Nonlinear Dynamic Modelling and Analysis of a 3-D Overhead Gantry Crane System with System Parameters Variation
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1 Noli Dic Modllig d Ali of 3-D Ovd Gt C St wit St t Vitio R.M.T. Rj Iil, M.A. Ad, M.S. Rli, F.R.M. Ridi Fcult of Elcticl d Elctoic Egiig Uiviti Mli g, k, 66, g, Mli Abtct - Ovd c widl ud i idut fo tottio of v lod. T tul w of c lod i dtitl to f d fficit otio. Howv, t c ccltio, quid fo otio, lw iduc udibl lod wig. Ti t dic odllig of 3-D ovd gt c t bd o clod-fo qutio of otio. T Lggi tod i ud to div t dic odl of t t. A dic odl of t t icootig lod d o lgt i dvlod. T t ffct of lod d o lgt o t o of t t dicud. Etiv ult tt vlidt t toticl divtio td i t ti d fquc doi. Kwod: 3-D ovd gt c, Lggi tod, dic odllig, t t vitio. I. INTRODUCTION A 3-D c t i ud fqutl to ov v cgo i fctoi d bou. Ovd c v o iou obl, i.. c ccltio o dcltio lw iduc udibl lod wig. Ditubc, uc wid d i, lo dc t c foc b dvl ffctig t c cotol. Futo, c oto ot ffctivl cotol c du to cl o lck of ic. Suc obl dc t wok fficic d i o c cu dg to t lod d cu ft ccidt. Tu, ffctiv d ci c otio cotol dig i quid fo id d oot utotic lod dlig (Cg d Cig, 8. Tfo, c o t dic odllig of t gt c t i cucil bfo cotoll c b iltd to t t. So tudi dvlod t dic odl d cotol dig fo 3-D ovd gt c t. T dvlot of odllig d cotol tod fo 3-D c b otd i L (1998. T lot of dic c odl to dti otil d tt iiizd lod wig b ivtigtd i Fg t l., (3 d izzi d Viioli (. T odllig d g bd oli cotol fo c lift b otd i Kkoub d Zibi (. I t tudi, col t dic qutio fo c ut b coidd fo t cotoll. Atiwigig cotol d otio lig fo 3-D gt c lo otd i Blj d Kolodzijczk (7 d Gido t l. (8. Moov, t tud of t dic odllig of t ot vibto t icootig lod i lo b otd. T tud o t ffct of lod o t dic bviou of t 3-D gt c t b td i Iil t l. (9 d t ivtigtio o t ffct of lod o t dic bviou of t two-lik flibl iulto t b td i Ad t l. (8,. Ti t glid odllig fwok tt ovid clod-fo dic qutio of otio of 3-D ovd gt c t. T wok lo t t ffct of lod o t dic bviou of t t. T Eul-Lgg icil i ud to div t dic odl of t t. T iultio lgoit tu dvlod i iltd i Mtlb. Ct oitio d wig gl of t t d t ow ctl dit (SD obtid i bot ti d fquc doi. To tud t ffct of lod d o lgt, t ult vlutd wit vig t two t of t gt c. Siultio ult ld i bot ti d fquc doi to t ccuc of t odl i tig t ctul t. T t of ti i tuctud i t followig. T t ctio ovid bif dcitio of t 3-D ovd gt c t coidd i ti IJSSST, Vol. 11, No. 9 ISSN: oli, it
2 tud. Sctio 3 dcib t dic odllig of t t divd uig Eul-Lgg foultio. Siultio ult wit t vitio otd i Sctio 4. Fill, cocludig k offd i t lt ctio. II. THE 3-D OVERHEAD GANTR CRANE SSTEM T 3-D ovd gt c t wit it lod coidd i ti wok i ow i Figu 1, w d dot t wig gl of t o, L i t lgt of o, d F i t ct div foc. Z Ril Ct F To div t dic qutio of otio of t 3- D ovd gt c t, t totl g ocitd wit t c t d to b coutd uig t Lggi oc (Sog, 1997; Sog t l., 6. T, t Eul-Lgg foultio i coidd i cctizig t dic bviou of t c t. Bd o Figu 1, t il, ct d lod oitio vcto giv b [,, ] c [,, ] [ L i i, L i co, L co ] w d t ct oitio i - d -dictio ctivl. T totl kitic g, K d ottil g, of t wol t giv b w K K il K ct lod K lod 1 1 K il, K ct c c, K lod J J, lod gl co L w J dot t ot of iti of t lod d g i t gvit ccltio. T, t totl kitic g d t totl ottil g obtid lod Figu 1 Dcitio of t 3-D ovd gt c. Fo ilicit, t ct fictio foc i igod. T ct d t lod c b coidd oit. T tio foc tt cu t o logt i lo igod. III. DNAMIC MODELLING OF THE 3-D OVERHEAD GANTR CRANE SSTEM Ti ctio ovid dtil dcitio o t odllig of t 3-D ovd gt c t, bi of iultio viot fo t tud o t ffct of t vitio to t t. T Eul-Lgg foultio i coidd i cctizig t dic bviou of t c t icoot lod d lgt of t o. K 1 ( 1 c L co i L co co ( L i co L i i 1 1 ( ( i L J L J gl co ( Uig t Lggi oc, t Lggi of t t i obti c (1 L K (3 T, t Eul-Lgg foultio i ud tt i d dt L L q i qi F i, i 1,,3,4. (4 w q i t tt vcto d F i t cotol vcto wic dfid IJSSST, Vol. 11, No. 1 ISSN: oli, it
3 w f d q F T T f f f dot t cotol foc iut ctig o t - d -dictio otio. Fo (4, t of fou qutio i obtid tt i d L L f dt d dt L L,, d L L f dt d dt L L Fo (5, t dic odl of t 3-D ovd gt c t c b d i t fo of tic uc tt w t tic (5 M( q q C( q, q q G( q F (6 M( 44 q, C( q, q G( t, d gvit, ctivl, d dfid 44, d 4 q t t iti, Ctitl-Coioli c M( q L co i L i co C( q, q c L co co L i i L co i L co co L J L i co Li i (7 L i J L i i L co co L i co L co i L i co L co co L i i L co i Li co L i co (8 L i co G( q gl i (9 T Aft gig (6 d ultilig bot id b o obti q M 1 ( Cq G F (1 1 M, 1 w M i gutd to it du to dt( M. Fo (1, t wig gl of t lod wit ct to Z-l, θ d wit ct to Z-l, θ obtid t 1 (t i i 1 (i co I ti tud, t oil vlu of t t dfid =.73 kg, c = 1.6 kg, = 6.4 kg, L =.7 d J =.5 kg- (Cg d Cig, 8. I od to ivtigt t dic bviou of t t icootig lod, i vid fo to 1 kg wil L i t to it oil vlu. O t ot d, to tud t ffct of t o lgt to t dic bviou of t t, t vlu of L i vid fo to 1 wil i t to it oil. T vlu of gvit ccltio i g = IV. SIMULATION RESULTS I ti ctio, iultio ult of t dic bviou of t 3-D ovd gt c t td i t ti d fquc doi. A bgbg igl of litud 1 N d 1 widt i ud iut foc, lid t t ct of t gt c. A bg-bg foc oitiv (ccltio d gtiv (dcltio iod llowig t ct to, iitill, cclt d t dclt d vtull to t tgt loctio. St o vifid b udtkig cout iultio uig t fout-od Rug-Kutt itgtio tod fo dutio of wit lig ti of 1. T wig gl o of t t wit t ow ctl dit (SD obtid d vlutd. A. St Ro wit lod Vitio To dott t ffct of lod o t dic bviou of t t, viou lod of u to 1 kg wigt w iultd. Figu d Figu 3 ow t o of t ct oitio i - d -dictio, ctivl fo viou lod. It i otd tt t vg fil oitio fo bot - d -dictio of t ct dc. Moov, t cttig of t fil oitio o vi fo 3 to 7 fo oitio i - dictio d. to 3.6 c fo oitio i -dictio. IJSSST, Vol. 11, No. 11 ISSN: oli, it
4 Tbl 1 uiz t ltio btw lod d t ct oitio o. Figu 4 d Figu 5 ctivl ow t wig gl wit ct to Z- d Z-l o, wit viou lod. It i ow tt t ocilltio of t wig gl dc wit icig lod. T ltio btw lod d t wig gl i lo ow i Tbl 1. Figu 6 d Figu 7 ow t coodig SD fo bot wig gl. It dott tt t oc od of vibtio of t t ift to ig fquci wit icig lod. Tbl uiz t ltio btw lod d t oc fquci of t t. Ti tud ow tt, i od to dc t ocilltio of t t, cotol dig c b ud fo vl t ltoug t v difft lod. Ti ult i kbl ic t t of cotoll do ot d to b digd if t lod i vid witi t cific g. ( c t io ḏi i io o it C t =.1 kg =.5 kg = 1. kg Ti ( Figu oitio of t ct i -dictio wit vitio of =.1 kg =.5 kg = 1. kg d ( Z-l ctto w ig R o Figu 3 oitio of t ct i -dictio wit vitio of =.1 kg =.5 kg = 1. kg Ti ( Figu 4 Swig gl of t o wit ct to Z-l wit vitio of d ( Z-l ctto g w i R o Ti ( =.1 kg =.5 kg = 1. kg Figu 5 Swig gl of t o wit ct to Z-l wit vitio of. ( tio c d i Ȳ i io o it C t Ti ( IJSSST, Vol. 11, No. 1 ISSN: oli, it
5 Tbl 1: Rltio btw lod wit ct oitio d wig gl o. lod Ct oitio ( Swig gl (d -dictio -dictio Wit ct Wit ct (kg Avg Ocilltio Avg Ocilltio to Z-l, θ to Z-l, θ ± ±.359 ±.3 ± ± ±.97 ±.711 ± ± ±.347 ±.66 ± ± ±.357 ±.1949 ± ± ±.38 ±.1631 ± ±.7.63 ±.39 ±.147 ± ± ±.34 ±.14 ± ± ±.78 ±.133 ± ± ±.51 ±.887 ± ± ±.1 ±.769 ± B (d Z-l ctto g i w o ft D S =.1 kg =.5 kg = 1. kg Fquc (Hz Figu 6 ow ctl dit of t wig gl wit ct to Zl wit vitio of B (d Z-l ctto g i w o ft D S =.1 kg =.5 kg = 1. kg Fquc (Hz Figu 7 ow ctl dit of t wig gl wit ct to Zl wit vitio of. Tbl : Rltio btw lod d oc fquci of t 3-D ovd gt c t. Roc fquc (Hz lod (kg Swig gl θ Swig gl θ Mod 1 Mod Mod 1 Mod B. St Ro wit Ro Lgt Vitio Siill, to dott t ffct of o lgt o t dic bviou of t t, viou lgt of o u to 1 w iultd. Figu 8 d Figu 9 ow t o of t ct oitio i - d - dictio, ctivl fo viou o lgt. Rkbl, t vitio of t o lgt do ot ffct uc o t vg fil oitio fo bot - d -dictio. Howv, t cttig of t fil oitio o vi fo.5 to 13 fo oitio i - dictio d.9 to 6 c fo oitio i -dictio. Tbl 3 uiz t ltio btw o lgt d t ct oitio o. Figu 1 d Figu 11 ctivl ow t wig gl wit ct to Zd Z-l o, wit viou o lgt. It i IJSSST, Vol. 11, No. 13 ISSN: oli, it
6 ow tt, t fo o lgt.1 d., t ocilltio of t wig gl ic wit icig o lgt. T ltio btw o lgt d t wig gl i lo ow i Tbl 3. Figu 1 d Figu 13 ow t coodig SD fo bot wig gl. It dott tt t oc od of vibtio of t t ift to ig fquci wit icig o lgt. Tbl 4 uiz t ltio btw o lgt d t oc fquci of t t. Ti ivtigtio ow tt, iil cotol dig c b ud fo vl t ltoug t v difft o lgt, i od to duc t ocilltio of t t. Ti ult i lo igifict ic o do ot v to dig cotoll if t o lgt i vid witi t cific g ( c t io ḏi i io o it C t ( tio c d i Ȳ i io o it C t L =.1 L =.5 L = Ti ( Figu 8 oitio of t ct i -dictio wit vitio of L L =.1 L =.5 L = Ti ( Figu 9 oitio of t ct i -dictio wit vitio of L. ctto g w i R o L =.1.4 (d L =.5 L = 1. Z-l.3. d ( Z-l ctto w ig R o B (d Z-l ctto g i w o ft D S Ti ( Figu 1 Swig gl of t o wit ct to Z-l wit vitio of L L =.1 L =.5 L = Ti ( Figu 11 Swig gl of t o wit ct to Z-l wit vitio of L L =.1 L =.5 L = Fquc (Hz Figu 1 ow ctl dit of t wig gl wit ct to Zl wit vitio of L. IJSSST, Vol. 11, No. 14 ISSN: oli, it
7 Tbl 3: Rltio btw o lgt wit ct oitio d wig gl o. Ro Ct oitio ( Swig gl (d lgt -dictio -dictio Wit ct Wit ct ( Avg Ocilltio Avg Ocilltio to Z-l, θ to Z-l, θ ± ±.89 ±.194 ± ± ±.88 ±.1885 ± ± ±.85 ±.19 ± ± ±.87 ±.43 ± ± ±.85 ±.519 ± ± ±.188 ±.795 ± ± ±.89 ±.17 ± ± ±.39 ±.1346 ± ± ±.494 ±.163 ± ± ±.597 ±.1899 ±.1475 B (d Z-l ctto g i w o ft D S L =.1 L =.5 L = Fquc (Hz Figu 13 ow ctl dit of t wig gl wit ct to Zl wit vitio of L. Tbl 4: Rltio btw o lgt d oc fquci of t 3-D ovd gt c t. V. CONCLUSION Ivtigtio ito t dvlot of dic odl of 3-D gt c t icootig lod d o lgt b td. A clod-fo fiit diiol dic odl of t t b dvlod uig t Eul-Lgg oc. T divd dic odl b iultd wit bgbg foc iut. T ct oitio d wig gl o of t gt t v b obtid d ld i ti d fquc doi. Moov, t ffct of lod d o lgt o t dic cctitic of t t v b tudid d dicud. T ult v lful d iott i t dvlot of ffctiv cotol lgoit fo 3-D gt c t wit vitio of lod d o lgt. REFERENCES Ro Roc fquc (Hz lgt Swig gl θ Swig gl θ (kg Mod 1 Mod Mod 1 Mod [1] Ad, M.A., Mod, Z. d Hbli, N. (8. Dic Modllig of Two-lik Flibl Miulto St Icootig lod. 3d IEEE Cofc o Idutil Elctoic d Alictio, [] Blj, W. d Kolodzijczk, K. (7. Motio lig d Cotol of Gt C i Cluttd Wok Eviot. IET Cotol To Alictio, Vol. 1, No. 5, [3] Cg, C.. d Cig, K.H. (8. Fuzz ojctio Cotol Lw d it Alictio to t Ovd C. Joul of Mctoic, Vol. 18, [4] Fg,., Dio, W.E., Dwo, D.M. d Zgoglu, E. (3. Noli Coulig Cotol Lw fo Udctutd Ovd C St. IEEE/ASME T. o Mctoic, Vol. 8, No. 3, [5] Gido, S., Abdi, M., Giz, A., Diz, R. d Blgu, C. (8. Ati-Swigig Iut Sig Cotol of Autotic Cotuctio C. IEEE T. o Autotio Scic d Egiig, Vol. 5, No. 3, [6] Iil, R.M.T.R., Ad, M.A., Rli, M.S. d Ridi, F.R.M. (9. Noli Dic Modllig d Ali of 3-D IJSSST, Vol. 11, No. 15 ISSN: oli, it
8 Ovd Gt C St wit lod Vitio. Tid UKSi Euo Soiu o Cout Modlig d Siultio, [7] Kkoub, M.A. d Zibi, M. (. Modllig d Eg Bd Noli Cotol of C Lift. IEE oc. Cotol To Alictio, Vol. 149, No. 3, [8] L, H.H. (1998. Modlig d Cotol of T-Diiol Ovd C. Joul of Dic St, Mut, d Cotol, Vol. 1, [9] izzi, A. d Viioli, A. (. Otil Dic-ivio-bd Cotol of Ovd C. IEE oc. Cotol To Alictio, Vol. 149, No. 5, [1] Sog, M.W. (1997. Udctutd Mcicl St, Cotol obl i Robotic d Autotio. Lodo: Sig-Vlg. [11] Sog, M.W., Hutcio, S. d Vidg, M. (6. Robot Modlig d Cotol. Nw J: Jo Wil. IJSSST, Vol. 11, No. 16 ISSN: oli, it
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