AC : FLEXIBLE MULTIBODY DYNAMICS EXPLICIT SOLVER FOR REAL-TIME SIMULATION OF AN ONLINE VIRTUAL DYNAMICS LAB

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1 AC : FLEXIBLE MULTIBODY DYNAMICS EXPLICIT SOLVER FOR REAL-TIME SIMULATION OF AN ONLINE VIRTUAL DYNAMICS LAB Mr. Haem M. Wasfy, Advanced Scence and Auomaon Corp. Haem Wasfy s he Presden of Advanced Scence and Auomaon Corp. (ASA) a company ha specalzes n he developmen of onlne vrual learnng envronmens, and advanced engneerng smulaons. He has helped desgn several neracve learnng envronmens ha nclude a CNC machnng course, a cenrfugal pump manenance course, an undergraduae physcs course, and a weldng course. He receved a B.S. (1994) and an M.S. (1996) n mechancal engneerng from he Amercan Unversy n Caro. Wasfy s research neress nclude advanced learnng sysems, cavaon modelng, compuaonal flud dynamcs, nernal combuson engne modelng and desgn, and AI rule-based exper sysems. Dr. Tamer M. Wasfy, Indana Unversy-Purdue Unversy, Indanapols Tamer Wasfy receved a B.S. (1989) n mechancal wngneerng and an M.S. (1990) n maerals engneerng from he Amercan Unversy n Caro, and an M.Phl. (1993) and Ph.D. (1994) n mechancal engneerng from Columba Unversy. He worked as a Research Scens a he Deparmen of Mechancal Engneerng, Columba Unversy ( ) and a he Unversy of Vrgna a NASA Langley Research Cener ( ). Wasfy s an Assocae Professor a he Mechancal Engneerng Deparmen a Indana Unversy-Purdue Unversy Indanapols (IUPUI). Dr. Wasfy s also he founder and charman of Advanced Scence and Auomaon Corp. (founded n 1998) and AscenceTuor (founded n 2007). Wasfy s research and developmen areas nclude: flexble mulbody dynamcs, fne elemen modelng of solds and fluds, flud-srucure neracon, bel-drve dynamcs, res mechancs/dynamcs, ground vehcle dynamcs, vsualzaon of numercal smulaon resuls, engneerng applcaons of vrual-realy, and arfcal nellgence. He auhored and co-auhored more han 70 peer-revewed publcaons and gave more han 65 presenaons a nernaonal conferences and nved lecures n hose areas. He receved wo ASME bes conference paper awards as frs auhor. He s he sofware archec for he DIS, IVRESS and LEA sofware sysems, whch are used by ndusry, governmen agences, and academc nsuons. Wasfy s a member of ASME, AIAA, SAE, and ASEE. Ms. Jeanne Peers, Advanced Scence and Auomaon Corp. Jeanne Peers receved a B.A. n mah/compuer scence from he College of Wllam and Mary. She worked a NASA Langley Research Cener n Hampon, Va. for more han 20 years as a Senor Programmer/Analys for George Washngon Unversy, Unversy of Vrgna, and Old Domnon Unversy. She co-auhored more han 70 journal and conference papers n he areas of compuaonal mechancs, fne elemen mehod, shells/plaes, compose maeral panels, and res. She has also worked on numerous projecs o creae advanced engneerng desgn and learnng envronmens whch nclude mulmodal user nerfaces for space sysems. As Vce Presden of Informaon Technology, Peers drecs he developmen of advanced vrual realy applcaons, ncludng scenfc vsualzaon applcaons and web-based mulmeda educaon/ranng applcaons. c Amercan Socey for Engneerng Educaon, 2012 Page

2 Flexble Mulbody Dynamcs Explc Solver for Real-Tme Smulaon of an Onlne Vrual Dynamcs Lab Absrac A hgh-fdely onlne vrual vdeo-game-lke Newonan dynamcs lab ha can be used as a eachng lab for unversy freshman physcs and sophomore engneerng dynamcs courses s presened n hs paper. The lab s drven usng a vrual-realy dsplay engne wh an negraed flexble mulbody dynamcs fne elemen explc real-me solver. The herarchcal scenegraph represenaon of he lab ha s used n he vrual-realy dsplay engne s also used n he solver. The mulbody sysem ncludes rgd bodes, flexble bodes, jons, frconal conac consrans, acuaors and conrollers. Flexble bodes are modeled usng sprng, russ and beam elemens. A penaly echnque s used o mpose jon/conac consrans. An aspery-based frcon model s used o model jon/conac frcon. A boundng box bnary ree conac search algorhm s used o allow fas conac deecon beween fne elemens and oher elemens as well as general rangular/quadrlaeral rgd-body surfaces. The followng expermens are modeled: mass-sprng sysems, pendulums, pulley-rope-mass sysems, ar-hockey, bllards, 1D and 2D frconal and frconless moon wh and whou gravy, roller-coasers, planeary moon, gears, cams, roboc manpulaors and lnkages. 1. Inroducon A flexble mulbody sysem s a sysem of nerconneced rgd and/or flexble bodes. The bodes are conneced usng varous ypes of jons ncludng sphercal, revolue, cylndrcal, prsmac, planar and screw jons. The bodes can come no conac wh one-anoher or wh he surroundngs. Vrual-realy applcaons requre a real-me mulbody dynamcs engne n order o allow users o nerac wh objecs n he vrual-envronmen (VE) n a physcally accurae way. In he presen paper we presen he applcaon of an advanced real-me flexble mulbody dynamcs solver o smulae a dynamcs lab ha can be used n freshmen unversy physcs and sophomore engneerng dynamcs courses. The suden can nerac wh he vrual lab expermens usng a user-frendly nerface. For example, a ypcal Newon 2 nd law expermen conssng of wo vercally suspended masses conneced by a rope and pulley s shown n Fgure 1. The suden can se he value of he masses, hen release he masses and observe he moon of he masses. The moon of he masses s also ploed n graph. The suden can copy he graph daa o a spreadshee o do furher analyss and o wre a lab repor. Anoher ypcal expermen s of a puck movng on 1-dmensonal ral s shown n Fgure 2. The suden can do varous expermens by seng he nclnaon angle of he ral, he frcon coeffcen beween he ral and he puck, he nal poson of he puck on he ral, and he nal velocy of he puck. For example, by seng he nclnaon angle and he frcon coeffcen o zero, he suden can observe he moon of he puck due o an nal velocy. In hs case he expermen can be used o prove Newon s frs law of moon. Alernavely, he suden can se he nclnaon angle of he ral o 30 degrees and sudy he moon of he puck under he acon of gravy. Then he suden can se he frcon coeffcen o say 0.3 and fnd he nclnaon angle a whch he puck sars movng, whch s he frcon angle. Page

3 Fgure 1. Two mass rope-pulley expermen used o llusrae Newon s 2 nd law. Fgure 2. Gravy expermen of a puck movng on an nclned rack. The leraure on compuaonal echnques used for modelng flexble mulbody sysems s very large. A revew of hs leraure up o 2003 s gven n Reference 1. Varous echnques have been developed o enable real-me smulaon of flexble mulbody sysems. Those nclude: he use of symbolc and sem-analycal echnques o formulae he equaons of moon 2-4, he use of relave coordnaes and recursve formulaons 5,6, he use of parallel GPUs 7, and he use of herarchcal boundng volumes for fas conac deecon 8. The explc me-negraon mulbody dynamcs solver used n he presen paper has he followng feaures: Explc me-negraon soluon procedure 9. The explc soluon procedure has he followng feaures: o For a sable soluon, he soluon me sep mus be less han a crcal me sep. The soluon speed can be conrolled usng he crcal me sep. The crcal me sep can be ncreased by reducng some jons/elemens sffness or by ncreasng some of he bodes nera. Thus, a radeoff can be acheved beween soluon accuracy and compuaonal speed. o The soluon cos per me sep s lnearly proporonal o he number of elemens. o The soluon procedure s embarrassngly parallel. Algorhm for accurae accounng for he rgd body roaonal moon 10. In hs algorhm, he oal roaon marx relave o he neral frame s used o measure he roaon he rgd bodes. The oal body roaon marx s updaed usng an ncremenal roaon marx correspondng o ncremenal roaons angles, whch are obaned by negrang he roaonal equaons of moon. Toal-Lagrangan lumped parameers 3D fne elemens ncludng sprng/russ, hn beam and hck beam elemens 9, Penaly echnque for modelng jon consrans ncludng sphercal, revolue, cylndrcal and prsmac jons 11. Penaly echnque for modelng normal conac Frconal conac modeled usng an accurae and effcen aspery-based frcon model 17. Page

4 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 12. Herarchcal objec-orened framework Ths paper s organzed as follows. In Secon 2 he ranslaonal and roaonal sem-dscree equaons of moon are presened. In Secon 3 he fne elemen formulaons for he russ and beam are presened. In Secon 4 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, we descrbe how he mulbody dynamcs solver s negraed n he herarchcal objec-orened framework of he vrual-realy dsplay engne for dsplayng and nerfacng wh he vrual dynamcs lab. In Secon 7 we demonsrae he applcaon of he real-me mulbody dynamcs solver o smulang varous expermens n a ypcal dynamcs lab. Fnally, concludng remarks are gven n Secon Equaons of Moon 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. Two ypes of fne elemen nodes are used: pon parcle nodes and rgd body nodes. Pon parcle nodes have 3 ranslaonal DOFs (degrees-of-freedom). The echnque for wrng and negrang he equaons of moon for spaal rgd bodes usng an explc fne elemen code was presened n Reference 10. In hs algorhm, a rgd body s modeled usng a fne elemen node locaed a s cener of mass. The node has 3 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 3 and 4 parameer roaon measures 1. 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 (1) K K s K a K where s he runnng me, K s he global node number (no summaon over K; K=1 N where N s he oal number of nodes), s he coordnae number (=1,2,3), 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 Page

5 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 o respec o he body frame. The orenaon of he body-frame s gven by R whch s he roaon K marx relave o he global neral frame a me 0. The roaonal equaons of moons are wren for each node wh respec o s body-fxed maeral frames as: s K a K (& θ K ( I & θ ) K I & θ = T + T ) (2) 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=1,2,3), TsK are he componens of he vecor of nernal orque a node K n drecon, and TaK 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. The rapezodal rule s used as he me negraon formula for solvng Equaon (1) for he global nodal posons x: x& = x& (&& x + & x ) (3a) x = x ( x& + x& ) (3b) where s he me sep. The rapezodal rule s also used as he me negraon formula for he nodal roaon ncremens: & θ = & θ (&& θ + & θ ) (4a) θ = 0.5 ( & θ + & θ ) (4b) where θ 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 = R K R( θ ) K where R( θ K ) s he roaon marx correspondng o he ncremenal roaon angles from Equaon (4b). (5) The explc soluon procedure used for solvng Equaons (1-5) 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): f ({ x}) 0 (6) Jon consrans (Secon 6): f ({ x}) = 0 (7) Prescrbed moon consrans: f ({ x}, ) = 0 (8) Page

6 3. Fne Elemens 3.1 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& 0 ) + (9) l0 l0 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 0 s he un-sreched lengh of he russ. 3.2 Thn Spaal Beam Elemen The orsonal-sprng beam elemen developed n Reference 11 s used for modelng hn beams. The elemen has 3 pon mass ype nodes (nodes whch have only ranslaonal DOFs). A beam elemen s shown n Fgure 3a The beam elemen connecs he pon p 1 (md-pon of 12) o pon p 2 (md-pon of 23). The slope of he beam a p 1 s angen o 12 and he slope of he beam a p 2 s angen o 23. The beam elemen consss of wo russ sub-elemens ( p 1 2 and 2 p ) and a 2 orsonal-sprng bendng sub-elemen ( p 12ˆ p ). The nernal force n a sub-russ elemen s gven 2 by Equaon (9). The nernal momen n he bendng sub-elemen s gven by: EI CI M = α + & α (10) L0 L 0 where I s he cross-seconal effecve momen of nera, L 0 s he oal un-sreched lengh of he bendng elemen whch s equal o he lengh of p 1 2 plus 2 p, and α s he change n angle 2 beween p 1 2 and 2 p from he unsressed confguraon. Fgure 3b shows how a beam s 2 dscrezed usng he 3-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 cross-secon axes are he same. Node Md-sde pon Deformed elemen (a) Undeformed elemen p 2 2 α 3 p 1 Truss Truss p 2 1 Bendng sub-elemen 3 Nodes (b) Beam elemens Fgure 3. (a) 3-noded beam elemen; (b) fne elemen dscrezaon of a beam usng he 3- noded beam elemen. Page

7 4. Conac 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 rgd bodes 10, 15. 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 Gp = x (16a) x & Gp = x& K (16b) where x and K x& are he poson and velocy vecors of conac node K. For rgd bodes he K global poson x Gp and velocy x& Gp of a conac pon are gven by: x Gp = XBF + RBF j xlp j (17a) x & Gp = X BF + RBF ( WBF xlp) (17b) 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: Fk = N k Fc (18) 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 = (19a) K Fc ( xlp RBF j Fc ) Lp j RBF j ( xgp XBF j j M = (19b) x = ) (20) 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 flexble-flexble, rgd-rgd and rgdflexble body conac. 4.1 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 15, 16 : Page

8 F normal = Ak p cp d& d + A sp cp d& d& 0 d& < 0 (21) d & = vn n (22) Conac surface v r d n r Conac pon v r rel v r n Conac body Fgure 4. 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 4); d & s he sgned me rae of change of d; sp s a separaon dampng facor beween 0 and 1 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 n s he velocy vecor n he drecon of n r. The normal conac force vecor s gven by: F n = n Fnormal (23) The oal force on he node generaed due o he frconal conac beween he pon and surface s gven by: F po n = F + Fn (24) 4.2 Aspery Frcon Model An aspery-sprng frcon model s used o model jon and conac frcon 17 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 angenal frcon conac force vecor ransmed o he conac body a he conac pon. I s gven by: F = Fangen (25) Page

9 Fangen µk Fnormal 0 Smple approxmae Coulomb frcon elemen vsk vangen Fangen Sprng wh a varable anchor pon. Fgure 5. Aspery sprng frcon model. The aspery frcon model s used along wh he normal force o calculae he angenal frcon force (Fangen) 17. 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 5 shows a schemac dagram of he aspery frcon model. I s composed of a smple pece-wse lnear velocydependen approxmae Coulomb frcon elemen (ha only ncludes wo lnear segmens) n parallel wh a varable anchor pon sprng. 4.3 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 search. A he nalzaon of he algorhm he followng seps are performed: Each slave polygonal conac surface s dvded no 2 blocks of polygons. The boundng box for each block of polygons s found. Then each of hose blocks of polygons s dvded no 2 blocks and agan he boundng boxes for hose blocks are found. Ths recursve dvson connues unl here s only one polygon n a box. For each maser conac surface he conac pons are dvded no 2 blocks. The boundng sphere for each block of pons s found. Then, each of hose blocks of pons s dvded no 2 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 0). Page

10 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 n realme. 5. Jon Consrans 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. 5.1 Connecon pon locaon If he connecon pon s a node hen Equaons (16a) and (16b) are used o fnd he global poson x Gp and velocy x& Gp of he connecon pon. If he connecon pon s a fxed pon on rgd body B hen Equaons (17a) and (17b) 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 Gp = N (ξ ) x (26) Gp J l J l J ( l J x & = N (ξ ) x& (27) where J s he local node number of he elemen, N ξ ) are he nerpolaon funcons of he elemen, ξ l are he naural elemen coordnaes of he fxed pon and local node J of he elemen relave o he global reference frame. 5.2 Sphercal jon consran force J x J s he poson vecor of A jon s defned by defnng he relaon beween connecon pons. For example, a sphercal jon beween wo connecon pons s defned as: xc1 = xc2 (28) where xc1 s he poson vecor of he frs pon and xc2 s he poson vecor of he second pon. Ths consran s mposed usng he penaly echnque as: Page

11 F & c = kp v + cp vv (29) v xc1 xc2 v& = x& c1 x& c2 (31) F c = Fc v (32) = (30) 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 (14a) and (14b). If he connecon pon s a node, hen he consran force s appled drecly o he node: F = F (33) K c If he connecon pon s a pon nsde a fne elemen, hen s appled o he nodes of he elemen usng: F J = N J (ξl ) Fc (34) where J s he local node number of he elemen, F s he force on local node J of he elemen J relave o he global reference frame. Usng Equaons (16, 17, and 26-34), 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 (1). Also, he consran momens are appled o he nodes by assemblng hem no he global srucural orques Ts n Equaon (2). 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. Herarchcal Objec-Orened FrameWork The mulbody sysem s modeled usng a se of objecs of varous ypes (or classes). The man classes of objecs used n he presen solver and vrual-realy engne are : Inerface objecs nclude user nerface wdges (e.g. label, ex box, buon, check box, slder bar, dal/knob, able, and graph). Those objecs can be used o buld vrual user nerfaces. Inerface objecs also nclude conaner objecs (ncludng Group, Transform, Bllboard, ec). The conaner allows groupng objecs ncludng oher conaners. Ths allows a herarchcal ree-ype represenaon of he vrual-envronmen called he scene graph. Page

12 Geomerc enes represen he geomery of he varous physcal componens. Typcal geomerc enes nclude unsrucured surfaces, boundary-represenaon sold, box, cone and sphere. Geomerc enes can be exured usng b-mapped mages and colored usng he lgh sources and he maeral amben, dffuse, and specular RGBA colors. Fne elemens are he elemens used o model he mulbody sysem. They nclude rgd body, sprng, russ, hn beam, hck beam, sold brck, jons, prescrbed moon, conac surfaces, acuaors and sensors. Suppor objecs conan daa ha can be referenced by oher objecs. Typcal suppor objecs nclude maeral color, physcal maeral, poson coordnaes and nerpolaors. For example, a sphere geomerc eny can reference a maeral color suppor objec. Objec ypes are furher dvded no sub-ypes, for example, jons ypes nclude: sphercal, revolue, cylndrcal and prsmac jons. Each ype has a se of sandard properes and mehods ha are nhered by all he sub-ypes. The nherance consruc allows new objec ypes o be easly creaed. Each objec ype has a se of properes. The user creaes he mulbody sysem by creang objecs of varous ypes and specfyng he value of he properes. Properes values whch are no specfyed by he user and lef a her defaul values. For example, when he user creaes a rgd body, s/he can specfy he poson, mass and momen of nera of he body. An objec propery can be a sngle neger or real number, an array of neger or real numbers, a reference o anoher objec, or references o an array of objecs. By allowng objecs o reference oher objecs or arrays of objecs, he model can be represened usng as a herarchcal ree. Ths ree s called scene graph n vrual-realy applcaons. Objecs also have mehods whch are funcons ha he objec can perform. Objecs also can encapsulae (conan) he code necessary o make he objec perform a desred funcon. For example, a rgd body objec encapsulaes he mahemacal models for movng he body and negrang he body s equaons of moon. The vrual-realy dsplay engne refreshes he dsplay screen abou 20 mes per second. For every dsplay refresh, he solver perform n me seps, where n s ypcally n he range from Ths means ha he dsplay me sep s 0.05 sec, whle he solver compuaonal me sep s sec. The explc solver s oulned n he nex sub-secon. Explc Soluon Procedure The soluon felds for modelng mulbody sysems are defned a he model nodes. Noe ha a rgd body s 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. Afer loadng he model, he nal condons for all he nodes are se. The explc soluon procedure mplemened n he presen real-me mulbody dynamcs solver s fully negraed n he model scene-graph. The procedure a each soluon me sep s oulned below: Page

13 1. Traverse he scene graph and se he nodal values a he las me sep o be equal o he curren nodal values for all soluon felds. 2. Perform 2 eraons (a predcor eraon and a correcor eraon) of he seps:. Traverse he scene graph and nalze he nodal forces and momens o zero.. Traverse he scene graph and calculae he nodal forces and momens for he fne elemens, he jons and he maser conac surfaces. Those forces are assembled no he global srucural forces ( F s K ) and momens ( M s K ) (needed n Equaons 1 and 2). Ths s he mos compuaonal nensve sep.. Traverse he scene graph and fnd he nodal values a he curren me sep usng he semdscree equaons of moon and he rapezodal me negraon rule (Equaons 1-5). v. Traverse he scene graph and execue he prescrbed moon consrans whch se he nodal value(s) o prescrbed values. v. Incremen he me by and go o sep Vrual Dynamcs Lab The mulbody dynamcs solver presened above s used o smulae a vrual-dynamcs lab. The solver s used o smulae n real-me he dynamc response of he expermens. The suden can also, neracvely change varous expermen parameers and observe n real-me he effecs of he changes. The suden can perform varous dynamcs expermens n he lab and collec he expermen measuremens smlar o wha s/he would do n an acual lab. The daa of he expermen can be coped from he vrual envronmen and pased n a spreadshee for furher analyss by he suden. The man ypes of expermens ncluded n he lab wll be presened n he res of hs secon. 7.1 Gravy Tower Fgure 6 shows a smple gravy expermen of a ball fallng vercally. The user can le he objec fall from varous heghs and gve he objec an nal vercal velocy. The user can also conrol he value of gravy as a facor of earh s gravy. So a facor of one means earh s gravy and a facor 0.16 means moon s gravy. Afer runnng he expermen he suden can copy he graph daa showng he vercal poson of he ball versus me o a spreadshee where s/he can plo he daa and do furher analyss such as calculae he acceleraon of gravy, he nal or he fnal velocy, or he nal poson of he ball D rack and Puck Fgure 2 shows an expermenal seup of a puck movng on an nclned 1-dmensonal rack. The suden can neracvely conrol he nclnaon angle (α), nal poson (x 0 ), nal velocy (v 0 ) and frcon coeffcen beween he puck and he rack (µ). The followng ypes of expermens can be performed usng hs expermenal seup: If α = 0 and µ = 0, v 0 0 he seup can be used o perform Newon s frs law expermens. If α 0 and µ = 0, he seup can be used o perform expermens of moon under he acon of gravy. If α = 0 and µ 0, v 0 0 he seup can be used o perform expermens moon under he acon of frcon. Page

14 If α 0 and µ 0 he seup can be used perform expermens o calculae he frcon angle. Fgure 6. Gravy expermen of a ball fallng vercally from a ower D rack and puck and sprng Fgure 7 shows an expermen of a puck on an nclned rack aached o a sprng. Ths expermen s used o llusrae he concep of a force. The user can conrol he nclnaon angle of he rack and he mass of he puck. Fgure 7. Force expermen of a puck aached o a sprng on an nclned rack. 7.4 Moon of objecs on an nclned plane Fgure 8 shows an nclned plane expermen. The user can le spheres of varous dameers, cylnders of varous lenghs and dameers, and boxes of varous szes move down he nclned plane. The user can vary he coeffcen of frcon for he varous objecs. Ths expermen s used o llusrae frcon and he concep of mass momen of nera. If he frcon coeffcen for he objec s zero hen all he objecs reach he boom of he plane a he same me and he spheres and cylnders don roll. If frcon s se such ha he spheres and cylnders roll whou sldng hen he spheres wll reach he boom of he plane frs because hey have a smaller momen of nera. The sudens can observe ha all he spheres reach he boom a he same me rrespecve of her dameer. Also, all he cylnders reach he boom a he same me rrespecve of her dameer or her lengh. Page

15 Fgure 8. Inclned plane expermen showng varous objecs movng due o gravy on an nclned plane. 7.5 Vercal Masses Rope-Pulley Expermen Fgure 1 shows wo bodes suspended vercal usng a rope and a pulley. The suden can conrol he values of he masses of he wo bodes. The masses/nera of he rope and pulley are neglgble compared o he masses of he suspended bodes. The moon of he bodes can be used o derve Newon s second law of moon. The rope s modeled usng russ elemens. The conac search algorhm s used o quckly deec conac beween he rope nodes and he pulley. The rope and pulley are very lgh compared o he masses of bodes A and B n order no o affec he resuls of he expermen. 7.6 Moon of a block on a frconal plane under he acon of a force Fgure 9 shows a block on a plane conneced o a vercal body usng a rope and pulley. The suden can conrol he values of he masses of block and suspended body, he frcon coeffcen and he conrac area of he block. The expermen can be used o derve he Coulomb law for frcon. I can also be used o prove ha he frcon force s ndependen of he conrac area. Smlar o he expermen n Fgure 1, he rope and pulley are very lgh compared o he masses of bodes A and B n order no o affec he resuls of he expermen. Fgure 9. Block on a frconal surface pulled by a mass usng a rope and pulley used o llusrae sac frcon. Page

16 7.7 Moon of a puck on an nclned rack under he acon of a force Fgure 10 shows a puck on an nclned rack conneced o a vercal body usng a rope and pulley. The suden can conrol he values of he masses of he puck and body, he nclnaon angle of he rack and he frcon coeffcen beween he rack he puck. The expermen can be used o derve Newon s second law and he Coulomb frcon law. Fgure 10. Puck on an nclned rack pulled by a mass usng a rope and pulley. 7.8 Moon of a puck on crcular rack Fgure 11 shows a ball movng on a crcular rack. The suden can conrol he nal angle of he ball. Ths expermen s used o llusrae he conceps of perodc moon, knec energy, poenal energy and conservaon of energy. The rack s modeled usng he nsde surface of a orus. Fgure 11. Puck movng on a crcular rack. Page

17 7.9 Moon of a puck on roller-coaser rack Fgure 12 shows a ball movng on a roller coaser rack. The suden can conrol he nal drop hegh of he ball. The expermen s used o llusrae he conceps of knec energy, poenal energy, conservaon of energy and conservaon of momenum Pendulum Perodc Moon Expermens Fgure 12. Puck movng on roller-coaser rack. Fgure 13 shows a smple pendulum expermen. The user can conrol he momen of nera, nal angle and nal angular velocy of he pendulum. Ths expermen s used o llusrae perodc moon, roaonal moon and conservaon of energy. Fgure 15 shows a double pendulum expermen whch s used o llusrae vbraon of mul-degree of freedom sysems. Fgure 13. Smple pendulum. Fgure 14. Double pendulum Mass-Sprng Vbraons Lab Fgures 15 and 16 show horzonal and vercal mass-sprng expermens. The user can conrol he values of he mass, sprng sffness, dampng, frcon force, nal deflecon, nal velocy Page

18 and appled force magnude and frequency. Those expermens are used o llusrae he conceps of smple harmonc moon, free undamped vbraons, free damped vbraons and forced vbraons of one degree-of-freedom sysems. Fgure 15. Horzonal mass-sprng expermen. Fgure 16. Vercal mass-sprng expermen Spnnng Top Fgure 17 shows a spnnng op. The suden can conrol momen of nera of he op, he nal angular velocy, he dsance beween he p and he cener of gravy (c.g.) and he offse of he c.g. from he roaon axs of he op. The expermen s be used o llusrae gyroscopc moon D Ar-Hockey Expermen Fgure 17. Spnnng op (for llusrang gyroscopc moon). Fgure 18 shows an ar-hockey expermen. The expermenal seup can be used o llusrae he followng conceps: Newon s collson and momenum conservaon laws for elasc and nelasc mpacs. Newon s frs law of moon. Page

19 7.14 Bllards Table Fgure 18. Ar-hockey expermen used o llusrae mpac. Fgure 19 shows a bllard able expermen. Ths expermen s also used o llusrae Newon s collson and momenum conservaon laws Planeary Moon Fgure 19. Bllard able expermen. Fgure 20 shows a model of he solar sysem. The model ncludes he sun and all he planes. The gravy forces beween each par of planes are ncluded n model. The user can speed up me and can urn-off/on some planes. The user can also conrol he nal poson, velocy and mass of an magnary plane n order o see he dfferen shapes of orbs. Ths expermen s used o llusrae he concep of gravaonal orbal moon. Page

20 Fgure 20. Solar sysem model Srng Vbraons Fgure 21 shows a vbrang srng expermen. Ths expermen s used o llusrae he conceps of ransverse waves and vbraon of connuous sysems. The srng s modeled usng russ elemens under a pre-enson. The user can conrol he srng pre-enson, axal sffness and mass per un lengh. In addon, he user can se he nal condons of he srng n order o exce he varous modes of he srng. Also, he user can pluck he srng a one end and see a ravelng wave along he lengh of he srng. Fgure 21. Srng vbraon expermen. Page

21 7.17 Gear Lab The dynamcs lab ncludes varous gear rans ncludng compound gear rans and planeary gear rans (e.g. Fgure 22). The suden can observe he moon of he gears and observe he angular velocy of he varous gears as a funcon of he npu angular velocy. For planeary gear rans he npu can be a sun gear, he arm or rng gear Cam Lab Fgure 22. Gear lab. The dynamcs lab ncludes four ypes of cam-follower sysems (Fgure 23): Cam wh a ranslaonal fla face follower. Cam wh a ranslaonal roller follower. Cam wh an oscllang fla follower. Cam wh a oscllang roller follower. The suden can specfy he poson dagram of he follower as a funcon of he cam angle, he cam base radus and he follower radus. Then, he suden can observe he moon of he cam and follower and plo he velocy, acceleraon and jerk dagrams for he cam. Fgure 23. Cams wh: ranslang roller follow (lef); ranslang fla follow (mddle) and oscllang fla follower (rgh). Page

22 7.19 Roboc Manpulaors The dynamcs lab ncludes a roboc manpulaor (Fgure 24). Sudens can specfy he moon of program he end effecor from one pon o anoher pon n a sragh lne. Also, sudens can specfy he endng angles and posons of he varous axes and observe he moon of he manpulaor. Sudens can plo he angles and he poson of he end effecor versus me Mechansm Lab Fgure 24. Roboc manpulaors. Fgure 25, 26 and 27 show he user nerface for specfyng he dmensons of varous ypes of lnkages ncludng: 4-bar, crank-slder and nvered crank slder. The suden can hen anmae he moon of he lnkage. They can plo he poson of varous pons on he lnkage. They can also observe he moon of he mechansm under he acon of appled orques and/or forces. Fgure 28 shows a ypcal 7-bar mechansm. Sudens can observe he moon of he mechansm due o an appled orque and he presence of he sprng. Fgure bar mechansm. Page

23 Fgure 26. Crank slder mechansm. Fgure 27. Invered crank slder mechansm. Fgure 28. A 7-bar mechansm wh a sprng. 8. Concludng Remarks A flexble mulbody dynamcs explc me-negraon parallel solver suable for real-me vrual-realy applcaons was presened. The mulbody sysem ncludes rgd bodes, flexble bodes, jons, frconal conac consrans, acuaors and prescrbed moon consrans. The solver has he followng characerscs/feaures: Page

24 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 3D fne elemens ncludng sprng/russ and beam fne elemens. Penaly echnque for modelng jon consrans ncludng sphercal, revolue, cylndrcal and prsmac jons. Penaly echnque for modelng normal conac. Frconal conac modeled usng an accurae and effcen aspery-based frcon model. 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. Herarchcal objec-orened framework. The herarchcal scene-graph represenaon of he model used for dsplay and user-neracon wh he model s also used n he solver. The applcaon of he real-me solver o a vrual dynamcs lab was presened. References 1. Wasfy, T.M. and Noor, A.K., Compuaonal sraeges for flexble mulbody sysems, ASME Appled Mechancs Revews, Vol. 56(6), pp , Uchda T. and McPhee, J., Trangularzng knemac consran equaons usng Gröbner bases for real-me dynamc smulaon, 1 s Jon Inernaonal Conference on Mulbody Sysem Dynamcs, Lappeenrana, Fnland, May Hdalgo, A., Callejo, A., and de Jalon, J.G., Usng mplc negraors and auomac dfferenaon o compue large and complex MBS n real-me, 1 s Jon Inernaonal Conference on Mulbody Sysem Dynamcs, Lappeenrana, Fnland, May Lugrs, U., Escalona, J., Dopco, D. and Cuadrado, J, Effcen and accurae smulaon of he cable-pulley neracon n wegh-lfng machnes, 1 s Jon Inernaonal Conference on Mulbody Sysem Dynamcs, Lappeenrana, Fnland, May Cuadrado, J., Dopco, D., Gonzalez, M. and Naya, M.A., A combned penaly and recursve real-me formulaon for mulbody dynamcs, Journal of Mechancal Desgn, Vol. 126(4), 2004, pp Perera H.S. and Romano, R. and Nunez, P., Auomaed mehods for converng a non real-me Caresan mul-body vehcle dynamcs model o a real-me recursve model, SAE Inernaonal 2006 World Congress, Dero, Mchgan, Aprl, Heyn, T. Mazhar, h., Tasora, A, Negru, D., Tracked vehcle smulaon on granular erran leveragng parallel compung on GPUs, 1 s Jon Inernaonal Conference on Mulbody Sysem Dynamcs, Lappeenrana, Fnland, May Hppmann, G., An algorhm for complan conac beween complexly shaped surfaces n mulbody dynamcs, IDMEC/IST, Lsbon, Porugal, July Wasfy, T.M. and Noor, A.K., Modelng and sensvy analyss of mulbody sysems usng new sold, shell and beam elemens, Compuer Mehods n Appled Mechancs and Engneerng, Vol. 138(1-4) (25 h Annversary Issue), pp , Wasfy, T.M., Modelng spaal rgd mulbody sysems usng an explc-me negraon fne elemen solver and a penaly formulaon, ASME Paper No. DETC , Proceedng of he DETC: 28 h Bennal Mechansms and Robocs Conference, DETC, Sal Lake, Uah, Wasfy, T.M., A orsonal sprng-lke beam elemen for he dynamc analyss of flexble mulbody sysems, Inernaonal Journal for Numercal Mehods n Engneerng, Vol. 39(7), pp , Page

25 12. Wasfy, T.M. and O Kns, J., Fne Elemen Modelng of he Dynamc Response of Tracked Vehcles, 7 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, San Dego, CA, Augus Wasfy, T.M., Hgh-fdely modelng of flexble mng bels usng an explc fne elemen code, ASME DETC , Proceedngs of he ASME 2011 Inernaonal Desgn Engneerng Techncal Conferences & Compuers and Informaon n Engneerng Conference (IDETC/CIE 2011), 8 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, Washngon, DC, Augus Wasfy, T.M., Wasfy, H.M, and Peers, J.M., Real-me explc flexble mulbody dynamcs solver wh applcaon o vrual-realy based e-learnng, ASME DETC , Proceedngs of he ASME 2011 Inernaonal Desgn Engneerng Techncal Conferences & Compuers and Informaon n Engneerng Conference (IDETC/CIE 2011), 8 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, Washngon, DC, Augus Leamy, M.J. and Wasfy, T.M., Transen and seady-sae dynamc fne elemen modelng of bel-drves, ASME Journal of Dynamcs Sysems, Measuremen, and Conrol, Vol. 124(4), pp , Wasfy, T.M. and Leamy, M.J., Modelng he dynamc frconal conac of res usng an explc fne elemen code, ASME DETC , 5 h Inernaonal Conference on Mulbody Sysems, Nonlnear Dynamcs, and Conrol, Long Beach, CA, Sepember Wasfy, T.M., Aspery sprng frcon model wh applcaon o bel-drves, Paper No. DETC , Proceedng of he DETC: 19 h Bennal Conference on Mechancal Vbraon and Nose, Chcago, IL, Wasfy, T.M. and Noor, A.K., Objec-orened vrual realy envronmen for vsualzaon of flexble mulbody sysems, Advances n Engneerng Sofware, Vol. 32(4), pp , March Wasfy, T.M. and Leamy, M.J., An objec-orened graphcal nerface for dynamc fne elemen modelng of bel-drves, 27 h Bennal Mechansms and Robocs Conference, ASME Inernaonal 2002 DETC, Monreal, Canada, Wasfy, T.M. and Wasfy, A.M., An objec-orened graphcal nerface for dynamc fne elemen modelng of flexble mulbody mechancal sysems, Proceedng of MDP-8, Caro Unversy Conference on Mechancal Desgn and Producon, Caro, Egyp, January 4-6, Wasfy, T.M., Objec-orened modelng envronmen for smulang flexble mulbody sysems and lqudsloshng, DETC , 26 h Compuers and Informaon n Engneerng (CIE) Conference, ASME DETC, Phladelpha, PA, Sepember Wasfy, T.M. and Wasfy, H.M., Objec-orened envronmen for dynamc fne elemen modelng of res and suspenson sysems, IMECE , 2006 Mechancal Engneerng Congress and Exposon Conference, Chcago, IL, November 5-10, Page