Proceedings of the 2nd WSEAS Int. Conference on Applied and Theoretical Mechanics, Venice, Italy, November 20-22,

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1 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, Dynamcs Modlng Analyss o th Mchansm Systm asd on Rgd body Moton and Elastc Moton YANG YUAN-MING, ZHAO ING CHEN CHUAN-YAO SONG TIAN-XIA. Collg o Cvl Engng.and Mch.,.Dpartmnt o Cvl Engng.. Huazhong Unv. o Sc. and Tch.,. Nanyang Insttut o Tch.. Wuhan, ,.Nanyang, Hnan, , P.R.CHINA Abstract: Dynamcs modlng o th mchancal systm wth lxbl dormaton and rgd body moton ar dscussd. Rgard gnralzd coordnat as rgd body moton dgr o rdom and lastc dormaton dgr o rdom, utlz th nt lmnt mthod to dscrb moton and dormaton o lastc connctng rod, us Kan quaton to drv th movmnt quaton o th lastc connctng rod organzaton. Ky words: Elastc Dormaton; Th Fnt Elmnt Mthod; Kan Equaton; Dynamcs Analyss Introduc Dynamc analyss o mchansm systm has bn basd on th assumpton that th lnks bhav as rgd bods. Strsss n th mmbrs ar assumd to b only du to nrta orcs and xtrnal loads. asd on ths strsss calculatons th mchansm s dsgnd, bult, and tstd. Ths dsgn procdur s rasonably accurat th lnks bhav as rgd bods. Howvr, as spds o opraton bcom hghr, th nrta orcs bcom qut larg and th lnks undrgo consdrabl dormaton. Undr ths condtons th rgd body assumpton s no longr vald. Th movmnts o mchansm systm can b accurat smulatd by takng nto account th lastcty o th lnks durng smulaton and dsgn procss. Snc lastc bhavor n mchansm systms cannot b compltly lmnatd, mchansm systms would hav to b actvly controlld n ordr to urthr mnmz cts du to lastc dlctons. For such control applcaton t s ncssary to dvlop accurat modls, whch mor ralstcally rprsnt th actual mchansm systms. Th modlng o mchancal systm wth th lastc lnks has bn pad attnton by popl all th tm. Th work can b dvdd nto thr rspcts at prsnt, accordng to ts modlng way: Th rst approach [-5] whch orgnatd arlr, modls th lastc lnks as contnuous systms possssng nnt dgrs o rdom. Th quatons o moton obtand ar nonlnar partal drntal quatons. Ths approach has bn usd to drv quatons o moton, analyz and dtrmn th dynamc rspons o th sldr crank mchansm, whch has an lastc connctng rod and rgd rod. In th scond approach [6-8], th lastc lnks rprsntd as dscrt systms possssng nt lastc dgrs o rdom by usng mthods lk th nt lmnt mthod. Th advantag o usng th nt lmnt mthod to modl th lastc lnks s that t provds a systmatc modlng tchnqu or complx mchansms and lays th groundwork or a gnral approach or th modlng o mchansms. In ths works th nt moton or th total moton o th systm s consdrd to b a suprposton o th rgd body moton and th lastc moton. A thrd approach [9-3] uss th Lagrang Multplr tchnqu to ncorporat ont constrants nto th quaton o moton. Ths approach s broadly applcabl to a larg class o dynamc systms rathr than ust mchansm systms and rsults n a ormulaton that s gnral concs, and convnntly mplmntd on th computr. In ths work, analyzs th manpulator systm o opraton wth lastcs n systmatc way, dynamcs modlng o th mchancal systms wth rgd body movmnt and lxblty dormaton ar dscussd. Rgard gnralzd coordnat as rgd body btwn dgr o rdom and lastc dormaton dgr o rdom, utlz nt lmnt mthod to dscrb moton and dormaton o lastc connctng rod, us Kan quaton to drv th quaton o moton o th lastc connctng rod. Ths knd o quaton o moton can b usd or analyzng ndustry's machnry oprats hands. caus us th nt lmnt mthod to modlng to th lastc pol, no mattr whch knd o complcatd orms t has. Dscrpton o dormaton o th lmnt As shown g., s an lastc mchansm systm,

2 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, oxyz s a systm o coordnats o th nrta, oxyz s th systm o coordnats xd at lnk. Th dormaton o th opraton manpulator systm, can xprss wth th ollowng rlaton: Y O X Z Fg. Dscrpton o dormaton o th lmnt r = T0r (whr T ) () 0 =T0 TT3 T-, T 0 rprsnts th homognous coordnat transormaton matrx rom lnk to lnk, s a 4 4 matrx that rprsnts th rgd body translaton and rotaton translaton o lnk wth rspct to th rrnc coordnat systm oxy z,and s o th ollowng orm: xoo cos x x cos cos x y x z T0 = yoo cos y x cos cos y y x z zoo cos z x cos cos z y z z I only xstd rotaton translaton btwn onts, w hav cosφ cosφ 0 T -, = 0 snφ cosφ Th rst tm drvatv o T -, s: φ sn cos 0 φ φ φ T -, = 0 φ cosφ φsnφ cosφ snφ 0 T -, = = Q φ snφ cosφ 0 T Thus T 0 = QT 0φ T T-, + + T0T Q-T -,φ Q TT T φ + + QTT T φ + + QTT T φ = 0 -, 0 -, 0 -, = Lnk = Q = w T (-a) φ T 0 0 R y T 0 r δ m (G') d G o r y x x Elmnt whr w = Q φ s th oprator matrxq s = th constant transormaton matrx. Drnt T 0 by tm t, w hav T = 0 = ( Q φ ) T 0 + = ( Q φ ) T = ( Q φ ) T + ( Q φ ) ( Q φ ) T (3) = 0 = = Whr φ can b wrttn by ollowng: To th opratng systm o th gnral manpulator, ts quaton o gomtry rstrans can b wrttn: φ ( ϕ, ϕ ) = 0 ( =,, N) (4) Hr ϕ, ϕ show dgrs o rdom o systm, show systmatc numbr o obct. Th partal drvatvs o th rgd body constrant valus φ wth rspct to th valu ϕ o th rgd body dgrs o rdom s dnt as ollowng: φ ' = δφ / δϕ.thror k k k = φ = φ ϕ (5) l k Formula (5) can rgard as th xprsson ormula o th gnralzd spd that s drvatd by nclusv condton o dynamcs, so w can b wrttn as ollowng w = Qφ ϕ = [ Qφ ] ϕ = W ϕ (6) k k k k k k k = k= k= = k= ' k = φ k = whr W Q. Thus 0 k k 0 k k 0 = k= k= k T = φ Q ϕ T = W ϕ T (-b) Th scond drvatv o drvatvs oφ s φ φ = ϕϕ k l + φϕ k k k= l= ϕk ϕ = φ ϕϕ k l +φϕ (7) k k l k= k = So ormula (3) can b shown as ollowng: T = 0 Q( φϕ kϕl+ φ kϕk) T + [ ( Q φ)][ ( Q φk)] T 0 k 0 = k= = k= = ( Q φϕ kϕl + Wϕk) T0 + WWkϕ ϕkt0 = k= (8) 3 Dscrpton o moton o th lmnt To th mchansm systm wth th lastc lnks, shown as g., th lmntal mass δm n lmnt on th lnk. Coordnat systm oxyz s th xd rrnc coordnat systm, oxy z s local coordnat systm, whch s usually th rgd body k 0 0

3 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, poston o ts cntr o gravty. s th lmntal coordnat systm bult n th cntr o gravty o th lmnt. G ndcats th rgd body poston o th lmntal. z y o x lnk y z o lnk x Fg. Systm wth th lastc lnks x y z lnkn A addtonal 4 4 transormaton matrx R s dnd or lmntal. R dpnds on th constant orntaton o lmnt coordnat systm x y z wth rspct to th lnk coordnat systm x.thus s a constant matrx and s o th y z R ollowng orm: cosxx cos cos 0 xy xz = R 0 cosyx cos cos yy yz 0 coszx cos cos zy zz Th product o th transormaton matrcs and R s usd to transorm th lastc T o dsplacmnt d whch s masur n th lmnt coordnat systm to th rrnc coordnat systm. Ths transormaton matrx rqurs only th rlatv angular orntaton o th lmnt coordnat systm wth rspct to th rrnc coordnat systm. Thror th rst dagonal lmnt o R s st to zro. Th poston o th lmntal mass δ m n th rrnc coordnat systm s spcd by vctor R. Vctor R s mad up o two componnts, th rgd body poston o δ m and th lastc dsplacmnt o δ m. Vctor r locats rgd body poston o δ m n lnk coordnat systm x. Vctor r s a constant y z vctor. Th rgd body poston o δ m n th rrnc coordnat systm. s rprsntd by Tr. 0 Vctor d s th lastc dsplacmnt o δ m n th coordnat systm. x. Th lastc dsplacmnt y z o δ m s rprsntd n th coordnat systm oxyz as th matrx product TRd.Th poston o δ m n 0 th rrnc coordnat systm s gvn by: R = T0r+ T0R d (9) asd on th thory o th nt lmnt mthod, th lastc dsplacmnt d o δ m n th coordnat systm x can b xprssd as a lnar uncton y z o th nodal lastc dsplacmnt vctor u as shown orm: d= N u (0) Matrx N contans th nt lmnt shap unctons, whch rlat th nodal lastc dsplacmnt, u th lastc dsplacmnt vctor d. Th nodal rgd body poston vctor p s dnd that s th rgd body poston o th nodal o th lmnt as masurd n th lnk coordnat systm x. Ths y z s a constant vctor as th rgd body poston o th nodal n th lnk coordnat systm x y z ar xd. N Usng ths vctor and th shap uncton, th rgd body poston o th mass δ m s xprssd as r = N p () Equaton () holds or soparamtrc nt lmnts, whch ar th most commonly usd typ nt lmnt. Substtutng quatons (0) and () n quaton (9) th poston o th lmntal mass δ m s xprssd as: R=T () onp +ToRNu Th abov quaton s drntatd wth rspct to tm to dtrmn th vlocty th lmntal mass δ m n th rrnc coordnat systm : R=TNp (3-a) 0 +TRNu 0 +TRNu 0 T s th tm drvatv o T and s th 0 0 u vlocty vctor o th lastc dgr o rdom masurd n th lmnt coordnat systm x y z. Th shap uncton N and th 4 4 matrx R ar constant matrcs and ar unactd by th drntaton wth rspct to tm. Usng quaton (), th abov quaton s xprssd as: R=wT Np +wt RNu +T RNu = = W kϕkt0n p + W kϕkt0r N u +T0R N u k= k= [ WT k 0( Np +RNu )] ϕk +TRNu 0 k = (3-b) Th scond drntaton o R can b drntatd wth rspct to quaton (3-a) R=TNp + 0 TRNu 0 + TRNu 0 +TRNu 0 (4) Substtutng quatons () and (8) n quaton (4), w hav: R = [( Q φϕ ϕ + W ϕ ) T + W W ϕ ϕ T ] N p + k l k 0 k k 0 = k= Q φϕ kϕl + Wkϕk T0 + WW kϕ ϕkt0 R N u = k= ϕ 0 0 (5) = [( ) ] + W T R N u +T R N u

4 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, Th gnralzd partal vlocty, th gnralzd partal acclraton and th gnralzd nrtal orc Th rgd body dgr o rdom ϕ and th lastc dsplacmnt dgr o rdom u ar rgardd as th gnralzd coordnat, and rgard ϕ and u as th gnralzd spd. Rsarch quaton(3-b)and(5) dtrmn gnralzd partal spds o th lmnt mass δ m : v P = WkT0 ( N p +R N u ) +T0R N (6) k = Th acclraton o th lmnt δ m a N = s o R gvn by ormula (5). Th gnralzd nrtal orc rlatd wth th th. N Gnralzd spd F = V F whr F = a N o dxdx so Th gnralzd nrtal orc s xprssd as blow F = V P a P o dxdx (7) Usng th quatons (5) and (6), St T T T T T TT =[TNP], =[TrNp], 0 0 NN pp TT 0 0 RNN p T T T TT =[TRN], th quaton (7) can b 0 0 RR NN xpandd shown as ollowng orm: l F = QW k φϕ kϕl[ TNP] k= = l= QW k φϕ kϕl k= l= = WW ϕ [ TNP] k k k= = k k k= = [ TrNpu ] WW ϕ [ TrNpu ] W W W ϕ ϕ [ TNP] k k k k= = k= Wk W Wkϕ ϕk[ TrNp] u dxdx k= = k= QW k φϕ kϕl[ TrNpu ] dxdx k= = T QW k φϕ kϕl[trn] uu dxdx k= = WW kϕk[ TrNpu ] dxdx k= = T WW kϕk[trn] uu dxdx k= k= WWW k kϕ ϕk[ TrNpu ] dxdx k = T WWW k kϕ ϕk[trn] uu dxdx k = ) WW ϕ [ TrNpu ] k k= = k k= = WW ϕ [TRN] uu k = W [ TrNp] u k W [TRN] u u ) k k = k l = Q φϕ ϕ [ TrNp] W ϕk[ TrNp] dxdx) k = WW kϕ ϕk[ TrNp] Q φϕ kϕl = k k k = T [TRN] u W ϕ [TRN] u WW kϕ ϕk[trn] u = W ϕ [TRN] u [TRN] u (8) Nglct th lttl quantty o th scond-ordr drntaton and kp th trm o ntrsct, hav : F = W [ TrNp] u k k = k k= = WW ϕ [ TrNpu ] WW ϕ [ TNP] k k k= = k k k= = WW ϕ [ TrNpu ] k k k= = k k = WWϕ [ TrNpu ] W ϕ [ TrNp] dxdx) W ϕ [TRN] u k k k = = [TRN] W ϕ [TRN] u u dxdx (9) Arrangmnt th abov ormula, w hav: () Th gnralzd nrtal orc obtand by th rgd body dsplacmnt ϕ : F = { k([ ] [ ] ) [ ]} ϕ k= = W Wk TrNp u ϕ dxdx k= = W W TNP + TrNp u + TrNp ) { [ ] [TRN]}

5 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, W{ Wk[ TrNp] [TRN]} uϕ = k= k = { W [ TrNp] [TRN]} u (0) k ()Th gnralzd nrtal orc obtand by th lastc dsplacmnt ϕ : F = { W [ TrNp]+[TRN]} u k = k k = k= W { W [ TrNp] + [TRN]} ϕ u k k k k= = = dxdx W { W ϕ [ TrNp] + Wϕ [ TrNp] + +ϕ k [TRN]} u W { W [ TNP] + [ TrNp]} k ϕ kdxdx () = k= 5 Th gnralzd actv orc and th dynamcs quaton o moton [3,33] y takng nto account an arbtrary pont o an arbtrary lmnt o body, th strss-stran rlaton can b xprssd as 3 ε = ( W + W +W W, ) α, β α, β β, α γ, α γ β r= () Whr W α W α β = W a rprsnts dsplacmnt, x β componnt whl X a s th postonal coordnat componnt. Usng quaton(), w hav 3 εα, β = [ wα, β + w β, α + ( w r, αwr, β + wr, αw r, β ] (3) r = Τ Th strss s dnotd by: σ = [ σ, σ, σ33, σ, σ3, σ 3]. Th gnralzd actv orc thus nducd (σε) can b wrttn as ollowng two comptnt: M (φ s th charactrstc vctor F = Φ T K Φu = M matrx) T F = Φ K.Whr s th lastc GΦu K G = stnss matrx, KG s th nonlnar constructv stnss matrx. Th gnralzd actv orc can b wrttn as blow: F= F+ F (4) G Accordng to Kan s quaton, th dynamc quaton s wrttn as: F +F=0. (5) 6 Conclusons Ths concluds th applcaton o Kan quaton to drv th quatons o moton at th lmnt. Ths dvlopmnt allows or th ntrdpndnc o th rgd body and th lastc moton. Th lastc lnks ar modld by usng th nt lmnt mthod. Ths quatons n thr nal orm can b usd or ralstc modlng o lnks mchansms wth th rgd body moton and th lastc moton havng closd and opnd loop multpl dgr o rdom chans and gomtrcally complx lastc lnks. Th systm quatons wll b publshd sparatly. Rrncs [] Nubaur A.H., Cohn R., and Hall, A. S., 966,An Analytcal Study o th Dynamcs o an Elastc Lnkag, ASME Journal o Engnrng or Industry, Vol.88, No., 3-37 [] Jasnsk P.W., L H.C., and Sandor G.N., 97,Vbraton o Elastc Connctng Rod o Hgh-Spd Sldr-Crank Mchansm, ASME Journal o Engnrng or Industry, Vol.93, No., [3] Chu S.C., and Pan K.C., 975,Dynamc Rspons o a Hgh-Spd Sldr-Crank Mchansm Wth an Elastc Connctng Rod, ASME Journal o Engnrng or Industry, Vol.97, No., [4] adlan M., and Klnhnz W., 979,Dynamc Stablty o Elastc Mchansms, ASME Journal o Mchancal Dsgn, Vol.0, No., [5] adlan M., and Madha A., 98,Mmbr Intal Ects on th Elastc Sldr-Crank Mchansm Rspons, ASME Journal o Mchancal Dsgn, Vol.04, No., [6] Tadbakhsh I.G., 98,Stablty o Moton o Elastc Planar Lnkags Wth Applcaton to Sldr CRANK mchansm, ASME Journal o Mchancal Dsgn, Vol.04, No., [7] Wnry R.C., 97,Elastc Lnk Mchansm Dynamcs, ASME Journal o Engnrng or Industry, Vol.93, 68-7 [8] Wnry R.C., 97,Dynamcs Analyss o Elastc Mchansms by Rducton Coordnats, ASME Journal o Engnrng or Industry, Vol.94, [9] Erdman A.G., and Sandor H.N., 97, Knto -Elasto dynamcs-a Rvw o th stat o th Art and Trnds, Mchansms and Machn Thory, Vol.7 [0] Iman I., Sandor G.N., and Kramr S.N., 973,Dlcton and Strss Analyss n Hgh-Spd Planar Mchansms wth Elastc Lnks, ASME Journal o Engnrng or Industry, Vol.95, No.4, [] Iman I., and Sandor G.N., 973,A Gnral Mthod o Knto-Elasto dynamc Dsgn o

6 Procdngs o th nd WSEAS Int. Conrnc on Appld and Thortcal Mchancs, Vnc, Italy, Novmbr 0-, Hgh-Spd Mchansms, Mchansms and Machn Thory, Vol.8, [] ahgat.m., and Wllmrt K.D., 976,Fnt Elmnt Vbratonal Analyss o Planr Mchansms, Mchansms and Machn Thory, Vol., 47-7 [3] Nath P.K., and Ghosh A., 980,Stady-Stat Rspons o Mchansms wth Elastc Lnks by Fnt Elmnt Mthod, Mchansms and Machn Thory, Vol.5, 99- [4] Mdha A., Erdman A.G., and Forhrb D.A., 979,A Closd-Form Numrcal Algorthm or th Prodc Rspons o Hgh-Elastc Lnkags, ASME Journal o Mchancal Dsgn, Vol.0, No., 54-6 [5] Nagannathan G., and Son A.H., 986,Nonlnar Modlng o Knmatcs and Flxblty Ect n Manpulator Dsgn, ASME Journal o Mchansms, Transmssons and Automaton n Dsgn, Vol.88, [6] Sunada W., and Dubowsky S., 98,Th Applcatons o Fnt Elmnt Mthods to th Dynamc Analyss o Flxbl Spatal and Co-Planr Lnkag Systms, ASME Journal o Mchancal Dsgn, Vol.03, No., [7] Turcc D.A., and Mdha A., 984,Gnralzd Equaton o moton or Dynamc Analyss o Elmnt Mchansm Systm, ASME Journal o th Dynamc Systms, Masurmnts and Control, Vol.06, [8] Turcc D.A., and Mdha A., 984, Dynamc Analyss o Elmnt Mchansm Systm, ASME Journal o th Dynamc Systms, Masurmnts and Control, Vol.06, [9] Song J.O.,and Haug E.J.,980,Dynamc analyss o Planar Flxbl Mchansms, Computr Mthod n Appld Mchancs and Engnrng,Vol.4, [0] Shabana A., and Whag R.A., 984,Spatal Transnt Analyss o Inrta Varant Flxbl Mchansms Systm, ASME Journal o Mchansms, Transmssons and Automaton n Dsgn, Vol.06, 7-78 [] Shabana A., and Whag R.A., 983,Varabl Dgr-o-Frdom Componnt Mod analyss o Varant Flxbl Mchansms Systm, ASME Journal o Mchansms, Transmssons and Automaton n Dsgn, Vol.05, [] Shabana A.A.,and akr E.M., 986, Gomtrclly Nonlnar o Mult body systms, Solds and Structurs Vol.3, No.6, 0- [3] Whag R.A., and Haug E.J., 98,Gnralzd Coordnat Parttonng or Dmnson Rducton n Constrand Dynamc Systms, ASME Journal o Mchancal Dsgn, Vol.04,47-55 [4] Nkravsh P.E., Haug E.J., 983, Gnralzd Coordnat Parttonng or Analyss o Mchancal Systm wth Nonholonomc Constrants, ASME Journal o Mchansms, Transmssons and Automaton n Dsgn, Vol.05, [5] Man N.K., Haug E.J., and Atknson K.E., 985,Applcaton o Sngular Valu Dcomposton or Analyss o Mchancal Systm Dynamcs, ASME Journal o Mchansms, Transmssons and Automaton n Dsgn, Vol.07, 8-87 [6] Sngh R.P., and Lkns P.W., 985, Sngular Valu Dcomposton or Constrand Dynamc Systms, ASME Journal o Appld Mchansms, Vol.5, [7] Lown G.G., and Jandrasts W.G., 97,Survy o Invstgatons nto th Dynamc havour o Mchansms Contanng Lnks wth Dstrbutd Mass and Elastcty, Mchansms and Machn Thory, Vol.7 [8] Erdman A.G., Sandor G.N., and Oakbrg G.R., 97,A Gnral Mthod or Knto Elasto dynamc Analyss and Synthss o Mchansms, Journal o Engnrng o Industry, Vol.94, No.4, [9] Lown G.G., and Chassaps C.C., 986,Th Elastc havour o Lnkag: An Updat, Mchansms and Machn Thory, Vol. [30] Gar C.W., and Ptzod L.R., 984,ODE Mthods or th Solutons o Drntal/Algbrac Systms, SIAM Journal o Numrcal analyss, Vol., No.4, [3] Drass D., and Kan T.R., 985,Equatons and Moton Govrnng th Dploymnt o a Flxbl Lnkags rom a Spaccract, Th Journal o Astronautcal Scnc, Vol.33, No.4, [3] Nagaraan S., and Davd A. T., Lagrangan ormulaton o th quatons o moton or lastc mchansms wth mutual dpndnc btwn rgd body and lastc motons. Part: Elmnt lvl quatons, Journal o dynamcs, masurmnt, and control, [33] Yang Yuanmng, Dynamc analyss o lxbl body wth dnt movng atttud, Appld mathmatcs and mchancs, 006, Vol.7, No., 9-6