Appl. Math. Mech. -Engl. Ed. 30(4), 489–501 (2009) DOI: 10.1007/s10483-009-0409-x c Shanghai University and Springer-Verlag 2009

Applied Mathematics and Mechanics (English Edition)

Nonlinear numerical simulation method for galloping of iced conductor ∗ Xiao-hui LIU ()1 , Bo YAN ()1 , Hong-yan ZHANG ()2 , Song ZHOU ()3 (1. Department of Engineering Mechanics, Chongqing University, Chongqing 400030, P. R. China; 2. Sichuan Electric Power Test and Research Institute, Chengdu 610071, P. R. China; 3. Sichuan Electric Power Industry Commission and Test Institute, Chengdu 610016, P. R. China) (Communicated by Shan-lin CHEN)

Abstract Based on the principle of virtual work, an updated Lagrangian finite element formulation for the geometrical large deformation analysis of galloping of the iced conductor in an overhead transmission line is developed. In numerical simulation, a threenode isoparametric cable element with three translational and one torsional degrees-offreedom at each node is used to discretize the transmission line. The nonlinear dynamic system equation is solved with the Newmark time integration method and the NewtonRaphson nonlinear iteration. Numerical examples demonstrate the efficiency of the presented method and the developed finite element program. A new possible galloping mode, which may reflect the saturation phenomenon of a nonlinear dynamic system, is discovered under the condition that the lowest order of vertical natural frequency of the transmission line is approximately two times of the horizontal one. Key words

iced conductor, galloping, geometric nonlinearity, numerical method

Chinese Library Classification O39 2000 Mathematics Subject Classification

70K75, 37M05

Introduction The iced conductor in an overhead transmission line section located in a steady airflow is subject to aerodynamic forces due to its non-circular cross-section. These forces may induce the conductor to oscillate with large amplitude, which is usually termed as galloping. It is noted that the severe galloping may cause cascading of the transmission line and disruption of the power supply, which may in turn lead to large economic loss. The galloping of the iced conductor has been investigated by many authors since the first observation of this phenomenon in the 1930s. Den Hartog[1] and Nigol and Clarke[2] originally proposed two famous galloping theories, i.e., the vertical oscillation mechanism and torsional oscillation mechanism, based on which several coupling theories have been presented by some authors. In recent years, the numerical method to simulate the galloping of the iced conductor ∗ Received Sept. 13, 2008 / Revised Feb. 17, 2009 Project supported by the Science Foundation of the State Grid Corporation of China (No. 2007-1-77) and the Natural Science Foundation Project of CQ CSTC of China (No. 2006BB6149 Corresponding author Bo YAN, Professor, Ph. D., E-mail: [email protected]

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has attracted more and more researchers because of the complexity of the galloping phenomenon and the improvement of computer hardware and software. Simpson[3] numerically analyzed the dynamic characteristics of a bundle conductor by means of the transfer matrix method and the dynamic stiffness method. Yu et al.[4] presented a model with three degrees-of-freedom to simulate the galloping of transmission line. Desai et al.[5] developed a three-node isoparametric cable element to model the conductor lines, and simulated the galloping of several transmission line sections by means of a mode superposition method, which is actually a linearization to the original nonlinear dynamic equation. Based on the mode superposition method, He and Qian[6] set up a mathematical model for the analysis of the galloping of the iced conductor, and numerically simulated the galloping of a triple bundle conductor in an overhead transmission line. However, by means of the mode superposition method, the nonlinearity of the galloping of the transmission line is linearized, which may give rise to some error. In addition, some authors such as Wang et al.[7] have analyzed the nonlinear galloping of the iced conductor with beam element, which may be unsuitable because of the usage of beam element with bending degree-of-freedom and the ignoring of the eccentric action of the ice covered on the conductor. To more realistically simulate the galloping of the iced conductor, the three-node isoparametric cable element, proposed by Desai et al.[5] , is used to establish the nonlinear dynamic finite element equation in the updated Lagrangian form by means of the principle of virtual work, and the nonlinear equation is solved with the Newmark time integration method and the Newton-Raphson iteration in this paper. This finite element formulation can be applied to simulate the galloping of the iced bundle conductor with little extension.

1

Finite element equation for galloping of iced conductor

The galloping of the iced conductor in an overhead transmission line is a typical nonlinear geometrical problem with large displacement and small linear elastic tensile deformation in the axial direction. In terms of the updated Lagrangian (UL) formulation, the principle of virtual work at time t + Δt is generally written in the following form[8] :    t t δui t+Δt ρt+Δt u ¨i t+Δt dV + D e δ e dV + τij δ t ηij t dV t ijkl t kl t ij t+Δt V tV tV    t+Δt t+Δt t+Δt t t+Δt t+Δt = t δu dS + ρ f δu dV − τij δ t eij t dV, (1) k σ k t+Δt k t+Δt k t+Δt S

t+Δt V

σ

tV

¨ are the density, displacement, and acceleration, respectively, of the where t+Δt ρ, ui , and t+Δt u iced conductor at time t + Δt; t Dijkl is the constitutive tensor; t ekl and t ηij are the linear and nonlinear parts, respectively, of Green strain at time t; t τij is the Cauchy stress tensor at time t, which is equal to the Kirchhoff stress tt Sij because the configuration at time t is chosen as t+Δt the reference configuration for the UL formulation; t+Δt t+Δt tk and t+Δt fk are the body force and t+Δt t+Δt dV and dSσ are the integrated volume and face traction at current time, respectively; element, respectively. 1.1 Element analysis Now the three-node isoparametric cable element with three translational and one torsional degrees-of-freedom at each node, proposed by Desai et al.[5] , is used to establish the UL finite element formulation. The spatial coordinates of a point in an element, as shown in Fig. 1, at times t and t + Δt are represented respectively by t xi and t+Δt xi , and the natural coordinate along the axis of the curved element is denoted by S. If the displacement increment of a point is ui and the torsional angle increment around S is θ from t to t + Δt, the nodal displacement vector q e of the element is defined as q e = [u11

u12

u13

θ1

u21

u22

u23

θ2

u31

u32

u33

θ3 ]T ,

(2)

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491

where the right superscript of a variable indicates the local node number in the element. Moreover, the displacement increment from t to t + Δt, at the center of the cross-section through a point along the element, is depicted as [u1

u2

θ]T = N q e .

u3

(3)

Here, the shape function matrix N is N = [N1 I

N2 I

N3 I] ,

(4)

and I is a 4×4 unit matrix. The shape functions Nk are expressed as[5] N1 =

2S 2 3S − + 1, le2 le

N2 = −

4S 2 4S + , le2 le

2S 2 S − , le2 le

N3 =

(5)

where le is the length of the element. (x11, x21, x31) 1

S

x2

2 (x21, x22, x32)

3 (x31, x23, x33)

x1 x3

Fig. 1

A three-node cable element

A local coordinate system (x1 , x2 , x3 ) with the origin at the center of the cross-section through a point along the axis of the iced conductor is set up as shown in Fig. 2. It is noted that the local coordinate system is parallel to the global coordinate system and moves with the conductor. It is assumed that the sag of the conductor line is small, so the displacement ui of a point on the cross-section of the cable can be determined with the displacement of the center point of the cross-section by [u1 u2 u3 ]T = HNq e , (6) x2' Configuration at t

x2'

Ice

Wind

r

α0 O

r

α

p(tx2, tx3)

p(t+Δtx2, t+Δtx3) tψ−θ

x3'

O

tψ

x3' Configuration at t+Δt

x2 Conductor x3

Fig. 2

Cross-section of the iced conductor

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where

⎡

1 H =⎣ 0 0

0 0 1 0 0 1

⎤ 0 −x3 ⎦ . x2

Neglecting the change of the density of the conductor during movement and substituting Eq. (6) into the first term on the left hand of Eq. (1), we obtain that    δui ρt+Δt u ¨i dV = ρδq eT N T H T HN t+Δt q¨e dAdS t+Δt V le A  = δq e N T µN t+Δt q¨e dS. (7) le

The non-zero elements of the symmetric matrix µ are defined by   ⎧ ⎪ ⎪ μ11 = μ22 = μ33 = μ = ρdA, μ24 = −Sx2 = − ρx3 dA, ⎨ A A   ⎪ 2 ⎪ ⎩ μ34 = Sx3 = ρx2 dA, μ44 = Iθ = ρ(x2 2 + x3 )dA. A

(8)

A

The axial Green strain εS at time t + Δt in the reference configuration at time t can be decomposed into the linear part εl and the nonlinear part εnl , which are determined by εS = εl + εnl ,

(9)

where εl =

∂ t xi ∂ui , ∂S ∂S

εnl =

1 ∂ui ∂ui . 2 ∂S ∂S

(10)

Because only the axial tension force and torque in the iced conductor are taken into account and the interaction between the elongation and torsion is so small that it can be ignored, the second term on the left hand of Eq. (1) becomes    t t δ t εl E t εl dV + δ t εθ GJ t εθ dS t Dijkl t ekl δ t eij dV = tV tV le  = Aδq eT tt B T D tt B q e dS, (11) le

where A, E, and GJ are the cross-section area, the Young’s modulus, and the torsional stiffness of the conductor, respectively. The strain matrix B is composed of three sub-matrices, B = [B1

B2

B3 ],

⎤ 0 ⎥ ∂Nk ⎦ , ∂S Moreover, the elastic matrix in Eq. (11) can be expressed as 

1 AE 0 . D= GJ A 0

where

⎡

∂ t x1 ∂Nk ⎢ ∂S ∂S Bk = ⎣ 0

∂ t x2 ∂Nk ∂S ∂S 0

∂ t x3 ∂Nk ∂S ∂S 0

k = 1, 2, 3.

(12)

(13)

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493

If the tension force along the axis of the conductor line at time t is tt T , the third term on the left hand of Eq. (1) can be written as    t t τij δ t ηij dV = δ t εnl T dS = δq eT GTtt T G q e dS, (14) tV

le

where G=

le

 ∂N3 m , ∂S

∂N2 m ∂S

∂N1 m ∂S

⎡

and

1 ⎢ 0 m=⎢ ⎣ 0 0

0 1 0 0

0 0 1 0

t tT

= tt T m,

(15)

⎤ 0 0 ⎥ ⎥. 0 ⎦ 0

It is assumed that the polar coordinates of point P on the cross-section through a point along the axis of the conductor at times t and t + Δt are (r,t ψ) and (r,t ψ − θ), respectively, then the displacement of P along the x2 -direction at t + Δt can be determined by u2 = u2 + r[sin(t ψ − θ) − sin t ψ].

(16)

Because the increment of the torsional angle of the conductor from t to t + Δt is very small, it is reasonable to assume sin θ ≈ θ. If the gravity is in the direction of −x2 , the second term on the right hand of Eq. (1) can be simplified as     − ρgδu2 dV = − ρgδu2 dV + δθρgr cos t ψ cos θdV + gδq e Lq e dS, (17) V

V

V

⎡

where

0 ⎢ 0 ⎢ L=⎣ 0 0 

and Sx 3 =

AT

0 0 0 0

ρx2 dA =

le

⎤ 0 0 0 0 ⎥ ⎥ 0 0 ⎦ 0 Sx 3 

ρr sin t ψdA.

AT

It is noted that the first term on the right hand of Eq. (17) represents the virtual work of the gravity as the iced conductor translates in the vertical direction, and the second term is the virtual work of the gravity as the torsion takes place. These two terms will vanish if the equilibrium state of the iced conductor under the action of its self-weight is chosen as the initial configuration. In addition, moving the third term on the right hand of Eq. (17) to the left, the integration form of the eccentric stiffness matrix of the iced conductor can be obtained. The last term on the right hand of Eq. (1) can be written as 

t    0 t t t t t t eTt T t T t dS, (18) τij δ t eij dV = ( σδ εl + τ δ εθ ) dV = δq t B 0 tt M tV tV le where the axial tension force tt T and torque tt M on the iced conductor at time t are defined by t tT

= AE t εS + T0 ,

t tM

= GJ

∂tθ . ∂S

(19)

Here, t εS , determined by Eq. (9), is the axial strain of the conductor at time t, and T0 is the initial axial tension force deduced by the self-weight of the iced conductor.

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Finally, the aerodynamic loads on the iced conductor, which depend on the geometric characteristics of the cross-section of the iced conductor and the attack angle α between the wind direction and the axis of the iced conductor, are represented as FD

[FL

Mθ ]T =

1 2 ρair Urel d[CL (α) 2

CD (α) dCθ (α)]T ,

(20)

where ρair and d are the density of the air and the diameter of the conductor, respectively. The three aerodynamic parameters Ci can be expressed in the form of Ci = ai0 + ai1 α + ai2 α2 + ai3 α3

(21)

by fitting the wind tunnel experiment data. The nodal force vector of an element can then be written in the form  le (22) N T [0 Fx2 Fx3 Mθ ]T dS. Fe = 0

1.2 Modeling of adjacent conductor and insulator string The adjacent conductors and insulator strings connected to the analyzed span of the conductor line may affect the mechanical behavior of the analyzed span of the conductor line. This effect can be modeled by simplifying the adjacent conductors and insulator strings to springs. The stiffness expressions of the springs were given out by Veletsos and Darbe[9] as p2y L3x L 1 + = , KST AE 12H 3   1 Wl , py L + Klx1 = Ll 2

(23) Klx3 = Klx1 +

2H , Lx

(24)

where KST is the effective stiffness of the spring modeling the adjacent conductor line; L as well as Lx are the length and the horizontal span, respectively, of the conductor line; the variable py is the gravity per unit length of the conductor, and H is the horizontal component of the tension force of the conductor in the static state; Klx1 and Klx3 are the stiffnesses of the springs modeling an insulator string in the x1 - and x3 -directions, respectively; Ll and Wl are the length and weight of the insulator string, respectively. 1.3 The updated lagrangian formulation Substituting Eqs. (7), (11), (14), (17), (18), and (22) into Eq. (1), and taking into account the damping of the iced conductor, the finite element equation in the UL form is obtained as M e t+Δt q¨e + tt C e t+Δt q˙ e + tt K e q e = t+Δt F e − tt Qe ,

(25)

where Me =

 0

le

N Tt+Δt µN dS,

 0 dS, 0 tt M 0  t e t e t e t e t K = t K t + t Kσ + t Kice = t e tQ



=

le

t T tB

t

tT

0

le

Att

B

T

D tt B

 dS + 0

le

GTtt T GdS

 −g

0

le

N T LN dS.

After assembling all the element matrices, the dynamic system equation is obtained as follows: M t+Δt q¨ + tt C t+Δt q˙ + tt Kq = t+Δt F − tt Q, (26)

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495

where tt C and tt K are the damping matrix and stiffness matrix, respectively, of the system at ¨ t+Δt q, ˙ and t+Δt F are time t; q is the nodal displacement increment from t to t + Δt; t+Δt q, t the nodal acceleration, velocity, and load vectors at time t + Δt, respectively; t Q is the internal load vector at time t. The damping of a conductor line can be depicted by Raleigh damping[10] , which can be expressed as follows: cij = βk1 mij + βk2 kij , k = 1, 2, (27) where mij and kij are elements of the structural mass matrix and stiffness matrix, respectively, and the coefficients βk1 and βk2 are defined by 2ξk1 ωk1 ωk2 , ωk2 + ωk1

βk1 =

βk2 =

2 (ξk2 ωk2 − ξk1 ωk1 ) . 2 − ω2 ωk2 k1

(28)

The parameters ξk1 and ξk2 in Eq. (28) are the damping ratios corresponding to the two lowest natural frequencies of the conductor.

2

Solution of system equation

The nonlinear system equation (26) can be solved with the Newmark time integration scheme and the Newton-Raphson nonlinear iteration strategy. The recurrence formula is written as[8] 1 δ C)q (l) M+ 2 αΔt αΔt  1 1 t+Δt (l) t 1 t t ˙ ¨ −1) q−( q = t+Δt F (l) −M ( q − q)− αΔt2 αΔt 2α 

δ δ δ t+Δt (l) t ˙ ( −1)Δtt q¨ − t+Δt q − q)−( −1)t q−( Q(l) −C t αΔt α 2α (tt K (l) +

(29)

and t+Δt (1)

q

− t q = q (0) ,

t+Δt (l+1)

q

− t+Δt q (l) = q (l) ,

t+Δt (0)

q

= t q,

q=

k 

q (l) ,

l=0

where k is the iteration times. At the end of each iteration step, the vectors t+Δt F (l) and t+Δt (l) Q and stiffness matrix are updated based on t+Δt q (l) and t+Δt q˙ (l) determined in the last t step. The iteration proceeds until the convergence criterion is satisfied.

3

Numerical examples

3.1 Example 1 To justify the presented method in this paper, a suspended conductor line with one span under dynamic loads, as shown in Fig. 3, is analyzed by both the presented method and the ABAQUS software. The span length of the line cable is 100 m and its sag is 4.05 m under the action of its self-weight. The diameter of the conductor is 27.63×10−3 m, the Young’s modulus 6.9×104 MPa, the Poisson’s ratio 0.3 and the density 2 519 kg/m3 . The line cable is discretized with 20 three-node cable elements as the method presented in this paper is used. It is demonstrated that the number of elements is enough to obtain a stable and convergent solution. Because there is no cable element with torsional degree-of-freedom in the ABAQUS software, 40 two-node beam elements are used to model the cable by releasing their two rotational degreesof-freedom corresponding to bending deformation at each node. The total numbers of the nodes used in the two models are the same. A vertical load and a horizontal load as well as a torque

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with different amplitudes, i.e., 1 N, 2 N and 1 N·m respectively, are applied at each node. These loads vary with time in the same law as shown in Fig. 4. As the ABAQUS software is used, the loads applied at each node are smoothed by setting the loads as “smooth step”. The same smoothing technique is employed in the developed program for the convenience of comparison. The dynamic responses of the displacements and the torsional angle at midpoint of the line cable obtained by both the presented method in this paper and the ABAQUS software are shown in Fig. 5. It is obvious that the results obtained by the both methods are consistent with each other, which justifies the method presented in this paper. 1.2 1.0

x2

x3

Amplitude

x1 Torque

Horizontal load

0.8 0.6 0.4 0.2

Vertical load 0.0 0.0

Fig. 3 1.0×10−2

Fig. 4 0.4

Results by the presented method Results by ABAQUS

0.3

6.0×10−3

0.2

4.0×10−3

0.1

u3/m

u2/m

8.0×10−3

Single span model of the line cable

2.0×10−3

−0.1

−2.0×10−3

−0.2 2

4

6

8

t/s (a) Displacements in the x2-direction 25

−0.3 0

10

t/s

2.0

3.0

Variation of loads with time

Results by the presented method Results by ABAQUS

0.0

0.0

−4.0×10−3 0

1.0

2

4

6 8 t/s (b) Displacements in the x3-direction

Results by the presented method Results by ABAQUS

20

θ/(°)

15 10 5 0 0

Fig. 5

2

4

6 t/s (c) Torsional angles

8

10

Variation of displacements and torsional angle at midpoint of the cable with time

10

Nonlinear numerical simulation method for galloping of iced conductor

497

3.2 Example 2 A numerical example discussed in Ref. [5] is now analyzed with the method presented in this paper. The span of the transmission line, covered by ice with the D-type cross-section, is 125.88 m and its sag is 1.38 m under the action of it self-weight. The other parameters are listed in Table 1. The wind with the velocity of 4.1 m/s is horizontally applied on the conductor line and the initial attack angle is 10o . The aerodynamic coefficients varied with attack angle, as shown in Fig. 6, are adopted from Ref. [5]. The adjacent conductors and insulator strings connected to the analyzed single span of conductor line are modeled by the effective springs as discussed in Section 1.2. The length and weight of each insulator string are 2.1 m and 490 N, respectively. It is demonstrated that 15 three-node elements discussed in this paper are enough to obtain a stable and accurate solution after the comparison of the results obtained by means of the models discretized with different numbers of elements. The numerically determined galloping trace and history of displacements at the mid-span of the conductor line are shown in Figs. 7 and 8, respectively. It is observed that the iced conductor oscillates in the vicinity of its static equilibrium position with small amplitude at inception, and the amplitude gradually increases with time until a stable galloping state with an approximate elliptic trace arrives. This can be understood as the so-called Den Hartog vertical galloping. The obtained amplitudes of the vertical and horizontal vibration in the steady galloping state are 1.544 m and 0.058 m, respectively. The relative errors with respect to those amplitudes, i.e., 1.58 m and 0.04 m, determined by the mode superposition method proposed in Ref. [5], are 2.3 % and 31.1 %, respectively. Table 1

Physical parameters employed to simulate galloping

Parameters

Notation AE GJ H d KST ξx 2 ξθ μ I Sx 2 Sx 3

Axial stiffness Torsional stiffness Horizontal component of tension Diameter of bare conductor Stiffness of neighboring conductor Damping ratio in the x2 -direction Damping ratio in the θ-direction Mass unit length Mass moment of inertia per unit length Mass moment of area about the x2 -axis per unit length Mass moment of area about the x3 -axis per unit length

4 Aerodynamic coefficients

Aerodynamic coefficients

2.5 1.5 0.5 −0.5 −1.5 −2.5 −60

Drag Lift Moment −40

−20

0

20 40 α/(°) (a) Example 2

Fig. 6

60

80

3

Unit 106

N N·m2 /rad 103 N 10−3 m 103 N/m 10−2 10−2 kg/m 10−4 kg · m 10−3 kg 10−3 kg

Example 2

Example 3

13.30 101 21.73 18.8 75.97 0.08 3.79 1.53 57.02 0.459 −0.145

25.53 153 15.0 23.5 75.97 1.60 5.70 1.66 1.56 2.722 −0.830

Drag Lift Moment

2 1 0 −1 −2 100 120 140 160 180 200 220 240 260 α/(°) (b) Example 3

Aerodynamic coefficients versus attack angle[5]

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Fig. 7

Fig. 8

Galloping trace at the mid-span in Example 2

Time history of displacements and rotation at the mid-span in Example 2

In the numerical model of Ref. [5], the constant terms of the aerodynamic loads determined by Eqs. (20) and (21) are applied on the conductor as static loads and the portion of the loads varing with time are then applied to simulate the dynamic response of the iced conductor, which is an approximation to the nonlinear dynamic problem. However, the aerodynamic loads are totally applied on the conductor line in one step in the simulation here. In addition, the initial static sag 1.38 m of the conductor line in Ref. [5] is under the action of both the iced conductor’s self-weight and the constant lift force, however, the sag in the model here is under the action only of the self-weight of the iced conductor. This difference may be one of the reasons leading

Nonlinear numerical simulation method for galloping of iced conductor

499

to the bigger error between the horizontal vibration amplitude obtained by Ref. [5] and that by the method presented in this paper. Moreover, the linearization to the system equation in Ref. [5] is another reason giving rise to the error. The numerically determined time history of the vertical displacement at the mid-span of the conductor line is compared with that observed in the experiment provided by Ref. [11], as shown in Fig. 9. It is shown that the two results are in good agreement. 1.0

Measured Predicted

x2/m

0.6 0.2 −0.2 −0.6 −1.0 0.0

Fig. 9

1.0

2.0 t/s

3.0

4.0

Numerically determined and measured vertical displacements at mid-span

3.3 Example 3 The span length of the conductor line discussed in this example is the same as that of Example 2, and the sag is 2.12 m under the action of its self-weight. The other physical parameters and aerodynamic coefficients are as shown in Table 1 and Fig. 8, respectively. In addition, the initial attack angle is set to be 180◦ in this example and the model is discretized

Fig. 10

Galloping trace at the mid-span in Example 3

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with 15 three-node elements too. Wind loads with different velocities are respectively applied on the iced conductor in the model, and it is discovered that a new galloping mode appears as the wind velocity is 5.5 m/s. The galloping trace and the time history of displacements at the mid-span of the iced conductor are shown in Figs. 10 and 11, respectively. It is observed that the iced conductor under the action of the wind load initially oscillates vertically around its equilibrium position and arrives at a steady state with the accumulation of energy due to the minus damping, as shown in Fig. 10(a). After arriving at the stable vertical galloping, the conductor gradually swings in the horizontal direction until a new steady state arrives, as shown in Figs. 10(b) and 10(c). It is discovered that the lowest order natural frequency of the conductor line in the vertical direction, 0.695, is approximately two times of that in the horizontal direction, say 0.378, by means of the mode analysis. This newly observed possible galloping mode may be understood as the saturation phenomenon of a nonlinear dynamic system, which cannot be revealed if the linearization technique is employed during the set up of the dynamic system equation and/or the solution of the nonlinear system equation.

Fig. 11

4

Time history of displacements at the mid-span

Conclusions

A UL formulation for the nonlinear galloping analysis of the iced conductor in an overhead transmission line is derived by means of the principle of virtual work. The efficiency of the presented method is demonstrated by numerical examples. A new possible galloping mode of the iced conductor, which has not been reported before as to the authors’ knowledge, is observed on the condition that the lowest order natural frequency of the conductor in the vertical direction is approximately two times of that in the horizontal direction, as the case in Example 3 presented in this paper. This type of vibration is just the saturation phenomenon of a nonlinear dynamic system, which may not be revealed by means of the linearization method, such as the mode

Nonlinear numerical simulation method for galloping of iced conductor

501

superposition method. However, the new type of galloping mode presented in this paper needs to be demonstrated through the observation in field test and/or real transmission lines.

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