Abstract
A dynamic model of a galloping transmission line able to describe for the coupling of its longitudinal, inplane, outofplane and torsional vibrations is established. It also considers the effects of geometrical nonlinearity and aerodynamic nonlinearity. By the static configuration, the reduced model is obtained. Then, the equations of motion are obtained through the Galerkin method. It contains two inplane, two outofplane and two torsional components. By numerical calculation, the maximum amplitudes at wind speeds are drawn and the galloping behavior of transmission line with thin ice accretions is analyzed. The obtained results show that the second galloping mode is more triggered. The doublemode galloping occurs in all motions, in which the maximum amplitude is bigger than in singlemode galloping. And the doublemode galloping presents the track of inclined ‘8’ in longitudinal direction.
1. Introduction
Galloping of iced transmission lines is a selfexcited vibration with low frequency and high amplitude, which is a serious problem in the safe operation of the entire power transmission system. Many efforts have been done to investigate the galloping. Den Hartog [1] established the vertical galloping model and proposed the vertical galloping mechanism. Luongo [2] and Jones [3] proposed the verticalhorizontal coupling model to account for the galloping behavior, which neglected the effect of torsion motion. Through a detailed aerodynamic wind tunnel study on iced transmission line, Nigol [4, 5] proposed that galloping was caused largely by the selfexcited torsional mechanism. Yu. P [6] investigated the verticaltorsional coupling system and proposed inertially coupled galloping mechanism. Hence, through experimental and theoretical analysis, it was verified that the torsion motion played a significant role in galloping behavior of transmission line. Zhang et al. [7] also established the model with verticaltorsional motions and studied the influence of the system parameters on the amplitude of galloping. Lou et al. [8] proposed a model to describe threedimensional galloping behavior and studied the bifurcation and stability behavior for nonlinear galloping of iced transmission lines. Huo et al. [9] proposed the model of a transmission line to describe for the coupling of its inplane, outofplane and torsional vibrations. Then, by the singularity theory, they investigated the effects of the parameters on system.
Most of the above studies used the Galerkin method, which is usually adopted to disperse the partial different equations and reserves the concerned modes. However, through the study on suspended cable, we can find that the symmetric inplane modes consist of antisymmetric vertical components and symmetric longitudinal components [10, 11]. Hence, the phenomenon of frequency crossover occurs, which causes the order exchange of symmetric and antisymmetric modes [12]. Based on this theory, Yu. P et al. [13] and Luongo et al. [14] adopted the galloping model of the coupling of inplane, outofplane and torsional vibrations, in which the longitudinal motion was simplified, to study the internal resonance behavior of conductor galloping and the effect of twist on nonlinear galloping respectively. However, the nonlinear dynamic investigations were only focused on the first mode, which neglected the higher mode.
Hence, the objective of the present paper is to investigate the effect of the higher mode on galloping. The dynamic model of the transmission line containing the coupling effects of the longitudinal, inplane, outofplane and torsional vibrations is established, which also considered the effects of geometrical nonlinearity and aerodynamic nonlinearity. Then, the first twomode equations of motion are obtained through the Hamilton’s principle and the Galerkin method. By numerical calculation, the galloping behavior is analyzed at wind speeds and the effects of the coupling mode on galloping behavior are also investigated.
2. The dynamic model
The transmission line is modeled as a body made of flexible cable with length $l$, in which the effect of the flexural rigidity is neglected due to a small sagtospan ratio. A crescentshaped thin ice accretion that is assumed to be uniform along the transmission line is considered. Both ends of the transmission line are considered to be fixed for the rigid body assumption of the tower [15]. The initial configuration of conductor, ${\mathrm{\Gamma}}_{0}$, which just goes by gravity of the conductor and the ice, and the dynamic configuration, $\mathrm{\Gamma}$, are shown in Fig. 1(a). The symbols $u\left(x,t\right)$, $v\left(x,t\right)$, $w\left(x,t\right)$, $\theta \left(x,t\right)$ denote the longitudinal, inplane, outofplane and torsional dynamic displacements respectively at time $t$. An infinitesimal length of the transmission line, dx, is considered and its dynamic displacement is illustrated in Fig. 1(b).
Fig. 1The schematic diagram of transmission line model: a) configuration, b) dynamic displacement
a)
b)
Through the multivariate Taylor series expansion, the dynamic strain can be obtained:
where $d{s}_{0}$ and $ds$ mean the undeformed arc length and the deformed arc length respectively. The catenary equation, ${y}_{0}$, is:
where $H$ is the horizontal initial tension of the transmission line; $m$ is the mass per unit length of the iced transmission line; $g$ is gravitational acceleration.
The potential energy of the transmission line is given by:
where ${T}_{0}$ is the initial tension of the transmission line, ${T}_{0}=H{\left(1+{y}_{0x}^{2}\right)}^{1/2}$; $EA$ and $GJ$ are the inplane and torsional stiffness. ${\epsilon}_{\theta}$ is the torsional strain of the transmission line, ${\epsilon}_{\theta}=\partial \theta /\partial {s}_{0}$.
By the aerodynamic model of the iced transmission line and the effect of the thin ice accretions [16], the kinetic energy and the virtual work are given by:
where $I$ is the mass moment of inertia per unit length at original point; ${F}_{u}$, ${F}_{v}$, ${F}_{w}$ and $M$ are the aerodynamic forces acting on the longitudinal, inplane, outofplane and torsional direction of the iced conductor respectively. Their expressions are obtained by the reference [6], given by:
where ${U}_{r}$ is relative wind velocity, ${U}_{r}=\sqrt{{\left(U\dot{w}\right)}^{2}+{\dot{v}}^{2}+D\dot{\theta}/2}$; $U$ and ${\rho}_{a}$ are the mean wind speed and the air density; $D$ is the diameter of a bare conductor; ${C}_{y}$,${C}_{z}$, and ${C}_{M}$ are the aerodynamic coefficients; and ${r}_{yi}$, ${r}_{zi}$ and ${r}_{Mi}$ ($i=$ 1, 2, 3) are the fitting coefficients obtained from the experimental data [17].
The equations of galloping are obtained through the Hamilton’s principle:
For a small $\theta $, the equations of motion are found to be:
3. Galerkin discretization
By the static configuration [6, 11], a reduced model of the transmission line containing the inplane, outofplane and torsional motions is derived from Eq. (7)(10). Then, a discrete model is found by applying Galerkin procedure:
where ${q}_{ij}\left(t\right)\left(i=v,w,\theta ,\mathrm{}\mathrm{}j=\mathrm{1,2}\right)$ are the dynamic displacements for the inplane, outofplane and torsional modes. ${\phi}_{ij}\left(x\right)$ are the trial functions characterized by orthogonal series [10, 11]. Then, the discrete equations of the reduced model are given by:
where ${c}_{ij},{\omega}_{g,ij}\left(i=v,w,\theta ,\mathrm{}\mathrm{}j=\mathrm{1,2}\right)$ are the damping ratios and natural frequencies respectively; ${g}_{ij,k}$ are the integral constants; ${F}_{{i}_{1}{j}_{1}}\left({\dot{q}}_{v1},{\dot{q}}_{v2},{\dot{q}}_{w1},{\dot{q}}_{w2},{q}_{\theta 1},{q}_{\theta 2},{\dot{q}}_{\theta 1},{\dot{q}}_{\theta 2}\right)\left({i}_{1}=v,w,\mathrm{}\mathrm{}{j}_{1}=\mathrm{1,2}\right)$ and ${M}_{\theta {j}_{1}}\left({\dot{q}}_{v1},{\dot{q}}_{v2},{\dot{q}}_{w1},{\dot{q}}_{w2},{q}_{\theta 1},{q}_{\theta 2},{\dot{q}}_{\theta 1},{\dot{q}}_{\theta 2}\right)$ are nonlinear aerodynamic terms of the inplane and outofplane and torsional motions.
4. Numerical results and discussions
The numerical calculation of Eq. (12)(17) is performed. The selected parameters are shown in Table 1 [17]. At the different wind speeds, the maximum amplitude curve of the first two modes is shown in Fig. 2. I1st, O1st and T1st denote the first mode of the inplane, outofplane and torsional galloping respectively. Meanwhile, I2st, O2st and T2st denote the corresponding second modes.
From Fig. 2, we can see that the first two modes in three directions all show the limitamplitude galloping. Taking the inplane galloping as an example, with the increase of the wind speed, the first two modes are triggered successively. Meanwhile, the doublemode galloping area is found (38 m/s). In addition, under the selected parameters in Table 1, the maximum galloping amplitude of the second mode is bigger than that of the first mode. Similarly, the galloping area triggered in second mode (3 m/s16 m/s) is wider than that in first mode (18 m/s).
The doublemode galloping should be got more attention. Then, the spatial galloping profiles at 5 m/s are shown in Fig. 3 containing the projections in three directions. Duo to the coupling galloping, no fixed nodes or peaks are presented in the galloping profiles. Considering that the second mode is on the node at $l$/2 span, only the first mode is contained, which is shown in Fig. 3(b), (c). However, there are two modes at 3$l$/4 span, including the first mode (0.550 Hz) and the second mode (0.909 Hz), which is shown in Fig. 3(d), (e). Furthermore, from Fig. 3(d), we can see that the amplitudes present the superposition or counteract, the maximum of which is bigger than that in singlemode area. Hence, the antisymmetrical mode of vibration is presented. Meanwhile, the galloping track shows the inclined ‘8’ motion in doublemode area while it shows the elliptical motion in singlemode area along the longitudinal direction
Table 1Parameters of transmission line
Parameter  Data  Parameter  Data  Parameter  Data 
$H$ (N)  30000  $A$ (mm^{2})  423.24  ${\rho}_{a}$ (kg/m^{3})  1.29 
$l$ (m)  125.88  $GJ$ (N m^{2}/rad)  101  ${\xi}_{vj}$ ($j=$ 1,2)  0.01, 0.012 
$m$ (kg/m)  2.379  $I$ (kg/m)  0.3344×10^{3}  ${\xi}_{wj}$ ($j=$ 1,2)  0.03, 0.036 
$E$ (N/m^{2})  4.78×10^{10}  the initial angle of ice (${\theta}_{0}$)  40°  ${\xi}_{\theta j}$ ($j=$ 1,2)  0.10, 0.12 
Note: ${\xi}_{vj}$, ${\xi}_{wj}$ and ${\xi}_{\theta j}$ denote the damping ratios of inplane, outofplane and torsional motions respectively 
Fig. 2The maximum amplitude of the first two modes
Fig. 3The doublemode galloping at 5 m/s: a) the spatial galloping profiles, b) the time history at l/2 span, c) the amplitudefrequency responses at l/2 span, d) the time history at 3l/4 span, e) the amplitudefrequency responses at 3l/4 span
a)
b)
c)
d)
e)
5. Conclusions
For the iced transmission line, the continuous elastic dynamic model is established, containing the longitudinal, inplane, outofplane and torsional motions. Then, through the static configuration and Galerkin Discretization, the nonlinear equations of motions are obtained, which contain two inplane, two outofplane and two torsional components. The galloping behavior is analyzed as follows:
1) The first two modes present the limitamplitude galloping. As the wind speeds increases, the first mode, the coupling mode and the second mode will be appeared in turn.
2) The galloping area of the second mode triggered by the wind speed is wider than that of the first mode. And the maximum amplitude is even bigger in doublemode galloping area than in singlemode galloping area.
3) Once the coupling mode occurs, the galloping track presents the inclined ‘8’ motion along the longitudinal direction. However, it shows the elliptical motion in singlemode area.
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The authors gratefully acknowledge the support of the National Natural Science Foundation of China (51479136 and 51808389), the Project of Tianjin Municipal Transportation Commission (2018B33), the Project of Tianjin Natural Science Foundation (17JCYBJC18700 and 18JCQNJC08000).