EXPERIMENTAL VALIDATION OF A MULTIBODY DYNAMICS MODEL OF THE SUSPENSION SYSTEM OF A TRACKED VEHICLE

Transcription

1 NDIA GROUND VEHICLE SYSTEMS ENGINEERING AND TECHNOLOGY SYMPOSIUM MODELING & SIMULATION, TESTING AND VALIDATION (MSTV) MINI-SYMPOSIUM AUGUST 9- DEARBORN, MICHIGAN EXPERIMENTAL VALIDATION OF A MULTIBODY DYNAMICS MODEL OF THE SUSPENSION SYSTEM OF A TRACKED VEHICLE Tamer M. Wasfy Advanced Scence and Auomaon Corp. Indanapols, IN James O Kns and Davd Ryan U.S. Army Tank Auomove Research Developmen and Engneerng Cener Warren, MI ABSTRACT A me-accurae mulbody dynamcs model of he suspenson sysem of a racked vehcle s expermenally valdaed usng a full-scale racked-vehcle on an N-pos moon smulaor. The expermens conss of harmonc excaons a varous ampludes and frequences and ramp excaons of he vehcle road-wheels (whou he rack), wh each road wheel under one lnear acuaor of he N-pos moon smulaor. A hgh-fdely mulbody dynamcs model of he vehcle along wh he N-pos moon smulaor s consruced. The mulbody dynamcs model consss of rgd bodes, jons, roaonal sprngs (ha nclude non-lnear roaonal sffness, dampng and frcon), acuaors and conac surfaces. The rgd bodes roaonal equaons of moon are wren n a body-fxed frame wh he oal rgd-body roaon marx updaed each me sep usng ncremenal roaons. Connecon pons on he rgd bodes are used o defne jons beween he bodes ncludng revolue, cylndrcal, prsmac and bracke jons. A penaly model s used o mpose he jon and normal conac consrans. The conac model deecs conac beween dscree pons on he surface of each wheel (maser conac surfaces) and a polygonal surface represenaon of he lnear acuaors dshpans (slave conac surfaces). A recursve boundng box/boundng sphere conac search algorhm s used o allow fas conac deecon. An aspery frcon model s used for he conac frcon forces. The governng equaons of moon are solved along wh jon/consran equaons usng a me-accurae explc soluon procedure. The me-hsores of he suspenson sysem roaonal deflecons are expermenally measured for he varous npu moon excaons. The expermenal measuremens are compared wh he resuls predced usng he compuaonal model. The comparson shows ha he model can predc wh reasonably good accuracy he es racked-vehcle s dynamc response.. INTRODUCTION Tracked vehcles are used n cvlan and mlary off-road ground vehcles. Tracks have beer mobly characerscs han res on off-road and slppery errans. Ths ncludes: Beer racon over frm errans (such as pavemen) ha s hghly sloped or slppery (e.g. we/cy pavemen). Beer racon and mobly over sof errans such as snow, sand, and mud. Beer posve/negave obsacle crossng capably. Tracked vehcles can go over larger obsacles han equvalen wheeled vehcles. Typcal obsacles ha ground vehcles are esed agans nclude sem-crcular and rapezodal bumps/poholes. Tracks can be used on vehcles of varous szes rangng from large racors and excavaor o small unmanned ground vehcles. Tracks can also be used n conjuncon wh res n order o mprove he vehcle s mobly. Tracks can be dvded no wo ypes: Connuous bel rack. The cross-secon of a connuous bel rack s smlar n consrucon o a re. I consss of a rubber marx renforced wh seel, Kevlar and/or polyeser wre/ply along he lengh and wdh of he rack. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed.

2 Repor Documenaon Page Form Approved OMB No Publc reporng burden for he collecon of nformaon s esmaed o average hour per response, ncludng he me for revewng nsrucons, searchng exsng daa sources, gaherng and mananng he daa needed, and compleng and revewng he collecon of nformaon. Send commens regardng hs burden esmae or any oher aspec of hs collecon of nformaon, ncludng suggesons for reducng hs burden, o Washngon Headquarers Servces, Drecorae for Informaon Operaons and Repors, 5 Jefferson Davs Hghway, Sue, Arlngon VA -. Respondens should be aware ha nowhsandng any oher provson of law, no person shall be subjec o a penaly for falng o comply wh a collecon of nformaon f does no dsplay a currenly vald OMB conrol number.. REPORT DATE 9 AUG. REPORT TYPE N/A. DATES COVERED -. TITLE AND SUBTITLE Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle 6. AUTHOR(S) Tamer M. Wasfy; James O Kns; Davd Ryan 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army RDECOM-TARDEC 65 E Mle Rd Warren, MI 897-5, USA Advanced Scence and Auomaon Corp. Indanapols, IN, USA 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) US Army RDECOM-TARDEC 65 E Mle Rd Warren, MI 897-5, USA 8. PERFORMING ORGANIZATION REPORT NUMBER 96. SPONSOR/MONITOR S ACRONYM(S) TACOM/TARDEC/RDECOM. SPONSOR/MONITOR S REPORT NUMBER(S) 96. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release, dsrbuon unlmed. SUPPLEMENTARY NOTES Presened a he NDIA Vehcles Sysems Engneerng and Technology Symposum 9- Augus, Dearborn, Mchgan, USA, The orgnal documen conans color mages.. ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT SAR a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed 8. NUMBER OF PAGES 8 9a. NAME OF RESPONSIBLE PERSON Sandard Form 98 (Rev. 8-98) Prescrbed by ANSI Sd Z9-8

3 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Segmened rack. The rack consss of rgd or flexble segmens conneced usng revolue jons. For rgd segmened racks, he segmens are made of seel wh a rubber layer on he rack-road conac surfaces. Flexble segmened racks have boh renforced rubber and seel segmens. The rubber segmens are used for conac beween he rack and he erran. The seel segmens are used o connec he rubber segmens. Tracked vehcles are flexble mulbody sysems nvolvng frconal conac, large roaons and large deflecons (of he flexble rack). A revew of flexble mulbody dynamcs modelng echnques ncludng deformaon reference frames, reamen of large roaons, dscrezaon echnques, fne elemens, consran and conac modelng, and soluon echnques s presened n []. A revew of racked vehcles models was presened n []. A fne elemen mulbody dynamcs model for racked vehcles, ha can handle boh flexble bel-ype racks and rgd segmened racks, was presened n []. The model uses he followng compuaonal echnques/elemens: Explc me-negraon solver accurae for long smulaon mes []. The algorhm ha s used o accoun for a rgd body roaonal moon [] uses he oal roaon marx relave o he neral frame o measure he roaon he rgd bodes. Ths avods sngulary problems assocaed wh and parameer roaon measures. Rgd body roaonal equaons of moon are wren n a body (maeral) frame. Thus, he nera ensor s consan. The equaons of moon are negraed o yeld he ncremenal roaons angles. The oal body roaon marx s updaed usng he roaon marx correspondng o he ncremenal roaons angles. Penaly formulaon for modelng jon consrans ncludng sphercal, revolue, cylndrcal and prsmac jons []. Normal conac modeled usng a penaly formulaon [5, 6]. Frconal conac modeled usng an accurae and effcen aspery-based frcon model [7]. General conac search algorhm for fndng he conac peneraon beween fne elemens and oher elemens as well as general rangle and quadrlaeral surfaces []. Toal-Lagrangan sprng, russ, beam and sold nonlnear fne elemens wh Caresan coordnae degrees of freedom [, 8-]. Ths allows arbrarly large elemen roaons wh no soluon drf due o he use of dsplacemen ncremens. The elemen DOFs are referenced o a global neral reference frame. The hn beam elemens use a orsonal-sprng formulaon wh -nodes per elemen [, 8] for modelng he bendng behavor of he beams. 8-node naural deformaon modes brck sold elemens are used [9, ]. Those elemens can also be used o model shells and beams. One elemen hrough he hckness s suffcen o accuraely model he membrane, shear, and bendng characerscs. The brck elemens do no exhb lockng or spurous modes (wdely used echnques o allevae lockng such as hourglass conrol lead o elemens ha do no manan soluon accuracy over very long soluon mes). Any maeral law can be used wh hose elemens ncludng: lnear elasc, hyper-elasc, and non-lnear laws. Those elemens are used o model he rubber marx of he connuous bel-ype racks. The rack renforcemens for hose racks can be modeled usng he hn beam orsonal-sprng elemen. Lnear/roaonal suspenson sprngs ha can nclude a prescrbed non-lnear force-deflecon relaon, a forcedeflecon rae relaon and a frcon coeffcendeflecon rae relaon. Fgure. Tracked vehcle on he n-pos moon smulaor. The purpose of hs paper s o parally expermenally valdae he racked vehcle model presened n []. The expermenal valdaon was carred ou usng a full-scale racked vehcle ha was nsrumened and mouned, wh he rack removed, on an n-pos moon smulaor (Fgure ). The n-pos moon smulaor consss of hydraulc lnear Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

4 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) acuaors under each road wheel ha can be ndependenly conrolled o smulae ypcal road maneuvers ha he vehcle s subjeced o such as gong over bumps/poholes. The me-hsores of he angular deflecons of he road arms are measured for varous npu moon excaons. The expermens were used o une he force angular velocy relaon (dampng) of he road arms orson bars. The resuls of he expermens are hen compared wh he resuls predced usng he compuaonal model. In addon, n hs paper ypcal smulaons of racked vehcles ha can be performed usng he presen model are also presened. Also, a lumped parameers hck spaal beam elemen s presened. The elemen has wo rgd body ype nodes wh each node havng ranslaonal DOFs and he roaon marx s used o measure he spaal roaon of he node. The elemen models each response componen ndependenly usng smpler sub-elemens ha nclude: hree muually orhogonal lnear sprng sub-elemens for modelng one axal and wo shear deformaons; hree muually orhogonal orsonal sprng elemens for modelng one orsonal and wo bendng deformaons of he elemen. Ths beam elemen can be used o model boh segmened and connuous racks. The res of hs paper s organzed as follows. In Secon he ranslaonal and roaonal sem-dscree equaons of moon are presened. In Secon he fne elemen formulaon for he russ, hn beam, hck beam and sold brck elemens are presened. In Secon he frconal conac echnques, ncludng penaly formulaon for normal conac, aspery frcon model and conac search algorhm, are presened. In Secon 5 he penaly algorhm for mposng jon consrans s presened. In Secon 6, he overall explc soluon algorhm s oulned. In Secon 7, he expermenal seup s presened. In Secon 8 he comparson beween he expermen and model s presened. In Secon 9 ypcal numercal smulaons of racked vehcles are presened. Fnally, n Secon concludng remarks are offered.. EQUATIONS OF MOTION In he subsequen equaons he followng convenons wll be used: The ndcal noaon s used. The Ensen summaon convenon s used for repeaed subscrp ndces unless oherwse noed. Upper case subscrp ndces denoe node numbers. Lower case subscrp ndces denoe vecor componen number. The superscrp denoes me. A superposed do denoes a me dervave. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Two ypes of fne elemen nodes are used: pon parcle nodes and rgd body nodes. Pon parcle nodes have ranslaonal DOFs (Degrees of freedom). The algorhm for wrng and negrang he equaons of moon for spaal rgd bodes usng an explc fne elemen code was presened n []. In hs algorhm, a rgd body s modeled usng a fne elemen node locaed a s cener of mass. The node has ranslaonal DOFs defned wh respec o he global neral reference frame and a roaon marx defned also wh respec o he global neral frame. The use of he oal body roaon marx o measure rgd body roaons avods sngulary problems assocaed wh and parameer roaon measures []. The ranslaonal equaons of moon for he nodes are wren wh respec o he global neral reference frame and are obaned by assemblng he ndvdual node equaons. The equaons can be wren as: M & x = F + F () K K s K a K where s he runnng me, K s he global node number (no summaon over K; K= N where N s he oal number of nodes), s he coordnae number (=,,), a superposed do ndcaes a me dervave, M K s he lumped mass of node K, x s he vecor of nodal Caresan coordnaes wh respec o he global neral reference frame, and & x& s he vecor of nodal acceleraons wh respec o he global neral reference frame, Fs s he vecor of nernal srucural forces, and Fa s he vecor of exernally appled forces, whch nclude surface forces and body forces. For each node represenng a rgd body, a body-fxed maeral frame s defned. The orgn of he body frame s locaed a he node ha s also he body s cener of mass. The mass of he body s concenraed a ha node and he nera of he body s gven by he nera ensor defned wh respec o he body frame. The orenaon of he body-frame o s gven by R whch s he roaon marx relave o he K global neral frame a me. The roaonal equaons of moons are wren for each node wh respec o s bodyfxed maeral frames as: s K a K ( & θ K ( I & θ ) K I & θ = T + T ) () where I K s he nera ensor of rgd body K, θ & and θ & are he angular acceleraon and velocy vecors componens for rgd body K relave o s maeral frame n drecon j (j=,,), Ts K are he componens of he vecor of nernal orque a node K n drecon, and Ta K are he componens of he vecor of appled orque. The summaon convenon s used only for he lower case ndces and j. Snce, he rgd body roaonal equaons of moon are wren n a body (maeral) frame, hus, he nera ensor I K s consan. Page of 8

5 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) The rapezodal rule s used as he me negraon formula for solvng equaon () for he global nodal posons x: Δ Δ x& = x& +.5Δ (&& x + & x ) (a) x Δ Δ = x +.5Δ ( x& + x& ) (b) where Δ s he me sep. The rapezodal rule s also used as he me negraon formula for he nodal roaon ncremens: & Δ Δ θ = & θ +.5Δ ( && θ + & θ ) (a) Δθ Δ =.5Δ ( & θ + & θ ) (b) where Δθ are he ncremenal roaon angles around he hree body axes for body K. Thus, he roaonal equaons of moon are negraed o yeld he ncremenal roaon angles. The roaon marx of body K (R K ) s updaed usng he roaon marx correspondng o he ncremenal roaon angles: R K Δ = R R( Δθ ) (5) K K where R( Δ θ K ) s he roaon marx correspondng o he ncremenal roaon angles from equaon (b). The explc soluon procedure used for solvng equaons (-5) along wh consran equaons s presened n Secon 7. The consran equaons are generally algebrac equaons, whch descrbe he poson or velocy of some of he nodes. They nclude: Conac/mpac consrans (Secon 5): Jon consrans (Secon 6): Prescrbed moon consrans: f ({ x}) (6) f ({ x}) = (7) f ({ x}, ) = (8). Fne Elemens. Truss/Sprng Elemen The russ elemen connecs wo nodes. The nernal force n a russ elemen s gven by: EA CA F = ( l l l& ) + (9) l l where E s he Young s modulus, C s he dampng modulus, A s he cross-seconal effecve area, l s he curren lengh of he russ, l s he un-sreched lengh of he russ.. Torsonal-Sprng Thn Spaal Beam Elemen The orsonal-sprng beam elemen developed n [8] s used for modelng hn beams. The elemen has pon mass ype nodes (nodes whch have only ranslaonal DOFs). A beam elemen s shown n fgure a The beam elemen connecs he pon p (md-pon of ) o pon p (md-pon of ). The slope of he beam a p s angen o and he slope of he beam a p s angen o. The beam elemen consss of wo russ sub-elemens ( p and p ) and a orsonal-sprng bendng sub-elemen ( p ˆ p ). The nernal force n a sub-russ elemen s gven by equaon (9). The nernal momen n he bendng sub-elemen s gven by: EI CI M = Δα + & α () L L where I s he cross-seconal effecve momen of nera, L s he oal un-sreched lengh of he bendng elemen whch s equal o he lengh of p plus p, and Δα s he change n angle beween p and p from he unsressed confguraon. Fgure b shows how a beam s dscrezed usng he -noded beam elemen. Ths hn beam elemen does no have a orsonal response along he axs of he beam. In addon, assumes ha he bendng momens of nera of he cross-secon around wo perpendcular crosssecon axes are he same. Node Md-sde pon Deformed elemen (a) Undeformed elemen p Truss Truss p Bendng sub-elemen Nodes (b) Δα Beam elemens Fgure. (a) -noded beam elemen; (b) fne elemen dscrezaon of a beam usng he -noded beam elemen. p Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

6 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS). Lumped Parameers Thck Spaal Beam Elemen A lumped parameer spaal beam elemen s used o model hck beams. The elemen has wo rgd body ype nodes. Therefore, each node has ranslaonal DOFs and he roaon marx s used o measure he spaal roaon of he node. The beam elemen local X-axs s n he drecon of he lne connecng he wo nodes. The Y and Z axes are normal o he X-axs and o each oher. They are defned as an average of he local Y and Z axes of he wo nodes. The elemen models each response componen ndependenly usng smpler elemens (fgure ): Axal response s modeled usng a zero-lengh lnear sprng-damper russ elemen a he cener pon beween he wo nodes. The nernal force along he X-axs for he russ elemen s gven by: F x EA CA = d x + d& () x l l where E s he Young s modulus, C s he dampng modulus, A s he cross-seconal area, l s he unsreched lengh of he beam and d x s he deflecon beween wo nodes along he beam s X-axs. Shear response s modeled usng wo zero-lengh lnear sprng-damper elemens locaed a he cener pon beween he wo nodes. One elemen produces a reacon force along he elemen Y-axs and he oher elemen produces a reacon force along he elemen Z-axs. The nernal forces for he Y-russ elemen along he Y-axs and for he Z-russ elemen along he Z-axs are gven by: F F y z GA CA = Κ y d y + d& y l l GA CA = Κ z d z + d& z l l (a) (b) where G s he shear modulus, K y and K z are he crosssecon shear facors, and d y and d z are he deflecons beween wo nodes along he beam s Y-axs and Z-axs, respecvely. Bendng response s modeled usng wo orsonal sprngdamper elemens locaed a he cener pon beween he wo nodes. One elemen produces he bendng response around he Y-axs and he oher elemen produces he bendng response around he Z-axs. The nernal momens for he Y orsonal elemen and for he Z orsonal elemen are gven by: M M EI CI yy yy y = α y + & α (a) y l l EI CI zz zz z = α z + & α (b) z l l where I yy and I zz are he second momens of nera of he cross-secon around he Y and Z axes, respecvely, α y and α z are he change n angles beween wo nodes around he beam s Y-axs and Z-axs, respecvely. Torsonal response s modeled usng a orsonal sprngdamper elemen locaed a he cener pon beween he wo nodes. The elemen produces he orsonal response around he X-axs. The nernal momens for he X orsonal elemen s gven by: M x EJ CJ = α x + & α () x l l where J s he orsonal momen of nera of he crosssecon, α x s he change n angle beween wo nodes along he beam s X--axs. The deflecons d x, d y and d z can be obaned as follows. The cener pon he elemen for each of he elemen nodes found ( XC, XC, =,,). Then, he vecor connecng he wo pons s found: U XC XC = (5) The do produc vecor U r wh he X-axs, Y-axs and Z- axs vecors of he beam gve d x, d y and d z, respecvely. The angle changes α x, α y and α z can be found usng he nodes roaon marces. The man feaures of he lumped parameers beam elemen are: I can model spaal beams ncludng bendng along he wo ransverse cross-secon drecons and orson. Shear deformaon s modeled. Snce he beam s modeled usng nodes havng boh ranslaonal and roaonal DOFs (herefore, hey can be hough of as rgd bodes), herefore, dealed D conac surfaces for he beam can be used. Ths allows he use of hs beam elemen o model he dealed geomery of rack segmens. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 5 of 8

7 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Y X Z All elemen srucural forces are algned wh he elemen edges, hus he elemen does no exhb membrane or membrane-warpng lockng. roaon rgd body modes ranslaon rgd body modes Fgure. Spaal lumped parameers -node beam elemen. Each node s modeled as a rgd body. The X, Y, Z local axes of each node are shown red, green and blue, respecvely. The elemen consss of russ elemens (for modelng he axal and shear response) and orsonal sprng elemens (for modelng he bendng and orson response) locaed a he cener pon of he elemen.. Naural-Modes Brck Elemen The 8-nodes naural-modes brck elemen developed n [9, ] s sraegcally desgned o model all he deformaon and rgd body modes (fgure ) of a brck elemen whle avodng lockng and spurous modes. All he elemen nodes are pon mass ype nodes wh only ranslaonal DOFs. Two ypes of sub-elemens are used o model he brck: wo-node russ elemen and four-node surface shear elemen. Twelve russ elemens along he welve edges of he elemen are used o model he membrane and bendng modes of he elemen. Sx surface shear elemens are nroduced a each of he sx elemen surfaces o model he shear and warpng modes of he elemen (fgure 5). The dervaons of he sffness characerscs and srucural forces generaed by he russ and he surface shear elemens are gven n. The brck elemen wh he followng characerscs: The deformaon elemen modes, ncludng, membrane, bendng, shear, and warpng are accuraely modeled. Thus he elemen does no have any spurous modes. The shear sresses are evaluaed as he average sresses over each of he elemen faces, hus he elemen does no exhb shear lockng. membrane (srech) modes 6 bendng modes asymmerc bendng modes shear modes warpng modes Fgure. rgd body and deformaon modes of a spaal 8-node brck elemen. russ elemens Twelve -node russ elemens Membrane and bendng modes are modeled usng russ sub-elemens along each of he welve elemen edges. 6 surface elemens Sx -node surface shear elemens Shear and warpng modes are evaluaed usng surface shear subelemens on each of he sx elemen surfaces. Fgure 5. Sub-elemens of he 8-node lumped-parameers brck elemen.. CONTACT MODEL The penaly echnque s used o mpose he normal conac consrans beween fne elemen nodes or pons on a rgd body and fne elemen surfaces or quadrlaeral surfaces of Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 6 of 8

8 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) rgd bodes [5, 6]. The frs sep s o fnd he poson and velocy of he conac nodes and pons. For fne elemen nodes he global poson x Gp and velocy x& Gp of a conac node relave o he global neral frame are readly avalable: x = x (6a) Gp Gp K x & = x& (6b) K where x and K x& K are he poson and velocy vecors of conac node K. For rgd bodes he global poson x Gp and velocy x& Gp of a conac pon are gven by: x = X + R x (7a) Gp BF BF j Lp j & = & (7b) x Gp XBF + RBF j ( WBF xlp) j where X BF and X & BF are he global poson and velocy vecors of he rgd body s frame, R BF s he roaon marx of he rgd body relave o he global reference frame, W BF s he rgd body s angular velocy vecor relave o s local frame, and x Lp s he poson of he conac pon relave o he rgd body s frame. The frconal conac force Fc a each conac pon/node (sum of he normal conac and angenal frcon forces) s ransferred as a force and a momen o he cener of he rgd body. The negave of hs force s ransferred o he conac surface elemen by dsrbung o he nodes formng he surface usng he elemen shape funcon: F = N F (8) k where N k are he surface elemen shape funcons a he conac pon and F k are he conac forces on node k of he surface elemen. In case he conac body s a rgd body, hen hs force can also be ransferred o he cener of he conacng rgd body as a force and momen: F Fc k c = (9a) M = xlp RBF Fc ) (9b) x Lp j ( j = RBF xgp XBF ) () j ( where F s he conac force a he CG of he conac rgd body (cener of he body frame), M n he conac momen on he conac rgd body, x Lcp s he poson of he conac pon relave o he rgd body s frame and x Gcp s he poson of he conac pon relave o he global reference frame. Thus, he conac algorhm suppors conac flexbleflexble, rgd-rgd and rgd-flexble body conac. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed.. Penaly Normal Conac Model The penaly echnque s used for mposng he consrans n whch a normal reacon force (Fnormal) s generaed when a node peneraes n a conac body whose magnude s proporonal o he peneraon dsance. In he presen formulaon, he force s gven by [5, 6]: F normal = c d& p Akp d + A sp cp d& Conac surface v r rel d& d& < () d & = vn n () v r v r n d n r Conac pon Conac body Fgure 6. Conac surface and conac node. where A s he area of he recangle assocaed wh he conac pon, kp and cp are he penaly sffness and dampng coeffcen per un area; d s he closes dsance beween he node and he conac surface (fgure 6); d & s he sgned me rae of change of d; sp s a separaon dampng facor beween and whch deermnes he amoun of sckng beween he conac node and he conac surface a he node (leavng he body); n r s he normal o he surface and v r he velocy vecor n he drecon of n r. The normal conac force vecor s gven by: F n = n Fnormal () The oal force on he node generaed due o he frconal conac beween he pon and surface s gven by: F po = F +. Aspery Frcon Model n F () An aspery-sprng frcon model s used o model jon and conac frcon [] n whch frcon s modeled usng a pece-wse lnear velocy-dependen approxmae Coulomb frcon elemen n parallel wh a varable anchor pon sprng. The model approxmaes aspery frcon where frcon forces beween wo rough surfaces n conac arse due o he neracon of he surface asperes. F s he n n s Page 7 of 8

9 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) angenal frcon conac force vecor ransmed o he conac body a he conac pon. I s gven by: F = Fangen (5) μk Fnormal Fangen search. A he nalzaon of he algorhm he followng seps are performed: Each slave polygonal conac surface s dvded no blocks of polygons. The boundng box for each block of polygons s found. Then each of hose blocks of polygons s dvded no blocks and agan he boundng boxes for hose blocks are found. Ths recursve dvson connues unl here s only one polygon n a box. Smple approxmae Coulomb frcon elemen vsk vangen For each maser conac surface he conac pons are dvded no blocks. The boundng sphere for each block of pons s found. Then, each of hose blocks of pons s dvded no blocks and agan he boundng spheres for hose blocks are found. Ths recursve dvson connues unl here s only one pon (wh a boundng sphere of radus ). Sprng wh a varable anchor pon. Fgure 7. Aspery sprng frcon model. Fangen The aspery frcon model s used along wh he normal force o calculae he angenal frcon force (Fangen) [7]. When wo surfaces are n sac (sck) conac, he surface asperes ac lke angenal sprngs. When a angenal force s appled, he sprngs elascally deform and pull he surfaces o her orgnal poson. If he angenal force s large enough, he surface asperes yeld (.e. he sprngs break) allowng sldng o occur beween he wo surfaces. The breakaway force s proporonal o he normal conac pressure. In addon, when he wo surfaces are sldng pas each oher, he asperes provde ressance o he moon ha s a funcon of he sldng velocy and acceleraon, and he normal conac pressure. Fgure 7 shows a schemac dagram of he aspery frcon model. I s composed of a smple pece-wse lnear velocy-dependen approxmae Coulomb frcon elemen (ha only ncludes wo lnear segmens) n parallel wh a varable anchor pon sprng.. Conac Search Conac deecon s performed beween conac pons on a maser conac surface and a polygonal surface called he slave conac surface. The conac pons of he maser conac surface can eher be pon mass nodes or pons on a conac surface of a rgd body ype node. The slave conac surface can be a polygonal surface connecng pon mass ype nodes or a polygonal surface on a rgd body ype node. Conac beween he conac pons of he maser surface and he polygons of he slave surface s deeced usng a bnary ree conac search algorhm whch allows fas conac Durng he soluon he followng seps are performed. For each maser conac sphere, he radus of he conac sphere s added o he sze of he boundng box, and hen we check f he cener pon of he sphere s nsde a boundng box. If he cener of he conac sphere s no nsde any boundng box, hen all he pons nsde ha sphere are no n conac wh he surface. If he cener of he conac sphere s nsde a boundng box hen he wo sub-boundng boxes are checked o deermne f he pon s nsde eher one. If s, hen he sub-conac spheres are checked. If a conac pon s found o be nsde he lowes level boundng box, hen a more compuaonally nensve conac algorhm beween a pon and a polygon s used o deermne he deph of conac and he local poson of he conac pon on he polygon. Ths search algorhm has a heorecal average compuaonal cos of log(m) log(n), where m s he number of pons of he maser surface and n s he number of polygons of he slave surface. I allows deecng conac beween surfaces conanng mllons of polygons and conac pons n near real-me. 5. JOINT CONSTRAINTS Each rgd body can have a number of connecon pons. A connecon pon s a pon on he body where jons can be locaed. A connecon pon does no add addonal DOFs o he sysem.the connecon pon can be: A pon mass ype node. A pon on a rgd body. An arbrary pon nsde a fne elemen. 5. Connecon pon locaon If he connecon pon s a node hen equaons (6a) and (6b) are used o fnd he global poson x Gp and velocy Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 8 of 8

10 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) x& Gp of he connecon pon. If he connecon pon s a fxed pon on rgd body B hen equaons (7a) and (7b) are used o fnd he global poson and velocy of he connecon pon. If he connecon pon s a pon nsde a fne elemen, hen x Gp and velocy x& Gp are gven by: x = N (ξ ) x (6) Gp Gp J J l l J x & = N (ξ ) x& (7) where J s he local node number of he elemen, N J ( ξ l ) are he nerpolaon funcons of he elemen, ξ are he naural l elemen coordnaes of he fxed pon and x s he poson J vecor of local node J of he elemen relave o he global reference frame. 5. Sphercal jon consran force A jon s defned by defnng he relaon beween connecon pons. For example, a sphercal jon beween wo connecon pons s defned as: xc xc J = (8) where xc s he poson vecor of he frs pon and xc s he poson vecor of he second pon. Ths consran s mposed usng he penaly echnque as: F c = kp v + cp vv& (9) v xc xc = () v& x& c x& c = () F c = Fc v () where Fc s he penaly reacon force on he connecon pon, kp s he penaly sprng sffness, and cp s he penaly dampng. The consran force s appled on he wo connecon pons n oppose drecons. Dependng on he ype of connecon pon he consran force s appled as follows. If he connecon pon s a pon on a rgd body, hen s ransferred o he node a he cener of he body as a force and a momen usng equaons (a) and (b). If he connecon pon s a node, hen he consran force s appled drecly o he node: F = F () K c If he connecon pon s a pon nsde a fne elemen, hen s appled o he nodes of he elemen usng: F = N (ξ ) F () J J l c where J s he local node number of he elemen, F s he J force on local node J of he elemen relave o he global reference frame. Usng equaons (6, 7, and 6-), he followng ypes pn jons can be modeled: Sphercal -jon beween wo rgd bodes. Sphercal-jon beween a rgd body and a fne elemen pon. Sphercal-jon beween a rgd body and a pon parcle ype node. Sphercal-jon beween wo elemen pons. Sphercal-jon beween an elemen pon and a pon parcle ype node. Sphercal-jon beween wo nodes. The consran forces are appled o he connecon pon node(s) by assemblng hem no he global srucural forces Fs n equaon (). Also, he consran momens are appled o he nodes by assemblng hem no he global srucural orques Ts n equaon (). Revolue jons can be modeled by placng wo sphercal jons along a lne. Oher ypes of jons such as prsmac, cylndrcal, unversal and planar jons can also be modeled by wrng he consran equaon, hen wrng he correspondng penaly forces and momens on he connecon pons. 6. EXPLICIT SOLUTION PROCEDURE The soluon felds for modelng mulbody sysems are defned a he model nodes. Noe ha a rgd body modeled as one fne elemen node. These soluons felds are: Translaonal posons. Translaonal veloces. Translaonal acceleraons. Roaon marces. Roaonal veloces. Roaonal acceleraons. The explc me negraon soluon procedure predcs he me evoluon of he above response quanes. An advanage of explc soluon procedures s ha hey are embarrassngly parallel. The procedure descrbed below acheves near lnear speed-up wh he number of processors on shared memory parallel compuers. The procedure s mplemened n a mulbody dynamcs fne elemen code and s oulned below: - Prepare he run: a. Se he nal condons for he soluon felds denfed above. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 9 of 8

11 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) b. Creae a ls of all he fne elemens (Those also nclude jons and maser conac surfaces whch are consdered elemens). c. Creae a ls of elemens ha wll run on each processor. Ths s done usng an algorhm whch res o make he compuaonal cos on each processor equal. d. Creae a ls of all he prescrbed moon consrans. e. Calculae he sold masses for each fne elemen node by loopng hrough he ls of fne elemens. Noe ha he masses are fxed n me. f. Loop over all he elemens and fnd he mnmum me sep for he explc soluon procedure. - Loop over he soluon me and ncremen he me by Δ each sep whle dong he followng: a. Se he nodal values a he las me sep o be equal o he curren nodal values for all soluon felds. b. Do eraons (a predcor eraon and a correcor eraon) of he followng:. Inalze he nodal forces and momens o zero.. Calculae he nodal forces and momens by loopng hrough all he elemens (and jons) whle calculang and assemblng he elemen nodal forces. Ths s he mos compuaonal nensve sep. Ths sep s done n parallel by runnng each ls of elemens denfed n sep.c on one processor.. Fnd he nodal values a he curren me sep usng he sem-dscree equaons of moon and he rapezodal me negraon rule (Eqs. -5). v. Execue he prescrbed moon consrans whch se he nodal value(s) o prescrbed values. v. Go o he begnnng of sep. o follow a ceran vercal dsplacemen me-hsory. Thus, he acuaors can be conrolled n such a way as o smulae vercal poson me-hsores of he wheels as he vehcle goes over a rough road ha has bumps and poholes. Noe ha he moon smulaor canno smulae he neral cenrfugal nera forces ha arse due o he me-varyng moon of he vehcle on he road. Fgure 8. Snapsho of he compuaonal model of he racked vehcle on he n-pos moon smulaor. 7. EXPERIMENTAL SETUP The model s valdaed usng a full-scale army racked vehcle. The rack was removed and he vehcle was placed on an n-pos moon base smulaor n he US army s Tank Auomove Research, Developmen and Engneerng Cener s (TARDEC) Smulaon Laboraory (TSL) (see fgures and 8). The n-pos moon smulaor consss of lnear hydraulc acuaors each placed under one of he vehcle road wheels. The road-wheels are conneced o he road arms usng revolue jons. The vehcle suspenson sysem consss of he road arms and orson bars beween he roads arms and chasss. The road arms are conneced o he chasss usng revolue jons. Each road arm s free o roae beween wo hard suspenson sops a degrees (when he arm s horzonal) and abou 6 degrees from he horzonal. Each acuaor has one degree of freedom along he vercal drecon and can be ndependenly commanded Fgure 9. Dshpans wh fron and back cylndrcal sops and sde guard rals. The vehcle n our expermenal seup has road wheels, 6 on each sde. We number he wheels from fron o back hrough 6. Each hydraulc acuaor s composed of a cylnder, pson and a dshpan on whch he wheel ress. The dshpans for wheels,, and 6 (colored slver) have fron Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

12 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) and back horzonal cylndrcal sops as well as sde guard rals (fgure 9). Those dshpans are used n order o ensure ha he vehcle does no slde off he moon smulaor. Fla dshpans are used for wheels and 5 (fgure ). In addon, a srap harness was aached o he op of he vehcle durng he expermens for safey reasons. The harness was loose so as no o nerfere wh he moon of he vehcle. excaon he fron and rear acuaors are 8 o ou of phase. The nermedae acuaors are moved n such as way ha a, a pon n me, he lne connecng he cener of he fron dshpan and cener of he rear dshpan passes hrough he cener of he nermedae dshpans. Ths way he vehcle s roang around s pch axs. The pch and shake excaons smulae he vehcle gong over bumps and poholes. For each moon ype, harmonc excaons a ampludes, 6 or 7 mm and frequences from.6 o.5 Hz are appled o he vehcle. Each excaon s appled for abou 5 sec wh he nal sec and he las sec used o ramp up and down he excaon amplude. Also, for each ramp moon ype, a ramp excaon of 9 mm s appled o he vehcle, where he vehcle s rased 9 mm n. sec, hen he acuaors dwell for 5 sec, fnally he vehcle s lowered 9 mm n. sec. 7. Wheel. Wheels and Torque (N.m).. Momen (N.m) Angle (Rad) Angle (Rad.) 9. Wheels and 5 7. Wheel Fgure. Fla dshpans. The angles of he roads arms are measured and sampled a a rae of 5 samples/sec. The expermen road arms angles measuremens are compared wh he resuls predced usng he compuaonal model. A camera mouned on he ground s used o record he moon of he vehcle. The camera s se o capure frames/sec. The camera vew s shown n fgure 8. The orson bars a he road wheels were charged such ha he road arms are nally half way beween he wo suspenson sops (.e. a abou degrees from he horzonal) when he vehcle s a res under gravy and he rack s off. Fgure shows he roaonal sffness (orque versus angle) of he orson bar a each road wheel. The orson bars roaonal sffness was expermenally obaned. Momen (N.m) Angle (rad.) Angle (rad.) Fgure. Expermeally measured roaonal sffness (orque versus angle) of he orson bar a each road wheel. Table. Marx of he racked vehcle expermens (8 expermens). Momen (N.m) The marx of excaons appled o he vehcle s shown n Table. Two man ypes of excaons are used: shake and pch. In he shake excaon all he acuaors are movng he same way (.e. her moons are n phase). In he pch Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

13 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) 8. VALIDATION STUDY The expermens n Table were smulaed usng he presen model. Frs he expermens were used o fnd an average dampng momen versus angular velocy for he suspenson sysem s orson bars (fgure ). Ths average dampng momen versus angular velocy curve was used for all he orson bars. The dampng momen versus angular velocy ypcally depends on he ol charge of he orson bar. Noe ha he curve s nonlnear. For very small angular veloces, he enson bar produces a large amoun of dampng. For larger values of angular velocy, he dampng momen s nearly lnear wh angular velocy, and gradually devaes from lneary and becomes sffer for very large angular veloces. Also, from he expermens we found ha here s small frcon momen n he enson bars of abou 5 N.m. Noe ha he presen model accouns for non-lnear sffness, dampng and frcon of he suspenson sysem. Also, he model allows he wheels o leave he dshpans. amplude of moon of he road arms exceeded ±6 o. A hose ampludes, he wheels where bumpng and clmbng over he dshpans cylndrcal sops. The expermenal resuls for hose runs canno be accuraely modeled because he poson of a wheel relave o he dshpans sops s no known and vares beween runs. For hose excaons he dfference beween he expermen and smulaon was on he average abou % wh a maxmum dfference of abou 5%. Also, noe ha he range of moon of a road arm s abou ±9 o, bu he hghes moon amplude ha we esed was abou ± degrees. We could no exce he vehcle beyond ± degrees road arm deflecons for safey reasons snce he wheels sared bumpng and gong over he sops a ±6 o. Asymmery of he sffness and dampng beween he wo sdes of he vehcle. The dfference beween he response a he wo sdes of he vehcle was on average abou 8%. Presence of he srap harness. Fgure. Dampng momen versus angular velocy for he orson bars. The graphs n Fgures - show he resuls of represenave runs from he se of 8 runs n he es marx shown n Table. Those fgures show comparsons of he road arm angles beween he expermen and he DIS smulaon. The average dfference n magnude beween he expermen and smulaon s abou 5%. The average response dfference beween he rgh and lef sdes of he vehcle was also abou 5%. The man sources of error beween he expermens and smulaon n order of mporance are: A he hgher frequency shake excaons (.,.5 Hz) and pch excaons (.,. and. Hz), he 9. TYPICAL SIMULATIONS The presen model can be used o predc he dynamc response of racked vehcle. Ths ncludes predcng he followng: Sably of he vehcle whle movng a a ceran speed over a rough erran or whle seerng. Maxmum posve or negave obsacles ha he vehcle can overcome. Ths ncludes nal speed and engne orque requred o overcome he obsacle. Inernal forces n he varous componens of he vehcle ncludng he rack whle gong over varous errans and obsacles. Those forces can hen be appled o fner fne elemen models of he componens for dealed sress analyss. The predced sresses can be used o deermne he wheher he componen wll fal under he loads as well as he fague lfe of he componen. Vehcle energy and power requremens for gong a a prescrbed speed over a known course. Furher expermenal ess are needed o valdae and une he model parameers o accuraely predc he above response quanes. Those expermens should nclude acual road ess usng an nsrumened racked vehcle. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

14 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Dsplacemen [mm] Inpu Moon: Acuors Dsplacemens vs. Tme Tme [s] L R L R L R L R L5 R5 L6 R6-8 Fgure. Inpu moon: harmonc shake. Hz 7 mm. Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L5 Expermenal R5 Expermenal L5 Numercal R5 Numercal Angles vs. Tme L6 Expermenal R6 Expermenal L6 Numercal R6 Numercal Tme [s] Tme [s] Fgure. Expermen and smulaon angles of he road arms versus me for npu moon: harmonc shake. Hz 7 mm. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

15 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Dsplacemen [mm] Dsplacemens vs. Tme Tme [s] L R L R L R L R L5 R5 L6 R6-8 Fgure 5. Inpu moon: harmonc shake.5 Hz 7 mm. 5 Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal 5 Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal 5 Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L5 Expermenal R5 Expermenal L5 Numercal R5 Numercal 5 Angles vs. Tme L6 Expermenal R6 Expermenal L6 Numercal R6 Numercal Tme [s] - Tme [s] Fgure 6. Expermen and smulaon angles of he road arms versus me for npu moon: harmonc shake.5 Hz 7 mm. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page of 8

16 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Dsplacemen [mm] Inpu Moon: Acuaors Dsplacemens vs. Tme Tme [s] L R L R L R L R L5 R5 L6 R6-8 Fgure 7. Inpu moon: harmonc pch.8 Hz 7 mm. 6 Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Angles vs. Tme L5 Expermenal R5 Expermenal L5 Numercal R5 Numercal Tme [s] Angles vs. Tme Tme [s] - Fgure 8. Expermen and smulaon angles of he road arms versus me for npu moon: harmonc pch.8 Hz 7 mm. L6 Expermenal R6 Expermenal L6 Numercal R6 Numercal Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 5 of 8

17 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Dsplacemen [mm] 8 6 Dsplacemens vs. Tme L R L R L R L R L5 R5 L6 R6 Tme [s] Fgure 9. Inpu moon: ramp shake 9 mm. Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Angles vs. Tme L Expermenal R Expermenal L Numercal R Numercal Tme [s] Tme [s] Angles vs. Tme L5 Expermenal R5 Expermenal L5 Numercal R5 Numercal Angles vs. Tme L6 Expermenal R6 Expermenal L6 Numercal R6 Numercal Tme [s] Tme [s] Fgure. Expermen and smulaon angles of he road arms versus me for npu moon: ramp shake 9 mm. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 6 of 8

18 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) (a) (b) Fne elemen models of a ypcal racked vehcle, wh a connuous bel-ype rack, are shown n Fgures and. In hose models he chasss, road arms, road wheels, dlers and sprockes are modeled usng rgd bodes. The suspenson roary sprng-dampers are modeled as orsonal sprngs. Roary acuaors along wh conrol laws are used o model he engne and seerng sysem of he vehcle. In Fgure he rack s modeled usng brck elemens for he rubber marx and hn beam for he renforcemens. In Fgure he rack s modeled usng he hck beam elemens presened n Secon.. Fgures a shows a snapsho of he racked vehcle gong over sem-crcular bumps. Fgure b shows a snap sho of he racked vehcle gong over a rough course. Fgure shows a snapsho of he racked vehcle undergong a lane-change. Fgure. Snapshos of he moon of he racked vehcle wh a connuous bel rack. The rack s modeled usng brck elemens for modelng he bel rubber and hn beam elemens for modelng he renforcemens. (a) vehcle gong over.sem-crcular bumps; (b) vehcle gong over a rough course. Fgure. Snapsho of he moon of he racked vehcle wh a connuous bel rack modeled usng hck beam elemens.. CONCLUDING REMARKS A fne elemen model for predcng he fully coupled dynamc response of flexble mulbody sysems for racked vehcles was presened. The model has he followng characerscs: Parallel explc me-negraon solver. Algorhm for accurae accounng for he rgd body roaonal moon. The rgd bodes roaonal equaons of moon are wren n a body-fxed frame wh he oal rgd body roaon marx updaed each me sep usng ncremenal roaons. Toal-Lagrangan lumped parameers D fne elemens ncludng sprng/russ, hn beam, hck beam, hck shell, and sold fne elemens. Penaly echnque for modelng jon consrans ncludng sphercal, revolue, cylndrcal and prsmac jons. Penaly echnque for modelng normal conac. An aspery frcon model s used o model jon and conac frcon. General fas herarchcal boundng box-boundng sphere conac search algorhm for fndng he conac peneraon beween pons on maser conac surfaces and polygons on slave conac surfaces. Connuous bel-ype racks are modeled usng brck elemens represenng he rubber marx and beam elemens along he wdh and crcumference of he bel/segmen for modelng he bel renforcemens. Connuous or segmened racks are modeled hck beam elemens. A valdaon sudy of he fne elemen model was carred ou usng a racked vehcle (whou he rack) mouned on Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 7 of 8

19 Proceedngs of he Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) an n-pos moon smulaor. The sudy shows ha he model can predc reasonably well, whn 5% dfference on average, he response of he vehcle. The maxmum dfference beween he expermens and smulaon was 5% for he large frequency excaons, manly due o he wheels clmbng over he dshpan sops n he expermens. ACKNOWLEDGEMENTS Suppor for hs work provded by he US Army RDECOM- TARDEC, Warren, MI under SBIR gran number W56HZV5C6 s graefully acknowledged. Reference heren o any specfy commercal company, produc, process or servce by rade name, rademark, manufacurer, or oherwse, does no necessarly consue or mply s endorsemen, recommendaon, or favorng by he Uned Saes Governmen or he deparmen of he Army (DoA). The opnons of he auhors expressed heren do no necessarly sae of reflec hose of he Uned Saes Governmen or he DoA, and shall no be used for adverzng or produc endorsemen purposes. REFERENCES [] T.M. Wasfy and A.K. Noor, Compuaonal sraeges for flexble mulbody sysems, Appled Mechancs Revews, vol 56(6), pages 55-6,. [] T.M. Wasfy and J. O Kns, Fne Elemen Modelng of he Dynamc Response of Tracked Vehcles, ASME DETC , ASME 9 Inernaonal Desgn Engneerng Techncal Conferences, 7 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, San Dego, CA, Augus 9. [] T.M. Wasfy and A.K. Noor, Modelng and sensvy analyss of mulbody sysems usng new sold, shell and beam elemens, Compuer Mehods n Appled Mechancs and Engneerng, Vol. 8(-) (5h Annversary Issue), pp. 87-, 996. [] T.M. Wasfy, Modelng spaal rgd mulbody sysems usng an explc-me negraon fne elemen solver and a penaly formulaon, ASME Paper No. DETC-575, 8 h Bennal Mechansms and Robocs Conference, DETC, Sal Lake, Uah. [5] M.J. Leamy and T.M. Wasfy, Transen and seady-sae dynamc fne elemen modelng of bel-drves, ASME Journal of Dynamcs Sysems, Measuremen, and Conrol, Vol. (), pp ,. [6] T.M. Wasfy and M.J. Leamy, Modelng he dynamc frconal conac of res usng an explc fne elemen code, ASME DETC5-869, 5 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, ASME DETC, Long Beach, CA, 5. [7] T.M. Wasfy, Aspery sprng frcon model wh applcaon o bel-drves, Paper No. DETC-8, Proceedng of he DETC: 9 h Bennal Conference on Mechancal Vbraon and Nose, Chcago, IL,. [8] T.M. Wasfy, A orsonal sprng-lke beam elemen for he dynamc analyss of flexble mulbody sysems, Inernaonal Journal for Numercal Mehods n Engneerng, Vol. 9(7), pp , 996. [9] T.M. Wasfy, Edge projeced planar recangular elemen for modelng flexble mulbody sysems, 9h Bennal Conference on Mechancal Vbraon and Nose, Paper No. DETC-85, 9 h Bennal Conference on Mechancal Vbraon and Nose, ASME Inernaonal DETC, Chcago, IL,. [] T.M. Wasfy, Lumped-parameers brck elemen for modelng shell flexble mulbody sysems, 8 h Bennal Conference on Mechancal Vbraon and Nose, ASME Inernaonal DETC, Psburgh, PA,. Expermenal Valdaon of a Mulbody Dynamcs Model of he Suspenson Sysem of a Tracked Vehcle, Wasfy, e al. Unclassfed: Dsrbuon Saemen A. Approprae for publc release; dsrbuon s unlmed. Page 8 of 8