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Galloping of bundled power line conductors
Iournal of Sound and Vibration (1980) 73(3), .363-377
GALLOPING
OF BUNDLED
POWER
LINE CONDUCTORS
Y. NAKAMURA Research Institute for Applied Mechanics, Kyushu Crniversity, Fukuoka, Japan (Received 25 January
1980, and in recised form 10 May 1980)
This palper describes an experimental and analytical study of the galloping of a twodimensional section model of a two-conductor bundle in which ice-accreted conductors are replaced by two identical square prisms, with both vertical and torsional movements allowed but the horizontal one blocked, in a uniform wind tunnel flow. Emphasis is placed on elucidating the vital role played by the aerodynamic coupling in the stability of bundled conductors. It is shown that, apart from galloping type flutter, two other types of instability, namely, torsional and classical type flutter, can also occur for bundled conductors. In particular. it is shown that the aerodynamic coupling can cause violent classical type flutter to occur when the resonant condition is approached.
1. INTRODUCTION
The low frequency and high amplitude galloping of conductors is a serious problem in the design and operation of overhead power line conductors. Over the years considerable effort has been made to identify the galloping mechanisms and to find a solution to this problem. A general review of the problem is given in reference [l] while the more up to date information can be found in references [2] and 131. It is generally agreed that conductor galloping is a dynamic aeroelastic instability: namely, flutter which can occur for conductors experiencing a severe icing storm. It occurs in winds ranging from about 5 to 15 m/s. The frequency is usually in the range of 0.1 to 1 Hz and the peak to peak amplitude in the vertical direction may be as large as 10 m. For bundled conductors, the instability involves the bulk motion of conductor-spacer arrangements. Thus, it is in contrast with the subspan oscillation which is another type of dynamic instability involving essentially the relative motion between two conductors, one being placed in the wake of another. Although various methods of control have been proposed for conductor galloping, none of them have yet proved to be fully satisfactory. This is mainly due to the sporadic occurrence and the unpredictable nature of the phenomenon. In earlier investigations it had been identified as vertical galloping, more commonly known as the Den Hartog instability [4], for which a quasi-steady aerodynamic theory, account being taken of the steady lateral aerodynamic force with a downward gradient with incidence, had been well established [.5]. Although field observations [6] indicated that many cases of conductor galloping, particularly for bundled conductors, involve not only vertical but also significant torsional and horizontal motions of conductors, it was generally considered [6,7] that the latter two were inessential to promote and maintain the former. However, the work of Simpson and Lawson [S] was the first that demonstrated both analytically and experimentally that a two-conductor bundle is, under certain conditions, prone to dynamic instability, not identified as the Den Hartog instability but rather akin 10 classical flutter, involving large vertical, horizontal and torsional bulk motions. 363 c? 1980Academic Press Inc. ILondon I I.lmitrd 002?-3hOX/80,'230363+ 15 $02.00/O
364
Y NAKAMCIKA
More recently, there has been a renewed interest in the significant role played by the torsional motion in conductor galloping 19, lo]. This has led several investigators [9, 11, 121 to propose new mechanisms for conductor galloping, and the torsiorztrl mechanisms among others, which have been advocated by Nigol [O], are particularly relevant to the present investigation. It should be added that some large-scale field trials [13, 141 have recently been in progress to test new devices for controlling the torsional motion of conductors in the hope of suppressing galloping. On the other hand, while investigating bending-torsion flutter of bridge deck sections, the present author [lS] has pointed out that coupled flutter of bluff structures may be classified into three types depending upon the mode of energy transfer from fluid to structure: the three types of flutter mentioned are referred to as single degree of freedom type flutter, classical type and intermediate type, respectively. Siltgle degree of freedom type flutter may include the galloping and torsional type flutter, each corresponding to transverse galloping and torsional flutter in a single degree of freedom. It appears that one of the torsional mechanisms of Nigol is equivalent to the torsional type flutter mentioned above, while the other one is equivalent to the classical type flutter. This paper describes a wind tunnel investigation into the flutter of a two-dimensional section model of a two-conductor bundle, undertaken in order to get insight into the basic mechanisms of bundled conductor galloping in the field. To make the matter as simple as possible, a system allowing both vertical and torsional movements but with the horizontal one blocked was adopted in the experiment, in which ice-accreted conductors were replaced by two identical square prisms.
2. EQUATIONS
OF MOTION
OF ICE-ACCRETED
BUNDLED
CONDUCTORS
Consider a two-dimensional section of ice-accreted bundled conductors which allows both vertical and torsional movements, y and 19,but with the horizontal one blocked, in a uniform flow of speed V. This system is illustrated in Figure 1. For simplicity, the system is assumed to be balanced mechanically about its midchord rotation axis, and the structural dampings are also neglected. The linearized equations of motion for the system are mj; + m(2rfYo)*y = L[y]+L[H],
1di+r(2rf0”)2s
where m and I are the mass and mass moment
Figure
1. Bundled
of inertia
conductors
=M[y]+M[8],
per unit span, respectively,
in uniform
flow
(I, 2) fvO and
BUNDLED
POWER
LINE
365
GAL.I>OPING
are the still-air frequencies in the y and 0 degrees of freedom, respectively, L[ ] and M[ ] are the aerodynamic lifts and moments about the midchord, respectively, which act on the ice-accreted conductors, in which [ ] means that the aerodynamic lifts and moments are functionals of the arguments, and the dot denotes the differentiation with respect to the time t. The aerodynamic lifts and moments are respectively expressed in terms of the aerodynamic coefficients as
fHo
L[yl +Uol
= ~PV2hGW
131
I + U@ll>
:31
M[yl+M~81=~pV2hZ{~.~~ylhl+C.~~HII~ where p is the air density, h is a representative length of ice-accreted conductors, CI,[ ] and C,,[ ] are the aerodynamic lift and moment coefficients, respectively. The equations of motion are then non-dimensionalized for convenience: V”+(r*V = P/87r2cL){C, O”+R*c*O
is1
[r/l +C,.Lel),
= ( ~‘/%T~v){C,&]+
and
&,[O]},
IhI
where q(T) ==y(t)/h is the reduced vertical displacement, T = 2n-ft is the reduced time, in which f is the frequency of oscillation in wind, CL= ml(ph *) is the reduced mass, v = I/Q/I’) is the reduced mass moment of inertia, R = fHo/fso is the uncoupled frequency ratio, (+ = f,o/f is the frequency ratio, v = V/(fh) is the reduced wind speed, and the dash denotes the differentiation with respect to T. As in flutter analysis of bridge deck sections [ 151, the solution of equations (5 1and (6 1is assumed in a form of exponentially modified oscillation: ~(‘r“) = 70 exp K(P/2~)
+
iIT),
OiTI=O,,exp{[(P/27~)+i]T+id},
17.81 ! 9’1
771,/o,, = x. where q. the phase As was modified oscillations
and O. are the amplitudes, p is the logarithmic rate of growth of oscillation, d, is angle between q(T) and O(T), and i = t--l)’ ‘. shown in reference [15], the aerodynamic lifts and moments for exponentially oscillations may be replaced by those which correspond to purely sinusoidal on the condition that I/3\ K 1. For example. Cr.[q] is given by
CJ~] = CC;,, +iC, ,,L)VIT), where CLllR and CL,,, are the real and imaginary parts of the frequency response function. both of which are functions of the reduced wind speed I? Introduction of equations (7)-(9) into equations (5) and (6) yields the following four equations, when the exponential function exp [iT] is eliminated from both sides of the equations and then the real and imaginary parts are separated: [(p2/4r2)
- 1 + f12]X = ( V2/S7rTT:‘~)(CLVKX + C,.,,, cos 6 - C,,I
(p’/&r*) - 1 + R*a* = ( ~*/87r2v)((CM,, pX=(v2/8~~)(Cr,lX+CLHR p = ( v2/8~v)[(-CM,R
sin 41,
cos d + CM,, sin d)X + CMO, 1. sincb+CLHI
sin d + &,,I
COS&J),
cos d)X + C,,,l.
ill! i 121 (131 (141
Equations. (ll)-(14) can be solved to give cr, X, /3 and 4 provided the aerodynamic coefficients are specified as functions of I? However, a full solution will not be sought in this paper because one cannot specify all the aerodynamic coefficients at present. Instead
366
Y.
NAKAML’KA
an attempt will be made to get some approximate formulae under restricted conditions to help understand the basic mechanisms of the instability. In particular, equations I1 I) and (12) may be replaced respectively by (-1 +r’)X
-1 + R2a2 = ( v2/8~2~)[(CM,, on the assumption
sin ~$1,
cos 4 + C,+,, sin 4)X + C,w,,],
that IP I<< 1.
3. THREE After
cos d -CL,,
= (v’/8~2~)(C,,,X+C,,,,
a little manipulation P = [v2/87+
TYPES OF COUPLED
of equations
FLUTTER
(13) and (14), one gets
+~LX~MC~I + (Cuv + Cw,,,)X~0s4 + G,rX*l
+ (-Gfv~ + G0,)X sin 41 (17)
=P1+P2.
where PI=
[v*/Sr(v
+ELX*W~~ +(CLHI+ G,,I)X cos 4 +&,X21,
p2 = [ v2/87r(v
+ pX2)](-CM,,R
+ C,,,)X
sin 4.
(18) (19,
It is easy to see from equations (18) and (19) that p1 is associated with the work done by the aerodynamic damping forces and moments while p2 is associated with the work done by the aerodynamic stiffness forces and moments. Also, these equations show in a simple way how phasing between the two degrees of freedom controls the stability of the system. Thus, a classification of coupled flutter [15] may be possible on the basis of equation (17): (1) single degree of freedom type flutter if p, >>p2; (2) classical type flutter if p1 <
which shows that the oscillation quasi-steady aerodynamic theory
is identical is applicable. P = PI=
with transverse galloping to which If X = 0, on the other hand, then
( ~2/hKwo~,
the
(21)
which shows that the oscillation is torsional flutter for which the quasi-steady aerodynamic theory may not work but the fluid memory effect may play an essential role [16]. Furthermore, if d = 7r/2 or -7r/2, p2 is usually dominant over pl, and this offers the most typical example of classical type flutter. Single degree of freedom type flutter can occur typically for bluff structures with separated flow, while flutter of an unseparated thin aerofoil is typically of the classical type. Since ice-accreted bundle conductors are very bluff, one may anticipate that galloping type and torsional type flutter can occur. However, one may also anticipate that classical type flutter can occur. This is because the uncoupled frequency ratio for bundled conductors is always close to one so that the aerodynamic coupling between the two degrees of freedom may become exceptionally large.
BUNDLED
4. EXPERIMENTAL
POWER
367
LJNE GAI.I.OPING
APPARATUS
AND PROCEDURE
The experiment was conducted in a low-speed wind tunnel with a 3 m high by 0.7 m wide working section. A two-dimensional section model of an ice-accreted two-conductor bundle consisting of two identical square prisms was used in the experiment. Standard full-scale conditions which were assumed for the design of the model are listed in Table 1, in which 1 is the conductor spacing and h is the length of the diameter of a conductor plus ice thickness. TABLE
1
Relevant parameters for assumed full-scale system m (kg/m)
1 (cm)
h (cm)
f,o (Hz)
fw (Hz)
V (m/s,
4.06
40
6
0.4
0.4
10
The model was i-scaled where I=20 cm and h = 3 cm, in which h is the length of the side of the square prism. The square prisms were 65 cm in length, and were constructed by shaping wooden backbones with lightweight foam plastics. The model had two rectangular end plates so as to make the flow around it as two-dimensional as possible. It was suspended horizontally at its two ends, the suspenders being placed outside the working section. As illustrated in Figure 2, the suspender consisted of a horizontal bar 0, a flexure spring 0, a vertical frame @, main and auxiliary coil springs 0 and @ and restraining wires 8. One end of the flexure spring was attached to the horizontal bar, and the other end to the vertical frame, the centre of the flexure spring being in line with that of the model. The horizontal springs were relatively hard so that the vertical frame could move only vertically with horizontal and torsional movements, if any, neglected. The model was thus allowed either vertical or torsional movement or both but the horizontal movement was blocked.
Figure 2. Model and suspender. 0, Model; a, horizontal bar; 0, coil spring; 8, auxiliary coil spring; 8, horizontal restraining wire.
Rexure spring;
@I, vertical
frame:
0,
main
368
1’. NAKAMIJRA
If the distance between the auxiliary coil springs was varied, the torsional frequency was changed but the frequency of the vertical translation remained unchanged. This controlled the uncoupled frequency ratio R. As noted earlier, one of the important characteristics of bundled conductors is that R = 1. Therefore, fYo= 1.56 Hz and fr,,, = 1.09 to 1.76 Hz, approximately, and fvo = 1.32 Hz and fHo = 1.78 Hz, approximately, were chosen in the present experiment. This resulted in R = 0.7 to 1.13 and R = 1.34, approximately. The range of wind speed was V = 5 to 20 m/s, approximately. Values of the relevant non-dimensional parameters for the assumed full-scale system and for the model are given in Table 2. The structural dampings of the model were in a range of S, = 0,017 to 0.022, and 8, = 0,020 to 0.042, approximately, while the experimental range of the Reynolds number, based on h, was Re = (1 to 4) x 104, approximately. The model was mounted in the working section with the front face normal to the flow: in other words, the mean incidence of the model was set equal to zero.
TABLE
Relevant non-dimensional
1x10’
Full-scale
parameters for assumed full-scale system and for model Y
P
Model
2
2.5 x 10’
1x10” 6x10”
R
v
1.0 0.7-1.13 and 1.34
416 106 to 410
In most experiments, the instability built up spontaneously from rest to such a large amplitude as would result in a failure of the apparatus unless suppressed by hand. Accordingly, attention was focussed on the build up and decay of oscillation at low amplitudes, mostly in a range of y. < 40 mm and 13~) < 10 deg, approximately. The vertical displacement of the model was detected by the use of an optical displacement follower (EMNEK 102) while the torsional one was measured by the use of the strain-gauges which were cemented on the flexure spring. The displacement signals were from the exp [At] type was mostly of exp [At] type where A = const., but departure sometimes observed in the torsional signal. The frequency and the rate of growth of oscillation corresponding to & = 3 deg, approximately, were determined from the displacement signals.
5. EXPERIMENTAL 5.1.
TRANSVERSE
RESULTS
AND
DISCUSSIONS
GALLOPING
A system allowing only vertical movement of the model was obtained with the flexure springs locked. As had been expected, the onset of transverse galloping was observed with this system. Figure 3 shows that the instability was rather mild and the rate of growth of oscillation gradually increased with v, while Figure 4 shows that the frequency of oscillation, which is denoted by f,, remained almost unchanged over a range of the wind speed tested. If it is assumed that C,,, then, on the basis of the quasi-steady
= (2~1 v’) dC&da, aerodynamic
theory in which dC,,/da
(22) is the gradient
BUNDLED
POWER
369
GAL1 OPING
I
I
I
200
300
400
I
O h* 100
LINE
v
Figure
3. Frequency
cs. reduced
wind speed for vertical
translation
I
I
I
I
1
I
0
5
I IO
I '5
20
system.
2.0
,
I.5 ;; I e
I.0 ‘7 I
1
v (m/s)
Figure
4. Rate of growth
of the lateral aerodynamic equation (20),
us. reduced
force
wind speed for vertical
coefficient
d&/da
(positive
= (4~ul GM
translation
downward),
+ 151,
system
one
obtains,
from
(23)
where the efIect of the structural damping is taken into accout. The variation of the measured dC,,/dcz with v is shown in Figure 5, where it is indicated that the quasi-steady assumption is satisfied for a range of v > 200, approximately, where dC,,/da is constant and equal to 2.0, approximately. 5.2. TORSIONAL
FLUTTER
A system allowing only torsional movement about the midchord of the model was obtained with the vertical frame locked. It was found that the oscillations were slightly unstable b-it with relatively low irregular amplitudes so that the measurement of the rate of growth of vscillation was difficult.
370
‘I’.
NAKAMURA
J-r 4-o -
s\
4
3.0-
\ j
c c1
0
Cl
0 ‘0
2.0-
‘\.+_
I.0 -
I O L h* 100
j
n
I 200
-
I
I
300
4cNJ
_
V
Figure
5. dC&da
I
0
obtained
by use of quasi-steady
aerodynamic
1
I
I
I
5
IO
15
20
theory.
V (m/s)
Figure
6. Frequency
us. wind speed for torsional
systems
with various
torsional
stiffnesses.
As is shown in Figure 6, the frequency of oscillation increased with the wind speed. The aerodynamic moment coefficient C MORwas obtained by using the relation Cm‘jR= (89r2v/ v2)(-1 + R2& where cre = fYo/fe, in which fe denotes the frequency of oscillation for the torsional system with an axis of rotation at the midchord. Figure 7 indicates the variation of CMORwith v; the magnitude of CMORdecreases slowly with V,suggesting that, unlike the behaviour for CLllh a constant value of CMORhad not yet been reached in the range of wind speed tested. It is useful for later consideration to see how stability was changed with the position of axis of rotation. It was found that the oscillation built up rapidly if the axis position was located downstream of the midchord, while it was damped rapidly if the axis position was located upstream of the midchord. Figure 8 shows the results of measurement of p for two such positions of axis of rotation.
BUNDLED
LINE
zoo
371
GALLOPING
I
I 400
I 300
I
I 100
0
POWER
v
Figure
7. C,,,,
obtained
from frequency
h
”
measurement.
Symbols
I/--_-L-i
1 100
200
correspond
300
to those used in Fipurc
h
400
v
Figure positions.
8. Rate of growth
5.3.COUPLED
t’s. reduced
FLUTTER
FOR
wind speed for torsional
systems
corresponding
to two different
axls
R < 1
It was found that there was a remarkable difference in the instability characteristics hetween the cases R < 1 and R > 1. The results for R = 0.8,which were representative of the case R < 1,were as follows. At low wind speeds, the instability was substantially equivalent to transverse galloping: namely, it was the vertical oscillation having a frequency CT= 1 with sporadic torsional movements ha.ving a frequency (T = uO. With an increase in v, a torsional movement with the same frequency as that of the vertical one manifested itself: that is, the oscillation became strongly coupled, as is indicated in the oscillogram of Figure 9. The first point of interest about the characteristics of this instability is that the rate of growth of oscillation p increased rapidly with v, as is shown in Figure 10. The values of p shown in Figure 10 are much larger than those of transverse galloping shown in Figure 3. The second point of interest is the variation of the frequency with v which is shown in Figure 11. It appears that the frequency variation was associated with the resonance wind speed v,, which is defined as the wind speed at which fH= f, ,,, and can be expressed, by putting (T = 1 in equation (24 1, ilS
372
Y.
Figure
9. Oscillograph
showing
NAKAMLIKA
build up of coupled
instability.
R = O~XO, v = 364
The frequency, which had been constant and equal tof,,,, approximately, began to increase at around vr,, following fairly closely along the curve of fo.In other words, the effect of the aerodynamic coupling on the frequency variation was found to be rather small. The third point of interest is the variation of the phase angle 4 with v, which is given in Figure 12 for three values of R. The phase angles were equal to about 7r/2 at low wind speeds but began to increase just before the resonance wind speed. In particular, the values of C$at high wind speeds for R = 0.92 were approximately constant and equal to ?r. As was mentioned earlier, coupled flutter may be classified by a comparison between /3, and & in equation (16). There is no doubt that the observed instability with C$= 7r/2 belongs to classical type flutter. However, it may not necessarily be concluded that the observed instability with 4 = 7r for R = 0.92 belongs to torsional type flutter. This is because p2 when sin C$is of small magnitude may often exceed pl. A comparison was made for the case where R = O-92 and v = 370. It was experimentally shown that /3 = O-40 and X = 13.0, approximately. On this basis, an experiment with a torsional system with an axis of rotation X = 13.3 downstream of the midchord of the model was added. As is indicated in Figure 8, the result was that PI = 0.27, approximately, which is roughly comparable with
Figure
10. Rate of growth
0s. reduced
wind speed for coupled
system
with R = 0.80
BUNDLED
POWER
LINE
373
GALLOPING
V (m/s)
Figure
‘, 1. Frequency
0
bq
vs. reduced
wind speed for coupled
’
’
100
system with R = 0.80. - - -, f+
--_----_L--.-------__i--2 300
200
400
v
Figure 12. Phas’e angle US.reduced R = 0~80; V, R = 17.92.
wind speed for three values of uncoupled
frequency
ratio. ‘3, R = 0.70; i’ .
p = 0.40 for the coupled system. This seems to justify the view that the observed for R = 0.92 at V = 370 belongs to torsional type flutter 5.4.
COUPLED
FLUTTER
FOR
instability
R >1
The frequency variations with e for R > 1 were found to be similar to those for R < 1. As is exemplified in Figure 13, the frequency of the torsional branch was nearly equal to f,,, while that of the branch of the vertical translation was nearly constant and equal to f,,,). Thus, the case 19 > 1 was characterized by the fact that the difference in frequency between the two branches increased with the wind speed so that there was no resonance wind speed. At low wind speeds, both of the two branches with different frequencies became unstable. In particular, when R was close to one, the instability built up with a beat, since the two frequencies were close to each other. Figure 14 shows the displacement signals for
374
Y.
NAKAMLJKA
I
2.0
I
1
I
-
,J
.j
/’
f, ,/’
I
I 20
J
_H _----
/j
I
I
I 5
0
I
IO
15 V (m/s1
Figure
Figure
13. Frequency
14. Oscillograph
us. wind speed for coupled
showing
system
build up of instability
with
with beat.
R = 1.34. - - -, fH
R = 1.04, v = 147.
R = 1.04. It is seen that the vertical oscillation built up with a beat while there was only a slight indication of it in the torsional signal because of the unsymmetric aerodynamic coupling. When the wind speed increased, the torsional branch became increasingly dominant so that the signal of the vertical translation became more distorted.
6. ANALYSIS
OF
COUPLED
FLUTTER
The experimental results described in the preceding section suggest that the characteristics of coupled flutter changed drastically as the wind speed increased through the resonance speed. An analysis is presented to shed some light on this situation.
BUNDLED One
may
start
with the instability
POWER
LINE
375
GALLOPING
with u = 1. Substituting
(24) into equation
equation
( 16), one gets ( V2/8r21/)(-CMVR
cos 4 + CM,,, sin d)X
= R*(v*-v~).
(261
Equation (26) !;uggests that, because X must be positive, the phase angle is sensitively dependent O!I the signs of CM-R, CM,,, and cr2- (T:. For CM,,R and Cm+ the following relations hold to a good approximation for most bluff bodies [16]: C&&$7= -(27rl
V)Ger,
C&IV1= (27rf wh,.
In the case where R < 1 and v < vr, it is assumed, as corresponding to the present experiment, that CM*, < 0 and u = 1 so that C,+I < 0 and a* - (T: = 1 - cri < 0. Therefore. 4 = ~12 with cr = 1, which is in accordance with the experimental observations, satisfies equation (26). lt follows that X = 47rvR*( 1 -a;
(271
)/ ~Z’,VU,K,
which shows that X decreases from infinity as v increases from zero. Correspondingly, substitution of equations (22) and (27) into equation (13 ) leads to p = (v/4p)(dC&da)
+ ~3C,,,C,,,j32.rr2~vR2(1
-a$).
(28)
When v exceeds vr, cr’-cr’, changes sign to be positive so that 4 = --r/2 with v = 1 is suggested for equation (26). It follows that equation (28) holds also for v < v,. Obviously, the second term on the right-hand side of equation (28) represents the effect of the aerodynamic coupling, and it becomes dominant over the first term as v increases. Moreover, it changes sign as vr is passed, although equation (28) fails at v,. Keeping in mind that CL, >>0 (this holds because CLeR = dC,,/dcy + C, for large values of v, where CD is the drag coefficient), one easily sees that the oscillation is strongly unstable for p < v, due to the aerodynamic coupling, while it is stable for v> v, due to the aerodynamic coupling. In short, the aerodynamic coupling can play a vital role in the stability of bundled conductors when the resonant condition is approached. A similar analysis may be made for the oscillation with CT= crH.Instead of equation (261. the following relation is considered: (-1 +a2)X
= (v2/8~‘~)(C’,,,
cos d -CL,,
Since CLeR > 0 and -1 + (+i > 0 for v < v,., it is obvious equation (29). Accordingly, X = ~*C,,,/87r*/.~(cr; which shows that X increases
from zero as v increases
/3 = ( ~2/87r~)C,,,
sin 4).
that 4 = 0 with u = (TVsatisfies
- l),
!30)
from zero. Equation
+ Q3C,,,C,,,/32~2~~~~~
(29)
- l),
(14) now gives (311
which indicates that although C MBI is slightly positive, the oscillation is eventually damped due to the aerodynamic coupling as v increases from zero. When e> vr, on the other hand, 4 = 7r is suggested. It follows that equation (31) also holds for e> v,, where it is indicated that the oscillation is strongly unstable due to the aerodynamic coupling. In the experiment, it was observed that the instability with (+ = 1 occurred initially, but that with a gradual increase in frequency it shifted near v, to the instability with u = rH, During this shift p increased continuously with v but an increase beyond limit, as indicated by equation (28), was avoided. The prediction of the instability characteristics near vr would require a more precise evaluation of both the frequency and the phase angle
376
Y NAKAMURA
which in turn would require the specification of some more aerodynamic coefficients concerned. The analysis of the instability for R > 1 may be made in the same way. It turns out that the oscillation with cr = ug is unstable mainly due to the aerodynamic coupling while the oscillation with v = 1, though it is unstable initially, becomes eventually stable due to the aerodynamic coupling as v increases. It appears that the trend predicted so far is in agreement with the experimental observations.
7. CONCLUSIONS In this paper an experimental and analytical study has been described of the galloping of a two-dimensional section model of a two-conductor bundle in which ice-accreted conductors were replaced by two identical square prisms, with both vertical and torsional movements allowed but the horizontal one blocked, in a uniform wind tunnel flow. Emphasis has been placed on elucidating the vital role played by the aerodynamic coupling in the stability of bundled conductors. It has been shown that as well as galloping type flutter two other types of instability, namely, torsional and classical type flutter, can occur for bundled conductors. In particular, it has been shown that the aerodynamic coupling can cause violent classical type flutter to occur when the resonant condition is approached. The elimination of this unfavourable effect of resonance should be one of the most urgent problems in the design of bundled power line conductors.
ACKNOWLEDGMENT The author is grateful to Messrs S. Sakabe and Y. Matsubayashi for supplying valuable information on the field observations. He is also grateful to Messrs K. Watanabe and H. Kamimura for their assistance in conducting the experiment.
REFERENCES 1. A. T. EDWARDS 1970 Electra 12, 3148. Conductor galloping. 2. Proceedings of the Institute of Electrical and Electronic Engineers Power Engineering Society Summer Meeting 1979. Symposium on Mechanical Oscillations of Overhead Conductors, 15 July 1979. 3. Proceedings of the Institution of Electrical Engineers Power Division 1979. 2nd International Conference on Progress in Cables and Overhead Lines for 220 kV and above, 4-6 September 1979. 4. J. P. DEN HARTOG 1956Mechanical Vibrations. New York: McGraw-Hill. See pp. 299-305. 5. G. V. PARKINSON 1965 Proceedings of the Conference on Wind Effects on Buildings and Structures 2, 581-609. Aeroelastic galloping in one degree of freedom. London: Her Majesty’s Stationery Office. 6. A. T. EDWARDS and A. MADEYSKI 1956 Institute of Electrical and Electronic Engineers Transactions on Power Apparatus and Systems PAS-75, 666-686. Progress report on the investigation of galloping of transmission line conductors. 7. J. J. RATKOWSKI 1963 Institute of Electrical and Electronic Engineers Transactions on Power Apparatus and Systems PAS-82, 661-669. Experiments with galloping spans. 8. A. SIMPSON and T.V. LAWSON 1968Proceedings of Symposium on Wind Effects on Buildings and Structures 2, Loughborough University of Technology, Paper 25. Oscillations of “twin” power transmission lines. 9. 0. NIGOL and G. J.CLARKE 1974 Institute of Electrical and Electronic Engineers Power Engineering Society Winter Meeting Paper C74 016-2. Conductor galloping and control based on torsional mechanism.
BUNDLED
POWER
LINE
GALLOPING
377
10. K. ANJO,!;.YAMASAKI,Y.MATSUBAYASHI,Y. NAKAYAMA, A. OTSUKI and T. FUJIMURA 1974 Conference Internationale des Grands Reseaux Electriques a Houte Tension, CIGRE 22-04. An experimental study of bundle conductor galloping on the Kasatori-yama test line for bulk power transmission. Il. H. TAKAHASHI,R.KASHIMURA, A. OTSUKI,K.NUMATA,Y. KOJIMA and K. KAWAI 1974 Fujikura Technical Report Special Issue on Overhead Transmission Lines, 63-78. Overhead transmission line galloping and preventive measures (in Japanese). 12. Y. MATSUBAYASHI,I.MATSUBARA and T. OKUMURA 1978 Sumitomo Electric Industries Ltd Technical Paper No. KEHL 698-A. Torsion controlling type of galloping damper iin Japanese). 13. D. G. HARVARD and J.C. POHLMAN 1979 Institute of Electrical and Electronic Engineers Power Engineering Society Summer Meeting, Paper A79 499-5. Field trials of detuning pendulum for controlling galloping of single and bundle conductors. 14. S. SAKABE,Y.MATSUBAYASHI,T.OKUMURA~~~K.SHIMOJIMA~~~O InstituteofElectric and Electronic Engineers Power Engineering Society Winter Meeting. Development of SO0 kV low-inductance overhead transmission line. 15. Y. NAKA~V~URA 1978 Journal of Sound and Vibration 57, 471-482. An analysis of binarv flutter of bridge deck sections. 16. Y. NAKA~V~URA 1979 Journal of Sound and Vibration 67, 163-177. On the aerodynamic mechanism of torsional flutter of bluff structures.
GALLOPING
OF BUNDLED
POWER
LINE CONDUCTORS
Y. NAKAMURA Research Institute for Applied Mechanics, Kyushu Crniversity, Fukuoka, Japan (Received 25 January
1980, and in recised form 10 May 1980)
This palper describes an experimental and analytical study of the galloping of a twodimensional section model of a two-conductor bundle in which ice-accreted conductors are replaced by two identical square prisms, with both vertical and torsional movements allowed but the horizontal one blocked, in a uniform wind tunnel flow. Emphasis is placed on elucidating the vital role played by the aerodynamic coupling in the stability of bundled conductors. It is shown that, apart from galloping type flutter, two other types of instability, namely, torsional and classical type flutter, can also occur for bundled conductors. In particular. it is shown that the aerodynamic coupling can cause violent classical type flutter to occur when the resonant condition is approached.
1. INTRODUCTION
The low frequency and high amplitude galloping of conductors is a serious problem in the design and operation of overhead power line conductors. Over the years considerable effort has been made to identify the galloping mechanisms and to find a solution to this problem. A general review of the problem is given in reference [l] while the more up to date information can be found in references [2] and 131. It is generally agreed that conductor galloping is a dynamic aeroelastic instability: namely, flutter which can occur for conductors experiencing a severe icing storm. It occurs in winds ranging from about 5 to 15 m/s. The frequency is usually in the range of 0.1 to 1 Hz and the peak to peak amplitude in the vertical direction may be as large as 10 m. For bundled conductors, the instability involves the bulk motion of conductor-spacer arrangements. Thus, it is in contrast with the subspan oscillation which is another type of dynamic instability involving essentially the relative motion between two conductors, one being placed in the wake of another. Although various methods of control have been proposed for conductor galloping, none of them have yet proved to be fully satisfactory. This is mainly due to the sporadic occurrence and the unpredictable nature of the phenomenon. In earlier investigations it had been identified as vertical galloping, more commonly known as the Den Hartog instability [4], for which a quasi-steady aerodynamic theory, account being taken of the steady lateral aerodynamic force with a downward gradient with incidence, had been well established [.5]. Although field observations [6] indicated that many cases of conductor galloping, particularly for bundled conductors, involve not only vertical but also significant torsional and horizontal motions of conductors, it was generally considered [6,7] that the latter two were inessential to promote and maintain the former. However, the work of Simpson and Lawson [S] was the first that demonstrated both analytically and experimentally that a two-conductor bundle is, under certain conditions, prone to dynamic instability, not identified as the Den Hartog instability but rather akin 10 classical flutter, involving large vertical, horizontal and torsional bulk motions. 363 c? 1980Academic Press Inc. ILondon I I.lmitrd 002?-3hOX/80,'230363+ 15 $02.00/O
364
Y NAKAMCIKA
More recently, there has been a renewed interest in the significant role played by the torsional motion in conductor galloping 19, lo]. This has led several investigators [9, 11, 121 to propose new mechanisms for conductor galloping, and the torsiorztrl mechanisms among others, which have been advocated by Nigol [O], are particularly relevant to the present investigation. It should be added that some large-scale field trials [13, 141 have recently been in progress to test new devices for controlling the torsional motion of conductors in the hope of suppressing galloping. On the other hand, while investigating bending-torsion flutter of bridge deck sections, the present author [lS] has pointed out that coupled flutter of bluff structures may be classified into three types depending upon the mode of energy transfer from fluid to structure: the three types of flutter mentioned are referred to as single degree of freedom type flutter, classical type and intermediate type, respectively. Siltgle degree of freedom type flutter may include the galloping and torsional type flutter, each corresponding to transverse galloping and torsional flutter in a single degree of freedom. It appears that one of the torsional mechanisms of Nigol is equivalent to the torsional type flutter mentioned above, while the other one is equivalent to the classical type flutter. This paper describes a wind tunnel investigation into the flutter of a two-dimensional section model of a two-conductor bundle, undertaken in order to get insight into the basic mechanisms of bundled conductor galloping in the field. To make the matter as simple as possible, a system allowing both vertical and torsional movements but with the horizontal one blocked was adopted in the experiment, in which ice-accreted conductors were replaced by two identical square prisms.
2. EQUATIONS
OF MOTION
OF ICE-ACCRETED
BUNDLED
CONDUCTORS
Consider a two-dimensional section of ice-accreted bundled conductors which allows both vertical and torsional movements, y and 19,but with the horizontal one blocked, in a uniform flow of speed V. This system is illustrated in Figure 1. For simplicity, the system is assumed to be balanced mechanically about its midchord rotation axis, and the structural dampings are also neglected. The linearized equations of motion for the system are mj; + m(2rfYo)*y = L[y]+L[H],
1di+r(2rf0”)2s
where m and I are the mass and mass moment
Figure
1. Bundled
of inertia
conductors
=M[y]+M[8],
per unit span, respectively,
in uniform
flow
(I, 2) fvO and
BUNDLED
POWER
LINE
365
GAL.I>OPING
are the still-air frequencies in the y and 0 degrees of freedom, respectively, L[ ] and M[ ] are the aerodynamic lifts and moments about the midchord, respectively, which act on the ice-accreted conductors, in which [ ] means that the aerodynamic lifts and moments are functionals of the arguments, and the dot denotes the differentiation with respect to the time t. The aerodynamic lifts and moments are respectively expressed in terms of the aerodynamic coefficients as
fHo
L[yl +Uol
= ~PV2hGW
131
I + U@ll>
:31
M[yl+M~81=~pV2hZ{~.~~ylhl+C.~~HII~ where p is the air density, h is a representative length of ice-accreted conductors, CI,[ ] and C,,[ ] are the aerodynamic lift and moment coefficients, respectively. The equations of motion are then non-dimensionalized for convenience: V”+(r*V = P/87r2cL){C, O”+R*c*O
is1
[r/l +C,.Lel),
= ( ~‘/%T~v){C,&]+
and
&,[O]},
IhI
where q(T) ==y(t)/h is the reduced vertical displacement, T = 2n-ft is the reduced time, in which f is the frequency of oscillation in wind, CL= ml(ph *) is the reduced mass, v = I/Q/I’) is the reduced mass moment of inertia, R = fHo/fso is the uncoupled frequency ratio, (+ = f,o/f is the frequency ratio, v = V/(fh) is the reduced wind speed, and the dash denotes the differentiation with respect to T. As in flutter analysis of bridge deck sections [ 151, the solution of equations (5 1and (6 1is assumed in a form of exponentially modified oscillation: ~(‘r“) = 70 exp K(P/2~)
+
iIT),
OiTI=O,,exp{[(P/27~)+i]T+id},
17.81 ! 9’1
771,/o,, = x. where q. the phase As was modified oscillations
and O. are the amplitudes, p is the logarithmic rate of growth of oscillation, d, is angle between q(T) and O(T), and i = t--l)’ ‘. shown in reference [15], the aerodynamic lifts and moments for exponentially oscillations may be replaced by those which correspond to purely sinusoidal on the condition that I/3\ K 1. For example. Cr.[q] is given by
CJ~] = CC;,, +iC, ,,L)VIT), where CLllR and CL,,, are the real and imaginary parts of the frequency response function. both of which are functions of the reduced wind speed I? Introduction of equations (7)-(9) into equations (5) and (6) yields the following four equations, when the exponential function exp [iT] is eliminated from both sides of the equations and then the real and imaginary parts are separated: [(p2/4r2)
- 1 + f12]X = ( V2/S7rTT:‘~)(CLVKX + C,.,,, cos 6 - C,,I
(p’/&r*) - 1 + R*a* = ( ~*/87r2v)((CM,, pX=(v2/8~~)(Cr,lX+CLHR p = ( v2/8~v)[(-CM,R
sin 41,
cos d + CM,, sin d)X + CMO, 1. sincb+CLHI
sin d + &,,I
COS&J),
cos d)X + C,,,l.
ill! i 121 (131 (141
Equations. (ll)-(14) can be solved to give cr, X, /3 and 4 provided the aerodynamic coefficients are specified as functions of I? However, a full solution will not be sought in this paper because one cannot specify all the aerodynamic coefficients at present. Instead
366
Y.
NAKAML’KA
an attempt will be made to get some approximate formulae under restricted conditions to help understand the basic mechanisms of the instability. In particular, equations I1 I) and (12) may be replaced respectively by (-1 +r’)X
-1 + R2a2 = ( v2/8~2~)[(CM,, on the assumption
sin ~$1,
cos 4 + C,+,, sin 4)X + C,w,,],
that IP I<< 1.
3. THREE After
cos d -CL,,
= (v’/8~2~)(C,,,X+C,,,,
a little manipulation P = [v2/87+
TYPES OF COUPLED
of equations
FLUTTER
(13) and (14), one gets
+~LX~MC~I + (Cuv + Cw,,,)X~0s4 + G,rX*l
+ (-Gfv~ + G0,)X sin 41 (17)
=P1+P2.
where PI=
[v*/Sr(v
+ELX*W~~ +(CLHI+ G,,I)X cos 4 +&,X21,
p2 = [ v2/87r(v
+ pX2)](-CM,,R
+ C,,,)X
sin 4.
(18) (19,
It is easy to see from equations (18) and (19) that p1 is associated with the work done by the aerodynamic damping forces and moments while p2 is associated with the work done by the aerodynamic stiffness forces and moments. Also, these equations show in a simple way how phasing between the two degrees of freedom controls the stability of the system. Thus, a classification of coupled flutter [15] may be possible on the basis of equation (17): (1) single degree of freedom type flutter if p, >>p2; (2) classical type flutter if p1 <
which shows that the oscillation quasi-steady aerodynamic theory
is identical is applicable. P = PI=
with transverse galloping to which If X = 0, on the other hand, then
( ~2/hKwo~,
the
(21)
which shows that the oscillation is torsional flutter for which the quasi-steady aerodynamic theory may not work but the fluid memory effect may play an essential role [16]. Furthermore, if d = 7r/2 or -7r/2, p2 is usually dominant over pl, and this offers the most typical example of classical type flutter. Single degree of freedom type flutter can occur typically for bluff structures with separated flow, while flutter of an unseparated thin aerofoil is typically of the classical type. Since ice-accreted bundle conductors are very bluff, one may anticipate that galloping type and torsional type flutter can occur. However, one may also anticipate that classical type flutter can occur. This is because the uncoupled frequency ratio for bundled conductors is always close to one so that the aerodynamic coupling between the two degrees of freedom may become exceptionally large.
BUNDLED
4. EXPERIMENTAL
POWER
367
LJNE GAI.I.OPING
APPARATUS
AND PROCEDURE
The experiment was conducted in a low-speed wind tunnel with a 3 m high by 0.7 m wide working section. A two-dimensional section model of an ice-accreted two-conductor bundle consisting of two identical square prisms was used in the experiment. Standard full-scale conditions which were assumed for the design of the model are listed in Table 1, in which 1 is the conductor spacing and h is the length of the diameter of a conductor plus ice thickness. TABLE
1
Relevant parameters for assumed full-scale system m (kg/m)
1 (cm)
h (cm)
f,o (Hz)
fw (Hz)
V (m/s,
4.06
40
6
0.4
0.4
10
The model was i-scaled where I=20 cm and h = 3 cm, in which h is the length of the side of the square prism. The square prisms were 65 cm in length, and were constructed by shaping wooden backbones with lightweight foam plastics. The model had two rectangular end plates so as to make the flow around it as two-dimensional as possible. It was suspended horizontally at its two ends, the suspenders being placed outside the working section. As illustrated in Figure 2, the suspender consisted of a horizontal bar 0, a flexure spring 0, a vertical frame @, main and auxiliary coil springs 0 and @ and restraining wires 8. One end of the flexure spring was attached to the horizontal bar, and the other end to the vertical frame, the centre of the flexure spring being in line with that of the model. The horizontal springs were relatively hard so that the vertical frame could move only vertically with horizontal and torsional movements, if any, neglected. The model was thus allowed either vertical or torsional movement or both but the horizontal movement was blocked.
Figure 2. Model and suspender. 0, Model; a, horizontal bar; 0, coil spring; 8, auxiliary coil spring; 8, horizontal restraining wire.
Rexure spring;
@I, vertical
frame:
0,
main
368
1’. NAKAMIJRA
If the distance between the auxiliary coil springs was varied, the torsional frequency was changed but the frequency of the vertical translation remained unchanged. This controlled the uncoupled frequency ratio R. As noted earlier, one of the important characteristics of bundled conductors is that R = 1. Therefore, fYo= 1.56 Hz and fr,,, = 1.09 to 1.76 Hz, approximately, and fvo = 1.32 Hz and fHo = 1.78 Hz, approximately, were chosen in the present experiment. This resulted in R = 0.7 to 1.13 and R = 1.34, approximately. The range of wind speed was V = 5 to 20 m/s, approximately. Values of the relevant non-dimensional parameters for the assumed full-scale system and for the model are given in Table 2. The structural dampings of the model were in a range of S, = 0,017 to 0.022, and 8, = 0,020 to 0.042, approximately, while the experimental range of the Reynolds number, based on h, was Re = (1 to 4) x 104, approximately. The model was mounted in the working section with the front face normal to the flow: in other words, the mean incidence of the model was set equal to zero.
TABLE
Relevant non-dimensional
1x10’
Full-scale
parameters for assumed full-scale system and for model Y
P
Model
2
2.5 x 10’
1x10” 6x10”
R
v
1.0 0.7-1.13 and 1.34
416 106 to 410
In most experiments, the instability built up spontaneously from rest to such a large amplitude as would result in a failure of the apparatus unless suppressed by hand. Accordingly, attention was focussed on the build up and decay of oscillation at low amplitudes, mostly in a range of y. < 40 mm and 13~) < 10 deg, approximately. The vertical displacement of the model was detected by the use of an optical displacement follower (EMNEK 102) while the torsional one was measured by the use of the strain-gauges which were cemented on the flexure spring. The displacement signals were from the exp [At] type was mostly of exp [At] type where A = const., but departure sometimes observed in the torsional signal. The frequency and the rate of growth of oscillation corresponding to & = 3 deg, approximately, were determined from the displacement signals.
5. EXPERIMENTAL 5.1.
TRANSVERSE
RESULTS
AND
DISCUSSIONS
GALLOPING
A system allowing only vertical movement of the model was obtained with the flexure springs locked. As had been expected, the onset of transverse galloping was observed with this system. Figure 3 shows that the instability was rather mild and the rate of growth of oscillation gradually increased with v, while Figure 4 shows that the frequency of oscillation, which is denoted by f,, remained almost unchanged over a range of the wind speed tested. If it is assumed that C,,, then, on the basis of the quasi-steady
= (2~1 v’) dC&da, aerodynamic
theory in which dC,,/da
(22) is the gradient
BUNDLED
POWER
369
GAL1 OPING
I
I
I
200
300
400
I
O h* 100
LINE
v
Figure
3. Frequency
cs. reduced
wind speed for vertical
translation
I
I
I
I
1
I
0
5
I IO
I '5
20
system.
2.0
,
I.5 ;; I e
I.0 ‘7 I
1
v (m/s)
Figure
4. Rate of growth
of the lateral aerodynamic equation (20),
us. reduced
force
wind speed for vertical
coefficient
d&/da
(positive
= (4~ul GM
translation
downward),
+ 151,
system
one
obtains,
from
(23)
where the efIect of the structural damping is taken into accout. The variation of the measured dC,,/dcz with v is shown in Figure 5, where it is indicated that the quasi-steady assumption is satisfied for a range of v > 200, approximately, where dC,,/da is constant and equal to 2.0, approximately. 5.2. TORSIONAL
FLUTTER
A system allowing only torsional movement about the midchord of the model was obtained with the vertical frame locked. It was found that the oscillations were slightly unstable b-it with relatively low irregular amplitudes so that the measurement of the rate of growth of vscillation was difficult.
370
‘I’.
NAKAMURA
J-r 4-o -
s\
4
3.0-
\ j
c c1
0
Cl
0 ‘0
2.0-
‘\.+_
I.0 -
I O L h* 100
j
n
I 200
-
I
I
300
4cNJ
_
V
Figure
5. dC&da
I
0
obtained
by use of quasi-steady
aerodynamic
1
I
I
I
5
IO
15
20
theory.
V (m/s)
Figure
6. Frequency
us. wind speed for torsional
systems
with various
torsional
stiffnesses.
As is shown in Figure 6, the frequency of oscillation increased with the wind speed. The aerodynamic moment coefficient C MORwas obtained by using the relation Cm‘jR= (89r2v/ v2)(-1 + R2& where cre = fYo/fe, in which fe denotes the frequency of oscillation for the torsional system with an axis of rotation at the midchord. Figure 7 indicates the variation of CMORwith v; the magnitude of CMORdecreases slowly with V,suggesting that, unlike the behaviour for CLllh a constant value of CMORhad not yet been reached in the range of wind speed tested. It is useful for later consideration to see how stability was changed with the position of axis of rotation. It was found that the oscillation built up rapidly if the axis position was located downstream of the midchord, while it was damped rapidly if the axis position was located upstream of the midchord. Figure 8 shows the results of measurement of p for two such positions of axis of rotation.
BUNDLED
LINE
zoo
371
GALLOPING
I
I 400
I 300
I
I 100
0
POWER
v
Figure
7. C,,,,
obtained
from frequency
h
”
measurement.
Symbols
I/--_-L-i
1 100
200
correspond
300
to those used in Fipurc
h
400
v
Figure positions.
8. Rate of growth
5.3.COUPLED
t’s. reduced
FLUTTER
FOR
wind speed for torsional
systems
corresponding
to two different
axls
R < 1
It was found that there was a remarkable difference in the instability characteristics hetween the cases R < 1 and R > 1. The results for R = 0.8,which were representative of the case R < 1,were as follows. At low wind speeds, the instability was substantially equivalent to transverse galloping: namely, it was the vertical oscillation having a frequency CT= 1 with sporadic torsional movements ha.ving a frequency (T = uO. With an increase in v, a torsional movement with the same frequency as that of the vertical one manifested itself: that is, the oscillation became strongly coupled, as is indicated in the oscillogram of Figure 9. The first point of interest about the characteristics of this instability is that the rate of growth of oscillation p increased rapidly with v, as is shown in Figure 10. The values of p shown in Figure 10 are much larger than those of transverse galloping shown in Figure 3. The second point of interest is the variation of the frequency with v which is shown in Figure 11. It appears that the frequency variation was associated with the resonance wind speed v,, which is defined as the wind speed at which fH= f, ,,, and can be expressed, by putting (T = 1 in equation (24 1, ilS
372
Y.
Figure
9. Oscillograph
showing
NAKAMLIKA
build up of coupled
instability.
R = O~XO, v = 364
The frequency, which had been constant and equal tof,,,, approximately, began to increase at around vr,, following fairly closely along the curve of fo.In other words, the effect of the aerodynamic coupling on the frequency variation was found to be rather small. The third point of interest is the variation of the phase angle 4 with v, which is given in Figure 12 for three values of R. The phase angles were equal to about 7r/2 at low wind speeds but began to increase just before the resonance wind speed. In particular, the values of C$at high wind speeds for R = 0.92 were approximately constant and equal to ?r. As was mentioned earlier, coupled flutter may be classified by a comparison between /3, and & in equation (16). There is no doubt that the observed instability with C$= 7r/2 belongs to classical type flutter. However, it may not necessarily be concluded that the observed instability with 4 = 7r for R = 0.92 belongs to torsional type flutter. This is because p2 when sin C$is of small magnitude may often exceed pl. A comparison was made for the case where R = O-92 and v = 370. It was experimentally shown that /3 = O-40 and X = 13.0, approximately. On this basis, an experiment with a torsional system with an axis of rotation X = 13.3 downstream of the midchord of the model was added. As is indicated in Figure 8, the result was that PI = 0.27, approximately, which is roughly comparable with
Figure
10. Rate of growth
0s. reduced
wind speed for coupled
system
with R = 0.80
BUNDLED
POWER
LINE
373
GALLOPING
V (m/s)
Figure
‘, 1. Frequency
0
bq
vs. reduced
wind speed for coupled
’
’
100
system with R = 0.80. - - -, f+
--_----_L--.-------__i--2 300
200
400
v
Figure 12. Phas’e angle US.reduced R = 0~80; V, R = 17.92.
wind speed for three values of uncoupled
frequency
ratio. ‘3, R = 0.70; i’ .
p = 0.40 for the coupled system. This seems to justify the view that the observed for R = 0.92 at V = 370 belongs to torsional type flutter 5.4.
COUPLED
FLUTTER
FOR
instability
R >1
The frequency variations with e for R > 1 were found to be similar to those for R < 1. As is exemplified in Figure 13, the frequency of the torsional branch was nearly equal to f,,, while that of the branch of the vertical translation was nearly constant and equal to f,,,). Thus, the case 19 > 1 was characterized by the fact that the difference in frequency between the two branches increased with the wind speed so that there was no resonance wind speed. At low wind speeds, both of the two branches with different frequencies became unstable. In particular, when R was close to one, the instability built up with a beat, since the two frequencies were close to each other. Figure 14 shows the displacement signals for
374
Y.
NAKAMLJKA
I
2.0
I
1
I
-
,J
.j
/’
f, ,/’
I
I 20
J
_H _----
/j
I
I
I 5
0
I
IO
15 V (m/s1
Figure
Figure
13. Frequency
14. Oscillograph
us. wind speed for coupled
showing
system
build up of instability
with
with beat.
R = 1.34. - - -, fH
R = 1.04, v = 147.
R = 1.04. It is seen that the vertical oscillation built up with a beat while there was only a slight indication of it in the torsional signal because of the unsymmetric aerodynamic coupling. When the wind speed increased, the torsional branch became increasingly dominant so that the signal of the vertical translation became more distorted.
6. ANALYSIS
OF
COUPLED
FLUTTER
The experimental results described in the preceding section suggest that the characteristics of coupled flutter changed drastically as the wind speed increased through the resonance speed. An analysis is presented to shed some light on this situation.
BUNDLED One
may
start
with the instability
POWER
LINE
375
GALLOPING
with u = 1. Substituting
(24) into equation
equation
( 16), one gets ( V2/8r21/)(-CMVR
cos 4 + CM,,, sin d)X
= R*(v*-v~).
(261
Equation (26) !;uggests that, because X must be positive, the phase angle is sensitively dependent O!I the signs of CM-R, CM,,, and cr2- (T:. For CM,,R and Cm+ the following relations hold to a good approximation for most bluff bodies [16]: C&&$7= -(27rl
V)Ger,
C&IV1= (27rf wh,.
In the case where R < 1 and v < vr, it is assumed, as corresponding to the present experiment, that CM*, < 0 and u = 1 so that C,+I < 0 and a* - (T: = 1 - cri < 0. Therefore. 4 = ~12 with cr = 1, which is in accordance with the experimental observations, satisfies equation (26). lt follows that X = 47rvR*( 1 -a;
(271
)/ ~Z’,VU,K,
which shows that X decreases from infinity as v increases from zero. Correspondingly, substitution of equations (22) and (27) into equation (13 ) leads to p = (v/4p)(dC&da)
+ ~3C,,,C,,,j32.rr2~vR2(1
-a$).
(28)
When v exceeds vr, cr’-cr’, changes sign to be positive so that 4 = --r/2 with v = 1 is suggested for equation (26). It follows that equation (28) holds also for v < v,. Obviously, the second term on the right-hand side of equation (28) represents the effect of the aerodynamic coupling, and it becomes dominant over the first term as v increases. Moreover, it changes sign as vr is passed, although equation (28) fails at v,. Keeping in mind that CL, >>0 (this holds because CLeR = dC,,/dcy + C, for large values of v, where CD is the drag coefficient), one easily sees that the oscillation is strongly unstable for p < v, due to the aerodynamic coupling, while it is stable for v> v, due to the aerodynamic coupling. In short, the aerodynamic coupling can play a vital role in the stability of bundled conductors when the resonant condition is approached. A similar analysis may be made for the oscillation with CT= crH.Instead of equation (261. the following relation is considered: (-1 +a2)X
= (v2/8~‘~)(C’,,,
cos d -CL,,
Since CLeR > 0 and -1 + (+i > 0 for v < v,., it is obvious equation (29). Accordingly, X = ~*C,,,/87r*/.~(cr; which shows that X increases
from zero as v increases
/3 = ( ~2/87r~)C,,,
sin 4).
that 4 = 0 with u = (TVsatisfies
- l),
!30)
from zero. Equation
+ Q3C,,,C,,,/32~2~~~~~
(29)
- l),
(14) now gives (311
which indicates that although C MBI is slightly positive, the oscillation is eventually damped due to the aerodynamic coupling as v increases from zero. When e> vr, on the other hand, 4 = 7r is suggested. It follows that equation (31) also holds for e> v,, where it is indicated that the oscillation is strongly unstable due to the aerodynamic coupling. In the experiment, it was observed that the instability with (+ = 1 occurred initially, but that with a gradual increase in frequency it shifted near v, to the instability with u = rH, During this shift p increased continuously with v but an increase beyond limit, as indicated by equation (28), was avoided. The prediction of the instability characteristics near vr would require a more precise evaluation of both the frequency and the phase angle
376
Y NAKAMURA
which in turn would require the specification of some more aerodynamic coefficients concerned. The analysis of the instability for R > 1 may be made in the same way. It turns out that the oscillation with cr = ug is unstable mainly due to the aerodynamic coupling while the oscillation with v = 1, though it is unstable initially, becomes eventually stable due to the aerodynamic coupling as v increases. It appears that the trend predicted so far is in agreement with the experimental observations.
7. CONCLUSIONS In this paper an experimental and analytical study has been described of the galloping of a two-dimensional section model of a two-conductor bundle in which ice-accreted conductors were replaced by two identical square prisms, with both vertical and torsional movements allowed but the horizontal one blocked, in a uniform wind tunnel flow. Emphasis has been placed on elucidating the vital role played by the aerodynamic coupling in the stability of bundled conductors. It has been shown that as well as galloping type flutter two other types of instability, namely, torsional and classical type flutter, can occur for bundled conductors. In particular, it has been shown that the aerodynamic coupling can cause violent classical type flutter to occur when the resonant condition is approached. The elimination of this unfavourable effect of resonance should be one of the most urgent problems in the design of bundled power line conductors.
ACKNOWLEDGMENT The author is grateful to Messrs S. Sakabe and Y. Matsubayashi for supplying valuable information on the field observations. He is also grateful to Messrs K. Watanabe and H. Kamimura for their assistance in conducting the experiment.
REFERENCES 1. A. T. EDWARDS 1970 Electra 12, 3148. Conductor galloping. 2. Proceedings of the Institute of Electrical and Electronic Engineers Power Engineering Society Summer Meeting 1979. Symposium on Mechanical Oscillations of Overhead Conductors, 15 July 1979. 3. Proceedings of the Institution of Electrical Engineers Power Division 1979. 2nd International Conference on Progress in Cables and Overhead Lines for 220 kV and above, 4-6 September 1979. 4. J. P. DEN HARTOG 1956Mechanical Vibrations. New York: McGraw-Hill. See pp. 299-305. 5. G. V. PARKINSON 1965 Proceedings of the Conference on Wind Effects on Buildings and Structures 2, 581-609. Aeroelastic galloping in one degree of freedom. London: Her Majesty’s Stationery Office. 6. A. T. EDWARDS and A. MADEYSKI 1956 Institute of Electrical and Electronic Engineers Transactions on Power Apparatus and Systems PAS-75, 666-686. Progress report on the investigation of galloping of transmission line conductors. 7. J. J. RATKOWSKI 1963 Institute of Electrical and Electronic Engineers Transactions on Power Apparatus and Systems PAS-82, 661-669. Experiments with galloping spans. 8. A. SIMPSON and T.V. LAWSON 1968Proceedings of Symposium on Wind Effects on Buildings and Structures 2, Loughborough University of Technology, Paper 25. Oscillations of “twin” power transmission lines. 9. 0. NIGOL and G. J.CLARKE 1974 Institute of Electrical and Electronic Engineers Power Engineering Society Winter Meeting Paper C74 016-2. Conductor galloping and control based on torsional mechanism.
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POWER
LINE
GALLOPING
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