Prodings of th 4th EUROPEAN COMPUTING CONFERENCE Mathatial Modling and Analysis of a Vhil Crash WITOLD PAWLUS, JAN EIVIND NIELSEN, HAMID REZA KARIMI, KJELL G. ROBBERSMYR Dpartnt of Enginring, Faulty of Enginring and Sin, Univrsity of Agdr, Srvibos 509, N-4898 Gristad, NORWAY (-ail: haid.r.arii@uia.no Abstrat: Baus of th fat that vhil rash tsts ar oplx and opliatd xprints it is advisabl to stablish thir athatial odls. This papr ontains an ovrviw of th inati and dynai rlationships of a vhil in a ollision. Thr is also prsntd basi athatial odl rprsnting a ollision togthr with its analysis. Th ain part of this papr is dvotd to thods of stablishing paratrs of th vhil rash odl and to ral rash data invstigation i.. ration of a Klvin odl for a ral xprint, its analysis and validation. Aftr odl s paratrs xtration a qui assssnt of an oupant rash svrity is don. Ky-Words: Modling, vhil rash, Klvin odl, data prossing. 1 Introdution Th ain objtiv of this projt is to stablish a athatial odl of a vhil ollision. Th purpos of this tas is to siulat how th rash loos li i.. what ar th ain paratrs dsribing th ollision without prforing any ral tst. Ral world xprints ar diffiult to raliz thr ar ndd appropriat failitis, asuring dvis, data aquisition pross, qualifid staff and of ours a ar. Thrfor it is justifid to propos a athatial odl of a ollision and analyz it instad of a ral xprint to approxiat its rsults. In our ain intrst it is to analyz in dtails a Klvin odl. Having nowldg onrning on suh a syst w ar abl to xtnd th odl.g. to a oupl of Klvin lnts in ordr to obtain a or aurat rspons (w an rprsnt ar lnts and onntions btwn th xatly by ultipl spring ass dapr odls. Many rsarhs hav bn don so far in th ara of vhil rash odlling. Yang t al. [1] prsntd a fasibility study of using nurial optiization thods to dsign strutural oponnts for rash. Th prsntd produr rquird svral softwar, whih inludd paratri odling (Pro/ENGINEER, autoati sh gnration (PDA PATRAN3, nonlinar finit lnt analysis (RADIOSS, and optiization progras. It was found that rash optiization was fasibl but ostly and that finit lnt sh quality was ssntial for sussful rash analysis and optiization. Mahood t al. [] hav dsribd in dtail a produr for rapid siulation and dsign of th fra of an autootiv strutur. Thy dvlopd a siplifid progra, alld V-CRUSH, for rapid siulation of th strutur. Corrlation btwn th xprintal and siulation rsults was vry good. Huang t al. [3] dsribd Ford s Enrgy Managnt Syst that usd CRUSH (Crash Ronstrution Using Stati History lupd ass odlling apability. Using th syst, barrir loads and passngr opartnt loads wr alulatd and opard to th tst rsults in a frontal rash. Abov brif ovrviw of th litratur has bn don aording to Ki t al. [4]. In this papr w ovr th spring ass dapr odling of th vhil rash. W start with an ovrviw of Klvin odl an lnt in whih ass is attahd to spring and dapr whih ar onntd in paralll. Subsquntly w giv inforation about fators whih dtrin rash svrity for an oupant during ollision. Th largst part of this wor is dvotd to answr th following qustion how to stablish a odl fro ral rash data? Aftr prsnting two thods for solution of this probl w prod to analysis asurnts fro ral ollision. Vhil ollision siulation Klvin odl A Klvin odl is shown in Fig..1. It ontains a ass togthr with spring and dapr onntd in paralll. This odl an b utilizd to siulat th vhil-tovhil (VTV ollision, vhil-to-barrir ollision (VTB as wll as for oponnt ipat odling. In ajority of ass th rspons of th syst is undrdapd thrfor w fous on this typ of bhavior. ISSN: 1790-5117 194 ISBN: 978-960-474-178-6
Prodings of th 4th EUROPEAN COMPUTING CONFERENCE. Coffiint of rstitution (COR Fig..1: Klvin odl In th ipat of th dynai syst th offiint of rstitution (COR is dfind as th ratio of rlativ sparation vloity to th rlativ approah vloity. During th dforation phas, th rlativ approah vloity drass fro its initial valu to zro du to th ation of th dforation ipuls, as shown in Fig..3..1 Undrdapd syst (1>>0 Equation of otion (EOM: α + ω α + ω α = 0 (.1 whr = and ω ω = Transint rsponss of th undrdapd syst ar: ω v0 t = sin( ω (. ω displant (dynai rush ω t (.3 = v0 [os( ω sin( ω] vloity ω = v0ω t t [ os( ω + sin( ω ] (.4 dlration W s that abov losd for rsults ar oplx. To obtain th rsponss of th Klvin odl w us Matlab Siulin softwar. In th analysis of th rash puls (dlration alongsid with vloity and displant graphs w ar abl to obsrv spifi rlationships btwn th and btwn two tiings: t ti of dynai rush and t f ti of rbound (or ti of sparation vloity. Thos dpndns ar shown in Fig... Th valus on th graph blow ar just for prsnting th prinipl thy do not o fro any xprint. Fig..: Rlationships btwn t, t f and alration, vloity, displant At t th orrsponding vloity is zro and th dynai rush rahs its axiu valu. At t f th orrsponding dlration is zro and vloity rahs its axiu valu. Plas not that t f is twi as long as t (in Fig.. t f =0.5s and t =0.5s. Fig..3: Dforation and rstitution phas during a rash At th ti whn th rlativ approah vloity is zro, th axiu dynai rush ours. Th rlativ vloity in th rbound phas thn inrass ngativly up to th final sparation (or rbound vloity, at whih ti th two asss sparat fro ah othr (or a vhil rbounds fro th barrir. At th sparation ti, thr is no or rstitution ipuls ating on th asss, thrfor, th rlativ alration at th sparation ti is zro [5]. To driv th rlationship btwn th offiint of rstitution and daping fator of th syst w us (.4. At th ti of sparation (t=t r =t f th rlativ dlration α = 0 Thrfor fro (.4: os( ω + sin( ω = 0 W rwrit it in th following for: tan( ω = fro Pythagoran Thor w gt: os( ω = (.5 COR=rlativ sparation vloity/rlativ approah vloity = v COR ωt [os( ω 0 sin( ω] = (.6 Thr ar thr spial ass: 1. No daping in th syst =0, thn COR = 1.. Critially dapd syst =1, thn COR = 0.135. 3. Highly ovrdapd syst =, thn COR = 0. ISSN: 1790-5117 195 ISBN: 978-960-474-178-6
Prodings of th 4th EUROPEAN COMPUTING CONFERENCE W an siplify (.6 by substituting sin( ω = so that w gt: aros( COR = (.7 3 Bass of oupant vhil odling In this stion w prsnt basi notions and trs ndd to asss th rash svrity for an oupant. As th rash puls approxiation w us an (Equivalnt Squar Wav. Fig. 3.1 shows an unbltd oupant in a vhil during a ollision. Fig. 3.1: Oupant during ollision [5] v 0 initial vhil rigid barrir ipat vloity v oupant to intrior surfa ontat vloity δ oupant fr travl spa (rstraint sla vhil dynai rush at ti t t ti whn oupant ontats rstraint t ti of dynai rush EOM for vhil: F x v = = M xv xv (3.1 = v0 t (3. 1 = v 0 t t (3.3 EOM for oupant: x o = [1 os( p + ωt sin( p] (3.4 x o = x v + [sin( p + ωt os( p] ω (3.5 xo = xv + δ + [1 os( p + ωt sin( p] ω (3.6 x o ax = [1 + ( ωt whr p = ω( t t for t t and ω = ] oupant (3.7 δ t = (3.8 rstraint ontat ti 3.1 Prdition of oupant dlration using DAF Lt us dfin dynai aplifiation fator as th ratio of axiu oupant hst dlration to th : x DAF = DAF whr o ax ] [1 + ( ωt = = ( ω t = (πft (3.9 ω = πf and f is rstraint natural frquny. Sin DAF = γ = (πft and w approxiat th rash puls by w an writ that th axiu oupant hst dlration is givn by: a0 = DAF = [1 + (πft ] whr v = t thrfor a = + + ( πfv (3.10 0 It is a oon prati to install in trus prtnsionrs. This is baus of th fat that th of a tru is highr that that of a ar. Thrfor if w want to dras th oupant dlration w nd to dras th rstraint sla and that is justifid by th DAF rlationship. 4 Obtaining paratrs of th Klvin odl fro tsts Fig. 4.1 prsnts a Klvin odl of a vhil-to-barrir ipat. Fig. 4.1: VTB ollision Klvin odl spring stiffns daping offiint ass of th vhil v 0 barrir initial ipat vloity 4.1 Mthod 1 - analytial To obtain strutural paratrs and first w nd to dtrin two othr paratrs: daping fator and f strutur natural frquny. Bfor w do that lt us first rind th ntroid ti onpt. Cntroid ti it is a ti at th gotri ntr of ara of th rash puls fro ti zro to th ti of dynai rush. W dfin it as follows: C t = (4.1 v 0 ISSN: 1790-5117 196 ISBN: 978-960-474-178-6
Prodings of th 4th EUROPEAN COMPUTING CONFERENCE W dfin noralizd ntroid ti and angular position at dynai rush as: α τ τ = tω = ω = v 0 1 τ = t ω = artan whr α is th axiu dynai rush. Aftr transforing abov two quations w gt rlativ ntroid loation: artan τ t = = (4. τ t artan τ = t ( πf so 1 ft = artan (4.3 π On w find th rlativ ntroid loation by dtrining t and t w an gt daping fator fro (4.. Aftr driving daping fator and nowing ti of dynai rush t w obtain th valu of strutur natural frquny fro (4.3. Having alrady valus of and f w dtrin strutural paratrs of th odl and : Sin ω = πf = and = ω = 4π f (4.4 = 4πf (4.5 In ordr to stiat th paratrs of th Klvin odl basing on th ral rash puls data w just nd ain inforation onrning th ollision: ti of dynai rush t, initial ipat vloity v 0, dynai rush C and ass of th vhil. Taing into onsidration th oplxity of th ollision phnona it is a signifiant advantag w an.g. asss th stiffnss and daping of a frontal strutur of a ar using sipl data ntiond abov. 4. Mthod Using Matlab Idntifiation Toolbox This Toolbox allows us to obtain th paratrs of th syst aording to th input and output data. As an xapl w ar going to us th Siulin odl of th sond ordr diffrntial quation (sond ordr osillating ln. Th foring fator is th xtrnal for ovr ass (alration initial onditions (vloity and displan ar st to zro. Data: F = 300N; = 100N/; = 5N-s/; = 3g; v 0 = d 0 = 0 Equation of sond ordr osillating lnt is [6] d y( dy( T + T + y( = Kx( (4.6 dt dt whr y( output and x( input. By taing Lapla transfor of (4.6 with zro initial onditions w gt: T s Y ( s + TsY ( s + Y ( s = KX ( s (4.7 Thrfor th transfr funtion of th syst givn by (4.7 is: Y ( s K G( s = = (4.8 X ( s T s + Ts + 1 Fro th EOM of th Klvin odl w hav: d y dy u ( = + + y (4.9 dt dt input u( is an alration. By taing Lapla transfor of (4.9 with zro initial onditions w obtain th following transfr funtion: Y ( s G( s = = (4.10 U ( s s + s + (4.8 and (4.10 ar dsribing th sa odl. Thrfor thy ar qual to ah othr if and only if: T = and K = and = and ω ω = With this nowldg w prod to Idntifiation Toolbox. W slt th appropriat typ of stiation in our as sin w us Klvin odl - an undrdapd syst with two pols. Paratrs obtaind fro stiation ar shown in Fig. 4.4. Fig. 4.4: Idntifiation Toolbox - rsults Aftr obtaining th valus whih ar dsribing th stiatd odl w h what ar th valus of T, K and for our rfrn odl and w opar th with thos ons fro th stiatd odl. For = 100 N/, = 5N-s/, = 3g w hav: 3 K = = = 0.03 / N ID toolbox: K = 0.03/N 100 3 T = = = 0. 1731s ID toolbox: T = 0.1731s 100 5 = = = 0.14434 ID toolbox: = 0.14434 ω 100 3 3 Th rsults of approxiation ar prft. Ti onstant T, daping offiint and gain K for both odls ISSN: 1790-5117 197 ISBN: 978-960-474-178-6
Prodings of th 4th EUROPEAN COMPUTING CONFERENCE rfrn and our stiatd ar th sa. It ans that w an us Idntifiation Toolbox to prisly dtrin what ar th offiints of th Klvin odl whn w ar givn an input and an output of th syst and th initial onditions ar st to zro. 5 Invstigation of ral rash data Lt us now analyz data fro th xprint. 5.1 Exprint produr [7] In th xprint ondutd by UiA [7] th tst vhil, a standard Ford Fista 1.1L 1987 odl was subjtd to a ntral ipat with a vrtial, rigid ylindr at th initial ipat vloity v 0 = 35/h. Mass of th vhil (togthr with th asuring quipnt and duy was 873g. Sh of th xprint is shown in Fig. 5.1. Fig. 5.3: Vloity obtaind fro asurd alration W s in Fig. 5.3 that th initial vloity is not qual to 35/h as it was statd in th xprint s dsription but is 5/h highr. This disrpany is a rsult of using raw data without filtring. Fro this plot w rad th valu of ti of dynai rush t = 0.11s. To gt th displant graph w prod in th annr dsribd abov w approxiat and intgrat th vloity urv fro Fig. 5.3. Th plot of displant is shown in Fig. 5.4. Fig. 5.1: Sh of th tst ollision [7] Vhil alrations in thr dirtions (longitudinal, latral and vrtial togthr with th yaw rat at th ntr of gravity wr asurd. Using noral-spd and high-spd vido aras, th bhavior of th obstrution and th tst vhil during th ollision was rordd. 5. Data prossing Sin w ar givn th alrations in 3 dirtions (longitudinal x, latral y, vrtial z w ar abl to propos 3 diffrnt Klvin odls for vry dirtion. Baus of th fat that w ar ostly intrstd in what happns in th dirtion in whih a ar hits th obstal, w ar going to analyz x dirtion (longitudinal. To approxiat th rash puls w us Curv Fitting Toolbox with Gaussian approxiation as it is shown in Fig. 5.. Fig. 5.: Curv Fitting Toolbox prparation of asurd data To obtain th vloity urv w intgrat th approxiatd puls th rsult is shown in Fig. 5.3. Fig. 5.4: Displant obtaind fro asurd alration Fro th plot w dtrin axiu dynai rush C = 0.84 at ti of dynai rush = 0.11s. 5.3 Coparison btwn odl and ral data aording to thod 1 Knowing valus of v 0 = 11/s, t = 0.11s, C = 0.84 and = 873g fro th ral tst, using thod dsribd in Stion 4.1 w dtrin paratrs: t, t /t,, f,, : C 0.84 t = = = 0. 076s v0 11 / s t 0.076s = = 0.69 and furthror fro (4.: t 0.11s = 0.05 and fro (4.3: ft = 0.4Hz-s 0.4Hz s 0.4Hz s ft = 0. 4Hz s so f = = =.Hz t 0.11s W alulat th paratrs of th Klvin odl: = 4π f = 4π (.Hz 873g = 166809N / spring stiffnss = 4π f = 4π.Hz 0.05 873g = 107N s / daping offiint Having paratrs of th Klvin odl w invstigat its rspons using th Siulin diagra with th initial vloity v 0 = 11/s. Th rsult is shown in Fig. 5.5. ISSN: 1790-5117 198 ISBN: 978-960-474-178-6
Prodings of th 4th EUROPEAN COMPUTING CONFERENCE 6 Conlusions Fig. 5.5: Vloity and displant vs ti of th Klvin odl with stiatd paratrs That is th rspons of th ass for 10 sonds. It is a typial on for th sond ordr osillating lnt also Klvin odl. In Fig. 5.6 you s th rspons in ti intrval usd in th tst data analysis (a agnifid part of abov plo. Fig. 5.6: Final analysis vloity and displant vs ti of th Klvin odl with stiatd paratrs in th rash intrval Although th approxiation of th vloity urv is not quit xat w do not s.g. a rbound, still th auray of approxiation is vry good. Ti of dynai rush t obtaind fro th odl is xatly th sa as in xprint: t = 0.11s and axiu dynai rush C = 0.74 is about 1% lss than that fro th ral tst. 5.4 Estiation of axiu hst dlration of oupant Knowing initial ipat vloity v 0 = 11/s, axiu dynai rush C = 0.84, ti whn it ours t = 0.11s and distan btwn an oupant and vhil (rstraint sla δ = 0.6 w alulat: v0 11 = 0.5 = 0.5 = 7 / s C 0.84 a0 δ DAF = = (πft, whr t = t is th ti whn oupant ontats rstraint, f is rstraint natural frquny (w assu following [5] a typial valu of f = 6Hz and a 0 is th axiu oupant hst dlration. 0.6 DAF = (π 6Hz 0.11s = 4.64 0.84 Maxiu oupant hst dlration: a = DAF = 4.64 7 / s = 334 / s 34g 0 = W hav anagd to prpar th rash data for analysis and xtrat th athatial odl fro it. Challngs hr wr to hoos an appropriat tst data approxiation and ti intrval in whih w want to invstigat th ollision. Having this don w an dtrin axiu rush of a ar, whn it ours, how th vloity hangs and what ar th hangs in alration of a ar during a rash. What is or w hav also stiatd th axiu oupant dlration that is on of th ain tass in th ara of rashworthinss study. Whn it os to th furthr wor, w plan to xtnd our sipl spring ass dapr odl to ultipl Klvin lnts syst. Thn w will obtain or aurat rsults and what is also iportant for partiular ar oponnts, not for a ar as a on lnt. Th othr thing whih ould iprov th rsults is using a Maxwll odl (a ass togthr with a spring and dapr onntd in sris for a vhil to rigid pol rash siulation. This syst givs bttr approxiation of offst ipats and loalizd pol ollisions baus it provids or aurat rspons for longr tis of axiu dynai rush. Th last iprovnt is to filtr th alrotr asurnts and to us or aurat typ of urv approxiation. Rfrns: [1] R.J. Yang, L. Tsng, L. Nagy, J. Chng, 1994: Fasibility study of rash optiization. In: B.J. Gilor, D.A. Hotzl, D. Dutta, H.A. Eshnaur: (ds. Advans in dsign autoation, DE-Vol. 69-, pp. 549 556. ASME [] H.F. Mahood, D.G. Whatly, M.A. El-Baily, J.Y. Baar, 1993: On th front nd dsign of autootiv vhil for rashworthinss. Crashworthinss and oupant prottion in transportation systs, AMD- Vol. 169/BED-Vol. 5, pp. 311 318. Nw Yor: ASME [3] M. Huang, R. Chn, B. Margolin, 1995: Us of an advand CRUSH odl in stiating vhil ipat loading and nrgy absorption. In: Crashworthinss and oupant prottion in transportation systs, AMD-Vol. 10/BED-Vol. 30, pp. 73 85. Nw Yor: ASME [4] C.H. Ki, A.R. Mijar, J.S. Arora,001: Dvlopnt of siplifid odls for dsign and optiization of autootiv struturs for rashworthinss. In: Strutural and Multidisiplinary Optiization, Vol., No. 4, pp. 307-31. Springr Brlin / Hidlbrg [5] M. Huang, 00: Vhil Crash Mhanis, Boa Raton: CRC Prss [6] J. Kowal, 003: Podstawy Autoatyi, to 1, Kraów: Uzlnian Wydawnitwa Nauowo Dydatyzn AGH [7] K. G. Robbrsyr, 004: Projt Rport 43/004, Calibration tst of a standard Ford Fista 1.1L, odl yar 1987, aording to NS EN 1767, Gristad: Agdr Univrsity Collg ISSN: 1790-5117 199 ISBN: 978-960-474-178-6