Hopf Bifurcation of Maglev System with Coupled Elastic Guideway

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1 Hopf Bfurcto of Mglev System wth Coupled Elstc udewy No Loghu She, Ho Wg, Dogsheg Zou, Zhzhou Zhg* d Wese Chg Egeerg Reserch Ceter of Mglev Techology, Chgsh Isttute of Techology, Chgsh 7, Hu, PRCh sheloghu@yhoocomc, zzz6@6com* ABSTRACT: I ths pper, the dymcl behvor of sgle mget suspeso system of mglev tr cosderg the elstc gude-wy s reserched through HOPF bfurcto theory Wth three sttes feedbck vrbles (gp, curret, velocty), the HOPF bfurcto codtos d correspodg vbrto frequecy of the coupled system s preseted uder dfferet flexble gudewy codtos, tht s, the gudewy prmeters re chgeble These results c provde wth beefcl help for mprovg the trdtol desg of flexble gudewy d cotrol lgorthms INTRODUCTION I the preset, the TR8 Shgh ofte ppers voletly flexble vehcle-gudewy-coupled vbrto whle rug t the steel gudewy the mtece depot or crossg the turoff t low velocty The CMS tr developed by Chgsh Isttute of Techology lso vbrtes serously whle suspedg o the flotg gudewy But the TR8 tr Shgh le does't vbrte o the cemet grder whose desty s 7 to per meter I order to solve the problem of vbrto pheome of mglev tr, we hve to propose more strct requremets for the precso d dymcl performce of the system, d vestgte the prctcl egeerg problem v oler system theory Erly 986, Ng d Mso [] hd pd tteto to the oler self-oscllto owg to eglectg elstc deformto the cotroller desg, d desged cotrol lgorthm whch c mke the vehcle susped stbly o the flexble gudewy d be sestve to the flexble codtos Youhe Zhou d Xojg Zheg [-] estblshed costt coeffcet dfferetl equto of gudewy dymcl equtos wth vrble coeffcets, d judged the dymc stblty of mglev system v Lypuov chrcterstc dex, d the studed the probblty the pperg chotc pheomeo uder the oler codto udog Lu d Loghu She [] gve the HOPF bfurcto codto of oler mglev cotroller, d dscussed the boud codto d vbrto frequecy whch c mke the cotroller hve self-oscllto Xohog Sh d Loghu She [-6] utlzed the umercl vlue clculto method of HOPF bfurcto to study the correspodg reltoshp betwee gudewy prmeters d HOPF bfurcto, d gve the trsc frequecy scope mkg the gudewy suspedg stbly They lso qulttvely expled the vrble reltoshp of system lod, the cotroller frequecy, the rgdty of secod system, the heret frequecy of gudewy d so o I ths pper, the reltoshp of vrble elstc prmeters of gudewy d the possblty of vehclegude-wy-coupled vbrto ws vestgted through three sttes feedbck vrbles (gp, curret, velocty) to determe the correspodg coupled frequecy The the coupled vbrto pheomeo ws mde cler by HOPF theory Of course, there exst some systems ot usg the stte feedbck cotrol methods But the de of ths pper s lso vluble, tht s, usg the method of fxed cotrol prmeters to study the fluece to the system wth vrble gudewy prmeters ELASTIC COUPLED VIBRATION MODEL OF SINLE MANET SYSTEM physcl model of suspeso system The ttrctve electromgetclly levtted tr s suspeded geerlly by my suspeso mgets

2 The HSST-L model Jp hs suspeso pots There re 6 suspeso pots CMS tr Ch The fluece of every suspeso pots c be eglected owg to usg the mechcl decouplg d essetl cotrol decouplg Therefore, ths pper wll mke sgle suspeso pot s reserch object to study the cotrol problem of flexble coupled vbrto betwee vehcle d gudewy Every suspeso pot hs the prcple structure s the Fg The gudewy s smple pvot, m s the mss uder the secod system, M s the mss bove the secod system, d the mss of electromget s cluded the mss of m If we select the r-sprg s secod suspeso system, the dymcl rgd coeffcet d dmp coeffcet betwee m d M wll be very smll, so the fluece of the two prmeters c be eglected whle studyg the elstc coupled vbrto I ddto, suppose tht the legth of electromget s much less th the legth of gudewy, so the electromget force c be dopted s cetrlzed model X z z x m M Fgure the prcple structure of suspeso pot I the Fgure, OZX s the coordte coected wth gudewy per, x s the horzotl coordte, Z O z s the vertcl dsplcemet of electromget the plce of x, z s the vertcl dsplcemet of gudewy the plce of x Wth respect to electromget, suppose the symbol prmeters s followg Here u represets the voltge of the electromget wdg, s the curret of the electromget, R s the resstce of the electromget, L d s the ductce of the electromget, F e s the electromgetcl force, C s the electromgetcl coeffcet, g s the ccelerto of grvty d t s the tme vrble Wth regrd to gudewy, suppose the symbol prmeters s followg Here E deotes YAN elstc vrble, I s the secto ert of gudewy, ρ s the le desty of the mss, l s the sp of gudewy, m s the mss of gudewy ( m ρl ), φ s the th vbrtol fucto, q s the the brod coordte, η s the dmp rto of the th model, ω s the heret frequecy of the th model d Q s the brod force Cosderg the smple grder, the Beroull-Euler vbrtol prtl dfferetl equto s dopted to solve the system equtos s follows (, ) φ ( ) ( ) () z x t x q t π φ ( x ) s x () m l q&& + η ω q& + ω q Q () π ω l E I ρ () π x s Q Fe () m l Wth regrd to the electromget, we c get the equtos s follows C Ld (6) z z ( ) C d C d ( z z ) u R + ( z z ) (7) dt ( z z ) dt Wth regrd to the dymcl system, the equtos c be gotte s follows ( ) F e C z z (8) ( M + m) g Fe mz && (9) Suppose tht the th coupled vbrto betwee vehcle d gudewy occurs, the equtos bove c be cocluded s follows ( M + m) g C mz && () z z C C u R + & ( z& z& ) z z () z z ( ) ( )

3 && z + η ωz& + ω z CC () z z The three equtos bove (-) mke up of complex coupled oler physcl model, where C s ew defed prmeter Here C mes the squre of mgtude of the utry vbrto fuctol ths plce π C s x m l () From the lyss bove, we c fd out tht for dfferet, tht s, there re dfferet order coupled vbrto betwee vehcle d gudewy, d the forms of fuctos (-) re uchgeble but the vlue of η, ω, C () Therefore, we c study the frst order coupled vbrto ths pper Stte spce model of the system d three stte feedbck close-loop cotrol The physcl meg dctes tht z, z&, z, z&, re depedet stte vrbles of the system bove At preset, the suspeso gp, the vertcl vbrtol ccelerto d the curret of the electromget c be mesured drectly by some sesors, whle z, z& re dffcult to be mesured Especlly wth the movg of vehcles t the gudewy, the horzotl coordtes of z, z& re chgble ceselessly So the osgulrty trsformto s desged s follows x z x z & x z () x z& x The ew set of depedet stte vrbles whch re esy to be mesured c be gotte s x z z, x z& z&, x z, x z &, x We c prove tht fter the o-sgulrty ler trsformto, the cotrollblty of the equlbrum pot s uchgeble After the ew stte vrbles re selected, cosderg the frst order coupled vbrto, the equtos (-) c be descrbed s tht x M + m C x g ( CC ) η ω x m m + x + η ω x ω x + ω x x () M + m C x g m mx x R x x x x + u x C C Here we tke suspeso gp, the dsplcemet d the curret of the electromget s the stte feedbck vrbles, tht s, u uec + ks ( x se ) + kb x + kc x (6) Here s e s the expected gp of the gudewy, k s s the feedbck prmeter of the suspeso gp, k b s the coeffcet of vertcl velocty of the electromget d k c s the feedbck coeffcet of curret RELATED HOPF THEORY I the trdtol HOPF theory [], for orml oler dfferetl equto f ( x, µ ), x R, µ R (7) the blce pot of the system s x x ( µ ), tht s f ( x ( µ ), µ ) After sutble trsformto, the blce pot x x ( µ ) c be swtched to the org I geerl, suppose the blce pot of the system s the org Suppose tht x d µ re resolved er to the org, d f ( x, µ ) whe µ s prt of rego cludg zero The Jcob mtrx [6] c be gotte s A( µ ) D x (, µ ),f () A( µ ) D x (, µ ) hs pr of complex roots, λ d λ Here λ ( µ ) α ( µ ) +ω ( µ ) where ω ( µ ) ω >, α ( µ ), α '( µ ) ; () the other egvlues of A ( µ ) hve egtve rel prt The the system wll hve HOPF bfurcto wth the prmeter µ µ,tht s, perodcl resoluto occurs er to the pot µ µ HOPF theory s mture d perfect, but f the order of the mthemtcl model s very hgh, the clculto wll be very complex d the result s dffcult to coclude Therefore, some reserchers brought forwrd some smple d effectve lgebr crtero Usg the Hurwtz determts, lgebrc crtero d correspodg computtol method for determg the Hopf bfurcto pot s proposed the referece [7] Ths method does ot eed to clculte ll the egevlues of Jcob

4 mtrx of the system for y prmeter d sves computer tme demd Usg the method, the crtcl speed of wheelset d the hutg re studed The lgebr crtero s descrbed s follows λ + ( µ ) λ + + ( µ ) λ + ( µ ) (8) The chrcterstc equto det( A( µ ) λi) s chged s λ + ( µ ) λ + + ( µ ) λ + ( µ ) (9) µ where ( µ ), ( ) s deoted s, (,, ) The Hurwtz determt c be costructed s follows by the coeffcets of equto (8) m m m m m M M M M M M m where f >,, ( m,,, ) Theory Rel coeffcet lger equto (8) hs pr of pure mgry roots d other ( ) roots ll hve egtve rel prt, f d oly f >, >,, > d (,,), where > s the Hurwtz determt of equto (8) Theory If rel coeffcet lgebr equto (8) hs pr of pure mgry roots ± ω d other ( ) roots ll hve egtve rel prt, the ω Theory If the egepolyoml equto (8) of the Jcob mtrx of the system (7) hs egtve rel prt, d Hurwtz determt s stsfed wth the codtos below, () ( ) µ c > Where µ c m{ µ µ : ( µ ) } µ, the equto (8) hs pr of pure mgry root ± ω t the pot of µ µ c d other egevlues ll hve egtve rel prt Suppose tht U d V re left d rght egevector to the egevlue ω c of the mtrx A ( µ c ) A( µ ) () Re( UBV ) C, where B, the there d µ µ µ c exsts HOPF bfurcto t the pot of µ µ c for the system (7) Tht s, er to the pot of µ µ c, the system (7) hs perodcl movemet HURWITZ CRITERION OF HOPF BIFURCATION IN THE APPLICATION OF MALEV COUPLED VIBRATIONAL SYSTEM I ths pper, bsed o the three stte feedbck cotrol (gp, curret, velocty), we dscuss the HOPF bfurcto codtos d correspodg vbrto frequecy of the coupled system, whch re gotte uder dfferet flexble gudewy prmeters Here we tke prctcl dt the mtce depot of Shgh hgh speed mglev le s prmeters of equtos bove, whch re summed up s follows Tble prmeters m M g R C vlues 98 By the HOPF theory d smulto, we c get tht the system s stble f the three cotrol prmeters d three gudewy prmeters re set up s follows Tble prmeters kc kb se ks ω η C vlues - 9π The we wll lyze the HOPF bfurcto d vbrto cses whe the prmeters ks kb kc re fxed d η, ω, C re chgeble Tht s, whe the tr s rug t the gudewy, the cotrol prmeters re uchgeble, but f the prmeters of gudewy re vrble, the coupled vbrto cse my occur For the equtos (6-7), the blce pot c be clculted s C ( M + m) g ( M + m) g ( x, x, x, x, x ) ( s,, se +,, ± se ) C e ω C Ad the stte of the blce pot re stsfed wth x s, e uec ( R kc) x The system hs two sgulr pots whose chrcterstc re sme, d they re oly determed by the prmeter k c Therefore, we wll lyze the cse oly whe x s postve, d smply deote x s x Here C s expressed s C ( M + m) g, d C the the blce pot s deoted s

5 C C (,,,, ) se se + s ec ω At the blce pot x, fter the system s lered, we c get the Jcob determt [8] s follows C C C C ( + CC ) ω η ω ω η ω ( + CC ) m se m s e A CC CC mse mse seks sekb se ( kc R) C C C C The the correspodg chrcterstc polyoml of the Jcob determt c be ferred s J ( λ) λ + λ + λ + λ + λ + where msekc + mser + mηω C mc e c e b e c c e c s b kscmcc ) mηω s k + mηω s R + C k C + ω mc mc ( C C R + ω ms R + C C k + C C C mk C C C mr ω ms k + k C C + ηω C k C + mc C Cηω kc + ω CkbC + ksccηω C Cηω R mc C Cω R + C Cω kc + ksccω mc The the Hurwtz determt c be bult s follows where + η ω η ω ω ω C η ω 878η ω + 87ω 79968ω I the followg, we wll dscuss the HOPF bfurcto cses of the coupled vbrto system From the Theory, f the egepolyoml ppers pr of pure mgry egevlues d the other chrcterstc roots hve egtve rel prt, t should stsfy >, >,, > d, > Tht s +η ω > η ω ω > ω C η ω > 878η ω + 87ω > 79968ω > We c see tht becuse of the three postve gude prmeters, the equtes >, >,, > re lwys stsfed Now we try to determe the three gudewy prmeters wth some sutble vlues, whch re useful to serch the HOPF pots Here let η, ω π be fxed costt vlues, d the we c get C 697 d 767 > by through the clculto of By verfcto, the prmeters bove re ll stsfed wth every codto of the Theory Bsed o the Theory, whe the system hs two pure mgry root d the other roots re ll hve egtve rel prts, we c get ω * 868 By the Theory, we c get ω The the pr of pure mgry roots re ± ω ± 86 By the Theory, we c verfy the results bove s follows, where left egevectors re [ *I, *I, *I, *I, e--768e-*I] The rght egevectors re [8e--977e-*I, 7669e e-*I, e--9666e-*I, 76e--78e-*I, *I] T 6 9* 66 da( k ) s B dks UBV *I

6 Becuse the rel prt of UBV s ot zero, the system hs theoretcl HOPF bfurcto pot uder the codto η, ω π, C d the vbrto frequecy s 86Hz The tble below shows the correspodg coupled vbrtol frequeces clculted uder dfferet prmeters of gudewy It should be poted out tht whe the selected heret frequecy ω s cresg, the coupled vbrtol frequecy ω clculted by the theores bove s er to ω, especlly whe ω > π Tble umber ω C η ω π π 6 6 π π 68 9 π 97 6 π 98 7 π 7 8 π π 86 6π 98 6π π π 9 CONCLUSIONS I ths pper, the vehcle-gudewy-coupled vbrto problem of sgle mget suspeso system of mglev tr s cosdered usg the HOPF bfurcto tool We dscuss the reltoshp of dfferet gudewy prmeters of the coupled system, d the the coupled vbrtol frequecy s clculted wth qutttve lyss It should be poted out tht oly three stte vrbles re chose s feedbck prmeters ths pper, but other sttes re lso c be ppled to feedbck cotrol prctce Next pl s to combe wth more stte vrbles uder dfferet elstc gudewy codtos The expected results should mprove the desg of flexble gudewy or cotrol lgorthms 6 REFERENCES Ng, M S N K R, C He Cotrol of mgetc suspeso to suppress the self-excted vbrto of flexble gudewy IEEE Trsctos o dustrl electrocs 986, (78): Wu Jju, Zheg Xojg, Zhou Youhe, He Lhog THE NONLINEAR DYNAMICAL CHARACTERISTICS OF EMS MALEV CONTROL SYSTEM WITH TWO SUSPENSIONS Chese Jourl of Sold Mechcs, (): 68-7 Zhou Youhe, Wu Jju, Zheg Xojg, He Lhog ANALYSIS OF DYNAMIC STABILITY FOR MANETICLEVITATION VEHICLES BY LIAPUNOVCHARACTERISTIC NUMBER Chese Jourl of Theoretcl d Appled Mechcs, (): - udog Lu, Loghu She Louts Crtero for Hopf Bfurcto d Alyss of Vbrto of Mglev System Jourl of Vbrto, Mesuremet & Dgoss, (): Xohog Sh Reserch o the udewy-vehcle Couplg Noler Dymc Problems of the EMS Hgh-Speed Mglev System doctor thess, Xohog Sh, Loghu She, Wese Chg THE BIFURCATION ANALYSIS OF THE EMS MALEV VEHICLE-COUPLED-UIDEWAY SYSTEM Chese Jourl of Theoretcl d Appled Mechcs, 6(): 6-69 Zeg Jg, Yg Yre, Zhg Jye AN ALORITHM CRITERION FOR HOPF BIFURCATION AND ITS APPLICATIONS IN VEHICLE DYNAMICS Chese Jourl of Theoretcl d Appled Mechcs, vol,no Sep, Zhe M, Ycg Zhou Costt dfferetl equto d stblty method Chese Scece Press,