The effectiveness of vibration damper attached to the cable due to wind action

Transcription

1 EPJ Web of Conferences 4, 009 (07) DOI: 0.05/ epjconf/ EFM 06 The effeciveness of vibraion damper aached o he cable due o wind acion Irena Gobiowska, Maciej Dukiewicz,* Deparmen of Building Consrucion, Universiy of Science and Technology in Bydgoszcz, ul. Prof. S. Kaliskiego 7, Bydgoszcz, Poland. Absrac. In he paper he analysis of vibraion of cable wih aached damper is performed. Source of vibraions comes from he wind. The level of reducion of vibraions is also analysed. Damper aached o he cable is characerized by mass, siffness and damping raio. The paper also presens he equaions of moion for he cable wih damper, aking ino accoun he iniial parameers: cable cross-secion and he locaion of he damper. The analysis is performed in ime domain. I is discussed ha in real siuaions he differen ypes of cable vibraion due o heir ampliudes and frequencies is me: shor-erm oscillaions of high ampliude, caused by a single impulse force, called bouncing cable, low ampliude and low wave lengh and he high frequency, called aeolian vibraion, large oscillaion ampliude, long wave lengh and low frequency, called galloping or dancing cables. Inroducion. Bridge cable vibraions Due o consrucion benefis and archiecural view in recen years a significan number of suspension and cable-sayed bridges were buil. From he poin of load and work of bridge, he wind load can be reason of vorex induced vibraions ha can adversely effec on srengh, durabiliy of cables and connecion failures [,]. In some cases local faigue cracks are formed and furher desrucion of he srucure is performed. Mehods of reducion vibraions are applied o avoid such siuaions as cables have a very low level of damping an low bending siffness []. Connecion of cables o each oher is anoher possible soluion bu his soluion is used very rarely because of he adverse visual impac. Applicaion of passive vibraion damper, usually mouned a a disance no exceeding 5% of he lengh from he anchorage cable consrucion plaform is anoher soluion ha is ofen used. Aerodynamic cable surface modificaion is anoher mehod of vibraion reducion []. The selecion of appropriae parameers of he vibraion damper is associaed wih he characerisics of he damped srucure working under he dynamic load. Cables perform uninerruped vibraions boh a minimum parameers of he damping sysem as well as when damping parameer is large, so i is essenial o perform he uning of he damping sysem [4].. Transmission lines vibraions Vibraions of ransmission lines due o wind acions are characerized by high frequency and small ampliudes of vibraions. They are called aeolian vibraions. Apar from hem, oscillaions of high ampliude and low frequency called galloping vibraions and secondary induced vibraions of average ampliude bu lower han aeolian frequency are also observed. Aeolian vibraions are in he range of o 50 Hz a wind speed from 0,5 o 7 m/s acing perpendicular o he cable, which usually happens in open plos, in he area wihou buildings and infrasrucure. The ampliude of oscillaion does no exceed.5 cm [5,6]. These vibraions are difficul o observe due o he fac ha ransission lines diameers are from 6 o 50 mm, which exceeds he ampliudes of vibraions. Galloping of cables appears a winds speed from 7 o 5 m/s. Cables oscillae according o firs or second form as one or wo half-waves. The frequency of oscillaion is small, less han Hz [5]. The maximum ampliude of vibraions occurs a resonance, and increases when he naural frequency is close o he forced frequency called Srouhal frequency. Vibraions can be conrolled by he use of differen dampers. The use of Sockbridge'a damper is a very popular soluion. Sockbridge damper is consruced from a flexible cable or rod and he masses a is ends []. The vibraions of he cable are passed hrough he clamp mouned he damper's rod o he cable and consequenly he masses on he ends of he damper begin o vibrae. By careful selecion of parameers of he damper such as mass a he ends of he rod and siffness of he damper's rod, one can une he opimal damping of he ransmission energy line. Firs Sockbridge damper held a he ends of he rod masses made of concree blocks. Nowadays seel masses a end of he damper's rod are used and galvanized seel is applied for damper's rod. In he modelling of Sockbridge damper he massless rod elemen is * Corresponding auhor: macdu@up.edu.pl The Auhors, published by EDP Sciences. This is an open access aricle disribued under he erms of he Creaive Commons Aribuion License 4.0 (hp://creaivecommons.org/licenses/by/4.0/).

2 EPJ Web of Conferences 4, 009 (07) DOI: 0.05/ epjconf/ EFM 06 analysed wih masses a he ends. From he poin of reducion of vibraions, he locaion of damper on he cable is significan [5]. The aeolian vibraions were analysed in he paper [8]. Analysis confirms ha use of damper do no cause a dissipaion of energy and reducion of vibraions in any case. In he paper [6] he aplicaion of Sockbridge damper caused he feedback of he damper wih he cable and disurbance of aeolian vibraions. Deerminisic or probabilisic models are ofen applied among compuaional models. In he paper [5] a probabilisic model of he aeolian vibraion wih regard o faigue srengh was performed. The purpose of he dampers is reducion of vibraion by absorbing energy from he wind and sabilizaion of he movemen of he cable o avoid deformaion ha can cause he damage. According o analysis performed in he paper [5] he value of safe limi deformaion of he cable is 00 microns/m. The effeciveness of he damper depends on is response o he wind force a a cerain frequency. Modelling and analysis of cable's vibraions. Assumpions of he analysed sysem The issue of inexensible cable's free vibraion was analyzed by Hajduk [7]. Vibraion equaions shown below fulfill he law of change of he amoun of moion, law of conservaion of mass and he law of change of he body deformaion under load. Numerical simulaion is carried ou for 0 seconds of vibraion exciaion, for span of he cable of 0 m. The cable is modelled as a srand, non-displaceably fixed o he srucure. I is assumed ha pah of he cable is inplane and he chord is horizonal. I is analysed he coninous disribuion of he mass along he lengh of he cable. The analysis is limied o verical vibraions occurring in he plane of he cable sag. According o Kirchhoff's hypohesis, because he sag of he cable is small, he effecs of horizonal displacemens is omied. I is assumed ha he maerial of he cable is linear-elasic and fulfill Hooke's law, i.e. he relaionship beween displacemens and deformaions is linear. Taking ino consideraion he above assumpions he soluion of dynamic displacemen is aken in he form: () is unknown funcion of ime, reaed as he coefficien of series: () and is he ampliude coefficien of he funcion, 'i' is he imaginary par of he complex number, is he forced frequency, - means ime of he analysed process. shown in he equaion no. is funcion ha mees all boundary condiions of he analysed problem: () This assumpion is made according o Galerkin's mehod for cable's vibraion of an infinie number of degrees of freedom [7]. I is assumed ha funcions, j =,,...,n, are linearly independen. Vibraion of he single damper is presened in he following equaion: (4) where is he ampliude coefficien of he funcion. Fig.. Sysem of he cable wih he damper The general idea of he sysem equipped wih he damper, ha is he subjec of he paper, is shown on figure. I is shown he cable of lengh 'l' under he load p n wih he sysem of dampers represened by 'n-h' damper wih parameers: siffness k n, damping c n, mass M n, m - mass of he cable disribued uniformly on he cable. The purpose of he paper is o demonsrae he need of proecion agains vibraions of cables acing as bearing srucures and proper selecion of vibraion dampers' parameers.. Soluion and discussion Taking ino accoun he approximaion () and equaions from () o (4), he equaion of moion of he sysem can be wrien in Langrange'a coordinaes: (5) where M, C, K are respecively marices: mass, damping and siffness, P () is he exernal load, i.e:,,

3 EPJ Web of Conferences 4, 009 (07) DOI: 0.05/ epjconf/ EFM 06, (6) Orhogonaliy condiion for he equaion describing he vibraions of he cable akes he following form: (7) E - modulus of elasiciy of he cable, H s - ension force in he cable, F - cross-secional area of he cable, l - he lengh of he cable, - drag coefficien of he air, D - cable diameer, - inernal damping coefficien of he cable, - siffness coefficien of he "n-h" damper, - damping coefficien of he "n-h" damper, - Dirac dela funcion. I was assumed he following load: The equaion for "n-h" dynamic damper akes he following form: where is he mass of "n-h" dynamic damper. Afer calculaions, he soluion of he sysem of equaions (5), aking ino consideraion equaion (), he deflecion funcion akes he form: In equaions () and () and denoe funcions dependen on he known parameers of siffness, damping, frequency domains, he mass of he cable and he masses of dampers. Numerical analysis Numerical analyses were performed for various parameers and he locaion of dampers. I was assumed he lengh of he seel cable l=0m, cross secion of he cable F =, cm, force of ension H s =0 kn. Figure shows ha he use of he dynamic vibraion damper reduces he dynamic displacemen from he value of 6. mm o he value of 0,5mm. Afer using he damper placed in he middle of he cable, he displacemen have been reduced by almos 6 imes. The above reducion of vibraion was achieved by choosing he opimal parameers as follows:,,. e+0 e+0 e+0 -e+0 -e+0 0 s -e (s) Fig.. The dependence of displacemen versus ime for he vibraion damper in he middle of he span. Curve - vibraion in he middle of he span of he cable, curve - he vibraion in he /4 of span of he cable, curve - vibraions close o he e+0 e+0 e+0 -e+0 w -e+0 0 s -e (s) Fig.. The dependence of velociy versus ime for he vibraion damper in he middle of he span. Curve - vibraion in he middle of he span of he cable, curve - he vibraion in he /4 of span of he cable, curve - vibraions close o he Figure shows he velociy versus ime for he parameers of he damper as in he case shown in Figure.

4 EPJ Web of Conferences 4, 009 (07) DOI: 0.05/ epjconf/ EFM 06 Figure 4 shows he acceleraion versus ime for he parameers of he damper as in he case shown in Figure. 4e+0 4e+0 e+0 e+0 e+0 -e+0 Displacemen,w,[mm] locaionofdamper,[m] -e+0-4e+0 0 s -e (s) Fig.4. The dependence of acceleraion versus ime for he vibraion damper in he middle of he span. Curve - vibraion in he middle of he span of he cable, curve - he vibraion in he /4 of span of he cable, curve - vibraions close o he Figure 5 shows he displacemen depending on he frequency of vibraion of cable wih damper and wihou damper. Displacemen,w,[mm] frequency[hz] Fig. 5. The dependence of he displacemen versus frequency, for a cable wih damper and wihou damper. Curve - vibraions of he cable wihou damper, curve - cable vibraion for opimal parameers of he damper,,, curve - cable vibraion for parameers of he damper,,. Figure 6 shows he funcion of displacemen versus he locaion of he vibraion damper on he cable. Fig.6. Dependence of displacemen as a funcion of he locaion of he damper, Curve - vibraion of he cable for he parameers of he damper,,, curve - vibraion of he cable for he parameers of he damper,,, curve - vibraion of he cable for he parameers of he damper,,. 4 Summary and Conclusion In he paper he imporance of cable srucures' analysis is emphasized. Cables in srucures play an imporan supporing funcion. They are par of he consrucion of suspension bridges and he power ransmission lines. From he poin of view of he srucural analysis, cables are imporan elemen in he design process o keep he sufficien ulimae and serviceabiliy limi saes. These consrucions have large spans and low rigidiy suscepible o deformaion wha was discussed in he paper. Because of low ampliudes ofen less han he diameer of he cable, hese vibraions are slighly percepible, bu may lead o maerial faigue and hen o failure. In he paper he influence of he parameers of he damper and he locaion on cable is shown and analysed. The proper locaion and opimizaion of damper's parameers is shown as requiremen o reduce he vibraion of he cable. References. I.Gobiowska, K.Peszyski, Reducion of Cable Vibraions Caused by Wind, Engineering Mechanics, h Inernaional Conference Engineering Mechanics, Svraka 06. I.Gobiowska, M.Dukiewicz, B.Usewicz, Sposoby umienia drga napowierznych linii elekroenergeycznych, TTS Technika Transporu Szynowego - 05,, B.M. Pacheco,Y. Fujino, A. Sulekh, Esimaion Curve for Modal Damping in Say Cables wih 4

5 EPJ Web of Conferences 4, 009 (07) DOI: 0.05/ epjconf/ EFM 06 Viscous Damper, J. Sruc. Engrg., ASCE, 9(6), (99). 4. K. Uno, S. Kiagawa, H. Tsusumi, A.Inoue, S.Nakaya, A simple mehod of designing cable vibraion dampers of cable-sayed bridges, J. of Sruc.l Eng., JSCE, Vol. 7A, pp , (99) 5. H. Q. Wang, J. C. Miao, J. H. Luo, F. Huang, and L. G. Wang, The free vibraion of long-span ransmission line conducors wih dampers, Journal of Sound and Vibr., 08(4):50 (997). 6. H. Meynen, P.Verma, P. Hagedorn, M. Schafer, On he numerical simulaion of vorexinduced vibraions of oscillaing conducors, J.of Fluids and Sruc., () (005). 7. J.Hajduk, J. Osiecki, Usroje cignowe, eoria i obliczanie, WNT Warszawa (970) 8. J.Vecchiarelli, I.G. Curie, D.G. Havard, Compuaional analysis of aeolian conducor vibraion wih a sockbridge-ype damper, J. of fluids and sruc. 4, (000) 5