VEHICLE DYNAMICS SIMULATION, PART 1: MATHEMATICAL MODEL

Transcription

1 INTERNATIONAL DESIGN CONFERENCE - DESIGN 2002 Dubrovnk, May 4-7, VEHICLE DYNAMICS SIMULATION, PART : MATHEMATICAL MODEL Iztok Cglarc and Ivan Prebl Keywords: Computatonal dynamc, multubody systems, vehcle dynamc. Introducton The method of multbody systems [Melzer, 996] s well establshed and accepted tool n the dynamc analyss of mechansms, machnes, vehcles etc. Computer based assembly of the knematcs and knetc equatons of moton n symbolc form for a multbody system s an mportant tool n modern dynamc analyss. The symbolc generaton of the equatons of moton s partculary attractve snce the equatons have to be generated only once and afterwards numercally analyzed for varous parameter values. Also the classcal engneerng analyss could be easly extended to perform desgn senstvty analyss or stablty analyss. Nevertheless, there are lmtatons n symbolc computng, n partcular f the multbody system s complex, wth many degrees of freedom. Drawbacks n ths case are that for complex multbody system a long and tedous symbolc expressons are generated, whch may be dffcult to handle by computer algebra systems such as MATHEMATICA [Mathematca, 996], and whch may be numercal neffectve. Therefore t s necessary to formulate such multbody system formalsm that s capable to crcumvent such problems. 2. Multbody system formalsm Multbody system s defned as a set of rgd and flexble bodes connected by jonts and massless force elements. Jonts are lower and hgher knematc pars that characterze relatve moton between bodes. Force elements are sprngs, dampers and others energy transformaton elements such as Coulomb dampng element. An example multbody system s depcted n fg.. Fgure. Multbody system ANALYSIS TECHNOLOGIES 35

2 In order to obtan mathematcal model of multbody system t s necessary to make sutable mathematcal formulatons of bodes, jonts and topology of the structure as a whole. 2. Body descrpton Body s a fundamental element of multbody system. A body N (rgd or flexble) n the multbody system s lnked to nertal body through the parent body by jont (nboard jont), and can have several chldren bodes N j attached through jonts j (outboard jonts). From mechancal pont of vew body N can be defned wth the followng attrbutes: Reference attachment pont O of nboard jont on body N. Reference attachment ponts Oj of outboard jonts j on body N. The center of mass G of bodes N. A movng coordnate frame ( O, e, e2, e3 ) fxed to body N and located at the pont O. The poston vectors ( O, e, e2, e3 ). G r of the centre of mass G wth respect to coordnate frame r j The poston vector of the outboard jonts j wth respect to coordnate frame ( O, e, e2, e3 ). Mass m and nerta tensor J wth respect to center of mass G. One has to note that besde these basc attrbutes, flexble bodes need addtonal descrpton that wll be descrbed n detal n the next chapter. 2.2 Flexble body descrpton Superelement approach and the so-called contnuum approach are specal multbody technques for ncorporatng flexble bodes nto multbody system. The former descrbes flexble body as a seres of rgd bodes nterconnected by elastc force elements. The latter s used n ths research and descrbes the flexble body n terms of shape functons. In ths case the knematcs of a flexble body s expressed as superposton of the reference confguraton body moton (gross moton of rgd body) and dsplacement due to deformaton, measured wth respect to the body reference confguraton [Valembos, 997]. The poston of an elemental mass dm located on the body can be descrbed wth respect to the nertal reference frame as R = D + T +, () dm [ r g r ]] [ where D are the poston vectors and T rotaton matrxes. Vectors r descrbe the dsplacement of dm due to rgd body moton and g[ r] represent the dsplacement due to deformaton of the body. Deformaton of the -th body s expressed as the product of varyng admssble shape functons φ [ r ] and weghtng coeffcent N g [ r ] = φ [ r ] δ. (2) = Ths research focuses on elastc transverse deformaton of Euler-Bernoull beams. In ths case the admssble shape functon of k-th mode s [Caron, 998] φ λ k x cosh l + c λ k x snh l + c λ k x sn l + c yk ( x, l ) = ck k 2 k3 k 4 λ k x cos l (3) 36 ANALYSIS TECHNOLOGIES

3 where l are the length of the -th body (beam) and x the poston along l. Coeffcents c kj and depend on boundary condtons of consdered elastc body modeled as an Euler-Bernoull beam. 2.3 Jont descrpton Jonts lnk the bodes and restrct relatve moton between them. The confguraton space G of relatve moton between two reference coordnate frames fxed on two bodes s Specal Eucldean group consstng of all rotatons and translatons n R 3 space. An element of ths Specal Eucldean group may be represented by the transformaton matrx [Kovo, 989]. In general a relaton on the tangent bundle T G characterzes the jont. Such a relaton s usually expressed n local coordnates as f ( q, q ) = 0 λ k (4) where or f k : TG R. Natural constrants almost always occur n the followng two forms f ( q) = 0 F( q ) q = 0 (5a) (5b) Note that constrants of form (5a) defne submanfold of G [Wasserman, 992] and restrct the relatve poston of two bodes. Constrants of ths form are often called geometrc constrants. Constrants of the form (5b) restrct the relatve velocty of two bodes and they are often called knematc constrants. More general and more convenent characterzaton of equaton (5b) s A( q ) p = 0 where A s lnear operator that could be represented n terms of matrx and p Ker[ A( q)]. In the case of smple knematc jonts where the moton axes are fxed n at least one of the bodes the operator A s ndependent of confguraton and therefore constant. In ths case the solutons of (6) are of the form p = Hq where H s called the jont map matrx. It s of full rank r = dm Ker[ A] and the equalty Ker [ A ] = Im[ H] s vald. The columns of H form a bass for Ker [A]. In the case when t s possble to defne the acton of a jont n terms of a sequence of smple knematc jonts, we call such jonts compound knematc jonts. In general, a compound jont s defned as a jont that can be characterzed as the relatve moton of a sequence of reference frames such that relatve moton between two successve frames s defned by a smple knematc jont. Every smple jont s characterzed by a jont map matrx H, =,2,3,,n 2.4 Topology descrpton As far as the topology s concerned the followng conventons are adopted (see fg. ): Bodes are numbered n ascendng order from the nertal body (ndex 0) to the termnal bodes. A jont, whch precedes a body, has the same ndex as the body tself. (6) (7) ANALYSIS TECHNOLOGIES 37

4 To establsh the equaton of moton the topologcal confguraton of multbody system has to be represented n mathematcal termnology. To represent the topologcal structure of the multbody system the pertnent nformaton s mapped onto a graph. Each body s put nto correspondence wth a vertex of a graph. The vertces are enumerated =,2,...,N v. A vertex labeled =0 s adjoned to the reference nertal body. Vertces are nterconnected by sngle branches, representng the nteractons of bodes upon each other. The branches are enumerated j=,2,...,n b It should be noted that each branch could represent more than one physcal lnk that connects two bodes (such as jonts, sprngs, dampers etc.) To gve a graph a mathematcal characterzaton we defne an ncdent matrx n terms of vertces that are adjacent to one another or n terms of vertces on whch an edge or branch s ncdent j [ s ], =,2,..., Nv, j =,2,..., Nb S = (8) In column j the value of + s entered at the row number correspondng to the vertex from whch edge j emanates. The value s entered at the row number correspondng to the vertex on whch edge j termnates. The remnder of the column s flled wth the element 0. Ths s done for all columns. 2.5 Mathematcal model-the equatons of moton The equatons of moton can be obtaned by usng Euler-Lagrange equatons d dt In (9) ( q q ) E kn L L = Q q q L( q, q) = Ekn ( q, q) E pot ( q), where, s knetc energy and ( q ) E pot s potental energy of the multbody system. In the case of multbody system t s easer to formulate the equatons of moton n terms of velocty varables whch can not be expressed as the tme dervatves of any correspondng confguraton coordnate from vector q. If M s the m-dmensonal confguraton manfold [Wasserman, 992] for Lagrangan system (9), and ( t) :[ t t ] q t s the tangent vector to the path at the q, 2 M s a smooth path then ( ) q M. In ths case t s always possble to express q ( t) pont ( t) vectors q = V( q)p Usng (0) and set equalty L( q, p ) = L( q, q ) 99] as d dt L L V p p n = p X where X = [ v, v ], [ v, v2 ],...,[ v, vm ] L V = Q q and (9) as lnear combnaton of tangent (0) one can express (9) usng Hamlton s prncples [Jan, T V () m k [ v, v j ] = aj ( q) v k k = (2) s lnear combnaton of lnearly ndependent vectors [ ] Expresson () can be reduced to the form v, v2,..., v m on confguraton manfold M. 38 ANALYSIS TECHNOLOGIES

5 M ( q) p + C( q, p) p + F( q) = Q p (3) In (3) M s system nerta matrx, calculated as T M = H F M0FH T (4) C and F rght hand sde of Poncare's equatons calculated T m Mp Mp C, = q 2 q = T T ( q p) V + V + p X V M (5) F E q T ( q) = V ( q) T pot (6) and Qp generalzed forces represented n the p -vector bass calculated as Q p = V T ( q)q Q denotes the generalzed forces n the q -vector bass. In (4) M 0 s the spatal nerta matrx and F s spatal velocty transformaton matrx. Note that the matrx F and product FH can be recursvely computed [Ramakrshan, 989]. Once system nerta matrx s calculated, one compute C ( q, p ), F (q ) and Q p usng (5), (6) and (7) respectvely, assumng that the potental energy q and the generalzed force vector Q are avalable. functon ( ) E pot (7) 3. Mechancal and mathematcal model of the road vehcle The proposed multbody system formalsm s general and could be used for any knd of mechancal system wth arbtrary degree of freedom and complexty. However ths general proposed tool has been used to handle vehcle dynamc to demonstrate ts effcency. It s well known that vehcle dynamc s one of the most complex dynamc phenomenon [Crolla, 995] and therefore sutable benchmark. 3. Mechancal model Fg. 2 shows the proposed mechancal model of the road vehcle. The model has up to 26 DOF. The vehcle chasss (unsprung mass) has 6DOF. Each of four suspensons has DOF that enable wheel vertcal dsplacement. Fgure 2. Vehcle mechancal model Fgure 3. Vrtual tyre combned slp model ANALYSIS TECHNOLOGIES 39

6 Suspenson could be modeled as lumped mass model (fg. 4), or swng arm model (fg. 5). For the lumped mass model the suspenson components form sngle mass (sprung mass). Mass s connected to the vehcle chasss at the wheel centre by translaton knematc par whch allows vertcal sldng moton of sprung mass. Swng arm model also represents sngle mass, connected to the vehcle chasss by usng revolute knematc par. Revolute knematc par allows sprung mass to swng relatve to the vehcle chasss at the nstant centre of the suspenson lnkage assembly. The swng arm model advantage over the lumped mass model s that t could smulate the change n chamber angle. From the moton pont of vew the wheels are modeled by usng Vrtual Tyre Combned Slp Model (fg. 3) developed by Prell [Mancosu, 999]. Each wheel s represent as a sngle mass and has a spn DOF. Addtonally each tre has one vertcal, one longtudnal and one axal degree of freedom relatve to the rm. Corols effects of the spnnng wheels are neglected. Fgure 4. Lumped mass model Fgure 5. Swng arm model Sx aerodynamc forces and moments are appled to the vehcle chasss at the centre of the mass. Each s quadratc wth speed. Coeffcents n those equatons are quadratc wth aerodynamc slp angle. From the force pont of vew, the suspenson elements such as sprngs and shock absorbers are represented wth smple lnear model F F d v k ε = k ε 2 c ε = c ε 2 ε 0 ε < 0, ε 0 ε < 0, (8) or more sophstcated models such as F F d v = = ( k Tanh( k ε + k ) Tanh( k )) ( c Tanh( c ε + c ) Tanh( c )), (9) proposed n the work [Cafferty, 995]. Vertcal tre forces are computed usng a lnear sprng model. If the tre leaves the ground, the force s zero. Tre sde forces and algnng moments are computng by usng Magc formula [Blundell, 2000] y [ ], ( x) = D Sn C ArcTan( B x E ( B x ArcTan( B x) )) (20) where ( X ) = y( x) + S and x = X S. Y + v h (2) 40 ANALYSIS TECHNOLOGIES

7 S s vertcal shft, S s horzontal shft, Y s ether the tre sde force F, the algnng moment v h z the longtudnal force F and X s ether the slp angle α or the longtudnal slp κ. x M or z The steerng wheel angle s related knematcaly to the road wheels by nonlnear lookup table (deg/deg) [Sayers, 996] that account for lnkage geometry (e.g., Ackerman). Wheel steer angle s specfed as a functon of tme. Propulson forces are calculated by usng engne torque and a table lookup nvolvng throttle and engne speed. Tables based on engne speed determne upshftng and downshftng of the transmsson. Throttle control s specfed as a functon of tme. A nonlnear table s used to determne brake force for each wheel as a functon of hydraulc pressure. Brake force control s specfed as a functon of tme. The road surface s an arbtrary surface, modeled wth the help of symbolc splne [Yoshhko 998] and measured control ponts. Surface s specfed for each tre as vertcal dsplacements functon ( G, v, b, t), =,2,3, 4 s = s depend on road surface geometry G, vehcle geometry parameters b, vehcle velocty v and tme t 0,τ. [ ] 3.2 Mathematcal model and ts verfcaton For proposed vehcle mechancal model a mathematcal model has been developed wth proposed multbody formalsm [Cglarc, 200]. The test procedure for the severe lane-change manoeuvre has been used to perform numercal analyss of the proposed vehcle mechancal model. Manoeuvre s defned n the nternatonal standard [ISO3888, 999]. Test represents a part of the vehcle roadholdng ablty of passenger cars. Standard specfy the dmensons of the test track for duble lane change manoeuvre, whch s a dynamc process consstng of rapdly drvng a vehcle form ts ntal lane to another lane parallel to the ntal lane, and returnng to the ntal lane, wthout exceedng lane boundares. The drvng forces should be constant (the throttle poston shall be held as steady as possble). The vehcle Golf I has been used as test subject. The vehcle headng speed was 00km/h. The tme hstory plot for steerng wheel nputs [Blundell 2000] s shown n fg. 6. (22) Fgure 6. Steerng wheel nput for the ISO 3888 lane-change maneuver The study nvolve the use of Golf I parameter values (the geometry of vehcle chasss, the geometry of vehcle suspensons, the mass and nerta tensors of vehcle chasss, the mass and nerta tensors of vehcle suspensons, the stffness of suspenson sprngs and tres n vertcal drecton, the damper characterstcs ) as summarze n the work [Cglarc, 200]. Also ths study nvolve the use of Golf I tyre test data necessary for Magc formula as avalable from communcaton wth Professor Henz Burg from IbB-expetsen. ANALYSIS TECHNOLOGIES 4

8 Wth the help of proposed vehcle mathematcal model, the tme hstory of vehcle poston, orentaton, velocty, angular velocty, acceleraton and angular acceleraton has been calculated. Results of calculaton done by mathematcal models have been presented n vrtual envronment as shown fg. 7. The vrtual envronment called VRMLPath wll be dscussed n more detal n the part 2 of ths paper. Fgure 7. Anmaton of the lane-change maneuver The fg. 8 and fg. 9 presents part of results obtaned wth smplfed mechancal model wth 8 DOF and belongng mathematcal model. Smplfed mechancal model consst of rgd chasss wth 6 DOF, four suspensons modeled as lumped mass and smplfed VTCSM model, where each tre has only vertcal degree of freedom relatve to the rm. Each wheel has also a spn DOF. Comparson of calculated and measured dynamc response [Blundell 2000, Cglarc, 200] of lateral acceleraton and roll angle show good agreement. Fgure 8. Lateral acceleraton comparson (measured n green, calculated n red) 42 ANALYSIS TECHNOLOGIES

9 Fgure 9. Roll angle comparson (measured n green, calculated n red) 4. Concluson The current state of work, presented n ths paper, enables quck and easy developng of road vehcle mathematcal models n a symbolc form. Numercal analyss of relatve smple mathematcal model show, good agreement between calculated and measured vehcle dynamc response. A full vehcle model whch use smplfed representaton of suspenson system and emprcal model Magc formula for tre force representaton s sutable for road vehcle pre-mpact dynamc analyss. For the road vehcle accdents reconstructon such vehcle models show benefts over sophstcated ndustral vehcle dynamc smulaton programs. The latest requres many nput parameters that are normally not avalable to road accdent reconstructon expert. Acknowledgement The authors would lke to acknowledge the support of Mechancal Dynamcs (producer of ADAMS) and professor Dr.-Ing. Henz Burg from IbB-Expetsen. References Melzer F., Symbolc Computatons n Flexble Multbody Systems, Nonlnear Dynamcs. Vol. 9, 996, pp Mathematca, Gude to Standard Mathematca Packages, Champagn, IL: Wolfram Research Inc., 996. Valembos R.E. et all, Comparson of varous technques for modelng flexble beams n multbody dynamcs, Nonlnear Dynamcs. Vol. 2, 997, pp Caron M. et all, Order-N Formulaton and dynamc of mult-unt flexble space manpulators, Nonlnear Dynamcs. Vol. 7,998, pp Kovo A. (989). Fundamentals for control of robotc manpulators, John Wley & Sons, 989. Wasserman R.H. (992). Tensors & Manfolds, Oxford unversty press, 992, ISBN Jan A. (99) Unfeld formulaton of dynamcs for seral rgd multbody system, AIAA Jrl. Gudance, Control and Dynamcs 4 (3), 99, p.p Ramakrshan J.V. and Sngh R.P. (989). Computer aded desgn envronment for the desgn of mult-body flexble structures, proc. AIAA Gudance, Navgaton and Control, 989, p.p Crolla D. A. (995). Vehcle dynamcs-theory nto practce. Automoble Dvson Charman s Address, emcee, 995. ANALYSIS TECHNOLOGIES 43

10 Mancosu F. et al. (999). Prell actvtes on dynamc analyss n ADAMS ncludng tyres. Internatonal ADAMS users conference, Berln, November Cafferty et al. (995). Charactersaton of automotve shock absorbers usng random exctaton. Proc. Instn. Mech. Engrs. Vol ImechE 995, p.p Blundell M. V. (2000). The modellng and smulaton of vehcle handlng. Part, 2, 3 and 4. Proc. Instn. Mech. Engrs. Vol. 23 part K and Vol. 24 part K. ImechE Sayers M.W. and Han D.S. (996). A generc multbody vehcle model for smulatng handlng and brakng. Vehcle system dynamc, Vol. 25, supplement 996. Yoshhko T. (998). Theory of Curves and surfaces. An ntroducton to classcal dfferental geometry by Mathematca. ISBN , wth CD_ROM. ISO3888, (999). Internatonal standard ISO3888. Passenger cars-test track for a severe lane change manoeuvre. Part : Double lane-change. Cglarc I. (200). Mathematcal models of human body and road vehcles consdered for traffc accdent reconstructon. PhD thess. Unversty of Ljubljana, Faculty of mechancal engneerng, 200. Dr. Iztok Cglarc Unversty of Ljubljana Faculty of Mechancal Engneerng CEMEK laboratory Aškerceva 6, Ljubljana, Slovenja Tel.: Emal: ztok.cglarc@fs.un-lj.s 44 ANALYSIS TECHNOLOGIES