THE VIBRATIONS OF TRANSMISSIONLINE CONDUCTOR BUNDLES

R. Claren G. Diana - F. Giordana - E. MassaSalvi S.p.A. Polytechnic University of Milan

Abstract - This paper deals with the response of bundles of severalcables subjected to harmonic exciting forces. It shows how the spacercharacteristics expressed by the spacer elastic matrix will impose par-ticular types of natural modes and how excessive spacer stiffness willcause severe bending strains to occur on the cable close to the spacerclamps.

The paper will analyze the behaviour of spacer-dampers and showshow their characteristics can be optimized.

Basical Analytical Method

Let us consider a span of "a" taut cables which are connected byn-l spacers to form n subspans each having a different length li.

The physical conditions existing in a generical cross section of acable at a distance x along a subspan i are assessed by the displacementy, the rotation 0, the moment M and the resultant of the forces Q actingin that cross section.

In order to assess, for each subconductor, the direction alongwhich above parameters will be expressed, we must refer them to twoorthogonal reference planes for each subconductor. If each subspan isassessed by a subscript i (i = 1, 2, . n), each cable by an apexa (a = 1, 2, . a) and each reference plane by a subscript s (s = 1, 2,... 2a), then yjx (x) will express the displacement of subconductor a, atdistance x on subspan i along the reference plane s. The same notationswill be used for the other three parameters, k41 (x), Mg (x), Q9 (x).

It is therefore evident that eight parameters will be needed toassess the physical conditions of any cross section of a subconductorand 8a parameters for any cross section of the bundle of "a" subcon-ductors.

If we use the conventional matrix terminology, [ , a state vectorZ- (x) the components of which are the 8a parameters y4' (x), Mt (x),A5 (x), and Q9s (x) will assess the physical conditions existing in agenerical cross section x of a subspan i of the bundle.

With matrix notation the equation

Zi (X) = Bi (x) I Zi (o)(1,shows how it is possible to obtain all the parameters of a genericalcross section x of subspan i, from their values for x = 0, by means of afield transfer matrix [ Bi (x)] .

In order to transfer the state vector Zi (1) from the end of subspani, where x = li, to the beginning of the next one (i + 1), where x = 0,over the spacer connecting point, a point transfer matrix [P] has to beused:

Zi+I (0)= [PI Zi (i) (2)Such a point matrix will depend only on the spacer character-

istics and, if the same spacers are used on the whole span, it is possible*to assess the following product matrix:

D= [Bn(ln) ]-[P] [Bnl (in l) ] .... [PI [B1 (11) 1 (3)

Paper 7lTP 158-PWR, recommended and approved by the Transmission andDistribution Committee of the IEEE Power Engineering Society for presentationat the IEEE Winter Power Meeting, New York, N.Y., January 31-February 5, 1971.Manuscript submitted February 16, 1970; made available for printing December 22,1970.

and therefore

Zn (ln) = [DI Z (o) (4)If the two span extremities are clamped

yas(0) =YS (ln) = 00a(0) = y (ln) = 0

and if they are pivoted

ya (0) = Y (ln) = 0Ma (O) = Ma (In) = 0I.s ns(n0

(5)

(6)

It therefore follows that for clamped or pivoted span terminationsthe expression (4) will give a homogeneous linear system of 4a equa-tions with 4a variables.

The field transfer matrix [Bi (x)] is obtained from the equationsof motion of a vibrating cable [2] and therefore contains the frequencyterm w.

The resonance frequencies 'or of the bundle are therefore thosethat will equate to zero the determinant of.the above mentioned system'of 4a equations, which is called the frequency determinant.

After having obtained all the resonance frequencies wr it is possibleto assess the relative magnitudes of the components of the state vectorZI (0) and by means of the transfer matrixes [Bi (x)] and of the pointmatrix [PI those of the generical state vector Zi (x), that is the deforma-tion and its related parameters of the bundle system in any locationalong the span.

Although the above exposed matrix method can be easily expressedin Fortran for use in a digital computer, a number of simplificationsand modifications are needed to make it suitable for most of the usualconditions found on normal transmission lines.

A detailed description of the various difficulties which the authors,had to face and to solve in order to obtain a reliable and suitable com-putation method, is exposed in [3] [4] [ 5] .

It can be very briefly hereby stated that no sensible error can be,made in the computations if the cables flexural stiffness EJ is ignoredall along the span except, obviously, close to the subspan extremities.Furthermore it was also found that the spacers flexural stiffness hadvery little influence on the bundle resonance frequencies and naturalmodes deformation. The possibility of ignoring these parameters forat least a large portion of the span brought a considerable reduction ofcomputer time and of the number of digits required.

The experimental tests which were performed to check the accur-acy of the computation results, [5] [6], showed an error of about 10%which the authors believe to be essentially due to the difficulty of re-producing during the tests the exact physical conditions assumed forthe computations.

In the following paper it will always be assumed that the sub-conductors of the bundle will be identical, that is that they will haveidentical mass m, flexural stiffness EJ, and will be subjected to thesame tensile load S.

Although the basical analytical method will still be valid in thecase of different cables, some of the simplifications used may no morebe valid in such case. In practice however bundles are always composed

1796

of cables having practically the same physical characteristics, and theusual slight differences in tensile load are too small to bear a consider-able influence in the results.

Bundles of Two Conductors - Principal Modes

Let us assume a span of two taut cables, placed in a vertical con-,figuration (fig. 1). A number of flexible spacers connectthe two cablesto form a number of subspans of length li.

II 12 ,IY inA PI

Fig. 1. Twin Vertical Bundle.

The analysis of this system can be greatly simplified in respect tothe basical method previously exposed. The subconductors are practi-cally uncoupled if the oscillations occur in a direction perpendicular tothe plane containing the two subconductors and the spacers, that is inthe horizontal plane. For a unit displacement of a subconductor in thisdirection the reaction force at the spacer is essentially governed by therotational stiffness of the other subconductor which, considering theusual subspan length, is practically negligible.

If, for simplicity sake, we assume the spacer to have no weight(O),the resonance frequencies and vibration modes of the bundle, vibratingin a horizontal direction, coincide with those of a single taut cable [2].

If we consider instead the oscillations occurring in a vertical plane,then the coupling due to the spacers is of basical importance.

If we call Kz the spacer longitudinal stiffness, that is the force re-quired to cause a unit elastic elongation of the spacer and K6 the spacerflexural stiffness, that is the torque required to cause a unit rotation ofone spacer extremity, with the other rigidly clamped, the spacer pointmatrix [PI has the form shown in table I.

1

0

0

kz0

0

0

0

1

kg0

0

0

ke/20

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

kz

0

0

ko/20

0

1

o

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

Table I - Point Transfer Matrix.

1 0 0 0

kz I - k, O.~ ~ ~ 0

0 0 1 0l

.~~~~~~~~~~~~~~~~~~

T II-o kz

Table II -Point Transfer Matrix.

1

(o) The effect of spacer wieght will be discussed later.

If we ignore the cable flexural stiffness and, as a consequence, thespacer flexural stiffness Ko, the spacer point matrix [PI has the formshown in table II.

The frequency determinant has thus eight rows and columns in thefirst case, four rows and columns in the second case, and the resonantfrequencies and cable deformation can be easily obtained as previouslyexposed.

In order to illustrate the results of such computations, fig. 2 showsthe resonance frequencies and deformations of the first 24 vibrationmodes of a span composed of two ACSR conductors having a diameterof 31.5mm, a spacing of 0.4 meters, divided in three subspans havingrespectively a length of 15, 16, 15.52 meters and m = 0.202 kgm s2, S = 4750 kg, EJ = 200 kg m2, Kz = 3000 kg m-1, Ko = 37 kg mrad1.

In fig. 2 the abscissae are proportional to the lengths, whilst on theordinates the displacements of the cables are relative to the same centerline. In other words in the figure the cable spacing has not been con-sidered and both cables center lines made to coincide. The two spacerslocations are shown by the two vertical lines dividing the span in threesubspans.

As it can be seen in fig. 2 the cable deformation can, in corres-pondence to the spacers, be subjected to a considerable distortion as aconsequence of the forces applied to the cables by the spacers.

There are also vibration modes where both cable deformationsare identical and are therefore shown by a single line. In this case thesystem behaves as a single cable.

For these vibration modes the bending of the cables at the spacerlocation is practically equal to the bending that would occur withoutspacers on a single cable and therefore it will be much smaller than thebending that occurs at the rigidly clamped span extremities [7].

For all the other vibration modes, the bending values at the spacerlocation can reach the value occurring at the rigidly clamped span ex-tremities if the spacer longitudinal stiffness is not low enough, as it willbe thoroughly discussed further on.

w=10.45 20.30 .27.09 w.30.44

31.38 31.74 41.86 X 52.37

54.67 61.04 w 62.93 c 63.64

ev=73'.49 S =83.10 = 84.10 w 91.94

co..94.80 = 95.83 w 105.50 w = 112.58

116.26 . 123.31 127.15 x 128.45

Fig. 2. Twin Vertical Bundle Principal Modes.1797

z

These last vibration modes characterized by antiphase displace-ments of the subconductors are typical of bundle systems coupled byspacers.

it is worth pointing to the fact that these modes show antinodevibration amplitudes which can be quite different in one subspan fromthose found in another subspan. A detailed discussion on this fact willbe done further on. It is however clear that the deformation of a sub-conductor over the whole span cannot be expressed by a relativelysimple analytical expression as done for a single conductor, but thesubconductor deformation has to be expressed for each of the subspans.

It is also worth mentioning that the behaviour of such a bundlewill not change if the conductors are placed in a horizontal plane. Withsuch a bundle, if the spacer weight is ignored, vertical vibrations willhave only natural frequencies and modes identical to those of a singlesubconductor, whilst horizontal oscillations will have such single con-ductor natural modes and frequencies plus those "typical of the bundle"

Bundles of Three Conductors - Principal Modes

Let us consider now a span of three taut cables. A number offlexible spacers connect the three cables to form a number of subspansof length 11, 12, . * * li, * * *.n

For each subconductor, if its stiffnes is ignored, the parameterswhich assess the oscillations, inS a plane x - y passing at a generical crosssection x of a subspan, are the displacement y (x) and the componentQy (x) (in the direction y) of the forces acting in the cross section x.

Considering the whole bundle of subconductors and with refer-ence to fig. 3, it is necessary to assume, for each subconductor, twoorthogonal reference planes to which will be related the components ofthe above parameters which will be shown as Ys (x) and Qs (x).

The index s, in accordance with fig. 3, will be 1 and 2 for the firstsubconductor, 3 and 4 for the second, 5 and 6 for the third.

The equation (1) will assess a field transfer matrix of the bundleBi (x), for the subspan i, which is related to the various transfer

26a r

a) b)Fig. 3. Three-bundle Hinged Type Spacer.

matrixes Ai (x) of each subconductor by the expression shown in'table III.

The matrix Ai (x) can easily be obtained from the equations ofmotion of a vibrating cable and is:

cos-yxAi (x) =

_-S'ysinsinyx/S'y 1cosyx Jx

(7)

where y = /is, m is the cable mass per unit length and S the tensileload.

The point matrix shown in equation (2) will depend only on thespacer characteristics, and more precisely on the spacer flexibilitymatrix.

The linear flexibility characteristics of a three-bundle spacer canin fact be expressed by means of a flexibility matrix K of the sixthorder expressed by:

F = [K] Y (8)where Y is a state vector whose components ys (s = 1 . . . 6) are thedisplacements of the three clamps along the six reference directions(fig. 3) and F is a state vector whose components fs (s =1 . 6) arethe forces which are developed at the clamps in the direction of s as aconsequence of the displacement ys. The forces are considered positive(+) if their direction is opposite to the direction of the displacements.

If krs is a generical component of the matrix [K] it will representthe force fr developed as a consequence of a unit displacement ys, allother displacements being zero.

In accordance with the theory of linear flexible systems krs = ksr.

0.871 -1.4i8 -0.016 0.044 0.855 -1.393-1.438 2.686 -0.044 0.119 -1.482 2.805

-0.016 -0.044 0.871 1.437 0.855 1.394

0.044 0.119 1.438 2.686 1.482 2.8050.855 -1.48 0.855 1.481 1.711 0

-1.394 2.805 1.394 2.805 0 5.611

Table IV - Flexibility Matrix of Spacer Fig. 3.

Ai (x): 0 0 0 0 0

o :A(x):0 :0 0 0

o 0 Ai (x): 0 0 0................ ............

O :A0 0 0Ai(x): 0: 0

0: 0 0 0::A(x): 0

0 : 0 : : () 0 : A, (x)

Table III - Field Transfer Matrix.

1kl,0

k210ks,

0

0

0k61Ok,s,

0

001

0000000

0

0 0

k12 01 0

k22 I

o 0 0 0 0 0 0 0

k,3 0 kls 0 kl5 0 k1, 0o 0 0 0 0 0 0 0

k23 0 k24 0 /c25 0 k2, 0o o 1 0 0 0 0 0 0 0

k32 0 k33 t k34 0 k35 0 k36 0o o 0 0 1 0 0 0 0 0

k42 0 k43 0 k44 t k5 0 k46 0o 0 0 0 0 0 1 0 0 0

c51 0 kS3 0 kAU 0 k55 I ka 0o 0 0 0 0 0 0 0 1 0k*, 0 k63 0 k44 0 ks 0 kX 1

Table V - Point Matrix.

1798

[Bi (x)]=

Ty=

Type I

Planes f >OFf L -7 _ I

oscillations

Y2

0.5

0.866

0

if

l

I--10

- 1

lII 91 5l

I

L- T-

- 1

0

7

JL-T-J

I1

-1.969

0.179

El

L-T- _i_L

..-I L

I~~I

1

_.1.732I

0.169y3 0.5 0 - 1 -1 1 1Y4 I-0.866 - 1 0 -1.969 1.732 0.169Y5 | 1 0 1 0 2 0

Y6 0 I 0 -3.939 0 0.338Table VI - Oscillation Types of a Bundle with Spacer as per Fig. 3.

The flexibility matrix of the spacer shown in fig. 3 with a torsionalstiffness of hinges of 15 kg m rad-1 is given in table IV.

The point matrix [PI can easily be obtained from the flexibilitymatrix [K] if it is assumed that the displacement is the same at bothsides of the spacer clamp:

Yi+1is (o) = Yi,s Oiand that the forces are:

(9)

authors "characteristics of the conductor". All the other natural modeswhich belonged to the other "types" numbered IV, V, VI in table VI,did cause spacer clamp relative motion, had their own particular reso-nant frequencies and had subconductor deformations similar to those

'3

(10)

For a three-bundle spacer the point matrix has the aspect shown intable V.

As the field transfer matrix and the point matrix have beenassessed, it is possible to compute the resonant frequencies and thenatural modes of the system. A detailed evaluation of a great numberof natural modes of three bundle systems which were obtained with thecomputing method previously exposed, showed that they could beclassified into six major groups or "types" of oscillations.

For a bundle equipped with the spacer shown in fig. 3 the oscilla-tion "types" are exposed in table I. All the oscillation modes belongingto a particular "type" have in common the fact that the planes inwhich each subconductor oscillates does not change with changingfrequency and, furthermore, the relative amplitude of oscillations of thesubconductors do not change either with changing frequency. With re-ference to table VI and "type" IV, and remembering the referenceplanes of fig. 3, it can be seen that if we assume the amplitude of oscil-lation along the reference plane 1 to have a unit value (y1 = 1), thenthe amplitudes of oscillation along the other reference planes will havethe'values shown in the column belonging to "type" IV.

It was furthermore found that all the natural modes, belonging tothree of the six "types", showed no relative spacer clamp motion and,as a consequence, their natural frequencies and conductor deformationwere identical to those found on a single taut cable. These three "types"which are numbered I, II and III in table VI, have been called by the

Fig. 4 Four-bundle Hinged Type Spacer.

1.4700.6030.6680.3010.9700.6031.1680.301

0.6030.4590.3010.1570.6030.3420.3010.040

0.6680.3010.1470.6031.1680.3010.9700.603

0.3010.1570.6030.4590.3010.0400.6030.342

0.9700.6031.1680.3011.4700.6030.6680.301

0.6030.3420.3010.0400.6030.4590.3010.157

1.1680.3010.9700.6030.6680.3011.4700.603

0.3010.0400.6030.3420.3010.1570.6030.459

Table VII - Flexibility Matrix of Spacer Fig. 4.1799

9.124 5. 132

Qj+ I'S (0) = Qj,S (I j) + fs

I 1 0t.l| +1 1 1|1 1 1 1 0 +1Y| - 1 0 0.442 | 1 0 +1 -2.24Y3 -1 0 .1 1 _1 -1 0 +1

Y |.1 1 0 0.442 | 1 0 .1 -2.24Y5 +. 0 _ 1 .1 _1 0 +1

Y6 -1 1 0 0.442 +1 0 -1 -2.24Y7 _1 0 -1 1 _ +1 0 + 1

.1 0 0.442 1 0 -1 -2.24

Table VIII - Oscillation Types of a Bundle with Spacer as per Fig. 4.

corresponding to the antiphase oscillations found on the two bundlesystem 'and exposed in fig. 2. These last "types" of oscillations werecalled by the authors "typical of the bundle".

Bundles of Four Conductors - Principal Modes

-The computing methods exposed in the previous paragraphs weredirectly extended to bundles of four conductors. It was thus found thatfour-bundle systems had eight oscillation "types", three of them being"characteristics of the conductor", and five "typical of the bundle".Figure 4 shows one of the four-bundle spacers considered, table VIIits flexibility matrix, for a torsional stiffness of hinges of 15 kg mrad-1, and table VIII illustrates the eight oscillation "types" whichwere found.

Flexibility Matrix Eigenvalues and Spacer Stiffness

A complete analytical investigation of the natural frequencies,principal modes and "oscillation types" of bundle systems, exposed indetail in [41 and [5], showed that the principal modes were directlyrelated to the eigenvalues of the spacer flexibility matrix, and moreprecisely each eigenvalue did assess a particular "type" of oscillation.Tables VI and VIII show for each "type" of oscillation the corres-ponding eigenvalue X. This mathematical fact can be visualized inphysical terms in the following way.

Let us consider the spacer shown in fig. 3, which is characterizedby having a linear flexibility characteristic expressed by the flexibilitymatrix of table IV. On the basis of the previous assumptions, suchspacer has no weight.

Let us connect now to each of the spacer clamps a weight. The,three weights are identical and have each a mass of value "m".

If we write down the equations of motion of this system which hassix degrees of freedom, we will obtain a homogeneous linear system ofsix equations with six variables.

If we consider now the determinant of such a system, we willnotice that it will be identical to the flexibility matrix exposed intable IV, except that the diagonal terms will also contain the term

-o 2m due to the inertia forces of the applied masses. To be clearer,in the determinant of the system the first term in the first row will be0.871 -c2m, the second term on the second row will be 2.686 -w2mand so on.

The resonant frequencies of the spacer with masses m, will ob-viously be those which will equate to zero the determinant.

If c

computed by considering one single taut cable connected to the groundby means of a number of springs having a stiffness equal to the aboveeigenvalues X and located exactly where the spacer had been placed inthe original system.

The true definition of a spacer stiffness is probably the mostimportant result of the research work performed as such a parameter,together with the possible natural modes, is directly related to theresponse of bundles to any type of exciting force and to the dynamicstrains which will be developed on the subconductors.

In a general form it can be stated that any bundle of "a" cablesconnected by spacers will have 2a oscillation types and 2aoo oscillationmodes. Three "types", and precisely those "characteristic of the con-ductor", will correspond to eigenvalues X = 0, whilst the other2m - 3 will have a value X * 0. In most cases there will be 2m - 3different eigenvalues, but some spacers having a particular symmetricalflexibility matrix might yield multiple eigenvalues. In such a case theoscillation "type" can be assessed only with an arbitrary choice of somevariables.

It is worth mentioning that the validity of what has been exposedis subject to the assumption that all the cables of the bundle have thesame physical characteristics and are subjected to the same tensile load.As small variations of the tensile load do not modify considerably theresults, it can be stated that what has been exposed is valid for the vastmajority of bundled transmission line conductors.

Bundle Damping and Response

In the analysis of the natural modes of bundle systems it was as-sumed that no damping forces were acting in the system.

In order to compute the absolute values of the deformation ofthe bundles, when forced to oscillate by a known harmonic force, it isnecessary to introduce now all the damping forces which might bepresent.

In the following it will be assumed that there are only two typesof damping forces, those due to the internal damping of the cables andthose due to the internal damping of the spacers.

The analysis will follow the same concepts exposed in (2) and isbased on some assumptions which have been justified by the experi-mental tests.

The procedure is herewith summarized:a) After having found, as shown previously, the natural fre-

quencies and the bundle deformation, the kinetic and potential energieswill be expressed by means of principal coordinates, the orthogonal pro-perties of which have already been discussed [2]. As explained pre-viously, the cable stiffness will be ignored;

b) The cables' and spacers' internal damping will be assumed to beof hysteretic type [2];

c) The damping function of the system will be also expressed bymeans of principal coordinates, the coupling terms ignored [2] andthe orthogonality property of the principal modes of the undampedsystem extended to the damped one;

d) The Lagrange equations will be then used to obtain the sys-tem response. As a consequence of the orthogonality assumptions, theequations of motion expressed in principal coordinates will not becoupled if the exciting force is a function of time only and not of thebundle deformation.

Principal Coordinates

Let us assume a bundle composed of "a" subconductors, num-bered 1, 2, ... a . a, connected by n-l spacers. The deformation ofeach subconductor will be expressed by two quantities yaf (x, t) andZa (x, t) which assess the displacement along two orthogonal directionsy and z of the conductor at point x along the span and at time t.

The quantities yx (x, t) and Za, (x, t) can be expressed by meansof a linear combination of the cable deflections which occur at the

various principal modes r, the shape of which is given by the spacefunctions Oyur (x) and :ozar (x) and the intensity of which is given bythe time function Pr (x) which is the principal coordinate of the mode r:

00

yct (x, t) = E, r Pr (t)C.0

zct (xI t) = E, r Pr (t)

4? yaj(x)(1 1)

4 Z(X)

The space functions 0 (x) have been- discussed previously and ithas been seen that the total cable deformation can be found by meansof the cable deformation of each of the cable subspans. The variable x-of the 0 (x) has therefore to be referred to each subspan, being 0 atone subspan extremity, li at the other subspan extremity and generical-ly xi along the subspan i. The space functions will therefore have to beexpressed for each subspan and therefore they will be shown as4yar (xi) and kzaxr (xi), where i is the subspan number (i = 1, 2, . . . n).

It has also been explained that the cable stiffness has little in-fluence on the computations of the natural frequencies, of the oscilla-tion types, and on the cable deformation except at some distance fromthe spacer clamps. It has also been explained and shown in [31 and[S how the strains occurring at the cable, close to the spacer clamps,can be computed with sufficient accuracy with simplified methods.

If the cable stiffness is ignored, the 4ycxr (xi) and Pzatr (xi) can beexpressed by harmonic functions of xi:

4?yar (xi) = Ayarsinyrxi+Byarcos'yrxi(12)

?z;tr (xi) = Azarsinyrxi+Bzarcosyrxiwhere yr = wrV'T and cr is the natural frequency of mode r of thesystem, m the mass for unit length of the cable and S the tensile load.

It has been explained that the actual planes along which the oscilla-tions of each subconductor will occur, are assessed by the spacer flex-ibility matrix and the vibration mode. For a given type of spacer there-fore:

(13)4'zcT (xi) / 4?yar (xi) = Caorand therefore the (12) becomes:

4yar (Xi) = Ayarsin Yrxi+ By&-cos'rxi(14)

4?zar (xi) = Cur 4?yar (Xi)The vibration amplitude at point xi of subconductor can therefore

be expressed by:

ua1 (xi,t) = E r(yor (Xi)Pr(t)0 1+Ca (15)Kinetic and Potential Energies

The kinetic energy of cable oa, at subspan i, at mode r, is:

Tairy rPr(t)2 4yar (xi) m (l+C2ar) dx0 iand if we introduce the first of the (12):

(16)

Tair 'yPr(t)m(l+C2ar) [A2yaxr k-47I7 sin2yrli)+1801

2+ B yoer (2+ 47sin 2-1,rli )+

+2 A B sin2 y (17)lr yar yar ~ ri

The kinetic energy of the whole span of subconductor a in themode r will be:

n

Tar = i Tair (18)

and the kinetic energy of the whole bundle of subconductors, if weignore the masses of the spacer clamps, will be:

a

Tr =Ea Tar (1 9)

The maximum value of the potential energy of the whole bundlein the mode r is equal to the maximum value of the kinetic energy inthe same mode, and therefore it is possible to express the potentialenergy as follows:

vr 2 Pr (t) rm2 r (20)where:

* 2Tr

P2r(t)and also

2Vrk r=

P r(t)(21)

*k

m rM

r

The quantities m* and k* will be better understood when ther rLagrange equations will be considered.

The Damping Function

On the basis of the assumptions b) and c) exposed in the intro-duction to this chapter and assuming a harmonic motion of frequencyQ2, the damping function D(q) of the subconductor, at subspan i andair bodco,a usa nmode r, can be expressed by

D( = 2- - P (t) ( + C2)2 52rCIA2ar 2 4 in 2yr1i) +

,yr

2 1 1+Byar( + -sin 2,yrIi) +

the conductor, we can assume hir = hr, that is the same coefficient forany subspan. Obviously, it is always presumed that the cables of thebundle are all identical. Furthermore we can also assume hr = HX-3

r rwhere H is a parameter of the cable which will depend on its size,geometry, on the cable tensile load [7] and X the wavelengths. Suchassumption can be easily accepted and proved when the aeolian range ofvibrations, 5 + 50 Hz, is considered. In the case of low frequency oscil-lations, usually called "subspan oscillations", the validity of suchassumptions must be more carefully weighed, taking into considerationthe deflection which will really occur at the subspan terminationswhich, as seen previously, will depend on the spacer flexibility. Any-how, the values of hr or hir or H can be experimentally found orevaluated from experience. It must be also considered that the con-ductor damping is only one part of the total damping of the systemwhen spacer-dampers or spacers and dampers are used and thereforethe computation errors due to a non correct assumption of the con-ductor damping may be quite acceptable from a practical engineringpoint of view.

The damping function due to the conductors of the whole bundlefor the mode r will be:

D4=Ya En aiD(c) = E a E- i D(c)

1 1(23)

In order to assess the damping function of the spacers, it is nownecessary to express the damping forces which will be developed at thespacer clamps as a consequence of their motion.

Let us call F(d). and F(d) the components of such forces along theyai zaireference planes y and z at the clamp which connects spacer i at thesubconductor a. At one spacer location there will be therefore 2adamping forces.

In the same way, as shown previously, it is possible to express adamping matrix of the spacer which correlates the above mentioneddamping force components to the components of the velocity of theclamp movements in the same reference planes.

If the spacer damping is hysteretical and the motion harmonic withfrequency Si:

Fr(d) =- [hi u (i9) (24)where r.(d) is a vector having "a" components F(d). and "a" com-ponents Fz9J and hi(d) is a vector the components o which will be the"a" quantities S(li,t) and the "a" quantities i (li,t) previously ex-plained.

The spacer damping matrix [h] will obviously depend on the spac-er design. If a constant relationship can be assumed between the damp-ing forces and the flexibility for any direction of the motion of thespacer clamps then it is possible to express the following:

[h] = p [K] (25)where p is a dimensionless damping constant of the spacer.

The power dissipated by the spacer i will therefore be

p(d) = F(d) 7j(d) = 2 D(d)1 1 1 ithat is

+ 2 Ayar Byar sin Yrli ]

where hir is the hysteretic damping coefficient per unit length oconductor in the subspan i.

As shown in [21, if the number of wavelengths contained i:subspan is large enough (>5 9 \r) to minimize the effect of thterminations on the value of the energy dissipated in the subspa

(22) p4l() = E FyajiYov(lji t) + Fzoa i (lI t) (26)If we now introduce the principal coordinates, the (24) becomes

p(d) = I[h Pr(t) r(li) (27)1802

where now the force vector is related to the mode r and the space we can writefunction expressed by the vector rOli) has, as components, theoyar(li) and Ozar(li) previously discussed and shown in the (12).

The damping function of the spacer i for the mode r will be:where HIrRE1

D(d) = Pr (t) [h] (28) force and i ll"r 2 r r(l1 ) TrOid If a loa

subconductorThe damping function of all the spacers for the mode r will nents in the y

therefore be The Lag

n-l rrDrd) = (29)

1 The osc

The damping function of the whole bundle for the mode r will be

(30)D(cd) = D(c) + Dfd)and to the equations (21 ) it is possible now to add the term:

2 D(cd)-rh = Qp2 (t)

The quantity hr will be understood in the following.

Forced Vibrations

As exposed in a preceding paper [2], it is possible to compute thesystem displacement due to any harmonic exciting force by means ofLagrange's equations.

The more general equation of motion is

d aT av aD+ + . 7r (32)

dt apr aPr aprwhere T, V, D are the equations of the kinetic, potential and dampingenergies, Pr the principal coordinate and 7rr the Lagrangian componentof the exciting force.

By means of (21) and (31), the equation (32) can be written:

Pmr* r P* wPr r + Pr r+prkr = 7rr (33)

If we assume

p = P eigt and 7r = (3ei4)r r rn Hri~we obtain

* * 2i*

r r rQ>2)2+h2 (35)If we assume again

Wt2 &22REr 2 -

Mr

hr*

IMr h*(c4

-22) +( r)mr

i can therefor

Pr = Hr (REr - i IM) (37)r is the component in phase or at 1800 with the exciting[rIMr is the component at 900. ialized driving force F=Foe"Ot is applied on a subspan i ofr at distance xi such a force will have two space compo-yand z directions F elQt and F ityrnand z dircmpon yofe a foze

n3rangian component of such a force will be= Foy Dyar(xi) + Foz yoyar (xi) Car (38)-illation amplitude at any point xi of subconductor a ofre be expressed by:

00

UOX (Xi) = E r 4'yoar (xi) /f14 . 11 (RE -',Md) (39)where 11r is shown in (38).

The computations based on one or more localized forces aregenerally needed to compare computation results with laboratory tests.

(31) In practical application, distributed forces per unit conductor lengthsare used.

If far (xi) = FOT (xi)eict is a force, per unit conductor length,distributed along the subspan i on subconductor a, in the direction ofthe oscillations occurring at vibration mode r, the total Lagrangiancomponent on the bundle for mode r is:

a n liHr= E ~f Far(xi) 4Dyw (xi) Vl+C dx1 1 0 (40)

The oscillation amplitudes along the span can again be computedby means of (39).

A complete computation program, with all the particular methods,routines and subroutines required to reduce the computer time and keepa sufficient flexibility for use with different conductors, spacers andspan lengths has been achieved by the research group.

With a UNIVAC 1108 the deformation of the bundle under aeolianconditions for all the interesting frequencies, and the strains occurringat the span extremities and at the spacers, can be obtained in about 5 to10 minutes. The same time is practically required with any given as-sumption of distributed forces.

The accuracy of the computed results has been evaluated by meansof laboratory tests. The laboratory set up and test procedures have beenexposed in detail in [5] and [6]. Figures 5 and 6 as well as table IXshow one typical test performed.

For this test, two spacer-dampers having the following character-istics were used: kz : 5 kg mm1;kI: 37 kg m racE1; hz = 0.63 kgmm1; ho = 3 kg m rad-1. The spacers were located respectively at15 and 31 meters on the 46.52 m. span. The conductors were twoACSR "Curlew" subjected to a tensile load of 4750 kg. each andspaced 400 mm. in a vertical configuration. With such low stiffnessspacer dampers, the overall energy dissipated in the bundle during thetests was about 40 to 50 times greater than would have been dissipatedby one only conductor at the same tensile load, frequency and maxi-mum antinode vibration amplitude. The energy dissipated by the twospacer-dampers was about 70 times greater than the energy dissipatedby the two subconductors. With such an amount of damping on a 46.52m. span, the harmonic force required to obtain 1 mm. maximum anti-node vibration amplitude was considerable, 1 kg. and the whole bundledeformation was such that it cannot be represented graphically in theusual two-dimensional way.

If we refer to equation (39) of the preceding chapter, we will notethat the subconductor displacement at point xi, which can be repre-

1803

m*

V..

144

0 5 10 15 20 25 30 35 40 45 m

T 2

',4

2 i~~~~~-.---!4S41;>X X -/ t1

10 15 20 25 30 35 40 45 m

Fig. 5. Max. Conductor Displacement vs Distance.

F kgF 0

w rad se&la E 10-2'a mm2'a 9p02"a mmis2"a 0

2"'a mm2' a 9p02'b mm2' b p02"b mm2'"b 90

2"'b mm2'"'b p0

IC l 1M d| C M150.14910

0.056-270.22-310.34-310.58+1770.62+1680.58+162

160

58.0414

0.06-240.20-330.30-330.5+1710.60+1680.50+168

3a3a3b3b4a4b5b6a6a6b6b7a7a7b7b8a8b

mm

0

'p

mm

0

E-10-8E10-6

mm

0

mm0

mm

'0

mm

90mm

mm

0.37-300.27+4816.616.316.5

1-81

1+750.27+510.25600.420.36

0.40-280.22+14898

1-90

1

+850.20+410.1850

0.330.33

Table IX - Test Results.

180a) f

120

60

0

_60

-120

-180

I A _ _ : F-l-

-ii _ 73IN

343 --__-1

_~~~~~1+- I_

S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

,I I ,,Il t0 5 10 15 20 25 30. 35 40 45 m

a)

l b)

Fig. 6. Phase Angle vs Distance along Span.

sented by the vector U,T(xi), is the vectorial summation of the compo-nents due to the various modes.

The great amount of damping involved will cause considerable,time-phase differences between the displacement vector of the variousspan locations x [2] and therefore the true system deformation has tobe represented not only by the maximum vibration amplitudes thatcan be reached at the various span locations x, but also by the time-phase shift between the displacement vectors at these locations and areference vector which has been herewith chosen to be the vector Frepresenting the exciting force.

The bundle deformation is therefore exposed in the figures Sa and5b which show respectively the maximum value, that is the modulus ofthe displacement vector as a function of span location of the upper andof the bottom conductor, and in the figures 6a and 6b which show thephase shift of the displacement vector, in respect to the driving forcevector F, as a function of span location, of the upper and of the bottomconductor.

In the four figures the continuous lines show the computed valuesand the crosses the measured values. Table IX shows again under columnM the measured displacements and their phases and under column Cthe computed ones, together with the computed and measured strains.The points where measurements were made are indicated in the figuresby numbers and their location on the span clearly assessed by the spanlength scale.

The bundle response exposed here has been purposely chosenamongst others because it clearly shows the distortion caused to thebottom conductor by the exciting force. As it can be seen both thevibration amplitudes and their phases are clearly different for the twosubconductors in the first subspan. Measurements 2', 2", 2"' maderespectively at 9, 10 and 11 meters from the left-hand span extremityon both subconductors are probably the most significant of all theImeasurements made. The excellent correlation between the computedand measured displacement values and in particular between the com-puted and measured phase shift of these displacement vectors in respectto the exciting force vector, is the conclusive proof of the accuracyof the computation methods used and of the various assumptions. It isworth noting that, during the tests, the force vector phase shift inrespect to the displacement of its attachment point to the conductor

1804

/7

0

a

1,ny

a

0,

0 5

-riII

was 600 against the 50 of the computed data and such a discrepancycan well account for part of the errors in the phases of the othermeasuring points. The ratios between the measured and the computedstrain values seem to fit well with slippage coefficient of 0.4 to 0.55usually found on these conductors.

Spacer Design

The previous chapters have shown how it is possible to simulatewith a computer the response of bundles to various types of excitingforces. For a given type of exciting force it is therefore also possible toinvestigate the influence of the spacer characteristics on the responseof the bundle.

Before discussing the results of some of the authors' investiga-tions, it is worth mentioning briefly the present knowledge about twotypes of wind induced exciting forces, those related to vortex sheddingwhich cause aeolian vibrations and those related to the variations inlift and drag coefficients of an oscillating conductor, lying in the wakeof another conductor, which cause the so-called "subspan oscillations".

Aeolian Vibrations

The vortex shedding phenomenon on single conductors has beeninvestigated and discussed by a great number of authors over manyyears, but up to now no definite and proved quantitative informationhas been published which would allow the introduction in responsecalculations of a reliable distributed force function.

Generally, response calculations are based on the energy balanceconcept at steady state vibration conditions. The energy input from thewind, however, has to be obtained from theoretical or wind tunnel test

5)

2

QW1-7 J__

u .1.7 ml/s

5

2" ethi - /5

%-4

Q. T_

results and from field experience. Theoretical and wind tunnel results;most currently used are those of Bate [ 8], Farquharson and McHugh [9]and Slethei [10] which are shown in fig. 7. There is a considerabledifference between Slethei's theoretical maximum power values andthose of Bate and Farquharson obtained from wind tunnel tests.

The authors of this paper are presently engaged in wind tunneltests and the results obtained up to now give considerably higher maxi-mum power values than those shown by Bate and Farquharson.

Furthermore field experience on large river crossings and very flatterrain, such as in Saskatchewan, would prove that on such terrain thewind power input is considerably greater than the values shown byBate and Farquharson.

The terrain influence on vibration intensity and therefore on themaximum power input has clearly been shown by Edwards [ I 1].

From these facts it is therefore evident that any investigation onthe response of bundles to aeolian vibrations and therefore any evalua-tion of the spacer characteristics has to be made taking into due con-sideration the terrain where the line will be erected.

A further point however has to be considered in the case ofbundle conductors, and that is the possibility that the wake of one sub-conductor might affect the power input from the wind on the othersubconductor and vice versa. No wind tunnel investigation results onthis subject have yet been published, but Libermann and Krukov [ 12]field measurements on twin horizontal bundles show a reduction ofvibration intensity in respect to a single conductor. Normal transmissionline practice however shows that even if twin horizontal bundles dovibrate somehow less than a single conductor, the reduction is notenough to avoid in most cases the use of vibration dampers if neededfor single conductor operation.

Libermann and Krukov [ 12] field measurements show also avery large reduction of vibration intensity (5 to 10 times) on triple andquadruple bundles, but this is due to the fact that measurements were,as usual, made only at the span extremities and cannot therefore berepresentative of the vibration of the whole span, nor can they bedirectly compared with measurements made on a single or on a twinhorizontal bundle.

Although there is generally a feeling that triple and quadruplebundles are somehow less subjected to vibration, there has been in thepast years' an increasing number of reported fatigue failures of con-ductors on triple bundles which do not seem to be connected withsubspan galloping.

These failures were essentially located at the spacer clamps andnot at the span extremities where, as usual, dampers had been installed.If we look at a triple bundle configuration, it is evident that the wakeeffect, if any, might affect the power input on two subconductors only.

Although present knowledge on the maximum power input onvibrating bundles is very small, nevertheless experience seems to provethat it might be sufficient to cause conductor damage.

In the investigation on the effect of the spacer characteristics, theauthors have ignored the wake effect and have assumed that each sub-conductor of the bundle would take from the wind the amount ofenergy compatible with the frequency of the vibrations and the vibra-tion amplitudes of the subspans. The total wind energy of the bundleis assumed to be the sum of the various energies introduced in each sub-span of each subconductor.

With reference to tables VI and VIII, for a horizontal wind, thewind energy input has been assumed to depend on the vertical com-ponent of the conductor displacement, whilst the energy dissipated bythe conductors and spacer-dampers will certainly depend on the ab-solute value of this displacement. The wind assumption has not yetbeen definitely proved by wind tunnel tests, but it has reasonablegrounds to be valid for relatively small angles between the plane ofvibration and the vertical.

It follows that all the vibration modes corresponding to the typesII and IV of table VI would show the greatest tendency to vibrate; themodes corresponding to type III should not be excited and the modes

1805

0.01 2 5 0.1 2 5 1.0 2Relative vibration amplitude YID:

Fig. 7. Wind Power Functions.

corresponding to the remaining three types should be less prone tovibrating.

With reference to the four bundles and spacer as per fig. 4, thegreatest vibration possibilities will be found for types III, IV, VI oftable VIII.

It is furthermore worth remembering that, for each type of vibra-tion, there is a definite ratio between the amplitudes of oscillations ofeach subconductor, thus one subconductor might vibrate at higheramplitudes and thus be subjected to higher strains.

As the types of oscillations, the eigenvalues, the typical stiffnessand the amplitudes ratios between subconductors depend on the spacerdesign, it is evident that investigations made on one design cannot bedirectly compared or generalized, however comparisons can be madebetween the modes corresponding to the types of oscillation whichgive the higher vibration amplitudes.

The authors expose in this chapter the effect of increasing stiff-ness X and increasing damping for the spacer shown in fig. 3.

Subspan Oscillations

The authors will not discuss here the investigations performed onthis phenomenon, as the energy input assumptions have been essential-ly theoretical pending the results of wind tunnel research. Howeversome conclusions based on the dynamic response of the bundle will bevalid also for this type of oscillation of the bundle.

From present knowledge and field experience it would seem that,for spacers as per figure 3, the oscillation types more easily excited bylift and drag coefficient vibrations would be probably the I, V and VIof table VI, and, for the four-bundle of fig. 4, the types I, V, VII andVIII of table VIII.

Spacer Stiffness

An investigation of the effect of spacer stiffness X has been per-formed by the authors for various types of bundle systems. As ex-plained previously, after having assessed the "type" of oscillations andthe related eigenvalue, the deformation of the cables in their own planesof vibration will be essentially governed by the stiffness X. The resultsshown below refer to a three-bundle system and in particular to the firstof the three oscillation types "typical of the bundle" which is char-acterized by vertical or nearly vertical planes of vibration (type IV oftable VI). Such a type of vibration is common to any three-bundlespacer design. The span was composed of three, 402 m. long, ACSR"Curlew" conductors in an equilateral spacing of 400 mm., and thetensile load of each cable was 3432 kg. The six spacers were locatedwith the following spacings: 29 m., 67 m., 72 m., 69 m., 68 m., 67 m.,30 m.

The diagrams herewith shown refer to amplitudes and strainswhich would occur on the bottom conductor which, as previously ex-plained, will show the highest amplitudes of vibrations.

In the investigations the authors have assumed spacers having60,829 kg m-1, 18,248 kg m-1, 10,341 kg m-1, 6,082 kg m-1 and912 kg m1 stiffnesses N.

Investigations were first performed on a spacer without any in-herent damping that is,p = 0.

The wind input function was assumed the one which would mostlikely occur on a large crossing, that is somewhere in-between theFarquharson and the Slethei values. The reason for such a choice wasthat higher wind power input and consequent high vibration amplitudeswill allow a better perception on the effect of changing the spacerparameters. The wind input function was always the same for allcomputations.

In order to reduce costly computer time, the computations didnot cover all the resonant frequencies and modes of oscillation of thetype concerned, but, purposely, one out of six with equal frequencyintervals of about 5 radians. sec-1.

It is evident that with such a discontinuous series of data somehigh or low peak values may have been neglected but, as it will be seenlater, this possibility does not affect the results.

One of the most interesting facts on bundle oscillations is thateach subspan might oscillate with antinode vibration amplitudes con-siderably different from those of another subspan of the same sub-conductor. We shall call this phenomenon the "subspan effect".

If we call Umin the minimum antinode vibration amplitude thatcan be found on the whole span of a subconductor for a given resonantfrequency and Umax the maximum one, then the ratio Umin/Umaxcan represent the "subspan effect" of the spacers. The Umin/Umax ofthe bundle system investigated expressed- as a function of resonancefrequencies Q2 (rad sec71) is shown in fig. 8 for ) = 912 kg m41 andX =6082kgnmf.

As it can be seen, the Umin/Umax is clearly dependent on fre-quency and on X values. It can thus be pointed out that for a given X.value the oscillations due to lift and drag coefficient variations whichgenerally occur at very low frequency will emphasize the "subspaneffect" thus explaing their definition of "subspan galloping".

It can further be seen that if the ratio Umin/Umax has to be in-creased, for a given frequency, then the X values has to be considerablydecreased. These results confirm the field experience of C.O. Frederickand M. D. Rowbottom [ 13].

If we take now the maximum and minimum values of Umin/Umax ratios which were found on the whole frequency range, for eachvalue of X, we obtain figure 9 which gives the range of Umin/Umaxthat can be expected as a function of N.

The fig. 9 gives reliable information on the possibilities of reducingthe vibration amplitudes on a span by means of dampers located at thespan extremities.

Dampers located at the span extremities will, of course, be quiteeffective in reducing the vibrations which will be related to the type IIof table VI, that is those "characteristic of the conductor", inasmuchas the bundle will behave like a single conductor.

0

X =912kgm-'

. X.k 6082 kgmn-'0 ,

.-

0

.*. _ * .

_~~~~~050 00 150 200 Qrad's-

Fig. 8. Umin/Umax vs Frequency.

a = Maximum UmInUmax

b = Minimum UminUmax

Fig. 9. Umin/Umax vs X Values.1806

These dampers will also be quite effective in reducing the vibra-tions related to the type IV under discussion, if the Umax will occur atthe end subspan or if at least at these subspans the antinode vibrationamplitudes will not considerably differ from the Umax. But if the Uminwill occur at the end subspans the damping possibilities will depend onthe Umin/Umax ratio and fig. 9 shows that such a ratio becomes toosmall even at relatively low values of X.

The damper energy in fact depends on the square of the vibrationamplitudes occurring in the subspan where the damper is located. Fora Umin/Umax ratio of 0.5 it is possible to use only 25% of the dampingcapacity that the damper would have had on a single conductor vibrat-ing all along the span at an antinode amplitude of Umax.

From fig. 9 a ratio of Umin/Umax of 0.5 is always available forX < 912. Such a value of X is very seldom found on spacers. C. 0.Frederick and M. D. Rowbottom [ 13] estimate such a value necessary,but only for subspan oscillations.

Coming now to figures 10, 11, 12 they show, for three differentvalues of X, the strain vs frequency relationship of the bundle systemunder investigation. More precisely, the continuous curve shows thestrains ei that would occur at rigidly clamped span extremities of asingle ACSR "Curlew" at 3432 kg. tensile load under the assumedwind conditions; the circles show the maximum of the 12 strain valueses occurring at the six spacers (one at each side of the spacer clamp) atthe plotted frequency, and the crosses show the maximum of the twostrain values eib occurring at the two span extremities of the bundledconductor.

Although, as shown in the previous chapters, single conductorsand bundle systems cannot be directly compared, the authors havethought it useful for transmission line engineers to compare the strainsthat can occur on bundles to those that can occur on single conductorson which considerable experience has already been obtained.

When considering these figures, it must be remembered that notall the resonance frequencies have been computed, but practically oneof every 6 and therefore one single high value cannot be ignored asmost probably there might be another 6 just as high or maybe higher.

If we consider fig. 10 which refers to a X = 60,829 kg m-1, arather high value, it can be seen that the es are quite high throughoutthe whole frequency band, whilst the eib are very low or zero, whenthere are no vibrations in the end subspans, and at two frequencies,when these subspans vibrate severely, they are higher than the corres-ponding ei. This is typical of large "subspan effect".

If we consider now fig. 1 1, which refers to a X = 10,341 kg m-1,a rather normal value, we note that the es have not changed considerablybut the Cib have strongly increased and a very high value has beenreached at 130Q.

If we now look at fig. 12, which refers to a X = 912 kg m-1, avery low value, we note a very strong change. The es have sharplydropped to very low values and nearly all the eib raised above the ei.

With such a low value of X one would have expected to see theEib tend towards ei but not above.

In order to have a better view of the effect of the X, the maximumvalues of Cib/ei (Max eib/ei) within the whole frequency band havebeen plotted as a function of N and the curve is shown in fig. 13 underNo. 1.

For high values of X the Max eib/ei ratio tends to 1, it then in-creases to about 1.62 at X = 8000 kg m-1 and then drops back to 1for X - 0.

The explanation of this curve can be found from two facts. Firstof all, fig. 9 shows that Umin/Umax values of 0O can well be reachedfor X > 6082 kg m-1 and even with X = 912 kg m-1 Umin/Umax values,of 0.5 can be obtained. In other words, except for XN 0, there willalways be frequencies for which each subspan will vibrate with differ-ent antinode amplitudes.

Now, for a single conductor, the energy balance condition isreached throughout the whole span and for a given frequency at a

definite value of antinode vibration amplitude; for lower antinodevibration amplitudes there will be an excess of wind power input andfor higher amplitudes an excess of dissipated energy.

On the bundles, the response conditions require a definite rela-tionship between the antinode vibration amplitudes of each subspan; itfollows that the energy balance conditions cannot be obtained at eachsubspan, but only for the whole bundle. In other words, on some sub-spans where the antinode vibration amplitudes are too low, there willbe an excess of energy which will be transferred to the other subspanswhere the antinode vibration amplitudes will be too high, thus dissi-pating not only their own wind power input, but also the one cominginto it from the other subspans.

When X is very high, Umin will be t 0, thus practically no energywill be introduced into subspans with Umin. Consequently the subspanswith Umax will reach steady state conditions at lower antinode ampli-tudes.

C

4001

300

200

100

CE

400

300

200

100

o' C0-CslCb

00

0 50 100 150 200 250 Q2rads'Fig. 10. Strain Values vs Frequency for N = 60,829 kg m-1,,A= 0.

+o4

4i

0 50 100 150 200 250 QradFig. 1 1. Strain Values vs Frequency for X = 10,341 kg m-1, A = 0.

CI

500+++

++

e.C.+CCib

0 50 V0 150 200 250 n2 reFig. 12. Strain Values vs Frequency for X = 912 kg m-1, ,u= O.

1807

y O p = 005(2) p - 0.10 u 02

0D02

Fig. 13. Max. cib/ci vs X Values for p= 0, , = 0.1, p = 0.2.

912 6082A234P 18248 60829 K kg/mFig. 14. Max.es/eivsXValuesfor,u=0,p=0.1,u='0.2.

With decreasing X and increasing Umin, the subspan effect on thepower balance will be more and more felt, thus increasing the Umax,and consequently the Cib until, below a given value of X, the Umin willtend towards the value at which the energy balance can be reached foreach subspan thus decreasing Umax and the Eib.

If we had traced the Max Cib/Ci vs X curve for the upper two con-ductors of the bundle, we would have found the same trend, but theratio values would have decreased by a factor of 1.783.

Curve 1 of fig. 14 shows the maximum values of the ratio es/fib(Max es/ei), that is of the maximum strains at the spacer clamps versussingle conductor span extremity maximum strain, within the wholefrequency band, as a function of K.

The curve needs little comments as it shows that Max es/ei ratioof 1 is already reached for the bottom conductor at about K = 18,248kg m-1. The upper conductors would obviously show only Maxes/ei 0.50.

These curves show very cleary that, for too high X values, strainsat the span extremities and at spacers can reach and even be higher thanthose that would be found at span extremities on a single conductorwith all other parameters being equal. For a three bundle, furthermore,at vibrations corresponding to those of type IV, the bottom conductorwill be considerably more stressed than the two upper ones.

It is also evident that for each individual point where bendingstrains can develop, high strain values can occur only for a number ofresonant frequencies, whilst for a single conductor they will occurwithin a more continuous frequency band.

Thus, for a definite period of time, and for a particular location,the accumulated number of strain cycles should be lower for a bundle

conductor than for a single one, but this could only increase the timerequired to reach fatigue if strains are above the fatigue limit.

Spacer Dampers

The effects of the inherent damping capacities of a spacer as perfig. 3 have been subsequently investigated by the authors. All para-meters, conductor, wind, typical stiffnesses of the spacer, were identicalto those used for the investigation of the effect of X. Dimensionlessdamping constants p > 0 were, instead, introduced as explainedpreviously at equation (25). Investigations were made for , = 0.05,,=O.l andu= 0.2.

The results are shown in figures 13 and 14 which show MaxeibICi and Max es/q for various values of X and different values of ,u.

It can be seen from these figures that with spacer-dampers somereduction of vibration amplitudes and strains is still obtained with aX = 60,829 kg m1., but it is quite evident that an optimum stiffnessexists for which the Max ratio of eib/ei can be brought practically tozero.

For X values greater than the optimum, the reduced relative dis-placement of the spacer clamps, which is related to the "subspaneffect", will reduce the energy that can be dissipated by the spacerdampers.

For X values smaller than the optimum, the reduced dampingforce which is, for a given g, proportional to X, will not be compensatedby greater relative clamp displacements. This behaviour is similar to theone which is found with dampers.

The optimization possibility is less evident on the Max ratioes/ei as for small values of X such a ratio tends to decrease even withP = 0.

Inertia Forces

The authors have, for simplification sake, assumed the spacers tobehave essentially as springs, that is with positive values of X stiffness.

In practice there are also inertia forces due to various masses ofthe spacer. The spacer clamps will obviously develop inertia forces, butthe total inertia forces developed by the spacer motion will depend onthe spacer construction.

The effect of some inertia forces has been investigated by simu-lating spacer clamps having masses, the inertia forces of which wouldbe equivalent to those developed by a real spacer.

The results are shown in fig. 15 where m is the mass of each clampof the spacer investigated.

The improvement of the Max es/ei at high X values was to be ex-pected as the inertia forces would just reduce the effective stiffness toa lower value K'

forces show their negative effect. If the inertia forces become greaterthan those due to the stiffness X, then A' will become negative and canreach very high negative values.

The "subspan effect" and Umin/Umax ratio will depend on theabsolute value of A' and therefore the Max es/ei and also Max cib/Eiratios will increase.

The damping forces however will still be related through p to thespring stiffness A and consequently will be very low as previously seen.

As a consequence, an increase of negative A' will bring a fasterincrease of Max es/ei than happens with an increase of positive A.

The effect of spacer clamps masses and, to be more correct, theeffect of the complete spacer mass, become quite important for thosevibration modes belonging to the "types" which had been called"characteristic of the conductor". For an ideal spacer having no mass,it was stated that the natural modes belonging to these types of oscilla-tions are identical to those of a single taut cable, inasmuch as thesemodes do not cause any relative movement of the spacer clamps. As aconsequence, no elastic force is developed by the spacer, which is con-sequential to the fact that these types have eigenvalues X = 0.

The fact that a real spacer has a weight which, distributed amongstthe subconductors, leads to a mass m applied to each subconductor,means that a negative spacer stiffness - w2m will affect those modeswhich, for a weightless spacer, would have had a X = 0. Such a negativestiffness, furthermore, will increase with increasing resonance fre-quencies.

A mathematical analysis of these modes can be done by using apoint matrix [P] which is a function of frequency, but the effect ofsuch masses can be easily seen in fig. 13 and 14, if we replace X with2m. At 20 cps, for example, a 6 kg. spacer on a three-bundle system

will be equivalent to a X = 3260 kg m-1.TIhe twin bundle systems are obviously affected by the same prob-

lem. It follows that on twin horizontal bundles the spacer weight has tobe considered when aeolian vibrations are analyzed, whilst the spacerstiffness, together with the spacer weight, will have to be taken intoaccount when subspan galloping is investigated.

Dynamic Matrix

The introduction into the analysis of spacers having masses at theirclamps, directly connected to the conductor, has not basically modifiedthe results exposed up to now. The fact that the inertia forces are devel-oped at the spacer clamps gives the possibility of superimposing theeffect of the spacer stiffness X and the effect of the inertia forces co2m.

On some spacers, however, and in particular on some spacer-dampers, inertia forces are developed inside the spacer structure as aconsequence of harmonic oscillation of masses which are connected tothe clamps and subconductors by means of a flexible structure.

The flexibility characteristics of such spacers can no more be ex-pressed by an elastic matrix, but a dynamic matrix is needed whichtakes into account the effect of all elastic and dynamic forces which aredeveloped by the spacer motion. It is indeed possible, under theseconditions, that the oscillations of the internal mass of the spacer causesome dissipation of energy even without any relative displacement of thespacer clamps.

It is not possible to generalize an analysis of a bundle with dynamicspacer matrixes as it depends too much on the design of a real spacer.

The authors have studied the dynamic matrixes of spacers ofdifferent design and have reached the following conclusions:

1) The internal masses have a very limited effect on the oscilla-tions "typical of the bundle", inasmuch as the relative clamp displace-ments are much greater than the oscillations of the internal mass.Dynamic matrixes, in most cases, can be replaced by the elastic matrixeswhen these types of oscillations are investigated;

2) On a spacer damper, internal masses will show their dampingeffect on the oscillations "characteristic of the conductor", but theireffect will depend on the value of the resonant frequency of the

spacer. The heavier the internal mass, the lower will be the frequencyfrom which the damping effect will be noticeable. In order to lowersuch frequency, it should be necessary to increase the internal massesto rather uncommon values (6 to 10 kg). Instead of increasing the valueof the internal masses, it could be possible to achieve the same resultsby lowering the stiffness of the flexible structure connecting the in-ternal masses of the spacer clamps.

This procedure however would be detrimental to the efficiencyof the spacer-dampers for the oscillations "typical of the bundle" if theresulting eigenvalues X of the elastic matrix will be below the optimumvalue previously exposed.

CONCLUSION

The present paper had not the aim of exposing results which wouldallow a definition of the "optimum spacer". The concepts exposed herehowever are valid for any type of spacer and can be summarized asfollows:

1) Unless the vortex shedding phenomenon which governs thewind energy input on vibrating conductors is affected by the bundleconfiguration, dynamic bending strains of about the same value thatwould be found on a single conductor, are to be expected on twin ver-tical, three- and four-bundled conductors, unless the spacer stiffness hasextremely low values. With the usual spacer weights, dynamic bendingstrains on twin horizontal conductor spacers will not be too severe ex-cept under rather severe conditions (high tensile load, high windpower input);

2) Dynamic bending strains can be just as severe at spacers as atspan extremities with spacers having too high X stiffness or too heavyweight;

3) With spacers having too high A values or too heavy weight,dampers located at the span extremities will not protect the conductorsfrom dynamic strains occurring at the spacers;

4) Spacer-dampers to be effective must have a A value reasonablyclose to the optimum one;

5) The determination of the spacer stiffness X for each of theseverest oscillation types is a fundamental step in the evaluation of aspacer or spacer-damper performance. All other "flexibility" valuesnormally used to represent spacer characteristics might lead to erron-eous conclusions.

ACKNOWLEDGMENT

The authors gratefully acknowledge the assistance of the ItalianConsiglio Nazionale delle Ricerche which granted a financial aid tothis research program.

REFERENCES

[1] E. Pestel - F. Leckie, Matrix methods in elastomechanics,McGraw Hill Book Co. 1963.

[2] Rodolfo Claren and Giorgio Diana, Mathematical analysis of trans-mission line vibration, IEEE Trans. Power Apparatus and Sys-tems, Vol. PAS-88 No. 12 December 1969.

[31 Emilio Massa and Giorgio Diana, Sui modi principali di vibraredei fasci binati di conduttori tesati: Pulsazioni proprie, deformate,sollecitazioni, Energia Elettrica - fascicolo 4 Vol. XLVI 1969.

[4] Giorgio Diana and Franco Giordana, Vibrazioni dei fasci trinati equadrinati di conduttori tesati: Frequenze proprie, modi principa-li, sollecitazioni, Energia Elettrica - fascicolo 9 Vol. XLVI 1969.

[5 Rodolfo Claren and Giorgio Diana, Mathematical analysis ofmechanical oscillations of cable bundles, CIGRE 22-70 (SC-13).

[6] Rodolfo Claren and Giorgio Diana, Ricerche sperimentali sul com-portamento dinamico di fasci di conduttori, Energia Elettrica 1970.

[7] Rodolfo Claren and Giorgio Diana, Dynamic strain distribution onloaded stranded cables, IEEE. Trans. Power Apparatus and Sys-tems, Vol. PAS-99 No. 1 1 November 1969.

[8] E. Bate and J. R. Callow, The quantitative determination of theenergy involved in the vibrations of cylinders in an air stream,Journ. Inst. Eng. Australia Vol. 6 1405 (1934).

1809

[91 F. B. Farquharson and R. E. McHugh, Jr., Wind tunnel investiga-tions of conductor vibration with use of rigid models, AIEETrans. Vol. 75 III (1956) pp. 871-78.

[10] T. 0. Slethei, Vibration on overhead lines - Wind energy and con-ductor self-damping, Thesis 1968 The Techn. Univ. of Norway,Trondheim.

[111 A. T. Edwards and J. M. Boyd, Field observations of mechanicaloscillations of overhead conductors, Terrain and other effects.

[12] A. J. Liberman and K. P. Krukov, Vibrations of overhead line con-ductors and protection against it in USSR, CIGRE paper 23-061968.

[13] C. 0. Frederick and M. D. Rowbottom, Subspan oscillations,Spacer design, C.E.G.B. reports RD/L/M 229 and RD/L/M 230.

as a general rule. Some may have no practical significance, depending onthe location of the spacer.

If the spacer is rigid, these out-of-phase modes may be treatedindependently according to the sub-spans into which the spacers dividethe whole span. Careful attention to boundary reaction is required. Ifthe spacer is very weak, meaning that the kinetic energy of the out-of-phase motion is not appreciably influenced by the spacer, then the in-phase modes can be used as a starting point to assess the out-of-phasemode shapes and frequencies. The intermediate case is the most diffi-cult to assess, especially if laboratory tests are set up to predict fullspan effects.

Nevertheless, some simplification is possible. The adjacent figureshows a test set-up suitable for the study of out-of-phase vibrationmodes.

Discussion

A. S. Richardson, Jr. (Research Consulting Associates, Lexington, Mass.):The authors should be congratulated for demonstrating that their math-ematical techniques correctly predict the complex dynamic response ofspacer-damper-conductor systems.

As the paper is concerned' primarily with the class of vibrationknown as aeolian vibration, I shall speak only of this class of vibrationin bundled conductors. The questions which I have are the following:

(1) What is the quantitative measure of a "stiff" spacer in termsof transmission line parameters?

(2) Where should the spacer-damper be located in the span foreffective control of vibration?

(3) What differences can be anticipated between full scale trans-mission lines and laboratory test spans?

(4) Can results obtained for twin bundles be applied to quadbundles?

So as to throw additional light on these questions - and to contri-bute in a positive way to the paper itself - I would like to discuss thetwin bundle conductor systems identified in the accompanying figureas, I-Full scale span, and, Il-Laboratory test span. Numerical values havebeen chosen to correspond closely to values used in the paper.

In both cases, the objective is to characterize the vibration possi-bilities in terms of the principal modes.

It is obvious that one set of principal vibrations are the so-calledin-phase modes which are identified in the paper as "characteristic ofthe conductor". These are the same as the single conductor modesbecause there is no relative movement of the spacer clamping points. Inote in passing that these modes are easily handled by conventionaldampers placed at the span extremeties. (I note further 'that suchmodes become particularly troublesome when the frequency is low -below the normal aeolian range and in the so-called sub-conductoroscillation range).

There are, of course, many more principal modes in a given fre-quency range, in the full span. If it is desired to simulate the full spanin the laboratory under the same tension, and under the same looplength, the test span length must be adjusted accordingly. Certain fre-quencies of the single conductor would thereby coincide with the samefrequencies and corresponding loop lengths in the full span. Intermedi-'ate frequencies of the full span cannot be obtained in the laboratory.Careful attention to end conditions in the test span is required for validsimulation.

The out-of-phase modes, identified as "characteristic of the bundle"in the paper are particularly difficult to simulate in the test span. Theseare the principal vibrations which cause either tension or compressionforce in the spacer. As energy is stored, then released by the spacerduring a cycle of vibration, these principal vibrations differ from the in-phase vibrations in both frequency (eigenvalue) and span shape (eigen-vector). There are just as many out-of-phase modes as in-phase modes,

System I

System II

Single conductor test span for the study of out-of-phase modes.

The vibration modes for this simple single conductor systemshould be the same as the out-of-phase modes of the system shown asII in the first figure. The savings in experimental complexity are readilyapparent.

As already noted, frequencies (and modes) are much more closelypacked on the full span as compared to the test span. From field ob-servation of aeolian vibration the frequency, at a given wind speed, isnearly constant while the amplitude is characterized by beats. Thissuggests that only a few adiacent modes are excited at a time. A suitablerepresentation of the full span might therefore involve the superposi-tion of two nearby modes, selected from the spectrum for the singleconductor, but accounting for the coupling effect of the spacer. Such arepresentation may also apply to the test span when the number of loopsin the test span is large, say, in the order of ten.

By considering only adjacent modes having an odd number ofloops per span, a representation of the out-of-phase modes for bothSystem I and System II has been worked out. The results are shownbelow.

The figures show the effect of spacer stiffness on the vibrationmodes in the neighborhood of the resonant frequency of 100 rad./sec.for I-Full Span, and 1I-Test Span. The loop length at this frequency isfour meters. Note the difference in the frequency scales, indicating aclose proximity of modes in the full span case.

It is seen that at low stiffness the character of the spectra aresimilar. That is, the force-frequency spectrum, though shifted slightly,is fairly uniform in both cases. Furthermore, it is found that the spacerforces are at a relatively low value, and are about equal in both systems.

At the next level of spacer stiffness, the two spectra are no longersimilar. While the full span spectrum remains regular, it is found that thespacer forces in the adjacent modes differ markedly. The test spanspectrum is by no means uniform, and the modes having relativelyhigher spacer forces are concentrated at the upper (high frequency)side. In all cases shown, the force levels are given per milli-meter ofmode displacement.

At the third level of stiffness, the full span spectrum is still regular,but the difference in the spacer forces between adjacent modes isgreater still. On the other hand, the test span spectrum is shifted muchmore to higher frequencies.

It seems clear from the above, and which merely amplifies theconclusion of the paper, that "stiff" spacers develop much higher forceson the conductor than do soft spacers. Furthermore, it is clear that testspan experiments should be set up with a cautious view to the inter-pretation of the obtained results.

While the above has considered only the effects of axial stiffness,which are, important in the case of vertically spaced bundles, similar

400m

i~~ ~~ - - -4- I

mkCurlew conductor at 3250 kg. tensions. Spacer located at mid-span.

Manuscript received February 10, 1971.

1810

TWIN CURLEW CONDUCTORSFULL SPAN - SPACER FORCE SPECTRASPAN: 400m ; TENSION: 3250 kg.

10

5,

0

10Fz

kg. /mm S0

10

5

0

rad. /sec.

TWIN CURLEW CONDUCTORSTEST SPAN - SPACER FORCE SPECTRASPAN: 40m ; TENSION: 3250 kg.

10

5

0

10

Fkg. /mm 5

0

10

5

0

rad. /sec.

1811

k,; 10,341kg./m

kz; 6,082kg./m

kz; 912kg./m

kz; 10,341kg .1.

kZ4 6,082kg./m

kz ; 912kg./m

s'rsitm (A

,1.0

ik- .5 .

EFFECT OF SPACER STIFFNESS ON Umin/UmaxSYSTEM (A) SIX SPACERS ON TRIPLE BUNDLESYSTEM (B) : ONE SPACER ON TWIN BUNDLESPAN: 402m ; TENSION: 3432 kg.CURLEW CONDUCTORS

t::.:f

tion dampers were installed at the supports and no subconductor oscil-lation was observed. Severe aeolian vibration in the interior subspanswould account very nicely for the difficulties.

Although we have some questions about the complete translationof the analysis to field practice, we hasten to add that these reserva-tions are a matter of degree rather than substance.

COMPARES ACTIVITY OF SINGLE CONDUCTOR AGS TOBUNDLED CONDUCTOR AGS AND HELICAL WIRE SPACER.

*-AGS B HELICAL. SPACER (22.2%TENSION AT 60F)0 - AGS (20.4% TENSION AT 60- F)

/SINGLE DRAKE

Fig. B. Major portion of field damage to date at bundled conductorspacers has consisted of abrasive wear beneath loose clamps.

0 200 250 300 350 ;TSTRAIN - INCHES/ INCH

Fig. A. Horizontal twin bundle has substantially less vibration activitythan lower tensioned single conductor.

For example, on horizontal twin-bundles, the authors assumeequal wind energy input into both the windward and leeward sub-conductors and, further, that there is negligible coupling through thespacer during aeolian vibration. The bundle would then act as twosingle conductors.

During field studies some years ago, we measured the vibrationof single conductors and horizontal twin-bundles located on the samedouble circuit towers. All the conductors were of the same size in con-struction, but the bundle subconductors had higher tension. Figure Ashows the results of the simultaneous vibration study. The bundle hasabout 40 percent less vibration than the single conductor even thoughthe bundle was at higher tension. Here it would seem that the assump-tion regarding spacer coupling or wind energy input, or both, wouldhave to be modified slightly.

At the same test site we did find the effectiveness of dampers onhorizontal twin-bundles was in accordance with the authors' analysis.That is, one damper per subconductor per span reduced the vibrationthroughout the entire span as determined by recorders placed at eachend of the span.

On the other hand, we are aware of a utility having the same gen-eral experience when the effectiveness of dampers was measured on athree-conductor bundle with one damper per subconductor per span.Here, dampers being effective throughout the span would have somedegree of variance with the theory, depending on the spacer stiffness,of course.

In the case of four-conductor bundles, however, the work ofEdwards and Boyd lends qualitative support to the theory. They re-ported their outdoor studies of four-conductor bundles in IEEE 63-1075,and found that: ".... Rigid spacers act as reflectors in a manner similarto suspension systems. Provision of damping in the end spans and sub-spans will therefore not provide effective damping for the othersubspans."

We would like to ask the authors whether they have had the op-portunity to make field measurements. If so, their results would bemost welcome in resolving this anomaly about damper effectiveness.

The authors' analysis of three-conductor bundles shows that thebottom single subconductor will vibrate much more severely than thetwo top subconductors, the ratio of severity being approximately twoto one. This predicted ratio is in substantial agreement with straingauge measurements we made in the field several years ago at the sus-pension clamps of a three-conductor bundle without dampers. Ourengineers noted in their field report of the study, "In every case, thestrains (bending strains at clamp mouth) recorded for the lower sub-'conductor exceeded the strain in the upper subconductors, the ratiogenerally being about two to one. " [Emphasis added.]

The auithors point to the possibility of conductor fatigue atspacer clamps during severe aeolian vibration. We have had conductorfatigue failures at bolted spacer clamps in some of our laboratory cablevibration tests. In our field inspections, however, most of the severe

damage we have seen to date has not been fatigue, but rather has con-sisted of wear and abrasion under loose clamps, as shown in Figure B.Furthermore, most of this damage has apparently been caused by sub-span oscillation rather than aeolian vibration, and these field exper-iences would seem to be at variance with the theory.

However, if the authors' analysis does turn out to be completelytransferable to field practice, a sobering and ominous possibilityemerges. That is, lines which are not exposed to substantial subcon-ductor oscillation and are presently free from damage are not necessar-ily safe over the long term. Severe aeolian vibration in the interior sub-spans of bundles would expose the subconductors to fatigue failures atspacer clamps, even if aeolian vibration dampers were installed adjacentto the suspension. Would the authors care to predict how much longerit would take conductor fatigue to occur at the clamps of conventionalspacers (not spacer dampers) on a bundled line in contrast to the sus-pension point of a single-conductor line without dampers built in thesame environment and at the same tension? Would it be twice as long,three times as long or how much longer?

Finally, a question regarding the universality of the analysis as it.applies to a staggered spacer location. In West Germany, for example, itis a practice to use two-conductor spacers on four-conductor bundles.The subconductors are tied together horizontally at one location.Twenty meters further on they are tied vertically, and this alternatingprocess is repeated every 20 meters throughout the entire span. Forcases of this sort, would the authors' analyses and computer programsbe applicable as they stand? Or, would modifications be necessary toaccount for the complexities introduced by having vertical constraintsand horizontal constraints separated by considerable distance?

A. T. Edwards and J. Chadha (Ontario Hydro, Toronto, Ont., Canada):We would like to congratulate the authors for an outstanding and pains-taking study of the dynamic characteristics of bundle conductors.From our examination of the paper, we have not found any assumptionsor other areas which might be modified to improve the usefulness or thequality of the study except possibly that local bending stiffness effectscould have been included. It is clear that the transfer matrix method,used by the authors, is a very powerful mathematical tool. Have theauthors considered the use and applicability of the finite element tech-nique for taking into account conductor bending stiffness effects at thespacers and at the span terminations. We believe it is possible to dividethe span up into a large number of small beam and string elements.

The example, provided by the authors, in determining the generaleffect of stiffness, mass and damping of spacers on the vibration re-sponse of bundle conductors, will undoubtedly lead to greatly improvedcontrol of the various types of mechanical oscillations which occur ontransmission lines.

Another application of the general technique, used by the authors,would be to the determination of the vibration response of a completetransmission line comprising many spans of single conductor. Sincethe response of a given span is markedly influenced by end effectssuch as adjacent spans, it may be possible to reduce the overall responseto the galloping type of excitation (resulting from ice coatings on con-

Manuscript received March 1, 1971.

1813

2.0

1.0

20

z

54wxL544

0

U

ductors) by optimising "the end effects such as the arrangement of thespans in terms of length for example. We would be interested to knowif the authors have applied the transfer matrix method to the overallgalloping response of transmission lines and whether they think thatsome optimisation of the line parameters is a practical approach to re-ducing their response.

R. Claren, G. Diana, F. Giordana, and E. Massa: We thank Mr. A. S.Richardson, Mr. J. C. Poffenberger, Mr. Doyle, Mr. A. T. Edwards andMr. J. Chadha for their comments and we shall answer to our best totheir questions.

In terms of transmission line parameters we believe that a spacer,having an eigenvalue to conductor tension ratio greater than 5 m-l.can be considered a "stiff" spacer.

As a consequence of the broad range of frequencies involved inthe aeolian vibration field we do not think that there might be anoptimum spacer distribution and location for a given span length andnumber of spacers installed. We would recommend a staggered distri-bution to avoid, for some frequencies, that all the spacers might fall onsome of those points which, in absence of spacers, would be callednodes.

The difference between full scale transmission line and laboratorytest spans is considerable. First of all the usual short length of labora-tory test spans would cause the system under test to be far moredamped than a transmission line if equipped with spacers. In the secondinstance test spans are excited by means of a localized force and thisresults in a quite different cable deformation as shown in figures 5 and'6 of the paper.

To our opinion a laboratory test span has to be used to verify thevalidity and accuracy of an analytical system which will have to be usedfor the computation of a full line response.

The results obtained for twin bundles have been applied to threeand four bundles as mentioned in the paper and can be extended tobundles having any finite number of sub-conductors.

We do not believe that there is any similitude between the systemsA and B mentioned in Mr. Richardson's comments. The apparentsimilitude of the Umin/Umax ratios versus frequency of the two sys-tems might be accidental. We however estimate that strain values morethan Umin/Umax ratios should be compared.

The non-dimensional parameter proposed by Mr. Richardson ishowever interesting inasmuch as it correlates correctly the spacer stiff-ness, the conductor tension and the wavelength (span length: numberof loops).

We understand Mr. Zaffanella and Mr. Doil worries as they arehandling systems which might have twenty four different "types" ofoscillations. We agree that many "types" will not be susceptible ofaeolian vibrations but we must know them before discarding them. Ifsub-span galloping phenomena are investigated those to be ignoredhowever might not be so evident as the phenomenon occurs under a'combination of different oscillation "types".

The authors are also engaged in various wind tunnel investigationspertaining to the vortex shedding and wake effect on lift and dragcoefficients.

We appreciate Mr. Poffenberger's comments based on field re-cordings but although, as mentioned in the paper, a wake effect mightreduce the wind power input of a bundle we believe that the recorded

Manuscript received March 29, 1971.

40% reduction of vibration intensity of the bundle as compared to thesingle conductor cannot be generalized. It is quite common to finddifferences between vibration intensities of the conductors of the samespan. We agree that on twin horizonta