Dynamic Characteristics of Bare Conductors

Transcription

1 Dyamic Characerisics of Bare Coducors By Ojo Evas Eshiemogie Sude No: I parial * Fulfillme of he requiremes for he degree Masers of Sciece i Elecrical Egieerig School of Elecrical Elecroics ad Compuer Egieerig Faculy of Egieerig Uiversiy of KwaZulu-Naal Supervisor : Professor N.M. Ijumba Co-supervisor : Professor D. Mufic Durba Souh Africa Jue 011 i

2 PREFACE The work described i his disseraio was carried ou hrough he school of Elecrical Elecroic ad Compuer Egieerig Uiversiy of KwaZulu-Naal UKZN. The eperimeal work used i his sudy was doe a Vibraio Research ad Tesig Cere VRTC locaed a he Wesville campus of Uiversiy of KwaZulu-Naal. The auhor hereby cofirms ha he maerial preseed i his disseraio is my ow work ecep where specific ackowledgme is made by ame or i he form of a referece. DECLARATION I Ojo Evas Eshiemogie declare ha: i The research repored i his disseraio/hesis ecep where oherwise idicaed is my origial work. ii This disseraio/hesis has o bee submied for ay degree or eamiaio a ay oher uiversiy. iii This disseraio/hesis does o coai oher persos daa picures graphs or oher iformaio uless specifically ackowledged as beig sourced from oher persos. iv This disseraio/hesis does o coai oher persos wriig uless specifically ackowledged as beig sourced from oher researchers. Where oher wrie sources have bee quoed he: a heir words have bee re-wrie bu he geeral iformaio aribued o hem has bee refereced; b where heir eac words have bee used heir wriig has bee placed iside quoaio marks ad v refereced. Where I have reproduced a publicaio of which I am a auhor co-auhor or edior I have idicaed i deail which par of he publicaio was acually wrie by myself aloe ad have fully refereced such publicaios. vi This disseraio/hesis does o coai e graphics or ables copied ad pased from he Iere uless specifically ackowledged ad he source beig deailed i he disseraio/hesis ad i he Refereces secios. Siged E. E. Ojo Dae As he cadidae s supervisor I agree/do o agree o he submissio of his disseraio Siged Prof. N. M Ijumba Supervisor Dae ii

3 ACKNOWLEDGEMENTS My sicere graiude goes o my supervisor ad co-supervisor Professor N.M. Ijumba ad Professor D. Mufic for heir ivaluable suppor ad guidace hroughou his sudy. I am highly graeful o my meor Professor M. A. E. Kauda for his immese suppor ad also for beig he source of my ispiraio. I ow my sicere graiude o a umber of idividuals whom i oe form or he oher coribued o he successful compleio of his projec amog hem Dr R. Loubser Prof E. Boje Dr M. Gafarri Cosmai Umbao Prof Da Ima Dr. K. Aruakiriahar ad ohers oo umerous o meio. I would like o hak he former busiess maager of High Volage ad Direc Curre HVDC cere Leea Rajpal for beig here o led me her suppor ad o he coordiaor of he VRTC Pravesh Moodley for his immese coribuio o his sudy. My haks go o all pas posgraduae admiisraors: Neeu Precious Nicolas ad Kerry-A. My graiude goes o he Egieerig Faculy Officer Fioa Higgiso for her guidace durig he course of my sudy. I graefully ackowledge he fiacial suppor from he HVDC cere ad VRTC durig he course of my sudy My sicere haks go o my all my colleagues ad frieds for heir moral suppor. I wa o use his opporuiy o remember my lae fried ad colleague Do Chamagii he bigges quesio ha sill remai uaswered why mus his happe. I believe i is oly God ha kows why such a ragic eve has o happe o a ioce perso like you res i peace my bosso fried. I dedicae his disseraio o my faher ad moher Mr. Moses ad Mrs. Magdalee Ojo ad o he eire Ojo s family especially Cyril for heir cosa source of ispiraio ad if o for hem i would have o bee possible for me o complee my sudies. Fially I would like o hak all hose who have coribued eiher direcly or idirecly o he preparaio research ad compilaio of his projec repor. iii

4 Absrac The dyamic characerisic of rasmissio lie coducors is very impora i desigig ad cosrucig a ew lie or upgradig a eisig oe. This cocep is a impedime o lie desig ad cosrucio because i ormally deermies he esio a which he lie is srug ad his i respec affecs he ower heigh ad he spa legh. Ivesigaios io he pheomeo of mechaical oscillaio of power lie coducors have bee eesively looked io by may researchers usig coceps from mechaics ad aerodyamics o ry ad predic he coducor dyamic behaviour. Fidigs have show ha precise predicio of coducor wididuced vibraio is very difficul i.e. o-lieariy. Over he years various aalyical models have bee developed by researchers o ry ad predic he mechaical vibraio of rasmissio lie coducors. The firs par of his disseraio cosiders he aalysis of he model describig he rasverse vibraio of a coducor as a log sleder simply suppored beam isoropic i aure ad subjeced o a coceraed force. The soluio of his beam equaio was used o obai he coducor aural frequecies ad mode shapes. Coducor self-dampig was obaied by he iroducio of boh eeral ad ieral dampig models io he equaio of moio for he beam. Ne also usig he same beam cocep was he applicaio of he fiie eleme mehod FEM for he dyamic aalysis of rasmissio lie coducors. A fiie eleme formulaio was doe o prese a weak form of he problem; Galerki s mehod was he applied o derive he goverig equaios for he fiie eleme. Assembly of hese fiie eleme equaios he equaio of moio for he rasverse vibraio of he coducor is obaied. A oe dimesioal fiie eleme simulaio was doe usig ABAQUS sofware o simulae is rasverse displaceme. The eigevalues ad aural frequecies for he coducors were calculaed a hree differe esios for wo differe coducors. The dampig behaviour of he coducors was evaluaed usig he proporioal dampig Rayleigh dampig model. The resuls obaied were he compared wih he resuls from he aalyical model ad he compariso showed a very good agreeme. A elecrical equivale for he coducor was developed based o he cocep of mechaicalelecrical aalogy usig he discree simply suppored beam model. The developed elecrical equivale circui was he used o formulae he rasfer fucio for he coducor. Malab sofware was used o simulae he free respose of he developed rasfer fucio. iv

5 Fially he eperimeal sudy was coduced o validae boh he aalyical model ad he FEM. Tess were doe o a sigle spa coducor usig wo esig mehods i.e. free ad force vibraio. The es resuls are valid oly for Aeolia vibraio. From he es resuls he coducor s aural frequecies ad dampig were deermied. The eperimeal resuls as compared wih he aalyical resuls were used o validae he fiie eleme simulaio resuls obaied from he ABAQUS simulaio. v

6 CONTENTS PREFACE... ii ACKNOWLEDGEMENTS.iii Absrac...iv CONTENTS...vi 1. INTRODUCTION 1.1 Backgroud Moivaio Research Quesio Hypohesis Imporace of This Sudy Disseraio Oulie LITERATURE REVIEW.1 Waves Mechaical Vibraio Liear ad No-liear Sysems Vibraio Aalysis Types of Mechaical Vibraio Beas ad Resoace 1.4 Dampig Effecs of Dampig Coducor Dampig Coducor Mechaical Vibraio Srouhal ad Scruo Numbers Classificaio of Coducor Oscillaio Aeolia Vibraio Coducor Gallopig Wake-iduced Vibraio Faigue Failure ANALYTICAL MODELLING 3.1 Iroducio Physical Aspec ad Modelig Cocep Coducor Ipu Power Wid Loadig Coceps of Bare Coducor Modelig The Equaio of Moio Soluio Equaio of Moio Aalyical Modelig of Coducor Self-dampig Free Vibraio Forced Vibraio Bedig Siffess EI Value...35 vi

7 4. MODEL VERIFICATION: FINITE ELEMENT ANALYSIS AND MECHANICAL EQUIVALENCE 4.1 Discree Modelig Discreisaio of he Domai io Fiie Elemes Fiie Eleme Aalysis Semi-discree FEM Fiie Eleme Formulaio Weak Formulaio The Sysem Mari Numerical Compuaio for Naural Frequecies Numerical Compuaio for Dampig Evaluaio of Dampig Elecrical Equivalece: Elecrical Equivale Circui The Ipu-Oupu Model TESTING AND RESULTS 5. Vibraio Research ad Tesig Cere VRTC Mehods of Tesig Free Vibraio Tesig Forced Vibraio Tesig Mehods of Forced Vibraio Tesig Shaker Coducor Coecio Resuls Eperimeal resuls for Free Vibraio: Ter Coducor Eperimeal resuls for Forced Vibraio Sweep Tes: Ter Coducor Eperimeal resuls for Free Vibraio: Aero-Z Coducor Eperimeal resuls for Forced Vibraio Sweep Tes: Aero-Z Coducor Fiie Eleme Aalysis FEA Resuls Equivale Circui Resuls Aalysis of Resuls CONCLUSIONS AND RECOMMENDATIONS 6.1 Coclusios Recommedaios 87 Refereces.. 89 Appedices. 9 vii

8 CHAPTER Backgroud INTRODUCTION Souh Africa is presely eperiecig power crises because he curre geeraio capaciy is less ha he load. This has promped he aio uiliy o icrease he power oupu from all aspecs of is operaio. The power sysem i he coury is divided io geeraio rasmissio ad disribuio. The rasmissio eworks are used o rasfer power from he geeraio saios o he disribuio eworks. Ecoomic ad eviromeal pressure coupled wih he difficuly i acquirig lies righ-of-way serviude has grealy iflueced he desig of overhead rasmissio lies. The pressure resulig from hese codiios has ecessiaed he cosrucio of log high-capaciy highvolage rasmissio lies [1]. Trasmissio lies are desiged o esure availabiliy reliabiliy of power safey for he public ad maieace persoel ad ca be cosruced a opimal cos. Lies i Souh Africa are desiged o mee sadards se i accordace wih he code of pracice described i he SABS docume []. This code of pracice specifies he miimum clearace of he coducor from he groud public roads railway lies as well as oher power lies uder varyig weaher codiios such as emperaure ad wid. Based o hese crieria; i is very epesive o cosruc a ew lie upgrade or maiai eisig oes. Hece he cos of desigig ad cosrucig rasmissio lies coiues o icrease over he years. Overhead power lies usually cosis of he followig: seel owers coducors isulaors ad associaed aachme hardware. The coducor whose fucio is o rasfer power i power lies is cosidered o be he mos epesive compoe. Therefore is coribuio owards he cos of he power lie is sigifica. Coducor coss maerial cos ad isallaio coss associaed wih he capial ivesme of a ew over head power lie coribues up o 40% of he oal capial coss of he lie [3]. Cosequely much aeio has o be give o he selecio of a coducor cofiguraio o mee boh prese ad prediced fuure load requiremes. Trasmissio lie compoes are usually eposed o dyamic forces mechaical power ad moios. Ou of all he lie compoes cables are ormally suscepible o forces ha cause boh saic ad dyamic acio due o is fleible srucure. The dyamic acio is mosly caused by wid loadig. 1

9 Probably o oher large srucure so coiuously eposed o he forces of he wid has as much of is mass i such a highly fleible form [1][4]. This makes he lie suscepible o he developme of susaied cyclic coducor moios. The coiuous eposure of coducors o mechaical power from wid ca possibly lead o damage or faigue failure of coducors ad also o oher lie compoes. 1. Moivaio I uiliies all over he world power is ormally rasferred usig overhead rasmissio lies from he geeraio saios which are usually sied i remoe areas ad are coeced o load ceers which are a few kilomeers o hudreds of kilomeers away passig hrough differe errais. This power is rasferred by bare coducors. Bare coducors are o-isulaed ad are made of umbers of alumium sads wih or wihou seel wires a he core. The coducors ca wihsad higher level of curre compared o he isulaed coducors of he same crosssecioal area ad his is why hey are widely used i log-spa rasmissio lie. I rasmissio lies he mos impora compoe used o covey power is he bare coducor ad before i is pu io service he lie desigers have o ascerai is elecrical properies hermal limis mechaical properies ad oher facors which affec he performace of he lie. This process will help he lie desigers o desig a opimal power lie ha will guaraee availabiliy reliabiliy of power safey o he public ad maieace persoel ad will also mee he cos/beefi of cosrucig he lie. As he lies passes hrough various errais hey are usually subjeced o mechaical loadig from wid which is dyamic i aure ad also a fucio of he ype of errai [1]. Loadig causes mechaical oscillaios i he high-volage rasmissio lies ad prologed eposure of coducors o vibraios will eveually resul i faigue freig ad oher failure modes. Over he years umerous research projecs have bee coduced o ry ad udersad he dyamic behaviour of coducors whe subjeced o mechaical loadig from wid. Ivesigaios io he pheomeo of mechaical oscillaio of power lie coducors have bee eesively looked io usig coceps from rigid body aalyical fluid mechaics ad aerodyamics o ry ad predic he coducor dyamic behaviour. Based o fidigs he mechaical vibraios of a coducor whe i receives loadig from wid ehibis a comple dyamic ad also he sysem respose is o-liear. Thus precise modelig of he dyamics

10 ivolved is difficul due o he fac ha he behaviour of he sysem is o-liear. Therefore i lie wih he above here is a eed o coiue o sudy his form of coducor moio i order o be able o adequaely predic he sysem respose o some degree of accuracy especially wih respec o self-dampig whe subjeced o he dyamic forces from wid. The aims ad objecives of his research projec are: To aalyse he model describig he rasverse vibraio of bare coducors To evaluae he self-dampig capabiliy of bare coducors To evaluae he aalyical model usig fiie eleme aalysis To develop a elecrical equivalece for he vibraig coducor To verify eperimeally he aalyical model 1.3 Research Quesio Based o he iferece made i [5] i which i was saed ha he accurae iformaio o how o deermie he power lie self-dampig capabiliy is very impora because i ca be used o assess a mehod of esimaig he maimum ampliude ha occurs o he lie. Coversely i he same repor i was suggesed ha if he value of he coducor self-dampig was small eough o be igored cosequely he aalysis of he priciple modes of he udamped sysem could he also be used for he damped sysems. However he dyamic aalysis of sysems shows ha he dampig force of a sysem may be cosidered small as compared o mass ad siffess bu is ifluece o he sysem s dyamic characerisic is very sigifica. Therefore i is imperaive o kow he accurae amou of coducor self-dampig before igorig i or is coribuio whe carryig ou dyamic aalysis of power lie coducors. I lie wih he above he research quesio ha eeds o be aswered is How ca we effecively deermie bare coducors self-dampig? 1.4 Hypohesis Wid-iduced vibraio due o is caasrophic aure ad egieerig implicaio o power has led o eesive research work over he years. Ever sice mechaical oscillaio of power lie coducors was oiced o rasmissio lies ivesigaios have bee carried ou by may 3

11 researchers. This has led o eesive sudies usig heoreical field ad also from boh oudoor ad idoor vibraio esig laboraory eperimeal sudies o ry ad predic he coducor dyamic behaviour. Based o fidigs emaaig from oher researchers various bodies such as IEEE Gigré ad IEC have come wih sadards i cosrucig power lies. These sadards ormally specify he srigig esio wih respec o he coducor ulimae esile sregh UTS ad also vibraio absorber suiable for cerai climaic codiios ad he aure of he errai. The mai goal of his sudy is o deermie dampig of lie coducors a esios higher ha ha sipulaed as is srig esio ad also ascerai wheher i is desirable o adop a higher coducor esio ha he oe currely beig used by lie desigers. Thus his will help deermie he coducor selfdampig a ha esio ad i respec help o ascerai is ifluece o he dyamic characerisics of he vibraig coducor. Coversely his will help deermie he ype ad amou of vibraio absorbers dampers ha will be eeded o he lies. 1.5 Imporace of This Sudy As highlighed above sadards are se by various bodies o he specific esio as a perceage of is UTS which he lie should be srug. I accordace desigig a overhead lie rasmissio lie egieers ormally srig he lie coducor a 5% of is ulimae esile sregh for alumiium coducor seel reiforced ACSR coducors based o recommedaio i he Gigré repor o Aeolia vibraio [6] ad 0% of is ulimae esile sregh for Aero-Z coducors based o he recommedaio i he IEC sadard [7]. These sadards have bee adoped by lie desigers based o oucomes from years of research ad eperiece wih overhead lies. Based o his i has bee asceraied ha usig hese sadards; he dyamic loadig from he wid ca be curailed hereby esurig a good faigue life for he coducor. I adopig hese sadards i desigig a lie coducor self-dampig capabiliy is usually igored. This is due o he fac ha he self-dampig by a coducor is assumed o be very small ad also he value is o ormally specified by coducor maufacurers. Thus lie desig is ormally doe igorig he coribuio of coducor self-dampig. However if he coducor self-dampig ca be adequaely asceraied ad also ca be deermied above he esio a which he lies are srug by a icrease of 5% above he value recommeded above wih 4

12 respec o is ulimae esile sregh i.e. 30% ad 5% respecively. This will resul i reducig sag wih he followig implicaios: 1 The lie ca be desiged wih a shorer ageial ower while he spa legh is kep he same. The lie ca be desiged wih loger spa legh while he heigh of he ower is sill kep he same. The resul will be a higher cos beefi o he par of power uiliies boh i desigig ad cosrucio of ew lies or upgradig eisig lies. 1.6 Disseraio Oulie The backgroud of his sudy has bee elaboraed o i his chaper. Chaper gives he lieraure review of he mechaical vibraio of sysems wih sigle degree-of-freedom SDOF ad how he aalysis of hese sysems ca be used o describe coducor s mechaical vibraios whe i receives loadig from wid. Chaper 3 describes he aalyical model describig he dyamic characerisics of bare coducors. Chaper 4 will be used for verificaio of he model usig fiie eleme mehod. Also i his chaper a elecrical equivalece was also developed for he coducor. The aim of developig he coducor equivale circui is o provide a aleraive i deermiig he coducor self-dampig by meas of elecrical elemes. Chaper 5 will describe he procedure used o carry ou ess i Vibraio Research ad Tesig Cere VRTC i lie wih IEEE sadard o coducor self-dampig. Also resuls from vibraio ess ad fiie eleme mehod were preseed ad aalyses of he resuls doe. I Chaper 6 cocludig remarks ad fuure scope of he sudy was discussed. 5

13 Displaceme m CHAPTER LITERATURE REVIEW.1 Waves Waves are pheomea ha are commo o mos aspecs of our physical world ad hey are everywhere i aure. Based o he cocep of physics waves are creaed whe a eeral eergy disurbace is imposed o a sysem ad his ses he sysem io back-ad-forh vibraios abou is equilibrium or res posiio. Whe a wave is se up i a sysem a pulse ravels hrough he sysem coiuously ad periodically rasporig eergy away from he poi he impulse was imposed o he sysem. Hece waves are said o be a eergy raspor pheomeo i a medium wihou rasporig he maer. There are basically wo ypes of wave wih regards o eergy rasporaio; elecromageic ad mechaical. A mechaical wave is a wave ha requires a medium o raspor eergy while a elecromageic wave does o require a medium. Waves are usually described by he followig properies: 1 Ampliude A Wavelegh λ 3 Frequecy f 4 Speed or Velociy V 5 Period T The wave properies are illusraed usig he diagram below. Period T Ampliude A Time s Wavelegh Figure.1 Graphical represeaios of wave moio ad is properies 6

14 I he sudy of waves ad vibraio of sysems he vibraio of he sysem is caused by he waves ha ravel hrough i ad also he ampliude of vibraio of he sysem is a fucio of he quaiy of eergy beig raspored hrough i i.e. he greaer he eergy he higher he ampliude of vibraio of he sysem. Therefore he quaiy of eergy ha passes hrough he sysem deermies he level of vibraio. For a sysem a vibraig body wih varyig wave properies as meioed above will o have effecs o he speed. Bu for ay sysem eperiecig chage i esio his chage will affec he speed of he wave. This is because a chage i esio will have effecs o he sysem s maerial properies such as he sysem s siffess ad desiy ad his ulimaely affec he speed a which he wave ravels hrough he sysem. I his sudy he physics of waves is very impora because he vibraio of he sysem i focus is due o he rasporaio of eergy hrough he sysem ad aalysis of he sysem will be doe a differe esios. The varyig of he esio of he sysem eds o have a effec o properies of he wave ha ravels hrough he sysem. As more eergy is added o he sysem due o loadig he more eergeic he vibraio becomes. This ca resul o a pheomeo called resoace which will be discussed laer i his chaper. This pheomeo is very impora because some aalysis ha will be doe i his sudy will be aroud he regio i which i occurs.. Mechaical Vibraio Vibraio is a flucuaig moio brough by flucuaig forces due o waves ravelig hrough a sysem. The cocep of vibraio is a commo pheomeo i mechaical sysems ad i is evidece i mos sysems i our physical world. Mechaical oscillaio of a sysem is he moio abou a equilibrium poi ad his oscillaio may be periodic or o-periodic. The cocep of vibraio i ay sysem may be beeficial bu i mos cases i is a limiig facor i egieerig desig ad careful desig usually miimises uwaed vibraios...1 Liear ad No-liear Sysems Vibraig mechaical sysems ca be aalysed as eiher liear sysems or o-liear sysems. Liear sysems vibrae i a periodic form ad ca be aalysed usig he superposiio priciple. 7

15 Therefore he equaios of moio are such ha a liear combiaio of ipu o he sysem leads o he same liear combiaio of oupus respose i.e. superposiio priciple. Eamples of equaios describig such sysems are:- m c k 0.1 l 0. g No-liear sysems are sysems ha oscillae i a o-periodic maer ad all properies of liear sysems are violaed by o-liear sysems. No-lieariy i a vibraig sysem is due o is mass siffess dampig ad geomery. For ime ivaria sysem o-lieariy ca be of wo ypes: o-liear dampig ad o-liear siffess [8]. Modal aalysis does o apply o oliear sysems because i depeds o super-posiio of soluios. Eamples of equaios describig o-liear sysems are:- 3 m m c 1 k 0 Va der Pol s equaio...4 I worhwhile o oe ha vibraios eperieced i ay real physical sysems are o-liear which herefore implies ha he assumpio of small agles of oscillaio resrics he sysem o liear case as well as simplifies he problem. This cocep will be used i all aalysis i his sudy..3 Vibraio Aalysis I he area of vibraio aalysis udersadig he respose of boh simple ad comple sysems ca be achieved by sudyig ad aalysig he simple mass-sprig damper models. The vibraio of his form of sysem ca be described by sigle-degree-of-freedom ad also he eciaio of he sysem ca be brough abou by a harmoic force f as show i figure.. The formulaio of he equaio of moio for his sysem ad is aalysis he followig will be used: Based o Newo s secod law of moio he force produced is proporioal o he acceleraio of he mass 8

16 The force applied o he mass by he sprig is proporioal o he legh he sprig is sreched. The proporioaliy cosa is he siffess of he sprig i N/m The eergy dissipaed by he sysem is proporioal o is velociy i.e. viscous dampig i Ns/m The respose of he sysem a = 0 is 0 iiial displaceme ad v 0 iiial velociy ad are kow as iiial codiios. k c k c Mass Mass f m a Figure. a Showig he mass-sprig-damper sysem wih oe degree of freedom ad figure. b showig is free body diagram b.3.1 Types of Mechaical Vibraio The sysem of mass-sprig-damper sysem show i figure. ca be used o aalyse he followig differe ypes of mechaical vibraios usually eperieced by vibraig sysems. Free vibraio wihou dampig Free vibraio wih dampig 9

17 Forced vibraio wihou dampig Forced vibraio wih dampig. Free vibraio wihou dampig occurs whe a mechaical sysem is se io vibraio wih a iiial ipu ad allowed o vibrae freely assumig dampig is egligible ad eciaio force is zero i.e. c ad f equal zero. From figure. he equaio describig he free vibraio for he sysem is give as: m k 0 or Where k m ad i is aural frequecy for he sysem The respose of he sysem is give as Asi.6 Where 0 v0 A ad a 1 v 0 0 For free vibraio wih dampig he vibraig sysem has dampig mechaism o dissipae eergy which makes he iiial ampliude decay wih ime. Also usig figure. he equaio for he sysem becomes m c k Where δ c = m c m The respose of he sysem is give as Ae si d..8 10

18 Where v0 0 0d d 1 A d ad a 1 v 0 0 d 0 A sysem is said o eperiece forced vibraio wihou dampig wih he assumpio ha he sysem dampig is egligible. Whe he sysem is se io vibraio by he applicaio of a eeral force he frequecy of he vibraio of he sysem is he same as he frequecy of he drivig force bu he magiude of he vibraio is srogly depede o he dyamic characerisics of he mechaical sysem iself. Usig figure. if he eeral force f which is harmoic i aure is applied he equaio describig he sysem becomes:- m k f si dr F si..9 dr Where dr frequecy of he drivig force F = m f The respose of he sysem is give as F Asi cosdr...10a dr or v F F 0 si 0 cos cos...10b dr dr For forced vibraio wih dampig whe se io vibraio by a harmoic force f he ampliude decays wih ime. The equaio of moio describig he sysem becomes m c k f si dr m F si Whose geeral respose is give as dr 11

19 Ae si X cos...1 d Where 0 X cos A si a 1 v 0 0 d 0 X cos a X cos X si 1 Ad X F.3. Beas ad Resoace For a sysem eperiecig forced vibraio wihou dampig as show i figure. as described by equaio.8. As eplaied by he auhor [9] for such a sysem o maiai a cosaampliude a force f has o be applied ha is harmoic i aure. As he frequecy of he drivig force is beig varied a codiio is reached i which he frequecy of he drivig force approaches he aural frequecy of he sysem ω ad wo very impora pheomea occur. The firs is he beas. Beas are rapid oscillaios wih slowly varyig ampliude ad hey occur whe he frequecies are slighly differe. Applyig he priciple of superposiio he summaio of heir displacemes a each isace wih ime equals o he oal displaceme a ha ime as show i figure.3 below. dr 1

20 Figure.3 Beas [9] The secod is resoace ad i occurs whe he drivig frequecy dr becomes eacly equal o he sysem s aural frequecy dr = ω. Whe a sysem is a resoace he sysem eperieces oscillaio which is a ampliude peak i.e. he ampliude of vibraio becomes uboud as show i figure.4. The frequecy a which his occurs is called he resoace frequecy. Figure.4 Forced respose of a sprig-mass sysem drive harmoically a is aural frequecy [9] 13

21 .4 Dampig Dampig is he erm used o defie he o-coservaive forces used by mechaical sysems o dissipae eergy. I is resposible for he decrease i ampliude wih ime as he sysem udergoes oscillaio. I real-world sysems here is a ihere mechaism i which imposed eeral eergy is removed from he sysem wihou which he sysem will coiue o oscillae wihou comig o res. There are hree mai ypes of dampig prese i ay mechaical sysem: Ieral dampig i which dampig is caused by a microsrucure defecs hermoelasic effecs of dislocaio i meals ec. This form of dampig ca eiher be viscoelasic or hysereic dampig. Srucural dampig where dampig occurs from rubbig fricio or coac bewee differe elemes i a mechaical sysem. Fluid dampig his occurs whe a maerial is immersed i a fluid ad here is relaive moio bewee he maerial ad he fluid ad he maerial is subjeced o drag force resulig i dampig. Usig he mass-sprig-damper sysems wih dampig described i secio.3.1 he soluio o hese equaios depeds o he amou of dampig. Sysems vibraig wih some form of dampig ca be eperiecig oe of he followig ypes of dampig: Criical dampig: This occurs whe he sysem o loger oscillaes bu reurs o is equilibrium posiio wihou oscillaio whe i is displaced ad released. Over dampig: I his codiio he sysem does o oscillae bu reurs o is equilibrium posiio more slowly ha wih criical dampig. Uder dampig: I his codiio he sysem oscillaes wih seadily decreasig ampliude. I his sudy he sysem ha was aalysed is a eample of a sysem ha eperieces uderdampig bu wih very low ihere dampig characerisics ha made i proe o vibraio. I aalysig vibraig mechaical sysems several forms of dampig models were available for modelig a paricular mechaical device or srucure. These are: 14

22 1 Viscous dampig Coulomb dampig 3 Hysereic dampig solid dampig or srucural dampig. I is commo o sudy dampig mechaisms by eamiig he eergy dissipaed per cycle uder a harmoic loadig. Ofe force versus displaceme curve or sress versus srai curves are used o measure he eergy los ad hece deermie a measure of dampig i he sysem. Eergy los ΔE per cycle is defied by Where F d is he dampig force ΔE = F d d Effecs of dampig The vibraio of sysems usually slows dow ad eveually dies ou wih he presece of dampig. I moder egieerig dampig is a desig parameer because he ampliude of vibraio will be deermied by he amou of dampig prese i he sysem. Referece [10] wih relaio o dampig gave he eplaaio of he effec of he oher wo parameers: mass ad siffess. This implies ha o opimise ay sysem for maimum dampig cosideraio should be give o he variaio of all hree parameers ivoved hese are dampig mass ad siffess. For sysems eecuig harmoic eciaio icreasig dampig will affec he respose oly while a icrease i siffess icrease i frequecy will resul i a decrease i ampliude. Also chage i mass of he sysem resuls i chage i frequecy bu lile or o chage i he respose of he sysem..4. Coducor dampig Geerally ay sysem or body ha is subjeced o mechaical vibraio ca damp-ou eergy i he followig ways: Firsly by ieral dampig where he dampig is by ieral fricio a he molecular level due o microsrucures- impuriies grai boudaries ec i he sysem. Secodly is by srucural dampig as a resul of ier-srad fricio rubbig fricio ad coac amog differe 15

23 compoes ad assemblies of he sysem. Thirdly by fluid dampig where he relaive moio bewee he wid ad he body subjecs he body o a drag force hereby reurig he eergy back o he wid. I rasmissio lies whe coducors are eposed o rasverse vibraio by mechaical loadig ha is dyamic ad harmoic i aure coducors ca damp ou his eergy by ieral fricio a he molecular level; by ier-srad fricio wihi he coducor; by rasferece o clamps dampers spacers spacer-dampers ad suspesio assemblies; by rasferece o adjoiig subcoducors i he case of budled coducors; or by reur of eergy o he wid. [1] [11] The ieral eergy losses a microscopic molecular level wihi he core ad idividual srads of he coducor are kow as meallurgical or maerial dampig [1]. Whe a coducor flees he srads of he coducor slip agais each oher; his relaive moio geeraes fricioal forces ha provide dampig [11]. The combiaio of hese eergy dissipaive processes by a coducor is kow as coducor self-dampig..5 Coducor Mechaical Vibraio I a overhead rasmissio lie coducors are used o rasfer power from oe poi o aoher. Whe hese coducors are eposed o aure s dyamic forces hey are se io vibraio. Wid loadig is he mos commo form of loadig ha causes mechaical oscillaios of coducor i high-esio rasmissio lies. This form of dyamic moio icludes hose ypes ha are repeiive or cyclic i moio ad here are hree major ypes of wid-iduced coducor moio ha affec he rasmissio lies. These are Aeolia vibraio Coducor gallopig ad Wake-iduced vibraio. I is impora o oe ha his moio ehibied by a coducor whe i is eposed o wid loadig is a fucio of he followig: a Wid velociy b Lie esio c Diameer of coducor d Temperaure e Coducor self-dampig f Terrai 16

24 .5.1 Srouhal ad Scruo Numbers Srouhal [13] was he firs perso o describe he vore-sheddig pheomeo resulig from wid flowig across a saioary cylidrical srucure. This pheomeo ca be described by a o-dimesioal umber kow as Srouhal umber. This umber is give as: Where f s d V f sd S `...14 V = he frequecy of vore sheddig = he diameer of he cylidrical srucure = velociy of he smooh flow of wid. The Srouhal umber is ake o be o 0. for vibraig coducors The Scruo umber [14] is aoher impora parameer whe cosiderig vore eciaio of wid-iduced vibraios of power lie coducors. This umber is give as: S c m d..15 where m ζ ρ d = is mass of cable per ui legh kg/m = dampig as raio of criical dampig = air desiy kg/m 3 ad = cable diameer m. This relaioship shows ha icreasig he mass desiy ad dampig of he coducor icreases he Scruo umber. Mos ypes of wid-iduced oscillaio o a coducor ed o be miigaed by icreasig he Scruo umber because he ampliude of he coducor oscillaios is iversely proporioal o he Scruo umber S c. Therefore icreasig he mass ad dampig of he coducor icreases he Scruo umber ad herefore reduces oscillaio ampliudes. Thus is value is a measure of he coducor dampig whe subjeced o aerodyamic eciaio vore sheddig a he Srouhal frequecy. 17

25 .5. Classificaio of Coducor Oscillaio As meioed earlier here are hree major ypes of wid-iduced coducor vibraio which are cyclic i aure ha will affec he rasmissio lies coducors: 1 Aeolia vibraio Coducor gallopig 3 Wake-iduced vibraio.5..1 Aeolia vibraio Aeolia vibraio [1][4] is caused by he flow of lamiar smooh sreams of wid over a coducor. Based o he research usig he cocep from fluid mechaics ad aerodyamics i occurs as he wid sream passes over he coducor causig a formaio of aleraig vorices eddies behid. This creaes vore iduced air pressure flucuaios i he dowsream wake side of he coducor which eds o produce moio a righ agle o he direcio of he wid as show i figure.6. Aeolia vibraio is characerised by a frequecy rage which is usually bewee 3 ad 00Hz ad he vibraio ampliude could be oe coducor diameer peak-opeak. I occurs a low wid velociies bewee 1 o 7m/s. Figure.6 Aeolia vibraio vore formaios [1] 18

26 .5.. Coducor gallopig Coducor gallopig [1][4] is a cyclic coducor oscillaio ha is commo o areas ha eperieces sow. I is usually caused by ice deposied o he coducor modifyig is crosssecioal circular shape o a asymmerically-iced coducor surface as show i figure.7 below. As wid blows across he coducor because he asymmerically-iced coducor is aerodyamically usable coducor gallopig does occur. I is a low frequecy from 0.1 o 1Hz high ampliude ± 0.1 o ±1 imes he sag of he spa form of vibraio ad i is usually caused by a moderae seady crosswid acig upo a asymmerically iced coducor surface. Gallopig akes oe of wo basic forms sadig waves ad ravelig waves or a combiaio of hem. Figure.7 Asymmeric iced coducor [1] [4] 19

27 .5..3 Wake-iduced vibraio Wake-iduced vibraio or budle coducor oscillaio [1][4] is associaed wih budle coducors of a rasmissio lie. I is caused by he aerodyamic shieldig of leeward coducors by widward coducors. The vibraio occurs i moderae o srog wids rage of 7 o 18m/s ad i akes place whe he wake from he widward lie iduces lower drag ad creaes lifig forces o he leeward lie. Also because he budle is held ogeher by a spacer a combiaio of moio ca occur due o spacer effec. The wake-iduced vibraio is observed whe he coducors are dry bu i also occurs durig icy ad raiy codiios. The major four budle coducor oscillaios are show i he diagram below. Fig.8 Wake-iduced vibraio [1][4] The able i he e page gives a summary of he compariso of he hree ypes of wid iduced vibraio ormally eperieced i overhead rasmissio lies 0

28 Table.1 Compariso of ypes of cyclic coducor moio [1] [4] Aeolia Vibraio Coducor Gallopig Wake-iduced Oscillaio Types of Overhead Lies Affeced All All Limied o lies wih budled coducors Appro. Frequecy Rage Hz 3 o o o 10 Appro. Rage of Vibraio Ampliudes Peak-o-Peak Epressed i coducor diameers Weaher Codiios Favorig Coducor Moio 0.01 o 1 5 o 300 Rigid-Body Mode: 0.5 o 80 Sub spa Mode: 0.5 o 0 Wid Characer Seady Seady Seady Wid Velociy 1 o7m/s o 15mph 7 o 18m/s 15 o 40mph 4 o l8m/s 10 o 40mph Coducor Surface Bare or uiformly iced i.e. hoarlros Asymmerical ice deposi o coducor Bare dry Desig Codiios Affecig Coducor Moio Lie esio coducor self-dampig use of dampers armor rods Raio of verical aural frequecy o orsio aural frequecy; sag raio ad suppor codiios Subcoducor separaio il of budle subcoducor arrageme sub spa saggerig Damage Appro ime required for severe damage o develop 3 mohs o 0 + years 1 o 48 hours 1 mohs o 8 + years Direc causes of damage Meal faigue due o cyclic bedig High dyamic loads Coducor clashig accelerae ae wear i hardware Lie compoes mos affeced by damage Coducor ad shield wire srads Coducor all hardware isulaors srucures Suspesio hardware spacers dampers coducor srads 1

29 .6 Faigue Failure Based o he early ivesigaio by a Germa egieer amed Augus Wöhler he was able o esablish ha faigue occurs if he aleraig sress was oly slighly less ha he saic sresses which would cause failure of he meal ad oly a few cycles of loadig were required o cause failure [15]. Faigue of coducor is caused by dyamic sresses ha resul from aleraig bedig of he coducors where heir moio is resraied. Faigue is ehaced whe boh sresses ad freig icrease wih he ampliude of bedig i.e. he greaer he ampliude he more quickly faigue occurs. Faigue failure of coducor ad is associaed lie hardware i overhead lie is he mos commo form of damage from wid iduced vibraio especially from Aeolia vibraio. This is because his form of wid iduced vibraio produces several umbers of sadig waves which iduces bedig sresses a pois which moio is resraied ad may millio cycles ca be accumulaed. Hece faigue of coducor srads occurs a pois where moio of a coducor is cosraied agais rasverse vibraio ad is occurrece a hese pois is direcly liked o he rigidiy wih which coducor moio is resraied [1][4]. These pois iclude suppor locaio suspesio clamps lie hardware. To couer faigue ad oher effecs resulig from wid-iduced vibraio durig he desig sage vibraio absorbers dampers are usually icluded. Eample of a vibraio absorber is he Sockbridge damper [4] show below i figure.9 which has bee prove o be very effecive agais Aeolia vibraio.

30 3.1 INTRODUCTION CHAPTER 3 ANALYTICAL MODELLING This chaper preses he mehodology used i he sudy of wid-iduced vibraio eperieced by coducors o overhead rasmissio lies. The cocep ha is addressed i his aspec of he sudy is divided io wo phases: free ad forced vibraio. These iclude he eplaaio of he physical aure of he pheomeo as well as he aalysis of he power lie cable model as a beam he fiie eleme mehod equaio formulaio ad he deermiaio of bedig siffess for he coducor. 3. PHYSICAL ASPECT AND MODELLING CONCEPT Geerally he soluio o ay egieerig problem ormally proceeds hrough four sages: firs a real-world problem is ideified; secod wih proper assumpios he problem is modeled; hird he model is aalysed ad las he resuls are applied o he origial physical problem. This process eables he predicio of he sysem respose hereby helpig i developig a mechaism o fid a soluio o his ideified physical problem. The firs hree sages i erms of he coducor eperiecig rasverse vibraio will be applied i his chaper ad i he e. The las sage will be addressed i chaper five. I he physical world wid-iduced coducor oscillaio ca be caused by wid eciaio ad he loadig which is disribued hroughou he coducor spa. This ca be reproduced i a wid uel eperime whe he coducor is subjeced o wid a varyig codiios. Based o eplaaios give for eperimes coduced i wid uels wih regard o fluid solid ieracio repored i publicaios ad i lieraures by researchers ad which deailed eplaaios ca be foud i refereces [1][][4][13][16][17][18]. I hese research publicaios i was esablished ha he wid ha flows across he coducor i he horizoal plae will cause he coducor o vibrae i verical plae perpedicular o he direcio of he wid. I modelig he above as discussed earlier corary o wha happeed i he physical world ad i wid uel eperimes i a idoor laboraory he vibraio is assumed o be caused by a effecive 3

31 or iegraed wid loadig ha is coceraed a a poi ad his ca produce he same effec as ha of he disribued loadig foud i he real world as illusraed i figure 3.1 show below. I he diagram a sigle spa coducor is eposed o a sream of wid i he horizoal plae. To aalyse his form of vibraio he resula force is assumed o be harmoic i aure hus epressed as a sigle frequecy siusoidal force. This assumpio is made because he ipu loadig from vibraor shaker is se a fied vibraio frequecy ad his is allowed o vibrae or sweep across he coducor wihi a cerai frequecy rage o deermie he various coducor resoace frequecies. Sigle Spa Trasmissio lie Coducor Poi Loadig io he Coducor Wid Loadig Fig 3.1. Shows he coducor wid loadig i he physical world ad he poi loadig cocep used i modelig. 4

32 3.3 Coducor Ipu Power Wid Loadig I he real world he vibraio of a coducor is caused by vore formaio as he wid flows across he coducor ad his effec is kow as Koma effec. For Aeolia vibraio he frequecy of vibraio is calculaed usig srouhal fomula as give by equaio.14. Ivesigaios io his fluid dyamic eciaio force causig his form of oscillaio has bee doe usig wid uel eperimes [17][18]. Fidigs from hese eperimes have helped produce a emperical formulae o calculae he wid ipu force ad his is specific for a paricular coducor. Based o hese fidigs referece [4] provides empirical formulae o calculae he wid ipu power o he coducor. The eiaio of a coducor i a idoor esig laboraory is differe compared o ha doe i he wid urel which eds o produce he same eciaio similar o ha of he real world sceerio disribued loadig. I he idoor esig laboraory a coceraed force F is used o cause he coducor o oscillae ad his will replicae he same effec as he disribued force which a coducor i wid urel eperime or i he physical world would be subjeced o. This poi loadig is simulaed wih a shaker vibraor i he laboraory ad ipuig his force o he coducor as modeled by damper-sprig sysem aached o he coducor illusraed i fugure 3. show below. a b Figure 3. a ad b.the mass-sprig-damper model for Shaker-Coducor fleible lik coecio 5

33 The fleible lik coecio is used i he laboraory o coec he shaker ad he coducor is modeled wih he siffess ad he dampig as show above. Figure 3.a represes he acual physical model i which o all he power is rasferred from he vibraor o he coducor. Due o he egligible eergy ha is los as well as he complicaio caused by he preseces of he dampig cosa wih regard o is phase shif i is he assumed ha o power is los bewee he vibraor ad he coducor. Thus seig he dampig cosa o zero he rasformig he model o a pure sprig couplig show i figure 3.b. This model will be used o calculae he ipu force for a sigle spa coducor or deermie he amou of force impaced by he shaker o he coducor. 3.4 Coceps of Bare Coducor Modelig Coducor vibraios have bee a subjec of iesive sudies for a log ime. Over he years various aalyical models have bee developed by researchers ad used o ry ad predic he mechaical vibraio of rasmissio lie coducors. These developed models are eiher based o modelig he coducor as beam or au srig. Also he beam or srig model is eiher cosidered a coiuous or lump mass. The au srig model of he vibraig coducor eglecs he bedig siffess of he cable ha is kow o have some effec o he dyamics of he coducor. I his sudy he modelig of he rasverse fleural vibraio of a bare coducor is doe usig he cocep of bedig vibraio of a beam. This cocep also cosiders he beam o be a disribued-parameer coiuous or ifiie-dimesioal sysems i.e. mass of he sysem is cosidered o be disribued hroughou he srucure as a series of ifiiely small elemes. Hece whe here is vibraio each of hese ifiie umbers of elemes move relaive o each oher i a coiuous maer [9]. To udersad he cocep of bedig vibraio of a beam see appedi A for he derivaio of simple beam equaio also kow as he Euler-Beroulli equaio. From lieraure mos researchers model he coducor as beam clamped a boh eds i.e. rigidly fied ad permis o moio. I [19] i was poied ou ha his assumpio was oly valid for earh wires. I rasmissio lies he coducors are aached o suspesio isulaors which permi some degree of moio i he logiudial direcio. Hece based o he above iferece 6

34 he rasverse vibraio of he coducor was modeled as a log sleder simply suppored beam isoropic i aure ad subjeced o a coceraed force The Equaio of Moio The equaio of moio of a coducor usig he beam as eplaied above eperiecig rasverse vibraio has bee ivesigaed by may researchers ad auhors. Figure 3.3 shows he saic profile of he beam sag by a give esile force S ha ca be approimaely deermied by he parabolic curve subjeced o wid loadig. From his profile he wid loadig is assumed o be a coceraed force f wih a sigle harmoic frequecy ad his is used as he basis o model he rasverse vibraio of rasmissio lie coducor The formulaio of he equaio of moio is based o he followig assumpios [1]: Uiformiy alog he spa legh ad sleder hi Beam heory Coducor is regarded as a solid cylidrical body composed of liear homogeeous physical properies hroughou is cross-secioal area i.e. esio fleural rigidiy crosssecioal area assumed uiform hroughou he coducor. Such ha he plae of symmery of he beam is also he plae of vibraio so ha roaio ad raslaio are decoupled. Hece he deformaio will be small ha he shear deformaio is much smaller ha he rasverse displaceme ad also he slope of lie of he age o he coducor ha is y / <<1 S θ Graviy L Z Y X S θ F Wid loadig Fig 3.3. Saic profile of he simply suppored beam subjeced o wid dyamic loadig ad he loadig is assumed o be a equivale coceraed force F wih he same resula effec as he wid 7

35 Based o he above assumpios he equaio of moio of a coducor wih aial loadig esioed a boh eds as described i research publicaios i erms of wid-iduced vibraio eperieced by rasmissio lies coducors as a beam documeed i publicaios[5][0][1] [] is give as EI 4 y S 4 y y A f. 3.1 for 0 l > 0 wih he boudary codiios a 0 y 0 0 simply suppored or pied ed = 0 l y l 0 simply suppored or pied ed = l Ad iiial codiios. y 0 y 0 a = 0 y 0 y 0 a = 0 Where E I EI S ρ = youg modulus = area mome of ieria = fleural rigidiy or coducor bedig siffess = he esio aial loadig = he coducor desiy y = rasverse displaceme posiio ime A = he cross-secioal area F = he eeral force Subsiuig he coducor mass per ui legh m A EI 4 y S 4 y m y f 3. Epressig his equaio for he rasverse vibraio i dimesioless form as epressed i referece [1] 8

36 If epressig he followig dimesioless form as Y y D X L f Also epressig he eciaio i Dirac dela fucio he equaio 3. becomes 4 Y Y Y 1 M p. S [ ] 4 p I p F X X X F X X EI. D Where M p 4 L S. D S p ad L Df I p g Where g γ FXτ F τ δx-x is he graviaioal cosa is he coducor weigh per ui legh i.e. mg represes he e rasverse force per ui legh acig o he coducor represes he h coceraed force acig rasversely o he coducor a locaio X represes he Dirac dela fucio 3.4. Soluio o he Equaio of Moio The soluio o equaio 3.1 whose geeral soluio is assumed o be he same as Euler- Beroulli equaio see appedi A. The paricular soluio o his equaio of moio of a coducor modeled as a beam wih aial load S eperiecig rasverse vibraio igorig he eeral force ad dampig ca be obaied as a series of produc of wo fucios. Usig separaio of variables Y X T Where he ormalized fucios X is mode shapes for equaio ad i saisfy he orhogoaliy codiio Hece subsiuig io equaio 3.1 resuls i wo equaios give as follows //// // EI X S X AX T T //// 4 d y // d y d y Where X X T 4 d d d ad is he cosa which equae he variable of ad 9

37 Assumig ha X Ze equaio 3.4 becomes Ze 4 EI S A Ze 0 Therefore EI S A 0 I relaio o he geeral soluio of he Euler-Beroulli equaio he soluio o he above equaio becomes S S 4 EI A EI S S 4EI A EI The values of ad are he soluios for he geeral equaio describig he rasverse vibraio of he coducor ad because a coducor is a eample of disribued-parameer sysems which has ifiie umber of soluios. Thus ad is ideed o be ad respecively S EI S EI m L f EI. 3.8 S S f m L EI EI EI Where f To fid he ifiie aural frequecies of he coducor disribued sysem is by solvig equaio 3.1 assumig ha he mode shape is he same as he pied-pied beam eigefucio mode shape ad o eeral force. Y si cos where l 30

38 31 0 cos si cos si cos si 4 l A l l S l l EI 0 cos si 4 l A S l A EI l The aural frequecies for he coducor is obaied as A EI l A S l 4 S EI L m S L L 1 i rad/s If F S EI L m S L F L 1 1 i Hz Aalyical Modelig of Coducor Self-dampig To model he damped equaio of moio for bare coducor is by iserig dampig models io he parial differeial equaio i.e. equaio 3.1 of he beam pied a boh eds. Referece [3] gave he eplaaio of he various dampig mechaism ha ca be foud i cailever beams wih ip mass a he free ed ad how i ca be used o evaluae he various coribuio each make o he oal dampig of he beam. I a similar maer he dampig models were icorporaed io he beam equaio of moio based o he aalysis of he wo ypes of dampig mechaism foud i a vibraig coducor as eplaied i secio.4. ad also o he cocep proporioal dampig i which eeral ad ieral dampig were disiguished as eplaied by he auhors [0] [1]. I modelig coducor dampig he followig coceps were used: The coducor ier-srad moio ad fluid dampig boh form he eeral dampig of he coducor is proporioal o is velociy ad i is represeed by viscous dampig model.

39 3 The ieral dampig is proporioal o he rae of srai i he coducor. Icorporaig he above io equaio 3.1 he equaio describig he damped model for he coducor becomes f y A y C y I y S y EI Where β ad C are dampig cosa parameers. The above equaio is he equaio of moio for he vibraig coducor wih he presece of aial load esio S wih viscous air dampig eeral ad srai rae of dampig ieral. The srai rae dampig is also called Kelvi-Voig dampig. Thus he above equaio is simply he damped equaio of moio for he coducor Free Vibraio The above equaio for modelig he coducor self-dampig ca be used o aalyse is free vibraio where he forcig fucio becomes zero as give below y A y C y I y S y EI Usig he separaio of variables of equaio 3.4 equaio 3.14 becomes 0.. //// // //// X AT T CX T X I T X S T X EI Where 4 4 //// d y d X // d y d X d y d T d dy T Usig he eigefucio of l X si

40 33 0 si si si si si 4 4 T l A T l C T l l I T l l S T l l EI 0 si 4 4 AT CT T l I T l S T l EI l 0 4 T l EI l S T C l I AT T l A EI l A S T A C l A I T Comparig equaio 3.15 wih equaio.7 4 l A EI l A S A C l A I 4 The soluio o he emporal equaio 3.15 becomes A e T d si 1 Or B B e T d d cos si Where 1 d The respose will be 1 1 si si d l A e y Or 1 1 si cos si d d l B B e y. 3.17

41 Forced Vibraio Whe a rasmissio lie coducor is eposed o loadig o he field from wid he acual sysem represes a disribued loadig o he eire spa of he coducor. As meioed before his is simulaed as poi loadig where he force is assumed o cocerae a a poi. Evaluaio of he acual respose of he coducor o his specific eciaio ca be achieved by solvig he equaio of moio for he damped case i.e. equaio 3.13 wih he eciaio fucio prese. This will help deermie he dyamic sress for a rage of ipus for eample harmoic moio. Solvig equaio 3.13 wih he drivig force usig he separaio of variable of equaios f y A y C y I y S y EI F AT CT T l I T l S T l EI l dr si si 4 4 F T l A EI l A S T A C l A I T dr si Comparig equaio 3.18 wih equaio.11 Hece cos si X Ae T d Where The soluio becomes cos sisi si X Ae l y 4 l A EI l A S A C l A I 4

42 3.6 Bedig Siffess EI Value Mos researchers i his area of wid-iduced vibraio ed o adop a cosa bedig siffess value for vibraig ACSR coducor based o he recommedaio by Cigrè Sudy Commiee [6]. This value is chose such ha he effecive bedig siffess EI eff is a cosa value bewee he miimum ad maimum values of bedig siffess. Bu i referece [4] he auhor preseed he cocep of how he EI value varies wih he legh alog he coducor ad o his basis he EI value of he coducor he depeds o coducor bedig curvaure. Also from his fidigs he EI value varies o-liearly wih wire helical geomery ierlayer fricio ad slip durig bedig. Due o he fac ha he liear cocep is used i his sudy he cosa bedig siffess value was used for he ACSR coducor. The equaios o calculae for boh he maimum ad miimum bedig siffess for ACSR coducors ca be foud i refereces [4] [6] [4] The miimum value EI mi is obaied by cosiderig he coducor as a budle of idividual wires free o move relaive o each oher The calculaio of EI mi is give as: 4 4 ds da EImi ses aea Wherei: s ad = umber of srads of seel ad alumiium respecively a Es ad E = modulus of elasiciy of seel ad alumiium a ds ad d = diameer of seel ad alumiium srads a The maimum value EI ma is obaied by cosiderig he coducor as a budle of idividual wires uable o move relaive o each oher due o coac pressure I he compuaio of EI ma he displaceme of each srad from he coducor ais mus be cosidered accordig o he formula: I = I c + AD

43 Where I I c A D = mome of ieria abou ew ais = mome of ieria abou origial ais = area = disace bewee origial ad ew aes For he srads wihi a give lay of he coducor he disace D may be compued as he sie of he agle as show i he diagram. The mome of ieria for each srad i a give lay becomes: d d I d d I R si R si I ma d R d Hece he calculaio of EI ma is give as: EI ma =

44 The firs coducor used for aalysis i his sudy is he ACSR coducor wih code ame Ter which cosiss of he seel wires a he core ad alumium wires a he ouer layers. The calculaio of his maimum bedig siffess is give i appedi B as well as is physical properies as obaied from Aberdare Power cables Caalogue [5]. For boh he aalyical aalysis ad he fiie eleme simulaio of he rasverse vibraio he value of he cosa bedig siffess ha was used is ha for calculaed maimum bedig siffess value. This value is chose i lie wih he recommeded value for bedig siffess suggesed i [6] for a vibraig ACSR coducor. The secod coducor used for his sudy is he Aero-Z coducor ad physical properies for he coducor were obaied from referee [7] ad he deails are give i able-b- i appedi B. This coducor cosis of wo pars; he roud wires a he core ad he z-shaped wires ha form he ouer layers. For his coducor because he eac value for he Youg s modulus ca be foud i [7] he value for he bedig siffess ca easily be calculaed. To obai he value of he bedig siffess he followig process was used wih regards o he mome of ieria. For he roud wires ha cosiue he cere wire ad he e wo ier layers equaio 3. was used o calculae for he mome of ieria while for he regio of he z-shaped wires he assumpio was made i which he wo z-shaped wire layers were combied ad reaed as a hollow circular solid. Combiig he above resuled o he equaio give as EI = EI + EI zs = EI +I zs 3.4 Where I = I zs The calculaed value for he bedig siffess of he Aero-z coducor is give i appedi B. 37

45 CHAPTER 4 MODEL VERIFICATION: FINITE ELEMENT ANALYSIS AND ELECTRICAL EQUIVALENCE 4.1 Discree Modelig As idicaed i chaper 3 i which he coducor rasverse vibraio was modeled as a bedig vibraio of a beam ad also cosidered o be a disribued parameer model he same cocep was used for he fiie eleme mehod FEM bu he beam was cosidered a lump mass model. Carryig ou he fiie eleme aalysis FEA ivolves coverig his disribued model o is equivale lump mass model. Hece by discree modelig of coducor vibraio meas coverig disribued mass o is equivale lump mass by a mehod kow as discreisaio. The fiie-eleme discreise model of he pied-pied simply suppored beam which is made up of is fiie eleme is show i figure 4.1 below. This model was used for he fiie eleme aalysis of coducor rasverse vibraio. The modelig of he dyamic behaviour of he coducor will ivolve he use of boh he saic properies such as siffess wih he dyamic properies such as mass dampig ad dyamic loadig o ry ad predic is respose Discreisaio of he Domai io Fiie Elemes FEA begis wih formulaio of domai for he discreise fiie eleme model; he fiie eleme formulaio cosideraio assumes a domai limied o geomery ad ime. The domai of he coducor is give as 0 l which is divided io umber of fiie elemes mesh of equal space as show below i figure 4.1 ad he coducor masses are lump a he odes u1 y 1 u3 y 5 u3 y 5 u5 y 7 u y 5 u y 1 3 u y 1 3 u y 1 ELEMENT 1 ELEMENT ELEMENT-1 ELEMENT y4 y 4 3 y5 1 y y 4 y 1 y 1 y Figure 4.1 The discreise model of beam 38

46 This model show i he previous page will be used for he fiie eleme equaio formulaio ad also i he aalysis of he sysem equaio. 4. Fiie Eleme Aalysis The cocep of fiie eleme mehod FEM has provided opporuiies for he developme of procedures for he evaluaio of boh saic ad dyamic sysems problems. This mehod is a compuaioal echique ha ca be employed o evaluae boh he saic ad he dyamic resposes of sysems. Alhough a coducor is a coiuous sysem o evaluae he aalyical model describig is rasverse vibraio usig he FEM mehod ivolves discreisig he sysem io fiie elemes i order o obai he equaio for each. Assemblig hese fiie eleme equaios icludig he boudary codiios was he used o formulae he equaio of moio for he sysem. This resula equaio is obaied as he equaio of ierial siffess ad applied force i mari ad vecor forms. To obai he dampig force i vecor form for mahemaical coveiece he mechaisms of he dampig models are icluded i he global fiie eleme equaio obaied by assembly of hese equaios for he fiie elemes. This was used o evaluae he oal dampig of he sysem or he sysem s self-dampig. Usig coceps from sysem dyamics ad vibraio modal aalysis he mass siffess ad dampig marices for he sysem will he be used o solve for he required sysem parameers such as mode shapes modal frequecies ad dampig. Hece usig hese parameers for he vibraig coducor is dyamic respose ca he be deermied ad aalysed. The soluio o he resula or he global equaio of he sysem ca be accomplished by usig he umerical iegraio mehod such as hose developed i refereces [8] [9] [30] o direcly solve he equaio. The umerical mehod ha was used for he FEM is he Galerki s mehod. Usig coceps from aalyical mechaics mari ad vecor ad vibraio hese sysem parameers obaied was he used o carry ou aalysis o he respose of he sysem wihi a rage i lie wih he aalyical model describig he same sysem Semi-discree FEM Some physical pheomea ca be described by differeial equaio ha relaes cerai quaiies o heir derivaives wih respec o ime ad space variables. Coducor rasverse vibraio is a eample of such pracical problems i which he posiio ad ime dimesios have o be 39

47 cosidered. The mehod of fiie eleme approimaio of his ype of ime depede equaio ime dimesio is he semi-discree mehod. I his mehod he ime dimesio is approimaed by fiie differece ad for accuracy wih respec o ime a higher-order approimaio is used i.e. cubic polyomial. To his regard he fiie eleme ierpolaio fucios as ime depede such ha u is approimaed by u y c N 3 c i i 1 c c3 c4 i1 u H u 1 u1 H u H 3 u3 H Where H 1 3 l l l H 3 l H 3 3 l l 3 3 H 3 4 l l 4.3 Fiie Eleme Formulaio The developme of he compuaioal echique used o aalyse he rasverse vibraio of power coducors sars wih he fiie eleme formulaio. This ivolves he formaio of he equaio for a fiie eleme from he discreise model of he coducor. The basic idea of he fiie eleme formulaio is o liearise he weak form of he equaio of he problem ad solve his equaio for he fiie elemes discreised domai. The weak form or weak formulaio of he problem is usually derived from he priciple of virual work. Therefore for he coducor he fiie eleme formulaio will be o rasform he parial differeial equaio of moio io is variaioal form ad he deermie he approimae soluio usig variaioal mehod Weak Formulaio The weak formulaio for he power lie cables is obaied by rasformig he equaio of moio describig he coducor rasverse vibraio io is iegral form usig he es fucio of equaio 4.1 ad equaio 4.. Thus usig he Galerki s mehod mehod of weighed 40

48 41 residual or variaioal mehod [31] [3] [33] he fiie eleme formulaio for he sysem ca he be developed i iegral form. The by applyig he Galerki s variaioal priciple o his ime depede problem ad cosiderig he boudaries codiios he fiie eleme model equaio was obaied. The weak formulaio for he equaio of moio was formulaed usig cubic displaceme fields wih respec o ime because of he degree of freedom of he model. The equaio for he rasverse vibraio as give by equaio 3.1: Where F f d si The homogeous par of he equaio is give as y A y S y EI Le i e X y 1 i The homogeous par is rasformed io X A d X d S d X d EI 4.3 Usig he same boudary ad iiial codiios used for he aalyical aalysis as give below. Boudary codiios: y EI y pied ed a = 0 0 l y EI l y pied ed a = l Iiial codiios: 0 0 y y a = 0 0 y 0 a = f y A y S y EI

49 The weak formulaio for he semi-discree fiie eleme mehod ca be obaied as follows. Firsly by muliplyig equaio 3.1 by he fiie eleme ierpolaio fucios defied by equaio 4.1 o obai l 0 4 y y y EI u S u A u 4 fu d The weak formulaio is he obaied by carryig ou he iegraio by pars wice o equaio 4.4 ad akig io accou he fiie-eleme discreise model show i figure 4.1 which defie he umber of fiie eleme i he sysem domai. Hece he resula equaio is obaied as EI y u y u y. d S. d A ud fud Su y 3 y 3 u u y. l 0 0 Where is he eleme domai 4.5 From he weak formulaio above he equaios for he fiie eleme i erms of he siffess mass marices ad force vecor are give as: y u y u K e EI. d S. d A. 4.6a y ud. 4.6b M e e F f H T. 4.6c From equaio 4. give ha [ H] [ H1 H H 3 H 4 ] The A H 1 H H 3 H 4 B N N N N

50 The equaio 4.6 becomes e T T K EI B B S A Ad.. 4.7a e y T M A ud A H H y e F f H T b.. 4.7c 4.4 The Sysem Mari The sysem or he global mari is usually obaied by he assembly of fiie eleme equaios defie by equaios 4.7 a b ad c. I assemblig all of he fiie elemes equaios requires he saisfacio of he boudary codiios ad from he diagram of he discreise domai figure 4.1 o saisfy he boudaries codiios of he simply suppored beam i.e. y y y y 0 Thus global mari will be i he form:- 1 1 ad [K] represe he sysem or srucure siffess mari {y} is he sysem displaceme vecor [F ] represes he sysem force vecor Numerical Compuaio for Naural frequecies Fidig he aural frequecies for he sysem is by makig force vecor i he above udamped equaio equal zero. Equaio 4.8 becomes y y e j If

51 44 The j e y y 0 j j e y K e y M 0 M K y e j This resul o eigevalues problem as give 0 M K M K Where ω = λ ω = aural frequecies λ = eigevalues Numerical Compuaio for Dampig The formulaio of he dampig mari for he coducor vibraio was also based o he cocep eplaied i secio.4. which was also used i he formulaio of he coducor damped equaio i secio 3.5. Hece usig equaio 3.13 ad also applyig he Galerki s mehod he weak form of he equaio becomes fud ud y A u y C u y I d u y S d u y EI 4.11 To obai he dampig mari ha coforms o he liear cocep used for his sudy he oliear eleme is removed by simply equaig he ime parial differeiaig variable o equal oe. This he resuls i dampig equaio for he fiie eleme as ud y C d u y I D e..

52 e T T D. I B B C H H y.4.1 Subsiuig equaio 4 a ad b io equaio 4.1 he dampig mari equaio becomes e y u C e e. S y u C e D. I. d M K. M A E E A e e. S T C e D K A A. M E E A Icorporaig he above equaio io equaio 4.8 he damped equaio for he sysem becomes Where [D] is he dampig mari Evaluaio of Dampig The evaluaio of he amou of dampig from he sysem was doe by he compariso of he dampig mari of equaio 4.13 o he proporioal dampig developed by Rayleigh. The proporioal dampig equaio is give as D = M + * K D = I * + * ф ad D = U T DU Where ad * = real scalar cosas I * = ideiy mari U = he orhoormal mari of eigevecor ф = he diagoal mari of eigevalues 45

53 The value of he dampig raio ca be evaluaed usig he proporioal dampig which is give as * δ = The values of ad * are usually evaluaed usig he values of aural frequecies obaied from eperimeal resuls. I comparig equaio 4.15 wih equaio 4.13 = ad * = E I chaper 3 i was saed ha coducor self-dampig is a coribuio of he ieral dampig modeled by Kelvi-Voig dampig model ad eeral dampig due o fluid o drag ad iersrad moio modeled by viscous dampig. Based o he classical work doe by H.H Cudey ad D.J. Ima [34] i which he values for each dampig model ad a combiaio of he boh models were evaluaed for a quasi-isoropic pulruded cailever beam. For he firs case is he ieral dampig or he srai-rae dampig i his case dampig is proporioal o he mass mari ad from equaio 4.15 * equal o zero. Thus he perceage of he criical dampig is iversely proporioal o he aural frequecy of each mode. This will give decreasig dampig as heir frequecies icrease. δ = * = Where * = 0 The secod case is for he dampig from fluid drag ad ier-srad moio; he dampig is proporioal o he siffess mari ad from equaio 4.15 equal o zero. Recogizig ha he higher modes of vibraio damp ou quickly his form of dampig is proporioal o frequecies i ormal modes. Hece δ = * = * Where = 0 46

54 The hird case is he combiaio of boh viscous ad srai-rae dampig which were evaluaed by equaio 4.15 o deermie he oal dampig from he sysem. The evaluaio of he dampig cosas ad * was doe by he leas squares mehod also kow as he pseudo-iverse rouie. To deermie he esimaed value for parameer whe * = 0 is by fidig he leas-square soluio o Also he value for * whe = 0 is obaied by *. 4.1 For he values of boh ad * is evaluaed by *

55 4.5 Elecrical Equivalece: Elecrical Equivale Circui Alhough he rasverse vibraio of coducor is a eample of a mechaical sysem ha is dyamic i aure he udersadig ad aalysis of is dyamics ca also be achieved by developig ad aalysig is elecrical equivalece. The elecrical equivalece is developed usig he mechaical-elecrical aalogy from which a correspodig equivale circui of he mechaical vibraio of he coducor is obaied. The elecrical equivalece has he advaage ha i ca be cosruced easily from which he resuls ca be coveiely obaied ad aalysed o predic he sysem s dyamic respose. This is based o he fac ha much work has bee doe wih respec o dyamics of elecrical circui especially i he area of resoace. Thus resuls obaied from he equivale circui aalysis ca be used o deermie he sysem parameers like aural frequecies mode shapes ad dampig of mechaical sysems. Also he mechaical-elecrical aalogy ca help i furher udersadig he comple aure of he coducor s dyamic characerisics. The coducor is a coiuous model bu is elecrical equivalece is developed from is lumpmass or discree model similar o ha of FEM. The discree model of mass-sprig damper sysem of he equivale coiuous model of he pied-pied beam is show i figure 4. a ad b below. I his model he firs few modes of vibraio are used o model he sysem respose as show i figure 4.b wih he resula force from he wid loadig perpedicular o he beam ha produces he verical oscillaio. To develop he elecrical equivale hree degrees of freedom of he lump mass was used o describe he sysem respose ad he geomery of is domai of he lumped mass model is used o derive he equaio of moio for he sysem. Wid loadig resula force i he verical direcio 0 L Beam pied suppored a boh eds a 48

56 b Fig 4. a Disribued model of a simply suppored beam b is equivale lump-mass model I lie wih he cocep meioed i chaper 3 wih regards o figure 4.b because i is assumed ha he disribued loadig o he coducor will have a resula effec a a poi F 1 hus F ad F 3 are equal o zero. Employig Newo s laws of moio he equaio for he discree sysem will be i he form a b c To obai he differeial equaio i elecrical form from equaio of moio for he mechaical sysem is doe by usig he elecrical aalogies for mechaical sysems lised i able 4.1. There are geerally wo ypes of mechaical- elecrical aalogies [35 36]: 1. The volage force or mass-iducace aalogy ad. The curre force or mass capaciace aalogy I geeral he followig rule is used for developig a elecrical equivale circui for mechaical sysems. If he forces ac i series i he mechaical sysem he elecric elemes 49

57 represeig hese forces are pu i parallel. Forces i parallel are represeed by elemes i he series i he elecric circuis [35]. The able 4.1 below shows he aalogies ha eis bewee mechaical sysems ad elecrical sysems i erms of volage-force aalogy ad curre-force aalogy. Table 4.1 Mechaical-elecrical aalogies [35] Mechaical Sysems Elecrical sysems Volage-force Aalogy Curre-force Aalogy D'Alember's priciple Kirchhoff s volage law Kirchhoff s curre law Degree of freedom Loop Node Force applied Swich closed Swich closed Force F N Volage v vol Curre I ampere Mass mkg Iducace L hery Capaciace C farad Displaceme m Charge q coulomb Φ = vd Velociy m/s Loop curre I ampere Node volage v vol Dampig c Ns/m Resisace Rohms Coducace 1/R mho Sprig k N/m 1/Capaciace 1/C 1/farad 1/Iducace 1/L 1/hery Couplig eleme Eleme commo o wo loop Eleme bewee odes From able 4.1 usig he volage-force or mass-iducace aalogy ad comparig his wih he mechaical equaio he elecrical equivale differeial equaios will be i he form give below a 1c3i3d=0. 4.4b c 50

58 Based o Kirchhoff s volage law ad usig elecrical differeial equaio give above he resula elecrical equivale circui for he vibraig coducor was cosruced as show i fig 4.5 below. L 1 R1 C L L 1 3 R R3 R4 V1 i 1 i i 3 C C3 C4 Fig 4.5. Equivale elecrical circui for a vibraig coducor 4.6 The Ipu-Oupu Model The ipu-oupu model for he circui above is derived by usig he ses of elecrical differeial equaios used o form he sysem elecrical circui i.e. equaios 4.9 a b ad c. The model is derived by applyig Kirchhoff s volage law o hese ses of differeial equaios. The dampig was evaluaed from he sysem s equivale circui based o he assumpio ha he oal dampig eperiece by he sysem as a resul of he ipu volage V 1 is he oupu volage across resisor R 4. Takig he Laplace rasform of he above equaio =0 51

59 Rearragig. 4.5a b c Puig i mari form i becomes = Where a 11 = a 1 = a 1 = a = a 3 = a 3 = a 33 = If D = 5

60 D 1 = D = D 3 = Usig Creamer rule The oupu volage across R 4 is give as = R 4 I 3 s Bu I 3 s =

61 Hece he rasfer fucio for he elecrical equivale circui will be give below as = By defiig he coducor s equivale circui resisace capaciace ad iducace i erms coducor characerisics Where K = siffess of he coducor M = he mass of he coducor c = he dampig cosa = elecromechaical facor m = The rasfer fucio developed above was implemeed i Malab sofware wih he calculaed circui eleme equivales as defied above ad he resul of he simulaio is preseed i Chaper 5 54

62 CHAPTER 5 TESTING AND RESULTS 5.1 Vibraio Research ad Tesig Cere VRTC I verifyig he aalyical model aalysed i chaper 3 eperimeal sudies vibraio ess were coduced. These ess were coduced a he Vibraio Research ad Tesig Cere VRTC siuaed a he Uiversiy of KwaZulu-Naal Wesville Campus. The VRTC was se-up i 004 i parership bewee Eskom ad he Uiversiy of KwaZulu-Naal. The cere which is a idoor esig faciliy is aimed a esig ad carryig research o power lie coducors wih regard o mechaical vibraio. The VRTC laboraory faciliies used for esig were desiged developed ad cosruced i lie wih he guidelie provided i IEEE sadard [37]. The laboraory cosiss of: A esig faciliy ad a uel wih emperaure corol show i figure 5.1 A cosa esio loadig device A spa 85m sigle coducor show i figure 5.1 A shaker elecro-dyamic i operaio which is used o provide he ipu power o he coducor as show i figure 5.3 A corol sysem which is used o aalyse he oupu from he es faciliy Because he VRTC is a idoor esig faciliy ess coduced a he cere were used o simulae he wid-iduced vibraio o coducors ha occurs i he real world. This is he used as he bases of compariso ad predics wha acually occurs i real rasmissio lies. The ess ha were coduced a he cere for his sudy were used maily o deermie he self dampig capabiliy of coducors accordig o he procedures described i IEEE sadard o he Measuremes of Coducor Self-dampig [38]. Also based o he eperimeal daa was he evaluaio of dyamic siffess of he coducor. 55

63 5. Mehods of Tesig Figure 5.1 Faciliy uel ad sigle spa coducor a he VRTC A he VRTC he eperimes ha were carried ou are o sigle spa coducors as show above. These were used o aalyse ad udersad he coducor dyamic characerisics. For his purpose wo ypes of esig were doe wih he aim of verifyig he aalyical model. These are: free ad forced vibraios Free vibraio esig The free vibraio of a coducor is he aural respose of he coducor o some form of impac or displaceme ad he ampliude of vibraio decay wih ime due o dampig. For his es he impac o he coducor was doe usig he impac hammer show i he figure 5. below 56

64 Fig 5. Impac hammer To evaluae he amou of dampig from he coducor eperiecig his form of vibraio he leas square mehod [39] was used ad he descripio of how his mehod was developed from log decreme echiques is give i appedi C 5.. Forced vibraio esig The forced vibraio of he coducor was doe o obai is respose whe he coducor is subjeced o a repeiive forcig fucio. This caused he coducor o vibrae a he frequecy of he eciaio. For forced vibraio es i is assumed ha he effecive loadig ipu ha causes he coducor oscillaio is coceraed a oe poi o he coducor. The eciaio by his coceraed force is duplicaed by a force a a sigle poi poi loadig ad his source of loadig was simulaed by he aerodyamic shaker. For his mehod of esig a sweep es resoace frequecies search was doe for frequecy rage bewee 6Hz ad 50Hz for he Ter coducor ad 5Hz ad 50Hz for he Aero-Z coducor. The coducor dampig was deermied for each mode usig he badwidh mehod [40] also he descripio of his mehod is give i appedi C Mehods of forced vibraio esig Two mehods were employed for forced vibraio esig a he VRTC i order o evaluae he coducor respose ad are eplaied i IEEE sadard o power dissipaio from coducors wih respec o Aeolia vibraio [37]. The wo mehods of forced vibraio esig ha were eplaied are he power mehod ad he sadig wave mehod. 57

65 The power mehod i which he coducor is forced io resoace by a elecro-dyamic shaker ad he power ipu io he sysem was deermied direcly from he produc of he eciaio force ad he resulig velociy a he poi of applicaio of he load. This represes he power dissipaed by he coducor provided he wo quaiies are siusoidal ad are i phase wih each oher. I assumes also ha he losses a he ermiaios are small compared wih he dissipaio wihi he coducor. The power mehod also permis he mechaical resisace per ui legh of he coducor o be deermied direcly from he raio of he force o he velociy. The sadig wave mehod i which he power rasfer P 1 from he vibraio geeraor owards he eds of he spa a ay paricular ode i.e. ode 1 is derived from he iverse or he reciprocal of he sadig wave raio ha is he raio of he odal ad aiodal ampliudes ad is give as follows V a1 P 1 = Sm Y Where Tm = he wave or characerisic impedace A very high frequecies his may be Y a 1 V S m modified due o he effec of he siffess of he coducor. = sadig wave raio = sigle ampliude velociy a aiode = coducor esio = coducor mass per ui legh The he power dissipaed bewee wo odes 1 ad is simply P = P 1 P 5... Shaker coducor coecio The shaker-coducor coecio for he forced vibraio ess was by he fleible lik coecio as show figure 5.3. This arrageme was used because i has he advaage of separaig he coducor resoace from ha of he shaker hereby decouplig hem from each oher. This se-up helps i seig he coducor aloe io resoace for he purpose of 58

66 deermiig is resoace frequecies coversely helpig i asceraiig he amou of dampig oly from he vibraig coducor. Figure 5.3 Shaker-Coducor fleible lik coecio Noe: 1. Before carryig ou ay es o he coducor he coducor was iiially srug o a esio 40% of he UTS for ACSR ad 30% of he UTS for he Aero-Z. The he coducors were allowed o rela for 3 days before he es was doe.. For boh forms of esig: he free ad he forced vibraio he poi of impac ad loadig was a 1.m from he load cell ed for boh coducors. 3. Two acceleromeers where used for his sudy ad hey were placed a half 1/ ad a oe-eigh 1/8 of he spa legh from he load cell side of he coducor. 4. Also for boh forms of esig ess were doe for esios 19.96KN 4.74KN ad 9.64KN which are 0% 5% ad 30% of he UTS respecively for he case ACSR. While for Aero-Z he ess were doe a.5kn 30.0KN ad 37.53KN which are 15% 0% ad 5% of he UTS respecively. 5. For boh he free ad he forced vibraio es he PUMA aalyser [41] was used. 59

67 5.3 Resuls For he eperieial sudy coduced o he idoor sigle spa coducor wo forms of esig were carried ou: free ad forced vibraios accordig o he eplaaio give i secio 5.. These were doe for wo coducors: he ACSR-Ter ad he Aero-Z coducors. The resuls of he laboraory eperimes for boh esig mehods ad for he wo coducors are give below. Ne he resuls also preseed are ha from he fiie eleme simulaio usig ABAQUS which was also doe a he hree differe esios wih he same values used i he eperimeal sudies for boh coducors. Fially resul for he free respose was obaied from he simulaio for he rasfer fucio developed from he elecrical equivale circui for he vibraig coducor Eperimeal Resuls for Free Vibraio: Ter Coducor For he eperimeal sudies doe o Ter coducor he graphs below are he free vibraio resposes for Ter coducor a esios of 0% UTS-19.64KN 5% UTS-4.64KN ad 30% UTS-9.71KN. The wo ables accompayig each graph are used o evaluae he coducor dampig usig he leas square mehod. The firs able is used o evaluae dampig for he firs decay regio ad secod able for he secod decay regio for each graph ad hey are labeled firs decay ad secod decay respecively. Also for each esio used wo graphs are preseed o evaluae he coducor dampig labeled graph 1 ad graph respecively. 60

68 Fig 5.4 Free vibraio for Ter coducor a 0% UTS-19.64KN graph 1 Table 5.1 Usig he leas square mehod o calculae dampig for 0% UTS firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5. Usig he leas square mehod o calculae dampig for 0% UTS Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The Log-deg = ad zea =

69 Fig 5.5 Free vibraio for Ter coducor a 0% UTS-19.64KN graph Table 5.3 Usig he leas square mehod o calculae dampig for 0% UTS-Ter firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.4 Usig he leas square mehod o calculae dampig for 0% UTS-Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

70 Figure 5.6 Free vibraio for Ter coducor a 5% UTS-4.47KN graph 1 Table 5.5 Usig he leas square mehod o calculae dampig for 5 %UTS-Ter firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = 0.03 Table 5.6 Usig he leas square mehod o calculae dampig for 5% UTS-Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

71 Figure 5.7 Free vibraio for Ter coducor a 5% UTS-4.47KN graph Table 5.7 Usig he leas square mehod o calculae dampig for 5% UTS-Ter firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = 0.0 Table 5.8 Usig he leas square mehod o calculae dampig fo 5% UTS-Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

72 Figure 5.8 Free vibraio for Ter coducor a 30% UTS -9.71KN graph 1 Table 5.9 Usig he leas square mehod o calculae dampig for 30% UTS-Ter firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.10 Usig he leas square mehod o calculae dampig for 30% UTS-Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

73 Figure 5.9 Free vibraio for Ter coducor a 30% UTS-9.71KN graph Table5.11 Usig he leas square mehod o calculae dampig for 30% UTS-Ter firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.1 Usig he leas square mehod o calculae dampig for 30% UTS-Ter secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

74 5.3. Eperimeal Resuls for Forced Vibraio Sweep Tes: Ter Coducor To deermie he dyamic respose for he coducor a sweep es was coduced for he Ter coducor. The graphs show below are he forced vibraio sweep es for Ter coducor a esios 0% UTS-19.64KN 5%UTS-4.64KN ad 30%UTS-9.71KN. The able accompayig each graph is used o record he firs e resoace frequecies ad also he evaluaio of he coducor dampig a each mode usig he badwidh mehod. Figure 5.10 Forced vibraio for Ter coducor a 0% UTS-19.64KN Table 5.13 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod Resoace frequecy R f Acceleraio ag a/ Half- Power F 1 Half- Power F Badwidh f qualiy facor Q dampig facor δ

75 Figure 5.11 Forced vibraio for Ter coducor a 5%UTS-4.47KN Table 5.14 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod Resoace frequecy R f Acceleraio ag a/ Half- Power F 1 Half- Power F Badwidh f qualiy facor Q dampig facor δ

76 Figure 5.1 Forced vibraio for Ter coducor a 30% UTS-9.71KN Table 5.15 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod Resoace frequecy R f Acceleraio ag a/ Half- Power F 1 Half- Power F Badwidh f qualiy facor Q dampig facor δ

77 5.3.3 Eperimeal Resuls for Free Vibraio: Aero-Z Coducor Similar o Ter coducor he same free vibraio es was also doe for Aero-Z coducor. The resuls are preseed i a forma similar o ha of he Ter coducor. Figure 5.13 Free vibraio for Aero-Z coducor a 15%UTS-.5KN graph 1 Table 5.16 Usig he leas square mehod o calculae dampig for 15% UTS-Aero-Z firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = -0.6 ad zea = Table 5.17 Usig he leas square mehod o calculae dampig for 15% UTS-Aero-Z secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

78 Figure 5.14 Free vibraio for Aero-Z coducor a 15% UTS-.5KN graph Table 5.18 Usig he leas square mehod o calculae dampig for 15% UTS-Aero-Z firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.19 Usig he leas square mehod o calculae dampig for 15% UTS-Aero-Z secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

79 Figure 5.15 Free vibraio for Aero-Z coducor a 0% UTS-30.0KN graph 1 Table 5.0 Usig he leas square mehod o calculae dampig for 0% UTS-Aero-Z firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = 0.03 Table 5.1 Usig he leas square mehod o calculae dampig for 0% UTS-Aero-Z secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

80 Figure 5.16 Free vibraio for Aero-Z coducor a 0% UTS-30.0KN graph Table 5. Usig he leas square mehod o calculae dampig for 0% UTS-Aero-Z firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.3 Usig he leas square mehod o calculae dampig for 0% UTS-Aero-Zsecod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

81 Fig 5.17 Free vibraio for Aero-Z coducor a 5% UTS-37.53KN graph 1 Table 5.4 Usig he leas square mehod o calculae dampig for 5% UTS-Aero-Z firs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.5 Usig he leas square mehod o calculae dampig for 5% UTS-Aero-Z secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

82 Fig 5.18 Free vibraio for Aero-Z coducor a 5% UTS-37.53KN graph Table 5.6 Usig he leas square mehod o calculae dampig for 5% UTS-Aero-Zfirs decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea = Table 5.7 Usig he leas square mehod o calculae dampig for 5% UTS-Aero-Z secod decay Ieger j Ampliudemm Y LNY j-1 LNY j-1 j Sum The log-deg = ad zea =

83 5.3.4 Eperimeal Resuls for Forced Vibraio Sweep es: Aero-Z Coducor Fig 5.19 Forced vibraio for Aero-Z coducor a 15% UTS-.5KN Table5.8 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod For 15% UTS Aero-Z Resoace Frequecy R f Acceleraio am/s a/ Half- Power F 1 Half- Power F Half- Power F qualiy facor Q dampig facor δ

84 Fig 5.0 Forced vibraio for Aero-Z coducor a 0%UTS-30.0KN Table 5.9 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod for 0% UTS Aero-Z Resoace Frequecy R f Acceleraio am/s a/ Half- Power F 1 Half- Power F Half- Power F qualiy facor Q dampig facor δ

85 Fig 5.1 Forced vibraio for Aero-Z coducor a 5%UTS-37.53KN Table 5.30 Showig Resoace frequecies ad calculaio for dampig usig badwidh mehod for 5% UTS Aero-Z Resoace Frequecy R f Acceleraio am/s a/ Half- Power F 1 Half- Power F Half- Power F qualiy facor Q dampig facor δ

86 5.4 Fiie Eleme Aalysis FEA Resuls The fiie eleme aalysis modelig of rasmissio lies coducor simulaio ca be foud i [7] i which he simulaios of coducor vibraios were doe usig ABAQUS sofware. Similarly i his sudy ABAQUS sofware was also used for he coducor vibraio simulaio bu oly for Aeolia vibraio. Thus fiie eleme aalysis for he mechaical oscillaio for rasmissio lie coducors was simulaed usig he code for beam properies. The values for physical properies of he es coducors used for he simulaio are foud i able-b-1 ad able-b- i appedi B. I simulaig coducor vibraio usig ABAQUS he eigevalues were searched ad compued i he frequecy rage for boh coducors. For he Ter coducor he firs e aural frequecies for he coducor bewee he frequecy rages of 6 o 50Hz were obaied ad are recorded i able 5.33 i compariso o he values obaied for he aalyical model a esios 0% 5% ad 30% of is UTS. Similar simulaios were also doe for he Aero-z coducor bu for he frequecy rages of 5 o 50Hz a 15% 0% ad 5% of is UTS. The resuls also i compariso wih he values obaied from he aalyical model are recorded i able To compue he dampig cosas ha were used for hese simulaios was by he derived equaio 4. i relaio o eperimeal values. Thus usig he eperimeal resoace frequecies values he leas squares mehod pseudo-iverse rouie was used o compue he dampig cosas usig Malab. The values obaied were he used as proporioal dampig cosas i ABAQUS o simulae he coducor vibraio for dampig. The ABAQUS simulaio resuls for he eigevalues ad he aural frequecies for boh coducors are preseed i appedi D. 79

87 Table Showig compariso Naural frequecies values obaied from he aalyical mode ad FEA for Ter coducor Naural Frequecy Hz 0% UTS Naural Frequecy Hz 5% UTS Naural Frequecy Hz 30% UTS Mode Aalyical Model FEA Ep. value Aalyical Model FEA Ep. value Aalyical Model FEA Ep. value

88 Table 5.3 Showig compariso Naural frequecies values obaied from aalyical model ad FEA for Aero-Z coducor Naural Frequecy Hz 15% UTS Naural Frequecy Hz 0% UTS Naural Frequecy Hz 5% UTS Mode Aalyical Model FEA Ep. Value Aalyical Model FEA Ep. Value Aalyical Model FEA Ep. Value

89 5.5 Equivale circui resuls To simulae he rasfer fucio for he elecrical equivale circui developed i chaper 4 for he vibraig coducor he equivale parameers for he elecrical circui i equivalece o is mechaical parameers have o be obaied. From [9] he value of he ui rasverse siffess of a pied-pied beam of modulus E area mome of ieria I ad ui legh for a load applied a a poi a from is ed is give as Wih he ui legh = 1.5m ad a =1.m The values of he mass per ui legh of boh coducors are give i able-b-1 i appedi B usig he dimesioless umbers as defied by he auhor [36] ad simply usig he assumed scale facor of oe o obai he correspodig elecrical equivale values. To deermie he values for he resisace i equivale dampig was by ieraive process. This process was used o obai he values of resisace which i equivalece correspods o he dampig value for free vibraio of 0% UTS of Ter. Usig hese elecrical equivale values obaied he rasfer fucio for he impulse respose for he circui was simulaed usig Malab ad he resul is show i figure 5. below. To simulae he impulse respose ha correspods o 5% ad 30% UTS a liear relaioship was adoped i which he siffess icreases wih icrease i esio. Dampig was evaluaed by he assumpio ha chage i resisace of he coducor is a fucio of emperaure. 8

90 Ampliude Impulse Respose Time sec Figure 5. Impulse resposes for he rasfer fucio The same cocep was also used o simulae he impulse respose for Aero-Z coducor bu a 15% 0% ad 5% of is UTS 5.6 Aalysis of Resuls I sudyig he vibraio of a body he acual reproducio of he es is very difficul; however i ay es coduced he vibraio es resuls are always specific for a paricular eciaio. This sudy coduced wih regards o wid-iduced vibraio eperieced by power lie coducor is of hreefold doe wih respec o he wo forms of vibraio: free ad forced. The firs is he aalysis of he aalyical model describig he rasverse vibraio of coducors. This ivolved aalysig he developed aalyical model of he rasverse vibraio of a coducor as a simply suppored beam i.e. fleural rigidiy of he coducor was o is igored. Alhough his model was used o predic he rasverse vibraio of he coducor he fac was ha a precise model was very difficul due o he o-liear respose of he coducor. I some lieraure [17][18] he eperimes coduced were doe i wid uels i which he cocep of 83

91 fluid-solid ieracio was used o deermie wid loadig o he coducor which disribued alog he coducor spa legh. This aspec was o covered i his sudy isead he loadig was assumed o be a poi loadig ad his ipu force was provided by a shaker. This model was used o obai he coducor aural frequecies ad is associaed mode shapes. The values of he e aural frequecies for he aalyical model for boh coducors a he hree differe esios are preseed i ables 5.31 ad 5.3. Also obaied is coducor self-dampig equaio. This was achieved by icorporaig wo dampig models io he equaio of moio of he coducor. The secod was he developme of fiie eleme mehod FEM ad he elecrical equivalece usig mechaical-elecrical aalogy for he coducor ad compared wih daa obaied from eperimeal resuls aural frequecies. The FEM eailed he formulaio of global equaio for he coducor. The sigle spa legh coducor was developed ad implemeed o a compuer program usig ABAQUS sofware. The program was simulaed usig beam properies ad he sysem dampig was simulaed by usig proporioal dampig. The cosa dampig values were calculaed usig equaio 4. ad he values was used for he simulaios. The compuer program was used o simulae he respose of he coducor ad he mode of vibraio was limied o he verical plae. From he compuer program simulaios he values for he aural frequecies were obaied. Table 5.31 ad able 5.3 showed he compariso bewee he aural frequecies obaied from he aalyical model ad ha from he FEA. Comparig he resuls here was a good agreeme bewee he aalyical ad he FEM frequecies for he wo coducors doe a he hree differe esios. Also boh ables are also used o compare he values of he resoace frequecies obaied from resoace search es. I compariso he firs few lower modes ed o be close o boh he aalyical ad FEM values bu as he mode umber icrease here were deviaios ad he deviaios also icreases as he mode icrease. Also obaied was he developme of he elecrical aalogy for he power lie coducor. Due o he fac ha he model for he fiie eleme mehod ad elecrical equivalece were dimesioally equal similar resuls were obaied from boh mehods. The impulse respose was obaied by he simulaio usig Malab ad he calculaio for dampig was by usig he raios of decayig ampliudes. The resuls were i good agreeme wih he eperimeal resuls for free vibraio. The fial aspec of his sudy was he eperimeal verificaio of he aalyical model. The eperimeal daa available for verificaio was for Aeolia vibraio. The verificaio eperimes were doe o he 84.6m spa coducor a hree differe esios for he wo coducors used for he eperimeal esig. The laboraory resuls have show ha coducor vibraio is characerised by a high mode desiy ad a arrow badwidh. For he free vibraio es a ime hisory domai was used o 84

92 prese he capure sysem s respose due o he impac loadig. This was he used o calculae he amou of dampig from he sysem while for he forced vibraio es; a frequecy hisory domai was used o deermie he resoace frequecies for he sysem caused by eciaio from he siusoidal force. Based o hese values he amou of dampig for each mode was calculaed. For he free vibraio es he average value for dampig for Ter a 0% UTS 5% UTS ad 30% UTS were ad respecively. Similarly for Aero-Z coducor he average value for dampig a 15% UTS 0% UTS ad 5% UTS were ad respecively. I chaper oe besides his sudy seekig o gai more udersadig io he mechaism of wididuced vibraio he mai goal was he deermiaio of self-dampig of he coducor ad his promped he research quesio. This ieio has bee achieved ad a mehodology esablished for he deermiaio of coducor self-dampig. I his vai his process will help o deermie a a paricular srigig esio he coducor self-dampig capabiliy ad o ascerai wheher he value for dampig a ha paricular esio should be igored or o. This will also helped i deermiig he ype ad umber of vibraio absorber ha will be required o couer he effec of dyamic loadig i order o keep he coducor safe. I has bee highlighed hrough he lieraure reviewed i chaper wo ha some parameers were paricularly impora as relaed o he Scruo umber o he predicio of he respose of a sysem o some form of eciaio. Some of hese parameers are mass desiy ad dampig. From all aspecs of he sudy for Ter coducor i was oiced ha he coducor dampig value decreases as he aial esio was icreased. Due o his very low ihere dampig a higher esio greaer ha he recommeded UTS value he coducor eds o be more proe o mechaical oscillaio. For Aero-Z coducor i was observed ha a higher esios greaer ha he recommeded UTS esio value he dampig value was okay o he srig lie. Bu srigig of his coducor is doe a lower esio based o he less capabiliy of he coducor o wihsad boh he dyamic loadig ad he saic esile sress due o he abse of seel i he coducor. Hece from his aalysis i was esablished ha Aero-z coducor has a higher dampig capaciy compared o Ter ad his was cofirmed i all aspecs of his sudy. Therefore i his sudy he values of he coducor self-dampig were deermied usig shorer spa idoor coducor. So far i his sudy wo mehods of dampig ideificaio have bee developed o eable i he esimaio of dyamic acio of Aeolia vibraios o overhead rasmissio lies. For he eperimeal sudies he free vibraio prese resuls for he dampig values for he sysem a hree differe esios while forced vibraio prese he dampig values a he resoace frequecies also a he same hree differe esios for free vibraio. Boh resuls show ha he 85

93 dampig decreases as he cable esio was beig icreased. I secio.4. he various dampig mechaisms i a vibraig coducor were eplaied. The eplaaio saed ha dampig ca be caused by relaive moio bewee coducor srads ad also bewee he coducor accessories. Thus his form of dampig is caused by he fricioal force ha eis due moio relaively bewee each srad ad also bewee members like he clamps dampers spacers spacer-dampers ad suspesio assemblies. As he aial esio was beig icreased he ier-srad moio was highly beig resriced. This cosequely led o he reducio i dampig from his dampig mechaism. For he oher meas his is maerial dampig which has o do wih he coducor propery i.e. he coducor siffess. As he esio was also icreased he coducor siffess was also icreased which ivariably led o reducio i dampig. Secio 3.5 eplais how hese wo mechaisms have bee modeled i order o deermie he coducor dampig: viscous ad maerial dampig models respecively. The eperimeal ad fiie eleme resuls have also show he grea ifluece of boh cable esioig ad siffess ad i was oiced ha dampig ed o decrease as he aial esio was beig icreased i boh cases. I he mahemaical model i secio 3.5 hese mechaisms were model i relaio o he dampig parameers βi ad C. These parameers ed o servers as he meas o deermie he coribuio of boh dampig mechaism o he oal dampig a various esios. Because he foregoig aalysis i his sudy lacks some impora issues which ca be addressed by coducig more ivesigaios o he bases of he fidigs wih regards o furher ivesigaios he hese dampig cosas ca he be used o ascerai he coribuios of each dampig mechaism o he oal dampig as he cable esio is beig icreased or varied. Hece furher sudies eed be carried ou ad also some form of o-liear coceps iroduced. The oucome ca he be used i aalysis of acual rasmissio lies o deermie how bes hey agree. This will ivolve usig hese geeraed resposes for he aalyical model for he sysem fiie eleme simulaio ad eperime daa from he laboraory ad compared wih daa from acual rasmissio lies. Based o he compariso bewee hese resposes ad ha o he real overhead power lies coclusios ca he be draw o how self-dampig occurred ad how i ca be icorporaed io rasmissio lies desig ad cosrucio. 86

94 6.1 Coclusios CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS From he research sudy as well as he ess coduced ad compiled i his repor ad also from he ifereces draw based o aalysis doe i he las chaper he followig were esablished from he heoreical aalyical model fiie eleme mehod ad eperimeal sudy Naural Frequecies of he vibraig coducor icreases wih he icrease i he aial esio of he coducor. Dampig decreases wih he icrease i esio. I compariso here was close agreeme of he values of he aural frequecies ad dampig obaied i heoreical FEM ad ha from he laboraory eperimes. The parameers obaied from he above o some degree of accuracy ca be used o predic he respose of coducors o wid loadig. The mai objecive of his repor was he evaluaio of bare coducor self-dampig. I his sudy wo mehods were used o evaluae dampig a differe esios: free ad forced vibraio. The eperimeal daa showed ha self-dampig of Aero-Z coducor is higher ha ha of Ter coducor. However because Ter has higher resisaces o srucural failure due o he presece of seel srigig a higher esio is recommeded. 6. Recommedaios From resuls compiled i his sudy i could be observed ha i order o improve fuure research relaig o his area of wid-iduced vibraio some modificaios are eeded. Some of hem are as follows: 87

95 I all aspecs of his sudy ragig from aalysis of he aalyical model fiie eleme aalysis ad elecrical aalogy he liear cocep was used. This cocep is acually corary o wha happes o coducors whe hey are eposed o he dyamic forces of aure like wid. I he real world coducor respose is o-liear; herefore o improve o he coducor modelig some cocep of o-lieariy should be iroduced. Also i modelig srucural dampig he viscous dampig model was used i his sudy. The hyseresis or dry or coulomb model should be cosidered i order o improve o he accuracy of he value of his ype of dampig. This is because hese forms of dampig models ca help represe i erms of o-lieariy wih regard o modelig dampig. Boudary codiios i boh aalyical models ad FEA ad effec of ed ermiaio should also be eamied o improve o he accuracy of resuls. Wih respec o aial loadig a he lower esio he wid power ipu io he coducor does o eceed he power loss. As he esio is icreased he ipu power eds o eceed he power loss from he coducor. Therefore based o he oucome of sudy he abiliy o evaluae selfdampig of bare coducors ca deermied. However furher sudy could be doe o deermie he criical poi i which he coducor dampig power loss balaces he ipu power a a give esio. This process will help deermie he ypes ad amou of damper eeded o he lie i order o preve damage ha could be caused by he coducor oscillaio. Furher sudy could also be doe o ascerai ad developed a mehod ha ca be used o obai a value for bedig siffess which varies alog he coducor legh i he area of fiie eleme aalysis. This ca he be used o formulae a compuer program ha ca be used as a desig parameer for desigig rasmissio lies. Fially because of he eperimeal daa available verificaio of he aalyical model was doe for Aeolia vibraio oly. Similarly hese hree coceps covered i his sudy should also be eeded o he aalysis of wake-iduced vibraio. All he above areas for fuure sudy will form he basis of my PhD sudies. 88

96 Refereces [1] E.S Duocy Eal Wid iduced coducor moio Trasmissio Lie Referece Book Elecrical Power Research Isiue 1979 [] SABS 080 Code of pracice for overhead power lies for codiios prevailig i Souh Africa [3] P. Du Plessis. Mechaical oscillaio o overhead rasmissio lie PhD i Mechaical Egieerig Rad Afrikaas Uiversiy Jue 1994 [4] EPRI Trasmissio Lie Referece Book: Wid-Iduced Coducor Moio Fial Repor November 006 [5] R. Clare ad G Diaa Mahemaical Aalysis of Trasmissio Lie Vibraio IEEE rasacios o power apparaus ad sysems vol. pas-88 No. 1 December 1968 [6] Cigrè Sudy Commiee -Workig Group o1 Repor o Aeolia Vibraio Elecra o 14 pp-101 May 1989 [7] IEC 619 Overhead Elecrical Coducors - Formed Wire Coceric Lay Sraded Coducors February 1 00 [8] K. Goveder Digial aalysis of Vibraio ad Noliear Sysems Msc i Elecrical Egieerig hesis Uiversiy of Naal 199 [9] J. I. Daiel Egieerig Vibraio Preice Hall [10] A.D. Nashif D.I. Joes ad J.P. Hederso Vibraio Dampig Wiley Caada [11] C. Hardy. Aalysis of Self-dampig Characerisic of Sraded Cables i Trasverse Vibraios CSME Mechaical Egieerig Forum [1] J. Vecchiarelli I.G. Currie ad D.G. Havard Compuaioal Aalysis of Aeolia Coducor Vibraio wih a Sockbridge-ype Damper Joural of fluids ad Srucures [13] V. Srohal o Aeolia oes A of Phys Ediio 5 pp [14] S. Kumarasema P.N. Joes P. Irwi ad P.Taylor Wid-Iduced Vibraio of Say Cables publicaio o. FHWA-HRT Augus 007. [15] R E.D Bishop Vibraio Cambridge Uiversiy press Eglad

97 [16] R. D. Blevis Flow - Iduced Vibraio. New York Va Nosrad Reihold 1977 [17] F.B Faquharo ad R.E McHugh Wid Tuel Ivesigaioof coducor Vibraio of Rigid Model IEEE Trasacio paper. Pp Ocober 1956 [18] G. Diaa ad M. Falco O he Forces Trasmied o a Vibraig Cylider b a Blowig Fliud Meccaica vol. 6 pp [19] F. Kiesslig P. Nefzger J. F. Nolasco ad U. Kaizyk Overhead Power Lies: Plaig Desig ad Cosrucio Spriger 003. [0] C.F. Beards Egieerig Vibraio Aalysis wih Applicaio o Corol Sysems Edwards Arold Lodo [1] L. Meirovich. Priciple ad Techiques of Vibraios Preice Hall [] F.J. Shaker. Effec of Ed Load o Mode Shape ad Frequecies of Beams Lewis research Cere Repor NASA-TN_ [3] H. T. Baks D.J Ima O Dampig Mechaisms i Beams Trasacos of he ASME pp. 716 Vol. 58 Sepember 1991 [4] K.O. Papailiou O he Bedig Siffess of Trasmissio Lie Coducors 610 Malers Swizerlad IEEE 1996 [5] Aberdare Power cables Divisio Aberdare cables Py Ld Souh Africa Caalogue of Coducors [6] C. K Güher The Fiie Eleme Modellig of he Dyamic Behaviour of a Trasmissio Lie Coducor Msc i Elecrical Egieerig hesis Uiversiy of Cape Tow 1996 [7] Neas Trade Py Ld [8] M.N. Newmark A Mehod of Compuaio for Srucural Dyamics Joural of he Egieerig Mechaics Divisio Proceedigs ASCE vol. 85 N0.EM3 July [9] E. L Wilso ad R. W. Clough Dyamic Respose by Sep-by-sep Mari Aalysis Proceedigs Symposium o he Use of Compuers i Civil Egieerig Paper No.45 Lisbo Porugal Ocober [30] H.M Hilber Th.J.R Hughes ad R.L Taylor Improved Numerical Dissipaio for Time Iegraio algorihms i Srucural Dyamics Earhquake Egieerig ad Srucural Dyamics 5 p

98 [31] O. C. Ziekiewcz The Fiie Eleme Mehod McGraw-Hill Book Compay 3 rd Ediio New York 1979 [3] J.N. Reddy A Iroducio o he Fiie Eleme Mehod McGraw-Hill Ieraioal Ediio [33] Y.W. Kwo ad H. Bag Fiie Eleme Mehod usig Malab d Ediio CRC Press [34] H. H Cudey ad D. J Ima Deermiig Dampig Mechaisms i a composie beams by Eperimeal Modal Aalysis Sae uiversiy of New York a Buffalo Ambers NY 1460 April [35] W. W. Seo Mechaical Vibraio Schaums oulie series McGraw Hill [36]. Measureme Aalysis Corporaio Elecrical Aalogies ad he Vibraio of Liear Mechaical Sysem Los Ageles Califoria. [37] The Isiue of Elecrical ad Elecroics Egieers Guide for Laboraory Measureme of he Power Dissipaio Characerisics of Aeolia Vibraio IEEE sd [38] The Isiue of Elecrical ad Elecroics Egieers Guide o Coducor Self-dampig Measuremes IEEE sd R00 [39] L. Meirovich Fudameals of Vibraios McGraw-Hill Ieraioal Ediio 001 [40] C. W. De silva Vibraio Moiorig Tesig ad Isrumeaio The Uiversiy of Briish Columbia Vacouver Caada CRC press 007 [41] PUMA Vibraio Corol ad Aalysis Sysems

99 Appedi A: Derivaio of he Simple Beam Equaio: Euler-Beroulli Equaio The Simple Beam Theory The beam is a eample disribue-parameer sysem ad such sysem has ifiie umber of aural frequecies. For his kid of sysem whe here is vibraio each of hese ifiie umbers of elemes moves relaive o each oher i coiuous maer. Vibraio of a beam i he perpedicular direcio o he legh is ofe referred o rasverse vibraio or fleural vibraio. Derivig he fleural vibraio of a uiform beam show i figure 3.1 wih crosssecioal area A fleural rigidiy EI maerial desiy ρ ad legh L. y L Q d F Fig A-1 A Uiform Beam. M M + M d Q + Q d Figure A- Differeial Eleme of he Beam 9

100 93 Cosiderig small eleme wih legh d as show i figure 3. which is subjec o a eeral force f Where M is he bedig mome Q is he shear force F is he applied force From mechaics of maerials y EI M... A1 Summaio of force i he y-direcio y d A f Q d Q Q... A Summaio of mome abou z-ais 0 d f d d Q Q M d M M 0 d f Q d Q M Because d is assumed o be small he d is assumed o be almos zero M Q. A3 Subsiuig Eq 3.3 io Eq 3. y d A f M d M M y d A f d M..... A4 Dividig hrough by d ad subsiuig Equaio 3.1 io Equaio 3.4 f y EI y A.. A5

101 If EI ad A are assumed cosa ad o eeral force applied f = 0 he Eq A5 becomes y C 4 y 4 A6 Where C is he wave speed for he beam ad is give as C = EI A Noe: Equaio A6 is he free vibraio for he beam also kow as Euler-Beroulli equaio. The soluios o equaio A6 of a simple beam require four boudary codiios ad wo iiial codiios o solve he free wave equaio. Usig separaio-of-variable where he soluio is assumed i he followig form Y X T. A.7 The Equaio 3.6 becomes C //// X T X T Where ω² is he separaio cosa The righ had side of eq 3.8 becomes. A.8 T T 0.. A.9 This is he emporal equaio ad because he beam perform a harmoic vibraio wih ime he soluio of equaio is T A1 si B1 cos A.10 The cosas A 1 ad B 1 are deermied from he iiial codiios of he equaio. Also he lef had side of equaio 3.8 becomes //// X X 0. A11 C This is kow as spaial equaio If 4 A..A.1 C EI If X m Ae herefore //// X Am 4 e m 94

102 //// 4 X X 0 Ζm 4 e ma - 4 Ζe ma 4 m 4 m =0 Am e Ae 0 Ζe ma m = 0 Ae m m 0 m 4 4 Ae 0 i implies ha m 0 m 4 - If m + 4 = m + m = 0 = 0 he m = - i he implies ha m = ±j Therefore X = C 1 si + C cos. A.13 Also if m =0 he m = i implies ha m = ± Therefore X = C 3 sih +C 4 cosh Hece he geeral soluio becomes X = C 1 si + C cos +C 3 sih +C 4 cosh... A.14 The cosas C 1 C C 3 ad C 4 are deermied from he boudary codiios. To deermie he aural frequecies ad mode shapes for he rasverse vibraio of a simply suppored or pied a boh ed eds as show below wih he four boudary codiios Subsiuig hese boudary codiio io equaio 3.14 A = 0 ad X"0 = 0 C 1 + C 3 = 0 A = 0 ad X 0 C 1 - C 3 = 0 Therefore C 1 = C 3 = 0 The eq 3.14 becomes X = C cos +C 4 cosh A = L ad X L = 0.. A.15 X L = C cos L + C 4 cosh L A = L ad X L = 0 X L = C cos L - C 4 cosh L Hece X = C si L - C 4 sih L = 0 95

103 Based o he fac ha sih L 0 C 4 =0. The equaio 3.14 becomes X = C si A.16 Bu sice C sihλl = 0 ad C 0 ad because X will be zero for values of he Si l l or 0... l l l l l So ha EI 0 l A l EI A 3 l EI A rad/s. A.17 96

104 Appedi B: Physical Properies of Tes Coducors Code ame: Ter [5] Coducor ype alumium coducor seel reiforced Sraded ACSR Coducor properies: Diameers: O. Dia - 7 mm Core dia 6.75 mm Sradig umber & wire diameers: Alumiium: mm Seel: 7.5 mm Cross secio: Alumiium: square millimeers Seel: 7.83 square millimeers Toal: square millimeers Mass per ui legh: kg/m Fig B-1 Ter Coducor Table B-1Physical Properies for Ter Coducor Quaiy Ui Radius of coducor m Toal cross-secioal area m Coducor desiy kg/m 3 Coducor Spa legh 84.6 m Coducor Chord Legh 85.0 m Coducor mass per legh Kg/m Poisso s raio 0.33 Maimum bedig siffess 1350 Nm 97

105 To calculae for he maimum ad miimum bedig siffess values for Ter. Give: E a = Nm E s = 0.68 Nm Usig equaio 5.1 ad values of youg modulus for seel ad alumiium give above he miimum EI value is give as EI mi = EI mi = 1.47Nm The value for maimum bedig siffess for Ter is calculaed usig equaio 5.4 ad aalysis is give i able 5.1 The able is used o calculae he EI ma for Ter coducor Seel Alumiiu d R E.07E+11.07E E E E+10 I 1.6E E E E E-08 EI.60E E E E E+0 EI ma = 1350Nm 98

106 Code ame: 445-A3F-6 [7] Coducor ype Aero-Z Coducor properies: Diameers: O. Dia 6.10 mm Core dia 14.5 mm Sradig umber & z-wires diameers: Roud alumiiu wires: 19.9 mm Z-shape alumiu wires 4.9mm Cross secio: Toal alumiium: square millimeers Mass per ui legh: 1.84 kg/m Fig B- Aero-Z Coducor Table-B- Physical Properies for Aero-Z 445-A3F-61 Coducor Quaiy Ui Radius of coducor m Youg s Modulus 5.4 N/m Toal cross-secioal area m Coducor desiy kg/m 3 Coducor Spa legh 84.6 m Coducor Chord Legh 85.0 m Coducor mass per legh 1.84 Kg/m Poisso s raio 0.35 Bedig siffess Nm 99