamz: P =.5 0. * I * S 0. * 0. * I * S * S * V 0.6 * V * N *V * N () Subjct to constrant: 50 I 50 7 V 5 5 N S 55 Matlab Optmzaton toolbo and th functon fmncon was usd for ths optmzaton problm. h fmncon functon uss SQP mthod for optmzaton. wo m-fl was wrttn namly objfun and confun. In th objfun th objctv functon was dfnd and n th confun th constrants wr st. Optmzaton rsult shows n ordr to attan th mamum wld pntraton th frst four factors must b at thr mamum valu and th last on must b at ts mnmum valu. So th mamum wld pntraton wll b attand whn th followng valus ar chosn for th fv factors: I = 50 amp V = 5 volt N = 5 mm = 0 dgr S = 5 cm/mn Usng ths valus th wld pntraton bcom P = 5.75 mm. 6. Rsults and dscussons In addton to th adquacy tst prformd, th valdty of th rsults of mathmatcal modlng for prdcton of dpth of pntraton was also tstd wth th hlp of a scattr dagram (Fg. ). As s vdnt from ths fgur thr s a farly good corrlaton btwn th obsrvd and prdctd valu of wld bad pntraton. h optmzaton rsult shows whn th ) Wldng currnt ) Arc voltag ) Nozzl to plat dstanc ) Elctrod to wor angl attan thr mamum possbl valu and 5) Wldng spd b at ts mnmum valu thn pntraton wll b mamzd. Now physcal rasons for th abov rsults wll b pland. Incras n wldng currnt (I) ncrass th dpth of pntraton (P). It s nown that moltn mtal droplts transfrrng from th lctrod to th plat ar strongly ovrhatd. It can b rasonably assumd that ths tra hat contrbuts to mor mltng of th wor pc. As currnt ncrass th tmpratur of th droplts and hnc th hat contnt of th droplts ncrass whch rsults n mor hat bng transfrrd to th bas plat. Incras n currnt rducs th sz but ncrass th momntum of th droplts whch on strng th wld pool causs a dpr pntraton or ndntaton. h ncras n pntraton as currnt ncrasd could also b attrbutd to th fact that nhancd arc forc and hat nput pr unt lngth of th wld bad rsultd n hghr currnt dnsty that causd mltng a largr volum of th bas mtal and hnc dpr pntraton. Incras n wldng (S) causs a dcras n dpth of pntraton (P). hs may b attrbutd to lssr hat nput at hghr spds pr unt lngth of th wld bad whch causd a smallr wld pool and dcrasd dpth of pntraton. Incras n arc wldng voltag (V) rsultd n an ncras n dpth of pntraton (P), bcaus of ncras n hat nputs rportd by McGlon and Chadwc [0 ]. Incras n lctrod-to-wor angl from 90 to 0 (.. for normal to bachand) had rsultd n ncras of dpth of pntraton. h bachand wldng tchnqu has bn rportd [] to provd th bst pntraton, as th arc forc ps th slag from runnng n front of th lqud pool. Incras n nozzl-to-plat dstanc (N) also causs an ncras n dpth of pntraton (P) whch may b du to hghr tmpratur of th droplts mpngng on th wld pool bcaus of mor rsstanc hatng of th wr at hghr nozzl-to-plat dstancs.

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