CHAIN DYNAMIC FORMULATIONS FOR MULTIBODY SYSTEM TRACKED VEHICLES

Transcription

1 2012 NDIA GROUND VEHICLE SYSEMS ENGINEERING AND ECHNOLOGY SYMOSIUM MODELING &SIMULAION, ESING AND VALIDAION (MSV) MINI-SYMOSIUM AUGUS 14-16, MICHIGAN CHAIN DYNAMIC FORMULAIONS FOR MULIBODY SYSEM RACKED VEHICLES Mchael Walln 1, Ahmed K. Aboubar 1, aramsothy Jayaumar 2 Mchael D. Letherwood 2, Ahmed A. Shabana 1 1 Department of Mechancal and Industral Engneerng, Unversty of Illnos at Chcago, 842 West aylor Street, Chcago, IL U.S. Army ARDEC, 6501 E. 11 Mle Road, Warren, MI ABSRAC hs paper s focused on the dynamc formulaton of mechancal onts usng dfferent approaches that lead to dfferent models wth dfferent numbers of degrees of freedom. Some of these formulatons allow for capturng the ont deformatons usng dscrete elastc model whle the others are contnuum-based and capture ont deformaton modes that cannot be captured usng the dscrete elastc ont models. Specfcally, three types of ont formulatons are consdered n ths nvestgaton; the deal, complant dscrete element, and complant contnuumbased ont models. he deal ont formulaton, whch does not allow for deformaton degrees of freedom n the case of rgd body or small deformaton analyss, requres ntroducng a set of algebrac constrant equatons that can be handled n computatonal multbody system (MBS) algorthms usng two fundamentally dfferent approaches: constraned dynamcs approach and penalty method. When the constraned dynamcs approach s used the constrant equatons must be satsfed at the poston, velocty, and acceleraton levels. he penalty method, on the other hand, cannot ensure that the algebrac equatons are satsfed at the acceleraton level. In the complant dscrete element ont formulaton, no constrant condtons are used; nstead the connectvty condtons between bodes are enforced usng forces that can be defned n ther most general form n MBS algorthms usng bushng elements that allow for the defnton of general nonlnear forces and moments. he new complant contnuum-based ont formulaton, whch s based on the fnte element (FE) absolute nodal coordnate formulaton (ANCF), has several advantages: (1) It captur es modes of ont deformatons that cannot be captured usng the complant dscrete ont models; (2) It leads to lnear connectvty condtons, thereby allowng for the elmnaton of the dependent varables at a preprocessng stage; (3) It leads to a constant nerta matrx n the case of chan le structure; and (4) It automatcally captures the deformaton of the bodes usng dstrbuted nerta and elastcty. he formulatons of these three dfferent ont models are compared n order to shed lght on the fundamental dfferences between them. Numercal results of a detaled traced vehcle model are presented n order to demonstrate the mplementaton of some of the formulatons dscussed n ths nvestgaton. : Dstrbuton Statement A. Approved for publc release.

2 ROBLEM DEFINIION Accurate formulaton of mechancal onts s necessary n the computer smulaton of multbody system (MBS) that represent many technologcal and ndustral applcatons. An example of these MBS applcatons, n whch accurate modelng of ont complance s necessary, s the traced vehcle shown n fgure 1. he lns of the trac chans of ths vehcle are connected by pn onts that can be subected to sgnfcant stresses durng the vehcle functonal operatons. Nonetheless, there are dfferent ont formulatons that can lead to dfferent dynamc models whch have dfferent numbers of degrees of freedom. hs study nvestgates the use of three dfferent methods for formulatng mechancal onts n MBS applcatons. hese three methods are the deal, the complant dscrete element, and the complant contnuum-based ont formulatons. hese three dfferent methods are descrbed below. Ideal Jont Formulaton he deal ont formulaton s based on a set of algebrac equatons that do not account for the ont flexblty; ths s regardless of whether or not the body s flexble. he algebrac ont equatons are expressed n terms of the coordnates of the two bodes connected by the ont. hese algebrac equatons are consdered as constrant equatons whch can be enforced usng two fundamentally dfferent methods; the constraned dynamcs approach and the penalty method. In the constraned dynamcs approach, the technque of Lagrange multplers or a recursve method s used. In ths case, the ont constrant equatons must be satsfed at the poston, velocty, and acceleraton levels. he number of degrees of freedom of the model n ths case s equal to the number of the system coordnates mnus the number of the algebrac ont constrant equatons. In the penalty method, on the other hand, the number of degrees of freedom of the model s not affected by the number of ont constrant equatons. hese ont algebrac equatons are enforced usng hgh stffness penalty coeffcents that ensure that the algebrac constrant equatons are satsfed at the poston level. he penalty method does not ensure that the constrant equatons are satsfed at the acceleraton level. Fgure 1: M113 traced vehcle model n SAMS/2000 age 2 of 22

3 Complant Dscrete Element Jont Formulaton In ths approach, no algebrac equatons are used to descrbe the onts between bodes n the system. he connectvty between bodes s descrbed usng force elements that have forms defned by the user of the MBS code. MBS system codes have bushng elements that can be used to defne general lnear or nonlnear force and moment expressons. he stffness and dampng coeffcents n the force and moment expressons can be selected by the user. he busng elements can be used to model the ont complance n the case of rgd and flexble body dynamcs. It s mportant, however, to pont out that addng bushng elements has no effect on the number of degrees of freedom of the model. Unle the penalty method, the use of bushng element does not requre the formulaton of algebrac ont equatons. Bushng elements allow for systematcally ntroducng three force components and three moment components. capture deformaton modes that cannot be captured usng complant dscrete ont formulatons. he use of the ANCF gradent coordnates allows for developng dfferent ont models wth dfferent numbers of degrees of freedom that allow for dfferent stran modes. 2. he use of the ANCF gradent coordnates allows for developng lnear ont constrant equatons. hese lnear algebrac equatons can be used to elmnate dependent varables at a preprocessng stage, thereby sgnfcantly reducng the model dmensonalty. 3. ANCF fnte elements can also be used to model the body deformaton n addton to the ont complance. Dstrbuted nerta and elastcty are used for both body flexblty and ont complance. 4. ANCF fnte elements lead to new types of FE meshes that have constant nerta, a feature that can be exploted to develop a sparse matrx structure of the MBS dynamc equatons. Complant Contnuum-Based Jont Formulaton he new fnte element (FE) absolute nodal coordnate formulaton (ANCF) allows for systematcally developng new ont formulatons that capture modes of deformaton that cannot be captured usng the dscrete ont models. It also allows for modelng body flexblty usng new FE meshes that have constant nerta and lnear connectvty condtons. Specfcally, the complant ANCF contnuum-based ont formulaton has the followng advantages: 1. ANCF fnte elements allow for developng new ont formulatons that age 3 of 22 It s the obectve of ths nvestgaton to provde a comprehensve study of dfferent ont formulatons and demonstrate the fundamental dfferences between them when appled to the analyss of complex traced vehcle system models. Smulaton models of the M113 traced vehcle wll be used to compare the numercal results obtaned usng the ont formulatons dscussed n ths paper. hese results are obtaned usng the general purpose MBS computer code SAMS/2000 [17] whch allows for systematcally modelng MBS applcatons usng the augmented formulaton, penalty method, bushng elements, and ANCF fnte elements. In the augmented formulaton, the technque of Lagrange multplers s used.

4 he computatonal algorthm used n SAMS/2000 ensures that the algebrac constrant equatons are satsfed at the poston, velocty, and acceleraton levels. RACKED VEHICLES: BACKGROUND Hgh moblty traced vehcle such as mltary battle tans and armored personal carrers are desgned for the moblty over rough and off-road terrans. Investgatons on the dynamc analyss of such traced vehcles shown n fgure 1 have been lmted because of the complexty of the forces resultng from nteracton between the vehcle components. hese forces are mpulsve n nature, and ther dynamc modelng requres sophstcated computatonal capabltes. Several two dmensonal models for the analyss of traced vehcle have been developed. Galtss [3] demonstrated that the dynamc trac tenson and suspenson loads n hgh speed traced vehcles developed by analytcal methods are useful n evaluatng the dynamc characterstcs of the traced vehcle components. Bando et al. [1] outlned a procedure for the desgn and analyss of rubber traced small-sze bulldozers, and presented a computer smulaton whch was used n the evaluaton of the vehcle performance. Both steel and contnuous rubber tracs are modeled by dscretzng them nto several rgd bodes connected by complant elements. he smulaton results ndcate that the vehcle has favorable characterstcs, such as less damage to the road surface, and reduced vbraton and nose. Murray and Canfeld [7] used general purpose multbody computer codes to model a smple trac ln and sprocet system. he behavor of the age 4 of 22 nteracton between the trac ln and the sprocet was llustrated graphcally and t was found that the computer tme can be sgnfcantly reduced by usng supercomputers. Naansh et al. [8] developed a two dmensonal contact force model for planar analyss of multbody traced vehcle systems. Modal parameters (modal mass, modal stffness, modal dampng, and mode shapes), whch are determned expermentally, are employed to smulate the nonlnear dynamc behavor of a multbody traced vehcle whch conssts of nterconnected rgd and flexble components. he equatons of moton of the vehcle are formulated n terms of a set of modal and reference generalzed coordnates, and the theoretcal bass for extractng the component modal parameters of the chasss from the modal parameters of the assembled vehcle s descrbed. A number of approaches have been proposed n the lterature for developng three-dmensonal MBS models. Cho et al. [2] developed the nonlnear dynamc equatons of moton of the threedmensonal multbody traced vehcle systems, tang nto consderaton the degrees of freedom of the trac chans. o avod the soluton of a system of dfferental and algebrac equatons, the recursve nematc equatons of the vehcle are expressed n terms of the ndependent ont coordnates. In order to tae advantage of sparse matrx algorthms, the ndependent dfferental equatons of the threedmensonal traced vehcles are obtaned usng the velocty transformaton method. hree-dmensonal nonlnear contact force models that descrbe the nteracton between the trac lns and the vehcle components such as the rollers, sprocets, and dlers as well as the nteracton between the trac

5 lns and the ground are developed and used to defne the generalzed contact forces assocated wth the vehcle generalzed coordnates. A computer smulaton of a traced vehcle n whch the trac s assumed to consst of trac lns connected by a sngle degree of freedom revolute ont s presented n order to demonstrate the use of the formulatons presented n ther study. Ryu et al. [13] developed complant trac ln models and nvestgated the use of these models n the dynamc analyss of hghspeed, hgh-moblty traced vehcles. he characterstcs of the complant elements used n ths nvestgaton to descrbe the trac onts are measured expermentally. A numercal ntegraton method havng a relatvely large stablty regon s employed n order to mantan the soluton accuracy, and a varable step sze ntegraton algorthm s used n order to mprove the effcency. he dmensonalty problem s solved by decouplng the equatons of moton of the chasss and trac subsystems. Recursve methods are used to obtan a mnmum set of equatons for the chasss subsystem. Several smulatons scenaros ncludng an accelerated moton, hgh-speed moton, brang, and turnng moton of the hghmoblty vehcle are tested n order to demonstrate the effectveness and valdty of the methods proposed. Oza and Shabana [9-10] evaluated the performance of dfferent formulatons usng a traced vehcle model that s subected to mpulsve forces. hey developed models for ont constrants and the mpulsve contact forces that result from the nteracton between the trac chans and the vehcle components as well as the nteracton between these chans and the ground. he nonlnear contact force models used n ther numercal study were developed, and the formulatons of the generalzed forces assocated wth the age 5 of 22 generalzed coordnates used n each of the formulatons were presented. Ryu et al [14] nvestgated the nonlnear dynamc modelng methods for the vrtual desgn of traced vehcles by usng MBS dynamc smulaton technques. he results nclude hgh oscllatory sgnals resultng from the mpulsve contact forces and the use of stff complant elements to represent the onts between the trac lns. Each trac ln s modeled as a body whch has sx degrees of freedom, and two complant bushng elements are used to connect trac lns. Effcent contact search and nematcs algorthms n the context of the complance contact model are developed to detect the nteractons between trac lns, rollers, sprocets, and ground for the sae of speedy and robust solutons. Rubnsten and Htron [12] developed a three-dmensonal multbody smulaton model for smulatng the dynamc behavor of traced off-road vehcles usng the LMS-DADS smulaton program. Each trac ln s consdered a rgd body and s connected to ts neghborng trac ln va a revolute ont. he road-wheel trac-ln nteracton s descrbed usng three-dmensonal contact force elements, and the trac-ln terran nteracton s modeled usng a pressuresnage relatonshp. SCOE AND OBJECIVES OF HIS INVESIGAION Whle some of the formulatons used n the analyss of traced vehcles requre the use of MBS algorthms that are desgned for solvng systems of dfferental/algebrac equatons arsng from nematc ont constrants, other formulatons do not requre the use of such algorthms but use penalty forces or complant elements to

6 model the chan dynamcs. he trac lns are subected to hgh contact forces that produce hgh stress levels, whch often cause damage to the trac lns durng the functonal operaton of the vehcle. he obectve of ths paper s to present and compare between dfferent ont formulatons that can be used n the dynamc modelng of complex traced vehcle systems. Better understandng of these formulatons can lead to more accurate, and possbly faster, computer smulatons that can be the bass for more relable performance evaluaton of the vehcles. he traced vehcles consdered n ths nvestgaton are assumed to consst of nterconnected bodes that can have arbtrary dsplacements. In the frst chan formulaton, referred to n ths paper as the deal ont chan model, nematc onts between the trac lns are descrbed usng nonlnear constrant equatons that lead to sgnfcant reducton n the number of vehcle degrees of freedom. hs ont model does not requre assumng stffness and dampng for the trac ln connectvty, and therefore, does not allow for flexblty between the tac lns; t requres, however, the soluton of a system of dfferental and algebrac equatons f redundant coordnate formulatons are used. Redundant coordnate algorthms based on the Lagrangan augmented form of the equatons of moton requre the use of Newton-Raphson method n order to ensure that the constrant equatons are satsfed at the poston level. Recursve and ont varables methods can also be used nstead of redundant coordnate formulatons n order to avod Newton- Raphson algorthm. Another approach that can be used to enforce the constrant equatons at the poston level s the penalty method. hs model does not lead to reducton n the number of the system degrees of freedom. wo other approaches that capture the ont complance wll also be consdered n ths study. he frst s the complant dscrete element method that employs MBS bushng elements to defne the connectvty between the trac lns. hs approach as n the case of the penalty method requres assumng stffness and dampng coeffcents at the connecton, and therefore, t allows for the flexblty between the trac lns. In the second, the complant contnuum-based ont formulaton that employs ANCF fnte elements s used. hs approach, whch captures new ont deformaton modes, leads to lnear connectvty condtons whch can be appled at a preprocessng stage allowng for an effcent elmnaton of the dependent varables, ths leads to a constant nerta matrx and zero Corols and centrfugal forces [18]. hs approach leads to new types of FE chan meshes that have desrable characterstcs. ALGEBRAIC CONSRAIN EQUAIONS In the methods of constraned dynamcs, there are two approaches that are often used to model deal mechancal onts that do not account for the effect of elastcty and dampng. hese two methods are the augmented formulaton that employs the technque of Lagrange multplers or the recursve formulaton whch allows for systematc elmnaton of the dependent varables usng the algebrac equatons. hese two formulatons are brefly dscussed n ths secton. age 6 of 22

7 Augmented Formulaton In the augmented formulaton, the constrant forces explctly appear n the dynamc equatons whch are expressed n terms of redundant coordnates. he constrant relatonshps are used wth the dfferental equatons of moton to solve for the unnown acceleratons and constrant forces. Whle ths approach leads to a sparse matrx structure, t has the drawbac of ncreasng the problem dmensonalty and t requres more sophstcated numercal algorthms to solve the resultng system of dfferental and algebrac equatons (DAE). Usng the generalzed coordnates, the equatons of moton of a body can be wrtten as [11, 15] M q Q e Q c Q (1) where M s the mass matrx of the body, q R θ s the vector of the acceleratons of the body wth R defnng the body translaton and θ defnng the body orentaton, Q s the vector of e external forces, Q s the vector of the c constrant forces whch can be wrtten n terms of Lagrange multplers λ as Q c C λ, C s the constrant Jacoban q q matrx assocated wth the coordnates of body, and Q s the vector of the nerta forces that absorb terms that are quadratc n the veloctes. he constrant equatons at the acceleraton level can be wrtten as C q q Q d, where Q d s a vector that absorbs frst dervatves of the coordnates. Usng equaton (1) wth the constrant equatons at the acceleraton level, one obtans M C q q Qe Q Cq 0 λ Qd (2) he matrces and vectors that appear n ths equaton are the system matrces and vectors that are obtaned by assemblng the body matrces and vectors. he precedng matrx equaton, whch ensures that the constrant equatons are satsfed at the acceleraton level, can be solved for the acceleratons and Lagrange multplers. In order to ensure that the algebrac nematc constrant equatons are satsfed at the poston and velocty levels, the ndependent acceleratons q are dentfed and ntegrated forward n tme n order to determne the ndependent veloctes q and ndependent coordnates q. Knowng the ndependent coordnates from the numercal ntegraton, the dependent coordnates qd can be determned from the nonlnear constrant equatons usng an teratve Newton- Raphson algorthm that requres the soluton of the system Cq q d d C, where qd s the vector of Newton dfferences, and C q d s the constrant Jacoban matrx assocated wth the dependent coordnates. Knowng the system coordnates and the ndependent veloctes, the dependent veloctes q d can be determned by solvng a lnear system of algebrac equatons that represents the constrant equatons at the velocty level. hs lnear system of equatons n the veloctes can be wrtten as C q C q C ; where C s the q d d q t constrant Jacoban matrx assocated wth the ndependent coordnates, and Ct C t s the partal dervatve of the constrant functons wth respect to tme. q age 7 of 22

8 Lagrange multplers, on the other hand, can be used to determne the constrant forces. For a gven ont, the generalzed constrant forces actng on body, connected by ths ont, can be obtaned from the equaton c Q C q λ F (3) where F and are the generalzed ont forces assocated, respectvely, wth the translaton and orentaton coordnates of body. Usng the results of equaton (3), the reacton forces at the ont defnton pont can be determned usng the concept of the equpollent systems of forces. Recursve Formulaton mnmum set of dfferental equatons from whch the worless constrant forces are automatcally elmnated [11, 17]. he numercal procedure used n solvng these dfferental equatons s much smpler than the procedure used n the soluton of the mxed system of dfferental and algebrac equatons resultng from the use of the augmented formulaton. In the recursve formulaton, the equatons of moton are formulated n terms of ont degrees of freedom. In ths formulaton, the multbody system s assumed to consst of subsystems, as n the case of the trac chans shown n fgure 2 where body correspondng to body 1. he absolute coordnates and veloctes of an arbtrary body n a subsystem are expressed n terms of the ndependent ont varables as well as the absolute coordnates and veloctes of body 1. If body s connected to body 1 through a revolute ont, whch s the case n ths subsystem, the relatve rotaton s the only degree of freedom represented between the bodes. he connectvty between bodes and body 1can then be descrbed usng the nematc relatonshps R A u R A u 0 1, 1 ω ω ω (4) where R s the global poston vector of the orgn of body ; A s the transformaton matrx that defnes the body orentaton and Fgure 2: Revolute ont can be expressed n terms of Euler 1 parameters; u and u are the local Another alternate approach for formulatng poston vectors of pont defned n the the equatons of moton of constraned coordnate systems of body and -1, mechancal systems s the recursve method, 1 respectvely, ω and ω are, respectvely, wheren the equatons of moton are the absolute angular velocty vectors of formulated n terms of the ont degrees of, 1 freedom. hs formulaton leads to a bodes and 1, and ω s the angular velocty vector of body wth respect to age 8 of 22

9 body -1 whch can be defned as, 1 1 ω v, wth v A v where 1 v s a unt vector along the axs of rotaton defned n the coordnate system of body 1, and s the angle of relatve rotaton. By dfferentatng the frst equaton n equaton (4) twce and the second once wth respect to tme, one obtans R ω ( ω u ) ( α u ) R ω ( ω u ) ( α α α v ( ω v 1 1 u ) 1 ) (5) In ths equaton, α s the absolute angular acceleraton vector of body. Usng the nematc equatons obtaned n ths secton, one can systematcally elmnate the dependent varables n order to obtan a number of dfferental equatons of moton equal to the number of the system degrees of freedom. Usng ths approach, one obtans a dense nerta matrx n a system of dynamc equatons that does not have constrant forces. A second, alternatve approach s to use the nematc equatons developed n ths secton to determne all the system absolute coordnates and veloctes. One can then construct equaton (2), whch can be solved for the acceleratons and Lagrange multplers. Usng the absolute acceleraton relatonshps of equaton (5), one can determne the relatve ont acceleratons. he ont acceleratons can be ntegrated forward n tme n order to determne the ont coordnates and veloctes. generalzed and the Cartesan moments [11, 17]. hs s mportant n nterpretng the reacton forces of the deal onts and also mportant n the mplementaton of the penalty method and bushng elements. Let F be a force vector that acts at a pont on a rgd body. If ths force vector s assumed to be defned n the global coordnate system, then the vrtual wor of ths force vector can be wrtten as W e F r, where r can be found usng the vrtual change n the poston vector of an arbtrary pont on rgd body as R r I A u G (6) θ In ths equaton, A s the transformaton matrx that defnes the body orentaton, u ~ s the sew symmetrc matrx assocated wth the vector u that defnes the local coordnates of the pont, and G s the matrx that relates the angular velocty vector ω defned n the body coordnate system to the tme dervatves of the orentaton coordnates, that s ω G θ. Note that snce u ~ ~ A ua, equaton (6) can be wrtten as r R u G θ. Usng ths equaton n the vrtual wor expresson, one obtans W e F R F u G θ, whch can be wrtten as GENERALIZED FORCES In defnng the ont forces between the trac lns, t s mportant to understand the relatonshp and dfferences between the age 9 of 22 e R W F R F θ (7) where F F, and F G u F. hese R equatons mply that a force that acts at an arbtrary pont on the rgd body s

10 equpollent to another system defned at the reference pont that conssts of the same force and a set of generalzed forces, defned by F G u F assocated wth the orentaton coordnates of the body [11, 17]. Snce u s a sew-symmetrc matrx, t follows that u u. Usng ths dentty, one can show that the generalzed moment can be wrtten as F G () u F, where Ma u F s the Cartesan moment resultng from the applcaton of the force F, and G s the matrx that relates the angular velocty vector ω defned n the global coordnate system to the tme dervatves of the orentaton coordnates, that s ω G θ. It follows that the relatonshp between the generalzed and Cartesan moment s F G M a. If the components of the moment vector are defned n the body coordnate system, one has F G M a, where M a A M a. Relatonshps developed n ths secton wll be used n the formulaton of the ont forces n the case of the penalty method. hese relatonshps wll also be used n the computer mplementaton of the bushng element n general MBS algorthms. JOIN FORMULAIONS In ths paper, four dfferent ont models are consdered; two models are based on algebrac equatons that requre the use of the methods of constraned dynamcs or the penalty method. In the case of constraned dynamcs, an alternatve to the use of Lagrange multplers s the use of the recursve methods, as prevously dscussed. When Lagrange multpler technque or the recursve methods are used the constrant equatons must be satsfed at the poston, velocty, and acceleraton levels. he penalty method, on the other hand, cannot satsfy the algebrac constrant equatons at the acceleraton level. Constrant Equatons he revolute ( pn) ont s used n ths secton as an example to demonstrate the formulaton of the algebrac constrant equatons. hs ont has been used n the lterature n the modelng of the trac chans. As shown n fgure 2, the trac chan can be assumed to consst of lns connected to each other by a pn ont that allows for the relatve rotaton between them. In general MBS algorthms, the nonlnear algebrac equatons that defne the pn ont are expressed n terms of the absolute coordnates of the two bodes and connected by the ont. he fve algebrac constrant equatons that elmnate fve degrees of freedom can be wrtten n terms of the absolute Cartesan coordnates of the two bodes as [17] C( q, q ) v 1 v v2 v v1 r v2 r r 0 (8) where v and v are two vectors defned along the ont axs on bodes and, respectvely; v, v1, v2 trad defned on body ; [17]; and form an orthogonal s a constant r r r R A u R A u (9) age 10 of 22

11 In ths equaton, A and A are the transformaton matrces that defne the orentaton of bodes and, respectvely, and u and u are the local poston vectors of ponts and wth respect to bodes and, respectvely. onts and are defned on the axs of the pn ont on bodes and, respectvely. One can show that the Jacoban matrx of the pn ont constrants s defned as v H1 v 1 H v H2 v 2H C q C C r H1 v 1 H v 1 H q q r H2 v 2H v 2H 2 2 r H r H (10) where C and q C q are the constrant Jacoban matrces assocated wth the coordnates of bodes and, respectvely; and other vectors and matrces that appear n the precedng equaton are ~ ~ H [ I A u G ], H [ I A u G ], v1 ( ) v 2 H1 A v 1, H2 ( A v2), q q q q H v q q A v Another alternate approach for formulatng the revolute ont constrants s to consder t as a specal case of the sphercal ont n whch the relatve rotaton between the two bodes s allowed only along the ont axs. If pont s the ont defnton pont, and v and v are two vectors defned along the ont axs on bodes and, respectvely,. the constrant equatons of the revolute ont can be wrtten as, 1 2 C q q r v v v v 0. he last two equatons n ths equaton guarantee that the two vectors v and v reman parallel, thereby elmnatng the freedom of the relatve rotaton between the two bodes n two perpendcular drectons. enalty Method he constrant equatons that descrbe the connectvty between the trac lns can be enforced usng the penalty approach. In ths case, these algebrac equatons are not satsfed at the acceleraton level. he penalty method does not lead to elmnaton of degrees of freedom, and therefore, t s conceptually dfferent from the case of Lagrange multpler technque or the recursve approach. In order to demonstrate the penalty approach, the volatons n the constrant equatons of a revolute ont can be wrtten as d r v 1 v v 2 v (11) Usng ths volaton d, a restorng force vector can then be defned as f d cd, where, and c are assumed penalty stffness and dampng coeffcents, respectvely, and d s the tme dervatve of the volaton vector d. he vrtual wor of ths restorng force f can then be wrtten as can be wrtten as W f d, whch age 11 of 22

12 W F r B F1 ( v1 v ) F2 ( v 2 v ) (12) proper selecton of the penalty coeffcents, these forces ensure that the constrant equatons are satsfed at the poston level. where r R u G θ R u G θ, p B B F r c r, () v 1 v v v1 v1 v, B p p v v v v v v, () d v 1 v F1 v 1 v c, and dt d v 2 v F2 v 2 v c. dt COMLIAN DISCREE ELEMEN JOIN FORMULAION Equaton (12) can be used to defne a set of generalzed forces actng on bodes and that mantan the connectvty between the two bodes as W Q R R Q Q R R Q (13) Whch can be rewrtten n a compact form as W Q q Q q, where q and q B B are the generalzed coordnates of bodes and, and Q R FB QB Q G u B FB G M1 G M2 Q R FB QB Q G u B FB G M1 G M2 (14) where M1 F1 v 1v, M1 F1 v 1 v, M2 F1 v 2 v, and M2 F1 v 2 v. Equaton (14) defnes the generalzed forces assocated wth the absolute Cartesan coordnates due to the revolute ont connecton between bodes and. Wth a age 12 of 22 Fgure 3: Bushng element he complant dscrete element ont formulaton allows for ntroducng ont deformatons. In ths approach, no algebrac equatons are enforced. In most MBS computer codes, the complant dscrete element ont method can be appled usng the standard MBS bushng element that allows for ntroducng three force and three moment components that can be lnear or nonlnear functons of the body coordnates. As shown n fgure 3, the poston vectors u 1 and u 2 of two ponts, 1 and 2 on body, can be used to defne one axs of the coordnate system of the bushng element as p p p p n u u u u, where one of the bushng axes defned n the body coordnate system. hs axs can then be n s

13 defned n the global coordnate system as n A n, where A s the transformaton matrx that defnes the orentaton of the coordnate system of body n the global system. Usng ths axs, one can defne the drectonal propertes of the bushng element wth the other two axes of the bushng coordnate system beng defned usng the b transformaton matrx A t 1 t2 n where t1 and t2 are the two unt vectors that complete the three orthogonal axes of the bushng element coordnate system. Assumng that body s a rgd body, the bushng coordnate system can be defned wth respect to the global coordnate system b b as A A A. Choosng ponts and 1 to ntally concde; one can defne the bushng deformaton and rate of deformaton vectors n the bushng coordnate system as b b b b δ A r, and δ A r, respectvely, where r r r s the poston vector of pont 1 pont. 1 wth respect to he rotatonal deformaton of the bushng element can be obtaned usng the transformaton matrx that defnes the orentaton of the bushng coordnate system on body wth respect to the bushng coordnate system on body. hs matrx s b b b b defned as A A A, where A s the orentaton matrx of the bushng coordnate b system on body, whle A s the orentaton matrx of the bushng coordnate system wth respect to the coordnate system b b of body that s defned as A A A. Assumng that the relatve rotatons between bodes and are small, the relatve rotaton matrx, b A can be used to extract three relatve rotatons defned n the bushng coordnate system, b b b b θ θx θ y θ z he relatve angular velocty between the two bodes defned n the bushng coordnate system b b can also be wrtten as ω A () ω ω, where ω and ω are the absolute angular velocty vectors of bodes and, respectvely, defned n the global coordnate system. he bushng stffness and dampng coeffcents are often determned usng expermental testng, and these coeffcents are defned generally n the bushng coordnate system. Let Kr and C r be the translatonal stffness and dampng matrces, respectvely, defned wth respect to the bushng coordnate system; and assume that the rotatonal stffness and dampng matrces are K and C, respectvely. In terms of translatonal and rotatonal stffness and dampng matrces, the force vector defned n the bushng coordnate system can be wrtten as F M b R b K r 0 0 δ K θ b b C r 0 0 b δ C b ω (15) hs force vector can then be defned n the global coordnate system, and the results can be used to defne the generalzed bushng forces and moments actng on the two bodes as prevously descrbed n ths paper. age 13 of 22

14 COMLIAN CONINUUM-BASED JOIN FORMULAION Fgure 4: ANCF beam element coordnates he complant contnuum-based ont formulaton allows capturng ont stran modes that cannot be captured usng the complant dscrete element ont method. When ANCF fnte elements are used, one can develop new FE meshes that have lnear connectvty and constant nerta [18]. hs allows for systematcally elmnatng dependent varables at a preprocessng stage, and as a result, there s no need for the use of ont formulatons n the man processor. hs approach can be used to develop new spatal chan models where the modes of deformatons at the defnton ponts of the onts that allow for rgd body rotatons between ANCF fnte elements can be captured. he dsplacement feld of an ANCF fnte element, as the one shown n fgure 4, can be wrtten as r( x, y, z,)( t, s,)() x y z e t where x, y, and z are the element spatal coordnates; t s tme; S s the element shape functon matrx, and e s the vector of element nodal coordnates. Usng ths dsplacement feld, age 14 of 22 the equatons of a pn ont between elements and can be wrtten usng the sx scalar equatons r r, r r, where s the coordnate lne that defnes the ont axs; can be x, y, or z or any other coordnate lne. he sx scalar equatons elmnate sx degrees of freedom; three translatons, two rotatons, and one deformaton mode. herefore, ths ont has fve modes of deformaton that nclude stretch and shear modes. hs ANCF 1 revolute ont model ensures C contnuty wth respect to the coordnate lne and 0 C contnuty wth respect to the other two parameters. It follows that the Lagrangan r r 1 2 s stran component contnuous at the ont defnton pont, whle the other fve stran components can be dscontnuous. he resultng ont constrant equatons are lnear, and therefore, can be appled at a preprocessng stage to systematcally elmnate the dependent varables. Usng these equatons, one can develop a new nematcally lnear FE mesh for flexble-ln chans n whch the lns can have arbtrarly large relatve rotatons. he comparatve numercal study presented n the followng secton wll be focused on three models; the deal constraned ont model, the deal penalty ont model, and the complant dscrete element model (bushng). he results of the ANCF ont model wll be consdered n future nvestgatons n order to allow for presentng a more meanngful study that compares dfferent ANCF ont formulatons.

15 SIMULAION RESULS components, whle more detals about the roller arrangement and the confguraton of the suspenson system of the M113 traced vehcle model are shown n fgure 6. able 1 shows the nerta prospertes for all the traced vehcle model components used n ths smulaton. More specfcatons of ths vehcle can be obtaned from open sources [5]. he vehcle has a suspenson system that conssts of road arms placed between the road wheels and chasss as well as shoc absorbers connected to each road arm. able 2 shows the stffness and the dampng coeffcents of the contact models as well as ther frcton coeffcents. Fgure 5: raced vehcle dfferent contact models Fgure 6: Suspenson system layout of M113 traced vehcle [6] In ths secton, numercal results, obtaned usng M113 traced vehcle model shown n fgure 1, are used to compare the dfferent ont formulatons presented n ths nvestgaton. he M113 traced vehcle s armored personnel carrer that conssts of a chasss, dler, sprocet, 5 road-wheels, and 64 trac lns on each trac sde (rght and left). Fgure 5 shows the engagement of the trac lns wth some of the vehcle age 15 of 22 In ths study, two dfferent smulaton scenaros, one wth the suspenson system and the other wthout the suspenson system, wll be consdered to study the effect of usng the suspenson system on the results. he road arms and the sprocets are connected to the chasss by revolute onts, and the road arms are connected to the roadwheels by revolute onts. he trac lns are connected to each other usng revolute onts, whch can be modeled usng the penalty method, bushng element, or ANCF fnte elements as prevously mentoned. ensoners are added to the system by connectng each dler to a tensoner wth a revolute ont and connectng the tensoner to the chasss wth a prsmatc ont to ensure only translaton between them.

16 Angular Velocty (rad/s) me (s) Fgure 7: Sprocet angular velocty he angular veloctes of the sprocets of the vehcle model consdered n ths numercal nvestgaton are assumed to ncrease lnearly after 1 sec untl t reaches a constant value of 25 rad/sec after 8 seconds as shown n fgure 7. Fgure 8 shows the chasss vertcal dsplacement usng the ont model wth and wthout the suspenson system. he results presented n ths fgure show that the model wth a suspenson system has less vbraton compared to the model wthout suspenson. he penalty method model and bushng element model shown n fgures 9-14 each have a stffness coeffcent of 10 9 N/m. In real lfe applcatons, the stffness coeffcent of bushng elements used n trac chan connectons are much lower; they are ncreased n ths case for better comparsons to the revolute ont and penalty models. he rotatonal stffness and dampng coeffcents about the axs of rotaton for the trac lns of the bushng element model are set to zero n ths case to allow for rotaton about the lateral axs. Fgures 9 and 10 show, respectvely, the chasss forward poston and velocty results obtaned usng dfferent ont models. Whle the results presented n these fgures show good agreement, the computatonal tme vares when these dfferent models are used. he constraned ont model taes less computatonal tme compared to the penalty and the bushng element models; the penalty model CU tme s twce that of the constrant model CU tme, whle the bushng model CU tme s four tmes that of the constrant model CU tme. hs ncrease n the bushng model CU tme s attrbuted to the hgh stffness coeffcents used n ths model. Vertcal oston (m) me (s) Fgure 8: Chasss vertcal dsplacement (--- wthout suspenson, wth suspenson) age 16 of 22

17 Forward oston (m) Fgure 9: Chasss forward poston ( Constraned ont model, enalty method model, Bushng element model) Forward Velocty (m/s) me (s) me (s) Fgure 10: Chasss forward velocty Fgure 11 shows the vertcal dsplacement of a trac ln n the global coordnate system usng dfferent ont models. Whle the results show good agreement the ont model has less vbraton compared to the penalty and bushng element models. Fgure 12 shows the moton traectory of a trac ln n the chasss coordnate system usng the constraned model, the penalty model, and the bushng element model, respectvely. Whle the results presented n these fgures show good agreement, t s clear that the constraned dynamcs model soluton exhbts less oscllatons. Vertcal oston (m) me (s) Fgure 11: Vertcal dsplacement of a trac ln ( Constraned ont model, enalty method model, Bushng element model) ( Constraned ont model, enalty method model, Bushng element model) age 17 of 22

18 Local Z-oston (Vertcal, m) Local Y-oston (Longtudnal, m) Fgure 12: raectory moton of a trac ln n the chasss coordnate system ( Constraned ont model, --- enalty method model, Bushng element model) Force (N) me (s) Fgure 13: Jont longtudnal forces ( Constraned ont model, --- enalty 9 method model, = 110 N/m) Fgures show the ont forces n the longtudnal and the vertcal drectons, respectvely, obtaned usng the constraned and penalty ont models. he penalty results are obtaned usng a stffness coeffcent of N/m and dampng coeffcent of N.s/m. he results show good agreement between the constraned and penalty ont models. Fgures 15 and 16 show the same results n the case of a 7 stffness coeffcent of 110 N/m and 4 dampng coeffcent of 510 N.s/m. Force (N) me (s) Fgure 14: Jont vertcal forces ( Constraned ont model, --- enalty 9 method model, = 110 N/m) age 18 of 22

19 Force (N) Force (N) me (s) Fgure 15: Jont Longtudnal forces ( Constraned ont model, --- enalty 7 method model, = 110 N/m) me (s) Fgure 16: Jont vertcal forces ( Constraned ont model, --- enalty 7 method model, = 110 N/m) he results of fgures 15 and 16, whch show sgnfcant dfferences between the two models, demonstrate the drawbac of the penalty method when the penalty stffness coeffcent s reduced. Smlar results can be expected n the case of the bushng element models where the forces obtaned also depend on the stffness and dampng coeffcent of the ont. Fgure 17 shows the ont deformaton predcted usng the penalty model usng dfferent stffness coeffcents. he penalty model wth a 9 stffness coeffcent of 110 N/m shows much less deformaton, less than 0.06 mm, whle the penalty model wth a stffness of N/m has much more deformaton, over 2.5 mm, between the trac lns. Jont Deformaton (m) me (s) Fgure 17: Jont deformaton usng penalty model ( = N/m, --- = 110 N/m) age 19 of 22

20 SUMMARY AND CONCLUSIONS In ths nvestgaton, dfferent MBS ont formulatons are presented and compared usng detaled traced vehcle models. hree man ont formulatons are dscussed; they are the deal ont formulaton, the complant dscrete element ont formulaton, and the complant contnuum-based ont formulaton. he deal ont formulaton s developed to elmnate the relatve dsplacement between the two bodes connected by the ont. hs can be acheved by enforcng a set of ont algebrac equatons usng a constraned dynamcs approach or by usng the penalty method. he constraned dynamcs approach elmnates degrees of freedom and ensures that the constrant equatons are satsfed at the poston, velocty, and acceleraton levels. he penalty method, on the other hand, does not reduce the number of degrees of freedom and ensures that the constrant equatons are satsfed at the poston level only provded that a hgh stffness coeffcent s used. he complant dscrete element formulaton, whch allows for ont deformatons, can be systematcally appled usng a standard MBS bushng element that allows for sx degrees of freedom of relatve moton. he complant contnuum-based approach can be used to develop new onts that capture deformaton modes that are not captured by the complant dscrete element ont formulaton. ANCF fnte elements can be used to systematcally develop new onts wth dstrbuted elastcty and lnear connectvty condtons. As dscussed n ths paper, t s mportant to choose the proper stffness and dampng coeffcents when the penalty method and the bushng elements are used. age 20 of 22 Numercal results were presented n order to compare between dfferent methods. he deal ont formulaton produces the desred ont nematcs and accurate ont forces. he same s true wth the penalty force based ont when large penalty stffnesses are used. enalty force based ont constructon has been shown to be senstve to the selecton of penalty stffness wth the hgher stffness coeffcents leadng to better overall results. However, hgher penalty stffness ncreases CU tme sgnfcantly due to hgher frequences. he penalty method and bushng element models each have much larger CU tmes than the deal constraned model due to these hgh stffness coeffcents. Bushng force based ont formulaton when used wth very large stffnesses has been shown to not automatcally equate to the penalty force based ont model. he ANCF ont formulaton s beng mplemented and wll be compared wth the prevous models n future wor. Acnowledgment hs research was supported by the U.S. Army an-automotve Research, Development and Engneerng Center (ARDEC) (Contract No. W911NF-07-D- 0001). REFERENCES [1] Bando, K., Yoshda, K., and Hor, K., 1991, "he development of the rubber trac for small sze bulldozers," SAE echncal aper , do: /911854, Mlwauee, WI. [2] Cho J. H., Lee H. C., and Shabana A. A., 1998, Spatal Dynamcs of Multbody raced Vehcles,

21 Vehcle System Dynamcs, Vol. 29, pp [3] Galatss, A. G., 1984, "A Model for redctng Dynamc rac Loads n Mltary Vehcles, ransactons of the ASME, Journal of Vbraton, Acoustcs, Stress, and Relablty n Desgn, Vol. 106, pp [4] Hamed, M. A., Shabana, A. A., Jayaumar,., and Letherwood, M. D., 2011, Nonstructural geometrc dscontnutes n fnte element/multbody system analyss, Nonlnear Dynamcs, Vol. 66, pp [5] M113, Image, A roop 4/12 CAV, 2003, Web, 11 July 2012, < ACAV/mages/M113-drawng.gf> [6] M113 Vehcle Famly Rubber rac Installaton Instructons, Ltho d n Canada, 2003, Web, 16 July 2012, < racinstallatona-654rb.pdf>. [7] Murray, M. and Canfeld,. R., 1992, "Modelng of a flexble ln power transmsson system," roceedngs of the 6 th ASME Internatonal ower ransmsson and Gearng Conference, Scottsdale, Arzona. [8] Naansh,., and Shabana, A. A., 1994, he Use of Recursve Methods n the Dynamc Analyss of raced Vehcles, echncal Report KMR , Department of Mechancal Engneerng, Unversty of Illnos at Chcago, Chcago. [9] Oza,., and Shabana, A. A., 2003, reatment of Constrants n Multbody Systems art I: Methods of Constraned Dynamcs, Journal age 21 of 22 of Mult-Scale Computatonal Engneerng, Vol. 1, pp [10] Oza,. and Shabana, A. A., 2003, reatment of Constrants n Multbody Systems, art II: Applcaton to raced Vehcles Journal of Mult-Scale Computatonal Engneerng, Vol. 1, pp [11] Roberson, R. E., and Schertasse, Rchard, 1988, Dynamcs of Multbody Systems, Sprnger-Verlag, Berln. [12] Rubnsten, D., Htron, R., 2004, A detaled mult-body model for dynamc smulaton of off-road traced vehcles, Journal of erramechancs, Vol. 41, pp [13] Ryu, H. S., Bae, D. S., Cho, J. H., and Shabana, A. A., 2000, A complant trac ln model for hghspeed, hgh-moblty traced vehcles, Internatonal Journal for Numercal Methods n Engneerng, Vol. 48, pp [14] Ryu, H. S., Huh, K. S., Bae, D. S., and Cho, J. H. 2003, Development of a Multbody Dynamcs Smulaton ool for raced Vehcles art I: Effcent Contact and Nonlnear Dynamcs Modelng, JSME Internatonal Journal, 46(2), pp [15] Shabana, A. A., Zaazaa, K. E., and Sugyama, Hroyu, 2008, Ralroad Vehcle Dynamcs: A Computatonal Approach, aylor & Francs/CRC, Boca Raton, Florda. [16] Shabana, A. A., 2008, Computatonal Contnuum Mechancs, Cambrdge Unversty ress, Cambrdge.

22 [17] Shabana, A. A., 2010, Computatonal Dynamcs, hrd Edton, John Wley and Sons, New Yor. [18] Shabana, A. A., Hamed, A. M., Mohamed, A. A., Jayaumar,., and Letherwood, M. D. 2012, Use of B- Splne n the Fnte Element Analyss: Comparson wth ANCF Geometry, Journal of Computatonal and Nonlnear Dynamcs, Vol. 7, pp [19] Yaoub, R. Y., and Shabana, A. A., 2001, hree Dmensonal Absolute Nodal Coordnate Formulaton for Beam Elements: Implementaton and Applcaton, ASME Journal for Mechancal Desgn, Vol. 123, pp able 1: Mass and nerta values for traced vehcle parts art Mass (g) I XX ( g.m 2 ) I YY ( g.m 2 ) I ZZ ( g.m 2 ) I XY, I YZ, I XZ (g.m 2 ) Chasss Sprocet Idler Road Wheel Road Arm rac Ln able 2: Contact arameters arameters Sprocet-rac Contact Roller-rac Contact Ground-rac Contact N/m N/m N/m c N s/m N s/m N s/m μ age 22 of 22