NEW FINITE ELEMENT / MULTIBODY SYSTEM ALGORITHM FOR MODELING FLEXIBLE TRACKED VEHICLES

Transcription

1 2011 NDIA GROUND VEHICLE SYSEMS ENGINEERING AND ECHNOLOGY SYMPOSIUM MODELING & SIMULAION, ESING AND VALIDAION (MSV) MINI-SYMPOSIUM AUGUS 9-11 DEARBORN, MICHIGAN NEW FINIE ELEMEN / MULIBODY SYSEM ALGORIHM FOR MODELING FLEXIBLE RACKED VEHICLES Paamsothy Jayakuma 1 Ulysses Conteas 2 Mike D. Lethewood 1 Ashaf M. Hamed 2 Abdel-Nasse A. Mohamed 2 Ahmed A. Shabana 2 1 U.S. Amy RDECOM-ARDEC & 2 Univesity of Illinois at Chicago ABSRAC he dynamic simulation of multibody tacked vehicles offes enginees a poweful tool with which they may analyze and design. Cuently, pats of these complex mechanisms ae intoduced to multibody algoithms as igid bods. hen in a follow-on stuctual analysis, the loads fom the multibody dynamic simulation ae input to calculate stains and stesses within the bods. he pesent investigation aims to establish appopiate means by which flexible theedimensional tack links, which allow lage elative otation between the elements, can be modeled. his will pave the way towads the incopoation of detailed flexible stuctual models into a multibody dynamic simulation envionment allowing fo an integated solution. In addition, a new fomulation fo the inteaction between the igid spocket teeth and flexible chain is pesented. Numeical esults ae intoduced to illustate the effects of flexible links on the dynamics of tacked vehicles. INRODUCION Cuent commecial implementations fo tacked vehicle modeling ae not ideal fo the analysis of thee-dimensional flexible chain tacked vehicles. Each tack chain may contain lage numbe of sepaate bods/elements which equie connectivity conditions be imposed at each time step. Non-multibody simulation models have had success in poviding enginees with a means with which to quantitatively compae tacked vehicles [1-3]. It is hoped, howeve, that multibody simulation models will povide the woking enginee tacked vehicle models based on fewe assumptions and geate flexibility of simulation. It is one of the objectives of this investigation to pesent the esults of incopoating the ecently developed Absolute Nodal Coodinate Fomulation (ANCF) C 0 /C 1 flexible chain into a tacked vehicle multibody model. he C 0 /C 1 chain model offes a numbe of advantages ove cuent chain models. Fist, the linea connectivity conditions esulting fom ANCF may be appld at a pepocessing stage [5]. his allows fo an efficnt elimination of the dependent vaiables. When used in conjunction with ANCF spatial finite elements, the linea connectivity conditions lead to a constant inetia matix, as well as zeo Coiolis and centifugal foces. Futhe, instead of consideing a ses of inteconnected igid o flexible bods/elements, the new fomulation allows fo the use of one finite element mesh fo the chains despite the finite otations of the tack links with espect to each othe and elative tanslation between the two tack chains. Numeical examples will be used to illustate the advantages of the C 0 /C 1 flexible chain fomulation as elevant to tacked vehicle models. Anothe objective of this investigation is to pesent a newly developed fomulation fo the contact between the C 0 /C 1 chain model and the spocket teeth. his inteaction fomulation will allow the building of tacked vehicle models that can moe accuately epesent the complex dynamics of tacked vehicles. he oganization of the pape is as follows. he fist section contains a summay of the absolute nodal coodinate fomulation (ANCF), as well as a pesentation of the beam element used in this investigation. In the next section, the constained multibody dynamic equations of motion ae pesented. he thid section summaizes the ecently developed finite element ANCF mesh which can be used to : Dist A. Appoved fo public elease

2 Repot Documentation Page Fom Appoved OMB No Public epoting buden fo the collection of infomation is estimated to aveage 1 hou pe esponse, including the time fo evwing instuctions, seaching existing data souces, gatheing and maintaining the data needed, and completing and evwing the collection of infomation. Send comments egading this buden estimate o any othe aspect of this collection of infomation, including suggestions fo educing this buden, to Washington Headquates Sevices, Diectoate fo Infomation Opeations and Repots, 1215 Jeffeson Davis Highway, Suite 1204, Alington VA Respondents should be awae that notwithstanding any othe povision of law, no peson shall be subject to a penalty fo failing to comply with a collection of infomation if it does not display a cuently valid OMB contol numbe. 1. REPOR DAE 09 AUG REPOR YPE N/A 3. DAES COVERED - 4. ILE AND SUBILE New Finite Element/Multibody System Algoithm fo Modeling Flexible acked Vehicles 6. AUHOR(S) Paamsothy Jayakuma; Ulysses Conteas; Mike D. Lethewood; Ashaf M. Hamed; Abdel-Nasse A. Mohamed; Ahmed A. Shabana 7. PERFORMING ORGANIZAION NAME(S) AND ADDRESS(ES) US Amy RDECOM-ARDEC 6501 E 11 Mile Rd Waen, MI , USA Univesity of Illinois, Chicago, IL, USA 9. SPONSORING/MONIORING AGENCY NAME(S) AND ADDRESS(ES) US Amy RDECOM-ARDEC 6501 E 11 Mile Rd Waen, MI , USA 5a. CONRAC NUMBER 5b. GRAN NUMBER 5c. PROGRAM ELEMEN NUMBER 5d. PROJEC NUMBER 5e. ASK NUMBER 5f. WORK UNI NUMBER 8. PERFORMING ORGANIZAION REPOR NUMBER SPONSOR/MONIOR S ACRONYM(S) ACOM/ARDEC/RDECOM 11. SPONSOR/MONIOR S REPOR NUMBER(S) DISRIBUION/AVAILABILIY SAEMEN Appoved fo public elease, distibution unlimited 13. SUPPLEMENARY NOES Pesented at the 2011 NDIA Vehicles Systems Engineeing and echnology Symposium 9-11 August 2011, Deabon, Michigan, USA 14. ABSRAC 15. SUBJEC ERMS 16. SECURIY CLASSIFICAION OF: 17. LIMIAION OF ABSRAC SAR a. REPOR unclassifd b. ABSRAC unclassifd c. HIS PAGE unclassifd 18. NUMBER OF PAGES 7 19a. NAME OF RESPONSIBLE PERSON Standad Fom 298 (Rev. 8-98) Pescibed by ANSI Std Z39-18

3 model flexible chains. he fouth section pesents the models used fo the inteaction between the diffeent pats of the tacked vehicle unning gea. Specifically, the fouth section will intoduce a new fomulation fo the simulation of spocket-flexible chain inteaction. 1. ABSOLUE NODAL COORDINAE FORMULAION In this investigation, the finite element absolute nodal coodinate fomulation is used in the modeling of flexible bods. he global position of an abitay point on element e of body i can be defined (figue 1) in an inetial global coodinate system XYZ as = S ( x, y, z ) e (1) whee S ( x, y, z ) is the space-dependent matix that defines the element shape functions, and e is the timedependent vecto of nodal coodinates. he vecto of nodal coodinates e consists of nodal displacements and gadnts. It can be defined as 1 2 e = [ (2) x y z x y z k 2 k 2 2 ] k In equation (2), k,,, and efe x y z to the displacements and gadnts at node k. It can be shown that the selection of nodal positions and gadnts as element coodinates leads to an exact modeling of igid body dynamics. he shape function fo the thee-dimensional beam element based on the absolute nodal coodinate fomulation used in this investigation is defined by s = [ s I s I s I s I s I s I s I s I ] (3) whee I is the 3x3 identity matix and the shape functions s, k = 1,...,8 ae given as follows [4]: k S = 1 3ξ + 2 ξ, S = l( ξ 2 ξ + ξ ), S = lη (1 ξ ) S = lς (1 ξ ), S = 3ξ 2 ξ, S = l( ξ + ξ ) S = lηξ, S = lςξ 7 8 (4) Figue 1: ANCF Beam Element Coodinates In equation (4), ξ = x l, η = y l, ς = z l, and l is the length of the element. It can be shown that the thee dimensional beam element pesented hee is capable of captuing the effects of shea defomation and otay inetia as well as the coupling between the coss section defomation and bending/extension. 2. CONSRAINED EQUAIONS OF MOION Using the pinciple of vitual wok, one may wite the finite element equations of motion in tems of the vecto of nodal coodinates as follows: In equation (5), elastic foces, ɺ Q (5) M e = Qk + Qe extenal foces, and element. e Q k is the vecto of genealized element is the vecto of genealized element M is the constant mass matix of the Using equation (5), one may wite the equations of motion of the constained multibody system in the following fom: Mqɺɺ + C = qλ Q C( q, t) = 0 z x y (6) New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles Page 2 of 7

4 whee M is the system mass matix, q is the vecto of the system coodinates (which includes the absolute nodal coodinates and othe coodinates used to descibe the motion of the igid bods), λ is the vecto of Lagange multipls, Q is foce vecto, C is the vecto of kinematic constaint equations, and C is the Jacobian matix of constaint equations. q elative otations between its constituent elements at the mesh joints. 3. NEW ANCF FINIE ELEMEN MESH Fully paameteized absolute nodal coodinate fomulation finite elements have 12 degees of feedom. hese degees of feedom include: thee igid body tanslations, thee otations, and six defomation modes. We may intoduce a new fomulation fo a pin joint that uses the following six scala equations defined at the joint node between bods i and j [5]: i j (7) = i j = α α In the above equations, α is the coodinate line that defines the joint axis. It can be eithe x, y, z, o any othe coodinate line since the use of ANCF makes easy the use of the gadnt tenso tansfomation. Conside, fo example, the gadnt tansfomation between u, v, w and x, y, z. his tansfomation is given by [ u v w ] = x y z A whee A is given by the constant matix shown below. x x x u v w y y y A = u v w z z z u v w he six scala equations epesented by equations (7) and (8) educe the numbe of degees of feedom fo the new pin joint to six; one otation, and five defomation mode degees of feedom. As a esult, we have C 1 continuity with espect to the α coodinate and C 0 continuity with espect to the emaining coodinates. he linea scala equations (7) and (8) can be imposed at a pepocessing stage, thus allowing fo the elimination of the dependent vaiables. hese equations can be used to develop a new kinematically linea finite element mesh. his would allow abitaily lage (9) (8) Figue 2: Geomety of contact between igid spocket teeth and the C 0 /C 1 chain 4. FLEXIBLE CHAIN-RIGID COMPONEN CONAC FORMULAIONS he new finite element ANCF C 0 /C 1 mesh fomulation that was pesented in the pevious section can be used to moe accuately descibe, with one mesh, a flexible tack. In this section the contact between the flexible tack links, spockets, olles, and idles will be descibed. he idles, spockets, and olles will be modeled as igid bods in this investigation. While pevious wok has focused on modeling flexible belt diven vehicles [6], the pesent investigation diffes in that it includes a fomulation fo the inteaction between the flexible chain and spocket teeth. his will allow fo futue investigations in which the dynamics of igid and flexible link tacked vehicles can appopiately be compaed. he fomulation fo the inteaction between the spocket teeth and the C 0 /C 1 chain is as follows (one can efe to figue 2 fo the definitions of the vaiables used): let N be the vecto that defines the local position of the cente of the pin joint in the spocket coodinate system, U is a vecto which defines the axis aound which the spocket otates, pin is the pin adius, l B is the backlash as a pecentage of the pin adius pin, nt is the cuent tooth numbe, and n is the numbe of teeth. Note that it is assumed that the spocket New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles Page 3 of 7

5 axis is paallel to the Y-axis. ake angle α as fom the cente line of the tooth to the vetical cente line of the spocket. he two angles α and β can be calculated fom geomety as α = ( 1) 2 n t 1 β = tan (10) (11) One can then find the vectos that define the location of the contact points between the spocket teeth and the pin as π n ( 1 + l ) B pin N With the tooth contact foce established, one also needs to include the contact foce due to the spocket base cicle and pin contact. Figue 3 illustates the geomety used to define the contact between the C 0 /C 1 chain and the cylindical igid body that constitutes the spocket base cicle [6]. he adial (nomal) penetation fo the configuation shown in figue 3 can be fomulated as follows: ( α β) s ( α β) p1 = p1 cos + 0 in + p2 = p2 cos ( α β) 0 sin ( α β) (12) (13) whee the magnitude of equation (14). p1 and p2 can be calculated fom 2 2 p 1 = p 2 = N l B pin (14) he distance between the pin and the tooth contact point is given by equations (15-16). n1 = p1 N n2 = p2 N he condition fo contact is given by ( ) ( ) 2 n1 pin n2 pin New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles (15) (16) o (17) If the conditions fo contact ae tue, the foce nomal to the contact suface is evaluated using a continuous foce model defined as follows F ( ) ˆ (18) n = kd + cdɺ V whee d can be found fom the following equation d = pin - n1 o d = n2 pin (19) and the vecto nomal to the contact suface is easily found fom V = U N (20) whee Page 4 of 7 Figue 3: Geomety of contact between igid cylinde and the C 0 /C 1 chain d = R c N (21) Rc is the adius of the cylinde and N is a vecto that defines the position of the contact point in the coodinate system of the cylinde. N can be defined as ( n ) N = n c c (22) he nomal foce fo the contact between the igid cylinde and the C 0 /C 1 chain can be then be found by using the appopiate penetation and ate of penetation in equation (18). he tangential contact foce is based on a tilinea Coulomb fiction law chaacteized by the following equations F = µ F t, V > v (23) t n t 0 F = v V t, V v (24) t s t t 0 whee V t is the tangential velocity, v s is a fiction paamete that defines the slope in the tansition aea, and the constant v0 is defined as (25) v 0 µ Fn = v s

6 he total foce will then be the summation of the nomal and tangential foces. In the case of the idles and olles, the contact foces will be calculated in a manne simila to the contact between the spocket base cicle and the C 0 /C 1 chain. 5. NUMERICAL RESULS he pevious section pesented the fomulations used in modeling the contact between the igid spockets, olles, idles, and the C 0 /C 1 tack o chain. In this section, numeical esults of a model of a ubbe tacked vehicle that includes only one tack ae pesented. he model contains a gound, chassis, spocket, idle, two olles, and a flexible tack. he mass of the diving spocket and the idle ae assumed to be 40 kg. he inetia of diving spocket and the idle about the X, Y, and Z axes ae espectively 0.4 kg.m 2, 0.8 kg.m 2 and 0.4 kg.m 2. he mass of the olles is 10 kg. he mass of the vehicle chassis is 100 kg. In the numeical example pesented in this section, the chassis is assumed to have zeo initial velocity. he angula velocity constaint that is appld to the spocket is defined by the following equation: 0 ( t 0.2) = t <= < t <= 1 ω (26) t > 0.2 whee ω is the otational speed of spocket in ad/s and t is the time in seconds. he diamete of the base cicle of the spocket is 0.38 m and has 10 teeth. he diamete of the idle and the olles ae espectively 0.38 m and 0.18 m. he chain is modeled using 24 ANCF elements that have a ectangula coss section of dimensions 0.02m 0.4m and density of 2000 kg/m 3. he incompessible Neo-Hookean model and non-linea damping model ae used to model the intenal behavio of the ubbe 8 chain. he Neo-Hookean constant is assumed to be 1 10, 9 the incompessibility constant is assumed to be 1 10, and the dilatation and deviatoic dissipation factos used fo the -4 damping model ae assumed to be 1 10 and espectively. A petension positive axial stain of value 0.05 is appld to the ubbe chain. A compliant foce model is used to descibe the contact between igid bods and the chain. he stiffness and damping coefficnts used in the igid bods and chain contact foce model ae given, , 7 espectively, by 9 10 N/m 3 3 and 2 10 N.s/m 3. angential fiction foces ae also intoduced using a coefficnt of dy fiction of 1.2. he fiction paamete that defines the slope in the tansition egion is assumed to be 10 6 N.s/m 3. Figue 4 shows the X, Y and Z coodinates of the position vecto of the cente of mass of the chassis. Figue 5 shows the velocity of the chassis in the X-diection. Note that in figue 6 the displacement deceases at cetain times; this is an indication of sliding behavio, which will continue until the ubbe elements in contact with the gound obtain zeo speed in the x diection. Figue 4: Position of Cente of Mass of the Chassis ( X coodinate; Y coodinate; Z coodinate) Figue 5: X velocity of the cente of mass of the chassis New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles Page 5 of 7

7 Figues 6 and 7 show, espectively, the X and Z coodinates of a point on the ubbe chain that is initially located at the top of the idle, as function of time. Figue 8 shows the motion tajectoy of this point elative to the moving cente of mass of the chassis. Figue 8: ajectoy of the position vecto of a point on the ubbe chain 6. SUMMARY AND CONCLUSIONS Figue 6: X coodinate of the position vecto of a point on the ubbe chain he pesent investigation poposed a method to model the contact between spatial tack chain links and igid unning gea components. he Absolute nodal coodinate fomulation was used to model the C 0 /C 1 tack o chain using linea connectivity conditions. his chain fomulation allows fo C 1 continuity in the diection of the pin joint definition and C 0 continuity with espect to the othe components. A new fomulation fo the compliant foce contact between thee dimensional flexible tack links and igid spocket was pesented. Along with the fomulation fo the contact between the igid olles, idles, and flexible links, these contact fomulations wee used to model a tacked vehicle with a single tack. his model may then be used in futue investigations to compae the dynamics of igid and flexible tacks in tacked vehicle models. ACKNOWLEDGEMENS his eseach was suppoted by the US Amy ank- Automotive Reseach, Development, and Engineeing Cente (ARDEC) (Contact # W911NF-07-D-001). Figue 7: Z coodinate of the position vecto of a point on the ubbe chain REFERENCES [1] Ma, Z. D., and Pekins, N. C. A ack-wheel-eain Inteaction Model fo Dynamic Simulation of acked Vehicles. Vehicle System Dynamics, 37:6, , 2002 [2] Muakami, H., Watanabe, K., and Kitano, M., A Mathematical Model fo Spatial Motion of acked Vehicles on Soft Gound. J. eamechanics, 29:71, 635, 1992 New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles Page 6 of 7

8 [3] Gao, Y., and Wong, J. Y., he Development and Validation of Compute-Aided Method fo Design Evaluation of acked Vehicles with Rigid Links. Jounal of Automobile Engineeing, 208D(3):207, 7, 1994 [4] Yakoub, R. Y., and Shabana, A. A., hee Dimensional Absolute Nodal Coodinate Fomulation fo Beam elements: Implementation and Applications. ASME J, Mech. Design, 123, , 2001 [5] Shabana, A. A., Hamed, A. M., Mohamed, A. A., Jayakuma, P., and Lethewood, M. D., Development of New-hee Dimensional Flexible- Link Chain Model Using ANCF Geomety. echnical Repot # MBS UIC, Depatment of Mechanical Engineeing, Univesity of Illinois at Chicago, Chicago, 2011 [6] Maqueda, L. G., Mohamed, A. A., and Shabana, A. A., Use of Geneal Nonlinea Mateial Models in Beam Poblems: Application to Belts and Rubbe Chains. ASME J. Comput. Nonlinea Dynam., 5:2, , 2010 [7] Choi, J. H., Lee, H. C., and Shabana, A. A., Spatial Dynamics of Multibody acked Vehicles, Pat I: Spatial Equations of Motion. Vehicle System Dynamics, 29, 27-49, 1998 [8] Ryu, H. S., Huh, K. S., Bae, D. S., and Choi, J. H., Development of a Multibody Dynamics Simulation ool fo acked Vehicles. JSME Int. J., 46C:2, , 2003 [9] Pedeson, S. L., Model of Contact Between Rolles and Spockets in Chain-Dive Systems. Appld Mechanics, 74:489, 508, 2005 New Finite Element / Multibody System Algoithm Fo Modeling Flexible acked Vehicles Page 7 of 7