he h Inenaonal onfeene Advaned opose Maeals Enneen OMA 8- Oobe Basov Roana MAHEMAIAL MODEL O HE DUMMY NEK INLUDED IN A RONAL IMPA ESIN SYSEM unel Sefana Popa Daos-Lauenu apan Vasle Unves of aova aova ROMANIA unelsefana@ahoo.o Unves of aova aova ROMANIA popadaoslauenu@ahoo.o anslvana Unves of Basov Basov ROMANIA.ovdu@unbv.o Absa: Boehans s essenall he sene ha uses ehanal laws n sudn boloal suues suh as usles laens jons and ohe bod suues. I s poan n undesandn he ehans of nju o he eval spne -7. e developed a aheaal appoah o ehanal behavo of eval veebae and ne-veebal dss n es of laboao esn. he pape pesens he aheaal odel based on Laane euaons fo a fonal pa esn anoeval sse. he seps fo he solvn hese euaons ae eposed n he pape. hese sudes wee obned wh EA and dna analss fo deenaon of onsans foes and funonal elaons. Resuls and onlusons ae analed n he fnal. Kewods: aheaal odel du fonal pa ne esn sse vual esn sse. INRODUION Boehans s essenall he sene ha uses ehanal laws n sudn boloal suues suh as usles laens jons and ohe bod suues. I s poan n undesandn he ehans of nju o he eval spne -7.e developed a aheaal appoah o ehanal behavo of eval veebae and neveebal dss n es of laboao esn.. PREPARIN HE MAHEMAIAL MODEL o hs sud was onsdeed he hee-densonal sse onssn of odel used o desn he es sse. he obned du ne base flanes onsdeed fed and wo spheal eal aps have ehanal and aheaal odel pesened n ue. he ene odel wos wh ousde foes: - A sse eenal foe e whh sulaes he foe an on he head pa foe of he fo: e e penu [ e e e penu j penu whee: e - au oponen ha as on OX as du head - he e fo whh he pa foe neases o he au - e fo whh he pa foe deeases o he nu -e peod afe he sse pa. - oes of av of he ehanal sse oponens le: v d v d v d v d f s sf whee: v he ass of eall ds d he ass of elas ds ubbe o slone 67
v he ass of eall ds d he ass of elas ds ubbe o slone v he ass of eall ds d he ass of elas ds ubbe o slone v he ass of eall ds d he ass of elas ds ubbe o slone a he ass of jon e s he ass of lowe spheal ap of he eleen sf he ass of uppe spheal ap of he eleen - un veo of OZ as. - Lonudnal foes due o elas o slone ubbe dss le: e e e e whee: lonudnal elas onsans oespondn o elas foes e lonudnal enealed oodnaes =. - ansvesal elas foes due o elas o slone ubbe dss le: hee: ansvesal elas onsans oespondn o elas foes ansvesal enealed oodnaes =. he ehanal odel of he suded assebl was shown n ue. ee hosen as enealed oodnaes he oodnaes:. he sses of oodnaes O O O s O ae fed on he eleens. Z=Z =Z =Z =Z Z e sf s a v d v d e X X Y v d e X X Y Y v d e Y X O Y X e ue : he ehanal odel of he suded assebl In ue he elas dss wee eplaed b d dss and b lonudnal and ansvesal spns and lonudnal onsans le and ansvesal onsans le o be deened b vual esn. 68
69 Elas ds ubbe o slone Meall ds alunu d v e ue : Mehanal euvalen of he elas dss o oban he Laane euaons s neessa o epess he oodnaes of he ene of ass dependn of enealed oodnaes pesened n ue [ [ [. hus an be wen: 5 whee: he oodnaes of he ass ene fo he eleen he oodnaes of he on O of he oposed eleen epoed o he wold sse O he oodnaes of he ass ene fo he eleen epoed o he own sse O u =osu he deo osnus of he u and as. X Z X X Z O O ue : he laeal vew of he eleen Known ha 6 an be wen: 7 In hs ase: 8 ha s an epess he oodnaes of ene of ass of a opound eleen : 9 an no aoun he hosen enealed oodnaes euaons 9 beoe: Afe he devaon of he euaon we oban:
7 Non: d v he ne ene of he eleen oespondn o he enealed oodnaes s ven b: Replan oodnaes devaves of he ass enes n he ne ene euaon we oban: [ [ o he alulaon of enealed foes s epessed fo he sa vual eleena ehanal wo : M 5 whee: - oal eenal foe an on he eleen - eleena vual dsplaeen of he poson veo M - oal eenal oue oenul an on he eleen - eleena vual dsplaeen of he anle. In he ase of he suded odel he oue an on eleen was onsdeed nvald so he vual eleena ehanal wo beae: 6 he eenal foe sse an on eleen s: d v [ 7 he eleena vual dsplaeen of he poson veo s: j 8 e an we: 9 And: In hs suaon we an epess he vual eleena ehanal wo fo eleen : } [ { d v he vual eleena ehanal wo fo eleen beae: Slal he alulus was ade fo he eleens and. San fo he euaon of he ne ene we alulae paal devaves le and hen d d. Known ha he Laane euaons has he ne shape [ [ [: Q d d whee: he ne ene of he ehanal sse he uen enealed oodnae Q he uen enealed foe. e an epess fo he sa he ne paal devaves: d d 5 Ne we an epess he es le : 6
7 enealed foes ae ven b he euaon Q : Q Q Q [ penu penu penu Q e e Q 7 Q Q [ penu penu penu Q e e Usn he pevous euaons we epess he Laane euaons: [ penu penu penu e e 8 [ penu penu penu e e. HE RESULS O HE SOLVIN O HE LARANE EQUAIONS. RESULS. ONLUSIONS he sse of dffeenal euaons 8 has he follown soluons fo needae ubbe dss onsans wee deened usn vual epeenal es based on fne eleen analss [ [5 pendulu ass = nal pendulu anle = pendulu ass and anle deene he foe pesened n ue : =5.7-6 sn.988 =5.7-6 sn.988 =5.7-6 sn.988 =.55 - sn.65 =-.575-6 sn9.985 9 =-.575-6 sn9.985 =-.575-6 sn9.985..5 7.775.775 7 sn.8. [.. sn. 5
Euaons 9 defne he ene neaal behavou of he epeenal esn sse bu we ae ve neesed b he behavou of he ene of ass of eleen. In he ne daas wee pesened he oponens of he posons veloes and aeleaons on OX and OZ as fo needae dss ade b ubbe. ue : OX and OZ poson of he eleen head of he du ue : OX and OZ veloes of he eleen head of he du ue 5: Aeleaon and he ajeo of he eleen head of he du Usn he sae aloh an be deened he euaons fo he behavo of he du ne usn slone dss. Also hese esuls wll be opaed wh epeenal daa obaned on a du head-ne esn deve. REERENES [ Buule M. Man M. Eleene de eana ehna Edua Unvesaa aova 99. [ Buule M. Rndeu P. Modelaea opuaonala a sseelo eane oble Edua Unvesaa aova 995. [ Duu Nolae Meanse spaale. Modelae neaa s dnaa pn eode opueae Edua Unvesaa aova 999. [ Adas Use ude and uoals -. [5 Soldos Use ude and uoals 995-. 7