A NON-LINEAR MODEL FOR STUDYING THE MOTION OF A HUMAN BODY. Piteşti, , Romania 2 Department of Automotive, University of Piteşti
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1 ICSV Carns ustrala 9- July 7 NON-LINER MOEL FOR STUYING THE MOTION OF HUMN OY Ncola-oru Stănscu Marna Pandra nl Popa Sorn Il Ştfan-Lucan Tabacu partnt of ppld Mchancs Unvrsty of Ptşt Ptşt 7 Roana partnt of utootv Unvrsty of Ptşt Ptşt 7 Roana s_doru@yahoo.co bstract In our papr w dvlopd a non-lnar odl for studyng th otons of a huan body. For our odl w obtand th quatons of oton usng th Nwton s quatons. W also tratd a fw aspcts concrnng th stablty of th qulbru and oton.. INTROUCTION Th chancal odl s capturd n fgur. W ad th followng notatons: - ass of th body C - wght cntr of th body J - nrtal ont of th body wth rspct to th wght cntr C - dstanc btwn and C asurd n th drcton of th as y C C l C l M C E C l hl C L F - lastc forc n th sprng btwn th bods and - lastc ont n th sprng btwn th bods and M - lastc ont n th sprng btwn th bods and lastc forc n th F sprng btwn th bods and lastc forc n th sprng btwn th bods and d - dstanc btwn C and C asurd n th drcton of th as and for unstrssd sprngs - couloban frcton forc btwn th bods and N - noral racton F f btwn th bods and N - noral racton btwn th bods and. W shall consdr that all lastc forcs and onts ar qual to zro for zro strans. F. ISOLTION OF THE OIES For th fourth body w obtan th followng quatons (fg. :
2 ICSV 9- July 7 Carns ustrala Fgur. Mchancal odl. ( y g V H Fgur. Isolaton of th bods. ( J H l cos V l sn M. ( For th thrd body w obtan: H H F ( l sn ( y g V V H ( ( M ( l cos H cos L ( l ( V sn V L l M F l sn l l J l sn ( ( cos. For th scond body and th frst body t follows
3 ICSV 9- July 7 Carns ustrala ( V g ( H µ N F N rspctvly ( F ( l F ( µ N F sn N g N. (. OTINING THE SECON ORER IFFERENTIL EQUTION SYSTEM It s asy to obtan th followng rlatons: J ( y g l sn M ( J l cos (5 [ F ( l sn ] l cos ( L l cos ( y g y g l sn ( y g( L l sn M ( M ( F ( l sn ( l l cos µ On th othr hand [ F ( l sn ] [ y y ( g] F ( ( F (6. (8 (7 cos l l sn y l sn l cos L cos L sn l cos l sn y L sn L cos l sn l cos (9 and rplacng n th quatons (5-(8 w obtan th syst ( whr l cos l cos L cos l cos L sn l sn Ll cos J l cos l sn J l ( ( l cos ( L l cos ( l cos ( L l cos ( l J L l l cos( l ( L l ( cos ( (
4 ICSV 9- July 7 Carns ustrala l cos L cos µ l sn µ L sn l cos µ l sn l cos L cos l cos ( ( sn( gl sn M ( sn( F ( sn l l cos gl sn M ( M ( ( sn µ cos L ( sn µ cos ( sn µ cos F ( l sn µ ( g F ( sn L sn l F ( Ll l L gl sn l l l sn hnc ( j j ( j. Th syst ( lads to (5 f( f( f ( f (. notng ξ ξ ξ ξ ξ5 ξ6 ξ 7 ξ8 (7 t rsults th frst ordr non-lnar dffrntal quatons syst (6 dξ dt ξ dξ dt f ( ξ K ξ (8 Lt us obsrv that th syst (8 s valuabl for N othrws th syst has fv dgrs of frdo. Our syst has four dgrs of frdo : and.. THE STUY OF THE EQUILIRIUM In ths cas w obtan. (9 W study th qulbru for th lastc forcs and onts gvn by lnar rlatons
5 ICSV 9- July 7 Carns ustrala F ( k ( l sn k( l sn M ( k ( M ( k ( F ( L th bars ar hoognous and l l It follows th syst L l µ.. ( Subtractng th thrd and th fourth quaton ( tr by tr t rsults (. ( W obtan a syst of thr quatons wth thr unknowns. ( notng ( on obtans th qulbru condtons or quvalntly (5 ( ( ( ( ( ( ( ( (. (6 5
6 ICSV 9- July 7 Carns ustrala It rsults two stuatons: thr or th dtrnant of th syst n th unknowns s qual to zro. Th scond stuaton would ply that th quatons ( ar not lnar ndpndnt whch s absurd. It rans (7 Fro th dfnton of th paratrs t follows th qulbru postons gvn by. (8 Th study of th stablty of ths qulbru poston wll b ad nurcally consdrng a dvatd poston charactrzd by ξ 8 wth ξ suffcntly sall. W slctd th followng valus: [ kg] 6.68 [ kg ] 6.6[ kg] L.7[] L.[ ] k [ ] 8 N for rad othrws k [ N ] rad 5[ N ] rad k N k 6[ ] rad N k 6[ N ] k 8[ ] N. 5.5[ kg] k for othrws [ ] Th stp t s t.[] s and th ntal valus ar π [ rad].[ rad ] 6 Th graphcs ar plottd n th nt fgurs. (9. s ( [] t[s] Fgur. T hstory for. It s asy to obsrv th quas prodcty of th dagras so th qulbru s a sply stabl on. Th qulbru bcos f k for <. Th doan of stablty ncrass around th qulbru poston (8 f th frcton coffcnt has a non-zro valu. 6
7 ICSV 9- July 7 Carns ustrala thta[rad] t[s] Fgur. T hstory for. 5. CONCLUSIONS In our papr w prsntd a four dgrs of frdo odl for study th oton oh a huan body. Ths odl charactrzs th bhavor of th huan body n a car. W obtand th quatons of oton and w study th stablty of th qulbru poston n a partcular cas. REFERENCES []. K. rrowsth and C. M. Plac C. M. n Introducton to ynacal Systs Cabrdg Unvrsty Prss 99. [] H. W. ror. Krauskopf and G. Vgtr (ds. Global nalyss of ynacal Systs Insttut of Physcs Publshng rstol and Phladlpha. [] J. Gucknhr and P. Hols Nonlnar Oscllatons ynacal Systs and furcaton of Vctor Flds Sprngr Vrlag Nw York rln Hdlbrg London Pars Tokyo Hong Kong 98. []. I. Lur nalytcal Mchancs Sprngr Vrlag rln Hdlbrg Nw York arclona Hong Kong London Mlan Pars Tokyo. [5] Ju. I. Nĩark and N.. Fufav ynacs of Nonholonoc Systs Translatons of Mathatcal Monographs vol. rcan Mathatcal Socty Provdnc Rhod Island 97. [6] L. Prko ffrntal Equatons and ynacal Systs Sprngr Vrlag Nw York rln Hdlbrg 996. [7] N. Pandra and N.-. Stănscu Mchancs dactcal and Pdagogcal Publshng Hous ucharst. [8] N.-. Stănscu L. Muntanu V. Chrou N. Pandra ynacal systs. Thory and pplcatons Roanan cady Publshng Hous ucharst 7. [9] N.-. Stănscu Nurcal thods dactcal and Pdagogcal Publshng Hous ucharst 7. [] P. P. Todorscu Mchancal systs. Classcal odls Tchncal Publshng Hous ucharst
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