Anti-sway Control Input for Overhead Traveling Crane Based on Natural Period

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1 Mmoirs of h Fculy of Egirig, Kyushu Uivrsiy, Vol.67, No.4, Dcmbr 7 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod by Pgfi GAO *, Mooji YAMAMOO ** d Yoshiki HAYASHI * Rcivd Novmbr 5, 7 Absrc For sf d ffici work of hdlig sk for hvy lod by ovrhd rvlig cr, swy of h suspdd lod should b voidd. Vrious mhods hv b proposd o rduc h swy, mosly by usig fdbck corol mhods. Howvr, mos mril hdlig sysm uss op loop yp srvo corollr whr rpzoidl or S-shp vlociy pr is ypiclly usd for h ipu of h corollr. Cosidrig us of such srvo corollr, his ppr proposs i-swy op loop corol mhod bsd o url priod of h ovrhd rvlig cr sysm d by usig rpzoidl or S-shp vlociy pr for h dsird ipu of h corollr. horicl lysis for h codiios of o swy by h rpzoidl d S-shp vlociy pr ipu is prsd. his mhod c b sily pplid for i-swy corol of h cr sysm wh ypicl srvo corollr is usd. h id is lso pplid for h cs of chgig rop lgh of h ovrhd rvlig cr sysm. Kywords: Ovrhd rvlig cr, Ai-swy corol, url priod, rpzoidl vlociy pr, S-shpd vlociy pr. Iroducio My yps of cr mchisms r widly usd for hdlig hvy lod. Espcilly, rvlig cr is grlly usd such s gry crs i por yrd d ovrhd rvlig crs i mufcurig fcoris. Whil hdlig work by h cr mchism, h lod is sy o swig bcus h lod is suspdd by sigl rop. h swig is dgr for workrs i h workspc of h cr d i lowrs h fficicy of h hdlig work, bcus i ks im o rduc h swig for posiioig of h lod. hus, los of sudis o i-swy corol hv b do by my rsrchrs. o rduc h swy of h suspdd lod of gry cr d ovrhd rvlig cr, my lir-fdbck corol hory bsd mhods hv b proposd -4. Mos of hm r combiio mhods of lir corol hory d compsio corol, bcus h cr sysm is ssilly olir sysm. For mpl, corol mhod by lir rgulor d * ** Grdu Sud, Dprm of Illig Mchiry d Sysms Profssor, Dprm of Illig Mchiry d Sysms

2 P.GAO, M.YAMAMOO d Y. HAYASHI compsio ipu for o-lir pr 5, mhod by mulipl opimum rgulors gis for chgig rop lgh 6, combiio mhod of lir fdbck d disurbc obsrvr 7 r proposd. o dl wih h o-liriy of h cr sysm dircly, som o-lir fdbck corol mhods 8- such s ipu-oupu lirizio mhod hv b proposd. Furhrmor, usig h opror's skill for h cr work, fuzzy or rul bs fdbck corol mhods - r lso proposd for i-swy of suspdd lod. hs corol mhods r s fdbck yp usig h s of h lod such s swig gl, whr som ssor sysms o dc h s r dd. h bcoms complicd corol sysm. O h ohr hd, rjcory followig corol by usig pr-dsigd opimum corol ipu c b lso pplid o h i-swy corol problm. For mpl, corol ipu dsig by mimum pricipl d ypicl ipu pr of lss swy r proposd 4-7. Combiio of ypicl ipu pr d lir fdbck corol r lso proposd 8-. I ddiio o cr mchisms, my posiioig d srvo mchisms us srvo corollr, whr dsirbl vlociy d rjcory r rlizd by ir fdbck loop isid of h corollr wh suibl dsirbl corol ipus r commdd o h corollr. hus usr is jus rqusd o ipu h dsirbl vlus for h corollr wihou fdbck of s of mchism. Wh such srvo corollr is islld s corol sysm, rpzoidl vlociy pr or S-shp vlociy pr r of usd s h dsirbl rjcory of h mchism. his corol is bsiclly op-loop yp i h viw poi of h usr wh usig h srvo corollr. For such cs, i hs b kow by pric h usig url priod of h mchism s h cclrio priod of rpzoidl vlociy pr lds o low swy of h sysm. Howvr, horicl bckgroud d mhmicl codiio hs o b clrd y. hus, cosidrig h fc h my srvo mchisms r corolld by rpzoidl vlociy pr or S-shp vlociy pr, h ppr proposs dsig mhod of o swy corol ipu for ovrhd rvlig cr sysm by h ypicl vlociy pr. his is mhod bsd o h rlio bw cclrio im of h ypicl vlociy pr d url priod of h cr sysm. h ppr givs horicl bckgroud for h priml kowldg of h url priod mhod, d lds o grl codiios for o swy usig vrious vlociy ipu prs. his ppr lso mks clr h hr is codiio of o swy wh S-shp vlociy pr is usd. Wh h rop lgh is chgd durig rvlig of h rolly, his id of o swy c o b pplid dircly, bcus url priod is coiuously chgd for h chgig rop's lgh. Howvr, h ppr shows h hr is o swy if h rop lgh is chgd cos vlociy irvl. h ppr, firsly, prss dymicl modl of h ovr hd rvlig cr sysm. h, o swy codiio usig rpzoidl vlociy pr wih d wihou dmpig is show. Furhrmor, o swy codiio usig S-shp vlociy pr wih d wihou dmpig is lso show. h cs of chgig rop lgh is discussd. Vlidiy of h proposd corol dsig mhod of o swy corol ipu is cofirmd by umricl simulios.. Dymicl Modl of Ovrhd rvlig Cr his scio shows dymicl modl of ovrhd rvlig cr. For simpliciy, i is ssumd h suspdd lod is poi mss, mss of rop is smll ough comprig h suspdd lod, h rop dos o gr bdig mom d h rop dos o g logr. h, h quio of moio of lod log cosrid rc by rop is m & s mg siφ m& cosφ

3 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod whr is rolly posiio,m is mss of h suspdd lod, g is grviiol forc, φ is swig gl of rop d s is h coordi vlu log cosrid rc by rop s Fig.. Sig l s rop lgh which c chg for im, iroducig viscous dmpig rm C d usig slφ, Equio lds C l& g && & & l & φ l φ l φ l his is scod ordr im vri sysm diffril quio. If rop lgh is o chgd, h quio bcoms & C g & φ & φ φ l l l whr h url priod for h sysm is l π 4 g rolly M ril rop l mcos mgsi lod m s mg m Fig. Modl of ovrhd rvlig cr.. rpzoidl Vlociy Pr d Nurl Priod Wh usig commrcil srvo corollr, rpzoidl vlociy pr is of usd for posiio corol of mchicl sysms. I hs b kow by pric h kig sm vlu for url priod of h mchism d h cclrio irvl of rpzoidl vlociy pr lds o o swy of h sysm. his scio, hus, discusss grl rlioship bw swy of h sysm d dsig of rpzoidl vlociy pr. I his scio, rop lgh is ssumd o b cos. h cs of chgig rop lgh is rd i h lr scio.. Cs of o dmpig his subscio discusss h cs whr h rolly is corolld by rpzoidl vlociy pr ssumig o dmpig i h cr sysm. I cul posiioig sk by ovrhd rvlig cr, h rvlig disc L is spcifid i dvc. By h limi of drivig sysm for rolly, mimum vlociy of h rolly v m is lso spcifid. h h rvlig disc L d cclrio d dclrio irvl d cos spd irvl hold h followig rlio L v m if L 5 v m

4 4 P.GAO, M.YAMAMOO d Y. HAYASHI Bcus L d v m is spcifid, is uomiclly drmid wh is spcifid by h codiio of o swy. By ssumig o chg of rop lgh, w clcul h rspos for h rpzoidl vlociy ipu i Fig. from o. Firsly, rwri h quio by sdrd form of scod ordr sysm s & φ & φ φ u 6 whr ; bcus h ypicl cr mchism hs vry low dmpig coffici is dmpig coffici, is url gulr frqucy d u is ipu for h sysm. By sig cos ipu u, h rspos is φ C cos C si 7 Usig iiil codiio ofφ d & φ,d ssumig o dumpig, φ cos 8 p v m - r d q Fig. rpzoidl vlociy pr. A h poi p i h Fig., φ d & φ for o swy, hus φ cos 9 & φ si his codiio lds o π,, L Usig h url priod of h sysm Equio 4,h codiio is rwri by π,,l his codiio ms h kig sm vlu for h cclrio irvl d ims of url priod lds o o swy of h cr sysm poi p i Fig.. h irvl of d is complly symmricl wih h irvl of. hus hr is o swy d if h

5 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod 5 Equio is sisfid. Furhrmor, hr is o swy durig cos spd scio, bcus zro ipu for h Equio 6 d φ, & φ. No h sm for cclrio irvl d dclrio irvl is o cssry. For mpl cclrio irvl is d dclrio irvl d is will lso b o swy poi p, q, r i Fig... Cs of wih dmpig Wh hr is dmpig ffc i dymicl quio 6, h codiio of o swy chgs. his subscio drivs h codiio. W firs rwri h Equio 7 usig h iiil codiio φ, & φ s φ si By h codiio of φ, & φ, si θ 4 whr θ cos θ si 5 θ. From quio 4 d 5, θ ξ cos 6 his is codiio of o swy o, howvr, i is o clr h hr is soluio of or o for h olir quio 6.. Cs of wihou cos spd irvl his subscio discusss h cs of L <. For such cs, cos spd irvl i v m Fig. is dispprd, d mimum spd of h rolly dos o rch o spcifid v m d holds L 7 Wh h is spcifid for h codiio of o swy, h cos cclrio is uomiclly drmid,bcus L is spcifid. h rspos for h irvl is s sm s h o i scio. d., bcus of sm rmil codiios d sm cos ipu. hus h codiio of o swy for h cs of wihou dmpig is

6 6 P.GAO, M.YAMAMOO d Y. HAYASHI π,,l 8 Ad for h cs of wih dmpig, o swy codiio is h Equio 6. p v m - d - d Fig. rpzoidl vlociy pr wihou cos spd irvl. 4. S-Shp Vlociy Pr d Nurl Priod Wh rpzoidl vlociy pr is usd for posiioig corol of mchicl sysms, udsirbl vibrio my b cid for h sysm bcus of discoiuiy for ipu cclrio swichig pois poi p d q i Fig.. hrfor, smoohr curv is dsirbl for h vlociy pr. h S-shp vlociy pr is of usd s dsirbl ipu for srvo corollr for such rso. Whr h vlociy is scod ordr polyomil fucio o im, which is show i Fig.4. his scio discusss o swy codiio usig h S-shp vlociy pr. 4. Cs of o dumpig Wh h vlociy pr is giv i Fig.4, h rlioship bw rvlig disc of rolly L d cclrio d dclrio irvl d cos spd irvl is L v if m L 9 v m Whr L d v m is spcifid i cul posiioig problm, hus cos spd irvl is uomiclly drmid if h irvl of is giv by h followig o swy codiio. h slop of ipu u i Fig.4 is giv by v m hus, for giv v m, h slop is uomiclly drmid by clculig by h followig o swy codiio. For h dymicl sysm quio 6, rspos of swy gl φ for h S-shp vlociy pr is prssd by h form of C cos C si A B φ whr C, C, A, B r coss drmid by iiil codiio d ipu. Assumig o dumpig, h rspos wih zro iiil codiio, h swy gl d vlociy of swy gl for h irvl r

7 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod 7 φ si & φ cos Usig rmil codiio i h Equios,, for h irvl, φ cos si 4 & φ si cos 5 p v q m r d - d Fig. 4 S-shp vlociy pr. No swy codiio im is giv by h Equios 4, 5 s; φ cos si 6 & φ si cos 7 hrfor h codiio of o swy p, q, r i Fig.4 wih S-shp vlociy pr is π,, L 4. Cs of wih dumpig h codiio of o swy for h cs of cosidrig dumpig is discussd. Bsd o h dymics quio 6, swy gl φ is dscribd usig h S-shp vlociy ipu i Fig.4 s, 8 C cos C si A B φ 9 whr, C C A B

8 P.GAO, M.YAMAMOO d Y. HAYASHI 8 h swy gl rspos for h irvl is si cos φ si cos φ & For h irvl, φ is giv by quio 9 whr si cos cos C si cos C A B 4 φ & is giv by quio 9 whr si cos cos C 5 si cos cos C 6 A B 7 h o swy codiio is obid by sig d by sig φ, for h bov quios. Howvr, h rlio is oo complicd o g closd form codiio for φ &. I is o clr h hr is codiio of o swy for h dumpig cs or o. 4. Cs of o cos spd irvl Wh m < v L, h cos spd irvl is dispprd s i Fig.5. h h rlioship bw h rvlig disc of rolly L d i Fig.5 is

9 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod 9 L 8 h irvl of is giv by h followig o swy codiio, hus h prmr is drmid by h bov quio. h h codiio of o swy for is sm s h Equio 8 for o dumpig cs, d sm s h Equios 9, -7 for dumpig cs, bcus of h sm codiio for < i scio 4.. v m d - d Fig. 5 S-shp vlociy pr wihou cos spd irvl. 5. Numricl Simulio o cofirm vlidiy of h proposd corol ipu dsig mhod for ovrhd rvlig cr for i-swy corol, umricl simulios r prsd i his scio. I h followig simulio, h rvlig disc for rolly is L5[m], mss of h suspdd lod m7[kg], iiil rop lgh l 5[m], mimum vlociy of rolly v m 6.[m/sc] for rpzoidl vlociy pr d v m 9.[m/sc] for S-shp vlociy pr. h url priod.67[sc] for iiil rop lgh. Firs simulio mpl is h cs of usig rpzoidl vlociy pr s h ipu of cr sysm. Figur - i Fig.6 is rpzoidl vlociy ipu wih sig h cclrio d dclrio irvl s sm vlu s url priod. h h rspos of swy - i Fig.6 is zro rmil im. h zro swy is lso chivd by sig s show i - d -4 i h figurs. Howvr, i is foud h sig. [sc] which is diffr from ims of url priod -5 d -6 i Fig.6 rsuls i rsidul swy. Scod simulio is h cs of usig S-shp vlociy pr s h ipu of cr sysm. Figur S- d S- i Fig.7 is h cs of proposd vlociy ipu dsig by S-shp vlociy pr. h figur of S-5 is o cosidrd o h url priod wh drmiig cclrio d dclrio irvl. As show i S-, S-4 d S-6 i Fig.7, proposd ipu dsig mhod rsuls i o swy rmil lso for h cs of S-shp vlociy pr. hird simulio is h cs of chgig rop lgh i Fig. 8. I h figur SR-, rop lgh rjcory is giv by l. Bcus chgig rop lgh is oly i h irvl of cos vlociy which ms zro ipu, hr is o rsidul swy for his cs. Figurs SC-, SC-, SC- d SC-4 show h cs of cosidrig dumpig coffici C i h Equio whr h rop lgh is fid wih iiil vlu l. By sig C. wih rpzoidl vlociy pr d sig, i is foud h lil swy cos vlociy irvl d rmil im. his is cusd by chg of url priod d diffr phs gl i

10 P.GAO, M.YAMAMOO d Y. HAYASHI Equio 4, 5 from h cs of o dumpig for swy gl d vlociy of swy gl. Vlociy [m/s] Swy gl φ [rd] im [s] im [s] im [s] Vlociy [m/s] Swy gl φ [rd] [sc] im [s] 5. im [s] im [s] Vlociy [m/s] Swy gl φ [rd] Fig. 6 rpzoidl vlociy pr by proposd ipu dsig mhod d is rspos. Vlociy [m/s] Swy gl φ [rd] 5. im [s].. -. S- S- Vlociy [m/s] im [s] Swy gl φ [rd] 5. im [s]. S S im [s] Vlociy [m/s] Swy gl φ [rd] -..[sc] S im [s] S im [s] Fig. 7 S-shp vlociy pr by proposd ipu dsig mhod d is rspos.

11 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod Fig. 8 Cs of chgig rop lgh. Fig. 9 Cs of cosidrig dumpig. Howvr h cs of smll dumpig coffici C which is mor prcicl for ypicl cr sysm, lil rsidul swy is foud.

12 P.GAO, M.YAMAMOO d Y. HAYASHI 6. Coclusio A corol ipu dsig for i-swy of ovrhd rvlig cr bsd o url priod hs b proposd. h ppr hs discussd h rlioship bw swy of h suspdd lod d h url priod of h cr sysm. W propos ipu dsig mhod whr h cclrio d dclrio irvl is s wih ims of url priod of h cr sysm wh rpzoidl or S- shp vlociy ipu is usd. I is lso prsd h h mhod c b pplid for chgig rop lgh durig h moio of rolly if h chg of rop is corolld durig cos vlociy irvl boh for rpzoidl d S-shp vlociy ipu. Numricl mpls show vlidiy of h proposd i-swy mhod. Rfrcs K. mur, S. sukw d M. Kobyshi; Corol of Ovrhd rvlig Cr Sysm wih Widig Mchism d h Applicio of Adpiv Idificio, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.54, No.54, pp , [i Jps]. K-S. Hog, J-H. Kim d K-I. L; Corol of Coir Cr: Fs rvrsig, d Rsidul Swy Corol from h Prspciv of Corollig Udr cud Sysm, Proc. of h Amric Corol Cofrc, pp N. Kodi,. l.; A Ai-Swy Corol of rvlig Cr Bsd o Hoo Corol hory, Proc. of h Aul Cofrc o Sysm Igrio Divisio of SICE, pp.45-46, [i Jps]. 4 Y. Hshimoo,. l.; Lod-Swig Supprssio Corol for Crs wihou Oscillio Fdbck, rscios of h Sociy of Isrum d Corol Egirs, Vol., No., pp , [i Jps]. 5. oi, K. Nomi d H. Okmur; A Mhod for Rlizig Lir Opiml Corol by im-ivri Corol Lw d Compsio Ipu, rscios of h Sociy of Isrum d Corol Egirs, Vol., No.,pp , [i Jps]. 6 I. Mur,. l.; Srvo-yp Ai-swy Corol of Crs wih rolly, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.6, No.58, pp , [i Jps]. 7 H. Aschm,. l.; Disurbc Esimio d Compsio for rjcory Corol of Ovrhd Cr, Proc. of h Amric Corol Cofrc, pp N. Yi, M. Ymmoo d A. Mohri; Fdbck Corol of Cr Bsd o Ivrs Dymics Clculio, rscios of h Sociy of Isrum d Corol Egirs, Vol.7, No., pp.48-55, [i Jps]. 9 M. Kobyshi d K. mur; Nolir Corol of Ovrhd rvlig Cr wih Widig Mchism, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.58, No.584, pp , [i Jps]. B. Vikrmdiy d R. Rjmi; Nolir Corol of rolly Cr Sysm, Proc. of h Amric Corol Cofrc, pp.-6. D-Y. H; Dsig of Coir Cr Corollr Usig Illigc Algorihm, Proc. of h IEEE Iriol Fuzzy Sysms Cofrc, pp M. Skumoo d. Hyshi; Applicio of Fuzzy Corol o Ai-Swy-Sysm of Coir Cr, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.58, No.55,

13 Ai-swy Corol Ipu for Ovrhd rvlig Cr Bsd o Nurl Priod pp , [i Jps]. M. Aro, Y. Sio d S. Kwji; Swig Dmpig Corol usig FIMC for Clig-Cr, Proc. of h Auls Cofrc o Roboic d Mchroics Divisio of Sociy of Mchicl Egirs, pp.a-g8, [i Jps]. 4 J. Shiri,. l.; Dvlopm of Elcroic Swy Corol Sysm for Coir Cr, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.59, No.56, pp , [i Jps]. 5 N. Ohir d. Hismur; Corol of Simpl Pdulum wih Vribl Rop Lgh, rscios of h Sociy of Isrum d Corol Egirs, Vol.9, No.9, pp , [i Jps]. 6. Ymgishi; Swy Corol for Cr, Proc. of h Joi Cofrc o Auomic Corol, pp , [i Jps]. 7 Z. N. Msoud d M. F. Dqq; A Grphicl Approch o Ipu-Shpig Corol Dsig for Coir Crs Wih Hois, IEEE rs. o Corol Sysm chology, Vol.4, No.6, pp Y. Iijim,. Okw d O Ymguchi; Ai-swig Corol Sysm for Coir Hdlig Cr, Proc. of h Aul Cofrc o Sysm Igrio Divisio of SICE, pp.45-46, [i Jps]. 9 I. Morishi; A Nw Corol Algorihm for h Grb-Swig Elimiio i h Auomic Oprio of rvlig Crs, rscios of h Sociy of Isrum d Corol Egirs, Vol.4, No.6, pp , [i Jps].. Ymmoo d N. Nomos; A Swig Pdulum Corol of h rolly Cr wih Vribl Rop Lgh, rscios o Jp Sociy of Mchicl Egirs, Sris-C Vol.65, No.6, pp , [i Jps].

EEE 303: Signals and Linear Systems

EEE 303: Signals and Linear Systems 33: Sigls d Lir Sysms Orhogoliy bw wo sigls L us pproim fucio f () by fucio () ovr irvl : f ( ) = c( ); h rror i pproimio is, () = f() c () h rgy of rror sigl ovr h irvl [, ] is, { }{ } = f () c () d =