Hindawi

Shock and Vibration

Volume 2017, Article ID 1634292, 16 pages
https://doi.org/10.1155/2017/1634292

Research Article
Study on Galloping Oscillation of Iced Catenary
System under Cross Winds

Guo Chen, Yiren Yang, Yang Yang, and Peng Li

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

Correspondence should be addressed to Guo Chen; apple88125@hotmail.com

Received 2 June 2017; Revised 29 August 2017; Accepted 6 September 2017; Published 22 October 2017

Academic Editor: Tai Thai

Copyright © 2017 Guo Chen et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper mainly aims at revealing the nature of the galloping oscillation of iced catenary system under cross winds. The
aerodynamic force on the iced catenary system is assumed to be quasi-steady, and then the quasi-steady aerodynamic lift and
drag coefficients are completed in FLUENT. By fitting the discrete simulation data, the expression of the vertical aerodynamic
force is further obtained. According to the Den Hartog vertical galloping mechanism, the stability of iced catenary is discussed
and the initial icing angle corresponding to the critical stability is obtained. On this basis, the dynamic model of the simple iced
catenary system under cross winds is established. The partial differential vibration equation of the system is converted into the
ordinary differential equation by the Galerkin method and then numerically solved. The condition of the unstable catenary motion
in simulation is in agreement with that from theoretical stability analysis. In addition, the effects of structural damping, initial icing
angle, and wind velocity on the system responses are investigated.

1. Introduction velocity and angle of attack, when the negative slope of

lift force curve was larger than the amplitude of drag force
High speed train is driven by electric locomotives which gain curve, the occurrence of instability phenomenon could cause
electrical power from pantograph-catenary system. There- the conductor galloping in the vertical direction. Nigol and
fore, proper functioning of catenary is one of the key factors Buchan further verified the galloping theory mentioned in
that guarantee the operating safety of high speed railway. [4] by the wind tunnel experiment. Additionally, Nigol and
With the wide application of electrified railway, the working Buchan [5] found that when the vertical galloping condition
environment is complicated and various. Especially for areas was not satisfied, the galloping would also be caused by the
of low temperature and high humidity and altitude, it is self-excited torsional vibration of iced conductor. Aiming at
extremely likely for catenary to be covered with ice. Covering the galloping mechanism, Yu et al. [6, 7] proposed the novel
ice can not only severely affect the current collecting quality, suggestion that the galloping was caused by the eccentricity
but also cause a large amplitude oscillation of catenary under inertia of cross section of iced conductor.
cross winds, which may lead to withdrawn train and equip- Taking a linearized lumped parameter system with two
ment damage [1]. Generally speaking, this kind of vibration degrees of freedom as the research object, Luongo and
is known as galloping, which is also called dancing [2]. Due Piccardo [8] analyzed the bifurcation characteristic of the
to the widespread existence of galloping phenomenon, it is system with galloping by the perturbation approach. Jones
of great significance for engineers to master the dynamic [9] studied the linearized coupled vertical-horizontal gal-
characteristics of catenary system with galloping oscillation. loping behavior analytically and drew the conclusion that
Investigations on iced conductor galloping have been of the coupled galloping criterion might be either more or less
interest to researchers for decades. Den Hartog [3] proposed stringent than Den Hartog’s criterion. In reference [10], a
the vertical galloping criterion. He found that after icing comprehensive analysis was presented for the galloping of an
changed the cross section of conductor, under certain wind oscillator that might vibrate both transversely and torsionally.
2 Shock and Vibration

Furthermore, a three-degree-of-freedom model for galloping

was developed and analyzed analytically in [11, 12].
In the previous work, the galloping model was usually
considered as a lumped model. Actually, the continuous char- dw
acteristics of multispan conductor are extremely important Dw
and should not be ignored. In view of this, Desai et al.
[13] described the galloping behavior of the multispan iced
transmission line by the finite element method, in which the
three-node isoparametric cable element included torsional (a) Without ice coating (b) With ice coating
and three translational degrees of freedom at each node. Since
then, the finite element method has been widely used in the Figure 1: Two kinds of cross section of contact wire.
galloping research. Based on the work mentioned in [13],
Desai et al. [14] developed an efficient perturbation method
Moreover, the effects of initial icing angle, damping, and wind
to calculate the galloping vibration of the multispan overhead
velocity on the galloping characteristics of catenary system
transmission line. Zhang el al. [15] established a computa-
are discussed in detail.
tionally efficient hybrid model, in which the mode shapes
were determined by the finite element method and the initial
condition, and analytically investigated the steady vibration 2. Quasi-Steady Aerodynamic
amplitudes of the system. Keutgen and Lilien [16] conducted a Force of Iced Contact Wire
wind tunnel test to obtain a full set of data of recent galloping
results, which were validated by the corresponding numerical As one of typical slender structures in the actual engineering,
simulation. By using the spatial curved beam theory, Yan et catenary system is mainly composed of messenger wire,
al. [17] considered the nonlinearity and bending stiffness of contact wire, and droppers. In certain circumstance, slender
iced conductor, and then presented the finite element model, wires may be covered with ice and then their cross sections
including three translational degrees of freedom and three would become asymmetric. As a result, average cross winds
rotational degrees of freedom. can cause the galloping vibration of the iced wires under
In the aspect of catenary galloping, Stickland et al. some certain conditions. Previous engineering cases have
[18] analyzed the aerodynamic characteristics of a series of illustrated that the mean lift and drag coefficients obtained
contact wires by experiment and determined the occurrence under static conditions can accurately describe the galloping
conditions of vertical galloping for different contact wires phenomenon [22]. Here the mean lift and drag coefficients are
based on the Den Hartog criterion. Meanwhile, the effects of the functions of the angle of attack. Therefore, in this paper,
mechanical damping on the occurrence conditions of gallop- quasi-steady aerodynamic force is used to describe the action
ing were introduced in [18]. Xie et al. [19] conducted a series of cross winds on the catenary system.
of wind tunnel tests on the 2 : 1 scaled model of iced contact The cross sections of contact wire without and with ice
wires and obtained the relation between aerodynamic force coating are shown in Figure 1. For the case of no ice coating,
coefficients and angles of attack in the different turbulent flow the contact wire has a circular cross section with two side
fields. Besides, the galloping stability of iced contact wire was grooves, which are used to install clamps. The diameter of
discussed by them. Song et al. [20] studied the wind-induced cross section of contact wire is 𝐷𝑤 = 14.4 mm [25].
vibration of the contact wire with different thickness of ice The ice coating shown in Figure 1(b) is assumed ideal
coating and concluded that the thicker the ice coating was, crescent, and then an ice shape coefficient [26] is introduced
the more serious the wind deviation of contact wire could get. to describe the relation between ice thickness and diameter
Meanwhile, literature [20] stated that the influence of average of cross section; namely,
wind load was so little while the fluctuating wind load had a 𝐷𝑤
significant effect on the current collecting quality. As for the 𝜆𝑤 = , (1)
𝐷𝑤 + 𝑑𝑤
wind-induced vibration of high speed catenary, the dynamic
behaviors of the pantograph-catenary under stochastic wind where 𝑑𝑤 is the thickness of ice coating and the value range
field were analyzed in reference [21]. of 𝜆 𝑤 is usually [1.2, 1.8]. In this paper, the coefficient is
Up to now, galloping as well as wind deviation of a single 𝜆 𝑤 = 1.6, indicating that the thickness of ice coating is
contact wire and fluctuating wind response of iced contact 𝑑𝑤 = 8.64 mm.
wire has been studied in detail. However, the attention that To simulate the aerodynamic force on the catenary
has been paid to the galloping behavior of the whole iced system, the flow is considered two-dimensional and uniform.
catenary is not sufficient. In this paper, the galloping behavior The cross section of contact wire is fixed in the flow field.
of iced catenary system under average cross winds is studied. According to [27], the drag force and the lift force, respec-
To this end, the quasi-steady aerodynamic force coefficients tively, satisfy
of iced contact wire are simulated by FLUENT. The condition 1
of critical instability obtained from the Den Hartog criterion 𝐹𝑙,𝑤 (𝛼𝑤 ) = 𝜌air 𝑈2 𝐷𝑟,𝑤 𝐶𝑙,𝑤 (𝛼𝑤 ) ,
2
is verified by simulating the system response. On this basis, (2)
the phenomenon of low frequency and large amplitude 1
𝐹𝑑,𝑤 (𝛼𝑤 ) = 𝜌air 𝑈2 𝐷𝑟,𝑤 𝐶𝑑,𝑤 (𝛼𝑤 ) ,
vibration of catenary system under cross winds is explained. 2
Shock and Vibration 3

Fl,w
Fl,w Wind Fl,w

Wind Wind
Fd,w
Fd,w
Fd,w

(a) 𝛼𝑤 = 0∘ (b) 𝛼𝑤 = 90∘ (c) 𝛼𝑤 = 180∘

Figure 2: Definition of angle of attack.

right boundary is flow outlet. The top boundary and bottom

Symmetry boundary are considered as symmetry. Besides, the wall of
Inlet cross section of iced contact wire is set as no-slip wall. Due to
the drastic changes of the flow field near the cross section wall
Wall
55.6D

11D
Interior D Interior and in the wake region, the grids should be further densified
Outlet in these two regions. The unstructured grids are applied all
11D 48.6D over the computational domain and the corresponding total
grid number is 140,000. In order to ensure the computational
Symmetry
precision, the boundary layer mesh is needed in the near-wall
region.
111D Since the quasi-steady aerodynamic forces can describe
(a) The boundary condition the galloping process accurately, the Reynolds-Averaged
Navier-Stokes (RANS) equations, in which the transient
quantity in the Navier-Stocks equations is decomposed into
mean and fluctuating quantities, are considered here. SST 𝑘-𝜔
model is recommended as the viscous model in this work. It is
a two-equation eddy-viscosity turbulence model which does
not need the wall-function and plays a good performance on
the adverse pressure gradient, such as flow around a circular
cylinder while the Reynolds number is large. SIMPLE-type
pressure-velocity coupling scheme is used and second-order
upwind scheme is applied to discretize the momentum,
turbulent kinetic energy, and the specific dissipation rate.
(b) The mesh of flow field During the simulation process, drag and lift coefficients of
the cross section wall are monitored. Wind velocity is set to
Figure 3: Schematic diagram of computational field in FLUENT. 15 m/s, time step is specified as 0.1 ms, and the total simulation
time is 1 s. The time series data of the first 0.5 seconds are
deliberately excluded from the total time series data to ensure
where 𝐹𝑑,𝑤 and 𝐹𝑙,𝑤 denote the drag force and lift force, 𝐶𝑑,𝑤 that the data used correspond to the steady state.
and 𝐶𝑙,𝑤 denote the drag coefficient and lift coefficient, 𝛼𝑤 The Reynolds number of the iced contact wire is Re =
denotes the angle of attack, 𝜌air denotes the air density, 𝑈 21110 which belongs to the subcritical region. In the sub-
denotes the wind velocity, and 𝐷𝑟,𝑤 denotes the maximum critical region, the vortex street of the wake region becomes
height of windward side of cross section and it is approxi- turbulent gradually while the separation of boundary layer
mately equal to the diameter of cross section. appears because of the strong inverse pressure gradient near
It is shown in Figure 2 that the range of angle of attack the iced section wall. Figure 4 is the velocity contour maps
is in [0∘ , 180∘ ]. To simulate the aerodynamic characteristics of the flow field near the wire section wall under different
of iced contact wire under cross winds more accurately, angles of attack. From the figures the turbulent flow and the
the increment of angle of attack is set to 10∘ in the CFD separation of boundary layer can be seen clearly.
simulations, meaning that there are 19 simulation cases. In accordance with the symmetry of cross section of iced
CFD simulations in this paper are carried out by FLUENT contact wire, the drag and lift coefficients follow the following
in which the Finite Volume Method is applied to spatially relationships.
discretize the governing equations. The boundary conditions
and mesh of the computational field for 𝛼𝑤 = 0∘ are 𝐶𝑑,𝑤 (𝛼𝑤 ) = 𝐶𝑑,𝑤 (−𝛼𝑤 ) ,
shown in Figure 3, where the computational field is divided (3)
into three parts. The left boundary is velocity inlet and the 𝐶𝑙,𝑤 (𝛼𝑤 ) = −𝐶𝑙,𝑤 (−𝛼𝑤 ) .
4 Shock and Vibration

2.00e + 01 2.52e + 01
1.92e + 01 2.42e + 01
1.84e + 01 2.32e + 01
1.76e + 01 2.21e + 01
1.68e + 01 2.11e + 01
1.60e + 01 2.01e + 01
1.52e + 01 1.91e + 01
1.44e + 01 1.81e + 01
1.36e + 01 1.71e + 01
1.28e + 01 1.61e + 01
1.20e + 01 1.51e + 01
1.12e + 01 1.41e + 01
1.04e + 01 1.31e + 01
9.60e + 00 1.21e + 01
8.80e + 00 1.11e + 01
8.00e + 00 1.01e + 01
7.20e + 00 9.06e + 00
6.40e + 00 8.05e + 00
5.60e + 00 7.05e + 00
4.80e + 00 6.04e + 00
4.00e + 00 5.03e + 00
3.20e + 00 4.03e + 00
2.40e + 00 3.02e + 00
1.60e + 00 2.01e + 00
8.00e − 01 1.01e + 00
0.00e + 00 0.00e + 00
(a) 0∘ (b) 40∘
3.36e + 01 2.35e + 01
3.22e + 01 2.25e + 01
3.09e + 01 2.16e + 01
2.96e + 01 2.07e + 01
2.82e + 01 1.97e + 01
2.69e + 01 1.88e + 01
2.55e + 01 1.78e + 01
2.42e + 01 1.69e + 01
2.28e + 01 1.60e + 01
2.15e + 01 1.50e + 01
2.02e + 01 1.41e + 01
1.88e + 01 1.31e + 01
1.75e + 01 1.22e + 01
1.61e + 01 1.13e + 01
1.48e + 01 1.03e + 01
1.34e + 01 9.39e + 00
1.21e + 01 8.45e + 00
1.07e + 01 7.51e + 00
9.41e + 00 6.57e + 00
8.06e + 00 5.63e + 00
6.72e + 00 4.69e + 00
5.37e + 00 3.75e + 00
4.03e + 00 2.82e + 00
2.69e + 00 1.88e + 00
1.34e + 00 9.39e − 01
0.00e + 00 0.00e + 00

(c) 90∘ (d) 150∘

Figure 4: Velocity contour maps under different angles of attack.

According to the time-averaged value of the calculated the pictures that the coefficients of drag and curves fit-
data, the relations between drag coefficient, lift coefficient, ted from CFD and wind tunnel tests have uniform ten-
and angle of attack are depicted in Figure 5. dency.
It is clear that when the angle of attack is near ±0∘ and
±180∘ , the drag coefficient reaches the minimum because that 3. Vertical Aerodynamic Force Model
is when the windward side is the smallest, while the drag
coefficient reaches the maximum when the angle of attack The galloping model of iced contact wire is shown in Figure 8,
is near ±90∘ because that is when the windward side is the in which 𝑈 is the wind velocity, 𝑦𝑤̇ is the vertical vibration
largest. As for the lift coefficient, the peaks appear when the velocity, and 𝑈𝑟 is the relative velocity between 𝑈 and 𝑦𝑤̇ .
angle of attack is near 40∘ , 170∘ , and −130∘ , and the troughs The angle between aerodynamic drag force and central
appear when the angle of attack is near −40∘ , −170∘ , and line of cross section of iced contact wire is described by 𝛼𝑤 ,
130∘ . The changing rules of drag and lift coefficients are in which is usually called angle of attack in the aerodynamics.
agreement with the wind tunnel test data in [16]. Additionally, 𝛼0,𝑤 denotes the initial icing angle, and 𝛼𝑟,𝑤
To further verify the results of CFD, aerodynamic coef- denotes the relative angle of attack. Therefore, the specific
ficients of bluff body section such as a D-shape section geometric relationship between them can be expressed as
and a square section of iced conductor are performed by
FLUENT. The model parameters of D-shape section and
𝛼𝑤 = 𝛼0,𝑤 − 𝛼𝑟,𝑤 , (4)
square section are referred to in [23] and [24], respec-
tively. Figures 6 and 7 show the comparison between the
𝑦𝑤̇ 𝑦̇
results obtained by CFD in this work and the wind tun- 𝛼𝑟,𝑤 = arctan ≈ 𝑤. (5)
nel test results obtained in [23, 24]. It can be seen from 𝑈𝑟 𝑈
Shock and Vibration 5

4.5 Assuming the relative angle of attack 𝛼𝑟,𝑤 is rather small

at the very beginning of galloping, (10) can be expanded at
3.5 𝛼𝑟,𝑤 = 0 by the Taylor expansion method, so that

2.5
󵄨 𝜕𝐶𝑦,𝑤 󵄨󵄨󵄨󵄨
𝐶𝑦,𝑤 (𝛼𝑟,𝑤 ) = 𝐶𝑦,𝑤 󵄨󵄨󵄨󵄨𝛼 + 󵄨󵄨 𝛼
1.5
𝑟,𝑤 =0 𝜕𝛼𝑟,𝑤 󵄨󵄨󵄨𝛼 =0 𝑟,𝑤
𝑟,𝑤

󵄨
1 𝜕 𝐶𝑦,𝑤 󵄨󵄨󵄨󵄨
2
0.5 2
+ 󵄨󵄨 𝛼𝑟,𝑤 (11)
2
2! 𝜕𝛼𝑟,𝑤 󵄨󵄨
󵄨𝛼 𝑟,𝑤 =0
−0.5
󵄨
1 𝜕 𝐶𝑦,𝑤 󵄨󵄨󵄨󵄨
3
3 4
−1.5 + 󵄨󵄨 𝛼𝑟,𝑤 + 𝑂 (𝛼𝑟,𝑤 ),
−180 −120 −60 0 60 120 180 3
3! 𝜕𝛼𝑟,𝑤 󵄨󵄨
󵄨𝛼 𝑟,𝑤 =0
 (∘ )
4 4
Cd where 𝑂(𝛼𝑟,𝑤 ) denotes the terms proportional to 𝛼𝑟,𝑤 and the
Cl higher powers of 𝛼𝑟,𝑤 which are ignored.
Combining (5) with (11), the vertical aerodynamic force
Figure 5: Relations between 𝐶𝑑 , 𝐶𝑙 , and 𝛼.
on unit length of iced contact wire can be obtained according
to (7).

Since the vertical galloping velocity is far less than the 𝐹𝑦,𝑤 = 𝑎3,𝑤 𝑦𝑤3̇ + 𝑎2,𝑤 𝑦𝑤2̇ + 𝑎1,𝑤 𝑦𝑤̇ + 𝑎0,𝑤 , (12)
wind velocity, 𝑈𝑟 is approximately equal to 𝑈. The component
of drag and lift forces on the axis 𝑦𝑤 can be expressed as where the expressions of 𝑎3,𝑤 , 𝑎2,𝑤 , 𝑎1,𝑤 , and 𝑎0,𝑤 are

𝜌air 𝐷𝑤 1 3 3 1 2
𝐹𝑦,𝑤 = −𝐹𝑑,𝑤 sin 𝛼𝑟,𝑤 + 𝐹𝑙,𝑤 cos 𝛼𝑟,𝑤 . (6) 𝑎3,𝑤 = [ 𝑑3 𝛼0,𝑤 + ( 𝑙3 + 𝑑2 ) 𝛼0,𝑤
2𝑈 6 2 6
(13)
1 1 1
By substituting (2) into (6), (6) can be further derived as + (𝑙2 + 𝑑1 − 3𝑑3 ) 𝛼0,𝑤 + 𝑙1 + 𝑑0 − 𝑑2 ] ,
6 2 6
𝜌air 𝐷𝑤 1 3 1 2
1 𝑎2,𝑤 = [− 𝑙3 𝛼0,𝑤 + (3𝑑3 − 𝑙2 ) 𝛼0,𝑤
𝐹𝑦,𝑤 = 𝜌air 𝑈2 𝐷𝑤 𝐶𝑦,𝑤 , (7) 2 2 2
2 (14)
1 1
+ (3𝑙3 − 𝑙1 + 2𝑑2 ) 𝛼0,𝑤 + 𝑑1 + 𝑙2 − 𝑙0 ] ,
where 𝐶𝑦,𝑤 obeys 2 2
𝜌air 𝐷𝑤 𝑈 3 2
𝑎1,𝑤 = − [𝑑3 𝛼0,𝑤 + (𝑑2 + 3𝑙3 ) 𝛼0,𝑤
𝐶𝑦,𝑤 = −𝐶𝑑,𝑤 (𝛼𝑤 ) sin 𝛼𝑟,𝑤 + 𝐶𝑙,𝑤 (𝛼𝑤 ) cos 𝛼𝑟,𝑤 . (8) 2 (15)
+ (𝑑1 + 2𝑙2 ) 𝛼0,𝑤 + 𝑑0 + 𝑙1 ] ,
Aiming at the range of angle of attack [−50∘ , 50∘ ], the drag
and lift coefficients are gradually extracted and then used in 𝜌air 𝐷𝑤 𝑈2 3 2
𝑎0,𝑤 = [𝑙3 𝛼0,𝑤 + 𝑙2 𝛼0,𝑤 + 𝑙1 𝛼0,𝑤 + 𝑙0 ] . (16)
the cube polynomial fitting; namely, 2
As for the iced messenger wire, its cross section is similar
𝐶𝑑,𝑤 (𝛼𝑤 ) = 𝑑3 𝛼𝑤 3 + 𝑑2 𝛼𝑤 2 + 𝑑1 𝛼𝑤 + 𝑑0 , to that of iced contact wire; that is, 𝜆 𝑚 = 𝜆 𝑤 . Here
(9) the subscript 𝑚 represents the messenger wire. Due to the
𝐶𝑙,𝑤 (𝛼𝑤 ) = 𝑙3 𝛼𝑤 3 + 𝑙2 𝛼𝑤 2 + 𝑙1 𝛼𝑤 + 𝑙0 , same working environment and same order magnitude of
diameters (𝐷𝑤 = 14.4 mm, 𝐷𝑚 ≈ 9.17 mm), the Reynolds
where the polynomial coefficients 𝑑0 , 𝑑1 , 𝑑2 , and 𝑑3 are of iced messenger wire and iced contact wire are in the same
shown in Table 1. Correspondingly, the fitting curves of drag interval (Re𝑤 = 21110, Re𝑚 = 13443).
coefficient and lift coefficient are shown in Figure 9. As a result, the drag and lift coefficients of iced contact
By substituting (4) into (9), (8) is further expressed as wire and iced messenger wire are assumed to be the same.
Similarly, the vertical aerodynamic force on unit length of
3 2 iced messenger wire can be written as
𝐶𝑦,𝑤 (𝛼𝑟,𝑤 ) = − [𝑑3 (𝛼0,𝑤 − 𝛼𝑟,𝑤 ) + 𝑑2 (𝛼0,𝑤 − 𝛼𝑟,𝑤 )
3 ̇ 3 + 𝑎2,𝑚 𝑦𝑚
𝐹𝑦,𝑚 = 𝑎3,𝑚 𝑦𝑚 ̇ 2 + 𝑎1,𝑚 𝑦𝑚
̇ + 𝑎0,𝑚 , (17)
+ 𝑑1 (𝛼0,𝑤 − 𝛼𝑟,𝑤 ) + 𝑑0 ] sin 𝛼𝑟,𝑤 + [𝑙3 (𝛼0,𝑤 − 𝛼𝑟,𝑤 ) (10)
2 where the coefficient forms of 𝑎3,𝑚 , 𝑎2,𝑚 , 𝑎1,𝑚 , and 𝑎0,𝑚 are the
+ 𝑙2 (𝛼0,𝑤 − 𝛼𝑟,𝑤 ) + 𝑙1 (𝛼0,𝑤 − 𝛼𝑟,𝑤 ) + 𝑙0 ] cos 𝛼𝑟,𝑤 . same as those shown in (13)–(16).
6 Shock and Vibration

3 3

2 2
Cd

Cd
1 1

0 0
0 10 20 30 40 50 60 0 10 20 30 40 50 60
 (∘ )  (∘ )

Discrete data Discrete data

Fitting curve Fitting curve
(a) Drag coefficients from CFD (b) Drag coefficients from wind tunnel tests [23]
0 0

−0.6 −0.6
Cl

Cl

−1.2 −1.2

−1.8 −1.8
0 10 20 30 40 50 60 0 10 20 30 40 50 60
 (∘ )  (∘ )

Discrete data Discrete data

Fitting curve Fitting curve
(c) Lift coefficients from CFD (d) Lift coefficients from wind tunnel tests [23]

Figure 6: Comparison of drag and lift coefficients from CFD and wind tunnel tests of a D-shape section.

Table 1: Polynomial coefficients of fitting curves.

Coefficient Value Coefficient Value

𝑑0 0.4238 𝑙0 0
𝑑1 0 𝑙1 1.9841
𝑑2 2.0079 𝑙2 0
𝑑3 0 𝑙3 −2.0687

4. Galloping Vibration of Catenary System According to the mechanical characteristics, the droppers
are simplified as the nonlinear springs, in which the tensile
In this section, the equations of vertical motion of the simple stiffness is far larger than the compression stiffness.
catenary system under cross winds are derived. As shown The infinitesimal method is used to derive the equations
in Figure 10, the contact wire and messenger wire can be of motion of the messenger and contact wires which are
described by the simply supported Euler-Bernoulli beams. simplified as Euler-Bernoulli beams [28]. Figure 11 is the force
Shock and Vibration 7

1.9 1.9

1.6 1.6
Cd

Cd
1.3 1.3

1 1
0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45
 (∘ )  (∘ )

Discrete data Discrete data

Fitting curve Fitting curve
(a) Drag coefficients from CFD (b) Drag coefficients from wind tunnel tests [24]
0 0

−0.2 −0.2
Cl

Cl

−0.4 −0.4

−0.6 −0.6
0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 45
 (∘ )  (∘ )

Discrete data Discrete data

Fitting curve Fitting curve
(c) Lift coefficients from CFD (d) Lift coefficients from wind tunnel tests [24]

Figure 7: Comparison of drag and lift coefficients from CFD and wind tunnel tests of a square section.

yw Fl,w

U 0,w z
o r,w
Ur Fd,w
yẇ w
r,w
U

Figure 8: Galloping model of iced contact wire in the vertical direction.

8 Shock and Vibration

2.5 1.5

2 1

1.5 0.5
Cd,w

Cl,w
1 0

0.5 −0.5

0 −1
−50 −40 −30 −20 −10 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 10 20 30 40 50
w (∘ ) w (∘ )

Discrete data Discrete data

Fitting curve Fitting curve
(a) Drag coefficient (b) Lift coefficient

Figure 9: Fitting curves.

Support Messenger wire

yc
U U U
x
z Dropper
··· ··· ···
yw

x
z
Contact wire

Figure 10: Schematic diagram of simple catenary system under cross winds.

diagram of the Euler-Bernoulli beam infinitesimal element in According to the mechanical characteristic and deforma-
which 𝑓(𝑥, 𝑡) denotes the vertical force per unit length, 𝑦(𝑥, 𝑡) tion relationship introduced in material mechanics, we can
the vertical displacement, 𝜌𝐴(𝑥) the mass per unit length, obtain that
𝐸𝐼 the bending stiffness, 𝑇 the tensile force, 𝑀 the bending 𝜕𝑦 (𝑥, 𝑡)
moment, 𝑄 the shear force, and 𝑑𝑥 the length of the beam 𝜃= ,
𝜕𝑥
element.
Derived from Figure 11(b), the force equation of motion 𝜕𝑀
𝑄= , (20)
in the vertical direction can be expressed as 𝜕𝑥
𝜕2 𝑦 (𝑥, 𝑡)
𝜕𝑄 𝜕𝜃 𝑀 = −𝐸𝐼 .
𝑄 − (𝑄 + 𝑑𝑥) − 𝑇 sin 𝜃 + 𝑇 sin (𝜃 + 𝑑𝑥) 𝜕𝑥2
𝜕𝑥 𝜕𝑥 Substituting (20) into (19), the equation of motion of an
(18)
𝜕2 𝑦 (𝑥, 𝑡) equal section Euler-Bernoulli beam can be written as
+ 𝑓 (𝑥, 𝑡) 𝑑𝑥 = −𝜌𝐴 (𝑥) 𝑑𝑥 .
𝜕𝑡2 𝜌𝐴𝑦̈ + 𝐸𝐼𝑦󸀠󸀠󸀠󸀠 + 𝑇𝑦󸀠󸀠 = −𝑓 (𝑥, 𝑡) . (21)
In which 𝑦̈ = 𝜕2 𝑦/𝜕𝑡2 , 𝑦󸀠󸀠󸀠󸀠 = 𝜕4 𝑦/𝜕𝑥4 , 𝑦󸀠󸀠 = 𝜕2 𝑦/𝜕𝑥2 .
Due to the reason that 𝜃 is relatively small, there exists the
For the iced messenger wire, the lumped external forces
assumption sin 𝜃 ≈ 𝜃. Therefore, (18) can be derived as
include the elastic forces acted by droppers and the elastic
force acted by masts, and the distributed external forces
𝜕𝑄 𝜕𝜃 include the damping force and the aerodynamic force. For
− 𝑑𝑥 + 𝑇 𝑑𝑥 + 𝑓 (𝑥, 𝑡) 𝑑𝑥
𝜕𝑥 𝜕𝑥 the iced contact wire, the lumped external forces include the
(19) elastic forces acted by droppers and the distributed external
𝜕2 𝑦 (𝑥, 𝑡) forces include the damping forces and the aerodynamic
= −𝜌𝐴 (𝑥) 𝑑𝑥 .
𝜕𝑡2 forces.
Shock and Vibration 9

y f(x, t)
T
A(x), EI(x) T+ dx
f(x, t) x

Q + dx
x
y(x, t)
M
x  M+ dx
x
x
M dx Q
L T Q+ dx
x
(a) Bending configuration of an Euler-Bernoulli beam [28] (b) Free-body diagram for a beam element

Figure 11: Diagram of the Euler-Bernoulli beam.

Thus, the vertical vibration equation of iced messenger According to the mechanical characteristic, the elastic
wire under cross winds can be derived as force of 𝑖th dropper can be written as
󸀠󸀠󸀠󸀠 󸀠󸀠
̈ + 𝐶𝑚 𝑦𝑚
𝜌𝑚 𝐴 𝑚 𝑦𝑚 ̇ + 𝐸𝐼𝑚 𝑦𝑚 + 𝑇𝑚 𝑦𝑚 𝑓𝑑,𝑖 = 𝑘𝑑,𝑖 (𝑦𝑚,𝑖 − 𝑦𝑤,𝑖 + Δ𝑙0 ) ,
𝑏 𝑝 (22)
= −∑𝐹𝑚,𝑖 − ∑𝐹𝑠,𝑖 + 𝐹𝑦,𝑚 , {𝑘𝑡𝑑 , (𝑦𝑚,𝑖 − 𝑦𝑤,𝑖 + Δ𝑙0 ) ≥ 0 (26)
𝑖=1 𝑖=1
𝑘𝑑,𝑖 = {
𝑘 , (𝑦𝑚,𝑖 − 𝑦𝑤,𝑖 + Δ𝑙0 ) < 0,
{ 𝑝𝑑
in which 𝑦𝑚 , 𝜌𝑚 , 𝐴 𝑚 , 𝐶𝑚 are the vertical displacement,
linear density, area of cross section, and the structural where 𝑘𝑡𝑑 and 𝑘𝑝𝑑 denote the tensile stiffness and compres-
damping coefficient, respectively. 𝐸𝐼𝑚 , 𝑇𝑚 , 𝐹𝑚,𝑖 , 𝑏, and 𝑝 sion stiffness, respectively, Δ𝑙0 is the initial elongation of
denote bending stiffness, the tensile force acted on the iced dropper, and 𝑦𝑚,𝑖 and 𝑦𝑤,𝑖 denote the vertical displacements
messenger wire, the sum of elastic force given by 𝑖th dropper of messenger wire and contact wire at the position of 𝑖th
and gravity of droppers and clamps, the number of droppers, dropper.
and the number of support springs. In order to obtain the dynamic characteristic of the
The elastic support force acted on the messenger wire (see catenary system, the Galerkin method is adapted to discretize
Figure 10) can be expressed as the partial differential equations. The main vibration mode
functions are assumed as 𝜑𝑚 (𝑥) and 𝜑𝑤 (𝑥), which satisfy the
𝐹𝑠,𝑖 = 𝑘𝑠 𝑦𝑚,𝑠,𝑖 × 𝛿 (𝑥 − 𝑥𝑠,𝑖 ) , (23) boundary conditions of contact wire and messenger wire.
where 𝑘𝑠 , 𝑦𝑚,𝑠,𝑖 , 𝑥𝑠,𝑖 denote the support stiffness, the vertical Therefore, the solutions to vibration equations of catenary
displacement of the messenger wire at 𝑖th support, and the system can be expressed as
location of 𝑖th support on the 𝑥 axis, and 𝛿 is the Dirac 𝑛
function. 𝑦𝑚 = ∑𝜑𝑚,𝑖 (𝑥) 𝑞𝑚,𝑖 (𝑡) (27)
By repeating the above steps, the vertical vibration equa- 𝑖=1
tion of iced contact wire under cross winds can be expressed 𝑛
as 𝑦𝑤 = ∑𝜑𝑤,𝑖 (𝑥) 𝑞𝑤,𝑖 (𝑡) , (28)
𝜌𝑤 𝐴 𝑤 𝑦𝑤̈ + 𝐶𝑤 𝑦𝑤̇ + 𝐸𝐼𝑤 𝑦𝑤󸀠󸀠󸀠󸀠 + 𝑇𝑤 𝑦𝑤󸀠󸀠 𝑖=1

𝑏 (24) where 𝑞𝑚,𝑖 (𝑡) and 𝑞𝑤,𝑖 (𝑡) are the mode coordinates of messen-
= −∑𝐹𝑤,𝑖 + 𝐹𝑦,𝑤 . ger wire and contact wire; 𝑛 is the modal truncation order.
𝑖=1 According to [25], the main parameters of catenary
system are shown in Table 2. In addition, the span of catenary
The coupling relations between contact wire and mes- is set to 10, the tensile stiffness of dropper is 𝑘𝑡𝑑 = 106 N/m,
senger wire are achieved by the droppers. For each dropper the compression stiffness of dropper is 𝑘𝑝𝑑 = 0.5 × 104 N/m,
shown in Figure 10, the elastic forces acted on the contact wire
and messenger wire, respectively, obey and support stiffness is 𝑘𝑠 = 2.5 × 107 N/m. Referring to
the structure design of actual catenary, the vector position of
1 dropper in each span obeys [3.375 10.125 16.875 23.625 30.375
𝐹𝑚,𝑖 = 𝛿 (𝑥 − 𝑥𝑖 ) ( 𝑚𝑖 𝑔 + 𝑓𝑑,𝑖 ) ,
2 37.125 43.875 50.625] (m).
(25) Since the catenary structure is extremely complicated, the
1
𝐹𝑤,𝑖 = 𝛿 (𝑥 − 𝑥𝑖 ) ( 𝑚𝑖 𝑔 − 𝑓𝑑,𝑖 ) , analytical expression of main vibration mode functions can
2 hardly be derived. Thus the finite element method is applied
where 𝑥𝑖 is the location of 𝑖th dropper on the 𝑥 axis; 𝑚𝑖 is the to analyze the catenary mode for obtaining the discretized
total mass of 𝑖th dropper and clamps. main vibration mode, as shown in Figure 12. The frequencies
10 Shock and Vibration

0 108 216 324 432 540 0 108 216 324 432 540
Length (m) Length (m)

Messenger wire Messenger wire

Contact wire Contact wire
(a) First mode (b) Second mode

0 108 216 324 432 540 0 108 216 324 432 540
Length (m) Length (m)

Messenger wire Messenger wire

Contact wire Contact wire
(c) Third mode (d) Fourth mode

Figure 12: Vibration mode shapes of the ten-span catenary.

Table 2: Parameters of catenary model.

Tensile force Linear density Cross section area Span length

𝑇 (N) 𝜌𝐴 (kg/m) 𝐴 (mm2 ) 𝑙 (m)
Messenger wire 14000 0.64 105.65 54
Contact wire 20000 1.43 260.58 54

are listed in Table 3. One can observe that, in the figures of first stiffness matrix, Q is the discrete force vector, and q is the
fourth modes, mode type in each span is the same. Indeed, modal coordinates vector.
mode type in each span is the same of every ten modes.
Correspondingly, frequencies of the ten-span catenary are 5. General Results and Discussion
distributed by groups of ten.
On this basis, (27) and (28) are substituted into (22) 5.1. Stability Analysis of Catenary System. As illustrated in
and (24), respectively. And then multiply both sides of [27], the aerodynamical stability of the dynamic system is
the equations by 𝜑𝑚,𝑗 (𝑥) and 𝜑𝑤,𝑗 (𝑥). Thus the ordinary mainly determined by the sign of the net damping coefficient.
differential equations of catenary system can be obtained by For the case that the net damping coefficient is greater
numerical integration. The detailed derivation is shown in the than zero, the catenary system is aerodynamically stable.
appendix. Otherwise, the catenary system is aerodynamically unstable.
Therefore, the discrete vibration equations of the catenary It can be seen from (22) and (24) that the unstable
system can be rewritten in the matrix form as follows: conditions of iced messenger wire and iced contact wire,
Mq̈ + (C𝑙 + C𝑛 ) q̇ + Kq = Q, (29) respectively, obey

where M is the mass matrix, C𝑙 is the linear damping 𝐶𝑤 − 𝑎1,𝑤 < 0, (30)
matrix generated by structural damping, C𝑛 is the nonlinear
damping matrix generated by aerodynamic force, K is the 𝐶𝑚 − 𝑎1,𝑚 < 0. (31)
Shock and Vibration 11

Table 3: Frequencies of catenary. 0.04

Number Frequency (Hz)

(1) 1.0495

Displacement (m)
(2) 1.0605 0.035
(3) 1.0780
(4) 1.1006
(5) 1.1267 0.03
(6) 1.1541
(7) 1.1800
(8) 1.2017
(9) 1.2161 0.025
0 4000 8000 12000 16000
(10) 1.2213 Time (s)
(11) 2.0981
(12) 2.1208 Figure 13: Vertical vibration response of the mid-point of 5th span
contact wire in the condition of 𝛼0,𝑚 = 43.3∘ and 𝛼0,𝑤 = 43.3∘ .
(13) 2.1562
(14) 2.2011
(15) 2.2520
(16) 2.3051 5.2. Effect of Initial Icing Angle. Considering that the main
(17) 2.3561 range of galloping frequency is [0.1 (Hz), 3 (Hz)] [13], the
modal truncation order of the ten-span catenary is set to 20
(18) 2.3998
in this paper. The Runge-Kutta method is used to calculate
(19) 2.4301
the dynamic response of iced catenary system numerically. In
(20) 2.4410 order to achieve the relatively accurate results, the integration
time step is specified as 0.0008 s and the steady-state response
of the mid-point of 5th span of iced contact wire is analyzed.
Due to the reason that the structural damping coefficients Keeping the parameters shown in Table 2, the effect of
are positive, the necessary condition for the occurrence of initial icing angle on the dynamic characteristic is conducted
galloping motion of iced contact wire can be derived from in this section, where the structural damping is not taken into
(15) and (30). consideration.
𝜌air 𝐷𝑤 𝑈 In the condition of 𝑈 = 15 m/s, when the initial icing
3 2
− [𝑑3 𝛼0,𝑤 + (𝑑2 + 3𝑙3 ) 𝛼0,𝑤 + (𝑑1 + 2𝑙2 ) 𝛼0,𝑤 angles are 𝛼0,𝑚 = 43.3∘ and 𝛼0,𝑤 = 43.3∘ , the vertical vibration
2 (32) response of the iced contact wire is shown in Figure 13. It
+ 𝑑0 + 𝑙1 ] > 0. is evident that the vertical vibration displacement gradually
reduces to the static vertical deformation, which is mainly
In the above equation, the air density 𝜌air , diameter of cross caused by the cross winds. Therefore, this phenomenon
section 𝐷𝑤 , and wind velocity 𝑈 are positive. Therefore, (32) shown in Figure 13 is called stable vibration.
can be further reduced; namely, When the initial icing angles are changed to 𝛼0,𝑚 = 43.4∘
and 𝛼0,𝑤 = 43.4∘ , the vertical vibration displacement of
3 2
− [𝑑3 𝛼0,𝑤 + (𝑑2 + 3𝑙3 ) 𝛼0,𝑤 + (𝑑1 + 2𝑙2 ) 𝛼0,𝑤 + 𝑑0 the iced contact wire is shown in Figure 14. By comparing
(33) Figure 14 with Figure 13, it can be seen that with the increase
+ 𝑙1 ] > 0. of initial icing angle, the vertical response progresses to a kind
of unstable vibration with low frequency and large amplitude
Obviously, the necessary condition for the occurrence of rather than static deformation. This is the moment when the
galloping motion of iced contact wire is only affected by the galloping vibration of iced contact wire occurs.
drag and lift coefficients and initial icing angle. According In the following, the effect of single initial icing angle
to the parameters in Table 1, the necessary condition of (initial icing angle of contact wire or initial icing angle of
instability of iced contact wire satisfies messenger wire) on the dynamic characteristic of the system
is discussed. For the case of 𝛼0,𝑚 = 40∘ and 𝛼0,𝑤 = 45∘ , only
󵄨󵄨 󵄨
󵄨󵄨𝛼0,𝑤 󵄨󵄨󵄨 > 43.4 .
∘
(34) the initial icing angle of contact wire meets the condition of
instability. Correspondingly, the vibration response of iced
The above inequality is also the sufficient condition for contact wire is shown in Figure 15, in which the system
the occurrence of galloping of iced contact wire when the maintains a small amplitude vibration after reaching the
structural damping is ignored. steady state. For the case of 𝛼0,𝑚 = 45∘ and 𝛼0,𝑤 = 40∘ ,
On the assumption that the drag and lift coefficients of the similar dynamic phenomenon is shown in Figure 16. The
messenger wire and contact wire are the same in Section 3, main reason of the above phenomena is that the relative
(34) is also the necessary condition of instability of iced oscillations between contact wire and messenger wire are
messenger wire. restricted by the droppers.
12 Shock and Vibration

0.1 0.05

0.04
Displacement (m)

0.05

0.03

0
0.02

−0.05 0.01
0 4000 8000 12000 1.198 1.199 1.2
Time (s) Time (s) ×104

Figure 14: Vertical vibration response of the mid-point of 5th span contact wire in the condition of 𝛼0,𝑚 = 43.4∘ and 𝛼0,𝑤 = 43.4∘ .

0.1 0.033
Displacement (m)

0.05 0.032

0 0.031

−0.05 0.03
0 2000 4000 6000 8000 7980 7985 7990 7995 8000
Time (s) Time (s)

Figure 15: Vertical vibration response of the mid-point of 5th span contact wire in the condition of 𝛼0,𝑚 = 40∘ and 𝛼0,𝑤 = 45∘ .

In addition, beating phenomenon in Figures 14–16 can be proportional damping in numerical calculations. Thus, the
observed. The similarity of frequencies of every ten modes proportional damping matrix can be expressed as
can cause the modal coupling while the catenary is vibrating
under the action of cross winds which directly leads to the C𝑙 = 𝛼M + 𝛽K, (35)
beating phenomenon. And since the neglecting of structure
damping in this case, the phenomenon is particularly obvi- where damping coefficients can be derived as [29]
ous.
1
When the initial icing angle of contact wire is equal to that 𝜔
[ 𝜔1 1 ]
of messenger wire, the relation between maximum vertical 𝜁1 [ ]
{
{ }
} [ ]
displacement and initial icing angle is depicted in Figure 17. It {
{ }
} [ 1 ]
is clear that the critical initial icing angle for the occurrence of {𝜁2 }
{ } 1[[𝜔 𝜔 ]
2] 𝛼
{ . } = [ 2 ]{ }, (36)
galloping motion is 43.4∘ , which exactly meets the condition {
{ .. }
} 2[ . .. ] 𝛽
{
{ } [ . .]
of instability of iced contact wire obtained in Section 5.1. At { } } [ . ]
[ ]
the interval of 40∘ –43.4∘ , the vibration of iced catenary system {𝜁𝑛 } [1 ]
is stable. Meanwhile, the wind deviation is small and changes 𝜔𝑛
𝜔
[ 𝑛 ]
little with the increase of initial icing angle. At the interval
of 43.4∘ –49∘ , the galloping vibration of iced catenary system in which 𝜔𝑛 is the circular frequency, 𝜁𝑛 is the damping
occurs and the max displacement becomes larger with the ratio, and 𝑛 is the number of modes which is set to 20. The
increase of initial icing angle. estimation of the Rayleigh coefficients can be performed by
solving the overdetermined equation (36) by least-squares
5.3. Effect of Structural Damping. As shown in expres- solution. According to [30], the damping ratio is set to 0.0013,
sion (34), the stability condition is closely related to the and the coefficients are 𝛼 = 0.0125, 𝛽 = 0.0001.
structural damping. For most of actual engineering struc- After considering the structure damping, the vertical
tures, the structure damping is usually characterized by the vibration responses of contact wire with different initial icing
Shock and Vibration 13

0.1 0.035

0.034
Displacement (m)

0.05

0.033

0
0.032

−0.05 0.031
0 2000 4000 6000 8000 7980 7985 7990 7995 8000
Time (s) Time(s)

Figure 16: Vertical vibration response of the mid-point of 5th span contact wire in the condition of 𝛼0,𝑚 = 45∘ and 𝛼0,𝑤 = 40∘ .

3 oscillation appears, as shown in Figure 19. Therefore, the new

critical initial icing angle with the effect of structural damping
2.5
is about 𝛼0,𝑚 = 44.8∘ and 𝛼0,𝑤 = 44.8∘ . Overall, the structural
damping has a positive contribution to the motion stability of
Displacement (m)

2
iced catenary system.
1.5
5.4. Effect of Wind Velocity. Taking the structural damping
1 into consideration, the effect of wind velocity on the vibration
43.4∘ , 0.373m
response of iced catenary system is further discussed in
0.5
this section. In the range of wind velocity 1 m/s–35 m/s, the
0 Reynolds numbers of both iced contact wire and messenger
40 41 42 43 44 45 46 47 48 49 wire are in the transcritical scope. When the wind velocity is
Initial icing angle (∘ ) in the range, the drag and lift coefficients are available.
For the case of 𝛼0,𝑚 = 45∘ and 𝛼0,𝑤 = 45∘ , the
Figure 17: Relation between maximum displacement and initial
vertical vibration response of iced contact wire is calculated
icing angle.
in the condition of 𝑈 = 12.4 m/s, as shown in Figure 20.
It is clear that the system response converges to a static
0.036 deformation, which is the deviation caused by the cross
winds. This phenomenon suggests that the iced catenary
system is stable at this moment. Keeping the initial icing
angles of contact wire and messenger wire constant, the wind
Displacement (m)

0.032 velocity is changed to 𝑈 = 12.5 m/s. Correspondingly, a

periodic vibration with low frequency appears in Figure 21,
which indicates the instability phenomenon.
0.028
The relation between maximum vibration in the steady-
state and wind velocity is further analyzed, as shown in
Figure 22. When the initial icing angle is 45∘ , the critical wind
velocity of stability is 𝑈 = 12.5 m/s. When the wind velocity
0.024 is smaller than the critical value, the iced catenary system is
0 0.5 1 1.5 2 2.5 3 3.5 4 stable. While the wind velocity is larger than the critical value,
×104
Time (s) the galloping vibration occurs and the vibration displacement
Figure 18: Vertical vibration response of the mid-point of 5th span turns to be larger with the increase of wind velocity.
contact wire in the condition of 𝛼0,𝑚 = 44.7∘ and 𝛼0,𝑤 = 44.7∘ .
6. Conclusion
angles are analyzed. As shown in Figure 18, the contact wire is In this paper, the galloping vibration of iced catenary sys-
still stable in the conditions of 𝛼0,𝑚 = 44.7∘ and 𝛼0,𝑤 = 44.7∘ . tem under cross winds has been investigated. By FLUENT
This phenomenon suggests that the structural damping can simulation, the lift and drag coefficients of quasi-steady
raise the critical initial icing for unstable station. aerodynamic force acted on the iced catenary system have
When the initial icing angles are 𝛼0,𝑚 = 44.8∘ and been calculated. By fitting the discrete simulation data, the
𝛼0,𝑤 = 44.8∘ , the system becomes unstable and the galloping expression of the vertical aerodynamic force has been further
14 Shock and Vibration

0.08 0.08

0.06 0.06
Displacement (m)

0.04 0.04

0.02 0.02

0 0

−0.02 −0.02
0 2 4 6 8 7.998 7.999 8 4
×104 ×10
Time (s) Time (s)

Figure 19: Vertical vibration response of the mid-point of 5th span contact wire in the condition of 𝛼0,𝑚 = 44.8∘ and 𝛼0,𝑤 = 44.8∘ .

0.05 Appendix
𝑛 𝑙𝑐
̈
𝜌𝑚 𝐴 𝑚 ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝑑𝑥) 𝑞𝑚,𝑖
0
Displacement (m)

𝑖=1
0.03
𝑛 𝑙𝑐
̇
+ 𝐶𝑚 ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝑑𝑥) 𝑞𝑚,𝑖
𝑖=1 0
0.01
𝑛 𝑙𝑐
󸀠󸀠󸀠󸀠
+ [𝐸𝐼𝑚 ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝑑𝑥)
𝑖=1 0
−0.01
0 5.6 11.2 16.8 𝑛 𝑙𝑐
×104 󸀠󸀠
Time(s) + 𝑇𝑚 ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝑑𝑥)] 𝑞𝑚,𝑖
𝑖=1 0
Figure 20: Vertical vibration response of the mid-point of 5th span
contact wire in the condition of 𝑈 = 12.4 m/s. 𝑏 𝑛
= − ∑ ∑𝑘𝑑,𝑘 𝜑𝑚,𝑗 (𝑥𝑑,𝑘 ) 𝜑𝑚,𝑖 (𝑥𝑑,𝑘 ) 𝑞𝑚,𝑖
𝑘=1 𝑖=1

derived. Then the necessary condition of system stability has 𝑏 𝑛

been further obtained based on the Den Hartog galloping + ∑ ∑𝑘𝑑,𝑘 𝜑𝑚,𝑗 (𝑥𝑑,𝑘 ) 𝜑𝑤,𝑖 (𝑥𝑑,𝑘 ) 𝑞𝑤,𝑖
mechanism. Moreover, the vibration equations of iced cate- 𝑘=1 𝑖=1 (A.1)
nary system have been solved by the Runge-Kutta method
𝑏
and the corresponding responses have been analyzed by the 1 𝑏
− ∑ 𝑘𝑑,𝑘 Δ𝑙0,𝑘 𝜑𝑚,𝑗 (𝑥𝑑,𝑘 ) − 𝑔 ∑ 𝑚𝑑,𝑘 𝜑𝑚,𝑗 (𝑥𝑑,𝑘 )
waveform. Meanwhile, the analysis of varying parameters, 𝑘=1
2 𝑘=1
for instance, the initial icing angle, structural damping, and
wind velocity, has been conducted. According to the analysis 𝑝 𝑛
results, the following conclusions can be obtained. − ∑ ∑𝑘𝑠,𝑘 𝜑𝑚,𝑗 (𝑥𝑠,𝑘 ) 𝜑𝑚,𝑖 (𝑥𝑠,𝑘 ) 𝑞𝑚,𝑖
𝑘=1 𝑖=1
(1) As for vertical galloping vibration of iced catenary
system, the critical condition of motion stability is mainly 𝑛 𝑛 𝑛 𝑙𝑐
determined by initial icing angle, wind velocity, and struc- + 𝑎𝑚,3 ∑∑ ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝜑𝑚,𝑙 𝜑𝑚,ℎ 𝑑𝑥)
tural damping. 𝑖=1 𝑙=1 ℎ=1 0

(2) For the case of stable vibration, the dynamic response 𝑛 𝑛 𝑙𝑐

of iced catenary system usually reduces to a static wind ̇ 𝑞𝑚,𝑙
⋅ 𝑞𝑚,𝑖 ̇ 𝑞𝑚,ℎ
̇ + 𝑎𝑚,2 ∑∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝜑𝑚,𝑙 𝑑𝑥)
𝑖=1 𝑙=1 0
deviation. For the case of unstable vibration, the galloping
motion occurs in the iced catenary system. 𝑛 𝑙𝑚
(3) Suspended on the messenger wire, the stability of ̇ 𝑞𝑚,𝑙
⋅ 𝑞𝑚,𝑖 ̇ + 𝑎𝑚,1 ∑ (∫ 𝜑𝑚,𝑗 𝜑𝑚,𝑖 𝑑𝑥) 𝑞𝑚,𝑖
̇
𝑖=1 0
iced contact wire is improved when the initial icing angles of
messenger and contact wires are different. 𝑙𝑚

(4) Increasing the structural damping can reduce the + 𝑎𝑚,0 ∫ 𝜑𝑚,𝑗 𝑑𝑥 = 0, 𝑗 = 1, 2, . . . , 𝑛,
0
chance that the galloping motion occurs. In the actual
𝑛 𝑙𝑤
engineering case, mechanical damping can be used to restrain ̈
𝜌𝑤 𝐴 𝑤 ∑ (∫ 𝜑𝑤,𝑗 𝜑𝑤,𝑖 𝑑𝑥) 𝑞𝑤,𝑖
the galloping oscillation of the catenary. 𝑖=1 0
Shock and Vibration 15

0.06 0.06

0.04 0.04
Displacement (m)

0.02 0.02

0 0

−0.02 −0.02
0 2 4 6 8 7.998 7.999 8
×104 ×104
Time (s) Time (s)

Figure 21: Vertical vibration response of the mid-point of 5th span contact wire in the condition of 𝑈 = 12.5 m/s.

𝑛 𝑙𝑤 𝑙𝑤
2.5
̇ + 𝑎𝑤,0 ∫ 𝜑𝑤,𝑗 𝑑𝑥
+ 𝑎𝑤,1 ∑ (∫ 𝜑𝑤,𝑗 𝜑𝑤,𝑖 𝑑𝑥) 𝑞𝑤,𝑖
𝑖=1 0 0
2
= 0, 𝑗 = 1, 2, . . . , 𝑛.
Displacement (m)

1.5 (A.2)

1 12.5 m/s, 0.034 m Conflicts of Interest

0.5 The authors declare that there are no conflicts of interest
regarding the publication of this paper.
0
1 5 9 13 17 21 25
Wind velocity (m/s) Acknowledgments
Figure 22: Relation between maximum vibration displacement and This work was supported by the National Nature Science
wind velocity. Foundation of China (Grants nos. 11372258 and 11702228)
and the Fundamental Research Funds for the Central Uni-
versities (2682017CX087).

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