tuned vibration control of overhead line conductors

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 48:1215}1239

Tuned vibration control of overhead line conductors

Alexander Tesar* and Jana Kuglerova

Institute of Structures and Architecture, Slovak Academy of Sciences, Dubravska Cesta, 84220 Bratislava, Slovak Republic

SUMMARY

Linear and non-linear theoretical and numerical analysis of ultimate response of overhead line conductors istreated in present paper. Interactive linear and non-linear conditions in ultimate response are considered.Numerical solution of non-linear problems appearing is made using the updated Lagrangian formulation ofmotion. Each step of the iteration approaches the solution of linear problem and the feasibility of the parallelprocessing FETM technique with adaptive mesh re"nement and substructuring for non-linear ultimatewave propagation and ultimate transient dynamic analysis is established. Some numerical results demon-strating current applicabilities and e$ciency of procedures suggested are submitted. Copyright ( 2000 JohnWiley & Sons, Ltd.

KEY WORDS: aeolic vibrations; FETM-method; parallel processing; Stockbridge damper; tuned vibrationcontrol; ultimate dynamics

INTRODUCTION

Conductors of high-voltage overhead lines (see Figure 1) are subjected to short-wave mechanicaloscillations of frequencies between 5 and 100 Hz due to action of the wind. This phenomenon isa matter of resonance aeolic vibrations being caused by von KaH rmaH n vortices arising at windvelocities from 0.5 up to 10 m/s.

Above environmental in#uences can produce dynamic forcing with a predominant frequencyconcept around structural resonance region of slender overhead line conductors. Such structuresare lightly damped, a condition described as &lively', and can undergo aeolic vibrations with smallamplitudes as well as large amplitude vibrations due to galloping when subjected to aboveforcing. Such vibrations can create a serviceability problem because initiating service limit statebehaviour including crackings, large stress and strains, dynamic instabilities or large dynamicdeformations of overhead line conductors studied.

The vibration motions produce an alternate bending stress in the conductor initiating tensileand #exural stresses in the suspension clamps. Consequently, this may lead to early fatiguefailures of conductors.

*Correspondence to: Alexander Tesar, Institute of Structures and Architecture, Slovak Academy of Sciences, DubravskaCesta, 84220 Bratislava, Slovak Republic

Received 8 February 1999Copyright ( 2000 John Wiley & Sons, Ltd.

Figure 1. Overhead line conductor.

Figure 2. Special type of Stockbridge damper developed for tuned vibration control of aeolic vibrations ofoverhead line conductors.

One measure against above vibrations is the application of tuned vibration control facilities inthe systems of overhead line conductors.

One type of Stockbridge damper developed recently in ELBA Group, Slovakia, is presented inFigure 2. The improvements of dynamic response obtained in such a way are interesting from thepoint of view of tuned vibration control of aeolic vibrations of overhead line conductors subjectedto the action of wind. In order to obtain e$cient tuned vibration control facility, the dynamicdampers are installed in the range of suspension clamps. Such dampers are made as resonancefacilities having a high damping capacity within a frequency range and wind velocities underconsideration.

Another possibility for tuned vibration control is the adoption of facilities allowing thevariability of axial prestressing forces in overhead line conductors studied. The change offrequency spectrum initiated in such a way enables e$cient and active tuned vibration control ofoverhead line conductors in all forcing situations possibly appearing.

The topic includes further the application of automatically working tuning joints, identi"cationof their parameters, formulation of ultimate dynamics, selection of target reliability and

1216 A. TESAR AND J. KUGLEROVA

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:1215}1239

development of optimal tuning by evaluation of ampli"cation curves for all tuned vibrationcontrol parameters studied.

The topic of tuned vibration control is simulated by assumption of &standing waves'with a localexchange of potential and kinetic energy appearing. The energy absorber is radiated by travellingwaves initiated by aeolic vibrations and damping appearing in such a way is transmitted into allparts of overhead line conductor studied. So the wave equation describes the activity of all kindsof tuned vibration control systems possibly adopted.

The complexity of tuned vibration control requires the adoption of new fast algorithmsfor parallel numerical approach. Theoretical backgrounds of such approach as well asnumerical treatment of tuned vibration control problems on overhead line conductors aresubmitted below.

ULTIMATE BEHAVIOUR OF OVERHEAD LINE CONDUCTORS

Theoretical, numerical and experimental analysis of ultimate dynamic behaviour of overheadline conductors has become the focus of intense e!orts because of pressing problems ofdisaster prevention of cables, suspension clamps and masts due to action of aeolic andgalloping vibrations appearing. Required is sophisticated analysis in order to answerquestions associated with ultimate response of overhead line conductors to above kinds offorcing.

Considerable work has been directed towards improving the computational e$ciency ofcomponents of "nite dynamic processes, such as temporal integration of incremental equationsof motion, optimization of numerical techniques or development of parallel processingalgorithms, that permit the analysis of complex overhead line systems with correspondingstructures to whatever degree of modeling desired. Material and geometric non-linearities andtheir interactions with time-dependent ultimate structural behaviour are to be treated for anessentially unlimited range of structural con"gurations, geometries and exploitation conditionsappearing.

Predicting the non-linear response of overhead line conductors due to ultimate forcing is one ofgreat challenges for the analyst in present branch. After de"ning the kinds of non-linearities whichare to be dealt with, the analyst is confronted with di$culty of expressing the non-linearitiesmathematically, with following treatment of corresponding di!erential equations, discrete simu-lation and solution of the problem de"ned. Concerned with the propagation of discretizationerrors, he is inclined to increase the number of equations and to decrease the step size in order tominimize the errors. Now, he is confronted with di$culty how to minimize the analysis cost inorder to produce an acceptably accurate solution.

One way to manage the above problems e$ciently is by adopting parallel processing algo-rithms using adaptive mesh re"nement and substructuring simulation approaches.

Recently, the requirements for development of computational algorithms of discrete streamlinetype were formulated, exploiting the parallel processing facilities of modern computers. One ofsuch algorithms is the combined "nite element versus transfer matrix approach (FETM-method)[1]; the numerical technique adopting dynamically variable regular and irregular mesh simulatedby moving elements. The discrete simulation of above problems is modelled during the motion oftypical element over the structure. Since a sti!ness formulation is adopted for the development ofsuch a typical element, the matrix inversion is required to eliminate the displacements of interior

TUNED VIBRATION CONTROL OF OVERHEAD LINE CONDUCTORS 1217


nodes before converting the governing equations for the periodic unit into transmission form. TheFETM-method has the streamline character suitable for parallel processing facilities adopted insuch cases.

The solution of non-linear ultimate response of overhead line conductors subjected to theaction of wind requires the consideration of a number of physical phenomena occurring ona macroscopic level. As most signi"cant there are to be mentioned the geometric imperfectionsand second order geometric e!ects, elastic}plastic material behaviour, local and total instabilitye!ects in critical and postcritical regions, non-linear sti!ness and damping parameters, etc. Allabove e!ects have to be taken into account in linear and non-linear general interactions possiblyappearing.

The solution of above problems below is concerned with the FETM simulation of overheadline conductors consisting of an assemblage of substructures in space and time. The mixedmultigrid schemes of discretization adopted in space and time allow the problem orientedvariability of substructure sizes and time steps in various space regions and time intervals of wavepropagation and ultimate dynamics of overhead line conductors. Mixed techniques for directtime integration of incremental equations of motion are used in combination with the FETM-method for computer simulation when adopting the substructuring approaches in space andtime [1].

Perfect overhead line conductors under ultimate behaviour, governed by a wave approachfor analysis of aeolic or galloping vibrations and provided with tuned vibration controljoints represented by special types of Stockbridge dampers or by facilities for variability ofaxial prestressing forces, as mentioned above, are optimal in the sense that their dampingor tuning capacity is generally larger than that of line conductors without facilites. Such casesoccur in ultimate dynamics of overhead line conductors with linear and non-linear resonancemodes acting in the interaction. The ultimate dynamic behaviour of such line conductors stronglydepends on the stability of the former equilibrium state.

Further, the overhead line conductors are often imperfection sensitive. The degree of imperfec-tion sensitivity, that is the di!erence between the bifurcation forcing of the respective perfectstructure and the limit state forcing of its imperfect realization, is essentially a function of theinitial slope of the perfect structure's postbifurcation path as well as of the magnitude and theshape of the imperfection mode appearing. The actual ultimate forcings are below their classicalcritical values even though imperfection levels may only be a fraction of the overhead lineconductor thickness assumed.

The objective of this paper is to demonstrate the application of some state of the art FETManalysis procedures for solution of some problems of ultimate dynamic analysis of overhead lineconductors and thus contribute to the body of knowledge available for suitable "nite modelling ofthe problems considered.

In this paper the following is presented below:

(a) the mathematical formulation of the problem of tuned vibration control of ultimatedynamics including incremental governing equations of motion for wave propagation ofoverhead line conductors.

(b) the brief description of the solution methodology adopted for numerical analysis ofultimate transient and wave dynamics of overhead line conductors studied,

(c) the applications of developed approach for analysis of actual problems of ultimate vibra-tions of overhead line conductors.



GENERALIZED ANALYSIS OF MOTION

The displacements of a thin "bre of overhead line conductor are considered as a family ofmappings from one region in space to another one. The current con"guration is completelyde"ned by locations of displacements in time. The variations of con"gurations are assumed to becontinuous and new boundaries will not arise during deformation. Each new position is de"nedin relation to a reference position assumed.

Assuming the Cartesian co-ordinates x, y, z and corresponding displacements u, v, w, the Greenstrain tensor is de"ned by

Exx"Lu

x/Lx#[(Lu

x/Lx)2#(Lu

y/Lx)2#(Lu

z/Lx)2]/2 (1)

Exy"[(Lu

y/Lx)#(Lu

x/Ly)#(Lu

x/Lx) (Lu

x/Ly)#(Lu

y/Lx)/(Lu

y/Ly)#(Lu

z/Lx) (Lu

z/Ly)]/2 (2)

etc.In order to establish the constitutive equations with Green strain tensor, a stress tensor with

the same reference is needed. A symmetric one will be advantageous in applications. The secondPiola}Kirchho! stress tensor denoted as S

ijhas the desired properties. The general equilibrium

equation for the deformed con"guration, expressed by second Piola}Kirchho! stress tensor, isgiven by

Sij"g(E

ij) (3)

where g is a single valued function of Green strain tensor Eij.

Considered is the overhead line conductor with volume, surface area and mass density in aninitial con"guration denoted by <, S and o

0, respectively. The body forces per unit mass are

denoted by F0,i

and surface tractions are speci"ed by force components ¹i.

The line conductor in equilibrium is subjected to a virtual displacement duiwhich is kinematic-

ally consistent with boundary conditions assumed. The balance of the work is given by

P SijdE

ijd<!P ¹

idu

idS!P P

idu

id<"0 (4)

with

Pi"o

0F0,i

(5)

Equation (4) states that among all kinematically admissible displacement "elds uithe actual one

renders the value of the total potential energy stationary.The incremental form of the variational principle is given by

P S (1)ij

dE (1)ij

d<!P ¹ (1)i

du(1)i

dS!P P (1)i

du (1)i

d<"0 (6)

P S(2)ij

dE (2)ij

d<!P ¹ (2)i

du(2)i

dS!P P(2)i

du(2)i

d<"0 (7)

where superscripts (1) and (2) denote the two con"gurations studied.



The components of surface tractions and body forces refer to the same reference con"gurationand may therefore be substracted directly to give

*¹i"¹ (2)

i!¹ (1)

i(8)

*Pi"P (2)

i!P (1)

i(9)

The variations of two displacement "elds are chosen to be the same

dui"du (1)

i"du (2)

i(10)

An incremental form of the virtual work equations is then obtained by subtracting Equations (6)and (7), giving

P (S (2)ij

dE (2)ij!S (1)

ijdE (1)

ij) d<!P *¹

idu

idS!P *P

idu

id<"0 (11)

and considering the virtual variations of both con"gurations analysed.The neglection of the higher-order strain energy terms can be done only if two con"gurations

studied are su$ciently close to each other. Equation (11) gives con"guration (2) from the knowncon"guration (1) and known load increments.

When the work done by inertial and damping forces over virtual displacements duiis added to

Equation (4), the virtual work principle for dynamic problems is given by

P Sij

dEijd<#P ou

idu

id<#P Cu

idu

id<!P ¹

idu

idS!P P

idu

id<"0 (12)

where o and C are mass and damping terms, respectively.

ULTIMATE DYNAMICS

The non-linearity in ultimate dynamics of overhead line conductors studied generally leads toa complex interactive resonance oscillation and dynamic stability behaviour that often is joinedby a snapping from one deformation mode to another one. The numerical treatment of suchprocess is done below using the parallel processing approach.

The motion is de"ned to be stable if deformations of the structure remain in the pre-ultimateregion without a transition to the post-ultimate range. The ultimate process there may beindicated by an increase of local or overall deformation modes. As proper measure for the state ofdeformation a norm of the deformation "eld is chosen. The procedure below is based ona comparison of the load-induced strain energy with critical strain energy that is necessary tocause ultimate behaviour of the structure studied, considering all modes of frequency spectrumappearing.

The rate equation of the energy balance is basis for investigation below, taking into account theinitial condition F

0"0 at time t"t

0. The symbols F, G and H denote time derivatives of total,



kinetic and potential energies of overhead line conductor system system studied, respectively. Itpays

F"G#H (13)

The geometrical non-linearities are taken into account as &moderate rotations'. The rate of totalenergy in ultimate range depends on displacements and slopes of the line conductor surface,velocities, integrated stress variables and forcing e!ects appearing. The ultimate motion is e!ectedby time-dependent loads and describes a trajectory in the phase plane of displacements andvelocities. The distance of the trajectory to the static equilibrium state is a measure of the energyinduced by initial conditions. The ultimate state of the motion is stated by a critical trajectorycalled separatrix. Such trajectory is the limit curve for all locally stable motions in the pre- andpost-ultimate range of response. For investigation of ultimate behaviour, the energy level *F

#3*5indicating the separatrix is of interest. Such energy level holds for all states of motion on theseparatrix. The criterion of stable motion is satis"ed if the state of motion at the end ofperturbation time is inside of separatrix since the energy level is lower than the critical one. Theintegration with respect to spatial co-ordinates is done by some reduced integration technique.This e!ects a smooth approximation of the solution without causing considerable oscillations.Equation (13) leads to a "nite description in the matrix formulation given by

F(z)"zT MMz#Cz#A(z0)z#A

N(0.5z) z!RN"0 (14)

with displacement vector

zT"[ux, u

y, u

z] (15)

with A(z0) as linear system matrix for initial state z

0and A

N(0.5z) as system matrix related to

geometrical non-linearities, with M and C as mass and damping matrices, respectively, and withR as vector of forcing acting.

For computing of critical energy level the stable equilibrium state is given by the minimum ofthe potential energy. A perturbation of this equilibrium state leads to a locally stable motion ofthe structure with the energy level *F. An increase from *F to *F

#3*5shifts the structure to

a saddle point which denotes the critical deformation state of the overhead line conductor studiedsince two locally stable motions in ultimate range meet here. A further increase of the energye!ects a motion in the whole ultimate range. The critical deformations are given by the distancefrom the state of stable equilibrium N to the saddle point S (see Figure 3). The decisive factor forthe stability of the equilibrium state N is the energy level that is given by the maximum of thestrain energy occurring. Hence, the "rst variation of the strain energy has to vanish in case ofcritical state

dG"0 (16)

Equation (14) leads to the system of non-linear equations that may be solved iteratively as a non-linear eigenvalue problem. The non-trivial solution describes the critical state *z(crit) and thestate of unstable equilibrium as well

dzTMAL(z)#A

N[0.5*z (crit)] *z (crit)N"0 (17)



Figure 3. Energy balance for line conductor in ultimate range.

The critical energy which may be used as criterion for ultimate state may be then calculated forthe state of stable equilibrium by integrating Equation (13).

TRANSIENT DYNAMICS

The solution of non-linear response in ultimate range of overhead line conductors subjected towind forcing is based on the application of updated Lagrangian formulation with reference statetaken as current con"guration which is continuously updated throughout the entire deformationprocess [1]. A new reference frame is established at each stage along the deformation pathstudied. A major advantage of the formulation is its simplicity which provides an easy physicalinterpretation of generalized non-linear behaviour of overhead line conductors studied.

An incremental form of the equations of motion for the case of transient dynamics of overheadline conductors is obtained by considering the dynamic equilibrium at two con"gurations a timestep *t apart. The increments of external loading then balance dynamic equilibrium at timet#*t as

Mt*a

t#C

t*v

t#K

t*u

t"R

t`*t!(< It#<D

t#<S

t) (18)

with inertia forces <It"M

tat, damping forces <D

t"C

tvt, elastic forces <S

t"K

tut

and withcorresponding accelerations, velocities and displacements a

t, v

tand u

t, respectively. The vectors of

nodal point accelerations and velocities are given as time derivatives of the vector of nodaldisplacements u

t. The mass, damping and sti!ness matrices M

t, C

tand K

t, respectively, are

constructed of element matrices established in incremental fashion directly for the simulationadopted. The subscript t denotes the current time and R is the vector of external loads. If thestructure is in equilibrium at time t, the right-hand side of Equation (18) there will be identicalwith the increment in external loads over the time step *t.

Increments in nodal displacements, velocities and accelerations are thus expressed by externalload increments and known physical property matrices. If however, these matrices changeduring the time steps, as is the case in ultimate response studied, then Equation (18) is only



approximately true. The vector of local approximation error given by

*<t`*t"R

t`*t!(< It`*t#<D

t`*t#<St`*t) (19)

is a measure of how close to equilibrium the solution has been increased by approximateexpression (18).

The governing incremental equation of motion for ultimate dynamics of line conductors isgiven in modi"ed form of Equation (18) by

Mt*a

t#C

t*v

t#P

t*u

t"*R

t(20)

where Pt*u

tis the vector of internal, deformation-dependent non-linear forces.

The pseudo-force method [1], as applied here, is de"ned by

Pt*u

t"K

t*u

t#N

t*u

t!*<

t`*t (21)

where Nt*u

tis the vector of non-linear terms (pseudo-forces) and *<

t`*t is the local approxima-tion error de"ned above. In the application of the pseudo force technique the term P

t*r

tis placed

on the right-hand side of Equation (20) and the vector of non-linear terms is treated as pseudo-force vector. At each time step an estimate of N

t*u

tis computed and iterations are performed

until *<t`*t becomes su$ciently small when compared to a prescribed tolerance norm. As an

estimate for Nt*u

tfor the "rst iteration at time step *t an extrapolated value from previous

solutions is used, e.g.

Nt*u

t"(1#a)N

t~*t *ut~*t!aN

t~2*t *ut~2*t (22)

where a is an extrapolation parameter ranging from 0 to 1.Because of the large computation e!ort required for non-linear ultimate analysis, it is desirable

to seek a strategy of optimal numerical calculations which may be de"ned in terms of a number ofcontrol parameters specifying the linearization techniques, the frequency of reformulation ofe!ective sti!ness matrix, convergence tolerances and limits on the maximum number of iterationsand adaptively change the time step size.

WAVE PROPAGATION

The simpli"ed wave equation for treatment of ultimate dynamics of a elastic medium of overheadline conductors is given by the vector equation [2]

kg(ut)#(j#k) grad (divu

t)#f"oL2u

t/Lt2 (23)

with j and k as Lame's constants, o as mass density, with Laplace operator g, with body forcevector f and with displacement vector u

t.

In terms of derivatives of the displacement components ut, the governing equation is given by

c2ut#(c2

1!c2

2)u

t#f

i/o"a

t(24)



with propagation velocities for dilatational waves

c1"J(j#2k)/o (25)

and distortional or shear waves

c2"Jk/o (26)

Strain and stress components are de"ned by

eij"0.5 (u

ij#v

ji) (27)

and

pij"je

kkdij#2k e

ij, i, j"1, 2, 3 (28)

where di,j

is the Kronecker delta function.Neglecting the body forces and using the divergence of each term of Equation (24), the

uncoupled scalar wave equation for dilatational waves is obtained, given by

g(/)"//c21

(29)

where

/"div (ut) (30)

Vector equation for distortional waves is given by

g(t)"t/c22

(31)

where

t"0.5 [rot (ut)] (32)

Numerical parallel processing FETM-approach was veri"ed in Reference [3] and is appliedbelow for further treatment of above governing equations for ultimate transient and wavedynamics of overhead line conductors.

FETM METHOD

The actual geometry of overhead line conductor con"guration studied is simulated over a multi-grid space mesh of micro and macro elements as developed in Reference [1]. The discrete modelassumed allows the simulation of general anisotropy of structural and material parameters asoccurring in large deformation and elastic}plastic regions of non-linear ultimate analysis ofoverhead line conductors. Various types of non-linearities are dealt with using various systemsof discrete simulation adopted. In the updated Lagrangian formulation of motion the major



rigid-body geometric non-linearities are embodied into co-ordinate transformations of themicroelement mesh used. The e!ects of physical non-linearities (non-linear damping, elas-tic}plastic material behaviour) are analysed on the level of macro element simulation mesh. Bothsystems are interactively coupled and may be varied and combined. The proposed multigridsimulation mesh of the FETM method uses the dynamically variable size of micro- andmacroelements in various regions of overhead line conductors studied and in various time andforcing steps of non-linear ultimate analysis performed.

Generalized transfer hypermatrices of the FETM-method (1) when applied for the analysis ofproposed multigrid simulation are constructed by diagonal set up of transfer matrices of theelements adopted. The transfer matrices of macroelements consist of diagonal assembly of transfersubmatrices for microelements used. Transfer submatrices are derived over inverse transformationsof linear and non-linear sti!ness matrices of microelements assumed. The details of vectorprocessing FETM approach adopted below are summed up, for example, in Reference [1, 3 or 4].

ADAPTIVE MESH REFINEMENT

The adoption of a suitable re"nement strategy for numerical analysis of ultimate response ofoverhead line conductors is required. Such strategy takes into account both important aspects,e$ciency and accuracy of computations made. Error indicators based on the weighted stressgradients yield good e$ciency for singularity type solutions. An alternative indicator controls thelinearization assumptions of the non-linear formulation which is especially important for "niterotations occurring. The output from the error estimation is a scalar measure of the error withineach of the elements. This information is used in advance to adapt the mesh in order to reduce theglobal error possibly appearing. In a posteriori error estimation in non-linear ultimate responsecomputations the principal mechanisms for control of the adaptive process are path control(selection of the step length and of the local parameters or of the corrector adjustment) andapproximation control (modi"cation of the mesh and element type).

The mesh may be modi"ed using so called h-methods and/or p-methods. When h-methods areemployed, the error information is converted into mesh re"nement indicators giving the charac-teristic size of elements in the new mesh as a function of space. Alternatively, the mesh can bere"ned by means of p-methods where the polynomial order p of the elements are increased andthe element size is "xed. The mesh can be rede"ned in di!erent ways after the desired mesh sizeparameters have been found. A totally new mesh is generated in such a way that new elements arede"ned by successive subdivision of old elements. In this case, extra nodal points are inserted intothe old mesh and all nodes of the old mesh are also present in the new mesh adopted.

From computational point of view it is necessary to organize the updated structural data inultimate region e$ciently, in order to compute the best "tting meshes automatically for everyimportant state of the non-linear ultimate analysis performed [4]. The volume of the data set adoptedmust be #exible enough to establish suitable element meshes and to modify them e$ciently.

MAPPING OF STATE VARIABLES

After the simulation model has been rede"ned according to a given error distribution at loadstep i, the values of state variables are to be mapped from the old mesh into the new one before the



FETM solution procedure can be continued. The mapping is performed at the previous load step(i!1) which is the latest state with acceptable errors and where the internal stresses are inequilibrium with external loads and forcings acting. This transformation process is referred to asrezoning and involve the steps given below:

1. The state variables in the old mesh are de"ned in terms of discrete variables located at thenodes. Smoothing is performed, if necessary, to obtain a continuous "eld.

2. The values of state variables are computed at location of the nodal points of the new FETMmesh adopted. Generally, this step involve a searching procedure and a parametric inversionwhere an element of the old mesh that matches the location of a certain node of the newmesh is to be speci"ed.

3. The distribution of the state variables is then de"ned within each new element usingcorresponding shape functions and necessary integration point data.

4. After the solution has been mapped into the new mesh, the equilibrium between internalforces and external loads may be violated. Additional equilibrium iterations may thereforebe necessary before continuing with the next load step.

The parametric inversion required for overhead lines when provided, for example, withStockbridge dampers for tuned vibration control, as mentioned earlier, may there be performedas a minimization of the distance between the new node and a parametric point on the elementsurface, using a second-order Newton iteration procedure.

FREQUENCY RESPONSE MEASUREMENTS

The basic test set-up required for performance of frequency response measurements of overheadline conductors studied, consists of mechanical vibration exciter, transducers for the measure-ment of dynamic inertial force and parameters of vibrating motion, computer, which is theexcitation source for driving signals of the exciter and analyser for data acquisition and signalprocessing operations and also digital processing oscilloscope. The system contains also powerampli"er with integrator and operating electronics of the exciter.

Present facilities were adopted for the frequency response measurements in rami"cation of thedevelopment of the Stockbridge damper ELBA, a.s., as shown in Figure 2 (see References [5] and [6]).

NUMERICAL EXPERIMENTS WITH DISCUSSION

The time response of the overhead line conductor having the length 100 m and the diameter60 mm is studied and it is subjected to the action of single transversal midspan impulse forcingwith constant amplitude 100 kN in interval 0}1.0 s (see Figure 4).

Tuned vibration control adopting the variability of prestressing axial forces in overhead lineconductor studied is treated numerically below.

The in#uence of the variability of prestressing axial force, assumed by values 0.1, 0.5 and1.0 MN, on time response has been investigated in order to gain the information on tunedvibration control of overhead line conductor studied.

The time response of #exural and axial midspan displacements (in m), velocities (in m/s) andaccelerations (in m/s2) for each of assumed values of prestressing axial forces is plotted in



Figure 4. Overhead line conductor studied.

Figure 5. Time response of rope*#exural displacement 1}5.

Figures 5}28. Figures 5}12 pay attention to above components for the value of axial prestressingforce 0.1 MN, Figures 12}20 pay attention to the value of axial prestressing force 0.5 MN and theFigures 21}28 pay attention to the value of prestressing axial force 1.0 MN. Above "guresillustrate the values of the components along the length of overhead line conductor studied in timeinterval 0}10 s, with assumed time step 1.0 s. In all above "gures the lines 1}5 specify the resultsappearing in times 1}5 s and the lines 6}10 specify the corresponsing results in times 6}10 s.




Figure 7. Time response of rope*#exural velocities 1}5.

The comparison of amplitudes calculated for assumed values of axial prestressing forcessubmitted in Figures 29}32 illustrate the in#uence of axial prestressing as one of e$cient toolsdistinctly in#uencing the tuned vibration control of overhead line conductors.




Figure 9. Time response of rope*#exural accelerations 1}5.




Figure 11. Time response of rope*axial displacement 1}5.





Figure 29. Flexural displacement midspan of overhead line conductor studied for three values of axialprestressing forces assumed: (A) 0.1 MN; (B) 0.5 MN; and (C) 1.0 MN.



Figure 30. Flexural velocities midspan of overhead line conductor studied for three values of axialprestressing forces assumed: (A) 0.1 MN; (B) 0.5 MN; and (C) 1.0 MN.

Figure 31. Flexural accelerations midpsan of overhead line conductor studied for three values of axialprestressing forces assumed: (A) 0.1 MN; (B) 0.5 MN; and (C) 1.0 MN.



Figure 32. Axial displacements midspan of overhead line conductor studied for three values of axialprestressing forces assumed: (A) 0.1 MN; (B) 0.5 MN; and (C) 1.0 MN.

CONCLUSIONS

The results obtained con"rm the e$ciency of the tuned vibration control system developed fortuning of ultimate aeolic and galloping vibrations of overhead line conductors. The principle isbased on the adoption of additional prestressing forces axially acting in the cables of the system ofoverhead line conductor.

Present theoretical approach allows the analysis of ultimate dynamic behaviour of overheadline conductors under consideration of non-linearities appearing. Numerical solutions presentedsubmit the image on linear and non-linear response in ultimate dynamic range of overhead lineconductors studied as well as on some possibilities of their e$cient tuned vibration control.

REFERENCES

1. Tesar A, Fillo L. ¹ransfer Matrix Method. Kluwer Academic Publishers: Dordrecht, Boston, London, 1988.2. Elmer KH. Dynamic analysis and high speed simulations with a fast vectorized algorithm. Structural Dynamics,

Proceedings of the European Conference on Structural Dynamics, E;ROD>N190, Bochum, 1990; 953}960.3. Tesar A, Svolik J. Wave distribution in "bre members subjected to kinematic forcing. Communications in Numerical

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tuned vibration control of overhead line conductors

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