ENERGIES METHODS FOR CONTROL OF VIBRATION OF TRANSMISSION LINES

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1 The 3 Intenatnal Cnfeence n Cmputatnal Mechancs an Vtual Engneeng COMEC OCTOBER 9, Basv, Rmana ENERGIES METHODS FOR CONTROL OF VIBRATION OF TRANSMISSION LINES Macel Mgalvc 1, Tu Seteanu 1, Eml Mate Vea 1, 1 Insttute f Sl Mechancs, Buchaest, ROMANIA, e-mal: macel_mgalvc@yahcm Abstact: The pblem f vbatn cntl f tansmssn lnes subjecte t lamna tansvese n, nucng statnay vbatns by Kaman effect s lage stue f lfetme an sevce eznes We cnse u hypthess f the cable mel as Eule-Benull beam, th vscus hysteetc ampng, hch etaches the cable mel f the beam mel The analytcal expessns f the fee vbatn mes f the cable, n u hypthess f the cable, ae puce The vscus, hysteetc Culmb ntenal ampng mels f the cable ae analyze The analytcal expessns f the knetc enegy f the cable s euce an cmpae th the amunt f sspate enegy, fune by expemental means, f the vbatn cntl f tansmssn lnes Keys: cntl f vbatn, cable, tansmssn lne, ampng, ptmzatn 1 INTRODUCTION We cnse the cable mel as Eule-Benull beam th vscus, hysteetc Culmb ntenal ampng The analytcal expessn f the fee vbatn mes an the esnance fequences equatn f the cable th clampe extemtes ae puce usng u hypthess f the cable Expemental eseach as pefme n a specalze stan ene th the vehea cnuct f clampe extemtes, alne th a chce f Stckbge ampes, munte n the extemtes znes f the cnuct The esnance fequences an vbatn mes ae als entfe theetcal an expementally n the cnuct f chce n the stan The pssblty f analyze the nfluence f the cncentate hamnc fce, apple n the mle f the cable an the nfluence f the aelan fce by the enegy agams an als the pssblty t use the enegy balance pncple t etemne the vbatn level f the cable at the esnance an f etemnatn f the ynamc benng stan f the cable, vesus fequences, n the man f nteest, s unelne The analytcal cnseatns n vscus, hysteetc y fctn (Culmb ntenal ampng n the cable mels ae puce th ntepetatn f the esults MATHEMATICAL MODEL OF CABLE WITH GENERAL DAMPING The fllng [1]-[],[8],[13] equatn s cnsee: (, H x t H * (, V c ( x, t ( x, t ( x, t m L = c (, x t c + T EI q x t + + (1 t ω t x x The equatn (1 escbe behav f the cable, excte by the fce q( x, t, apple tansvesal n the cable, hch actn n the pnt f abscssa x, at the tme t, n vscus ampng hypthess by the cnstant ceffcent c, n hysteetc ampng hypthess by the cnstant ceffcent f the fm ch / ω an y fctn V (Culmb ampng hypthess, expesse by the ceffcents c H* The ceffcent c * s cnstant n pecese as a functn f tme t an the sgn f them s such that the sgn f the ampng fce ch* ( x, t t be ppste t that f the velcty & ( x, t at the any tme t H 37

2 We ente ω the esnance fequency f the e f ampe fee vbatn, cable, EI the benng gty f the cable, T the tensn n the cable, (, m L the mass unt length f y x t cespnng vetcal splacement f the cable f vbatn me f e f the cable an L the span length f the cable Fstly, e seach the stablze fee tansvese vbatns f the cable thut ampng an th clampe extemtes The fee vbatns thut ampng ae f stanng aves fm: ( x, t = ( xsn ( ω t+ ϕ ( In fmula ( ω = π f an f s the esnance fequency f the cable n fee vbatns thut ampng; ϕ s the phase angle beteen the ntal mpulse an splacement We use [3] the fllng ntatns: 1/ 1/ 1/ 1/ TL π ml f L α α α α α =, β =, δ = + + β, ε = + + β, ξ = x / L, EI EI The expessns α, β, δ, ε vefy the elatnshps: δ,,, ε = α δ ε = β δ α δ β = ε + α ε β = (3 We seach the slutn f (1 usng the cntn, pefme by the cable e n the cases stue n the lteatue, hch epesents u hypthess that etaches the cable mel f the beam mel [1]: δ e ( F the clampe cable analyze n ths pape, the equatn f esnance fequences [1] s: α sn ε β cs ε = (5 The analytcal expessn f the fee vbatn mes f unampe fee vbatns f the cable, n mentne case f bunay cntns s efne by: ( 1 ( 1 ( e sn cs e sn e x C δ ξ δ δ δ ξ δ ξ = + εξ εξ ε+ cs ε, ξ = x / L (6 ε ε The fact C s a cnstant Anyne can vefy that any Eule-Benull beam mel can be substtute by u cable mel f suffcent hgh fequences because δ an thus e δ We specfy the fllng patcula slutns ( x f the cable mel that efne fmula ( n the case q( x, t =, cv =, ch =, ch* =, slutns that efne als patcula slutns f the beam mel: δ ξ δ ξ ( x = e, ( x = e, ( x = sn ε ξ, ( x = cs ε ξ, ξ = x / L (7 The enttes fm (3 can be use t justfy the patcula slutns (7 The vbatn me f unampe vbatn, expesse by elatns ( an (6, s a slutn f equatn (1 hee q( x, t =, cv =, ch, H* = c =, because ( x fm (6 s a lnea expessn f patcula slutns fm (7 The vbatn me ( vefes als the bunay cntns mpse In the case f ampe vbatns escbe by equatn (1, the slutn s seache n the fm ( x, t = X ( x T ( t, hee X ( x efne a vbatng me f e fm (6 The equatn euce fm equatn (1 f unknn functn T ( t s as flls: T ( ( * V H t T t H c c Ω ΩH* H * + c + ( c T ( t =, c = +, ( c = ω + c / ml (8 t t ml mlω The equatn (8 s euce usng the elatnshp: X ( x X ( x EI T = ω ( ml X x (9 x x In equatn (8 ω s esnance fequency f fee unampe vbatn f the cable an ω s esnance fequency f fee ampe vbatn f the cable Ω < ( If c H * c ω < then the geneal slutn f equatn (8 es nt escbe u physcal H c c * / ml ω H c c * / ml mel It s necessay t take nt accunt the nequalty ( If the ntal cntns f seache slutn f the fm ( x, t = X ( x T ( t (th fxe nex f equatn (1 (hee q( x, t = ae chsen ( x, t = D, ( x, t = V th X ( x a vbatng t me efne by (6, then e take the expessn f the vbatn me that s escbe bel 38

3 X ( x c ( (, t t ( D V sn ( cs x t = e c + ω t t + D ω ( t t, X ( x ω ω ω { ω * ( } 1/ = + c / m c = 1,, H L The fmula (1 specfes that nfluence f the hysteetc an y fctn ampng ae neglgble f vbatn V me, f ths me s suffcent hgh, because ω ω an c c / ml /, f suffcent hgh fequences, but the nfluence f vscus ampng s mantane Ths ppety f the cable s cnfme an expementally The fmula (1 can be use als f pefmng the bjectve functn efee t unknn H H* V paametes EI, c, c, c expesse by theetcal an expemental ay We use cable splacements an the eghte least squae meth f entfy the specfe paametes The expessn f the bjectve functn s: f ( EI, cv, H, H * ( c (, exp (, c c = x t x t (11, x, t We use theetcal splacements c ( x, t an expemental splacements exp ( x, t, theetcal calculate expemental measue n sme pnts f abscses n the cable, ente genec x, als f sme tmes fm the evlutn f the cable mtn an f sme fequences n the man f nteest an hee the eghts has the values 1/( exp ( x, t that assue mensnless f the bjectve functn (1 3 THE ENERGY BALANCE PRINCIPLE The enegy balance pncple s expesse by the fllng equatn []: E = E + E (1 C D th E, the enegy nuce by the n, E C an E D the enegy sspate by the cable f any me f vbatn, espectvely by the ampes, munte n the extemtes znes f the tansmssn lne span The enegy balance pemts us t etemne the ampltue f the vbatn me analyze an the ynamc benng stan f the cable, hch s the key fact n evaluatng the ynamcal stess/stan n the cable, an thus, the pstn egang the fatgue amage The ynamc benng stan ε f the cable, at the gly clampe extemtes, can be expesse by the fllng empcal elatnshp: ε = k y ( x (13 ε b In (13 x = 89m, s the amete f ute stan f the cable, y ( x s the benng ampltue f the b me f the cable vbatn an k ε empcal ceffcent Cncenng the n enegy, thee ae seveal kes [1] hch has stue the n enegy mpate t the cnuct (see fg1 F example, ae euce expementally theetcal the elatnshps: P / f 3 / D = E / f / D = fnc ( y / D = 1z (1 hee P s the n pe n the unt length f cnuct, E s the n enegy n the unt length f cnuct, f s fequency, D s the cnuct amete, y s the antnes ampltue, y / D s efne as mensnless ampltue, fnc ( y / D s name a euce pe ( euce enegy f the n an cul be taken by pe z f the fm z= an X n, X = lg 1( y / D n b Fgue 1: The empcal pe nuce by the n 39

4 F the expemental agam f the n enegy f the Dana (Italy e ppse the fllng analytcal 9 expessn f the euce pe, efne by z= an X n, X = lg 1( y / D th values a,, a9 ne a = 16575, a = , a = 186, a = 17859, a = 316, n= 1 3 a = 3756, a = 31583, a = 1319, a = 96931, a = (15 THE MATHEMATICAL DEDUCTION OF THE KINETIC ENERGY OF THE CABLE We cnse the analytcal expessn (1 f slutn ( x, t = X ( x T ( t (th fxe f ampe fee vbatns f equatn (1 hee q( x, t = Ths slutn s f clampe cable t the extemtes an th D, V ntal cntns Thus knetc enegy f the cable, pe cycle, s: 1 1 t π L ω c L L ( x, t t + E = m T& ( t X ( x x t = m T& ( t X ( x x t (16 The knetc enegy s evaluate usng the values x = L /, t =, V = In ths case, D s the ampltue f the vbatn me The ntegal efee t vaable x s fstly calculate We euce t as bel: L 3 α 5δ L α α δ X ( x x= L+ L( + + ( + + sn ε ( ( + δ ε ε ε ε δ + ε δ + ε (17 δ α δ δ α + L( + cs ε = L kl ε ε ( δ + ε δ + ε δε Als, e calculate: π π 1 ( ( c t c ω ω + ω ω X ( x ω T& ( t t= D e sn( t t (18 We get fm (18, the fllng ntegal value: π ω π 16 X ( x ( ω c π c ω D (( c ( (1 ( c T& + ω ( t t= (1 e (19 D (( c ( (1 ( c ω T& π + ω ( t t ( X ( 3 ( x ω In the ntal cntns f the ampe fee vbatn ( x, t, hen D s the vental ampltue f the vbatn, e can calculate the maxmum knetc enegy ( pe, theefe, the ttal enegy f the cable, usng these elatns: π 1 t+ L ω π m LLkLD c + ω c c = β & L 3 t 8 X ( x ( ω (( ( (1 ( E m T ( t [ X ( x x] t δ + ε δ ( = ( / = csε sn ε ε ε ε X x X L α The calculatn f E c pemts t cmpae the pecentage f the sspate enegy, btane expementally, fm the ttal enegy f the cable 5 EXPERIMENTAL DEDUCTION OF THE DISSIPATED ENERGY The expemental etemnatn [], [5]-[7], [9], [11] f the self-ampng chaactestc f a cable s base n the ecmmenatns fm specalty papes F measung the sspate enegy pe cycle by a cable that vbates n ne f the fee mes, thee ae seveal meths, hch can be gupe n t man categes: the meth f the fee vbatn an the ne f (1

5 the fce vbatn The fce vbatn meth can be apple n t vaants: the statnay ave meth an the pe meth In the case f the pe meth, the cnuct cable s excte als t the esnance, the nuce pe beng etemne ectly by the puct beteen the cncentate fce ampltue n the exctng pnt an the ampltue f the velcty f the esultant splacement In the statnay state, ths pe equals the selfsspate pe f the cable, th the cntn that the t quanttes t be hamnc an n quaate (a phase shft f π / F such a egme, e have, n the pnt f the exctatn applcatn the fmulas escbe bel ( ( = W sn π f t θ, v= V sn π f t θ, F = F sn ( π f t ( 1 θ = θ ( f, θ = θ ( f, V = π f W (3 1 1 Thus, the sspate enegy pe cycle an the pe sspate, has the same magntue as the nuce nes by the excte Ths meth s me ap an smple than the the, but f g esults t s necessay t mnmze the lsses f enegy at the ens, s, the cable nee t be fmly embee n vey sl clam 6 NUMERICAL RESULTS As an example, a cllectve fm u nsttute, etemne the self-ampng f a cnuct cable, n the expemental ste, ene th the cnuct Al-St 3/69, havng the next chaactestcs ([9], [1]: Table I: Paametes f the cnuct Chaactestcs values mensn Oute amete 515 mm Beakng fce 19,63 N Mass pe unt length 1389 Kg/m Span length 8 m tactn 35 N Fstly, e have calculate the values f the ttal pe f the stue cable, f the esnance fequences, usng H H* the elatn (1 n vscus ntenal ampng hypthess ( c =, c = Als e have pefme expementally the sspate pe f the cable (Fg an the pecentage f the sspate pe fm the ttal pe f the cable such that the pecentage f the sspate pe ncease th nceasng f the esnance fequences value Fgue : Expementally sspatn pe 1

6 7 CONCLUSIONS The gnal analytcal cnseatns abut the efntn f the cable n vscus, hysteetc y fctn ntenal ampng hypthess, usng u cable mel etache fm the Eule-Benull beam mel, pemts us t pefm the analytcal vbatn mes f the cable an cntl f these vbatns Fm the last expemental esults f the cable e ntce that th the ncease f the esnance fequences value, the pecentage f the sspate pe als ncease, tenng t each almst 1% f hghe fequences, ta the lmt f the cmmn ange, pvng that the self ampng s a vey mptant cmpnent n the veall ampng assessment Analytcally e emak that thee s the pssblty f the cnuct cable t cnse smultaneus the nfluence f vscus, hysteetc an y fctn ntenal ampng n the cable, but the hysteetc an y fctn ampng ae neglgble f vbatn me f the cable suffcent hgh hle the nfluence f vscus ampng s mantane REFERENCES [1] Tmshenk S: Thee es vbatns a l usage es ngeneus, Lbae plytechnque Ch Beange, Pas et Lege, 1937 [] Claen R, Dana G: Mathematcal Analyss f Tansmssn Lne Vbatn, IEEE Tansactns, vl PAS, 88, , 1969 [3] Nack, W: The ynamcs f the elastc systems (n Rmanan, tanslatn fm Plsh, Techncal Publshng Huse, Buchaest, 1969 [] Chen SS, Wambsganss MW: Paallel Fl-Inuce Vbatn f Fuel Rs, Nuclea Engneeng an Desgn, Vl18, 53-78, 197 [5] * *: Gue n self-ampng measuements, IEEE, St 563, 1978 [6] Agelnk, G: Results f vehea lne cnuct self-ampng measuements, CIGRE, S -81, 1981 [7] Ralns, CH: Ntes n the measuements f cnuct self-ampng, ALCOA Labs, Rept N , 1983 [8] Thmsn WT: They f vbatn th applcatns, Allen & Unn, Syney, 1988 [9] Seteanu T, Vea EM: Self-ampng chaactestc etemnatn f the cables Al St, use t the vehea lne cnucts, Cntact IFTM ICEMENERG, 1988 [1] WG1, SC: Rept n Aelan Vbatn, (CIGRE ept, Electa, n1, 1-77, 1989 [11] * *: Cnuct self-ampng, CIGRE, SC, WG11, TF1, 199 [1] Vea, EM: The electc analgy f the cmplex ynamc phenmena fm the vehea electc cables excte by the n an meths t mtgate the vbatn level, PhD thess, Unvesty Pltehnca f Buchaest, 1995 [13] Petsch G an all: Pantgaph/ Catenay Dynamcs an Cntl, Vehcle System Dynamcs, 8, , 1997 [1] Mgalvc M, Seteanu T, Vea E: Abut the aelan vbatn cntl f vehea lne cnucts, Pceengs f the ASME-PVP Cnfeence, San Deg, USA, 366, 119 1, 1998 [15] Mgalvc M: Cntl f vbatn f vehea lne cnucts, Tpcs n apple mechancs, vlii, cap8, -61, Rmanan Acaemy Publshng Huse, Buchaest, [16] Mgalvc M, Baan Danela: Cntl f vbatns f sme ynamcal systems, BREN Publshng Huse, Buchaest, 6 [17] Vea, EM: Cap 15 fm Vl Reseach Tens n Mechancs (es Dnel Ppa, Vetua Chu, Ileana Tma, Rmanan Acaemy Publshng Huse, Buchaest, 8

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state):

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state): Electc ptental enegy Electstatc fce des wk n a patcle : v v v v W = F s = E s. Ptental enegy (: ntal state f : fnal state): Δ U = U U = W. f ΔU Electc ptental : Δ : ptental enegy pe unt chag e. J ( Jule)