VEHICLE DYNAMIC MODELING & SIMULATION: COMPARING A FINITE- ELEMENT SOLUTION TO A MULTI-BODY DYNAMIC SOLUTION

Transcription

1 21 NDIA GROUND VEHICLE SYSTEMS ENGINEERING AND TECHNOLOGY SYMPOSIUM MODELING & SIMULATION, TESTING AND VALIDATION (MSTV) MINI-SYMPOSIUM AUGUST DEARBORN, MICHIGAN VEHICLE DYNAMIC MODELING & SIMULATION: COMPARING A FINITE- ELEMENT SOLUTION TO A MULTI-BODY DYNAMIC SOLUTION Paramsohy Jayakumar, PhD Dynamcs & Srucures Modelng & Smulaon, US Army RDECOM-TARDEC Warren, MI Tamer Wasfy, PhD Advanced Scence and Auomaon Corp. Indanapols, IN ABSTRACT The dynamc response of wo mulbody sysems, a planar mechansm and a spaal robo, are generaed usng an explc me negraon fne elemen code and a mul-body dynamcs code. Comparsons are made of he dynamc soluon ncludng body moon, jon consran forces, conservaon of energy, and CPU me. Whle fne-elemen smulaon offers accurae modelng of srucural flexbly, mulbody dynamc smulaon demonsraes he capably o produce accurae and effcen resuls. INTRODUCTION The Army rounely performs mulbody dynamc smulaons of s ground vehcles, boh racked and wheeled, n order o assess her mobly performance and mprove durably. Tradonally hese smulaons are performed usng rgd mulbody dynamcs sofware. However, o mprove he accuracy of modelng flexble srucures and he desre o have an negraed smulaon envronmen, fne elemen sofware have laely been consdered for performng he same smulaons. There are advanages and dsadvanages o such an approach. Ths paper addresses he soluon accuracy of hs approach by comparng soluons of wo benchmark problems usng an explc fne elemen approach o he mul-body dynamc smulaon. EXPLICIT FINITE ELEMENT (FE) CODE FORMULATION The ranslaonal equaons of moon are wren wh respec o he global neral reference frame and are obaned by assemblng he elemen equaons. The fne elemens use only ranslaonal degrees-of-freedom (DOF) and no roaonal DOF. The ranslaonal DOF ncludes he rgd-body ranslaon and he fne elemen nodal posons. The equaons of moon can be wren as: M K x F F (1) K s K a K where s he runnng me, K s he global node number (no summaon over K; K=1N 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, Fs s he vecor of nernal srucural forces, and Fa s he vecor of exernally appled forces whch nclude surface forces and body forces. Each rgd body s represened by a body-fxed maeral frame whose orgn s locaed a he body s cener of mass. The mass of he body s concenraed a hs cener-of-mass node, and he nera of he body, gven by he nera ensor I j, s defned wh respec o he body frame. The orenaon o of he body-frame s gven by R whch s he roaon K marx relave o he global neral frame a me. The roaonal equaons of moons are wren for each rgd body wh respec o s body-fxed maeral frame as: Kj Kj s K a K K Kj Kj K I T T ( I ) (2) where I K s he nera ensor of rgd body K, Kj and Kj are he angular acceleraon and velocy vecor componens for rgd body K relave o s maeral frame n drecon j, Ts K s he componen of he vecor of nernal orque a node K n drecon, and Ta K s he componen of he vecor of appled orque a node K n drecon. The Ensen

2 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) summaon convenon s used only for he lower case ndces and j. Consran equaons are algebrac equaons, whch descrbe Prescrbed moon consrans: f ({ x}, ) (3) Jon consrans: f ({ x}) (4) Conac consrans: f ({ x}) (5) The penaly echnque s used for mposng he jon and conac consrans. In ha echnque a normal reacon force s generaed beween wo pons x p1 and x p2 n order o sasfy he jon or conac consran. The consran penaly force s gven by: Fc ( kp d cp dˆ d ) dˆ (6) j d xp1 xp2 d x p1 x p2 (7a) (7b) dˆ d / d (7c) where k p s he penaly sffness and c p s penaly dampng. The magnude of he penaly force s gven by Equaon (6) for jon consrans. For conac consrans he penaly force s gven by Equaon (6) f pon 1 s nsde he volume of he body ha pon 2 belongs o, oherwse s zero. An aspery-sprng frcon model s used o represen jon and conac frcon n whch frcon s modeled usng a pecewse lnear velocy-dependen approxmae Coulomb frcon. An explc soluon procedure such as he Newmark negraon formula s used o solve he equaons of moon (1, 2) along wh he consran equaons. In explc soluon echnques he me sep () used mus be less han a crcal me sep n order o oban a sable soluon. The crcal me sep s less han half of he smalles characersc wave propagaon me n an elemen. Ths procedure s mplemened n a commercal sofware [1] ha s used n he presen paper. MULTIBODY DYNAMIC (MBD) CODE FORMULATION A commercal sofware descrbed n [2] s he mulbody dynamcs code used n hs paper. In, he equaons of moon for a rgd mulbody sysem are combned wh he consran acceleraon equaons and wren n he followng form: j M q q T q Q where q s he vecor of generalzed coordnaes of he sysem, M s he generalzed mass marx, Q s he vecor of appled forces, are he consran equaons, s he vecor of Lagrange mulplers of he consrans, and s he rgh hand sde vecor of he consran acceleraon equaons. In addon, he consran equaons and consran velocy equaons are, respecvely. ( q, ) (9) q q (1) Equaons of moon (8) along wh he knemac consran equaons (9) and he consran velocy equaons (1) yeld a mxed sysem of dfferenal-algebrac equaons of moon for he mulbody sysem. These equaons can be solved usng mplc negraon mehods such as he Backward Dfferenaon Formula (BDF). For sff sysems, he superor sably characerscs of he BDF mehods allow hem o ake much larger sepszes han would be possble wh explc mehods. BENCHMARK PROBLEMS Two problems ha were used as benchmark problems n Mulbody Sysems Handbook [3] are solved usng he explc fne elemen sofware and he mulbody dynamcs sofware. A comparson of he smulaon resuls s presened below. Benchmark Problem 1: 7-Lnk Planar Mechansm The 2-D mechansm shown n Fgure 1 was used as one of he mulbody dynamcs benchmarks n Mulbody Sysems Handbook [3]. A descrpon of he mechansm and a ls of parameers such as he mass, momens of nera, lnk lenghs, and nal lnk posons are also gven n [3]. The mechansm consss of 7 rgd lnks conneced by frconless revolue jons. The mechansm has one lnear sprng ha connecs pon D on lnk K 3 and pon C on he ground. In he nal poson, he sprng s under compresson. The mechansm s drven by a moor orque ha s appled o lnk K 1 and gven by: (8) Page 2 of 1

3 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) T K 1.33 N.m.1.1 The drve orque s removed a me.1 sec n order o assess he energy conservaon capably of he smulaon code. The smulaon s run for a oal me of.5 sec. The 2-D mechansm has 1 degree-of-freedom. angles, angular veloces, and sprng force, one can conclude ha boh FE and MBD predc he same lnk moon. Table 1: Comparson of MBD and FE me seps and CPU mes for he 7-lnk planar mechansm for a oal smulaon me of.5 sec. Runs were performed on a DELL OPTIPLEX 76 Inel Core 2 Duo 3.16GHz. MBD FE FE Tme sep (s) 1.E-5.4E-5.1E-5 CPU me (s) Poson/velocy error Tolerance 1E Fgure 1: 7-lnk planar mechansm [3]. Table 1 shows he mulbody dynamc smulaon CPU me, fne elemen smulaon CPU me, and he correspondng me seps used. Two runs wh wo dfferen me seps ( =.4E-5,.1E-5) are used for FE, whle he MBD me sep chosen was 1.E-5. I wll be shown laer (Fgure 7) ha for comparable accuracy beween he wo codes, FE requres he fner me sep of.1e-5 s. Fgure 2 shows snapshos of he frs revoluon of he mechansm smulaed usng FE. Table 2 shows a comparson of MBD and FE resuls agans Handbook [3] resuls usng he Kane s Mehod for he roaon angle of lnk K 1. Boh MBD and FE resuls mach well wh Handbook resuls. Fgures 3 and 4 show he me-hsores of he roaon angles and angular veloces of lnks K 1, K 3, and K 5 generaed usng FE and MBD. Fgure 5 shows he me-hsory of he sprng force generaed usng FE and MBD. The sprng force s a funcon of he lnk posons. Comparng hese resuls, he roaon Fgure 2: Snapshos of he moon of he 7-lnk planar mechansm smulaed usng FE ( =.4E-5). Page 3 of 1

4 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Table 2: Comparson of MBD and FE ( =.4E-5) agans Handbook [3] resuls for he roaon angle of lnk K1 of he 7-lnk planar mechansm. % Dfference beween Handbook Tme (s) Bea (rad) & Handbook [3] MBD FE MBD FE.E E E E E E E E E E E E E E E E Fgure 3: Tme-hsores of he roaon angles of lnks K 1, K 3, and K 5 generaed usng FE ( =.4E-5) and MBD. Page 4 of 1

5 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Fgure 5: Tme-hsory of he sprng force generaed usng FE ( =.4E-5) and MBD. Fgure 6 shows he me-hsores of he X and Y reacon force componens a jon B (beween lnk K 3 and ground) generaed usng FE ( =.4E-5) and MBD. Fgure 6 shows ha he reacon forces predced usng FE have hghfrequency oscllaons. However, he me average of he forces predced usng FE s very close o he forces predced by MBD. Also, he amplude of he FE oscllaons decreases wh me. Fgure 7 adds o he plos n Fgure 6 FE smulaon resuls wh he me-sep reduced by a facor of 4: =.1E-5. From Fgure 7, we see ha he amplude of he oscllaons predced wh a FE me-sep of.1e-5 s less han he amplude usng a me sep of.4e-5. In addon, he amplude decreases more quckly wh me bu he frequency of he oscllaons s hgher a he smaller me sep. Fgure 4: Tme-hsores of he angular veloces of lnks K 1, K 3, and K 5 generaed usng FE ( =.4E-5) and MBD. Page 5 of 1

6 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Fgure 6: Tme-hsores of jon B (beween lnk K3 and ground) X and Y reacon force componens generaed usng FE ( =.4E-5) and MBD. Fgure 7: Same as Fgure 6 bu wh he addon of FE resuls for a reduced me sep of 1e-6 sec. The reason why he forces predced usng fne-elemen sofware have hgh-frequency oscllaons s because of he penaly echnque used o model he jons. Each revolue jon n he mechansm s modeled usng a penaly sprngdamper. The sffness and dampng of he penaly sprngdamper are se by FE such ha hey are he maxmum allowable for a sable soluon for he me-sep used causng he oscllaons. As he me-sep s reduced, he sffness and dampng of he penaly sprng-damper ncrease. So, he jon becomes sffer ncreasng he oscllaon frequency furher. Though hs approxmaes a perfec (nfnely sff) jon beer, also causes he hgher-frequency oscllaons. The penaly sffness n FE may be vewed as he physcal jon Page 6 of 1

7 Jon B Y-Force (N) Jon B X-Force (N) UNCLASSIFIED: Ds A. Approved for publc release Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) sffness. Mulbody dynamc sofware on he oher hand models he jon usng an algebrac consran ha enforces he jon consran whn olerance whou he use of sff penaly sprngs. Thus, jons n MBD are near-perfec jons whou he use of sff sprngs. Explc fne-elemen sofware such as [1] canno solve for sac equlbrum as he way MBD solves usng an mplc solver. Hence, o avod he hgh-frequency oscllaons n he FE soluon, he nal drve orque and he sprng compresson can be gradually appled whou beng sep npus. To demonsrae hs fac, anoher FE run s performed wh he npu orque lnearly rampng from me o me.5 sec: T K Also, he sprng lengh s lnearly ramped from he undeformed lengh o he nal compressed lengh n.5 sec. Fgure 8 shows he resulng reacons forces a jon B. These reacon forces exhb no hgh-frequency oscllaons and are close o he MBD reacon forces seen n Fgure 7, excep ha he new resuls are shfed n me by.5 sec. Fgure 9 shows he me-hsory of he mechansm s poenal energy predced usng FE and MBD. Fgure 1 shows he me-hsory of he mechansm s oal energy (sum of knec energy and poenal energy). The drve orque s removed a me.1 sec, so he oal energy mus reman consan afer ha me. Boh FE and MBD predc he correc mechansm oal energy. The speed wh whch a soluon looses energy wh me ha s supposed o have a consan oal energy s a measure of he soluon drf. Thus, he energy conservaon capably of a code s a measure of he capably of he code o manan an accurae soluon wh no soluon drf over me. Jon B X-Force on Lnk K3 (FE) Fgure 9: Tme-hsory of he mechansm poenal energy predced usng FE ( =.4E-5) and MBD Jon B X-Force on Lnk K3 (FE) Tme (s) Jon B Y-Force on Lnk K3 (FE) Jon B Y-Force on Lnk K3 (FE) Tme (s) Fgure 8: Tme-hsores of he X and Y reacon force componens a jon B predced usng FE ( =.4E-5) by applyng a ramped drve orque and a ramped nal sprng deflecon. Fgure 1: Tme-hsory of he oal sysem energy of he mechansm predced usng FE ( =.4E-5) and MBD. Page 7 of 1

8 Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Benchmark Problem 2: Spaal Roboc Manpulaor The 3-D roboc manpulaor shown n Fgure 11 was used as one of he benchmarks n Mulbody Sysems Handbook [3]. A descrpon of he sysem and a ls of sysem parameers are gven n [3]. The manpulaor consss of 3 rgd bodes conneced by frconless cylndrcal and revolue jons. The mass, momens of nera, and nal posons of he bodes are gven n [3]. Each jon acuaor of he manpulaor s drven by a prescrbed force and/or orque such ha he end-effecor races a sragh lne wh a rapezodal velocy profle over a smulaon me of 2 sec. The manpulaor sysem has 5 degrees-of-freedom. Table 3: Comparson of MBD and FE me seps and CPU mes for he roboc manpulaor. Runs were performed on a DELL OPTIPLEX 76 Inel Core 2 Duo 3.16GHz. MBD FE Tme sep (s) E-5 CPU me (s) Fgure 12 shows snapshos of he moon of he manpulaor smulaed usng FE. Fgures show he me-hsores of jon acuaor moons generaed usng FE and MBD and compared wh Handbook [3] resuls. These fgures show ha he moon predced by FE and MBD are praccally he same hough small oscllaons are noceable n he FE resul n Fgure 16. Fgure 11: Spaal roboc manpulaor [3]. Table 3 shows he MBD and FE smulaon CPU me and correspondng me sep used. For mplc codes he CPU me ncreases as he square of he number of coordnaes whereas for explc codes CPU me ncreases lnearly wh he number of coordnaes. The moon of he robo s slow and does no nvolve hgh-speed roaon, hus a large me sep can be used wh an mplc code. However, explc codes mus use a me sep ha s smaller han he crcal me sep even for slow movng sysems. Ths s because he me sep depends prmarly on he penaly sffness of he jon consrans and no on he speed of he moon. For problems nvolvng a small number of coordnaes and a slow moon, mplc codes are always more compuaonally effcen han explc codes. However, as he number of coordnaes ncreases and/or as he speed of he moon ncreases explc codes become more compuaonally effcen relave o mplc codes. Fgure 12: Snapshos of he moon of he manpulaor smulaed usng FE. Page 8 of 1

9 Body2 Y (m) Body2 Y_Do (m/sec) Body1 Z (m) Body1 Z_Do (m/sec) Body1 Z-Angle (deg.) Body1 Z-Angle Do (deg/sec) UNCLASSIFIED: Ds A. Approved for publc release Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Z1_Angle_FE Z1_Angle_MBD Z1_Angle_Handbook Z1_Angle_Do_DIS Z1_Angle_Do_DADS Z1_Angle_Do_AUTLEV Fgure 13: The roaon angle of body 1 around he z-axs (GA1). Fgure 16: The angular velocy of body 1 abou he z-axs (GA1_do) Z1_FE Z1_MBD Z1_Handbook Z1_Do_FE Z1_Do_MBD Z1_Do_Handbook Fgure 14: The z-coordnae of body 1 (Z1). Fgure 17: The velocy of body 1 along z-axs (Z1_do) Y2_FE Y2_MBD Y2_Handbook Y2_Do_FE Y2_Do_MBD Y2_Do_Handbook Fgure 15: The y-coordnae of body 2 (Y2). Fgure 18: The velocy of body 2 along y-axs (Y2_do). Page 9 of 1

10 Base Force Y (N) Base Force X (N) Base Force Z (N) UNCLASSIFIED: Ds A. Approved for publc release Proceedngs of he 21 Ground Vehcle Sysems Engneerng and Technology Symposum (GVSETS) Fgure 19 shows he me-hsores of he X, Y, and Z reacon force componens a he base of he robo. Smlar o he prevous mechansm benchmark problem, Fgure 19 shows ha he reacon forces predced usng FE have hghfrequency oscllaons. However, he me average of he forces predced usng FE s very close o he forces predced usng MBD. The reason for he hgh-frequency force oscllaons n FE s agan because he jons are modeled usng penaly sprng-dampers and a he same me he jon acuaor orques and forces have jump dsconnues a mes,.5, and 1.5 sec. In order o elmnae he force oscllaons n FE, he drve forces and orques need o be made connuous whch s also physcal BaseForceZ_FE -2 BaseForceZ_MBD Fgure 19: X, Y, and Z componens of he reacon force on he base of he roboc manpulaor Base_ForceX_FE Base_ForceX_MBD BaseForceY_FE BaseForceY_MBD CONCLUSIONS Two mulbody dynamcs benchmark problems were solved usng an explc fne-elemen code and an mplc mulbody dynamcs code. The wo codes predc praccally he same sysem moon. However, he jon reacon forces predced by FE have hgh-frequency oscllaons. These oscllaons are due o he penaly sprng-dampers used for modelng he jons and he presence of dsconnues n appled force/momen. If appled forces/momens were connuous and he smulaon were sared from a sac equlbrum confguraon, hen he FE soluon would no exhb he hgh-frequency force oscllaons. An explc fne-elemen code such as [1] offers he advanage of modelng non-lnear srucural flexbly, flud-srucure neracon, and hermal effecs. However, mulbody dynamcs code such as [2] offers accurae and effcen soluons REFERENCES [1] Wasfy, T. M., Modelng Spaal Rgd Mulbody Sysems Usng An Explc-Tme Inegraon 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, 24. [2] Haug, E. J., Compuer-Aded Knemacs and Dynamcs of Mechancal Sysems, Volume I: Basc Mehods, Boson: Allyn and Bacon, [3] Schehlen, W. (ed.), Mulbody Sysems Handbook. Berln: Sprnger-Verlag, 199. Page 1 of 1