UNIVERSITATEA TRANSILVANIA DIN BRA$OV Catedra Design de Produs (i Robotic*
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1 UNIRSIAA RANSILANIA DIN BRA$O Catda Dsgn d Pdus Rbt* Spznul na9nal u patpa ntna9nal: PRtaa ASIstat: d Calulat P R A S I C ' 0 l. II - Ogan d an. anss an 7-8 Nb Bav Râna ISBN ON H FINI LMNS ASSMBLING IN H LASO-DYNAMIC ANALYSIS OF H MCHANISMS Sn LAS Unvsty RANSILANIA f Bav Abstat: Whn th dyna analyss f a ult-bds syst hans) s ad n ptant stp s t assbl th tn uatns. In ths uatns appa as unknwns bth ndal dsplants and ndal fs lasn fs). h pap psnt thd f lnatng ths lasn atns that appa n th tn uatns f a hanal syst analyzd by fnt lnt thd. h sult s ptant t btan th fnal dffntal uatns n ts f fnt lnts wth ndal ndpndnt dnats and wthut lasn ndal) fs. In ths way th dffntal -algba syst a tansfd nly n a dffntal syst wthut lasn fs as unknwns. Ky wds: fnt lnt thd hans lasn fs. 1. Intdutn In any ass whn a study f a ult-bds syst s pf th bas hypthss usd s that all lnts a gd. In alty th lastty f th pnnts an b lag nugh s that th dyna spns an b nt nly uanttatv but als ualtatv dffnt. F ths asn n s applatns patulaly n th fld f bts and hgh-spd vhls s nssay t nsd th lastty f lnts and t us spndnt dls. Gnally th ult-bds systs hav a gat plty and a stng nn-lnaty. study suh systs wth th lass hans ths s nt a patal task baus th tn uatns hav gnally n analytal slutns. F ths asn s nssay t us nual thds and th fnt lnt thds FM) ans n f th st ptant tls [1]-[4][6]- [8][9][1]. Usng fnt lnt thd n ptant stp s t ad th assbly f th tn uatns f ah lnt wttn n a lal dnat syst t btan th fnal uatns n th gnal dnat syst. In th pap a stablshd th nntal tn uatns f a gnal ult-bds syst wth last lnts bng n a th-dnsnal tn and s psntd a pdu t lnatng th lasn fs whn th assblng pss s ad.. Mtn uatns In th fllwng w wll stablsh th tn uatns f an last fnt lnt wth a gnal tn tgth wth an lnt f th syst. h btand uatn a patal f th sa f f w nsd a plan lnt a n-dnsnal lnt. h typ f th shap funtn s dtnd by th typ f th fnt lnt and wll dtn th nt valus f th uatns ffnts. W wll nsd that
2 158 th sall dfatns wll nt afft th gnal gd tn f th syst. W nsd that f th all lnts f th syst w knw th fld f th vlts and f th alatns. W f th fnt lnt t th lal dnat syst Oyz bl and havng a gnal tn wth th pat f syst nsdd Fg.1). W dnt wth v X& Y& Z& ) th vlty and wth a X&& Y&& Z& ) th alatn f th gn f th lal dnat syst. h tn f th whl syst s f t th gnal dnat syst O XYZ. By [R] s dntd th tatn s pss by at. h pstn vt { M { { { { [ R]{ ' '. M W ntd by { ' th pstn vt MM wth th pnnts n th gnal dnat syst O XYZ. h knt ngy f th fnt lnt nsdd s 1 1 v d { vm ' { vm 'd 4) wh F s th ass dnsty. h latns btwn stans and fnt dfatns a { [ a]{ f wh [a] s a dffntatn pat and th dfatn ngy s p 1 { [ k ]{ d wh [ k ] 5) s th gdty at f th lnt [ k ] [ a][ N] ) [ D] [ a][ N ]d. 6) h gnalzd Hk s law hav th f { [ D]{. th dstbutd fs vt th tnal wk f ths s If w nt wth { p { p y z ) {{ p f d {[ p N ] d { 7) W Fg. 1. Fnt lnt n a th dnsnal tn M h pnt M has a dsplant { f { { [ R] { ' { f ) M and b ' 1) wh { M ' s th pstn vt f pnt M wth th pnnts pss n th glbal fn syst. h ntnuus dsplant fld f y z t ) s appatd n FM by { { f [ N y z )]{ t ) ) wh th lnts f at [N] th shap funtns) a dtnd by th typ f th fnt lnt hs. h vlty f pnt M wll b { v { [ R& ]{ [ R& M ]{ f [ R ]{ f & ' & ' { & [ R & ]{ [ R & ][ N ]{ [ R][ N ]{ & '. 3) and th ndal fs { wk W { { pdu an tnal. 8) h Lagangan f th nsdd lnt s btan wth th latn L p W W. 9) W apply th Lagang s uatns d L L 0. dt & Rgd bdy hypthss ps th fllwng latn [ R & ] [ R] f th angula vlty pat n th glbal syst f fn and
3 [ R & ] [ R] [ R& ] [ R& ] wh s th angula alatn pat wttn n th glbal fn syst. W wll dnt by and 159 th angula vlty pat and th angula alatn pat n spt wth th lal dnat syst. It sts th latns: [ R] [ R] [ R] [ ] [ R]. Aft s algba patns w btan th tn uatns f a sngl fnt lnt und th f { N N d & N R R& N d {& [ k ] [ N ] [ R] [ R& ][ N ] d { { [ N ]{ p d & [ N ] d [ R]{ & [ N ] [ R] [ R&& ]{ ' d. 10) [ N ] If w nt by N 1 ) N ) N 3) w btan th lns f at [ N ] [ N ] d N N N N N N )d 1) 1) ) ) 3) 3) [ N ] [ R] [ R& ][ N ] d [ N ] [ N ] d [ ] [ ]) [ ] [ ]) [ ] ) 3 3 y z 1 1 [ N ] [ R] [ R&& ][ N ] d [ ] [ ]) [ ] [ ]) [ ] ) 3 3 y z 1 1 ) ) ) y z 11 z y 33 [ ] [ 1] ) y [ 3] [ 3] ) yz [ 31] [ ]) z 1 13 [ N ] [ R] [ R&& ]{ ' d [ ] [ ]) [ ] [ ]) [ ] [ ] 3y z y 1z 3 z 1y ) ) )[ ] )[ ] y z 1 z y y 3z [ ] [ ]) [ ] [ ]) [ ] [ ] z ) z 1 y y z 3y y z 3 1. In ths latns w hav dntd: { [ N ) ] d { y [ N ) ] y d { z [ N ) ] zd [ j ] N ) N j) d * { [ N ] { p d [ ] [ N ] d. If w dnt: [ ] [ ] [ ] [ ] ) N ' { {d [ )] [ N] [ N ] d [ k ) ] [ N] [ N ] d [ k )] [ N ] [ N ] d { ) { N ' d t sult th tn uatns f th fnt lnt analyzd n a pat f [ ]{ && [ ]{ & [ k ] [ k ) ] [ k )] * { { { ) { ) [ ][ R] {& ){ 11) wh psnt th angula vlty and th angula alatn wth th pnnts n th lal dnat syst. hs tn uatns a fd t th lal dnat syst and th ndal dsplant vt { and th ndal f vt { { * a pss n th sa dnat syst. h tn uatns a tu f th nstantanus pstn f th syst. W nsd that th syst s fzn f th nt nsdd. h pssn 11) ntan s akabl ts: [ ]{ & - psnt th Cls alatns and th aus f ths s th latv vlty {& f th ndal dnats [ k ) ] [ k )] - dfy th stffnss at and th aus a th latv tn pss by th angula vlty and angula alatn { ) { ) - psnt th nta ffts du t th tatn f th lal dnat syst [ ][ R] {& - psnt th nta ffts du t th tanslatn f th fnt lnt. Whn th tw-dnsnal fnt lnt s n a plan tn f w us th sa pdus w btan th tn uatns f ths as ){ [ ]{ && [ ]{ & [ k ] [ k ] [ ] * { { { ) { ) [ ][ R] {&. 1)
4 In th as f a n-dnsnal fnt lnt th sts s spal fs f th dfatn ngy. W ust tak nt aunt that th snd d ffts ak stff th lnt whn ths pf a tn wth a hgh spd. Fnally w an btan f ths stuatn th tn uatns G [ ]{ && [ ]{ & [ k ] [ k ) ] [ k )] [ k ] { { { ) { ) [ ][ I ]{ [ ][ R] { & ){ * 13) G wh at [ k ] 160 tak nt aunt th snd d ffts and th t [ ][ I]{ dsb th nflun f th tatn nta. h shap funtns wll dtn th fnal f f th at nsdd n ths uatns. 3. Assblng Pdus and Lasn Fs lnatng 3.1. Knats In th fllwng th auths psnt an analyt thd t justfy th assblng thds usd f ths typ f systs. h unknwns n th lastdyna analyss f a hanal syst wth lasns a th ndal dsplants and th lasn fs. By assblng th tn uatns wttn f ah fnt lnt w ty t lnat th lasns fs and th tn uatns wll ntan nly ndal dsplants as unknwns. h lasn btwn fnt lnts a alzd by th nds wh th dsplants an b ual an sts th typ f funtnal latns btwn ths. Whn tw fnt lnts blng t tw dffnt lnts bds) th lasn alzd by nd an ply latns platd btwn ndal dsplant and th dvatvs. Gnally th latns btwn th fst d dvatv f th ndal dsplants an b pssd by th lna fulas and th dsplants pssd n a syst { th lal bl dnat syst { { { 16) R wh nd dnt th -th lnt. 3.. Dyna dsptn f th syst F a sngl fnt lnt that blng t an last pnnt f th syst that has a gnal th-dnsnal gd tn wth th angula vlty and th angula alatn and n th bl dnat syst) w nsd th tn uatns btand by th latn 11). F th th ass th pdus a th sa. h uatns a pssd n th lal bl fn syst. If w wt ths uatns n th glbal f dnat syst thy kp th f ){ [ M ]{ && [ C ]{ & [ K ] [ K )] K ) * { { { ) { ) [ Q Q Q Q M ][ R] { & 17) and w an btan fnally th tn uatns f th whl stutu fd t th glbal dnat syst und th f ){ [ M ]{ && [ C]{ & [ K ] [ K )] K ) t * t { Q { Q { Q lg { Q nta. 18) If w tak nt aunt th latns 18) and 0) w an wt [ M ] [ A]{ && [ A& ]{ & ) [ C][ A]{ & [ K ] [ K )] K ))[ A]{ t * t { Q { Q { Q lg { Q nta. 19) {& [ A ]{ & 14) wh by { w hav ntd th ndal dsplant vt and by { th ndal ndpndnt dsplants. By dffntatn 14) w btan {& [ A ]{ && [ A& ]{ &. 15) h tansfatn latns btwn th dsplants pssd n th glbal f dnat 3.3. Wk f lasn fs It an b shwn [10] [11] that th wk f th lasn fs f th syst an b wttn lg lg dl { & { Q dt { & [ A] { Q dt. 0) But th wk du t th lasn fs s null f an dal syst [5] [14] and th ndpndn f th ndal dnats ff th latn
5 [ A ] { Q lg 0 1) that s th bas latn n th fllwng Mtn uatns assblng W nsd latn 19) and w p-ultply ths wth [ A]. It sult [ A] [ M ][ A]{ && [ A] [ M ][ A& ] [ C][ A] ){ & [ A] [ K] [ K )] [ K )])[ A]{ t * t lg [ A] { Q [ A] { Q [ A] { Q * lg nt [ A] { Q { Q ). ) If w tak nt aunt th latn 1) th lasn fs th ndal fs) vansh and t sult a syst f uatns wthut lasn fs and th unknwn a nly th ndal dsplants. hs sult justfy th assblng thds usd n th as f th hanal systs wth lasns analyzd va fnt lnt thd [ A] [ M ][ A]{ && [ A] [ M ][ A& ] [ C][ A] ){ & [ A] [ K ] [ K )] [ K )])[ A]{ t * t [ A ] { Q [ A] { Q [ A] { Q nta. 3) h syst f dffntal uatns btand s nnlna th at f th lft t dpndng n th nfguatn f th ult-bdy syst. hs uatns ntan th gd tn f th syst and f ths thy hav n sngulats. slv th uatns th gd tn ust b lnatd. h pdu psntd justfy th lasn fs lnatng n th tn uatns. Rfns 1. Bahgat B.M. and Wllt K.D. 1976). Fnt lnt batnal Analyss f Plana Mhanss. Mh.and Mah.hy vl.11 p Bljwas ). h Sulatn f last Mhanss Usng Knats Cnstants and Lagang Multpls. Mhans and Mahn hy vl.16 N.4 p Clghn W.L. Fntn.G. abak K.B. 1981). Fnt lnt Analyss f Hgh-Spd Flbl Mhanss. Mhans and Mahn hy vl.16 N.4 p dan A.G. Sand G.N. Oakbg A. 197). A Gnal Mthd f Knt- lastdyna Analyss and Synthss f Mhanss. Junal f ngnng f Industy ASM ans. Nvb p IabC. 1980). Mana tt9. d. Ddat: Pdagg: Buut. 6. Mdha A. 1978). Fnt lnt Appah t Mathatal Mdlng f Hgh Spd last Lnkag. Mhanss and Mahn hy vl.13 p Manu D. Ga I. las S. asu O. 1990). pntal Chkngs n th last-dyna analyss f hans by usng fnt lnts. Int.Cngss On p. Mhans Lyngby Dnak p Nath P.K. Ghsh A. 1980). Knt- lastdyna Analyss f Mhans by Fnt lnt Mthd. Mh. and Mahn hy vl. 15 p las S. 1985). lastdynash Analys d Mhanshn Syst duh d Mthd d Fntn lnt. Bul. Unv. Bav p las S. 1987). A Mthd f lnatng Lagangan Multpls f th uatns f Mtn f Intnntd Mhanal Systs. Junal f Appld Mhans ASM tans. vl.54 n las S. 1987). lnatn f Lagangan Multpls. Mhans Rsah Cunatns vl. 14 p las S. 1994). Mdlng f Multbdy Systs wth last lnts. Zwshnbht. ZB-86 hnsh Unvstät Sttutgat. 13. las S.199). Fnt lnt Analyss f th Plana Mhanss: Nual Aspts. Appl. Mh lsv p las S. 1999). h Lasn Fs lnatng n th last-dyna Analyss f Mhanal Intnntd Systs va Fnt lnt Mthd. h Cnfn ARA Lg.
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