Dynamic modeling, simulation and control of a hybrid driven press mechanism

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1 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 Dynamc modlng smulaon and conrol of a hybrd drvn prss mchansm Mhm Erkan Küük Lal Canan Dülgr bsrac Hybrd drvn mchansm combns h moon of a larg consan vlocy moor h a small srvo moor and a mchansm. Th consan vlocy moor provds h man orqu and moon rqurmns hl h srvomoor conrbus o modulaons on hs moon. Hybrd drvn mchansm has hghr flxbly n producng moon. Dynamc modlng and smulaon of h hybrd sysm s prsnd n hs sudy. moon scnaro s dsgnd o b usd on mal formng procss. Th quaon of moon s drvd by Lagrangan chnqu. cuaor dynamcs for boh axs ar ncludd h a PID conrol algorhm. Th sysm smulaon s prformd h an xplc mhod; h fourh ordr Rung-Kua mhod as an ngraon chnqu o g an approxma soluon. Th smulaon rsuls ar prsnd. Kyords dynamc analyss hybrd prss sysm smulaon PID conrol I. INTRODUCTION HE man prncpl of hybrd drvn sysm s o drv a mchansm h a srvo moor and a consan spd moor Tncludng a flyhl. [1]. Thus s possbl o ak advanag of boh drv sysms. Non-unform moon oupus can b oband by combnng hs moors h such faurs as flxbly and rducd por rqurmn. [ 3]. Th frs hng o do n hybrd sysm analyss s o dsgn knmacs characrscs of h oupu lnk sldr). moon scnaro as dsgnd. I s a quck rs and slo rurn moon. I s dffrn from radonal mal formng moon oband by a sldr crank mchansm. Ths dffrnc s shon n fg.1. fg. 1 sldr moons svn bar lnkag sysm h o dgrs of frdom as slcd as h confguraon. Th schmac llusraon of h hybrd drvn mchansm s gvn n Fg.. Havng dsgnd h sldr moon nvrs knmacs analyss s carrd ou o g knmacs proprs of srvo moor. Th npu paramrs of nvrs chnqu ar knmacs of sldr and h crank drvn by consan vlocy moor. [46]. Th auhors hank o Mnsry of Scnc Indusry and Tchnology for h fnancal suppor undr SNTEZ projc Numbr: 14.STZ.1-I) and Unvrsy of Gazanp RM.1.1 projc) M.Erkan Küük s h Dparmn of Mchancal Engnrng Gazanp Unvrsy Turky corrspondng auhor o provd phon: ; fax: ; -mal: mkuuk@ganp.du.r). L. Canan Dülgr s h Dparmn of Mchancal Engnrng Gazanp Unvrsy Turky -mal: dulgr@ganp.du.r F B BD CD FE ED fg. a -DOF planar mchansm ISSN:

INTERNTIONL JOURNL OF MECHNICS Volum 1 16 Th crank r s drvn by a consan vlocy moor. Th crank r s drvn by a srvo moor. ll nvrs and forard knmacs analyss and moon dsgn crra ar gvn h all dals n [4].

2 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 Th crank r s drvn by a consan vlocy moor. Th crank r s drvn by a srvo moor. ll nvrs and forard knmacs analyss and moon dsgn crra ar gvn h all dals n [4]. lso command graphs usd n Smulaon Rsuls ıv. scon) ar drcly akn from [4]. I s also possbl o smula h hybrd mchansm h SmMchancs hou drvng knmacs xprssons as gvn n fg. 3 [6]. cranks ar akn as h gnralzd coordnas; q1 = q =. Equaon of moon for o axs ar gvn n 3-4). d L L = 3) d d L L = 4) d Ffh lnk s drvn by a consan vlocy moor hras h scond lnk s drvn by a srvo moor as sn n fg... Knc Enrgy of h Sysm Th vlocs of h mass cnr of vry lnks should b oband o calcula h knc nrgy of h mchansm. and Y ar rspcvly h horzonal and vrcal mass cg cg cnr posons of h h lnk of h hybrd mchansm. s h locaon of h cnr of mass of h h lnk. r cg Lnk lnghs of h mchansm s gvn n Tabl 1 n mm. Tabl 1. Lnk Lnghs fg. 3 3D SmMchancs smulaon [6] dynamc modl plays a vry mporan rol n h rms of sysm valdaon analyss and conrol sysm. I s vry ssnal o sudy h dynamc modlng of h hybrd mchansm n ordr o provd h dsrabl prformanc [7 8]. In hs sudy dynamc modl of hybrd sysm s drvd by Lagrangan approach and solvd. DC moor s usd n smulaon of Moor-load modl and PID chnqu s usd n conrol. II. DYNMIC MODELING Lagrangan mchancs s formd by dffrnaon of h nrgy xprssons h rspc o h varabls of sysm and m. Lagrangan xprsson s oband by subracng h ponal nrgy of h sysm from h knc nrgy of h sysm. L=T-V 1) Th quaons of moon ar hn drvd by akng nrgy xprssons of h sysm. d L L = Q d q q ) Q s h gnralzd orqu or forc and q s h gnralzd coordna. Q and q n lnar moons ar h forc F) and h lnar dsplacmn x) rspcvly. Q and q roaonal moons ar orqu ) and h angular dsplacmn ) rspcvly. [8]. Th roaon angls of h r 1 r r 3 r r r Inral valus ar gvn n Tabl. Tabl. Inral Paramrs Paramrs Lnk Lnk 3 Lnk 4 Lnk Lnk6 Sldr mass kg) Mass Momn of Inra kg m ) cg = cos rcg ) Ycg = Y sn rcg 6) cg = 3 rcos rcg cos 3 3 7) Ycg = Y 3 rsn rcg sn 3 3 8) cg 4 cos r6cos 6 rcg cos 4 4 9) Ycg 4 sn r6sn6 rcg sn 4 4 1) cg cos cg 11) Ycg sn cg 1) cg 6 cos rcg cos 13) Ycg 6 sn rcg sn 14) sldr = 1) Y = S s 16) sldr ISSN:

3 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 Vlocy and acclraon of cnr of masss of h lnks ar oband by akng frs and scond ordr drvavs of -16) h rspc o m rspcvly [4]. Vlocs and acclraon of mass cnrs of h lnks ar; V = Y 17) cg cg cg a = Y 18) cg cg cg Knc nrgy of h sysm; T = mv cg I ) msldrvsldr 19) = hr I s mass momn of nra of h h lnk of h mchansm. Th lnks usd n h mchansm ar akn as sold cylndrs [9]. 1 I = m r 3 d ) ) 1 hr m r and d ar mass lngh and radus of h h lnk rspcvly. Toal knc nrgy of h mchancal sysm s gvn n 1). T= m3 r cos rcg 3 3cos 3) r sn rcg 3 3sn 3) ) / m6 r cos rcg cos 6) r sn rcg sn 6) )) / m4 r cos rcg 4 4cos4 r cos 6) r sn rcg 4 4sn4 r sn6) )) / m rcg )) / m rcg )) / msldr r cos r 4 4cos4 r cos 6) ) / m 3 d r )) / 4 m d3 r3 )) / 4 m d4 r4 )) / 4 m 3 d r )) / 4 m 3 d6 r6 )) / 4 1) B. Ponal Enrgy of h Sysm By lookng a fg. pon F s h rfrnc pon n h calculaon of h ponal nrgy of h sysm. Thn h ponal nrgy of h sysm s; 6 V = m gycg m sldr gysldr ) = Th oal ponal nrgy of h hybrd drvn mchansm s gvn n 3). V= mgr 6 sn rcg sn ) mgy 3 rsn rcg sn 3 3) mgy 3 rsn rcg sn 3 3) mgr 4 sn rcg sn 4 4 r6sn 6) msldr gr sn r4sn4 r6sn 6) mgy rcg sn ) mgy rcg sn ) 3) III. MOTOR-LOD MODELING Th lcrc quvaln crcu of h armaur and h frbody dagram of h roor ar shon n fg. 4. Th roor and shaf ar assumd o b rgd [111]. fg. 4 schmac rprsnaon of a DC moor Gar rao of h spd rducr b a N b n = = = a b N a = ) 4) hr = and for spd rducr of srvo and consan vlocy moor rspcvly. and ar h angular vlocs of h npu shaf a and a b h oupu shaf b rspcvly. Na and N b ar h numbrs of h of h gars a and b. I s assumd ha h npu of h sysm s h volag sourc V ) appld o h moor's armaur h oupu s orqu b ). R and L rprsn h armaur currn rssanc and nducanc rspcvly. Shaf b s conncd o h crank of h mchansm n ordr o drv h mchansm. B s h vscous frcon coffcn. l s consan mchancal orqu causd of frcon of gar and brush. J s h oal mass momn of nra ncludng flyhl rducr and roor of moor. J = J J J ) hr J f f r m s h momn of nra of h flyhl momn of nra of h spd rducr and J m J r s h s h momn of nra of h moor roor. Krchhoff volag la s appld around ndngs of h armaur [11]; d V = R L 6) d hr s back lcromov forc of h moor. Nonan quaon s usd o g h orqu quaon blo [11]; da b = n ) oal l B a J 7) d oal and ar hr oal s h magnc moor orqu. dfnd as. oal = K 8) = K a 9) hr K s h moor orqu consan and K s h moor volag consan. In gnral a = n b = nq 3) d V R nk q = 31) d L ISSN:

4 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 = nk n n Bq n Jq 3) b l b for srvo moor s hch s gvn n 3) and b for consan vlocy moor s hch s gvn n 4). Hybrd drvn prss mchansm has go o dgrs of sysm. Equaon 3) s rn for o axs nd and h lnk). Th xprssons oband ar hghly nonlnar and coupld. ll dals ar gvn n [4].. Conrol of Hybrd Drvn Sysm PID conrollrs us a conrol sgnal ncludng poson vlocy and ngral of poson rror funcons. Th mahmacal rprsnaon of sandard PID conrol s dscrbd as follos; [111314]; V = kp q ) ) ) q k a d q q k a I q q d a 33) hr k s h proporonal gan consan k s h p drvav gan consan and k I s h ngral gan consan. Th rqurd angular dsplacmn and angular vlocy of h crank ar q and q. Th acual angular dsplacmn and angular vlocy of h crank ar q a and q a rspcvly. So h moor currn quaon s modfd as follos; kp q ) ) ) q k a d q q k a I q q d R a nk q a d = d L 34) B. Sa Spac Rprsnaon Th dynamc bhavour of h hybrd drvn sysm s sudd by usng a numrcal mhod o g an approxma soluon. Th fourh ordr Rung-Kua s usd as h ngraon chnqu. I s an xplc mhod usd n ngraon of nonlnar sysms. I s ncssary o form saspac rprsnaon. Th am n sa-spac rprsnaon s o nroduc a suabl s of sa varabls. Thn h quaons of moon of h sysm ar formd as a sysm of frs ordr dffrnal quaons [1]. Th angular dsplacmns h angular vlocs and h currns ar rad as h sa varabls. d α and α xprssons ar akn from 3) and 4) rspcvly. Thr xplc forms h addon of h moor dynamcs ar gvn n [4]. IV. SIMULTION RESULTS Modlng s carrd ou o undrsand h dynamc bhavour of h complx sysms. I s usd o nvsga h rlaonshp bn h sysm varabls. Th mahmacal modl of h sysm s dscrbd by a s of mahmacal xprssons. Thn smulaon s prformd on a compur [7]. Th proprs of h moors usd ar gvn n Tabl III and Tabl IV. Th gar rao for DC moor s 6. Srvo moor s drcly drvn o h crank r. flyhl s coupld o DC moor. Toal mass momn of nra of h flyhl and h spd rducr s kgm Jr J ) f. Th m rqurd for on cycl s 6 sconds. Incrmnal m of smulaon s akn as.166 sconds. Vscous frcon coffcn B and h consan mchancal orqu l ar assumd as zro n calculaons. Smulaon rsuls ar prsnd as; Tabl III. DC Moor Paramrs Rad Por P ) 18. KW Rad Volag V ) 44 V Rad Currn ) Rad Spd/Maxmum Spd W max ) 6 rpm Momn of Inra J ) 1.7 kg m Momn of Inra m J J ) kg m r f Gar Rao n ) 6 Wndng Rssanc R ).973 Ω Wndng Inducanc L ) 19.9*1-3 H Moor Volag Consan K ) 6. V/rad/s Moor Torqu Consan K ).667 Nm/ Tabl IV. DC Srvo Moor Paramrs 1 = = 3 = 4 = = Rad Por P ) 3 KW 6 = Maxmum Spd W max ) 3 rpm Momn of Inra J m ) 6.8*1-4 kg m α 1 Wndng Rssanc R ).8 Ω 4 Wndng Inducanc L ).8*1-3 H α 3 = Moor Volag Consan K ) k p 1) k ) 1) / d k I d R n K L V/rad/s Moor Torqu Consan K ).76 Nm/ 6 k p 3) k 4) 3) 6 4 / d k I d R n K L Th nal condons of h angular dsplacmn and 3) vlocy of h srvo moor h angular dsplacmn and vlocy of h consan vlocy moor and h currns of h ISSN:

5 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 srvo moor and consan vlocy moor ar gvn as c 1 = [-.96rad) -1.64rad/sc) 4.1rad) -1.47rad/sc) ) )] rspcvly. Th gan consans o conrol h sysm ar found by usng Zglr Nchols mhod [11 14]. Thn hy ar modfd by ral and rror. Th proporonal k p ) drvav k d ) and ngral consan k ) for h srvo moor ar akn as 1 1 and rspcvly. Th proporonal gan consan k p ) for h consan vlocy moor s 4. Snc s consan vlocy moor no drvav and ngral gan consans ar usd. Th smulaon rsuls of Moon ar shon n fg. 4. Th vsbl lns n fg. 4 rprsn h rspons rsuls of h smulaon. Thy ar h acual rsuls. Th hddn lns n fg. 4 ar for command valus. Thy ar oband from h knmacs analyss of h hybrd drvn mchansm [4]. Fg.4 s composd of four pars. Fg.4a and fg.4b ar for angular dsplacmn and vlocy for srvomoor rspcvly. Fg.4c and fg. 4d ar for angular dsplacmn and vlocy for consan vlocy moor rspcvly. Rspons dsplacmn valus akn from fg. 4a and 4c ar gvn o forard knmacs quaon as npu daa o g rspons for sldr dsplacmn. Command dsplacmn valus akn from fg. 4a and 4c ar gvn o forard knmacs quaons as npu daa o g command for sldr dsplacmn. I s gvn n fg.. Dsplacmn rror for sldr s shon n fg.6. Command ln s h dal moon dsgnd. c) d) fg. 4 smulaon rsuls fg. sldr smulaon rsuls a) fg. 6 dsplacmn rror b) V. CONCLUSION Dynamc modl of h hybrd sysm s drvd h Lagrangan approach n hs sudy. Knmacs rlaonshp n h dynamc analyss s basd on forard knmacs hch o cranks ar usd as ndpndn npu paramrs. Gnralzd angular acclraon and orqu xprssons ar oband as a rsul of Lagrangan xprssons. Bcaus h ISSN:

6 INTERNTIONL JOURNL OF MECHNICS Volum 1 16 sysm has such a faur of hghly nonlnar hr ar a lo of coupld quaons n h analyss. Th fourh ordr Rung-Kua hch s an xplc mhod usd n ngraon of nonlnar sysms s usd as h ngraon chnqu o g an approxma soluon. Sa spac rprsnaon of h sysm s carrd ou o xprss h sysm n frs ordr dffrnal quaons. conrol sysm s sudd o succd dsgn objcvs. PID conrol chnqu s appld on h sysm. Conrollably of h sysm s dmonsrad by hroughou h smulaon rsuls. Malab s usd o g compl analyss and dsgn sags. REFERENCES [1] L. C. Tokuz Hybrd machn modllng and conrol Ph.D. hss Lvrpool Polychnc Lvrpool England 199. [] L.C. Dülgr and. Krçc sudy on a hybrd machn Mchansm and Machn Thory 3 pp [3] B. Sh Programmabl hybrd mchansms n Proc. of h 11h World Congrss n Mch. and Machn Scnc Tanjn Chna 4. [4] M. E. Küük Hybrd machn sysms: analyss and conrol Msc. hss Gazanp Unv. Gazanp Turky 13. [] M. E. Küük L. C. Dülgr hybrd prs sysm: moon dsgn and nvrs knmac ssus Engnrng Scnc and Tchnology an Inrnaonal Journal o b publshd. [6] M. E. Küük R. Halıcıoğlu L. C Dülgr Knmacs and smulaon of a hybrd mchansm: Malab/Smmchancs Journal of Physcs: Confrnc Srs 74 IC-MSQURE 14 Madrd Span1. [7] L. C. Dülgr. Krçc M. Topalbkroglu Modlng and Smulaon of a Hybrd cuaor Mchansm and Machn Thory 38 pp [8] S. B. Nku Inroducon o Robocs: nalyss Sysms pplcaons. 1. Edon. Prnc Hall. 1. [9] J.E. Shgly J.J. Uckr Thory of Machns and Mchansms.. Edon. McGra-Hll Company [1] H. L and Y. Zhang Svn bar mchancal prss h hybrd drvn mchansm for dp drang; par 1: knmacs analyss and opmum dsgn par : dynamc modlng and smulaon Journal of Mchancal Scnc and Tchnology 4 11) pp [11] Dorf R.C. and Bshop R.H. Modrn Conrol Engnrng. 11. don. Parson Prnc Hall 8 [1] H. L Y. Zhang H. Zhng Dynamcs modlng and smulaon of a n nn-bar prss h hybrd drvn mchansm Journal of Mchancal Scnc and Tchnology pp [13] H. Ka Dsgn and conrol of a conrollabl hybrd mchancal mal formng prs Ph.D. Thss. Th Chns Unvrsy of Hong Kong 8. [14] K. H Y. Luo C. T. Kong R. Du Trajcory Plannng Opmzaon and Conrol of a Hybrd Mchancal Prss WSES Transacons on Sysms 8 pp [1] K. Ogaa Modrn Conrol Sysms. 4. Edon. Parson Prnc Hall. ISSN:

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm