Electro-Mechanical Modelling and Load Sway Control Of Gantry Cranes

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1 Eleco-Mecancal Modellng and Load Sway Conol Of Gany Canes F.S. Al-Faes, T. G. A. Al-Fq, H.R. Al-Mubaak and M.S. Al-Ajm Pof., Poducon Eng. Dep., Faculy of Eng., An Sams Unvesy, Cao, Egyp, and Now w e Dep. of Pod. Tec., College of Tec. Sudes, Auoy of Appled Educaon, Kuwa, fq@yaoo.co.uk.. Assan Pof. Teace; Dep. of Poducon Tecnology, College of Tec. Sudes, Auoy of Appled Educaon, Kuwa. ABSTRACT n wok, a nonlnea model epesenng e dynamcs of e conane sway of a gany cane deved. Te dynamcs of e nducon moos ae also aken no consdeaon n addon o e smulaneous avellng, olley and ong moons. Te daa of an acual 45-on gany cane used o smulae e beavo of e conane unde an acual anspoaon plan. A smulaon example en pesened o llusae e unconolled sysem esponse. Te conane sway found o be g suc a a conol sceme as o be mplemened o suppess e load sway. A feedback conol sceme developed o suppess load sway. Te feedback conol max cosen suc a e poles of e closed-loop sysem ae abay assgned. Te conol sceme appled and e smulaon esul llusae e effecveness of e poposed sceme. Keywods: Gany Canes, Dynamc Modellng, Eleco-Mecancal Sysems, An-Sway Conol.. NTRODUCTON Gany canes ae wdely used n po, and ae nsalled a quaysde o andle eavy conanes. Te necessy o ncease e avellng, olley and ong speed geneally nduces undesable sway of e conane, and seous damage could occu dung load anspoaon. Teefoe, a safacoy conol sceme, based on an accuae dynamcal model, desable o suppess load sway. n e las decades, seveal nvesgaos consdeed e modellng and conol of gany canes as a nonlea conol poblem [- 4]. Mos of exng appoaces cons of a wo-sage pocedue: off-lne ajecoy/pa plannng, caed ou n accodance w pope opmaon cea, and on-lne ackng by adonal conolles. Opmal conol ecnques ave been wdely used o addess e pa plannng poblem. Specfc pa mnmng avelng me, enegy consumpon o pope pefomance ndexes lnked o e swng angle and devave ave been poposed n e leaue. Neveeless, due o model unceanes and many oe mplemenaon facos, ofen appens a e acual cane beavo sgnfcanly dffes fom e opmal, desed, one. Te usual goal o aceve zeo swng only a e end of e anspo, and a wo-sage conol sucue ofen used: a

2 ackng conolle dung e load ansfe, and a sablzng one o be swced on wen a suable vcny of e desnaon pon aceved. Bameswa e al. [5] poposed a nonlnea conol saegy fo e olly cane sysem usng Lyapunov meod wou consdeng e sway angle dynamcs n e sably analys. Fang e al. [6] desgned a popoonal devave (PD) conolle o egulae e oveead cane sysem o e desed poson w naual dampng of sway oscllaon. Lu e al. [7] developed a fuzzy logc conol w sldng mode conol fo an oveead cane sysem. Fang e al. [8] developed a nonlnea couplng conol law o sablze a 3-DOF oveead cane by usng Lasalle nvaance eoem. Howeve e paamees mus be known n advance. Bug e al. [9] used e vaable ansfomaon meod o egulae e cane sysem. d'andea-novel and Bousany [] poposed an adapve feedback lneazaon meod fo mecancal sysems of e oveead cane ype. Kakoub and Zb [] used e pssvy popey of mecancal sysem n e desgn of e nonlnea conol of oveead cane. sde e al. []consuc a fuzzy back-popagaon neual newok based conol. Howeve e esul ave sown a e speed of e olly lage a e desed desnaon.yu e al. [3] developed a nonlea ackng fo load poson and velocy. Howeve e esul ae sown fo sway angle dynamcs muc fase an e ca moon dynamcs. Anoe appoc developed by Ks e al. [4] n wc an oupu feedback PD conolle used o sablze a nonlnea cane sysem. Bendjeb and Gsnge [5] compaed a fuzzy logc conol sysem w Lnea Quadac Guassan conol (LQG) fo an oveead cane.y e al. [6] developed a fuzzy conolle fo an-swng and poson conol fo an oveead cane based on Sngle npu Rule Modules (SRMs) dynamcally conneced fuzzy nefeence model. M. El- Raeb suded e effec e cable flexbly on e load. found a cable flexbly as neglgble effec on flexual esponse. Howeve, e above wok negleced e dynamcs of e dvng eleccal moos, wc could lead o consdeable eos. Also, convenen o use e npu of e dvng moos as conol vaables. n wok, a nonlnea dynamc model of a gany cane deved. Te dynamcs of e nducon moos as well as e smulaneous avel, ansvese and ong moons ae aken no consdeaon. Te deved model en lneazed abou e nomal opeang condons, and a smulaon example en pesened o llusae e conane sway unde an acual anspoaon plan. found a e sway elavely g. Teefoe a conol sceme poposed o suppess e conane sway. Te oques of e dvng cane moos ae adjused by feedng back e sway angles and e devaves. Te feedback conol max cosen suc a e poles of e closed-loop sysem ae abay assgned. Te conol sceme appled and e smulaon esul llusae e effecveness of e poposed sceme.. DYNAMCAL MODELLNG Te dynamcal model of e gany cane sown n Fg. b wll be deved. Te cane avels paallel o e quaysde on e alway wle e olley moves n e ansvese decon cayng e conane. Te ong of e conane akes place dung anspoaon. Te conane, wc can be assumed as a suspended load fom pon O, wll be assumed as a gd body. Cane moos ae nducon moos fo e smplcy and elably. n e followng analys, e avel, ong and olley moos wll be modelled usng e d-q ecnque []. Te knec and poenal eneges of e compound elecocmecancal sysem wll be obaned, and e dynamc equaons assocaed w e genealzed coodnaes wll be deved, usng e Lagangan appoac. T wll be deved and pesened n e followng subsecons.. Lagangan Funcon of e Sysem Te coodnaes of e conane x c, yc and zc can be expessed n ems of e coodnaes of e cane, u, v and w, e

3 conane sway angle, θ and θ, and e leng of e we l as follows; x c = u + l, y c = v + l, and z c = w+ l Te knec enegy of e mecancal sysem epesen e knec enegy of e suspended conane of mass M, e olley and e avellng gany cane. Te nomenclaue can be found a e end of e pape, and e knec enegy can be wen as follows; K. E. m = M ( x c + y c + z c ) + m u + α θ cos θ + m ( u + v ) + + ( θ + θ sn θ ) v + + l 3 3 u () Te poenal enegy of e mecancal subsysem epesen e poenal enegy of e suspended conane, and e soed enegy n e we opes of sffness Q. Te followng expesson of e poenal enegy can be obaned; P. E m = Qφ Mg l () Te equvalen knec enegy of e cane nducon moos; namely e avellng moo, olley moo and e ong moo can be expessed n e followng equaons; L ( ) + L K. E. = + Lm + (3) L ( ) s s s + L K. E. = + Lm s + s (4) L ( ) + L K. E. = + Lm + (5) Now, e knec and poenal enegy of e sysem can be expessed n e eqs. (-5), and e Lagangan funcon can be obaned as follows; L = K.E. m + K.E. + K.E. + K.E. P.E. m wc can be wen as; L = L + L + Lm + + mu Qφ + Mgl + Ls s s + L + Lm s + + M ( x c + y c + z c ) + m ( u + v ) + L + L + Lm + + cos + ( + αθ θ θ θ sn θ ) u v l [ ( )] [ ( )] [ ( )] s (6) Te above Lagangan equaon wll be used o deve e dynamc dffeenal equaon of e sysem.. Modellng of e Eleccal Sub-Sysem n e followng, e Lagangan funcon deved n eq. (6) wll be dffeenaed o oban e dynamc equaon govenng e ansen of one of e cane nducon moos. Tese moos ae e avellng moo olley moo, and e ong moo and e subscp wll efe o one of em. Fs, dffeenae w espec o e sao cuen, e followng equaon can be obaned; dl d = L + L =ψ (7) m Te g and sde of eq. (7) e sao flux. Te above equaon can be dffeenaed w espec o me o oban e followng equaon;

4 d d dl d = V R jω ψ = k d d { ψ } (8) Regadng e oo sde, e above seps can be epeaed o oban e followng equaons; dl d d d = L + L =ψ (9) dl d = V j m ( ω ω ) ψ = { ψ } k R d d () Equaon (8) and eq. () epesen e sae equaon of e moo, wee e flux vecos ae consdeed as saes. Howeve, convenen o use e pmve macne o e dec-quadaue pase quanes, d-q epesenaon. n e followng, e above moo veco quanes ae epesened by d-q componen; Te above equaons epesen e sae equaons of e moo, and can be wen n max fom as follows; X = G X + V,,, (3) = T wee; X [ ψ d ψ q ψ d ψ q ] T V = [ V V ], and =, (4) d q G = α ω α ( ω ω ) k k 4 ω α ( ω ω ) k α k 4 α α 3 α α 3 Te dynamcs of e oo and e moo oque ae ncluded n e followng ff equaon; F α [ ψ ψ ψ ψ ] 5 = d q d q (5) and f lneazed; δf α [ ψ doψ qo ψ do`ψ qo ] X 5 = (6) o n veco noaon; δ F = E X,,, (7) =.3Modellng of e Mecancal sub-sysem ψ = ψ d + jψ q, ψ = ψ d + jψ Lagange's equaon wll be used o oban q e dffeenal equaons assocaed w e = d + j q, = d + j q () genealzed coodnaes. Fve genealzed V = Vd + jvq, V = Vd + jvq coodnaes ae assocaed w e mecancal sde of e compound o be noed a e volage on e oo elecomecancal sysem, wle welve zeo;.e. V = coodnaes ae assocaed w e eleccal pa. Te deals of e sae equaons of e Subsung e above elaonsps no eq. nducon moos ae gven n secon., (8) and eq. (), e followng sae equaons wle e devaon of e sae equaons fo can be wen; e mecancal sde ae gven n e pesen secon. Te followng fve equaons ae ψ d = α ψ d + ω kψ q + α ψ d + V assocaed w e mecancal genealzed d coodnaes; ψ q = ω ψ d α k ψ q + α ψ q + V d () ψ d = α 4ψ d +{ ω k ω } ψ q + α 3 ψ d A u l 3 4θ + C l 35 θ + C l 34 θ (8) ψ q = { ω k ω } ψ d + α 4 ψ q + α 3 ψ d + C θ θ + C θ + C θ θ = A + C 45 θ θ + C 55 + C 44 v θ θ + C 45 4 A u v 33 + C 5 θ l + C θ l θ + C l 34 θ θ l = 35 θ = Mg F 3 F F (9) ()

5 A 4 u 4 v 44θ + C434l θ + C θ + D = () A u v θ + C l θ () + C 545θ θ + D55θ = Qφ wee A j epesen e elemen of e effecve nea max A, wc can be found n Appendx A. On e oe sde, e elemen C jk epesen e coeffcen of e dynamc foce (cenpeal o cool) a coodnae due o e veloces a j and k. Tese elemen can be obaned fom e C ( q, q ), wc can be found also n Appendx A. Regadng e gavy loadng max D, evey dagonal elemen D epesen e gavy loadng a jon. All e elemen of max ae zeo excep D 44 = Mgl. Te above equaons can be wen as follows; ( q) q + C( q q ) + D( q) F A, = (3) 3. SYSTEM SMULATON Cane acceleaons u, v and l ae decded accodng o e anspoaon plan. Teefoe, e moo npu can be calculaed by usng e nonlnea elecomecancal model. Te above equaons wll be used o smulae e esponse of e consdeed cane fo a specfed anspoaon plan. Te poposed plan o move e gany w a consan acceleaon of.3 m / sec, and e olley w a consan acceleaon of.5 m / sec fo en seconds. Ten, e gany and olley moves w a consan velocy fo anoe en seconds. Fnally, e gany moves w a consan deceleaon of.3 m / sec, and e olley moves w a consan deceleaon of.5 m / sec fo ano en seconds. Te poposed plan llusaed on Fg.-3, and e esponse pesened on Fg Snce n pacce e suspenson opes ae no wound wle e cane n moon fo safey consdeaons [4], no-ong of e load wll be consdeed n case. clea a e sway angles ae nceasng fo bo θ and θ. can be also noced a e fequency of oscllaon of θ abou.7 Hz, wc can be pedced fom e sysem equaon 33. Fom Fg.4-5, can be noced a θ eaces.35 adans, and θ exceeds.6 adans, wn 3 seconds. T means a e conane load of 45 on vbaes w an amplude of 7 cm., and oaes abou ax w an angle of abou 6 degees, wn 3 seconds. T dangeous sae ndcaes e necessy of applyng a conol sceme o suppess e load sway. T wll be consdeed n e nex secon. Te daa of e consdeed gany ae pesened n Table. 4. LNEARZED MODEL Te am of e pesen wok o suppess e load sway angles θ and θ dung e anspoaon peod. n ode o faclae e analys of e above sysem, and e applcaon of feedback conol sceme, ecommended o lneaze e above model abou e nomnal values of e saes. Te nomnal values of e saes can be obaned by seng e devaves of e sway angles o zeo. Teefoe, e nomnal saes, θ and θ, can be obaned as follows; θ = ( u + v ) g (4) o o / θ = an ( u / v ) (5) o o Teefoe, e equaons 8- of e mecancal subsysem, afe neglecng e ge ode ems and subsung fo e foces fom equaons 5-7, can be wen as follows; A δu δ l δθ δθ C l δθ + C l δθ = δf = E X A δv δ l δθ δθ C l δθ + C l δθ = δf = E X (6) (7) A δ u δv δ l = δf = E X (8)

6 A A 4 5 δu 4 δv 44 δθ + C + D δu 5δv 55δθ + C l δθ Qφ = l δθ δθ = (9) (3) Accodng o e assumpon of small peubaons abou e nomnal values, e non-olonomc consan; φ θ =, assocaed w sysem can be educed o be; δ φ = δθ. Teefoe, equaon 3 can be ewen as follows; A 5 δu 5 δv 55 δθ + C535l δθ (3) + Qδθ = Te cane acceleaons u, v and l ae decded accodng o e anspoaon plan. Teefoe, e moo npu can be calculaed by usng e elecomecancal model. Now, e followng vecos wll be noduced; T Ψ = [ δu, δv, δl] Θ T = δθ, δθ ] (3) [ wee e fs veco Ψ epesen e ganay moons wc ae decded accodng o e poposed anspoaon plan, and e second veco Ψ epesen e conolled saes of e consdeed sysem. Now e sysem dynamc equaon of moon can be wen as follows; J Θ + D Θ + K Θ = B Ψ (33) J E X ] Θ + D Θ + B Ψ = [ E X, E X, J Θ + D Θ + B Ψ = ; (34) F E Te paamees of e maces can be found n Appendx A. 5. PROPOSED CONTROL SYSTEM Te above lneazed sysem wll be used o apply a conol sceme suc a e load sway can be suppessed. Te consdeed sysem wll be sown o be conollable, befoe applyng e conol sceme. Equaons can be educed o e followng fom; T J Θ + D Θ + K Θ = B B J Θ BB D Θ + B B wc can be ewen as follows; Θ + ξ Θ + Ω Θ = βf E F E (35) wc as e followng sae space pesenaon; Χ = Α Χ + ΒΕ (36) wee; Α = Ω Θ ξ, Β =, Χ = β Θ, Ε = F E Te above sysem wll be conollable f e followng conollably max; = 3 [ Β, ΑΒ, Α Β, Α Β ] as e full ank. f e max A cyclc, a ank feedback conol sceme can be appled wee e wo npu model educed o sngle npu model. T conol sceme fxes e ao of e wo npu o e moos accodng o e followng [8]; Ε = Ζ χ (37) wee Z a veco of consan o be cosen suc a e educed sysem conollable fom e new sngle npu χ. Now e sysem equaon 36 can be wen as follows; Χ = Α Χ + ( Β Ζ) χ (38) Te feedback conol wll be caed ou by assumng e conol acon o be; T χ = Κ Χ (39) wee K a gan veco o be cosen suc a all egenvalues of e feedback conol sysem ae assgned as desed. wll be now sown a e max A cyclc and en o selec a veco Z a ensues e conollably of e educed sysem. Te max A wll be cyclc f e max s-a,

7 s e Laplace ansfom vaable, as one non-uny nvaan polynomal [8]. Te nvaan polynomals of e of s-a can be obaned as follows; Λ 4 = ε k s k = 4 k =,,3 = 4 (4) Fom e above equaon, and e value of e sysem paamees, clea a las nvaan polynomal non-uny and consequenly e max A cyclc. Te exence of a veco Z wc capable o make e educed sysem equaon (38) conollable ensued by e followng Lemma (see [8] fo e poof). Lemma: f e mul-npu sysem descbed by equaon (36) conollable and sysem max A cyclc, en almos any veco Z wll make e educed sngle-npu sysem descbed by eq. (38) conollable,.e. e conollably max of sysem = 3 [ ΒZ ΑΒZ, Α ΒZ, Α ΒZ ] as e full ank., (4) 5.. Selecon of Feedback Gan Te caacec polynomal of e sysem wou feedback can be wen as follows; 4 3 s + ε s + ε s + ε s + ε = (4) 3 f e desed egenvalues of e sysem w feedback ae assgned as λ, λ, λ3, λ4, e closed caacec equaon can be consdeed as follows; 4 3 s + µ s + µ s + µ s + µ = (43) 3 Te feedback gan veco a yelds e desed closed loop egenvalues can be deved as follows [8]; T Κ = [ ] Γ ( µ ε) (44) 4 4 wee Γ lowe angula Teoplz max w e fs column [, ε ] T, ε, ε 3 ; e conollably max defned n eq. (4), and e vecos µ and ε ae defned as follows; µ = [ µ, µ, µ µ ] T and ε = [ ε, ε, ε ε ] T 3, 4 3, Te above esul ave been mplemened o suppess e sway of e load. Te poles of e closed loop ae placed a.5 ± j. 75 and a 4 ± j8, fo e fs en seconds and e las en seconds. Regadng e second en seconds, e poles ae placed as.5 ± j. 75. Fg. 6 sows e block dagam of e conol sceme. Smulaons ave been caed ou o es e pefomance of e feedback conol sceme as appled o e consdeed gany cane. Te esul ae pesened n Fg. 7. clea a e sway angle δθ now exb a decayng vbaon caaecs and a maxmum sway angle now of.5 ad as been obaned and zeo sway wll be obaned. Regadng e sway angle δθ, suffcen o foceδθ o zeo o suppess e sway of e conane. CONCLUSONS A nonlnea dynamcal model of e elecomecancal sysem of gany cane as been developed o epesen e load sway dynamcs, usng e Lagangan appoac. T dynamcal model consdes e nduced sway of e suspended load due o e smulaneous moons of e gany, olley and e ong of e load. Te dynamcs of e nducon moos ae also aken no consdeaon n addon o e afomenoned moons. Te consdeed sway nduced no only n e plane of moon, bu also n e plane deemned by e opes and e vecal ax oug e suspenson pon. Te deved model adoped o smulae e esponse of an acual gany cane. Te daa of a 45-on gany cane used o smulae e beavo of e conane unde an acual anspoaon plan. A smulaon example en pesened o llusae e unconolled sysem esponse. Te conane sway found 4

8 o be g suc a a conol sceme as o be mplemened o suppess e load sway. A feedback conol sceme developed o suppess load sway. Te oques of e dvng cane moos ae adjused by feedng back e sway angles and e devaves. Te feedback conol max cosen suc a e poles of e closed-loop sysem ae abay assgned. Te conol sceme appled and e smulaon esul llusae e effecveness of e poposed sceme. Acknowledgemen Te auos acknowledge and appecae e fnancal suppo of e Publc Auoy fo Appled Educaon and Tanng unde e Reseac Gan TS-6-4. Refeences [] Sakawa,Y. and SndoY.,"Opmal Conol of Conane Canes", Auomaca, Vol.8-3, pp 57-66, 98. [] Bule,H., Honded, G., Ameongen, J.V. "Model efeence adapve conol of a gany cane scale model", EEE Conol Sysems, pp. 57-6, 99. [3] Kenon, M. and Sngose, W. "npu sape desgn fo double pendulum plana gany canes", MECE Conf., pp 55-59, 999. [4] Km, Y-S.; Hong, K-S.and Seung- K, An-sway conol of conane canes n. J. of Conol, Auomaon and Sysems v n 4 p , 4. [5] Bameswa Vkamadya, Rajes Rajaman. Nonlnea conol of a olley cane sysem. Amecan conol confeence, Ccago, L.. p [6] Fang Y, Zegeoglu E, Dxon WE, Dawson DM. Nonlnea couplng conol laws fo an oveead cane sysem. EEE n. Conf. on conol applcaons, Mexco.. p [7] Lu D, Y J, Zoa D. Fuzzy unng sldng mode conol of anspong fo an oveead cane. nd n. Conf. on macne leanng and cybenecs, X an, Cna. 3. p [8] Fang Y, Dxon WE, Dawson DM, Zegeoglu E. Nonlnea couplng conol laws fo a 3-DOF oveead cane sysem. EEE nenaonal confeence on decon and conol, Olando, FL, USA.. [9] Bug T, Dawson D, Ran C, Rodes W. Nonlnea conol fo an oveead cane va e sauang conol appoac of eel. n: Poceedngs of e EEE nenaonal confeence on obocs and auomaon, Mnnesoa, USA [] d Andea-Novel B, Bousany F. Adapve conol fo a class of mecancal sysems usng lneazaon and Lyapunov meods. A compaave sudy on e oveead cane example, EEE n. Conf. decon and conol, UK. 99. p. 5. [] Kakoub MA, Zb M. Modellng and enegy based nonlnea conol of cane lfe. EE Poc-Conol Teoy Appl ;49(3): 9 6. [] sde T, Ucda H, Myakawa S. Applcaon of a fuzzy neual newok n e auomaon of cane sysem, 9 fuzzy sysem symposum. 99. p [3] Yu J, Lew FL, Huang T. Nonlnea feedback conol of a gany cane, Amecan conol confeence, Wasngon, USA p [4] Ks B, Levne J, Mullaup P. A smple oupu feedback PD conolle fo nonlnea canes, 39 EEE conf. on decon conol, Ausala.. [5] Bendjeb A, Gsnge GL. Fuzzy conol of an oveead cane pefomance compaon w classc conol, Conol Eng Pacce 995;(): [6] Y Janqang, Yubazak Naoyos, Hoa Kaou. An-swng and posonng conol of oveead avelng cane. nfom Sc 3;55:9 4. [7] El-Raeb; M.; Effec of cable flexbly on ansen esponse of a beam pendulum sysem ; J. of Sound and Vbaon, V.37, ssues 3-5, 6 Nov. 7, P [8] Van De Vege, Feedback Conol Sysems Pence-Hall, 986. [9] Tecncal daa of 45-on Gany Cane, Tomen Copoaon and Mub Heavy nd. Ld., Japan, (). [] Begamude, R.D., "Eleco- Mecancal Enegy Conveson w Dynamcs of Macnes", Wley 988.

9 l θ (x c, y c, z c ) θ Fg. a) Acual gany cane a Suwak po n Kuwa, b) Fee body dagam fo gany cane Table. Daa of e Consdeed Cane. m = kg., m = kg., M = 45 kg., N.m/ad, Te daa of e moos can be found n [9]. = 4 kg. m, α = 5 kg. m, l = m., Q = 5 α u v Tme, sec. Fg.. Tanspoaon Plan of u. Fg. 3. Tanspoaon Plan of v θ θ Tme, sec. Fg. 4. Sway angle of θ wou conol Tme, sec. Fg. 5. Sway angle of θ wou conol.

10 Desed Sway Angles Sgnal Tansduce +_ Gan Max Cane Moos Gany Cane Sensos Fg. 6. Block Dagam of e Conol Sceme. θ Fg. 7. Sway angle of θ w conol NOMENCLATURE A sysem effecve nea max, A,A,A sysem submaces, B sysem max, B,B sysem submaces, b, b, b j, b m sysem vecos, c: subscp denoes e conane coodnae [C ], [C ] sysem dampng maces, subscp denoes e dec coodnae n e d-q sysem of coodnaes, d E veco ncludes e moo seady sae flux, E max maps e moo effec o e mecancal pa, E max epesen e moos effec, F x,f y plane foces acng on e gde and olley[n], F H ope enson foce [N], g gavaonal acceleaon [m/sec ], G, G g, G, G, G m, G sysem maces, subscp denoes e ong,, mass momen of nea of e objec n e axal and ansvese decons, [kg.m ],, and 3 mass momen of nea of e gde moo, olley moo and ong moo especvely [Kg.m ], = M + equvalen mass momen of nea, s, d, sd, ec. cuen, and subscp ae defned as e locaons [Amp.], j j = (-).5, [K ], [K ] sysem maces, dance fom e uppe pon of e unwound pa of e ope o e mass subscp, cene of e load, m M suspended objec mass [Kg], M x = M + m + m + ( / ) equvalen mass [Kg], M y = M + m + ( / ) equvalen mass [Kg], M = M + ( 3 / 3 ) equvalen mass [Kg], m, and m gde and olley masses especvely [Kg], [M ], [M ], [M ] sysem mass maces, m: subscp denoes e mecancal sysem Q:sffness of e we ope[n.m/ad] adus of gde moo pnon [m],

11 adus of olley moo pnon [m], 3 adus of dum [m], subscp denoes e oo, R esance, e subscp ae o cecked [Om], subscp denoes e sao, s [T] ansfomaon max, me [sec], subscp denoes e olley, U veco denoes e npu o e man sysem, U veco denoes e npu e subsysem, U volages on e sao [Vol], u,v,w: cane coodnaon x x- coodnae of e ope end [m], X veco denoes e saes of e moo, X sysem veco, y y- coodnae of ope end [m], Y sysem veco, z z- coodnae of ope end [m], z G z- coodnae of e objec [m], Z veco denoes e sae of e ole sysem, Z m veco denoes e sae of e mecancal pa, scala dummy vaable,,,, scala dummy vaables fo flux, 3 4 ß dummy vaable fo equaons 34-35, ; egenvalue, egnveco, m, vecos epesenng pa of e egenveco,, sway and oaonal angles especvely (Fg.), : sao flux : oo flux w angle of e ope, P osonal sffness of e ope. APPENDX A SYSTEM MATRCES A ( q) M u = Mθ Ml Mlθ Mθ M Ml Mlθ v Mθ Mθ M l Ml θ Ml Mlθ Mlθ θ C l θ θ + l θ + l θ θ l θ θ l θ θ l θ θ + l θ l θ θ l θ θ l θ θ [( ) ] ll θ + l θ θ ll θ θ + Ml + / M θ θ θ ( ) = q, q M l( θ θ + θ ) θ Ml Mgl J =, D =, K = θ Ml, θ Q Ml B = Mlθ Ml Ml θ

12 B M u = Mθ M v Mθ Mθ Mθ, M l F T E = [ E X, E X, E X ] Ml Mlθ J = Ml Mlθ, Ml Ml θ D = Ml Ml θ M. El-Raeb suded e effec e cable flexbly on e beam pendulum esponse. found a even fo lage pendulum swngs, cable flexbly as a neglgble effec on flexual esponse. Mcael El-Raeb a, a Oakfoes Lane, Pasadena, CA 97, USA Receved Mac 7; eved 3 June 7; acceped 8 July 7. Avalable onlne 9 Augus 7.

13 [] El-Raeb; M.; Effec of cable flexbly on ansen esponse of a beam pendulum sysem ; Jounal of Sound and Vbaon, Volume 37, ssues 3-5, 6 Nov. 7, P Jounal of Sound and Vbaon Volume 37, ssues 3-5, 6 Novembe 7, Pages Resul l pevous < 5 of 36 > nex Absac Te effec on beam pendulum esponse of cable flexbly suded. Te sysem foced a base by a pescbed damped peodc oscllaon. Response fom cable enson esmaed a a delayed me sep fom known vaables compued a a pevous me sep n a lnea modal analys. Te effec of base excaon and foce fom cable flexbly ae ncluded adopng e sac dynamc supeposon meod. Two dnc non-dmensonal paamees κ o and µ, conol e lnea modal esponse of beam and pendulum. Unlke peodc excaon wee e pendulum may be used as an absobe of enegy, n ansen esponse e condons leadng o absopon do no apply. Even fo lage pendulum swngs, cable flexbly as a neglgble effec on flexual esponse consdeng a cable enson domnaed by g fequences lage an e sea foce Q xxl ansm a e beam-fee end. Conay o effec on flexue, cable flexbly nduces a g-fequency axal foce compaable o Q xxl. Tle: An-sway conol of conane canes: nclnomee, obseve, and sae Feedback Auo:Km, Yong-Seok; Hong, Keum-Sk; Sul, Seung-K Copoae Souce:Scool of Mecancal Engneeng Pusan Naonal Unvesy, Gumjeong-gu, Busan , Sou Koea Souce:nenaonal Jounal of Conol, Auomaon and Sysems v n 4 Decembe 4. p Publcaon Yea:4 SSN: Tle: An-sway conol of conane canes: nclnomee, obseve, and sae Feedback Auo:Km, Yong-Seok; Hong, Keum-Sk; Sul, Seung-K Copoae Souce:Scool of Mecancal Engneeng Pusan Naonal Unvesy, Gumjeong-gu, Busan , Sou Koea Souce:nenaonal Jounal of Conol, Auomaon and Sysems v n 4 Decembe 4. p Publcaon Yea:4 SSN: Tle: A Gany Cane Poblem Solved Auo:O'Conno, Wllam J. Copoae Souce:Depamen of Mecancal Engneeng Unvesy College Dubln Naonal Unvesy of eland, Belfeld, Dubln 4, eland Souce:Jounal of Dynamc Sysems, Measuemen and Conol, Tansacons of e ASME v 5 n 4 Decembe 3. p Publcaon Yea:3 yazed@omal.com