A System Identification Algorithm for Vehicle Lumped Parameter Model in Crash Analysis

Transcription

1 Intrnational Journal of Modling and Optimization, Vol., No., Jun A Systm Idntification Algorithm for Vhicl Lumpd Paramtr Modl in Crash Analysis Javad Marzbanrad and Mostafa Pahlavani Abstract This study has invstigatd a vhicl lumpd paramtr modl (LPM in frontal crash. Thr ar svral ways for dtrmining spring and dampr charactristics and typ of problm shall b considrd as systm idntification. This study us gntic algorithm (GA procdur, bing an ffctiv procdur in cas of optimization issus, for optimizing rrors, btwn targt data (xprimntal data and calculatd rsults (bing obtaind by analytical solving. In this study analyzd modl in 5DOF thn compard our rsults with 5DOF srial modl. In this papr, th solution mthod of crash quations for lumpd paramtr is invstigatd in discrt analysis mthod and prsntd a gnral solution with th hlp of numrical solution. Indx Trms Vhicl, Lumpdparamtr modl, Gntic algorithm, Optimization, Systm idntification. I. INTRODUCTION Historically, thos considrations about th mannr of dcorating matrials and ncssitis about physical structur of vhicl hav rsultd in dsigning structur and its body. Gnrally, final dsign of a vhicl is th product of a long trm procss, bing drivd by svral tsts and supportd by simpl linar stiffnss ways. By dvloping softwar and hardwar, it is possibl to us mor analytical facilitis, making svral tools, for analytical dsigning modrn structur of vhicl. Thrfor, nginrs ar abl to mt thir growing nds and bttr prformanc of crashworthinss and saf driving. Ths tools includ lumpd paramtrs modls (LPMs, Bam lmnt modls, hybrid modls and finit lmnts modls (FE. Although, thy ar diffrnt in cas of complxity, but ach is dsignd, on th basis of structural mchanics, nding into stability of mass, momntum and nrgy. During rcnt yars, auto industry has confrontd with most rqusts of customrs, law makrs and mdia, for production saf vhicls. This progrss includs improving crashworthinss of structur in crashs, for xampl, Fdral Motor Vhicl Safty Standard (FMVSS, Nw Car Assssmnt Program (NCAP, Insuranc Institut for Highway Safty (IIHS, consistncy tst and insuring from kping childrn and th short adults. In othr words, it is ndd to us vast pictur from crashworthinss of vhicl. In 97, Kamal [] prsntd a simpl and strong modl for simulation of crashworthinss in frontal crash. As, providing accptabl rsults, data was usd by crash nginrs, widly. Not, spring charactristics wr xprimntally dtrmind in static dampr. Also in 988, Mag [] prsntd a modl from crashing with barrir. This study usd actual crashs information, for dtrmining proprtis of springs, masss and braking modls. This modl has bn dsignd with considring th proprtis of loadmoving springs, for obtaining bst consistncy of acclrations, its pak and crash schduling. Chva t al. [3] prsnts a on dimnsional lumpd mass modl. This usd simulations of finit lmnts, for dtrmining spring proprtis. Also, rcordd normalizd acclration in tst and simulation of lumpd mass hav had dsird consistncy with ach othr. Alxandra t al. [4] has prsntd a lumpd mass modl in frontal/offst crash, in national Highway Traffic Safty Administration (NHTSA, bing dirctly xtractd structural proprtis of vhicl from data, rlatd into crash tst. Thomas and Josph [5] dsignd a modl, bing indicatd in Figur. Also, Tomassoni [6], considrd a modl, prsntd in Figur. Figur. LPM as statd by Thomas and Josph in sid crash. [6] Manuscript rcivd April 5, ; rvisd May 5,. J. Marzbanrad is assistant profssor in School of Automotiv Enginring, Iran Univrsity of Scinc and Tchnology, , Thran,Iran(corrsponding author to provid phon: ;fax: ; marzban@iust.ac.ir,Thra n, Iran M. Pahlavani is MS graduat of Iran Univrsity of Scinc and Tchnology, Thran, Iran. H is now taching som courss in Automotiv filds. ( pahlavani.s@gmail.com. Figur. LPM as statd by Tomassoni in sid crash. [7] 63

2 Intrnational Journal of Modling and Optimization, Vol., No., Jun Kim and Arora [7] studid on linar and nonlinar systms for vhicl crash. It was prformd, thortically, for th purpos of stating th importanc of numrical ways in modling and analyzing crash. Excpt finit lmnt modls, bing tim consuming and difficult, hybrid modls of finit lmnt and Lumpd mass modls ar paid attntion by analysts and dsignr, such as Hollowll rsarch, in 986 [8, 9]. In 986, Ni and Song [] dscribd 3 mthods for simulating vhicl structurs in crashs. In 8, Db and Srinivas [] focusd on lumpd mass modl in sid crash, prsnting a simpl and comprhnsiv modl. Thir studis wr prformd, on th basis of attractd nrgis comparisons, in insid impact of cart and vhicl and lumpd mass modl with obtaind rsults, from finit lmnt simulations. This study prsnts a simpl and comprhnsiv modl with linar spring and dampr, for modling a frontal crash. Also, it shall considr diffrncs of dclration s pak and dclration on occupant. It is ncssary to us dampr, bcaus of masuring amount of vhicl structur damping, bing mad by xisting injcting foams. II. EQUION AND ANALYZING SOLUTION ALGORITHM Jonsén t al. [] had a rsarch about hybrid modl of LPMs and FEMs. In this sction, Jonsén s mthod in LPMs quations solving is prsntd thn a similar mthod will b approvd. Figur 3 shows a ndgr of frdom LPM and Equation ( dscribd in continuous tim by a scondordr diffrntial quation: M X + CX + KX = bf ( Figur 3. Ondimnsional mass spring dampr modl. whr m, c and k rprsnt n n mass, damping and stiffnss matrics, rspctivly. Th displacmnt x is an n vctor, and x and x ar both vctors of th sam siz with vlocitis and acclrations. Th matrix b is th n r input matrix and f is an r vctor of input xcitations. By introducing th stat vctor X as [ x x] T, th systm can b writtn in th firstordr matrix form as: X = AX Y = HX + Bf + Df whr A N N,B N r,h M N ar th timinvariant continuous tim systm matrics, whr N = n and M dnots th numbr of outputs Y. Multiplying first lin of Equation ( with At and intgrating yilds: f ( X t At t ( t = X ( + Bf ( τ dτ τ (3 Equation (3 is th analytical solution for th continuous modl in Equation (. In th digitizing of Equation (3, th assumption of zroordr hold, i.. f is assumd to b constant during ach tim stp, T. In th Equations (4, th discrt formulation is prsntd. X[ k] = X ( kt X[ k] = AkT X[ k + ] = k+ T X[ k + ] = X ( + + kt kt τ X ( + AkT ( k+ T kt + T τ ( k+ T ( k+ T τ X ( + Bf ( τ dτ kt kt τ Bf ( τ dτ Bf ( τ dτ Bf ( τ dτ Th xprssion nclosd by th brackts in Equation (4 can b rcognizd as X [k]. Th scond trm can b simplifid by introducing υ as: (4 υ = kt + T τ (5 and assuming that th xcitation vctor f is constant ovr th intgration intrval rsults in: T Aυ X[ k + ] = X[ k] + d υ Bf ( k (6 Equation (6 rprsnts th xact solution to th digitizing problm. Th matrix xponntial in Equation (6 is dfind as: s = ( (7 s= s! and it can b approximatd by a firstordr Taylor sris around T = as: = I + (8 Substituting Equation (8 into Equation (6 yilds: ( Xk [ + ] ( I+ Xk [ ] + IT+ Bf( k (9 For approving similar kind of algorithm and solution, it may b compard th analytical solution of systm quation of motion in dgr of frdom and corrsponding paramtrs optimizations with its rsults. By assuming systm as statd in figur 4, th modl quation of motion shall b qual to Equation (. m x + ( c + c x cx + ( k + k + k3 x ( k + k3 x m x cx + cx ( k + k3 x + ( k + k3 x = = ( 64

3 Intrnational Journal of Modling and Optimization, Vol., No., Jun Figur 4. Tow dgr of frdom modl. Th initial conditions ar x = x = and x = x = 4. Equation ( may b writtn in th matrix form as follows: x x x M + C + K = x x x whr: m M =, m c + c C = c c c k + k + k, K = k k3 Stat spac matrixs ar as following: A= ( K + K + K3/ m ( K + K3/ m B =, H = ( K + K / m ( K + K / m c / m 3 ( c + c / m, D = k k3 k + k3 + c / m c / m ( ( (3 It is also can rwrit th quations as blow to gnraliz th formation: [] n n [] I n n [ M] [ K] [ M] [ C] x x A =, B = [] n, H = [] I n n, D = [] (4 n Th following valus wr usd in this part for simulation: m =8kg, m =8kg, c =N.s/m, c =N.s/m, k =N.m, k =6N/m and k 3 =7N/m. Th main rason of considring k and k 3 in paralll way is tsting th stat of optimization in cass. Th first, w hav k +k 3 =86; it mans k and k 3 hav any amounts in abov mntiond condition, on th othr hand, it is anticipatd that k =k 3 =43. In this cas, dcision variabls ar c, c, k, k and k 3. Also, activ paramtrs such as m =8kg and m =8kg, wr considrd as vhicl and occupant masss, rspctivly. Th scond, according to bing paralll of springs and similarity of k 3 as a cofficint of k, it must b considrd th rat of k =6.875k 3 of solution condition. In this cas, dcision variabls ar c, c, k and k. m and m ar as sam as last cas that was discussd about it. Th function attmpts to find th constraind minimum of a scalar function of svral variabls. A typical problm can b formulatd as: min f(θ (5 whr θ dnots th unknown dsign variabls, which, in this cas ar th masss, damping and stiffnss constants in th modl. Th cost function f(θ is rfrrd to objctiv function, which is to b optimizd. In this study, th cost function is th Root Man Squar (RMS of diffrncs btwn th masurd and calculatd dclration for th load cass. Th Gntic Algorithm is usd for optimization of cost function. Th aim is to minimiz th cost function valu. Th cost function is dfind as: Z = RMS a / man ( abs ( a (6 ( xp hr man is avrag of data and a can b rprsntd as follows: a = a (7 i a xp that a is th dclration rror that is calculatd by diffrnc btwn i th mass dclration and targt dclration (a xp which obtaind from xprimntal tsts. Th rsults of optimization aftr itration with random initial population ar shown in Tabls and. TABLE I. Mthod Analytical a Optimization Error % OPTIMIZION RESULTS OF DOF MODEL IN STE K +K 3 = 86. c (N.s/m c (N.s/m k (N/m k +k 3 Error % a. Paramtrs ar assumd as activ. TABLE II. Mthod Analytical a Optimization Error % k (N/m k 3 (N/m OPTIMIZION RESULTS OF DOF MODEL IN STE K =6.785K 3. c (N.s/m.9. a. Paramtrs ar assumd as activ. c (N.s/m k (N/m k (N/m Th prvious algorithm for optimization mthod is usd hr to calculat th spring and dampr cofficint. Tim history of dclration is shown in figur 5. Figur 5. Dclration on m in dgr of frdom modl obtaind from analytical solution. 65

4 Intrnational Journal of Modling and Optimization, Vol., No., Jun Figur 6. Error of dclration on m in dgr of frdom modl obtaind from optimization undr k +k 3 =86. Tim history of rror valus shows th diffrncs btwn dclration of m in analytical and optimization solution which is occurrd aftr itration. It is illustratd in Figurs 6 and 7. As on can s in ths Figurs, in both cass (k +k 3 =constant and k =6.875k 3 dclration rror of m undr th wors conditions is lss than.m/s which is considrd as.8% rror quals to max dclration of 7.44m/s. Th obtaind optimizd valus indicat that thy ar accptably clos to accurat paramtrs aftr nough itration, so it can solv complx modls. W should considr that amount of corrsponding btwn dsird and targt dclration dpnds on dgr of frdom. So, solution algorithm will b dsignd in such way which will b stoppd aftr 5 itrations with qual valu of cost function that is assumd. as its limit. Figur 8 shows procdur of solving problm. III. 5DOF MODELING AND RESULTS Figur 7. Error of dclration on m in dgr of frdom modl rsultd from optimization undr k =6/875k 3. Initial Paramtrs Start TABLE III. Srial Modl Mass No. m Mass PROPORTION FOR 5DOF SERIAL MODEL. Lumpd Componnts Mass (kg Radiator 5 m Suspnsion and Front Rails Modling m 3 Engin and Shotguns 3 m 4 Fir Wall and Part of Body on Its Back 8 Diffrntial Equation m 5 Occupant 8 Tim History from Tst Rsults TABLE IV. MASS PROPORTION FOR 5DOF LH MODEL. Stat Spac Modling LH Modl Mass No. Lumpd Componnts Mass (kg m Engin and Radiator 3 Tim Domain of Simulation m Suspnsion and Front Rails m 3 Engin Cradl and Shotguns 5 Slct of Nw Paramtrs by GA Calculation of Diffrnc m 4 m 5 Fir Wall and Part of Body on Its Back Occupant 7 8 No Cost Function Equal to Prvious CFs No < Lim x 5 Ys Ys End Figur 8. Algorithm of problm solving. Figur 9. 5DOF LH modl. 66

5 Intrnational Journal of Modling and Optimization, Vol., No., Jun In this sction, a 5DOF modl is analyzd as a vhicl in crash and th spring and dampr spcifications ar dtrmind by using th optimization algorithm as indicatd in Figur 8. In compltion of Kamal s modl, w analyzd this modl in 5 dgr of frdom which shown in Figur 9 and thn compard our rsults with 5DOF srial modl as Figur. Tabls 3 and 4 ar show proportions of lumpd paramtrs. x x x 3 x 4 x 5 Figur 4. Diffrnt btwn optimizd (v 5 and xprimntal (v xp for vlocity in 5DOF LH and simpl modls. Figur. 5DOF srial modl In this rsarch, th occupant dclration of a Dodg Non vhicl tst is usd as th goal data to b critria for optimization. Th obtaind rsults, compard and shown in Figur to 4. TABLE V. COMPARISON BETWEEN DECELERION ERRORS OF FOUR MODELS. Modl Error 5 DOF LH Final Root Man Squar of Dclration Error (g 4.64 Maximum Dclration Error (g DOF Srial Figur. Optimizd (a 5 and xprimntal (a xp rsults for dclration in 5DOF LH and simpl modls. TABLE VI. Modl Error COMPARISON BETWEEN VELOCITY ERRORS OF FOUR MODELS Final Root Man Squar of Vlocity Error (m/s Maximum Vlocity Error (m/s 5 DOF LH 5 DOF Srial TABLE VII. VALUE OF PARAMETERS OF BOTH MODELS Paramtr LH Modl Srial Modl c c c Figur. Diffrnt btwn optimizd (a 5 and xprimntal (a xp for dclration in 5DOF LH and simpl modls. c 4 c 5 c c c c k k k k k k Figur 3. Optimizd (v 5 and xprimntal (v xp rsults for vlocity in 5DOF and simpl modls. k 7 k 8 k

6 Intrnational Journal of Modling and Optimization, Vol., No., Jun IV. COMPARISON BETWEEN TWO MODELS In Tabl 5, w prsnt final Root Man Squar of dclration rror, and maximum dclration rror in tow modls and also prsntd final Root Man Squar of vlocity rror and maximum vlocity rror in Tabl 6. Figur dats rvals thos rsponss of occupant dclration in both 5DOF LH and Srial modls ar sam and follows from xprimntal tst data rasonably. In Figur dats, w foundd that rror of LH modl dclration is lowr than of Srial modl. Figurs 3 and 4 rval tim history of vlocity and vlocity rror in both 5DOF modls. Paramtrs valu of both modls prsntd in Tabl 7. V. CONCLUSION In this rsarch w hav invstigatd LPMs. Solution mthod and algorithm ar valuatd by 5 dgr of frdom srial modl and rsults of dclration and vlocity ar prsntd and thn compard to xprimntal data and 5DOF modl. Rsults of comparison show that typ of spring arrangmnt in modl has significant ffct and w hav concludd that w should not only addrss dclration changs bhavior but also total rror of dclration and vlocity and maximum rror and vlocity bhaviors to mphasizd modl. Th following nots considrd in this rgard: Numbr of dgr of frdom should b dtrmind accuratly, so it shouldn t b too much quation that yilds convrgnc quations confront with rror nor will fw to dclration bhavior and rrors b insufficint. In ths modls way of spring s arrangmnt is vry important so w should analyz svral modls by th sam dgr of frdom to achiv dsird rsults. 3 Whras forcdisplacmnt bhavior of componnts is not availabl in optimization, algorithm could st on or mor paramtr too littl, so it don t mak our solution wrong bcaus som of paramtrs may b xtra and bcoming too littl don t mak mistak in our solving. 4 As mntiond, calculatd bhavior data ar not match with xprimntal data, so numbr of itration quals numbr of rror for stopping of itration is mor logic than rror is bing zro in solution algorithm. REFERENCES [] M. M. Kamal, Analysis and simulation of vhicl to barrir impact. SAE papr 744, 97. [] C. L. Mag, Dsign for crash nrgy managmnt prsnt and futur dvlopmnts. In Procdings of th Svnth Intrnational Confrnc on Vhicl structural mchanics, Dtroit, Michigan, USA, SAE Intrnational, Warrndal, Pnnsylvania, 988. [3] W. Chva, T. Yasuki, V. Gupta, and K. Mndis, Vhicl dvlopmnt for frontal/offst crash using lumpd paramtr modling. SAE papr 96437, 996. [4] C. C. Alxandra, G. M. Stuart, and R. R. Samaha, Lumpd paramtr modlling of frontal offst impacts. SAE papr 9565, 995. [5] J. T. Thomas and N. K. Josph, Occupant rspons snsitivity analyss using a lumpd mass modl in simulation of cartocar sid impact. SAE papr 85689, 985. [6] J. E. Tomassoni, Simulation of a twocar obliqu sid impact using a simpl crash analysis modl. SAE papr 84856, 984. [7] C. H. Kim, and J. S. Arora, Nonlinar dynamic systm idntification for automotiv crash using optimization: A rviw. Struct Multidisc Optim, Vol 5, pp. 8, 3. [8] W. T. Hollowll, Adaptiv tim domain, constraind systm idntification of nonlinar structurs. Symposium on Vhicl Crashworthinss Including Impact Biomchanics, AMDVol. 79/BEDVol., pp. 5 3, ASME, 986. [9] W. T. Hollowll, Adaptiv tim domain, constraind systm idntification of nonlinar structurs. Doctorat Dissrtation, Th Univrsity of Virginia, Charlottsvill, VA, 986. [] C. M. Ni, and J. O.Song, Computraidd dsign analysis mthods for vhicl structural crashworthinss. In: Proc. of Symposium on Vhicl Crashworthinss Including Impact Biomchanics AMDVol. 79/BEDVol., pp. 5 39, ASME, 986. [] A. Db, and K. C. Srinivas, Dvlopmnt of a nw lumpdparamtr modl for vhicl sidimpact safty simulation. Journal of Automobil Enginring. Vol., pp.7938, 8. [] P. Jonsén, E. Isaksson, K. G. Sundin and M. Oldnburg, Idntification of lumpd paramtr automotiv crash modls for bumpr systm dvlopmnt. Intrnational Journal of Crashworthinss, Vol. 4, No. 6, pp , 9. Javad Marzbanrad is an associat profssor in Automotiv Enginring at Iran Univrsity of Scinc and Tchnology. H was born in 964 in Thran, Iran. H got his Ph.D s Dgr in from Tarbiat Modarrs Univrsity. H was in th Clarkson Univrsity, NY, U.S. in as visiting scholar. His rsarch intrsts ar Automotiv, Control and Vibration and Solid Mchanics. Now, h is th faculty of School of Automotiv Enginring in Iran Univrsity of Scinc and Tchnology. H has mor than 5 publishd paprs in th confrncs and journals. H also publishd thr books in mchanical softwar filds. H has also rgistrd six invntions in Iran. Mostafa Pahlavani tachs in Univrsity of Applid Scinc (NirooMohark Branch and Payam Noor Univrsity (Qazvin Branch in th fild of vhicl mchanics. H was born in 983 in Rasht, Iran. H got his Mastr s Dgr in from Iran Univrsity of Scinc and Tchnology, School of Automotiv Enginring. His rsarch intrsts ar body and structur and dsign filds in Automotivs. 68