-
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