Electric Power Systems Research, 1 (1977/78) 269 - 282
269
© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands
Mathematical Analysis of Transmission Line Vibration Data
P. W. DAVALL, M. M. GUPTA and P. R. U K R A I N ETZ
Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan S7N OWO
(Canada)
(Received June 19, 1978)
SUMMARY
In this paper three methods of analyzing
transmission line vibration data are discussed,
and possible applications which could give a
better insight into the problems of vibration
failures are investigated. The analysis procedures are shown to complement each other
and together may be used in line design, and
in evaluating the performance o f line dampers.
A summary of current measurement standards
is presented and the hardware requirements of
an analysis system are briefly discussed.
1. INTRODUCTION
In recent years increasing interest has been
shown by power companies in the reliability
aspects of electrical overhead transmission
lines. In recognition o f this, the Transmission
and Distribution Committee o f the IEEE
power group presented a study on line vibration measurement problems in 1965. Their
recommendations are to be found in ref. 1.
Much of the early work o f this committee is
based on research conducted by Ontario
Hydro [2 - 4], and a recommended vibration
analysis approach is to be found in ref. 4.
With the development of more sophisticated
vibration measurement techniques (notably
the HILDA system developed by S.E.D. Systems Ltd.) and with the lowering o f costs for
computer hardware, it has become possible
to apply more sophisticated analysis procedures to the measurement of line vibration.
However, the advantages o f using alternative
analysis procedures to those recommended in
ref. 4 have not been investigated. Indeed, even
the original approach has yet to become widely
accepted as an industry standard [5, 6].
The main contention is whether bending
amplitude or c o n d u c t o r strain, and hence
stress, should be measured [5, 6]. Conductor
strain is more closely related to line fatigue.
However, the latter requires the placement of
strain gauges on the line near the clamp support and this is not a simple procedure if the
line is live. In addition, problems such as
strain gauge temperature sensitivity, nonuniform strain in individual strands o f bundle
conductors, and the occurrence of fatigue
failures on inner strands, where the strain cannot be measured, make the measurement of
line displacement a more attractive proposition. In the light of this, the Ontario Hydro
and the HILDA recorders have been developed
[4, 7] to measure line bending amplitudes;
the latter recorder produces a cassette tape of
line displacement vibration data which may
be processed by computer.
Measurements of the line bending amplitude (defined as the relative line displacement
measured at a point 3.5 in. from the last point
o f contact o f the clamp} may be used to estimate the bending strain at the clamp [4]
(Figs. 1 - 3). Although the procedure for normalizing the bending strain is only approximate, the approach appears to give satisfactory results and is suitable as a measurement
standard.
The aims o f vibration analysis remain illdefined. The findings of refs. 2 and 4 suggest
that adherence to a safety limit of 150 u-strain
peak-to-peak ensures that the line will not
fatigue. The limit is somewhat arbitrary and,
clearly, a more accurate picture of fatigue life
would be obtained from a series of S/N curves
for transmission lines for various core diameters and types of clamps utilized. An initial
study to obtain the required failure curves is
reported in ref. 6.
Accepting the limit recommended by the
IEEE Committee o f 150 ~-strain for want of
suitable S/N curves, the engineer still requires
precise information on how to achieve accep-
270
24
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20
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16
z
z
-
z
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~z~
[2
s
uJ
o.
u_
0
I
I
~
0.05
i
i
OI
DIAMETER
I
~
015
Og CONDUCTOR
i
i
0,2
i
i
0.25
AND ARMOUR ALUMINUM
l
i
05
i
0.55
~
i
0.4
WIRES IN INCHES
Fig. 1. Suggested safe b e n d i n g a m p l i t u d e s for A C S R a n d a l u m i n u m c o n d u c t o r s [ 4 ] . F o r bare or a r m o u r e d cond u c t o r s in s t a n d a r d s u s p e n s i o n c l a m p s t h e 3.5 in. is m e a s u r e d f r o m t h e last p o i n t o f c o n t a c t b e t w e e n t h e c o n d u c t o r or a r m o u r a n d t h e clamp. T h e curve t h e n s h o w s t h e b e n d i n g a m p l i t u d e s at w h i c h t h e c o n d u c t o r has a b e n d i n g
s t r a i n o f 150 p i n / i n peak-to-peak. T h e curve also s h o w s t h e b e n d i n g a m p l i t u d e s at w h i c h a r m o u r rods have a
b e n d i n g strain o f 150 p i n ] i n , b u t t h e significance o f t h i s w i t h r e s p e c t t o r o d b r e a k a g e has n o t y e t b e e n established.
T h e s t r a i n o f 150 p i n / i n p e a k - t o - p e a k has b e e n selected as t h e suggested safe level o n t h e basis t h a t t h e b e n d i n g
a m p l i t u d e r e q u i r e d t o p r o d u c e t h i s s t r a i n (5.5 rail) in 7 9 5 MCM 5 4 / 7 A C S R (wire d i a m e t e r 0 . 1 2 1 4 in) has n o t
p r o d u c e d fatigue d a m a g e in O n t a r i o H y d r o c o n d u c t o r s , in c o n v e n t i o n a l s u s p e n s i o n clamps, in t h e field a f t e r 30
years o f service. It is likely t h a t t h e value o f 150 p i n / i n is s o m e w h a t c o n s e r v a t i v e a n d t h a t s t r a i n s o f 209 or
3 0 0 p i n / i n p e a k - t o - p e a k m a y well p r o v e t o be safe in p r o p e r l y fitted s u s p e n s i o n clamps. O t h e r factors, such as t h e
e f f e c t s o f v i b r a t i o n o n t o w e r s a n d d a m p e r s , also h a v e t o b e c o n s i d e r e d w h e n selecting t h e safe v i b r a t i o n level for
a particular situation.
In t h e case o f a r m o u r g r i p s u s p e n s i o n t h e a b o v e c o m m e n t s a p p l y e x c e p t t h a t t h e 3.5 in. is m e a s u r e d f r o m t h e
c e n t r e line o f t h e c l a m p . T h e curve d o e s n o t a p p l y t o t h e special " b i r d - c a g e d " a r m o u r r o d s used w i t h t h i s suspension u n i t .
3
2
uJ
£i]
I
J
w
0
I
0
I
2
DISTANCE
3
FROM L A S T
4
5
6
7
POINT OF CONTACT IN INCHES
Fig. 2. B e n d i n g curve for c o n d u c t o r s w i t h a n d w i t h o u t a r m o u r r o d s in s t a n d a r d s u s p e n s i o n c l a m p s a n d p i n - t y p e
i n s u l a t o r s [4].
E x a m p l e : If t h e r e c o r d e r driving a r m is a t t a c h e d 4.5 in. f r o m t h e last p o i n t o f c o n t a c t t h e r e c o r d e d b e n d i n g
a m p l i t u d e s s h o u l d b e divided b y 1.45 t o s t a n d a r d i z e t h e results.
N O T E : This curve m a y b e used for a n y c o n d u c t o r size.
271
10
8
fL
6
z
w
m 4
.J
t~
2
o
0
2
4
6
8
JO
12
I
14
INCHES FROM CLAMP CENTRE LINE
Fig. 3. Bending curve for conductors in armourgrip suspension clamps [4 ].
Example: If the recorder driving arm is attached 12 in. from the centre line of the AGS clamp the recorded
bending amplitudes should be divided by 8.75 to standardize the results.
NOTE : This curve may be used for any conductor size.
table levels of line vibration. This may be
forthcoming in terms of suggested positioning
of line dampers, a modification to the type of
line used, or even a change in the tower design.
Analysis of line vibrations for various line configurations and locations could conceivably
supply some of this required information.
In this paper, methods of data collection
are reviewed and various procedures for
vibration analysis are compared. Applications
of each procedure are discussed in turn and
some recommendations for future study are
made.
2. TRANSMISSION
COLLECTION
LINE
VIBRATION
DATA
It has been shown experimentally [1] that
the relative bending amplitude, as measured
on the line at a point 3.5 in. from the last
point of contact on the clamp, is approximately linearly related to the bending strain
induced at the clamp (Fig. 1). If the line displacement is measured at a point other than
3.5 in. from the clamp, readings may be
normalized to obtain the bending amplitude
(Figs. 2 and 3).
Experiments have also shown that, to
obtain an accurate picture of line vibrations,
recordings must be taken over an interval of
at least two weeks. In practice this is usually
extended. Over this interval, there is a high
probability that the full range of line vibration
amplitudes and modes will be encountered.
As the mode of line vibration is usually only
slowly varying, being a function of changing
climatic and wind conditions, it is sufficient
to sample the vibration data at regularly
spaced intervals of 15 minutes for a period of
1 second. The Ontario Hydro and HILDA
recorders are designed to do this with options
for longer sample intervals or for continuous
monitoring. The line bending amplitudes
encountered depend upon the line tension.
Although this varies on a daily basis over the
test period, the overall measurements will
reflect expected line amplitudes for a line
tension in accordance with the mean ambient
temperature encountered over the test duration. To predict vibration levels for varying
line tensions, the following equations may be
used:
i Tne w ~ 2
New amplitude = old amplitude X ~ Told / (1)
272
where Tne w = new tension (lb), To1d = old
tension (lb).
The line tension may be calculated from
either of the following two equations:
T (lb)=
conductor wt.(lb/ft) × [span (ft)] 2
8 × sag at centre (ft)
instances, the results of this analysis may be
sufficient. However, it is anticipated that
future studies will require a more accurate
and detailed comparison of line vibration
measurements and, with this in mind, the
applications of spectral analysis and peaktrough analysis are also discussed.
(2)
3. 1. Cross-over frequency~amplitude analysis
or, if a pulse is transmitted down the span and
is reflected back in S seconds, by
T (lb) =
conductor wt.(lb/ft) × [span (ft)] 2
8.05(ft/s 2 )[S(s)] 2
(3)
Equation (1) holds for conductor sizes up to
336.4 mcm.
In general, wind-induced oscillations occur
in the frequency range 0
100 Hz, with
bending amplitudes of up to 0.2 in. peak-topeak. Usually amplitudes will be considerably
less than this. For a typical test over a 2-week
interval, up to 3000 one-second samples may
be recorded. To obtain sufficient accuracy it
is imperative t h a t the analysis be performed
using a computer. This involves sampling,
digitizing, and then storing the data for later
analysis. It has been found [7] t h a t a sampling
frequency of 500 Hz is sufficient for the
analysis procedures used in this paper.
The full range of data collection and analysis programs [7] may be run using a 16 Kword mini-computer equipped with an A/D
interface for data digitization, a mass storage
device for logging the digitized data and
storing the analysis results, and, for ease of
interpretation, a graphical o u t p u t terminal.
The cost of such a system at today's prices
(1976) would be approximately $20000.
Clearly, this cost m a y not be justified for
vibration analysis alone. However, for most
major power companies this presents little
problem since a mini-computer is usually
available for other applications.
3. METHODS OF ANALYSIS OF TRANSMISSION
LINE VIBRATION DATA
Three methods of data analysis are considered here. The first is essentially the
approach that is recommended in ref. 1 for
adoption as an industry standard. In many
Cross-over frequency analysis is the one
recommended as an industry standard by the
IEEE Committee in their report [1]. In the
analysis it is assumed that the line is vibrating
sinusoidally at constant amplitude over a
given sample interval. Thus the frequency of
vibration can be determined from the number
of zero-crossings and the amplitude from the
peak-to-peak excursion. It is further assumed
t h a t this mode of vibration is maintained
until the next sample is recorded, which is
usually at 10 - 15 minute intervals. Over the
complete test duration, a measure of the
number of cycles of line vibration at any
given frequency and amplitude is obtained
(Table 1).
From the data presented in Table 1, the
engineer can determine the number of times
vibration amplitudes exceed the recommended
safety margin of 150 p-strain peak.to-peak
and the frequency range in which this occurs.
If dampers are to be placed on the line, they
must be spaced so that the d o m i n a n t high
amplitude vibrations are sufficiently attenuated. If the line exhibits a natural frequency
of fN Hz, a standing wave at this frequency
will have a loop length of
D(fN) -
1
32.2 × T
2fN
W
(4)
where T = line tension (lb), W = conductor
wt. (lb/ft).
To attenuate this frequency the damper
should be placed at or near a vibration antinode, a distance D/2 from the clamp. If the
line exceeds the recommended safety levels
in a frequency range f i -+ A f i (A >1 2.0),
then a single damper may not be effective as
the even harmonics will n o t be attenuated.
Under these circumstances an alternative
means of vibration attenuation is called for.
The limit of 150 p-strain as a safe bending
amplitude is based on the experience of
Ontario Hydro. In other climates, and with
the development of alternative line designs,
20.22
1
28.033
13.828
5
7
5
22
12
30
23
47
34
6
1
4
2
12
4
22
26
20
35
24
44
35
4
1
237
94
138
32
58
56
28
31
8
9
4
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
3
1
(Hz)
2
A m p l i t u d e (mil p e a k - t o - p e a k )
Freq.
8.128
1
2
4
24
10
22
21
31
21
0
4
4.147
3
0
2
10
8
16
15
20
1
1
5
1.999
1
2
2
9
4
11
5
2
2
6
1.037
0
5
1
5
2
5
0
7
0.412
1
0
0.536
1
1
2
4
9
0
1
2
2
8
0.176
2
1
3
10
0.028
1
11
12
13
14
15
0.837
O.664
1.660
O.650
0.162
3.237
2.130
4.427
3.051
5.465
3.729
0.457
0.O45
0.000
0.000
0.O55
Megacycles
per day
(× 1 0 - 2 )
Results o f cross-over f r e q u e n c y / a m p l i t u d e analysis o n 2 3 0 k V line f i t t e d w i t h S t o e k b r i d g e d a m p e r s
Test p e r i o d 17 days
V i b r a t i o n e x c e e d s I E E E r e c o m m e n d e d m a x i m a o f 150 m i c r o - s t r a i n P-P 0 . 8 4 % o f t h e time.
V i b r a t i o n exceeds I E E E r e c o m m e n d e d m a x i m a o f 1 5 0 m i c r o - s t r a i n P-P in t h e f r e q u e n c y r a n g e 5 Hz t o 11 Hz w i t h c e n t r e f r e q u e n c y 8 Hz, n o r m a l i z a t i o n f a c t o r
o f 35 ~ - s t r a i n / m i l deflection.
TABLE 1
274
1.660
1.185
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-.257
-.711
-Li85
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0.0
.08
t
.17
.26
I
I
.55
.44
TIME (SECS)
.55
I
I
i
.62
.?l
.80
2.538
1.813
j
1.088
.582
'V'V 'V
V
-.382
-I.088
-I.813
-2~39
oo
o.
~
I
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I
~
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.62
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.71
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.80
TIME (SECS)
F i g . 4 . S a m p l e records o f line vibration recorded on a 2 3 0 k V bundle conductor,
this limit may not represent an accurate safety
limit. To predict fatigue life, a series o f S/N
failure curves is required covering the range of
line diameters and clamp configurations that
are generally in use on major transmission
lines. Given these, the cross-over analysis can
be used to predict fatigue life. Assuming that
the Palmgren-Miner hypothesis is valid, a
275
measure o f the cumulative damage incurred
due to narrow-band variable amplitude loading
is given b y
k
D = ~ N(Sk)/Nf(S~)
(5)
i=l
where D = cumulative damage, 0 < D ~< 1.0;
N ( S k ) = number o f cycles of vibration amplitude Sk ; Nf(Sk ) = number of cycles to failure
at amplitude Sk.
Any calculation of this sort requires S / N
failure curves which are as y e t unavailable for
transmission lines.
3.2. P e a k - t r o u g h analysis o f line vibration
The cross-over analysis presented in § 3.1
assumes that line vibrations are sinusoidal
with constant amplitude. From consideration
of an example of line vibration as shown in
Fig. 4, it is evident that this is not always the
case. In general, line vibrations are narrowband random in nature with several frequency
modes superimposed. To assess the fatigue
damage incurred under such loading patterns,
it has been suggested [8, 9] that the distribution of successive p e a k - t r o u g h pairs in the
vibration loading plays a very important role.
Certainly peak or range distributions play a
key role in determining the expected fatigue
life [101.
The peak-trough probability distribution
may be estimated by sampling the vibration
data and recording the amplitudes of successive significant peak-trough pairs in the signal.
The resulting distribution is found to be
representative of the true distribution for the
line, if the IEEE recommendations on testing
procedure are met {i.e., a test duration of a
minimum of 15 days, sampling one-second
records every 15 minutes). In obtaining the
distribution, peak-trough pairs of less than a
minimum threshold amplitude may be discounted for t w o reasons. Firstly, the measurement signal is subject to external low amplitude noise perturbations due to recording,
playback, and digitization noise and, secondly,
it is reasonable to assume that fatigue is only
incurred above a threshold stress level. Again,
little is known about this safe threshold level
for transmission lines. In practice, amplitude
levels of less than 0.001 in. zero-to-peak are
ignored. A typical peak-trough analysis of
line vibration data is shown in Table 2. The
peak-trough matrix is by definition upper
triangular (peak amplitude being greater than
the associated trough) and may be used to
estimate the damage due to fatigue. If the
vibration pattern is narrow-band random,
then using the distribution shown in Table 2
N
N+I--I
D = ~
1=1
~_,
N(I,J)/Nf(I,J)
(6)
J=l
where D = cumulative damage, 0 <~ D ~< 1;
N(/, J) = number of peak-trough pairs encountered
with peak/trough
amplitudes
centred on the interval (/, J); Nf(I, J) = number
of cycles to failure for the line at a constant
amplitude loading equivalent to the peak-totrough excursion (/, J).
The calculation of eqn. (6) requires a
measurement of the S / N failure curves for the
line. If it is assumed that the failure curve
takes the form
N~(S) = aS-b
(7)
where Nf(S) = number of cycles to failure,
S = peak-to-peak amplitude of vibration, a =
a constant related to the lateral position of
the S / N curve, b = Basquin's exponent; the
inverse slope o f a plot of log 8 against log N~,
assumed straight; then, on substituting for
Nf(I, J) in eqn. (6)
N
D = ~,
I=1
J)
N + 1 -- I N(L
Z
- -
J= l
Szj °
(8)
a
It can be shown that eqn. (8) is the bth
m o m e n t of the cumulative peak-trough distribution shown in Table 2 about the axis o f
zero stress (peak amplitude = trough ampli.
tude).
In general, the total number of times a
given peak-trough amplitude has occurred is
not known, as this would involve continuous
monitoring from the date of installing the line.
Defining the peak-trough probability distribution by
p(I, J) = N(I, J ) / N T
(9)
where p(I, J) = probability of obtaining a
p e a k - t r o u g h pair in the interval centred on
(I, J), N(I, J) = number of occurrences o f
peak-trough pairs in interval (/, J), NT =
total number o f peak-trough pairs measured;
then an estimate o f the damage potential o f
a vibration level is obtained from
--0
---0
--0
--0
--0
--0
--0
--2
1
0
0
0
0
0
0
0
0
0
0
1
--0
----0
--0
--0
--1
---0
--0
0
1
0
0
0
0
0
0
0
0
0
0
--5.9
--0
--i
--3
--0
--4
--3
0
0
0
0
0
0
0
0
0
0
0
0
--5.3
--0
--0
--3
--5
--4
4
0
0
0
1
0
0
0
0
0
0
0
--4.6
--0
--0
--0
--2
--9
7
7
1
0
0
0
0
0
0
0
0
--3.9
--0
--0
--1
2
7
39
44
7
6
0
0
0
1
1
0
--3.1
--0
--0
0
1
5
36
155
140
33
2
1
1
0
0
--2.4
--0
0
0
0
3
2
116
585
336
26
3
7
0
--1.8
0
0
1
0
2
2
25
281
1658
816
62
0
--1.1
0
0
0
0
0
0
1
8
491
3571
0
--0.4
0
0
0
0
0
0
0
4
21
0
0.3
0
0
0
0
0
0
0
1
0
1.1
0
0
0
0
0
0
0
0
1.8
0
0
0
0
0
0
0
2.4
0
0
0
0
0
0
3.1
0
0
0
0
0
3.8
0
0
0
0
4.6
0
0
0
5.3
0
0
5.9
0
6.6
2 = 0 . 1 7 2 8 5 6 3 × 104
6 =0.9061975X105
3 = 0 . 3 3 6 6 5 0 9 × 104
7 =0.3560051x106
4 = 0 . 8 4 3 2 1 4 5 x 104
8=0.1507728×107
R a n d o m vibration i n d e x = 0 . 8 6 8 5 6 7 × 102 (micro-strain2).
V i b r a t i o n severity, m o m e n t s 2 t o 9 n o r m a l i z e d to micro-strain
5 = 0 . 2 5 6 7 2 4 9 x 105
9=0.6730378x107
N o r m a l i z a t i o n f a c t o r = 35 p-strain]rail d e f l e c t i o n .
M a x i m u m a m p l i t u d e peaks and t r o u g h s (mils) = 7 . 0 0 0 0 0 0 ; No. o f cells = 20; t r o u g h a m p l i t u d e o n h o r i z o n t a l scan - - V to +V; peak a m p l i t u d e on vertical scan
+V t o --V.
Negative indices indicate regions w h e r e vibration e x c e e d s IEEE r e c o m m e n d e d m a x i m a .
6.6
5.9
5.3
4.6
3.8
3.1
2.4
1.8
1.1
0.3
--0.4
--1.1
--1.8
--2.4
--3.1
--3.9
--4.6
--5.3
--5.9
--6.6
--6.6
Rough
C5~
P e a k - t r o u g h analysis for 230 kV line w i t h s p a c e r d a m p e r s (20-day t e s t p e r i o d )
Peak
bO
TABLE 2
277
N
Fb ~ ~
1=1
N+I--I
~
p(I, J)Sls b
(10)
J=l
where Fb is the bth moment of the peaktrough probability distribution about the
peak-trough zero stress axis. The exponent
b is dependent on the stress raising qualities
of the clamp and on the stress distribution
qualities of the line. For a low notch factor
at the clamp b may be equal to 5 or 6, while
for a badly clamped line b may be as low as
2 or 3. The latter values of b are equivalent to
obtaining steep S / N curves. The values of F b
for b = 2 to 9 are shown in Table 2.
The calculations of eqns. (8) and (10) have
the advantage of not requiring exact knowledge of the failure curves for the line. However, to find suitable values for the coefficient b (eqn. ( 1 0 ) ) f o r transmission lines
requires an extended study and analysis of
vibration data obtained from common line
and clamp configurations. These data are not
currently available.
The cumulative peak-trough distribution
shown in Table 2 may be used to measure the
"randomness" of line vibrations. In general,
the engineer wishes to reduce the amplitude
of standing wave vibration on the line. This
is achieved by ensuring the energy input to
the line is not concentrated in a narrow
frequency band. If the energy spectrum of
line vibrations is spread out in the frequency
domain, then the vibrations will appear more
random in nature. A measure of the random
vibration index is obtained by taking the
second moment of the peak-trough distribution (Table 2) about the axis peak amplitude
trough amplitude. Vibrations occurring
along this axis are sinusoidal. The random
vibration index is given by
=
- -
N
the rms vibration level. Under these conditions, the probability of the occurrence of a
large amplitude standing wave vibration mode
is low. The effectiveness of installing dampers
can then be measured by comparing the
various indices obtained from the peak-trough
distribution before and after installation.
Lastly, if peak-to-trough amplitudes exceed
the IEEE recommended maximum of 150 ustrain, this may be indicated on the peaktrough distribution by appending a negative
occurrence index (Table 2). Thus the region
on the peak-trough distribution in which
peak-trough amplitudes exceed 150 p-strain
is clearly distinguishable.
The calculation of the peak-trough distribution gives a more exact picture of the
nature of line vibrations in terms of amplitudes and sequencing effects. However, no
information is gained on the frequency content of the line vibrations.
3.3. Spectral analysis o f line vibrations
The power spectrum analysis of line vibrations is perhaps the most significant and, in
many ways, the most useful. Let the vibration
signal on each sample interval t E Ti (i = 1, M)
be given by xi(t). Then the discrete Fourier
transform ofxi(t) is given by
1N--1
Xi(jkcoo) = -~ ~,
xi(tt)exp(jklwo)
(12)
l=O
(k = O, N / 2 ; i = 1, M)
where w0 = frequency resolution (= 2n/Ti) ,
and each record xi(t ) is sampled in blocks of
N data points at equi-spaced intervals t~.
An estimate of the power spectrum of
line vibrations on each interval Ti is given by
cbTix(k w o ) = 2[Xi(jkw o ) [2
N+I--I
Rv = ~,
~,
I~1
J=l
p(I, J)S2D (I, J)
(11)
where Rv = random vibration index, SD =
mean amplitude level associated with the
peak-trough interval (I, J). Combining the
indices associated with the damage potential
of eqn. (10) and the random vibration index
of eqn. (11), it is apparent that an overall
safe level of line vibration is guaranteed if the
damage potential indices are low and the
random vibration index is high in relation to
(1 <~ k <. N / 2 )
(13)
= IXt(jk¢oo)12
(k = 0)
An overall estimate of the power spectrum
of line vibrations may be obtained by estimating
1
M
Cxx(ko~o)
(14)
278
.060
2.d
LINE
RESONANCE
AT
IOHZ.
Isl LINE
RESONANCE
AT 7 H Z
.052
.045
N
I
~...o34
d
uJ
.026
o
6.
.017
/
0.0
0.0
.008
]
I1.11
~
I
22.22
55.55
I--
I
44.44
55.55
FREQUENCY
--
'
66.66
,
I
I
I
77.77
88.88
99.99
(RZ)
Fig. 5. The average powerspectrumofline vibration on a 230 kV bundle conductor measured over a test duration
of12 days.
where M sample records are obtained over the
test duration.
Peaks in the averaged spectrum Sxx(kw0)
indicate the preferred modes o f resonance for
the line (Fig. 5). However, the d o m i n a n t frequencies for individual sample intervals may
differ considerably both in amplitude and
frequency content. As an example of this,
Fig. 6 shows two sample estimates of the line
vibration spectrum whose overall spectrum is
shown in Fig. 5.
The presence o f a fundamental frequency
plus significant harmonics can be determined
analytically by calculating the cepstrum,
which is given by
Cx~(F) = IF -1 [ l n ~ ( w ) ] I
(15)
where F -1 indicates the inverse transform and
the independent variable, F, is measured in
units of time. Peaks in the cepstrum (Fig. 7)
indicate the presence o f harmonics in the
power spectrum. The fundamental frequency
is given by
f0 = l i f o
spectrum shown in Fig. 8, curve (a). The
d o m i n a n t peak in the cepstrum indicates the
presence of harmonics with a fundamental
frequency o f approximately 14 Hz. A secondary peak would indicate a 7 Hz sub-harmonic;
however, this is difficult to distinguish in the
spectrum plot, as shown in Fig. 8, curve (a).
From a measurement of the frequencies in
the vibration spectrum, the location of line
dampers can be determined using eqn. (4).
If the cepstrum indicates the presence of significant harmonic content, the placement of
a damper to reduce the fundamental amplitude may not be sufficient to reduce the overall amplitude of vibrations to within acceptable
limits. For instance, it can be seen in Fig. 8
t h a t the placement of dampers to remove the
fundamental has led to an increase in the
amplitude of 2nd harmonic oscillations.
To measure the effectiveness of a line
damper system, the attenuation factor can be
estimated. This is given by
A T ( k W o ) = 10(log[(~x2 (kcoo)] --
(16)
where the peak in Cx~(F) occurs at F0. Figure
7 shows the cepstrum of the line vibration
- log[$1,(k~o)]}
(0 <~ k
<~ N / 2 )
(17)
279
.189
.162
.135
I
.Io8
.J
•
o
.081
.054
.027
0.0
-"
0.0
li.II
22.22
33.33
44.44
FREQUENCY
1.600
55.55
66.66
77.77
88.88
99.99
(HZ)
-
1,371
1.145
N
~.
%
.914
_J
.685
0
o_
.457 -
.228
-
0.0
0.0
ll.il
22.22
33.33
44.44
55.55
66.66
77.77
ee.ee
99.99
FREQUENCY ( H Z )
Fig. 6. Power spectrum of individual one-second sample records of line vibrations taken at different time intervals.
where $~x(kco0) = the overall power spectrum
o f line vibrations before the installation of
dampers, and (~2(kco0) = the power spectrum
of line vibrations with dampers installed.
For the dampers to be effective, the
attenuation A T must be highly negative in the
frequency range of interest. Figure 9 shows
the attenuation factor o f the damper system
280
1.000
PEAK DUE TO HARMONICS OCCURRING
IN POWER SPECTRUM WITH FUNDAMENTAL
OF APPROXIMATELY 14HZ
.857
,714
.571
D
I--
2
,i{
.428
SECONDARY PEAK SUGGESTS
A 7HZ FUNDAMENTAL
.286
.142
0.0
i %
0.0
.02
.05
.08
.11
.14
.i~
.2o
,2~,
~
-'
77
'~--~
88
.26
T I M E (SECS)
Fig. 7. Cepstrum of line vibration spectrum.
.028
-,~ (o) WITHOUT
DAMPERS
.024
.020
N
I
"~-
.016
J
'5
.012
o
:'..
' ""~ (b) WITH DAMPERS
.001
.00~
0.0
'
0.0
II
22
33
44
55
66
1
99
FREQUENCY (HZ)
Fig. 8. Example showing the change in the averaged power spectrum of line vibrations following the installation
of line dampers.
281
16
t2I48
0
|
-4
z
o
,%
#
I
I
~"
%"
~ ~ -- -- l
!
i
~
~
:
-8
,~ -12
~
|
,
4
J
-16
-24
I
!
o!
!
I
!
-28
-32
!
-36
!
-40
I
I
I
I
I
I
I
22
33
44
55
66
7-(
88
-44
0
FREQUENCY
(HZ)
Fig. 9. Attenuation performance of line dampers as calculated from Fig. 8.
from the results of Fig. 8. It should be emphasized that in order to obtain meaningful
results the test conditions (i.e., tests conducted with and without dampers) should be
as nearly identical as possible.
4. CONCLUSIONS
The significance of the three methods for
analyzing transmission line vibration data has
been discussed. Although the cross-over analysis, recommended by IEEE as an industry
standard, provides some useful information, it
is doubtful whether this analysis alone gives
sufficiently accurate information to form a
basis for comparison of different line and
damper configurations.
The peak-trough distribution can be used
to predict fatigue life. Although the prediction theory is open to considerable question
in terms of the assumptions of a log-linear
approximation to the S/N failure curve, the
distribution moments, nevertheless, give some
indication o f vibration severity and damage
potential.
The power spectrum analysis may also be
used to determine vibration severity. However, the averaging process inherent in estima-
ting the power spectrum means that damage
due to occasional large amplitude vibration
modes will be underestimated. The performance of line dampers and the measurement
of preferred line vibration frequencies is most
easily obtained from a spectral analysis of
line vibrations.
At present, too little experimental information is available to present direct comparisons of various damper designs or different
line suspension systems. This is partially due
to the unavailability, until recently, of a
suitable vibration monitoring system capable
of producing a reproducible output for computer analysis. With this problem solved, it is
hoped that a comprehensive range of vibration recordings will be gathered, thus enabling
a more detailed study of line vibration failures
to be undertaken. The results provided by the
analysis procedures described here together
with information on climatic conditions, the
nature of the surrounding terrain, and line
specifications provide the data base for further
research.
ACKNOWLEDGEMENT
This work is supported by S.E.D. Systems
Ltd. o f Saskatoon.
282
REFERENCES
1 Task Force on the Standardization of Conductor
Vibration measurements of Towers, Poles and
Conductors subcommittee of the Transmission
and Distribution Committee, IEEE report,
Standardization of conductor vibration measurements, IEEE Trans., PAS-85 (Jan) (1966) 10 - 23
(with discussion).
2 J. E. Sproule and A. T. Edwards, Progress towards
optimum damping of transmission conductors,
Trans. Amer. Inst. Electr. Eng. (Power Appar.
Syst.), 78 (Oct) (1959) 884 - 852.
3 A. T. Edwards and J. M. Boyd, Ontario Hydro
live-line vibration recorder for transmission conductors. IEEE Trans., PAS-82 (June) (1963)
269 - 274.
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Hydro Conductor Vibration Recorder, manufactured by General Instrument Company, Toronto,
Ontario.
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in transmission line conductors near the suspen-
6
7
8
9
10
sion clamp, Proc. Conf. on Large High Tension
Electric Systems (CIGRE), Paris, June 1968.
W. Philipps, W. Carlshem and W. Biickner, The
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Int. Conf. on Large High Tension Electric
Systems (CIGRE), Sept. 1972.
Transmission line vibration analysis. Handbook
for the HILDA Monitoring System, S.E.D. Systems Ltd., Saskatoon, 1977.
F. Sherratt and P. W. Davall, Accelerating random
sequence fatigue tests by response compensation
and cycle deletion. Proc. of Use of Computers in
the Fatigue Laboratory. ASTM STP 613, 1976,
pp. 104 - 125.
W. T. Kirkby and P. R. Edwards, A method of
fatigue life prediction using data obtained under
random loading conditions. RAE Tech. Rep.
66023, Royal Aircraft Establishment, Farnborough, England, 1966.
R. C. Linsley and B. M. Hilberry, in Probabilistic
Aspects of Fatigue. ASTM STP 511, 1971, pp. 156
167.
-