# Study on Galloping Oscillation of Iced Catenary System under Cross Winds

Authors: Guo Chen; Yiren Yang; Yang Yang; Peng Li
Journal: Shock and Vibration
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

High speed train is driven by electric locomotives which gain electrical power from pantograph-catenary system. Therefore, proper functioning of catenary is one of the key factors that guarantee the operating safety of high speed railway. With the wide application of electrified railway, the working environment is complicated and various. Especially for areas of low temperature and high humidity and altitude, it is extremely likely for catenary to be covered with ice. Covering ice can not only severely affect the current collecting quality, but also cause a large amplitude oscillation of catenary under cross winds, which may lead to withdrawn train and equipment damage [1]. Generally speaking, this kind of vibration is known as galloping, which is also called dancing [2]. Due to the widespread existence of galloping phenomenon, it is of great significance for engineers to master the dynamic characteristics of catenary system with galloping oscillation.Investigations on iced conductor galloping have been of interest to researchers for decades. Den Hartog [3] proposed the vertical galloping criterion. He found that after icing changed the cross section of conductor, under certain wind velocity and angle of attack, when the negative slope of lift force curve was larger than the amplitude of drag force curve, the occurrence of instability phenomenon could cause the conductor galloping in the vertical direction. Nigol and Buchan further verified the galloping theory mentioned in [4] by the wind tunnel experiment. Additionally, Nigol and Buchan [5] found that when the vertical galloping condition was not satisfied, the galloping would also be caused by the self-excited torsional vibration of iced conductor. Aiming at the galloping mechanism, Yu et al. [6, 7] proposed the novel suggestion that the galloping was caused by the eccentricity inertia of cross section of iced conductor.Taking a linearized lumped parameter system with two degrees of freedom as the research object, Luongo and Piccardo [8] analyzed the bifurcation characteristic of the system with galloping by the perturbation approach. Jones [9] studied the linearized coupled vertical-horizontal galloping behavior analytically and drew the conclusion that the coupled galloping criterion might be either more or less stringent than Den Hartog’s criterion. In reference [10], a comprehensive analysis was presented for the galloping of an oscillator that might vibrate both transversely and torsionally. 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 characteristics of multispan conductor are extremely important 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 and three translational degrees of freedom at each node. Since then, the finite element method has been widely used in the galloping research. Based on the work mentioned in [13], Desai et al. [14] developed an efficient perturbation method to calculate the galloping vibration of the multispan overhead transmission line. Zhang el al. [15] established a computationally 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 amplitudes of the system. Keutgen and Lilien [16] conducted a wind tunnel test to obtain a full set of data of recent galloping results, which were validated by the corresponding numerical simulation. By using the spatial curved beam theory, Yan et al. [17] considered the nonlinearity and bending stiffness of iced conductor, and then presented the finite element model, including three translational degrees of freedom and three rotational degrees of freedom.In the aspect of catenary galloping, Stickland et al. [18] analyzed the aerodynamic characteristics of a series of contact wires by experiment and determined the occurrence conditions of vertical galloping for different contact wires based on the Den Hartog criterion. Meanwhile, the effects of mechanical damping on the occurrence conditions of galloping were introduced in [18]. Xie et al. [19] conducted a series of wind tunnel tests on the 2 : 1 scaled model of iced contact wires and obtained the relation between aerodynamic force coefficients and angles of attack in the different turbulent flow fields. Besides, the galloping stability of iced contact wire was discussed by them. Song et al. [20] studied the wind-induced vibration of the contact wire with different thickness of ice coating and concluded that the thicker the ice coating was, the more serious the wind deviation of contact wire could get. Meanwhile, literature [20] stated that the influence of average wind load was so little while the fluctuating wind load had a significant effect on the current collecting quality. As for the wind-induced vibration of high speed catenary, the dynamic behaviors of the pantograph-catenary under stochastic wind field were analyzed in reference [21].Up to now, galloping as well as wind deviation of a single contact wire and fluctuating wind response of iced contact wire has been studied in detail. However, the attention that has been paid to the galloping behavior of the whole iced catenary is not sufficient. In this paper, the galloping behavior of iced catenary system under average cross winds is studied. To this end, the quasi-steady aerodynamic force coefficients of iced contact wire are simulated by FLUENT. The condition of critical instability obtained from the Den Hartog criterion is verified by simulating the system response. On this basis, the phenomenon of low frequency and large amplitude vibration of catenary system under cross winds is explained. Moreover, the effects of initial icing angle, damping, and wind velocity on the galloping characteristics of catenary system are discussed in detail.


## 2. Quasi-Steady Aerodynamic Force of Iced Contact Wire

As one of typical slender structures in the actual engineering, catenary system is mainly composed of messenger wire, contact wire, and droppers. In certain circumstance, slender wires may be covered with ice and then their cross sections would become asymmetric. As a result, average cross winds can cause the galloping vibration of the iced wires under some certain conditions. Previous engineering cases have illustrated that the mean lift and drag coefficients obtained under static conditions can accurately describe the galloping phenomenon [22]. Here the mean lift and drag coefficients are the functions of the angle of attack. Therefore, in this paper, quasi-steady aerodynamic force is used to describe the action of cross winds on the catenary system.The cross sections of contact wire without and with ice coating are shown in Figure1. For the case of no ice coating, the contact wire has a circular cross section with two side grooves, which are used to install clamps. The diameter of cross section of contact wire is Dw=14.4mm [25].Figure 1
Two kinds of cross section of contact wire.
(a)
Without ice coating
(b)
With ice coatingThe ice coating shown in Figure1(b) is assumed ideal crescent, and then an ice shape coefficient [26] is introduced to describe the relation between ice thickness and diameter of cross section; namely,(1)λw=DwDw+dw,where dw is the thickness of ice coating and the value range of λw is usually 1.2,1.8. In this paper, the coefficient is λw=1.6, indicating that the thickness of ice coating is dw=8.64mm.To simulate the aerodynamic force on the catenary system, the flow is considered two-dimensional and uniform. The cross section of contact wire is fixed in the flow field. According to [27], the drag force and the lift force, respectively, satisfy(2)Fl,wαw=12ρairU2Dr,wCl,wαw,Fd,wαw=12ρairU2Dr,wCd,wαw,where Fd,w and Fl,w denote the drag force and lift force, Cd,w and Cl,w denote the drag coefficient and lift coefficient, αw denotes the angle of attack, ρair denotes the air density, U denotes the wind velocity, and Dr,w denotes the maximum height of windward side of cross section and it is approximately equal to the diameter of cross section.It is shown in Figure2 that the range of angle of attack is in 0°,180°. To simulate the aerodynamic characteristics of iced contact wire under cross winds more accurately, the increment of angle of attack is set to 10° in the CFD simulations, meaning that there are 19 simulation cases.Figure 2
Definition of angle of attack.
(a)
α
w
=
0
°
(b)
α
w
=
90
°
(c)
α
w
=
180
°CFD simulations in this paper are carried out by FLUENT in which the Finite Volume Method is applied to spatially discretize the governing equations. The boundary conditions and mesh of the computational field forαw=0° are shown in Figure 3, where the computational field is divided into three parts. The left boundary is velocity inlet and the right boundary is flow outlet. The top boundary and bottom boundary are considered as symmetry. Besides, the wall of 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 and in the wake region, the grids should be further densified in these two regions. The unstructured grids are applied all over the computational domain and the corresponding total grid number is 140,000. In order to ensure the computational precision, the boundary layer mesh is needed in the near-wall region.Figure 3
Schematic diagram of computational field in FLUENT.
(a)
The boundary condition
(b)
The mesh of flow fieldSince the quasi-steady aerodynamic forces can describe 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. SSTk-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. During the simulation process, drag and lift coefficients of the cross section wall are monitored. Wind velocity is set to 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 that the data used correspond to the steady state.The Reynolds number of the iced contact wire isRe=21110 which belongs to the subcritical region. In the subcritical region, the vortex street of the wake region becomes turbulent gradually while the separation of boundary layer appears because of the strong inverse pressure gradient near the iced section wall. Figure 4 is the velocity contour maps of the flow field near the wire section wall under different angles of attack. From the figures the turbulent flow and the separation of boundary layer can be seen clearly.Figure 4
Velocity contour maps under different angles of attack.
(a)
0°
(b)
40°
(c)
90°
(d)
150°In accordance with the symmetry of cross section of iced contact wire, the drag and lift coefficients follow the following relationships.(3)Cd,wαw=Cd,w-αw,Cl,wαw=-Cl,w-αw.According to the time-averaged value of the calculated data, the relations between drag coefficient, lift coefficient, and angle of attack are depicted in Figure5.Figure 5
Relations betweenCd, Cl, and α.It is clear that when the angle of attack is near ±0° and ±180°, the drag coefficient reaches the minimum because that is when the windward side is the smallest, while the drag coefficient reaches the maximum when the angle of attack is near ±90° because that is when the windward side is the largest. As for the lift coefficient, the peaks appear when the angle of attack is near 40°, 170°, and −130°, and the troughs appear when the angle of attack is near −40°, −170°, and 130°. The changing rules of drag and lift coefficients are in agreement with the wind tunnel test data in [16].To further verify the results of CFD, aerodynamic coefficients of bluff body section such as a D-shape section and a square section of iced conductor are performed by FLUENT. The model parameters of D-shape section and square section are referred to in [23] and [24], respectively. Figures 6 and 7 show the comparison between the results obtained by CFD in this work and the wind tunnel test results obtained in [23, 24]. It can be seen from the pictures that the coefficients of drag and curves fitted from CFD and wind tunnel tests have uniform tendency.Figure 6
Comparison of drag and lift coefficients from CFD and wind tunnel tests of a D-shape section.
(a)
Drag coefficients from CFD
(b)
Drag coefficients from wind tunnel tests [23]
(c)
Lift coefficients from CFD
(d)
Lift coefficients from wind tunnel tests [23]Figure 7
Comparison of drag and lift coefficients from CFD and wind tunnel tests of a square section.
(a)
Drag coefficients from CFD
(b)
Drag coefficients from wind tunnel tests [24]
(c)
Lift coefficients from CFD
(d)
Lift coefficients from wind tunnel tests [24]


## 3. Vertical Aerodynamic Force Model

The galloping model of iced contact wire is shown in Figure8, in which U is the wind velocity, y˙w is the vertical vibration velocity, and Ur is the relative velocity between U and y˙w.Figure 8
Galloping model of iced contact wire in the vertical direction.The angle between aerodynamic drag force and central line of cross section of iced contact wire is described byαw, which is usually called angle of attack in the aerodynamics. Additionally, α0,w denotes the initial icing angle, and αr,w denotes the relative angle of attack. Therefore, the specific geometric relationship between them can be expressed as(4)αw=α0,w-αr,w,(5)αr,w=arctan⁡y˙wUr≈y˙wU.Since the vertical galloping velocity is far less than the wind velocity,Ur is approximately equal to U. The component of drag and lift forces on the axis yw can be expressed as(6)Fy,w=-Fd,wsin⁡αr,w+Fl,wcos⁡αr,w.By substituting (2) into (6), (6) can be further derived as(7)Fy,w=12ρairU2DwCy,w,where Cy,w obeys(8)Cy,w=-Cd,wαwsin⁡αr,w+Cl,wαwcos⁡αr,w.Aiming at the range of angle of attack-50°,50°, the drag and lift coefficients are gradually extracted and then used in the cube polynomial fitting; namely,(9)Cd,wαw=d3αw3+d2αw2+d1αw+d0,Cl,wαw=l3αw3+l2αw2+l1αw+l0,where the polynomial coefficients d0, d1, d2, and d3 are shown in Table 1. Correspondingly, the fitting curves of drag coefficient and lift coefficient are shown in Figure 9.Table 1
Polynomial coefficients of fitting curves.
Coefficient
Value
Coefficient
Value
d
0
0.4238
l
0
0
d
1
0
l
1
1.9841
d
2
2.0079
l
2
0
d
3
0
l
3
−2.0687Figure 9
Fitting curves.
(a)
Drag coefficient
(b)
Lift coefficientBy substituting (4) into (9), (8) is further expressed as(10)Cy,wαr,w=-d3α0,w-αr,w3+d2α0,w-αr,w2+d1α0,w-αr,w+d0sin⁡αr,w+l3α0,w-αr,w3+l2α0,w-αr,w2+l1α0,w-αr,w+l0cos⁡αr,w.Assuming the relative angle of attackαr,w is rather small at the very beginning of galloping, (10) can be expanded at αr,w=0 by the Taylor expansion method, so that(11)Cy,wαr,w=Cy,wαr,w=0+∂Cy,w∂αr,wαr,w=0αr,w+12!∂2Cy,w∂αr,w2αr,w=0αr,w2+13!∂3Cy,w∂αr,w3αr,w=0αr,w3+Oαr,w4,where O(αr,w4) denotes the terms proportional to αr,w4 and the higher powers of αr,w which are ignored.Combining (5) with (11), the vertical aerodynamic force on unit length of iced contact wire can be obtained according to (7).(12)Fy,w=a3,wy˙w3+a2,wy˙w2+a1,wy˙w+a0,w,where the expressions of a3,w, a2,w, a1,w, and a0,w are(13)a3,w=ρairDw2U16d3α0,w3+32l3+16d2α0,w2+l2+16d1-3d3α0,w+12l1+16d0-d2,(14)a2,w=ρairDw2-12l3α0,w3+3d3-12l2α0,w2+3l3-12l1+2d2α0,w+d1+l2-12l0,(15)a1,w=-ρairDwU2d3α0,w3+d2+3l3α0,w2+d1+2l2α0,w+d0+l1,(16)a0,w=ρairDwU22l3α0,w3+l2α0,w2+l1α0,w+l0.As for the iced messenger wire, its cross section is similar to that of iced contact wire; that is,λm=λw. Here the subscript m represents the messenger wire. Due to the same working environment and same order magnitude of diameters (Dw=14.4mm, Dm≈9.17mm), the Reynolds of iced messenger wire and iced contact wire are in the same interval (Rew=21110,Rem=13443).As a result, the drag and lift coefficients of iced contact wire and iced messenger wire are assumed to be the same. Similarly, the vertical aerodynamic force on unit length of iced messenger wire can be written as(17)Fy,m=a3,my˙m3+a2,my˙m2+a1,my˙m+a0,m,where the coefficient forms of a3,m, a2,m, a1,m, and a0,m are the same as those shown in (13)–(16).


## 4. Galloping Vibration of Catenary System

In this section, the equations of vertical motion of the simple catenary system under cross winds are derived. As shown in Figure10, the contact wire and messenger wire can be described by the simply supported Euler-Bernoulli beams. According to the mechanical characteristics, the droppers are simplified as the nonlinear springs, in which the tensile stiffness is far larger than the compression stiffness.Figure 10
Schematic diagram of simple catenary system under cross winds.The infinitesimal method is used to derive the equations of motion of the messenger and contact wires which are simplified as Euler-Bernoulli beams [28]. Figure 11 is the force diagram of the Euler-Bernoulli beam infinitesimal element in which f(x,t) denotes the vertical force per unit length, y(x,t) the vertical displacement, ρA(x) the mass per unit length, EI the bending stiffness, T the tensile force, M the bending moment, Q the shear force, and dx the length of the beam element.Figure 11
Diagram of the Euler-Bernoulli beam.
(a)
Bending configuration of an Euler-Bernoulli beam [28]
(b)
Free-body diagram for a beam elementDerived from Figure11(b), the force equation of motion in the vertical direction can be expressed as(18)Q-Q+∂Q∂xdx-Tsin⁡θ+Tsin⁡θ+∂θ∂xdx+fx,tdx=-ρAxdx∂2yx,t∂t2.Due to the reason thatθ is relatively small, there exists the assumption sin⁡θ≈θ. Therefore, (18) can be derived as(19)-∂Q∂xdx+T∂θ∂xdx+fx,tdx=-ρAxdx∂2yx,t∂t2.According to the mechanical characteristic and deformation relationship introduced in material mechanics, we can obtain that(20)θ=∂yx,t∂x,Q=∂M∂x,M=-EI∂2yx,t∂x2.Substituting (20) into (19), the equation of motion of an equal section Euler-Bernoulli beam can be written as(21)ρAy¨+EIy′′′′+Ty′′=-fx,t.In which y¨=∂2y/∂t2,y′′′′=∂4y/∂x4,y′′=∂2y/∂x2.For the iced messenger wire, the lumped external forces include the elastic forces acted by droppers and the elastic force acted by masts, and the distributed external forces include the damping force and the aerodynamic force. For the iced contact wire, the lumped external forces include the elastic forces acted by droppers and the distributed external forces include the damping forces and the aerodynamic forces.Thus, the vertical vibration equation of iced messenger wire under cross winds can be derived as(22)ρmAmy¨m+Cmy˙m+EImym′′′′+Tmym′′=-∑i=1bFm,i-∑i=1pFs,i+Fy,m,in which ym, ρm, Am, Cm are the vertical displacement, linear density, area of cross section, and the structural damping coefficient, respectively. EIm, Tm, Fm,i, b, and p denote bending stiffness, the tensile force acted on the iced messenger wire, the sum of elastic force given by ith dropper and gravity of droppers and clamps, the number of droppers, and the number of support springs.The elastic support force acted on the messenger wire (see Figure10) can be expressed as(23)Fs,i=ksym,s,i×δx-xs,i,where ks, ym,s,i, xs,i denote the support stiffness, the vertical displacement of the messenger wire at ith support, and the location of ith support on the x axis, and δ is the Dirac function.By repeating the above steps, the vertical vibration equation of iced contact wire under cross winds can be expressed as(24)ρwAwy¨w+Cwy˙w+EIwyw′′′′+Twyw′′=-∑i=1bFw,i+Fy,w.The coupling relations between contact wire and messenger wire are achieved by the droppers. For each dropper shown in Figure10, the elastic forces acted on the contact wire and messenger wire, respectively, obey(25)Fm,i=δx-xi12mig+fd,i,Fw,i=δx-xi12mig-fd,i,where xi is the location of ith dropper on the x axis; mi is the total mass of ith dropper and clamps.According to the mechanical characteristic, the elastic force ofith dropper can be written as(26)fd,i=kd,iym,i-yw,i+Δl0,kd,i=ktd,ym,i-yw,i+Δl0≥0kpd,ym,i-yw,i+Δl0<0,where ktd and kpd denote the tensile stiffness and compression stiffness, respectively, Δl0 is the initial elongation of dropper, and ym,i and yw,i denote the vertical displacements of messenger wire and contact wire at the position of ith dropper.In order to obtain the dynamic characteristic of the catenary system, the Galerkin method is adapted to discretize the partial differential equations. The main vibration mode functions are assumed asφmx and φwx, which satisfy the boundary conditions of contact wire and messenger wire. Therefore, the solutions to vibration equations of catenary system can be expressed as(27)ym=∑i=1nφm,ixqm,it(28)yw=∑i=1nφw,ixqw,it,where qm,it and qw,it are the mode coordinates of messenger wire and contact wire; n is the modal truncation order.According to [25], the main parameters of catenary system are shown in Table 2. In addition, the span of catenary is set to 10, the tensile stiffness of dropper is ktd=106N/m, the compression stiffness of dropper is kpd=0.5×104N/m, and support stiffness is ks=2.5×107N/m. Referring to the structure design of actual catenary, the vector position of dropper in each span obeys [3.375 10.125 16.875 23.625 30.375 37.125 43.875 50.625] (m).Table 2
Parameters of catenary model.
Tensile forceT (N)
Linear densityρA (kg/m)
Cross section areaA (mm2)
Span lengthl (m)
Messenger wire
14000
0.64
105.65
54
Contact wire
20000
1.43
260.58
54Since the catenary structure is extremely complicated, the analytical expression of main vibration mode functions can hardly be derived. Thus the finite element method is applied to analyze the catenary mode for obtaining the discretized main vibration mode, as shown in Figure12. The frequencies are listed in Table 3. One can observe that, in the figures of first fourth modes, mode type in each span is the same. Indeed, mode type in each span is the same of every ten modes. Correspondingly, frequencies of the ten-span catenary are distributed by groups of ten.Table 3
Frequencies of catenary.
Number
Frequency (Hz)
(1)
1.0495
(2)
1.0605
(3)
1.0780
(4)
1.1006
(5)
1.1267
(6)
1.1541
(7)
1.1800
(8)
1.2017
(9)
1.2161
(10)
1.2213
(11)
2.0981
(12)
2.1208
(13)
2.1562
(14)
2.2011
(15)
2.2520
(16)
2.3051
(17)
2.3561
(18)
2.3998
(19)
2.4301
(20)
2.4410Figure 12
Vibration mode shapes of the ten-span catenary.
(a)
First mode
(b)
Second mode
(c)
Third mode
(d)
Fourth modeOn this basis, (27) and (28) are substituted into (22) and (24), respectively. And then multiply both sides of the equations by φm,jx and φw,jx. Thus the ordinary differential equations of catenary system can be obtained by numerical integration. The detailed derivation is shown in the appendix.Therefore, the discrete vibration equations of the catenary system can be rewritten in the matrix form as follows:(29)Mq¨+Cl+Cnq˙+Kq=Q,where M is the mass matrix, Cl is the linear damping matrix generated by structural damping, Cn is the nonlinear damping matrix generated by aerodynamic force, K is the stiffness matrix, Q is the discrete force vector, and q is the modal coordinates vector.


## 5. General Results and Discussion

### 5.1. Stability Analysis of Catenary System

As illustrated in [27], the aerodynamical stability of the dynamic system is mainly determined by the sign of the net damping coefficient. For the case that the net damping coefficient is greater than zero, the catenary system is aerodynamically stable. Otherwise, the catenary system is aerodynamically unstable.It can be seen from (22) and (24) that the unstable conditions of iced messenger wire and iced contact wire, respectively, obey(30)Cw-a1,w<0,(31)Cm-a1,m<0.Due to the reason that the structural damping coefficients are positive, the necessary condition for the occurrence of galloping motion of iced contact wire can be derived from (15) and (30).(32)-ρairDwU2d3α0,w3+d2+3l3α0,w2+d1+2l2α0,w+d0+l1>0.In the above equation, the air density ρair, diameter of cross section Dw, and wind velocity U are positive. Therefore, (32) can be further reduced; namely,(33)-d3α0,w3+d2+3l3α0,w2+d1+2l2α0,w+d0+l1>0.Obviously, the necessary condition for the occurrence of galloping motion of iced contact wire is only affected by the drag and lift coefficients and initial icing angle. According to the parameters in Table1, the necessary condition of instability of iced contact wire satisfies(34)α0,w>43.4°.The above inequality is also the sufficient condition for the occurrence of galloping of iced contact wire when the structural damping is ignored.On the assumption that the drag and lift coefficients of messenger wire and contact wire are the same in Section3, (34) is also the necessary condition of instability of iced messenger wire.


### 5.2. Effect of Initial Icing Angle

Considering that the main range of galloping frequency is[0.1 (Hz), 3 (Hz)] [13], the modal truncation order of the ten-span catenary is set to 20 in this paper. The Runge-Kutta method is used to calculate the dynamic response of iced catenary system numerically. In 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.Keeping the parameters shown in Table2, the effect of initial icing angle on the dynamic characteristic is conducted in this section, where the structural damping is not taken into consideration.In the condition ofU=15m/s, when the initial icing angles are α0,m=43.3° and α0,w=43.3°, the vertical vibration response of the iced contact wire is shown in Figure 13. It is evident that the vertical vibration displacement gradually reduces to the static vertical deformation, which is mainly caused by the cross winds. Therefore, this phenomenon shown in Figure 13 is called stable vibration.Figure 13
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=43.3° and α0,w=43.3°.When the initial icing angles are changed toα0,m=43.4° and α0,w=43.4°, the vertical vibration displacement of the iced contact wire is shown in Figure 14. By comparing Figure 14 with Figure 13, it can be seen that with the increase of initial icing angle, the vertical response progresses to a kind of unstable vibration with low frequency and large amplitude rather than static deformation. This is the moment when the galloping vibration of iced contact wire occurs.Figure 14
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=43.4° and α0,w=43.4°.In the following, the effect of single initial icing angle (initial icing angle of contact wire or initial icing angle of messenger wire) on the dynamic characteristic of the system is discussed. For the case ofα0,m=40° and α0,w=45°, only the initial icing angle of contact wire meets the condition of instability. Correspondingly, the vibration response of iced contact wire is shown in Figure 15, in which the system maintains a small amplitude vibration after reaching the steady state. For the case of α0,m=45° and α0,w=40°, the similar dynamic phenomenon is shown in Figure 16. The main reason of the above phenomena is that the relative oscillations between contact wire and messenger wire are restricted by the droppers.Figure 15
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=40° and α0,w = 45°.Figure 16
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m = 45° and α0,w=40°.In addition, beating phenomenon in Figures14–16 can be observed. The similarity of frequencies of every ten modes can cause the modal coupling while the catenary is vibrating under the action of cross winds which directly leads to the beating phenomenon. And since the neglecting of structure damping in this case, the phenomenon is particularly obvious.When the initial icing angle of contact wire is equal to that of messenger wire, the relation between maximum vertical displacement and initial icing angle is depicted in Figure17. It is clear that the critical initial icing angle for the occurrence of galloping motion is 43.4°, which exactly meets the condition of instability of iced contact wire obtained in Section 5.1. At the interval of 40°–43.4°, the vibration of iced catenary system is stable. Meanwhile, the wind deviation is small and changes little with the increase of initial icing angle. At the interval of 43.4°–49°, the galloping vibration of iced catenary system occurs and the max displacement becomes larger with the increase of initial icing angle.Figure 17
Relation between maximum displacement and initial icing angle.


### 5.3. Effect of Structural Damping

As shown in expression (34), the stability condition is closely related to the structural damping. For most of actual engineering structures, the structure damping is usually characterized by the proportional damping in numerical calculations. Thus, the proportional damping matrix can be expressed as(35)Cl=αM+βK,where damping coefficients can be derived as [29](36)ζ1ζ2⋮ζn=121ω1ω11ω2ω2⋮⋮1ωnωnαβ,in which ωn is the circular frequency, ζn is the damping ratio, and n is the number of modes which is set to 20. The estimation of the Rayleigh coefficients can be performed by solving the overdetermined equation (36) by least-squares solution. According to [30], the damping ratio is set to 0.0013, and the coefficients are α=0.0125,β=0.0001.After considering the structure damping, the vertical vibration responses of contact wire with different initial icing angles are analyzed. As shown in Figure18, the contact wire is still stable in the conditions of α0,m=44.7° and α0,w=44.7°. This phenomenon suggests that the structural damping can raise the critical initial icing for unstable station.Figure 18
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=44.7° and α0,w=44.7°.When the initial icing angles areα0,m=44.8° and α0,w=44.8°, the system becomes unstable and the galloping oscillation appears, as shown in Figure 19. Therefore, the new critical initial icing angle with the effect of structural damping is about α0,m=44.8° and α0,w=44.8°. Overall, the structural damping has a positive contribution to the motion stability of iced catenary system.Figure 19
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=44.8° and α0,w=44.8°.


### 5.4. Effect of Wind Velocity

Taking the structural damping into consideration, the effect of wind velocity on the vibration response of iced catenary system is further discussed in this section. In the range of wind velocity 1 m/s–35 m/s, the Reynolds numbers of both iced contact wire and messenger wire are in the transcritical scope. When the wind velocity is in the range, the drag and lift coefficients are available.For the case ofα0,m=45° and α0,w=45°, the vertical vibration response of iced contact wire is calculated in the condition of U=12.4m/s, as shown in Figure 20. It is clear that the system response converges to a static 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 velocity is changed to U=12.5m/s. Correspondingly, a periodic vibration with low frequency appears in Figure 21, which indicates the instability phenomenon.Figure 20
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofU=12.4m/s.Figure 21
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofU=12.5m/s.The relation between maximum vibration in the steady-state and wind velocity is further analyzed, as shown in Figure22. When the initial icing angle is 45°, the critical wind velocity of stability is U=12.5m/s. When the wind velocity is smaller than the critical value, the iced catenary system is stable. While the wind velocity is larger than the critical value, the galloping vibration occurs and the vibration displacement turns to be larger with the increase of wind velocity.Figure 22
Relation between maximum vibration displacement and wind velocity.


## 5.1. Stability Analysis of Catenary System

As illustrated in [27], the aerodynamical stability of the dynamic system is mainly determined by the sign of the net damping coefficient. For the case that the net damping coefficient is greater than zero, the catenary system is aerodynamically stable. Otherwise, the catenary system is aerodynamically unstable.It can be seen from (22) and (24) that the unstable conditions of iced messenger wire and iced contact wire, respectively, obey(30)Cw-a1,w<0,(31)Cm-a1,m<0.Due to the reason that the structural damping coefficients are positive, the necessary condition for the occurrence of galloping motion of iced contact wire can be derived from (15) and (30).(32)-ρairDwU2d3α0,w3+d2+3l3α0,w2+d1+2l2α0,w+d0+l1>0.In the above equation, the air density ρair, diameter of cross section Dw, and wind velocity U are positive. Therefore, (32) can be further reduced; namely,(33)-d3α0,w3+d2+3l3α0,w2+d1+2l2α0,w+d0+l1>0.Obviously, the necessary condition for the occurrence of galloping motion of iced contact wire is only affected by the drag and lift coefficients and initial icing angle. According to the parameters in Table1, the necessary condition of instability of iced contact wire satisfies(34)α0,w>43.4°.The above inequality is also the sufficient condition for the occurrence of galloping of iced contact wire when the structural damping is ignored.On the assumption that the drag and lift coefficients of messenger wire and contact wire are the same in Section3, (34) is also the necessary condition of instability of iced messenger wire.


## 5.2. Effect of Initial Icing Angle

Considering that the main range of galloping frequency is[0.1 (Hz), 3 (Hz)] [13], the modal truncation order of the ten-span catenary is set to 20 in this paper. The Runge-Kutta method is used to calculate the dynamic response of iced catenary system numerically. In 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.Keeping the parameters shown in Table2, the effect of initial icing angle on the dynamic characteristic is conducted in this section, where the structural damping is not taken into consideration.In the condition ofU=15m/s, when the initial icing angles are α0,m=43.3° and α0,w=43.3°, the vertical vibration response of the iced contact wire is shown in Figure 13. It is evident that the vertical vibration displacement gradually reduces to the static vertical deformation, which is mainly caused by the cross winds. Therefore, this phenomenon shown in Figure 13 is called stable vibration.Figure 13
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=43.3° and α0,w=43.3°.When the initial icing angles are changed toα0,m=43.4° and α0,w=43.4°, the vertical vibration displacement of the iced contact wire is shown in Figure 14. By comparing Figure 14 with Figure 13, it can be seen that with the increase of initial icing angle, the vertical response progresses to a kind of unstable vibration with low frequency and large amplitude rather than static deformation. This is the moment when the galloping vibration of iced contact wire occurs.Figure 14
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=43.4° and α0,w=43.4°.In the following, the effect of single initial icing angle (initial icing angle of contact wire or initial icing angle of messenger wire) on the dynamic characteristic of the system is discussed. For the case ofα0,m=40° and α0,w=45°, only the initial icing angle of contact wire meets the condition of instability. Correspondingly, the vibration response of iced contact wire is shown in Figure 15, in which the system maintains a small amplitude vibration after reaching the steady state. For the case of α0,m=45° and α0,w=40°, the similar dynamic phenomenon is shown in Figure 16. The main reason of the above phenomena is that the relative oscillations between contact wire and messenger wire are restricted by the droppers.Figure 15
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=40° and α0,w = 45°.Figure 16
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m = 45° and α0,w=40°.In addition, beating phenomenon in Figures14–16 can be observed. The similarity of frequencies of every ten modes can cause the modal coupling while the catenary is vibrating under the action of cross winds which directly leads to the beating phenomenon. And since the neglecting of structure damping in this case, the phenomenon is particularly obvious.When the initial icing angle of contact wire is equal to that of messenger wire, the relation between maximum vertical displacement and initial icing angle is depicted in Figure17. It is clear that the critical initial icing angle for the occurrence of galloping motion is 43.4°, which exactly meets the condition of instability of iced contact wire obtained in Section 5.1. At the interval of 40°–43.4°, the vibration of iced catenary system is stable. Meanwhile, the wind deviation is small and changes little with the increase of initial icing angle. At the interval of 43.4°–49°, the galloping vibration of iced catenary system occurs and the max displacement becomes larger with the increase of initial icing angle.Figure 17
Relation between maximum displacement and initial icing angle.


## 5.3. Effect of Structural Damping

As shown in expression (34), the stability condition is closely related to the structural damping. For most of actual engineering structures, the structure damping is usually characterized by the proportional damping in numerical calculations. Thus, the proportional damping matrix can be expressed as(35)Cl=αM+βK,where damping coefficients can be derived as [29](36)ζ1ζ2⋮ζn=121ω1ω11ω2ω2⋮⋮1ωnωnαβ,in which ωn is the circular frequency, ζn is the damping ratio, and n is the number of modes which is set to 20. The estimation of the Rayleigh coefficients can be performed by solving the overdetermined equation (36) by least-squares solution. According to [30], the damping ratio is set to 0.0013, and the coefficients are α=0.0125,β=0.0001.After considering the structure damping, the vertical vibration responses of contact wire with different initial icing angles are analyzed. As shown in Figure18, the contact wire is still stable in the conditions of α0,m=44.7° and α0,w=44.7°. This phenomenon suggests that the structural damping can raise the critical initial icing for unstable station.Figure 18
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=44.7° and α0,w=44.7°.When the initial icing angles areα0,m=44.8° and α0,w=44.8°, the system becomes unstable and the galloping oscillation appears, as shown in Figure 19. Therefore, the new critical initial icing angle with the effect of structural damping is about α0,m=44.8° and α0,w=44.8°. Overall, the structural damping has a positive contribution to the motion stability of iced catenary system.Figure 19
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofα0,m=44.8° and α0,w=44.8°.


## 5.4. Effect of Wind Velocity

Taking the structural damping into consideration, the effect of wind velocity on the vibration response of iced catenary system is further discussed in this section. In the range of wind velocity 1 m/s–35 m/s, the Reynolds numbers of both iced contact wire and messenger wire are in the transcritical scope. When the wind velocity is in the range, the drag and lift coefficients are available.For the case ofα0,m=45° and α0,w=45°, the vertical vibration response of iced contact wire is calculated in the condition of U=12.4m/s, as shown in Figure 20. It is clear that the system response converges to a static 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 velocity is changed to U=12.5m/s. Correspondingly, a periodic vibration with low frequency appears in Figure 21, which indicates the instability phenomenon.Figure 20
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofU=12.4m/s.Figure 21
Vertical vibration response of the mid-point of 5th span contact wire in the condition ofU=12.5m/s.The relation between maximum vibration in the steady-state and wind velocity is further analyzed, as shown in Figure22. When the initial icing angle is 45°, the critical wind velocity of stability is U=12.5m/s. When the wind velocity is smaller than the critical value, the iced catenary system is stable. While the wind velocity is larger than the critical value, the galloping vibration occurs and the vibration displacement turns to be larger with the increase of wind velocity.Figure 22
Relation between maximum vibration displacement and wind velocity.


## 6. Conclusion

In this paper, the galloping vibration of iced catenary system under cross winds has been investigated. By FLUENT simulation, the lift and drag coefficients of quasi-steady aerodynamic force acted on the iced catenary system have been calculated. By fitting the discrete simulation data, the expression of the vertical aerodynamic force has been further derived. Then the necessary condition of system stability has been further obtained based on the Den Hartog galloping mechanism. Moreover, the vibration equations of iced catenary system have been solved by the Runge-Kutta method and the corresponding responses have been analyzed by the waveform. Meanwhile, the analysis of varying parameters, for instance, the initial icing angle, structural damping, and wind velocity, has been conducted. According to the analysis results, the following conclusions can be obtained.(1) As for vertical galloping vibration of iced catenary system, the critical condition of motion stability is mainly determined by initial icing angle, wind velocity, and structural damping.(2) For the case of stable vibration, the dynamic response of iced catenary system usually reduces to a static wind deviation. For the case of unstable vibration, the galloping motion occurs in the iced catenary system.(3) Suspended on the messenger wire, the stability of iced contact wire is improved when the initial icing angles of messenger and contact wires are different.(4) Increasing the structural damping can reduce the chance that the galloping motion occurs. In the actual engineering case, mechanical damping can be used to restrain the galloping oscillation of the catenary.



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*Source: 1634292-2017-10-22.xml*