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Journal of Chongqing University (English Edition) [ISSN 1671-8224]

Vol. 8 No.1

March 2009

50

Article ID: 1671-8224(2009)01-0050-07

To cite this article: YE Zhi-xiong, LI Li, LONG Xiao-hong. Aeolian vibration control measures for ground wires of Han River long span transmission lines [J]. J Chongqing

Univ: Eng Ed [ISSN 1671-8224], 2009, 8(1): 50-56.

Aeolian vibration control measures for ground wires of Han Riverlong span transmission lines

YE Zhi-xiong1,2,†

, LI Li1,2

, LONG Xiao-hong1,2

1 School of Civil Engineering and Mechanics, HuaZhong University of Science and Technology, Wuhan 430074, P. R. China

2 Hubei Key Laboratory of Control Structure, HuaZhong University of Science and Technology, Wuhan 430074, P. R. China

Received 17 June 2008; received in revised form 5 September 2008

Abstract: Long span ultra-high voltage (UHV) transmission lines have serious aeolian vibration problems. To control these

vibrations, we improved the energy balance method in the following aspects: the wind power input, the conductor self-damping,

and the damper dissipated power. Meanwhile, we built a theoretical mechanical model of β wire dampers and derived energy

dissipation calculation formulae. This permits the vibration energy dissipated by β wire dampers can be considered in the energy

balance method. Then, we developed a computer program based on the improved energy balance method using Matlab, and

analyzed UHV long span ground wires of the Han River long span project in P. R. China. The results show that the combination

of β wire dampers and Stockbridge dampers can reduce vibration of UHV long span transmission lines, which provides a

reference for research and construction of UHV engineering projects.

Keywords: ultra-high voltage transmission lines; aeolian vibration; energy balance method; Stockbridge dampers; β wire

dampers

CLC number: TM752 Document code: A

1 Introduction a

Researchers have investigated aeolian vibrations of

electrical transmission lines (also called conductors) for

many years. Many researchers have demonstrated that

suitable damping devices can reduce the induced

dynamic bending strain values of the conductors,

thereby protecting conductors from fatigue failure.

Zheng [1] and Ervik et al. [2] used the energy balancemethod to analyze aeolian vibrations of an overhead

line. They evaluated the vibration level by finding the

balance between the ambient wind energy input and the

energy dissipated by both the conductor itself (namely

self-damping) and the added dampers. However, the

†YE Zhi-xiong (叶志雄): [email protected].

∗ Funded by the Natural Science Foundation of China (No.

50878093).

result of this method has some errors because the windenergy input and conductor self-damping are uncertain

and the accuracy mostly depends on experimental

precision.

Recently, Diana et al. [3], Lu and Chan [4]

developed a novel approach based on impedance

transfer plus forced vibration to analyze aeolian

vibration of a single conductor with multiple dampers.

However, there is no method for calculating the powerdissipated by wire dampers even though the calculation

theory for Stockbridge dampers is comparatively

mature [5-6]. The installation of wire dampers usually

is based on qualitative analysis and simulation testing.

The electrical demands increased dramatically in P.

R. China as its economy developed rapidly. Therefore,

the State Grid Corporation of China planned and

constructed an ultra-high voltage (UHV) grid. Due to

their cross sections, the height of suspension and span

distance of long span UHV transmission lines increase,

and their vibration conditions become more serious. In


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Z. X. Ye, et al.

Aeolian vibration control measures

J. Chongqing Univ. Eng. Ed. [ISSN 1671-8224], 2009, 8(1): 50-56 51

this paper, we improved the energy balance method,

built a theoretical mechanical model of β damper, and derived energy dissipation calculation formulae. Basedon the improved energy balance method, we then

developed a computer program using Matlab software.

We also analyzed UHV ground wires crossing the Han

River. The results show that the algorithm is very

efficient and it meets the analytical requirement of

UHV transmission lines. Therefore, the combination of

β wire dampers and Stockbridge dampers can provide

anti-vibration for UHV long span transmission lines.

Our work serves as a reference for research and

construction of UHV engineering projects.

2 Improvement of the energy balance method

When conductor self-damping alone cannot ensure

the safety of transmission lines, special dampers need

to be mounted. The following energy balance holds for

a given frequency (or mode) over the span:

W C DP P P= + , (1)

whereW

P is the ambient wind power input over the

free span;C

P is the power dissipated by the conductor

itself over the free span; andD

P is the power dissipated

by the dampers. The Stockbridge damper, the wiredamper and the spacer damper are commonly used

damping devices. DP is the sum of their consumed

energy.

D d b sP P P P= + + , (2)

whered

P is the power dissipated by Stockbridge

dampers, P by wire dampers, ands

P by spacer

dampers. Obtaining each power is the key step of the

energy balance method. Therefore, we improved this

method in three aspects: the wind power input, the

conductor self-damping and the damper dissipated power.

2.1 Improvement of the wind power input

The wind power inputW

P normally is computed

from aerodynamic forces obtained by wind tunnel

experiments which are carried out by measuring the

drag and lift forces on the oscillating cylinders and

flexible rods or cables under uniform laminar cross-

flow in the wind tunnel (as shown in Ref. [7]). The

power (per unit length) imparted to a conductor by

wind can be expressed by the following empiricalequation [4]:

3 40

W ( ) ( ) A

P F V F f D D

′= , (3)

where V is the velocity of wind; D is the conductor

diameter; A0 is the steady state displacement amplitude;

f is the vibration frequency of conductor; and F and F ′ represent the functions, respectively.

Terrain and ground objects show significant effects

on reducing the wind power input values as compared

to that in laminar winds. Kraus and Hagedorn [8]

carried out field tests on an actual transmission line,and compared the results with that of the wind tunnel

data. Then they pointed out that the wind power input

strongly depends on the turbulence level and the

component of the wind velocity parallel to the cable.

Therefore, the wind power input obtained from wind

tunnel experiments should be verified for an actual line

according to terrain categories at the site. Considering

the terrain category, we applied a reduction factor

suggested by the CIGRE (conseil international des

grands réseaux électriques ) to reduce the wind energy

due to turbulence for the appropriate terrain. This

reduction factor can be expressed as

w2

L

1

1 ( / )V I I β =

+, (4)

wherew

β is the wind energy reduction factor due to

turbulence;L

I is the lock-in index and is used to

reflect the lock-in effect in aeolian vibration, which

may take the value of 0.09;V I is the turbulence

intensity of the wind and

m/V V I V σ = , (5)

in which V σ is the standard deviation of wind speed;and

mV is the mean wind speed.

The turbulence intensity can be determined by the

statistics of local wind. However, this information is

lacking. Then an empirical equation in Ref. [2] may be

used to estimate the value.

2.2 Improvement of conductor self-damping

When a conductor flexes, its strands slip against

each other. This relative motion generates frictional


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Z. X. Ye, et al.



forces which provide damping. In addition, internal

losses are incurred within the core and individualstrands of the conductor at the microscopic level,

which is known as the material damping. The

combination of these dissipative effects is referred to as

conductor self-damping, which depends on material

composition, manufacturing technology, actual

tensions, ambient temperature, vibration frequency and

amplitude, and other factors. Conductor self-damping

thus is determined mainly by experimental

measurement. It is usually approximated in the

following form:

0

C

( ) y

P Kf D

β α = , (6)

where0

y is the amplitude of vibration; K , α and β are

coefficients determined experimentally.

At the preliminary design stage, if no cable sample is

available and the cable type has not yet been measured,

some hypotheses are required. It would be convenient

to have available rules that predict internal losses of

cables. Noiseux [9] derived the earliest calculation laws

of internal damping of stranded cables based on the

basic cable differential equation. Lu [4] suggested asemi-empirical equation for calculating conductors

self-damping as:

4.5 6 2.56

C eq S 0 5

c

1.07 10 D

D f AP E k k K L

V

β +

= × , (7)

where A is the displacement amplitude; L is the length

of a conductor;S

k is the reduction factor on the

maximum conductor bending stiffness andS

0.5k = ;

Dk is an empirical factor, and

D0.54k = for an

aluminum conductor steel reinforced (ACSR)

conductor, whileD

0.65k = for a stranded steel

conductor;c

V is conductor’s wave velocity, which is

related to the conductor’s catenary constant C , and

cgV C = , in which g is the gravity constant; eq E is

the equivalent Young’s modulus of the conductor; 0K

is a constant for conductors, which can be determined

by test.

By curve fitting the Kinectrics’ data, the following

results were obtained [10].

1) For an all-aluminum conductor (AAC) Arbutus

conductor:

K 0 = 0.003 5 and β = −0.433 2;

2) For an ACSR Drake conductor:

K 0 = 0.004 2 and β = −0.425 6.

We modified the semi-empirical equation theoretical

calculations referring to the test results of Xu and

Wang [11] and realized the improvement of conductor

self-damping. Certainly, experiment tests might be

necessary to obtain the exact self-damping values of

some types of wire.

2.3 Improvement of power dissipated by dampers

We applied an improved calculation method [12] to

optimize the damper location. First, the probable

distribution of transmission line vibration frequencies

were obtained according to the frequency distribution

of the wind speed. Then the weighted amplitude ratio

was calculated to determine the optimal installation

position of dampers on aerial conductors. This method

can take into consideration the dangerous vibration

frequency range and the occurrence probability of each

vibration frequency. Finally, an advanced formula [13]

was used to calculate the power characteristics of

damper, which can take into account the damper

installation position and the conductor bending

stiffness and tension.

When dampers are fixed on overhead transmissionlines, the characteristics of the coupled vibration of

dampers and transmission lines create a complicated

problem involving a structure-flow interaction. There is

a significant error if we do not consider the distortion

due to the dampers when we study the response of

transmission lines. Therefore, finding the ratio of the

amplitude at the damper’s clamp to that at the free span

is necessary.

We found different approaches in the literature

solving for the amplitude ratios. For example, Li et al.

[14] indicated that the finite element method can

analyze the dynamic characteristics of conductor-damper coupling systems and account for damper

position, a conductor’s bending stiffness, the rotational

effect of dampers, and the possible support flexibility

at the end of a conductor span. We used this approach

to carry out modal analysis of the coupling system and

find the amplitude ratios for each vibration mode.

3 Calculation of power dissipated by wire

dampers

Wire dampers are very complex vibration


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Z. X. Ye, et al.



elimination systems. There is no complete systematic

method for designing and calculating the power theydissipated. Therefore, their installation is usually based

on qualitative analysis and simulation testing. Wire

dampers (Bretelles) consist of a length of conductor

which is similar to the main conductor in the span.

They are mounted under the main conductor and

attached by a type of parallel groove clamp. Three- and

four-loop dampers of one piece construction (festoons)

mount around the attachment clamps reaching out into

the span. The energy dissipation mechanism of wire

dampers is similar to the combinations of many

Stockbridge dampers. Festoons of varying lengths have

a damping action for different vibration frequencies.

They take many forms, such as the modified Bate

damper, the cross wire damper, the Christmas tree

formed damper, and so on.

Wang [5] pointed that β wire dampers combined

with Stockbridge dampers are reasonable and suitable

for the vibration damping of long span transmission

lines. This anti-vibration measure has been applied in

more than 20 long span projects in P. R. China.

Laboratory testing, the field vibration measurement,

and running state detection all have shown that it is

effective and practical. Therefore, we mainly studied

this measure in the paper.

3.1 Mechanical model of wire damper

We assumed that two ends of a wire damper are

rigidly fixed on an overhead conductor. The average

wire pressure is T ; the mass per unit length of wire is m;

the wire bending stiffness is EI ; and the wire damping

ratio is η . Suppose the vertical displacement of wire

dampers is y, the governing equation for the free

motion of a wire damper then can be deduced as

4 2 2

4 2 20

y y y y EI T m

t x x t

η ∂ ∂ ∂ ∂

− + + =

∂∂ ∂ ∂

, (8)

where x is the abscissa and represents the length from

the damper location to the supporting clamp; and t is

the time. Solving Eq. (8), the natural vibration

frequencies of wire damperscω can be obtained:

22 2 2 2

c 2 2

π π1

2

n T n EI

m L m L T

η ω

⎛ ⎞ ⎛ ⎞= + −⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠, (9)

where n is a natural number. Each main vibration shape

is

( )2 c cπ

( , ) e cos sin sint

mn

y x t A t B t x L

η

ω ω −

= + , (10)

where A and B are constants.

Suppose the connection points of a wire damper are

x1 and x2 respectively. According to the conditions for

matching displacement, velocity and acceleration of the

two points, the ratio of the amplitude of wire damper

( A ) to that of free span conductor ( c A ) is obtained.

( )

( )

2

2 12 12

b c b c

2 1c

b

ππ πsin cos sin

πsin

n x xn x n x

A L L L

n x x A

L

−⎛ ⎞−⎜ ⎟

⎛ ⎞ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎜ ⎟

⎝ ⎠2

1

c

πsin

n x

L

⎛ ⎞⎜ ⎟⎝ ⎠

, (11)

where Lc is the length of conductor; L b is the full arc

length of wire damper.

3.2 Power dissipation calculation formulae for wire

dampers

When a wire damper vibrates due to external forces(such as aeolian excitation), the vibration equation may

be approximated as:

4 2 2

ω4 2 2( , )

y y y y EI T m F x t

t x x t η

∂ ∂ ∂ ∂− + + =

∂∂ ∂ ∂, (12)

whereωF represents the dynamic force acting on the

wire damper due to vortex shedding. Multiplying Eq.

(12) by y

t

∂

∂ and integrating it over one cycle and over

one span length leads to

0

0

4 2 2

4 2 20d d

L t

t

y y y y y EI T m t x

t t x x t

τ

η + ⎡ ⎤∂ ∂ ∂ ∂ ∂− + + =⎢ ⎥∂ ∂∂ ∂ ∂⎣ ⎦

∫ ∫

[ ]0

0ω

0 ( , ) d d

L t

t

yF x t t x

t

τ + ∂

∂∫ ∫. (13)

The left side of Eq. (13) corresponds to the energy

dissipated by the wire damper over one cycle and over

the span length. The right side of Eq. (13) refers to the

energy imparted to the wire damper span by external

forces over one cycle. Thus the power dissipated by the

wire damper can be found by the left-hand side of Eq.


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Z. X. Ye, et al.



(13) after incorporating Eq. (10) into Eq. (13).

5 4 2 3 2

b c b c c16π

EI P EI A fL A m

T δ λ δ ω −= + , (14)

where λ is the vibration wavelength; andc

δ is the

material damping ratio of wire damper.

Generally, after a suspended wire damper is

subjected to its own weight, its assumes a catenary

configuration. The curve equation can be written as

cosh( / ) y C x C = . Assuming the span length of wire

damper is Ld and the sag of wire damper is S , the full

arc length of wire damper can be approximated by

2

b d

d

83S L L L

= + .

The material damping ratio of conductor can be

empirically expressed as follows.

Free span:

0.52

2 6D

c 0 2

c

(2π) 102

k D f AK f

V

β δ ⎡ ⎤

= ×⎢ ⎥⎣ ⎦

,

and span terminal

0.56D

c 02π 10

2

k DmK f fA

EI

β δ ⎡ ⎤= ×⎢ ⎥⎣ ⎦

.

The energy dissipation calculation formula for wire

damper then can be expressed as:

4.5 6 2.56

b eq D S 0 b 5

c

1.07 10 D f A

P E k k K LV

β +

= × +

1.5 3.5 2.54 0.25 0.75

eq D 0

c

8.7 10 D f A

E k K mV

β +

× , (15)

in which c gV T = .

4 Anti-vibration design of ground wires

We mainly introduced the anti-vibration measure

design for ground wires because the aeolian vibration

of ground wires generally is more severe. We took the

Han River long span project as an example. The Han

River long span transmission lines of the 1 000 kV

Jindongnan-Nanyang-Jingmen UHV AC transmission

project are 1 650 meters long. The type of ground wire

is JLB20B-240 and its diameter is 0.020 m. The mass

per unit length is 1.595 5 kg m−1, and the average

running tension is 48 945.8 N.In the project, we chose the wind power curve of

Diana and Falco to compute the wind energy. Taking

into consideration the effect of wind turbulence

intensity, we theoretically calculated the value of

ground wire self-damping and advanced it by referring

to test results of other types. Based on the improved

energy balance method, we developed a computer

program using Matlab software. Then we computed the

deflection and bending strain in a conductor without a

damper. According to the standard of danger vibration

[15], when the bending strain (ε ) exceeds 200 × 10−6, itis dangerous for aluminum steel-covered strands

JLB20B-240. Therefore, the dangerous vibration

frequency range is from 7.4 Hz to 62.9 Hz. According

to the calculation method in Ref. [12], we can

determine the maximum vibration proof efficiency ( M )

at the location of the corresponding x (Fig. 1).

It can be seen from the Fig. 1 that the best location

of FR4 dampers (one type of Stockbridge dampers) are

1.01 m, 2.34 m and 3.70 m away from the supporting

clamp when the key protection frequency range is from

6 Hz to 65 Hz. The results of analysis after dampers

were mounted in these positions are shown in Fig. 2.

Fig. 1 Maximum vibration proof efficiency ( M ) with its

corresponding length from the location of damper to the

supporting clamp ( x) after installing FR4 type dampers

The aeolian vibration of the Han River UHV long

span project ground wires is severe (Fig. 2). The

dangerous frequency range is 8 Hz to 64 Hz, which is

wide. Three Stockbridge dampers are needed at each

end. Fig. 2 shows that Stockbridge dampers of two FR3


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Z. X. Ye, et al.



type plus four FR4 type in the span achieve better

vibration reduction than 6 FR4 ones. Therefore,combined use of Stockbridge dampers with different

power characteristics better suppresses aeolian

vibrations. However, ε exceeds 200 × 10−6 whenvibration frequencies are between 45 Hz and 52 Hz.

Test calculations indicate that additional Stockbridge

dampers have no obvious effect. Therefore, we chose

to use Stockbridge dampers combined with β wire

dampers.

Fig. 2 Bending strain (ε ) of the ground wires with and

without dampers

Suppose β wire dampers using the same material as

the ground wire, the cable sag is 1/10 to 1/5 times its

full length. Referring to Eq. (9), the expression for

natural frequencies of a wire damper (n

ω ) can be

simplified as:

2 2

n 2

π π1

n T n EI

L m L T ω

⎛ ⎞= +⎜ ⎟

⎝ ⎠. (16)

Inserting each parameter into (16) gives:

22 2

n 2

π π π1 26.92

n T n EI n

L m L L T ω

⎛ ⎞ ⎛ ⎞= + =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠. (17)

The minimum resonant frequency for a 4.0 m single

festoon wire damper is approximately 3 Hz, about

42 Hz for a 1.0 m one, and roughly 66 Hz for a 0.8 m

one.

Considering the deficiency of energy consumption in

high frequency of Stockbridge dampers, we arranged 2

to 3 0.8 meter-long festoons wire dampers. After

computing and optimizing structural forms of wire

dampers, β wire dampers are eventually designed ascombinations of a 4.0 m single festoon wire damper, a

3.0 m one, a 2.0 m one, and two 0.8 m ones in the span

(5 festoons in total). The nearest mounting point is

0.3 m away from the supporting clamps of the

conductor. Fig. 3 shows the mounting configuration

and Fig. 4 shows the analysis results.

Fig. 3 Installation scheme of β dampers and FR dampers

Fig. 4 Bending strain (ε ) with combination anti-vibration

measures

It can be seen from Fig. 4 that β wire dampers

combined with Stockbridge dampers can respond to

most of the frequencies experienced over the range of

wind speeds causing vibration, and can keep the bending strain under the required standard. Short

festoon wire dampers have very good energy

dissipation capacity at high frequencies. If they are

decreased in weight by layer-stripping, a broader range

of frequencies possibly could be suppressed. The

results show that this damping device can satisfy the

vibration prevention requirements of UHV long span

transmission lines. Nevertheless, the accuracy of

numerical calculation results mainly depends on

parameters of wind input power and conductor self-

damping, which need the confirmation of field testing.


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Z. X. Ye, et al.



5 Conclusions

1) An improved energy balance method can account

for the effect of terrain and ground objects, support

boundary conditions at the span end of conductors,

average running tension of the conductor, and the

damper position. Therefore, the method is applicable in

aeolian vibration suppression design and calculation.

2) The derived energy dissipation calculation

formulae of β wire dampers are effective theoretically

and practically.

3) The results of numerical calculation show that β

wire dampers combined with Stockbridge dampers are

effective at suppressing aeolian vibrations. This

damping device can satisfy the vibration preventionrequirements of UHV long span transmission lines,

which can be used for the vibration damping of long

span transmission lines.

4) Theoretical results must be checked and verified

by field testing because of the particularity and

importance of UHV. Vibration model testing and field

vibration measurement absolutely are necessary for

construction. It is possible to design transmission lines

with more operational safety using these methods.

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