Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS Dnamc Anass Of An Off-Road Vehce Frame ŞTEFAN TABACU, NICOLAE DORU STĂNESCU, ION TABACU Automotve Department, Apped Mechancs Department, Automotve Department Unverst of Pteşt Târgu dn Vae,, Pteşt,, Argeş ROMANIA stefan.tabacu@upt.ro, s_doru@ahoo.com, on.tabacu@upt.ro Abstract: The paper presents computatona methods apped for the stud of an off road vehce frame. The structure s anazed frst ug fnte eement method. Interna energ consumed b the pastc hnges s determned and then the jont stffness s computed. Aso a mutbod mode s deveoped and anazed. Jont data are echanged between the numerca modes and the resuts are compared. Kewords: Vehce frame fnte eements, mutbod method, pastc hnge characterzaton INTRODUCTION Past and modern constructon soutons for off road vehces ncudes a frame that s used as a support for the vehce bod, powertran, aes and other components and t aso has an mportant roe n consumng the mpact energ. Tpca t ma contrbute n consumng the vehce nta knetc energ, b deformaton, about -% of the tota mpact energ. Therefore the confguraton of the frame must comp wth ts functona purpose but t must be aso an mpact effcent structure. The confguraton must account the number, poston and pastc hnges mechansm. Pastc hnge s a coapse mode of the structures and ths process must be ver carefu anazed and understood n order to obtan the mamum of performances from a structure that ma have a roe n goba vehce structura crashworthness. Anatca technques are avaabe and appcabe for a wde varet of modes [,,,]. VEHICLE FRAME MODEL The frame of an off road vehce w be anazed ug a both fnte eement method and a mutbod mode. Fgure present the CAD mode of the entre frame structure. Fg.. Vehce s frame structure The objectve s to stud the functon of the fronta structure of the frame durng a fu fronta mpact. Therefore from the assemb on the frst secton of the frame w be anazed ug numerca methods. Fg.. Front sde frame structure. Numerca mode. MECHANICAL SYSTEM Mechanca sstems ma be represented as a coecton or rgd or febe bodes joned together b knematc jonts and force eements [9]. Mutbod sstems consstng n n b nterconnected bodes are modeed ug a set of n c ndependent coordnates connected through knematc constrans. The number of sstem degrees of freedom s defned to be the number of tota degrees of freedom mnus the number of ndependent constrant equatons.. Generazed coordnates and knematca constrants The confguraton of a mutbod mode s dentfed b a number of varabes caed coordnates or generazed coordnates that must defne compete ISSN: 79-769 8 ISBN: 978-96-7-9-9
Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS the ocaton and orentaton of each bod beongng to the sstem. For D sstems each separate bod beongng to the sstem can be compete defned ug three ndependent coordnates: two coordnates descrbng the ocaton of the orgn of the bod oca coordnate frame and one coordnate descrbng the orentaton of the oca coordnate frame wth respect to the goba or fed coordnate frame. For an arbtrar pont P on the bod () the poston wth respect to the goba coordnate frame (XYZ) s: rp R A u () where R s the poston of the oca coordnate frame (z) attached to bod ; A s the transformaton matr from oca coordnate sstem to the goba coordnate sstem and u s a vector defnng the poston of pont P wth respect to the oca coordnate frame. The vector of generazed coordnates w be q denoted wth and for panar anass the vector s defned as: q [ ] () where and are the coordnates of the oca reference frame and s the rotaton of the oca reference frame about Z as. Fgure presents a panar revoute jont. The two boded share the common pont P. Y R O u R j P u j j O j Fg.. Jont defnton Wth respect to the goba frame the coordnates of pont P beongng to both bodes are the same. Thus one ma wrte that: j r P rp () where denotes the base bod whe j denotes the foower bod n the sstem. Ug equaton and wth respect to equaton the foowng dentt ma be stated [9]: j j j R A u R A u () X j The set of equatons defned between the boded n the sstem ma be defned n condensed form as: Φ( q, q, K qn, t) Φ( q, t) () b Therefore one ma wrte that: Φ ( q) (). Dnamc equatons The constrant equatons are wrtten as: Φ ( q) () whe the veoct constrants are C q (6) Ug equaton Φ C q C q and the sstems constrants Φ ( q) the foowng w be obtaned: C q C q (7) Equaton of moton for a dnamca sstem s: T M q Q D λ (8) Where: M s the nerta matr, Q s the sum Q Q e Q v of groscopc terms ( Q e ) and the eterna apped forces and moments ( Q v ) and λ s the vector of Lagrange mutpers. The equatons of moton of the constraned mutbod sstem ma be wrtten as one matr equaton: T M C q Q C λ C (9) q. Numerca mode The mechanca sstem to be anased has a number of 6 knematca jonts( fg. ). Bod s reated to the ground b a transatona constrant. There are two constraned degrees of freedom, name the vertca dspacement and bod rotaton. Referrng to the net secton () the ground has assocated the goba coordnate frame. The mechanca sstem ma be anased and the tota number of constraned equatons dentfed. As there are 6 knematca jont and n ths case each one of these jons constrans a number of two degrees of freedom the tota number of constrant equatons s. Y ground O (, ) O (, ) (, ) (, ) O O X (, ) O ground Fg.. Equvaent mechanca sstem The sstem s coordnates are defned as foows: ISSN: 79-769 9 ISBN: 978-96-7-9-9
for bod of the sstem the coordnates are defned as foows (6) () For the thrd bod the equatons are: (7) for the second bod the coordnates as defned as foows: For the forth bod the equatons are: () (8) for the thrd bod the coordnates as defned as foows: Or (9) () for the forth bod the coordnates are defned as foows: () Whe for the ast bod of the sstem the equatons are: for the ffth bod of the sstem the coordnates are defned as foows: () () The transatona jont between bod and the ground w add the foowng constrant equatons: The frst tme dervatve of the geometrca constran equatons w defne the veoctes of the bodes of the sstem and the defnton of knematc constrans w be competed. () A number of constrant equatons are obtaned. Combnng these equatons matr w be defned (equaton ). C Therefore the frst bod the foowng constrans are defned: () or wth For the second bod the equatons are: () Once the knematc constraned were defned the rest of the terms presented n equaton (9) must be defned. Durng the mpact the structure s deformed. Man, the energ s consumed b the bendng of the curved secton of the frame. Therefore jonts stffness eements must be added. In ths case the jont stffness s consdered to have a constant vaue obtaned from the numerca anass of the mode ug fnte eement method. The jo t resstant torque defned as n s ( ) ( ) ( ) j j j k j M () Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS ISSN: 79-769 ISBN: 978-96-7-9-9
Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS where k j stffness n s the measured near or nonnear jont Nm rad ; j are the rotatons of two adjacent sectons measured n radans whe are the nta rotaton of the sectons. Y, Fg.. Mutbod mode. Generazed forces. Where the resstant torques are: M k ( Δ Δ ) M k ( Δ Δ ) Μ Μ X (), j. Inta condtons In order to obtan the correct and requred moton of the mechanca sstem correct nta condtons must be defned [, 8]. Inta condtons ncude the correcton of the sstem s geometr and aso the correcton of transatona and rotatona veoctes. The correcton of geometrca postons and veoctes must be performed n accordance wth the constrant equatons defned. Otherwse nconsstent condtons w be set that w ead to a voaton of both geometrca and knematc constrants. Geometrca constrans apped to ths mode w ead to the foowng equates: L L L L () L L L Consderng the current tme the foowng equatons must be apped: L L L L (6) L L L In order to defne the actua postons of the bodes Newton Raphson method s used to determne the correct postons of boded and. The nonnear geometrca constran functons are defned as foows: f(, ) L ( L L L ) (, ) (7) f L ( L L ) The sstem s Jacoban s defned as foows f f J (8) f f Or n an epct form: L J (9) L The souton of the nonnear sstem of equatons s obtaned teratve as foows: J f ( ) () Startng from the nta confguratons: Bod [ ] Bod [. ] Bod [.6. ] Bod [.699.67 ].9. Bod [ ] Ug the Newton Raphson method to fnd the correct and nta poston of the bodes, the foowng sets of coordnates s obtaned: nta corrected Bod [.679. ] [.699.67 ] Bod [.96. ] [.9. ] Fgure 6 presents both nta out of poston confguraton and the fna corrected confguraton. Fg. 6. Mutbod mode. Correcton of the sets of coordnates. Whe for the geometrca confguraton of the sstem s obtaned sovng a nonnear equatons sstem the nta veoctes of the bodes are obtaned sovng the foowng sstem. Cd C v d () I v v where the Jacoban matr C d s parttoned as: C d [ C d C ] () or as presented n an epct form: As for the poston an teratve computatona method was used n ths case the souton s obtaned drect n one step. vd Cd C () v I v Startng from the vector of nta veoctes where the vaues for the transatona O veoct of bod and anguar veoctes of boded and, parameters that were set as ndependent coordnates the vector of a transatona and anguar veoctes of the sstem s bodes are we defned. ISSN: 79-769 ISBN: 978-96-7-9-9
Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS T [ V ] () RESULTS AND DISCUSSION The vehce structure w be tested under a constant apped force condton. Frst a numerca smuaton ug fnte eement method s performed and the resuts are anazed. The tota smuaton tme s of. s and. s for mutbod and a second run ug the fnte eement mode. Fg. 7. Smuaton mode Fgure 8 presents the deformed structure at the end of the smuaton tme. As the jont w be represented as a nonnear torson sprng the stffness ma be computed ug equaton (): U k () Where: -U s the computed nterna energ; -k s the jont stffness; - s the bendng ange measured from the nta confguraton. Anatca technques can be used to descrbe the pastc hnge consderng the secton dmensons and the matera mode []. The pastc moment s cacuated and then apped n equatons that ma defne the reacton torque [6]. The jont stffness dependence s used for the mutbod mode of the frame. The resuts of the smuaton ug fnte eement method are presented n fgure, whe fgure presents the souton ug the mutbod mode. Fg. 8. Deformed structure Smuaton data can be post processed n order to obtan pastc hnges data. Tpca structure sectons can be represented as a coecton of rgd bodes connected b sprng tpe eements. These eements ma be defned as near or nonnear dependng matera, geometr, deformaton pattern, user needs and appcaton. Fgure presents the jonts stffness as a functon of the reatve bendng ange. Fg.. Frame structure fnte eement mode Fg.. Frame structure mutbod mode Fg. 9. Jont stffness It mat be notced that there s a good agreement between the fnte eement mode and the mutbod mode s obtaned. Regardng the jonts stffness, the vaues used for the mutbod mode can be verfed b anag mode s resuts as the transatona veoct of the frst bod. If the jont response for the mutbod mode s the one requred then the resuts (fnte eements mode and mutbod) shoud match. Otherwse f a hgher stffness s used the bod w ncrease ts ISSN: 79-769 ISBN: 978-96-7-9-9
Proceedngs of the 8th WSEAS Int. Conf. on NON-LINEAR ANALYSIS, NON-LINEAR SYSTEMS AND CHAOS veoct sower. Fgure presents the veoct of the frst bod compared wth the resuts obtaned from the fnte eements smuaton. Fg.. Resuts anass. Mutbod vs. fnte eements mode Fgure presents the veoct hstor of the frst bod when a hgher vaue for the jonts stffness s used. Fg.. Resuts anass. Mutbod vs. fnte eements mode Anatca mode to stud the pastc hnges are avaabe n the terature []. B defnng the requrements of such a jont / pastc hnge the geometrca dmensons can be defned. CONCLUSIONS Numerca method are necessar toos for structura performance mprovement. Fnte eement method s a ver detaed, and structures can be anazed n a ver detaed manner. The man shortcomng of the method s that t requres specazed software to run the smuatons. On the other hand mutbod methods can be used ver eas because s based on anatca structure descrpton s t does not requre sophstcated sover. The paper presents both fnte eement and mutbod method apped for the stud of a vehce frame. Athough some of the nput data for the mutbod mode are based on the fnte eement smuatons the pastc hnges frame sectons can be aso anazed ug phsca testng. Further work w contan optmzaton methods [7] apped to the structure and mpact smuatons [] wth nta veoct condtons and nterna mpact/contact forces [,,]. References [] Ambroso, J.A.C., Contact and mpact modes for vehce crashworthness smuaton, Int. J. Crashworthness, 8 (), pp.7-8; [] Ambroso, J.A.C., Vehce structura mpact and occupant bomechancs n a mutbod ntegrated envronment, Int. J. Crashworthness, (999), pp.9-8; [] Bake, G., Jezeque, L., Smpfed crash modes ug pastc hnges and arge curvature descrpton, Mutbod Sst. Dn., 9 (), pp.-7; [] Bajer, W., Emnaton of constrant voaton and accurac aspects n numerca smuaton of mutbod sstems, Mutbod Sst. Dn., 7 (), pp.6-8; [] Chang, C.C., Peng, S.T., Impusve moton of mutbod sstems, Mutbod Sst. Dn., 7 (7), pp.7-7; [6] Consger, L., Pres, E.B., Anatca approach for the evauaton of the torques ug nverse mutbod dnamcs, Mutbod Sst. Dn., 8 (7), pp.7-8; [7] Etman, L.P.F., Campen, D.H., Schoofs, A.J.G., Desgn optmzaton of mutbod sstems b sequenta approach, Mutbod Sst. Dn., (998), pp.9-; [8] Nkravesh, P., Inta condton correcton n mutbod dnamcs, Mutbod Sst. Dn., 8 (7), pp.6-8; [9] Shabana, A.A., Mutbod Sstem Dnamcs, Cambrdge Unverst Press, 6. [] Sharf, I., Zhang, Y., A contact force souton for non-codng contact dnamc smuaton, Mutbod Sst. Dn., 6 (6), pp.6-9; [] Tabacu, S., Pandrea, N., Numerca (anatca-based) mode for the smuaton of the fronta coson, Int. J. Crashworthness (8) pp. 87-. ISSN: 79-769 ISBN: 978-96-7-9-9