Boundary Control of a Tensioned Elastic Axially Moving String

Transcription

1 ICCAS5 June -5 KINTEX Gyeonggi-Do Korea Boundary Conrol of a Tensioned Elasi Aially Moving Sring Chang-Won Kim* Keum-Shik Hong** and Hahn Park* *Deparmen of Mehanial and Inelligen Sysems Engineering Pusan Naional Universiy San 3 Jangjeon-dong Gumjeong-gu Busan Korea. Tel.: wonnyday@pusan.a.kr hpark97@pusan.a.kr **Shool of Mehanial Engineering Pusan Naional Universiy San 3 Jangjeon-dong Gumjeong-gu Busan Korea. Tel.: Fa: kshong@pusan.a.kr Absra: In his paper an aive vibraion onrol of a ensioned elasi aially moving sring is invesigaed. The dynamis of he ranslaing sring are desribed by a non-linear parial differenial equaion oupled wih an ordinary differenial equaion. A ime varying onrol in he form of righ boundary ransverse moions is proposed in sabilizing he ransverse vibraions of he ranslaing oninuum. A onrol law based on yapunov s seond mehod is derived. Eponenial sabiliy of he losed-loop sysem is verified. The effeiveness of he proposed onroller is shown hrough simulaions. Keywords: Aially moving sring boundary onrol eponenial sabiliy hyperboli parial differenial equaion yapunov mehod nonlinear sring. INTRODUCTION The onrol problems of aially moving sysems our in various engineering areas: For eample he srips in hin meal-shee produion lines he ables bels and hains in power ransmission lines he magnei apes in reorders he band saws e. The dynamis of hese sysems an be differenly modeled depending on he lengh fleibiliy and onrol objeives of he sysem onsidered. For insane he dynamis of a moving able of an elevaor an be desribed by a sring equaion bu ha of a rubber bel in he radiional mill an be well represened by a bel equaion. The differene beween a sring and a bel lies in wheher he longiudinal elongaion is onsidered or no. In aially moving sysems he ransverse (laeral vibraion of he moving maerial ofen auses a serious problem in ahieving good qualiy. I is also known ha hese vibraions are ofen aused by he eenriiy of a pulley and/or an irregular speed of he driving moor and/or a non-uniform maerial propery and/or environmenal disurbanes. Sine he qualiy requiremen as well as he produiviy in a produion line is geing srier an aive or a semi-aive vibraion onrol is nowadays seriously onsidered. In his paper he vibraion onrol mehod o redue he vibraion whih ours during a oninuous ho-dip zin galvanizing proess is onsidered. In order o ahieve he uniformiy of he zin deposi on he srip surfaes and o redue he zin onsumpion he srip should pass a equidisane from eah of he air knives. Bu due o he shifing and vibraion of he srip a disrepany beween he averaged deposied masses on he lef and righ srip surfaes and a non-uniformiy of he deposied mass aross he srip our. These variaions in deposied mass will degrade he qualiy of he produ. Depending on he hikness of he srip and he disane beween wo suppor poins Aially moving sysem an be modeled as one ou of hree models: a moving beam a moving sring and a moving bel. In he zin galvanizing line he disane beween wo-suppor poins is quie large ompared o he srip hikness and widh is small. Therefore he modeling as a sring is invesed. In he given sysem in-flu and ou-flu mass eis and a he ime varying boundary whih he onrol fore is given by he moving mass work is ourred. So we use Hamilon s priniple for sysem of hanging mass o derive nonlinear equaion of moion. The previous researhes abou aially moving sring didn onsider he elasiiy of he sring and used more han wo sensors. ee and Moe [] derived an opimal boundary fore onrol law ha dissipaes he vibraion energy of an aially moving sring and in [5] analyzed he wave haraerisis of he beam and derived opimal boundary damping laws as a funion of linear veloiy linear slope and linear fore. The onribuions of his paper are he following. Firs boundary onrol law abou non-linear sring o onsider he onsan ension and elasiiy of he sring is derived. Seond he derived boundary onrol law requires only one sensor o apply iself. Addiionally he damping oeffiien of he auaor is designed.. EQUATIONS OF MOTION Fig. shows a shemai (model of he aially moving sring onsidered whih will be used in deriving equaions of moion and a boundary onrol law. The sring is assumed o ravel a a onsan speed. The lef boundary is fied in he sense ha he boundary iself does no have any verial (ransversal movemen bu i allows he maerial o move longiudinally. However he righ boundary permis a ransversal movemen of he sring under a onrol fore and is ime varying. e be he ime be he spaial oordinae along he longiude of moion v be he aial speed of he sring w ( be he ransversal displaemen of he sring a spaial oordinae and ime and be he lengh of he sring. Then he absolue veloiy of he sring a spaial oordinae is given by v = v i Dw( D j = v i { w ( vw ( j ( where D( / D = ( / v ( / and ( = ( ( = ( denoe he parial derivaives in ime and spaial oordinae respeively. The kinei energy of he aially moving sring inluding he auaor and he poenial energy are wrien as follows: T = ρ A{ v ( w d mw ( = EA U Po d ε ε (3 where T is he kinei energy U is he poenial (srain energy ρ is he mass per uni volume (maerial densiy A is he ross-seional area m is he mass of he auaor

2 ICCAS5 Y June -5 KINTEX Gyeonggi-Do Korea ds v w( d Conroller A a P P r P s w w w d P r Fig. An aially moving srip under a righ boundary onrol fore. (ouh roll E is he elasi modulus of he sring ε is he srain due o he ension P. The poenial energy is proporional o he inrease in sring lengh ds when ompared o he sring a res. For small slopes see Fig. he following relaionship for sring elongaion is valid: ds d ( ε = w w( = d. ( Therefore (3 is rewrien as follows EA U = P w w d. (5 Now o derive he equaions of moion Hamilon s priniple for he sysems wih hanging mass is uilized as follows: δ ( T U W.... = n Wr b d (6 where W n.. is he non-onservaive work W r.b. is he virual momenum ranspor a he righ boundary (no variaions a he lef boundary. The variaions of he non-onservaive work and he virual momenum ranspor a he righ boundary are δwn.. = F ( δ w( dw δ w( (7 δ Wr. b. = { w δ w( ( where F ( is he onrol fore and d is he damping oeffiien of he auaor. Now he variaions of ( and (5 respeively are δ T = ρa ( w ( δw vδw mw δw (9 EA 3 δ U = P wδwd wδwd. ( The subsiuion of (7-( ino (6 yields ( δ T δu δw.. n δwr b d = A( w vw w d d ρ δ ( ρ Avw w δw d d EA 3 P w w δ w d d [ mw δ w δw( ]d d Fig. Shemai of sring elongaion for small slopes. { F ( d w δ w( d { Avw w δw( = ρ. ( And he inegraion of ( by pars yields [( ρ Aw w δw] d ( ρ Aw w δw d d [( ρ Avw w δw] d ( w w δwdd EA 3 P w w w d δ 3EA P w w w wdd δ [ mw δw( ] { mw F ( d w δw( d { Avw w ρ δw( d. ( Noe ha δ w( = beause he lef end is fied (i.e. w ( =. Therefore ( is rewrien as follows: ( ρ Aw w P ρ Av w 3EA δwdd wδwdd { mw F ( d w δw( d EA P w w w( d = δ. (3 Sine δ w is arbirary eep for he requiremen ha he lef end is fied (i.e. δ w( = he following governing equaion and a boundary onsrain a he righ end are derived as follows: 3EA ρaw w P ρ Av w w = ( F ( = mw d w

3 ICCAS5 EA P w w (5 where w is he loal aeleraion in he ransversal direion of he sring w is he Coriolis aeleraion and v w is he enripeal aeleraion. Remark: For a linear sysem whih is he ase ha 3EAw / = he soluion of ( an be obained hrough he mehod of separaion of variables. In his ase he naural nπ frequeny is given by ω n = ( v n = 3 where = P ρa is alled he wave veloiy (see []. The naural frequeny dereases as he raveling speed inreases. If he raveling speed is equal o he wave veloiy he naural frequeny beomes zero and a divergene of he soluion ours. In his sense is alled he riial speed v r. Hene he following is also assumed in his paper. < v < vr = P ρa. (6 If using he parameers in Table we an alulae = P / ρ A = m/se. v r 3. BOUNDARY CONTRO AW In his seion we design a boundary onroller inluding a damper o redue he ransversal vibraion of a longiudinally moving elasi sring. A posiive definie funion in he form of oal mehanial energy of he sring eluding he auaor is firs onsidered as follows: Vs ( = ρ A( w EA P w w d (7 where he subsrip s sands for sring. emma : Consider a funional V ~ ~ V ( = Vs ( V ( ( wih he seond (omplemenary erm is defined by V ( = ρ A w ( w (9 where > is a onsan. Then (7 and ( are equivalen ha is here eis onsans > and < C < saisfying ~ ( C VS ( V ( ( C VS (. ( Proof: The eisene of suh and C will be proved. Firs (9 beomes V ( = ρ A w ( w where ρa w d P ρ A w P ( w d d ( w vw d EA w ρ A EA d C V ( ( S June -5 KINTEX Gyeonggi-Do Korea ρa C = min( P ρa EA >. ( Hene he following holds: CV s ( V ( CV s (. (3 By adding V s ( a boh sides of (3 we obain ( C ~ V S ( V ( ( C V S (. ( In order o C > he range of is resried by min( P ρa EA < <. (5 ρa Verifying he range of using he parameers in Table < < / =.5 is obained. In his paper =.3 is seleed and C =. 6 is alulaed. emma is proved. Now wih emma he following yapunov funion andidae V ( whih is basially equivalen o he oal mehanial energy of he sring and he auaor is proposed. ~ V ( = V ( Va ( (6 where he addiional auaed-relaed erm V a ( is defined as m Va ( = { w ( v w. (7 To derive he ime derivaive of (6 a fied onrol volume is inrodued as in Fig. 3. Volume II represens he par of he sring ha oupies he inner par of he onrol volume a an arbirary ime whereas I and III represen he influ and efflu of he sring a d respeively. Using Reynolds ranspor heorem he ime-derivaive of (6 is given by dv ( / d = V / v V /. ( The firs erm in he righ-hand side of ( means he ime rae of he equivalen energy wihin he onrol volume and he seond erm is he ne energy flu ino he onrol volume. Noing ha V inludes V s in (7 V in (9 and V a in (7 individual erms are evaluaed as follows: Vs ( / = ρ A( w ( w EA P w w w d = = v ( [ ] ( P [ ] P w w w EA 3 [ ] [ ] 3EAv ww w [ w ] onrol volume (9 = I II III v d Fig. 3 Conrol volume of an aially moving sring sysem wih a ime varying righ boundary. d

4 ICCAS5 v Vs ( / = v ρ A( w ( w EAv 3 v P wwd wwd Av vp = ρ [( w ] [ ] EAv w [ w ]. (3 Using (9 and (3 dv s ( / d beomes v ( [ ] ( P [ ] s ( / d = P w w w EA 3 [ w ] Av 3EAv w ρ [ w ] [ ] w Av ρ [( w ] vp vw [ w ] EAv [ w ] = P w w vp { w w ( EA 3 EAv w w { w w ( dv. (3 For V ( he following are derived: V ( / = ρ A w ( w ρ A w ( w (3 v V ( / = vρ A w ( w vρ A w ( w vρ A w ( w. (33 Using (3 and (33 dv ( / d beomes dv ( / d = vρ A ( w w w w w w Av ρ wwd ρ A w w d w ( ρaw w w ρ Av w d. (3 To rewrie (3 he following inegraions by pars are uilized. ( w w w w ww = [ w w ] = w( w (35 wwd = w w d (36 ww d = w w d. (37 Also using he equaion of moion ( he following equaion is derived. w ( ρaw w w 3EA = w P w w d. (3 Therefore he subsiuion of (35-(3 ino (3 yields dv ( / d = w w w w d P P w w d June -5 KINTEX Gyeonggi-Do Korea 3 EA 3EA ρa w w d w ρa w d Av w ρ d. (39 Also he ime-derivaive of (7 is dv a / d = m w v w ( { ( { w ( v w. ( Finally he main par in his paper is saed as follows: Theorem: Consider he following aially moving sysem 3EA ρaw w P ρ Av = w w w ( = mw d w EA P w w( = F ( w = w ( w ( = w& (. ( ( ( If he onrol fore F and he damping oeffiien d are given by F ( = Kw ( ρ A / < d < vρa /( v (3 where K = m( v is he onrol gain hen he losed-loop sysem is eponenially sable. Proof: The subsiuion of ( ino (5 yields mw = d w EA P w w m( v w.( The subsiuion of ( ino ( yields dv ( / d = w v w a { ( EA 3 d w P w w. (5 Therefore ombining (3 (39 and (5 he ime-derivaive of he yapunov funion andidae (6 is given by EA dv ( / d = ( P w w ρa ( P wd w d 3EA E Av w d vp ( w w ( ρa d w ρ Av d ( v w w { ( P w EA w ρa ( P wd w d 3EA E Av w d vp ( w w ( A d w P w ρ ( ρa d w ρ Av d ( v w w. (6 { Sine v is under he riial speed (see (6 P > is saisfied. Noe ha he firs and eighh erms afer he equaliy

5 ICCAS5 sign have been spli ino wo halves. Noe also ha he firs eigh erms followed by he inequaliy sign are all negaive whereas he las hree erms are ombined o make hem negaive as a whole. The following noaions are inrodued. φ = ( P ρ Av / (7 > ( d ρ A { ρ Av d ( v φ = / ( ψ =. (9 From ( and (9 if d saisfies he range d ρ A d and > ( v d d < ρ Av (5 hen he las hree erms in (6 saisfies he following inequaliy. φ w ψ w w φ w min{ φ ψ / φ ( w w. (5 Therefore from (6 and (5 he asympoi sabiliy of he losed-loop sysem is assured. Now he eponenial sabiliy is shown wih furher manipulaion of he erms. (6 an be rewrien by spliing he hird erm ino wo pars as EA dv ( / d w vp w ( E Av w ( ( P wd ρa ( P ( vw w d v 3 EA ( w d P ( w ρa d w min{ φ ψ ϕ{ w ( w EA E Av w vp w ( w ( ρ A ( w d P Av w ρ d ρa ( P min w d ( vw v 3 EA ( w d P ( w ρa d w min{ φ ψ ϕ{ w ( w. (5 By using he following inequaliy w d ( vw ( w (53 and eliminaing he firs four and las (negaive erms (5 an be rewrien as ρa ( P dv ( / d min ( w v 3EA ( P wd w d ( P w ρa d w June -5 KINTEX Gyeonggi-Do Korea ( P ( P min 3 P P EA ρa wd wd ( w ( P ρa min d m( v m m { w ( v w. (5 If using (7 and (6-(7 (5 an be epressed as ( P ( P ~ dv ( / d min 3 V ( P ( P ρa min d V ( A m( v m ( P ( P min 3 P ( P ρa d m( v m ( V ~ ( VA( = λv (. (55 (55 means he following relaionship λ Voal ( V e (56 where V = V ( and λ is given by ( P ( P λ = min 3 P ( P ρa d >. (57 m( v m Therefore all he variables inluded in (6 onverge eponenially o zero.. IMPEMENTATION AND SIMUATION Implemening ( and (3 requires wo hings: measuremen of and saisfaion of he range d < d < d w. Sine he damping range is relaed o a design problem of he auaor i mus be answered ahead. Using he parameers in Table wih =.3 (5 is verified as follows: 7.67 < d < 7.7. (5 Therefore he eisene of suh a damping range is assured. The implemenaion of w an be ahieved by bakwards differening of w measured a eah sep. To demonsrae he performane of he losed loop sysem ompuer simulaions using he finie differene sheme have been performed. The plan parameers used for simulaions are gahered in Table. Wih =.3 given in (5 he onrol gain is alulaed as follows: K = m( v = 5(.3 = 39. (59 For simulaion purpose le d = 5. e he iniial ondiions be w ( = sin(3π [m] w ( = [m/s]. (6

6 ICCAS5 June -5 KINTEX Gyeonggi-Do Korea Fig. The ransverse displaemen wih onrol gain K = 39 and damping oeffiien d = 5 w ( where = m. Fig. 5 The ransverse displaemen wih onrol gain K = 39 and damping oeffiien d = 5 w where = m. Fig. and Fig. 5 show he ransverse displaemen a = / and he onrol fore a = respeively. The designed onroller eliminaes he ransversal vibraion in 3 seonds. Fig. 6 shows he variaion of boundary onrol inpu a =. 5. CONCUSIONS This paper invesigaed a ransverse vibraion suppression sheme of an aially moving non-linear sring and disussed he developmen of an effiien aive onroller and is implemenaion. In previous researhes he sring was mosly modeled as a linear sring. However we derived a non-linear sring PDE equaion wih auaor dynamis. The boundary onrol law was derived by yapunov mehod. Eponenial sabiliy of he losed-loop sysem was proved. The effiieny of he designed onroller was shown by simulaions. ACKNOWEDGEMENT This work was suppored by he Minisry of Siene and Tehnology of Korea under a program of he Naional Researh aboraory gran number NR M J--3-. REFERENCES [] Hong K. S. Kim C. W. and Hong K. T. Boundary Conrol of an Aially Moving Bel Sysem in a Thin-Meal Produion ine Inernaional Journal of Conrol Auomaion and Sysems Vol. No. pp Fig. 6 The onrol fore used in Fig. 5. Table The plan parameers for simulaion. Symbols Definiions Values A Cross-seion area.5. 5 m engh of sring m P Tension of he sring 9 kn m Mass of he auaor 5 kg v Sring moving speed m/s ρ Mass per uni area 7 5 kg/m d Damping oeffiien 5 Ns/m [] Kim C. W. Hong K. S. and Park H. Boundary Conrol of an Aially Moving Sring: Auaor Dynamis Inluded Journal of Mehanial Siene and Tehnology Vol. 9 No. pp [3] Khalil H. Nonlinear Sysems Prenie Hall nd ediion 996. [] ee S. Y. and Moe C. D. Vibraion onrol of an aially moving sring by boundary onrol ASME Journal of Dynami Sysems Measuremen and Conrol Vol. No. pp [5] ee S. Y. and Moe C. D. Wave haraerisis and vibraion onrol of ranslaing beams by opimal boundary damping Journal of Vibraion and Aousis Vol. No. pp [6] ee S. Y. and Park S. G. Free Vibraion Charaerisis of a Sring wih Time-Varying engh Korea Soiey for Noise and Vibraion Engineering Vol. 9 No.5 pp [7] i Y. Aron D. and Rahn C. D. Adapive vibraion isolaion for aially moving srings: heory and eperimen Auomaia Vol. 3 No. 3 pp [] Ryu D. H. and Park Y. P. Transverse vibraion onrol of an aially moving sring by veloiy boundary onrol Transaions of KSME A Vol. 5 No. pp [9] Ying S. and Tan C. A. Aive Vibraion Conrol of he Aially Moving Sring Using Spae Feedfoward and Feedbak Conrollers Journal of Vibraion and Aousis Vol. No. pp