research article hysteresis phenomenon in the galloping of...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 784239, 11 pageshttp://dx.doi.org/10.1155/2013/784239

Research ArticleHysteresis Phenomenon in the Galloping of the D-ShapeIced Conductor

Bin Liu, KuanJun Zhu, XinMin Li, and XuePing Zhan

China Electric Power Research Institute, Beijing 100192, China

Correspondence should be addressed to Bin Liu; [email protected]

Received 5 June 2013; Revised 16 October 2013; Accepted 19 October 2013

Academic Editor: Stefano Lenci

Copyright © 2013 Bin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is well known that there is a hysteresis phenomenon in the amplitude variation in the iced conductor galloping with the windvelocity, which will have more obvious disadvantages to the overhead transmission lines. But hysteresis characteristics in theconductor galloping have not received much attention. In this paper, a continuum model of the conductor galloping with D-shape ice is derived by using Hamilton principle, where the initial deformation, the geometric nonlinearity caused by the largedeformation, and the aerodynamic nonlinearity are considered. The aerodynamic forces are described by using the quasi steadyhypothesis, where the aerodynamic coefficients are expanded by the polynomial curves with a third order and a ninth order,respectively.Thehysteresis phenomenon is analyzed by using the approximate solutions of theGalerkin discretized equation derivedfrom the continuum model by means of the harmonic balance method. The influence of the different factors, dynamic angle ofattack, span length, initial tension, and conductor mass, is obtained in different galloping instability intervals. And two importantaspects about the point of the hysteresis phenomenon onset and the size of the hysteresis region over thewind velocities are analyzedunder different conditions.

1. Introduction

It is well known that iced conductor galloping is a typical low-frequency self-excited vibration phenomenon [1]. Conductorgalloping represents a classical motion instability mechanismin the steady flow over a noncircular cross-section caused byice accretion on the conductor. This conductor motion ischaracterized by large amplitude (possibly > 10) and lowfrequency (approximately 0.1–3Hz) [2]. Although conductorgalloping trace has usually an elliptical orbit on field obser-vations, the predominant motion in galloping is vertical.Galloping can cause various kinds of structural and electricaldamages in overhead lines, which can have devastating eco-nomic and social consequences [3, 4].

Conductor galloping has been studied extensively over ahalf century since DenHartog [5] firstly established the verti-cal gallopingmechanism using the quasi steady hypothesis todescribe the linearized aerodynamic forces based on a simplesingle degree-of-freedom (DOF) model. Whereafter Nigoland Buchan [6, 7] proposed a torsional galloping mechanismbased on a two-DOFs model coupled with vertical andtorsional oscillations. And Yu et al. [8, 9] obtained a torsional

feedback mechanism using a three-DOFs model coupledwith vertical, horizontal, and torsional oscillations. Andlater, with the development of linear and nonlinear motioninstability theories, galloping phenomenon has been studiedwidely by using nonlinear vibration methods [10–14].

At present, the Den Hartog theory and the torsionaltheory are still the two important and dominant mechanismsto explain the phenomenon of conductor galloping. Thetorsional theory shows that the torsionalmotion has a consid-erable effect on the instability and that its coupling with thevertical motion is responsible for most cases of the conductorgalloping phenomenon [15], which is the main differencewith the Den Hartog theory. So it can be excited based onDen Hartog mechanism when the torsional motion doesnot occur in the process of conductor galloping. And if thetorsional motion is observed, the conductor galloping maybe initiated by torsional theory. Especially for the bundledconductor, vertical-torsional coupling is usually the mostsignificant to lead to galloping because of the close proximityof the natural frequencies between vertical and torsionalmotion. This has been verified experimentally by a numberof investigators [16]. In this study, only the vertical vibration

2 Mathematical Problems in Engineering

is considered in order to discuss the hysteresis phenomenonmore conveniently for the more complex nonlinear dynamicproblem after the conductor galloping has appeared. And theinitial conditions of conductor galloping are obtained by DenHartog mechanism.

There are numerous examples of fundamental solutionsto the conductor galloping problem as a nonlinear motioninstability process [10–14]. The main objectives are usuallythe prediction of the stability of such aeroelastic systems ata range of flow conditions, the amplitudes, and frequencies ofthe Limit Cycle Oscillation (LCO) that may be encountered.And most of the efforts in galloping vibration researcheshave been focused on blunt bodies with regular square [17],rectangular [18, 19], or triangular cross-sections [20, 21]. Butthe iced-coated conductor on the transmission linesmay haveasymmetric complicated shapes that can usually be simplifiedas D-shape, U-shape, or other complex shapes, in which D-shape iced conductor is well known to induce high gallopingamplitudes [4, 22].

In fact, it may still present more complex nonlineardynamic characteristics such as bifurcation, hysteresis, andchaos, after the conductor galloping phenomenon starts off,which is significant equally to understand the gallopingmechanisms [13, 17, 23–27]. Qin et al. [14] analyzed the bifur-cation phenomenon for D-shape iced conductor gallopingwith a two-DOFs model by singularity theory. And theyconcluded that hysteresis phenomenon is probably obtainedafter LCO appeared based on appropriate parameters. Luoet al. [17] studied the square cylinder galloping phenomenonwith different Reynolds number. And the results showed thatthe existence of intermittent shear layer reattachment leadsto the existence of a hysteresis region. Barrero-Gil et al.[27] revealed the existing link between the hysteresis phe-nomenon and the number of inflection points at the aero-dynamic force coefficient curve by means of the method ofKrylov-Bogoliubov. And the bluff body was modeled by alinear oscillator of one DOF. Alonso et al. [28] demonstratedthat hysteresis takes place at the angles of attack where thereare inflection points in the lift coefficient curve based on theisosceles triangular cross-section bodies.

The hysteresis effect is observed in some field observa-tions after the onset of conductor galloping. In a specific rangeof wind velocity, the galloping amplitude and mode presentsevere variation. For instance, the galloping amplitude thatsuddenly increases a few timeswill causemore serious impactto the transmission lines. It is apparent that the hysteresis phe-nomenon presented in the galloping response will be a moresevere disadvantage to the overhead transmission lines thanthe only LCO onset. So understanding of the hysteresis phe-nomenon in the conductor galloping is important andmean-ingful not only in theory, but also in the development of theantigalloping methods for the transmission lines. Althoughmany researchers have been focused on these galloping fea-tures, the hysteresis phenomenon in the conductor gallopinghas not been given enough attention to date. In this paper,in order to analyze the hysteresis phenomenon in conductorgalloping, a continuum model that describes approximatelythe conductor galloping phenomenon is constructed by usingHamilton principle and considering the initial location and

the geometric nonlinearity caused by the large deformationand the aerodynamic nonlinearity caused by the flow. Theaerodynamic forces of D-shape iced conductor are describedby using the quasi-steady hypothesis. And as a contrast, theaerodynamic coefficients are expanded by the polynomialcurves with a third order and a ninth order, respectively. Thehysteresis phenomenon is analyzed by using the approximatesolution by means of the harmonic balance method. Thenthe influence of the different factors, dynamic angle of attack,span length, initial tension, and conductor mass, is proposedin different galloping instability intervals. And two importantaspects about the point of the hysteresis phenomenon onsetand the size of the hysteresis region over the wind velocitiesare analyzed under different conditions.

2. Construction of the Model

2.1. Continuum Model. The schematic diagram of the trans-mission line under initial tension 𝑇

0is shown in Figure 1(a).

The initial configuration 𝑦0(𝑥) can be expressed simply using

parabolic shape function as

𝑦0 (𝑥) = −

4𝑏𝑥 (𝑙 − 𝑥)

𝑙2

, (1)

where 𝑏 is the sag at the lowest point of the line and 𝑙 is thespan length. The line is modeled as a flexible centerline withrigid cross-sections shown in Figure 1(c) which is orthogonalto the axis. This configuration is planar that belongs to thevertical plane (𝑥- 𝑦). And 𝛼 is the modification of the angleof attack introduced by the vertical velocity ̇𝑦.

The displacement of an arbitrary line element 𝑑𝑥 isdescribed schematically in Figure 1(b). 𝑃𝑃

1is the initial

position with the line vector 𝑑 ⃗𝑟0and 𝑃𝑃

1is the movement

position with the changed vector 𝑑 ⃗𝑟 taken from the line atthe time Δ𝑡. And suppose that the displacements of the startand the end in 𝑥 and 𝑦 directions are 𝑢(𝑥, 𝑡), V(𝑥, 𝑡), and𝑢+𝑢𝑥𝑑𝑥, V+V

𝑥𝑑𝑥, respectively. Using arc length formula and

neglecting the higher-order terms of the axial deformation,the expression of the line element strain, 𝜀

𝑠, can be derived as

𝜀𝑠= 𝑢𝑥+

𝑑𝑦0

𝑑𝑥

V𝑥+

1

2

V2𝑥, (2)

where the right subscript “𝑥” indicates differentiation withrespect to coordinate 𝑥.

Then the equations of the line motion are obtained byemploying Hamilton principle [29], for example,

∫

𝑡2

𝑡1

(𝛿𝑇 − 𝛿𝑉 + 𝛿𝑊) 𝑑𝑡 = 0, (3)

where 𝛿 is the first-order variational operator, 𝑇 and 𝑉 arethe total kinetic and strain energies, respectively, and 𝑊 is

Mathematical Problems in Engineering 3

y

o

l

xdx

x

y0(x)

T0

P P1

(a)

P

P1

d−→r0

d−→r

Δ−→r1

Δ−→r2

P1

P

dx

(b)

y

FL

U o

𝜃0 𝜃z

Ur

−𝛼ẏ

U

FD

(c)

Figure 1: Model of the transmission line.

the work done by the nonconservative forces. And the 𝑇, 𝑉,and𝑊 are given by

𝑉 = ∫

𝑙

0

(𝑇0𝜀𝑠+

1

2

𝐸𝐴𝜀2

𝑠)𝑑𝑥,

𝑇 =

1

2

∫

𝑙

0

𝑚(�̇�2+ V̇2) 𝑑𝑥,

𝑊 = ∫

𝑙

0

(𝐹𝑦V − 𝑐0V̇V) 𝑑𝑥,

(4)

where𝑚 is the mass per unit length, 𝐸 is the elastic modulus,𝐴 is the cross-section area, 𝐹

𝑦is the external load done by the

aerodynamic force, 𝑐0is the damping coefficient, and a dot

superscript indicates differentiation with respect to time 𝑡.Substituting (2.1), (2), and (1) into (3) and omitting axial

inertial force, the line motion equation can be obtained [14,30] as

𝑚V̈ − 𝑇0V −

𝐸𝐴

𝑙

(

−8𝑏

𝑙2+ V𝑥𝑥)

× ∫

𝑙

0

(

4𝑏𝑙 − 8𝑏𝑥

𝑙2

V𝑥+

1

2

V2𝑥)𝑑𝑥 + 𝑐

01V̇ − 𝐹𝑦= 0.

(5)

From (5), it is evident that the line galloping equation intro-duces the initial stress, the geometric nonlinearity caused

by the large deformation, and the aerodynamic nonlinearitycaused by the flow. So it is difficult to solve the continuummodel directly.

2.2. Aerodynamic Force Model. Figure 1(c) shows a typicalshape of a typical ice accretion on the conductor surface,which is observed frequently in the case of freezing rain. InFigure 1(c), 𝐹

𝐷and 𝐹

𝐿are the aerodynamic lift and drag due

to wind actions. These forces depend on the dimensionlessdrag coefficient 𝐶

𝐷and lift coefficient 𝐶

𝐿, respectively, and

have the expressions:

𝐹𝐷=

1

2

𝜌𝑈2𝐷𝐶𝐷, 𝐹

𝐿=

1

2

𝜌𝑈2𝐷𝐶𝐿, (6)

where 𝜌 is the density of the flow, 𝑈 is the absolute windvelocity perpendicular to the conductor axis in the horizontaldirection, and 𝐷 is the reference length of the iced cross-section that is usually substituted by the conductor diameter.Then from Figure 1(c), considering the angle of attack as arelatively small amount, the aerodynamic force in 𝑦 directioncan be calculated as

𝐹𝑦= 𝐹𝐿cos (−𝛼) − 𝐹

𝐷sin (−𝛼) ≈ 𝐹

𝐿− 𝐹𝐷

V̇

𝑈

. (7)

The wind forces acting on the iced conductor can bemeasured in wind tunnels on stationary models by resorting


to the quasi-steady theory. This theory assumes that theaerodynamic forces acting on the iced conductor at anyinstant in motion are identical to that on the stationaryconductor under the same flow condition. And the use ofthe quasi-steady theory is justified because the frequencies ofconductor galloping aremuch lower than the vortex sheddingfrequencies.

So the curves of the aerodynamic coefficients are obtainedby the wind tunnel tests based on the stationary iced con-ductors at different angles of incidences. In order to establishthe aerodynamic model conveniently, the polynomial curveis applied to fit onto the force coefficients versus the angle ofattack by using the experimental data. Then drag coefficient𝐶𝐷and lift coefficient 𝐶

𝐿can be expressed as

𝐶𝐿=

𝑘

∑

𝑖=0

𝑐1𝑖𝜃𝑖, 𝐶

𝐷=

𝑘

∑

𝑖=0

𝑐2𝑖𝜃𝑖, (8)

where 𝑐1𝑖and 𝑐2𝑖are the fitting coefficients of the lift and

drag coefficient curves, respectively, and 𝑘 is the order ofthe polynomial. From Figure 1(c), 𝜃 is the dynamic angle ofattack based on the quasi-steady theory, which is relative tothe initial static offset angle 𝜃

0of the iced conductor (initial

ice position on the conductor) and the instantaneous angle of𝛼, and it can be expressed as

𝜃 = 𝜃0− (−𝛼) = 𝜃

0− arctg V̇

𝑈

≈ 𝜃0−

V̇

𝑈

. (9)

In the major applications of the existing literatures, thethird-order polynomial fitting formula is most widely used.But Parkinson et al. [23, 24] studied to fit by the differentorder polynomials. And they found that the seventh-orderpolynomial has superiority over the fifth order, because theformer can capture the inflection point of the aerodynamiccoefficient curve which is probably the pivotal factor to excitehysteresis phenomenon.Thework ofNg et al. [25] has alreadydemonstrated that a seventh-order polynomial curve used inthe quasi-steady theory was sufficient in revealing the square-cylinder galloping characteristics including the hysteresisphenomenon.

In this study, assume that the eccentric ice distributedalong the line is uniform and neglect the aerodynamic cou-ples. And the third-order and the ninth-order polynomialsare all selected to fit on the aerodynamic coefficient curves ofD-shape eccentric iced conductor.Then the contrast by usingdifferent-order polynomials can be obtained.

Substituting (6)–(9) into (5), the motion equation can betransformed:

𝑚V̈ − 𝑇0V𝑥𝑥−

𝐸𝐴

𝑙

(

−8𝑏

𝑙2+ V𝑥𝑥)

× ∫

𝑙

0

(


𝑙2

V𝑥+

1

2

V2𝑥)𝑑𝑥 + 𝑐

01V̇

−

1

2

𝜌𝑈2𝐷[

𝑘

∑

𝑖=0

𝑐1𝑖𝜃𝑘−

𝑘

∑

𝑖=0

𝑐2𝑖𝜃𝑘(

V̇

𝑈

)] = 0.

(10)

Neglecting the terms higher than third order, (10) is expandedinto

𝑚V̈ − 𝑇0V𝑥𝑥−

𝐸𝐴

𝑙

(

−8𝑏

𝑙2+ V𝑥𝑥)

× ∫

𝑙

0

(


𝑙2

V𝑥+

1

2

V2𝑥)𝑑𝑥

+ 𝑐1V̇ + 𝑑1V̇2 + 𝑑

2V̇3 = 0.

(11)

When 𝑘 = 3, there are

𝑐1= 𝑐01+

1

2

𝜌𝑈𝐷[𝑐11+ 𝑐20+ (2𝑐12+ 𝑐21) 𝛼0

+ (3𝑐13+ 𝑐22) 𝛼2

0+ 𝑐23𝛼3

0] ,

𝑑1=

1

2

𝜌𝐷 [ − 𝑐21− 𝑐12− (3𝑐13+ 2𝑐22) 𝛼0

−3𝑐23𝛼2

0] ,

𝑑2=

1

2𝑈

𝜌𝐷 (𝑐13+ 𝑐22+ 3𝑐23𝛼0) .

(12)

When 𝑘 = 9, there are

𝑐1= 𝑐01+

1

2

𝜌𝑈𝐷

× [𝑐11+ 𝑐20+ (2𝑐12+ 𝑐21) 𝛼0+ (3𝑐13+ 𝑐22) 𝛼2

0

+ (4𝑐14+ 𝑐23) 𝛼3

0+ (5𝑐15+ 𝑐24) 𝛼4

0

+ (6𝑐16+ 𝑐25) 𝛼5

0+ (7𝑐17+ 𝑐26) 𝛼6

0

+ (8𝑐18+ 𝑐27) 𝛼7

0+ (9𝑐19+ 𝑐28) 𝛼8

0+ 𝑐29𝛼9

0] ,

𝑑1=

1

2

𝜌𝐷

× [−𝑐12− 𝑐21− (3𝑐13+ 2𝑐22) 𝛼0− (6𝑐14+ 3𝑐23) 𝛼2

0

− (10𝑐15+ 4𝑐24) 𝛼3

0− (15𝑐

16+ 5𝑐25) 𝛼4

0

− (21𝑐17+ 6𝑐26) 𝛼5

0− (28𝑐

18+ 7𝑐27) 𝛼6

0

− (36𝑐19+ 8𝑐28) 𝛼7

0− 9𝑐29𝛼8

0] ,

𝑑2=

1

2𝑈

𝜌𝐷

× [𝑐13+ 𝑐22+ (4𝑐14+ 3𝑐23) 𝛼0+ (10𝑐

15+ 6𝑐24) 𝛼2

0

+ (20𝑐16+ 10𝑐25) 𝛼3

0+ (35𝑐

17+ 15𝑐26) 𝛼4

0

+ (56𝑐18+ 21𝑐27) 𝛼5

0+ (84𝑐

19+ 28𝑐28) 𝛼6

0

+36𝑐29𝛼7

0] .

(13)

2.3. Discrete Model. Due to solving (11), it will be approx-imated by using the Galerkin procedure and expanding


the displacement field in a series of suitable functions. Soaccording to the Galerkin method, the displacement V(𝑥, 𝑡)can be expressed with the expansion:

V (𝑥, 𝑡) =𝑛

∑

𝑖=1

𝑞𝑖 (𝑡) 𝜙𝑖 (𝑥) , (14)

where 𝑞𝑖(𝑡) is the generalized amplitude time law, 𝜙

𝑖(𝑥) is a

set of shape functions that satisfy the geometric boundaryconditions (𝜙

𝑖(0) = 𝜙

𝑗(𝑙) = 0), and 𝑛 is the number of shape

functions.In the displacement expansion, the use of the eigenfunc-

tions is often the best choice in the aspects of convergenceand accuracy [12, 31, 32]. But in order to concentrate thestudy of the hysteresis phenomenon of the vertical conductorgalloping, a simplified Galerkin procedure is presented byusing a sine series of the antisymmetric in-plane modes asassumed shape functions [33]; that is, 𝜙

𝑖(𝑥) = sin(2𝑖𝜋𝑥/𝑙).

And for the sake of decreasing the number of motionequations and achieving an analytical expression to discussthe nonlinear behavior of the hysteresis phenomenon, thefirst-order discretization is selected. Then the discretizedequation from (11) can be derived as

̈𝑞1+ 𝐾𝑞1+ 𝐶 ̇𝑞1+ 𝑁1𝑞2

1+ 𝑁2𝑞3

1+ 𝑁3̇𝑞2

1+ 𝑁4̇𝑞3

1= 0, (15)

where

𝐾 =

𝑇0𝜋2

𝑙2𝑚

+

512𝑏2𝐸𝐴

𝜋2𝑙4𝑚

,

𝐶 =

𝑐1

𝑚

, 𝑁1=

24𝑏𝜋𝐸𝐴

𝑙4𝑚

, 𝑁2=

𝜋4𝐸𝐴

4𝑙4𝑚

,

𝑁3=

8𝑑1

3𝜋𝑚

, 𝑁4=

3𝑑2

4𝑚

.

(16)

From (15), both quadratic and cubic nonlinearities appear inthe displacement and velocity terms.

3. Analytical Solution ofthe Galloping Equation

In the work of Vio et al. [13], six common methods werecompared with each other to predict bifurcation and LCOamplitudes of the transverse galloping for a square sectionbeam.They found that two methods, higher-order harmonicbalance and numerical continuation, can fully and accuratelycharacterize the problem. So in this study, the harmonicbalance method is employed to solve (15).

Assume the response solution of (15) is

𝑞1(𝑡) =

𝐽

∑

𝑗=1

𝐴𝑗+ 𝐵𝑗sin 𝑗𝜔𝑡, (17)

where 𝐴𝑗and 𝐵

𝑗are unknown coefficients which are correl-

ative with the static offset and the dynamic amplitude.Conductor galloping is a low-frequency, high-amplitude

wind induced vibration with a single or a few loops ofstanding waves per span. So assuming the galloping performs

a harmonic oscillation, as the initiative approximation thefirst term of (17) was used in this study. Substituting (17) into(15) and omitting the higher harmonic terms, the followingexpressions can be derived by making the constant term, thecoefficients of sin𝜔𝑡 and cos𝜔𝑡, respectively, to zero:

𝐵1(4𝐶 + 3𝑁

4𝐵2

1𝜔2) = 0, (18a)

2𝑁2𝐴3

1+ 2𝑁1𝐴2

1+ 2𝐾𝐴

1+ 3𝑁2𝐴1𝐵2

1

+ (𝑁1+ 𝑁3𝜔2) 𝐵2

1= 0,

(18b)

𝐵1(12𝑁2𝐴2

1+ 8𝑁1𝐴1+ 3𝑁2𝐵2

1+ 4𝐾 − 4𝜔

2)

= 0.

(18c)

From (18a), the expression of the coefficient 𝐵1can be

obtained as

𝐵1= 0, or 𝐵

1=

1

𝜔

√−

4𝐶

3𝑁4

, (

4𝐶

3𝑁4

< 0) . (19)

When 𝐵1= 0, from (18b), the coefficient 𝐴

1can be

obtained as

𝐴1= 0, or 𝐴

1=

−𝑁1± √𝑁

2

1− 4𝐾𝑁

2

2𝑁2

,

(𝑁2

1− 4𝐾𝑁

2≥ 0) .

(20)

When 𝐵1= (1/𝜔)√−4𝐶/3𝑁

4, from (18c), the coefficient

𝐴1can be obtained as

𝐴1=

𝑁1𝑁4𝜔4− (𝑁1𝑁4𝐾 + 6𝑁

2𝑁3𝐶)𝜔2− 5𝑁1𝑁2𝐶

−3𝑁2𝑁4𝜔4+ (2𝑁

2

1𝑁4− 6𝑁2𝑁4𝐾)𝜔2+ 15𝑁

2

2𝐶

.

(21)

Then substituting (19) and (21) into (18b), a higher-orderlinear equation about the unknown parameter 𝜔 can bederived as

𝑓 (𝜔) = 𝑍1𝜔12+ 𝑍2𝜔10+ 𝑍3𝜔8+ 𝑍4𝜔6

+ 𝑍5𝜔4+ 𝑍6𝜔2+ 𝑍7= 0,

(22)

where 𝑓(𝜔) is a 12th order polynomial expression (onlyincluding even order terms) with respect to 𝜔, 𝑍

1− 𝑍7are

the coefficients, and there are

𝑍1= 9𝑁2

2𝑁3

4,

𝑍2= 3𝑁2(9𝐾𝑁

2𝑁3

4− 3𝑁2

1𝑁3

4) ,

𝑍3= −81𝐶𝑁

3

2𝑁2

4,


𝑍4= 3𝑁2

× (−36𝐶𝐾𝑁1𝑁2𝑁3𝑁2

4+ 3𝐾2𝑁2

1𝑁3

4

+ 6𝐶𝑁2

1𝑁2𝑁2

4− 12𝐾

3𝑁2𝑁3

4+ 8𝐶𝑁

3

1𝑁3𝑁2

4

−18𝐶𝐾𝑁2

2𝑁2

4− 36𝐶

2𝑁2

2𝑁2

3𝑁4) ,

𝑍5= 3𝑁2(45𝐶2𝑁3

2𝑁4+ 72𝐶𝐾

2𝑁2

2𝑁2

4

−48𝐶𝐾𝑁2

1𝑁2𝑁2

4+ 8𝐶𝑁

4

1𝑁2

4) ,

𝑍6= 3𝑁2(−135𝐾𝐶

2𝑁3

2𝑁4+ 45𝐶

2𝑁2

1𝑁2

2𝑁4) ,

𝑍7= 225𝐶

3𝑁5

2.

(23)

Solving (22) and considering the engineering practice, theeffective values of 𝜔 are obtained. Returning 𝜔 into (19) and(21), the values of 𝐴

1and 𝐵

1can be also obtained.

4. Analysis for Hysteresis Phenomenon

In this section, without loss of generality, a typical conductortype ACSR LGJ-400/35 is selected to analyze the hystere-sis phenomenon in the conductor galloping. And physicalparameters of the line are tabulated in Table 1. The typicalD-shape eccentric ice accretion covered on the conductor isselected in this study. Figure 2 shows the schematic diagramsof the iced conductor cross-section. The ice thickness ischosen to be equal to 15mm shown in Figure 2.

4.1. Instability Regions for Conductor Galloping. Figures 3and 4 give the quasi-steady, aerodynamic force coefficientsmeasured in a wind tunnel for the D-shape iced conductorat the different angle of attack 𝜃. A third-order polynomialis used to fit the experimental data in the range of −85∘ ≤𝜃 ≤ 85

∘ and a ninth-order polynomial is also used to fitin the range of −175∘ ≤ 𝜃 ≤ 175∘. The fitting intervalby third-order polynomial is less than that by ninth-orderobviously. From the figures, there is a certain discrepancybetween experimental data and the third-order fitting results,but no significant difference with the ninth-order fit. And thethird-order polynomial cannot capture the data versus wholeangles of attacks.

The instability regions relatively to the angle of attack thatcan excite conductor galloping need to be obtained before thehysteresis phenomenon is studied. According to Lyapunovstability theory, the occurrence of the unsteady solutionsof (15) should satisfy the condition, 𝐶 < 0, presented inthe literature [1]. Figure 5 presents the different instabilityregions of the angle 𝜃 based on third-order and ninth-orderfitting curves shown in Figures 3 and 4. For the D-shapeiced conductor, the angle interval of galloping occurrenceobtained by using third-order fitting curve is only a part ofthe results by using ninth-order fitting. The three instabilityregions derived from the ninth-order fitting curve are I =[−130

∘, −119

∘], II = [−18∘, 18∘], and III = [119∘, 130∘],

in which region I is symmetrical distribution with 0∘. And

Table 1: Physical parameter of the conductor type LGJ-400/35employed in this analysis.

Parameters Notation Units DataElastic modulus 𝐸 GPa 76.5Diameter of conductor 𝐷 mm 26.82Horizontal initial tension 𝑇

0kN 25.92

Span length 𝑙 m 500Cross-section area 𝐴 mm2 425Mass of unit length 𝑚 kg/m 1.349Damping ratio 𝜉 0.004

U0∘

D

𝜃

15 mm

Figure 2: Schematic diagrams of the D-shape iced conductor.

the only one region obtained by third-order fitting curve isI = [−18∘, 18∘].

4.2. Influence of the Different Factors on the Hysteresis Phe-nomenon. It is very known that the hysteresis phenomenonis characterized by the existence of a solution that canalternately reach different cycles limits in theory. In thisstudy, the hysteresis region in the conductor galloping isanalyzed by the solutions of the dynamic amplitude 𝐵

1versus

the horizontal wind velocity 𝑈 obtained by (18a), (18b),and (18c). The hysteresis characteristics are also investigatedwith respect to different factors such as dynamic angle ofattack, span length, initial tension, and conductor mass. Andtwo interesting aspects about the point of the hysteresisphenomenon onset and the size of the hysteresis region areproposed under the different conditions.

(1) Influence of the Dynamic Angle of Attack. Luo et al. [17]and Alonso et al. [28] studied the cause of the hysteresisphenomenon in transverse galloping of the square andthe triangular cross-section bodies, respectively.Their studiesrevealed that the cause of the hysteresis phenomenon isrelated to the point of inflection that exists in the aero-dynamic force coefficient curve versus the angle of attack.And Barrero-Gil et al. [27] later proved that the hysteresisphenomenon is related to the number of inflection points.

From Figure 4, three inflection points can be observedusing the ninth-order fitting curve, which are evenly dis-tributed in the three instability ranges. And one inflectionpoint is found from the third-order fitting curve in its

Mathematical Problems in Engineering 7D

rag

coeffi

cien

t

4.0

3.5

3.0

2.5

1.5

2.0

Angle of attack, 𝜃 (rad)3210−1−2−3

D-shape

Data from the wind tunnel test3th order polynomial

9th order polynomial

Figure 3: Drag coefficients of the D-shape iced conductor.

Lift

coeffi

cien

t

3

2

1

0

−1

−2

−3

3210−1−2−3Angle of attack, 𝜃 (rad)

D-shape

Inflection pointData from the wind tunnel test3rd order polynomial

9th order polynomial

Figure 4: Lift coefficients of the D-shape iced conductor.

only one instability range. Then hysteresis characteristicsin the conductor galloping are obtained in each instabilityrange based on the physical parameters presented in Table 1.Figure 6 shows the amplitude 𝐵

1versus the horizontal wind

velocity 𝑈 with several typical angles of attack in the insta-bility interval I. From Figure 6, we can see that the hysteresischaracteristic presents a symmetric distribution based on themiddle of the instability range. At the two ends of the angleinterval, 𝜃 = −130∘ and 𝜃 = 119∘, the velocities of thehysteresis phenomenon occurrence are much higher thanin the middle of the interval and the sizes of the hysteresisregions are also broader than in the middle. In addition, thecritical velocity to excite hysteresis phenomenon is very highin comparison with the point of galloping instability. Fromthe results of this example, the critical wind velocities to excitehysteresis phenomenon are all greater than 40m/s. So underthe actual engineering conditions, it is difficult to achieve forthe hysteresis phenomenon.

I II III

3rd order fitting/D-shape9th order fitting/D-shape

I

−100−150 −50 0 50 100 150Angle of attack , 𝜃 (deg)

Figure 5: Instability regions for the D-shape iced conductor basedon the third-order and ninth-order fitting curves of the aerodynamicforce coefficients.

B1

30

25

20

15

10

5

0

U

0 20 40 60 80 100 120

𝜃 = −130∘

𝜃 = −127∘𝜃 = −124∘

𝜃 = −119∘

Figure 6: Amplitude 𝐵1versus horizontal wind velocity 𝑈 in the

instability range I using the ninth-order polynomial.

Figure 7 shows the amplitude 𝐵1versus the horizontal

wind velocity 𝑈 with several typical angles of attack inthe instability interval II. From Figure 7, at the two ends ofthe angle interval, 𝜃 = −18∘ and 𝜃 = 18∘, the velocities of thehysteresis phenomenon occurrence are much higher than inthe middle of the interval. The velocities to excite hysteresisphenomenon are lower than in the angle interval I except theends of the interval.Therefore it is easy to excite the hysteresisphenomenon in this angle interval.

Figure 8 shows the amplitude 𝐵1versus the horizontal

wind velocity 𝑈 with several typical angles of attack in theinstability interval III. Table 2 gives the size of the hysteresisregionwith respect to different angles of attack. FromTable 2,the hysteresis phenomenon obtained in this interval is similarto the characteristics in the interval I presented in Figure 5.


Table 2: The size of the hysteresis region with different angles of attack.

Ninth-order fitting curve Third-order fitting curveAngle interval I Angle interval II Angle interval III Angle interval I

Angle (∘) Hysteresis range(U, m/s2) Angle (∘) Hysteresis range(U, m/s2) Angle (

∘) Hysteresis range(U, m/s2) Angle (∘) Hysteresis range(U, m/s2)

−130 (87.1, 97.9) −18 (43.5, 47.9) 119 (73.9, 86.7) −18 (39.5, 45.5)−127 (54.3, 63.9) −12 (21.1, 23.9) 123 (45.9, 55.9) −12 (21.5, 25.1)−124 (46.7, 56.7) 0 (15.1, 17.9) 125 (43.1, 53.5) 0 (15.9, 19.1)−119 (65.1, 81.5) 12 (15.9, 19.5) 130 (59.5, 75.9) 12 (18.7, 22.7)

18 (31.1, 39.5) 18 (33.1, 40.7)

B1

3028262422201816141210

86420

U

0 10 20 30 40 50

𝜃 = −18∘

𝜃 = −12∘

𝜃 = 0∘

𝜃 = 12∘

𝜃 = 18∘

Figure 7: Amplitude 𝐵1versus horizontal wind velocity in the

instability range II using the ninth-order polynomial.

Figure 9 shows the hysteresis phenomenon by usingthird-order aerodynamic force fitting curve with severaltypical angles of attack in the interval I presented in Figure 5.This angle interval I is the same of the angle interval IIobtained by ninth-order fitting curve shown in Figure 5.From Table 2, the hysteresis region obtained by using thethird-order fitting curve is consistent with the result in thesame angle of attack from the ninth-order fitting curve.

(2) Influence of the Span Length. In actual engineeringstructures of overhead transmission lines, the span lengthhas a wide selection range from 100m to 1200m, and thecommon range is from 300m to 700m. Select the physicalparameters presented in Table 1. Without loss of generality,select ninth-order fitting curve and 𝜃 = 0∘ presented inFigures 3 and 4 as a typical example.

The effects of span lengths to the hysteresis phenomenonin the conductor galloping are shown in Figure 10. FromFigure 10 we can see that when 𝐿 < 440m, there is nohysteresis phenomenon occurrence on the process of theconductor galloping. And after the hysteresis phenomenon

B1

20

15

10

5

0

U

0 20 40 60 80 100

𝜃 = 119∘

𝜃 = 123∘𝜃 = 125∘

𝜃 = 130∘


instability range III using the ninth-order polynomial.

appears when 𝐿 = 440m, the horizontal wind velocity ofthe hysteresis onset point increases with the much largerspan length, and the hysteresis range makes much widerwith the increase of the span length. And in the same windvelocity, before the occurrence of hysteresis phenomenon,the conductor galloping amplitude decreases as the spanlength increases and the difference is not obvious. But theamplitude jumps abruptly over the hysteresis range andincreases obviously as the span length increases.

In addition, when the span length reaches 590m, amore complex nonlinear vibration phenomenon is excitedin the conductor galloping. In the low wind velocity, a sec-ondary bifurcation is presented with an arch-shaped curve.It is likely to be a trend to the chaos. And with the increase ofspan length this phenomenon is more obvious.

(3) Influence of the Initial Tension. Figure 11 shows the ampli-tude 𝐵

1versus the horizontal wind velocity 𝑈 with different

initial tensions. The span length is 500m and the angle ofattack is 0∘. Other parameters are shown in Table 1. FromFigure 11 we can see that the hysteresis phenomenon willbe excited with the decrease of the initial tension. And

Mathematical Problems in Engineering 9B1

20

15

10

5

0

U

10 20 30 40 50

𝜃 = −18∘

𝜃 = −12∘

𝜃 = 0∘

𝜃 = 12∘

𝜃 = 18∘


instability range I using the third-order polynomial.

when 𝑇0= 28 kN, the hysteresis phenomenon starts to

occur in the conductor galloping.The hysteresis range whichis characterized by the horizontal wind velocity increasesobviously as the initial tension decreases. And in the samewind velocity, before the hysteresis phenomenon occurrence,the conductor galloping amplitude increases as the initialtension increases. But the amplitude jumps abruptly overthe hysteresis range and increases obviously as the initialtension decreases. Therefore, it is an approach to control theconductor galloping by selecting a proper initial tension ofthe transmission line.

In this example, as the initial tension reaches 23 kN, amore complex vibration bifurcation phenomenon is excitedin the conductor galloping from Figure 11. It also presents asecondary bifurcation phenomenon. And with the decreaseof initial tension, this phenomenon is more obvious.

(4) Influence of the Conductor Mass. Figure 12 shows theamplitude 𝐵

1versus the horizontal wind velocity 𝑈 with

different conductor masses. The span length is 500m. Theangle of attack is 0∘ and the initial tension is 25.9 kN. Otherparameters are shown in Table 1. It should be pointed out thatthe different masses represent the different conductor types.

From Figure 12 we can see that the hysteresis phe-nomenon will be excited with the increase of the conductormass. And when 𝑚 = 1.2 kg/m, the hysteresis phenomenonstarts to occur in the conductor galloping. The hysteresisrange which is characterized by the horizontal wind velocityincreases obviously as the conductor mass increases. And inthe same wind velocity, before the occurrence of hysteresisphenomenon, the conductor galloping amplitude decreasesas the conductor mass increases. The amplitude jumpsabruptly over the hysteresis range and increases obviously asthe conductor mass increases. As the conductor mass equals1.6 kg/m, a more complex vibration bifurcation phenomenonis excited in the conductor galloping from Figure 12. It also

B1

3028262422201816141210

86420

U

5 10 15 20 25 30

L = 300mL = 440m

L = 520mL = 590m

Figure 10: Amplitude 𝐵1versus horizontal wind velocity with

different span lengths.

B1

22201816141210

8642

U

5 10 15 20 25 30 35

T0 = 33kNT0 = 28kN

T0 = 25kNT0 = 23kN


different initial tensions.

presents a secondary bifurcation phenomenon. And with theincrease of mass, this phenomenon is more obvious.

5. Conclusions

In this paper, the hysteresis phenomenon in the iced conduc-tor galloping is studied.The analysis is based on a continuummodel of the D-shape iced conductor derived by Hamiltonprinciple, in which the initial deformation, the geometricnonlinearity caused by the large deformation, and the aerody-namic nonlinearity are considered. The aerodynamic forcesare described by using the quasi-steady hypothesis, wherethe aerodynamic coefficients are expanded by the polynomialcurves with a third order and a ninth order, respectively.

10 Mathematical Problems in EngineeringB1

2422201816141210

8642

U

5 10 15 20 25 30 35 40

m = 0.9 kg/mm = 1.2 kg/m

m = 1.4 kg/mm = 1.6 kg/m


different conductor masses.

The hysteresis characteristics are obtained by solving theapproximate solutions of the Galerkin discretized equationderived from the continuummodel bymeans of the harmonicbalance method. And the influences of the different factorsare proposed in different galloping instability intervals.

For the aerodynamic force coefficient curves of the D-shape iced conductor, the fitting interval by the third-orderpolynomial is less than by the ninth order obviously. Theangle interval of galloping occurrence obtained by the third-order fitting curve is only a part of the results by the ninth-order fitting. There are three galloping instability regionsderived from the ninth-order fitting curve and there is onlyone instability region from the third-order fitting curve.Three inflection points are obtained from the ninth-orderfitting curve, which are evenly distributed in the threeinstability ranges. And one inflection point is found from thethird-order fitting curve in its only one instability range.

Hysteresis characteristics in the conductor galloping areobtained in each instability range. For the three instabilityintervals from the ninth-order fitting curve, the velocitiesof the hysteresis phenomenon occurrence at the two endsof the instability angle interval are much higher than in themiddle. Velocities to excite hysteresis phenomenon in theangle interval II are much lower than in the angle intervalsI and III expect the ends of the intervals. So under theactual engineering conditions, it is difficult to achieve forthe hysteresis phenomenon in the intervals I and III, but easyin the interval II.

The horizontal wind velocity of the hysteresis onsetpoint increases with the much larger span length, and thehysteresis range makes much wider with the increase ofthe span length. The hysteresis phenomenon will be excitedwith the decrease of the initial tension, and the hysteresisrange which is characterized by the horizontal wind velocityincreases obviously as the initial tension decreases. Andthe hysteresis phenomenon will be excited with the increase

of the conductor mass and the hysteresis range increasesobviously as the conductor mass increases.

Finally it should be pointed out that only the verticalvibration is considered in order to discuss the hysteresisphenomenon in this study. The torsional motion and thehorizontal motion can also have important effects in relationto aerodynamicmechanisms that causes galloping, whichwillbe presented and discussed in our subsequent studies.

Acknowledgment

The authors are grateful for financial support from theNational Natural Science Foundation of China (51008288).

References

[1] CIGRE SCB2,WG 11, and Task Force 02.11.06, “State of the art ofconductor galloping,” Electra 322, Convenor: J. L. Lilien, 2007.

[2] R. D. Blevins, Flow-Induced Vibration, Van Nostrand Reinhold,New York, NY, USA, 1990.

[3] Y. L. Guo, G. X. Li, and C. Y. You, Galloping of the TransmissionLine, China Electronic Power Press, Beijing, China, 2003,(Chinese).

[4] C. B. Rawlins, “Transmission line reference book-wind-inducedconductormotion,” inGalloping Conductors, vol. 792, chapter 4,EPRI Research Project, 1979.

[5] J. P. Den Hartog, “Transmission line vibration due to sleet,”Transactions of AIEE, vol. 51, part 4, pp. 1074–1086, 1932.

[6] O. Nigol and P. G. Buchan, “Conductor galloping part I: denHartogmechanism,” IEEETransactions onPowerApparatus andSystems, vol. 100, no. 2, pp. 699–707, 1981.

[7] O. Nigol and P. G. Buchan, “Conductor galloping part II: tor-sional mechanism,” IEEE Transactions on Power Apparatus andSystems, vol. 100, no. 2, pp. 708–720, 1981.

[8] P. Yu, A. H. Shah, and N. Popplewell, “Inertially coupled gallop-ing of iced conductors,” Journal of Applied Mechanics, Transac-tions ASME, vol. 59, no. 1, pp. 140–145, 1992.

[9] P. Yu, Y. M. Desai, A. H. Shah, and N. Popplewell, “Three-degree-of-freedom model for galloping. Part 1: formulation,”Journal of EngineeringMechanics, vol. 119, no. 12, pp. 2404–2425,1993.

[10] P. Yu, N. Popplewell, and A. H. Shah, “Instability trends of iner-tially coupled galloping. Part II: periodic vibrations,” Journal ofSound and Vibration, vol. 183, no. 4, pp. 679–691, 1995.

[11] C. Ziller and H. Ruscheweyh, “A new approach for determiningthe onset velocity of galloping instability taking into accountthe nonlinearity of the aerodynamic damping characteristic,”Journal of Wind Engineering and Industrial Aerodynamics, vol.69-71, pp. 303–314, 1997.

[12] A. Luongo and G. Piccardo, “Non-linear galloping of saggedcables in 1:2 internal resonance,” Journal of Sound andVibration,vol. 214, no. 5, pp. 915–936, 1998.

[13] G. A. Vio, G. Dimitriadis, and J. E. Cooper, “Bifurcation anal-ysis and limit cycle oscillation amplitude prediction methodsapplied to the aeroelastic galloping problem,” Journal of Fluidsand Structures, vol. 23, no. 7, pp. 983–1011, 2007.

[14] Z. H.Qin, Y. S. Chen, X. P. Zhan, B. Liu, andK. J. Zhu, “Researchon the galloping and anti-galloping of the transmission line,”International Journal of Bifurcation and Chaos, vol. 22, no. 2,Article ID 1250038, 2012.


[15] K. G. McConnell and C.-N. Chang, “A study of the axial-torsional coupling effect on a sagged transmission line,” Exper-imental Mechanics, vol. 26, no. 4, pp. 324–329, 1986.

[16] K. E. Gavronski,Non-linear galloping of bundle-conductor trans-mission lines [Ph.D. thesis], Clarkson College of Technology,1977.

[17] S. C. Luo, Y. T. Chew, and Y. T. Ng, “Hysteresis phenomenon inthe galloping oscillation of a square cylinder,” Journal of Fluidsand Structures, vol. 18, no. 1, pp. 103–118, 2003.

[18] P. Hémon and F. Santi, “On the aeroelastic behaviour of rectan-gular cylinders in cross-flow,” Journal of Fluids and Structures,vol. 16, no. 7, pp. 855–889, 2002.

[19] T. Tamura and Y. Itoh, “Unstable aerodynamic phenomena ofa rectangular cylinder with critical section,” Journal of WindEngineering and Industrial Aerodynamics, vol. 83, pp. 121–133,1999.

[20] G. Alonso, J. Meseguer, and I. Pérez-Grande, “Galloping insta-bilities of two-dimensional triangular cross-section bodies,”Experiments in Fluids, vol. 38, no. 6, pp. 789–795, 2005.

[21] G. Alonso and J. Meseguer, “A parametric study of the gallopingstability of two-dimensional triangular cross-section bodies,”Journal of Wind Engineering and Industrial Aerodynamics, vol.94, no. 4, pp. 241–253, 2006.

[22] V. D. Pierre and L. Andre, “Galloping of a single conductorcovered with a D-section on high-voltage overhead test line,”Journal of Wind Engineering and Industrial Aerodynamics, vol.96, pp. 1141–1151, 2008.

[23] P. Parkinson and N. P. H. Brooks, “On the aeroelastic instabilityof bluff cylinders,” Journal of AppliedMechanics, vol. 28, pp. 252–258, 1961.

[24] G. V. Parkinson and J. D. Smith, “The square prism as an aeroe-lastic non-linear oscillator,” Quarterly Journal of Mechanics andApplied Mathematics, vol. 17, no. 2, pp. 225–239, 1964.

[25] Y. T. Ng, S. C. Luo, and Y. T. Chew, “On using high-orderpolynomial curve fits in the quasi-steady theory for square-cylinder galloping,” Journal of Fluids and Structures, vol. 20, no.1, pp. 141–146, 2005.

[26] J. A. Dunnmon, S. C. Stanton, B. P. Mann, and E. H. Dowell,“Power extraction from aeroelastic limit cycle oscillations,”Journal of Fluids and Structures, vol. 27, no. 8, pp. 1182–1198, 2011.

[27] A. Barrero-Gil, A. Sanz-Andrés, and G. Alonso, “Hysteresis intransverse galloping: the role of the inflection points,” Journal ofFluids and Structures, vol. 25, no. 6, pp. 1007–1020, 2009.

[28] G. Alonso, A. Sanz-Lobera, and J. Meseguer, “Hysteresis phe-nomena in transverse galloping of triangular cross-sectionbodies,” Journal of Fluids and Structures, vol. 33, pp. 243–251,2012.

[29] R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, NY, USA, 1975.

[30] K. S. Wang and G. J. Tang, “Response analysis of nonlinearvibration of overhead power line under suspension chain state,”Journal of Vibration and Shock, vol. 22, no. 2, pp. 69–72, 2003.

[31] H. M. Irvine and T. K. Caughey, “The line theory of free vibra-tions of a suspended cables,” Proceeding of the Royal Society ofLondon A, vol. 341, pp. 299–315, 1974.

[32] A. Luongo, D. Zulli, and G. Piccardo, “A linear curved-beammodel for the analysis of galloping in suspended cables,” Journalof Mechanics of Materials and Structures, vol. 2, no. 4, pp. 675–694, 2007.

[33] L.Wang andG.Rega, “Modelling and transient planar dynamicsof suspended cables withmovingmass,” International Journal ofSolids and Structures, vol. 47, no. 20, pp. 2733–2744, 2010.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article hysteresis phenomenon in the galloping of...

Documents