Fatigue equivalent loads for visualization of multimodal dynamic simulations

Transcription

1 Vst the SIMULI Resouce Cente fo oe custoe exaples. atgue equvalent loads fo vsualzaton of ultodal dynac sulatons Henk Wentzel 1, Gwenaëlle Genet 1 Coespondng autho Scana Coecal Vehcles B SE , Södetälje, Sweden Tel: bstact: Tansent dynac sulatons gan potance n the autootve ndusty and oden fatgue postpocessos ae apt to evaluate the fatgue daage. Howeve, addtonal nsght nto a stuctue s behavou ay be obtaned fo obsevng the dsplaceents. Dsplaceent pattens ae potant fo desgn engnees n ode to pove the stuctue. Wth popotonal statc loads t s tval to dsplay and undestand the dsplaceents, but the dsplaceents n dynac sulatons ae often vey coplcated. Ths pape descbes a novel ethod fo vsualsng the tansent dsplaceents of ode-based analyses. Based on the odal dsplaceents a new sple, yet fatgue equvalent, odal dsplaceent s coputed and vsualzed nstead. pplcatons fo coecal vehcles ae used as exaples. Keywods: odal dynacs, equvalent load, fatgue, post-pocessng, Basqun s law 1. Intoducton The dsplaceent hstoy of a fnte eleent odel of an abtay stuctue ay be expessed as a lnea cobnaton of fxed dsplaceent vectos ultpled wth scala te vayng functons. In any engneeng applcatons t s possble to dscen a few dsplaceent vectos that adequately span the ente dsplaceent space. Typcally the fst few egenodes of the stuctue ae suffcent. Ths pape s concened wth fatgue loads on vehcle coponents and fo ths type of applcaton a lnea epesentaton wth only a few egenodes s often suffcent to epesent the ente dsplaceent hstoy. Moden fatgue post-pocessos, whethe stan- o stess based, ae capable of evaluatng the fatgue load fo a te vayng lnea cobnaton of fxed vectos. The fatgue daage s coputed as a local daage on a ctcal plane n evey pont/node of the stuctue. s an exaple; fo a nube of vectos n the ode of 10 1 and a nube of nodes n the odel n the ode of 10 6, the te fo fatgue evaluaton s counted n hous on an odnay copute. Ths type of fatgue evaluaton of lnea dynac sulatons s cuently a egula outne at any copanes n the vehcle ndusty. Howeve, the fatgue lfe of a coponent s not alone suffcent fo developng new engneeng solutons. If only a weak pont s ndcated the eedy s usually to add oe ateal (ead cost baqus Uses Confeence Vst the SIMULI Resouce Cente fo oe custoe exaples.

2 and weght) to that egon. Most often the cleve soluton to a poble eques a good undestandng of the ctcal dsplaceent pattens. Ths undestandng s dffcult to obtan n tansent (non-statonay) dynacs. It s hee poposed to fo a fatgue equvalent haonc statonay dsplaceent and use ths dsplaceent to copehend the ctcal coponents of the tansent dynacs. 2. atgue daage of tansent dynacs What s sought s a dsplaceent hstoy that s easy to vsualze. ssung that the actual dsplaceent ay be appoxated wth ( x t) = q ( t) ( x) u, Φ, (1) whee q s the odal apltudes and Ф the odal vectos. o lnea elastc stuctues t then follows that the stan s gven by and the stess by σ j D jkl kl ε = ε. j ( x, t) q ( t) Φ ( x) =, j Suppose that the fatgue stess s of unaxal type, such that fo each ateal pont, (o at least fo the ctcal ateal pont) the fatgue daage of a stess cycle s a onotonc functon of the stess apltude easued n a sngle fxed spatal decton b, efeed to as the ctcal decton. Suppose futheoe that anflow countng and Palgen-Mne s suaton ule ae vald technques fo coputng the daage of coplex load hstoes. The stess pojected on b s ( t) B jσ j = q Bj DjklΦ k, l, (3) whee B s a constant tenso that depends only on the ateal pont. It s useful to splfy Equaton (3) wth Bjσ j = q ( t) C, (4) C = Bj DjklΦ k, l whee s the ctcal stess coponent of ode. Intoducng now the noalzed weght factos c such that c C C = ( ) Bjσ j = C q t gvng c. (5) anflow countng of the stess hstoy gves a set of load cycle apltudes (2) baqus Uses Confeence

3 RC ( ) { σ } B j σ. (6) j Howeve, because the anflow countng s a popotonal opeato the noalzed weght factos ntoduced n Equaton (5) ay be used: ( B ) C RC c q () t RC jσ j =. (7) o a sngle load cycle t s assued that the fatgue daage s a functon of the stess apltude only n exaple ay be Basqun s law, ( σ ) cycle cycle =. (8) cycle ( σ ) σ = σ o, (9) whee σ o and ae ateal paaetes. o a coplex load hstoy the daage of the ndvdual cycles ae sued accodng to the Palgen-Mne hypothess cycle = ( σ ). (10) Wth Basqun s law we ay ewte the fatgue daage usng Equaton (7, 9-10) C = RC c σ o q () t, (11) and thee should be no abguty as to the ntepetaton of the notaton of RC ased to the powe of, c.f. Equaton (6). In the geneal case no specfc knowledge of the stuctue s avalable and thee s no chance of knowng whch lnea cobnaton of the egenodes that s ctcal. Howeve, usng the noalzed weght factos allows fo apd evaluaton of all lnea cobnatons. The axu daage of any stuctue s sue to be located on the suface S S : c = 1 (12) Consde as an exaple a stuctue wth two elevant egenodes Ф 1 and Ф 2 wth coespondng odal dsplaceents q 1 (t) and q 2 (t). Dffeent ateal ponts n the stuctue wll suffe dffeent daages. Ths s because the odal stesses ae dffeent at dffeent ponts and also the ctcal decton b s dffeent. o soe ponts only the fst ode s potant. The daage on these ponts s evaluated wth [c 1, c 2 ] = [±1, 0]. o soe othe ponts the two odes ae equally potant, these ponts ae captued wth [c 1, c 2 ] = [1, ±1]/ 2. It s qute possble that the stuctue n queston contans no ponts that ae nfluenced by only the second egenode. We have no baqus Uses Confeence

4 knowledge of ths, but we copute the daage also fo the lnea cobnaton [c 1, c 2 ] = [0, 1]. When all lnea cobnatons on the suface ae evaluated all the RC of all possble ponts n the stuctue ae evaluated. Dscetzng the S n n ponts allows expessng the fatgue daage fo these ponts wth Equaton (11). The fatgue daage of pont on the suface S beng C = RC c σ o q () t. (13) The nuecal evaluaton of the RC s apd, thanks to oden anflow countng algoths. In the patcula case of two egenodes, the angle θ = atan(c 2 /c 1 ) s a useful epesentaton S. When all angles θ n [0, π] have been evaluated the RC s detened fo all ponts. n exaple of the anflow count on S fo the case of two egenodes usng odal dsplaceent data fo a dynac sulaton of a spae wheel s pesented n gue 1(b and c). gue 1. a) Scheatc of a coecal vehcle seen fo the sde. The spae wheel s enccled. b) Sulated tansent odal dsplaceents fo the two egenodes that ae ost potant to the spae wheel backet. Mode 1 ndcated wth a black lne and ode 2 wth a gey lne. c) Non-noalzed anflow count of the dsplaceent fo ponts on the suface S usng Basqun s law wth exponent = atgue equvalent haoncs It s hee poposed to seach a fatgue equvalent dsplaceent n the fo u ~ ( x, t) = sn( ωt + ϕ ) Φ ( x), (14) whee s the fatgue equvalent odal apltudes and φ s the phase shft of ode. Ths dsplaceent s clealy easy to vsualze and undestand because all the odal dsplaceents have the sae fequency and constant apltude. It eans to show f and how the paaetes and φ ay be chosen to assue fatgue equvalence baqus Uses Confeence

5 Inseton of Equaton (14) nto Equaton (2-11) gves ~ 2π T C 1 1 = axt c t sn ω σo 2 2 ( ωt + ϕ ) n c sn( ωt + ϕ ) (15) ~ 2πT C = apl c sn ϕ ω σ o ( ) ωt +. (16) Equaton (16) s a useful epesentaton because t allows copang the fatgue daage of the ognal dsplaceent wth that of the equvalent dsplaceent. The elatve daage at the pont on S s ~ ρ = apl c sn 2πT = ω RC cq ( ωt + ϕ ) () t. (17) The fatgue equvalent dsplaceent has a ato ρ = 1 fo evey pont. Ths poble s ovedetened wth a nube n evaluated ponts on the suface S and a nube 2R unknown paaetes, whee R s the nube of egenodes used. The unknown paaetes ae the apltudes of the R egenodes, the R - 1 phase shfts φ and the nube of cycles 2πT/ω. Usng nuecal nzaton the best fttng fatgue equvalent apltudes and phase shfts ae sought. Once they ae found the fatgue equvalent dsplaceent (and stess) ay be wtten to an output database usng the odal vectos and odal stesses. The equvalent dsplaceent s haonc so only one cycle needs to be wtten. In ths wok the equvalent dsplaceents was wtten to an baqus output database usng the baqus python scptng nteface. 4. Exaple of applcaton Consde the spae-wheel of a coecal vehcle n gue 1a. dect fatgue evaluaton usng a coecal fatgue post-pocesso wth the tansent sulaton n gue 1b togethe wth the odal stesses, gue 2, gves a daage dstbuton pesented n gue 3a. It should be eaked that ths daage dstbuton coelates well wth expeental esults. alues have been obseved on and aound the top left and the botto ght fae-bolts and also aound the ea pessng. close exanaton eveals that the two ost potant egenodes ae donated by bendng n the vetcal and hozontal decton, espectvely. vsualzaton of the odal stesses, gue 2, shows that ndvdually none of these egenodes can be esponsble fo the daage dstbuton; t s necessay to consde the sultaneously baqus Uses Confeence

6 gue 2. a) Modal stess dstbuton of the vetcal ode. b) Modal stess dstbuton of the hozontal ode. gue 3. a) Daage dstbuton usng efat of the two egenodes wth tansent data. b) Daage dstbuton usng efat of the two egenodes wth equvalent dsplaceent. Usng the poposed ethod gves an equvalent haonc dsplaceent n tes of the apltudes of the two odes, the phase shft and the nube of cycles, efe to gue 4a. The tansent- and the equvalent dsplaceents ae also copaed n a phase-daga, and the daage of the equvalent dsplaceent s coputed fo all ponts on the suface S, gues 4b and 4c. It s noted that fo no pont on S s the dscepancy n anflow count lage than 18 pe cent. Howeve, the daage dstbuton of the equvalent dsplaceent ay also be coputed n efat, gue 3b. close exanaton and copason of gues 3a and 3b eveals that the daage dstbuton s slghtly dffeent between the ognal and the equvalent dsplaceents. Ths dffeence s due to seveal factos. stly, the equvalent dsplaceent s coputed fo the ostly daaged pont esdng on S wth a specfc fatgue exponent (hee = 7 s used). Because the equvalent baqus Uses Confeence

7 dsplaceent s stongly nfluenced by the fatgue exponent t ay be a poo epesentaton of the load fo ateal ponts wth dffeent exponents. Secondly, the coecal fatgue post pocesso uses dffeent assuptons about the daage. In patcula t pobably uses oe sophstcated ules than Basqun s law coputed on the stess n a fxed ctcal decton. Nevetheless the two daage dstbutons do have slates that ae potant. The asyetc daage dstbuton whch ndcates a geate sk of falue aound the top left and the botto ght fae-bolts, and aound the nne ght wheel nut, would be dffcult to obtan f the odal stesses wee egaded sepaately. vsualzaton n fo of a ove of the equvalent dsplaceent shows that the spae wheel follows an nclned ellpse n the x,z plane. The vsualzaton also povdes uch needed undestandng of the ctcal dsplaceent of the stuctue and allows fo a faste developent of stength nceasng odfcatons. gue 4. a) Equvalent odal dsplaceent fo the two egenodes n the te doan, ode one black lne, ode two gey lne. b) Phase plot of the two egenodes usng tansent data - thn gey lne, and coputed equvalent data - thck black lne. c) Non-noalzed anflow count of the tansent data sold lne, and the equvalent data squaes, fo all possble ponts on S. 4. Suay The pesented ethod of fatgue equvalent odal dsplaceent s estng on athe stct hypotheses, notably a un-axal daagng stess state, Basqun s law and lnea daage accuulaton ae assued vald. Despte these goss splfcatons the ethod povdes a useful tool to undestand and vsualze coplex tansent te hstoes fo a fatgue pespectve baqus Uses Confeence Vst the SIMULI Resouce Cente fo oe custoe exaples.