# Investigation on the Accuracy of Approximate Solutions Obtained by Perturbation Method for Galloping Equation of Iced Transmission Lines

Authors: Xiaohui Liu; Shuguang Yang; Guangyun Min; Mengqi Cai; Chuan Wu; Yantao Jiang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Perturbation method is a commonly used method to solve galloping equation of iced transmission lines, but few scholars have studied the influences of perturbation method on the accuracy of approximate solutions of the galloping equation. In order to analyze the accuracy of approximate solutions obtained by perturbation method for galloping equation of iced transmission lines, the partial differential galloping equation of iced transmission lines with quadratic and cubic nonlinear terms is obtained firstly. Then, the partial differential galloping equation is transformed into ordinary differential galloping equation by Galerkin method. Finally, the approximate solutions of the partial differential galloping equation are obtained by averaging method and first-order, second-order, third-order, and fourth-order multiple scales methods, and the results obtained by these methods are compared systematically. By comparing the numerical solutions and the approximate solutions obtained by averaging method, it can be found that, with the increasing in wind velocity and Young’s modulus of iced transmission lines, the nonlinearity of the system would strengthen and the drift of the vibration center of the system would also increase. The larger the drift is, the greater the error between the approximate solutions obtained by averaging method and the numerical solutions will be. And when the wind velocity reaches 32 m/s, the error would arrive at 17.321%. By comparing the numerical solutions and the approximate solutions obtained by the first-order, the second-order, the third-order, and the fourth-order multiple scales methods, it can be concluded that the first-order multiple scales method is less complex computationally. The accuracy of approximate solutions obtained by the fourth-order multiple scales method is better than that obtained by the first-order, the second-order, and the third-order multiple scales methods, and the error between the approximate solutions obtained by the fourth-order multiple scales method and the numerical solutions is less than 0.639%. The conclusions obtained in this paper would be helpful to the solutions of galloping equation of iced transmission lines and could also give some references to practical engineering.


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## Body

## 1. Introduction

As an important energy transmission channel, high-voltage transmission lines are related to national social and economic development as well as people's lives closely, so it is necessary to ensure the effective and normal operation of high-voltage transmission lines [1]. The galloping of iced transmission lines has always been a hot topic and iced transmission lines study is of great application value [2, 3]. As is known to us, the cross section of transmission lines would change from circular cross section to crescent or D-shape cross section under complex climatic conditions such as snow-rime, downburst, typhoon, and hurricane [3–5]. And, when iced transmission lines with noncircular cross section are subjected to horizontal wind loads, the iced transmission lines would be galloping. And the galloping of iced transmission lines is a self-excited vibration with low frequency and large amplitude [6], which may cause flashover, short circuit, frequent tripping, broken strand or wire, and other accidents [7, 8]. In order to investigate the galloping of iced transmission lines, as early as 1932, Den Hartog [9] proposed the vertical galloping mechanism and found that the horizontal vibration amplitude of iced transmission lines was much smaller than the vertical amplitude, indicating that galloping mainly occurred in the vertical direction. Based on this concept, only the galloping in the vertical direction for iced transmission lines is considered in this paper.In fact, the span length of transmission lines is much larger than the diameter of the transmission lines, so the transmission lines belong to the cable element structure. And since the transmission lines belong to the cable element structure, the partial differential galloping equation of transmission lines is also derived based on the cable element structure [10]. The partial differential vibration equation of cable element structure can be transformed into ordinary differential vibration equation by Galerkin method [11, 12]; then, Runge-Kutta method, averaging method, and multiple scales method can be used to solve the ordinary differential vibration equation. By solving the vibration equation, the vibration characteristics of cable element structure can be studied. In addition, vibration characteristics of cable element structure can be studied through experiments [13, 14]. However, for studying the nonlinear vibration characteristics of cable element structure, the perturbation method is used mostly. For example, Luongo [15] used the multiple scales method to obtain the periodic and quasiperiodic solutions of the suspension cable and studied the influences of turbulences on structural characteristics. Nielsen used averaging method to obtain the approximate solutions of cable subjected to superharmonics excitation and discussed the influences about combined harmonics and superharmonics effect on vibration characteristics of cable [16]. For the nonlinear dynamic behavior of a string-beam coupled system subjected to parametric and external excitation, the case of 1 : 2 internal resonance between the modes of the beam and string—principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string—is considered in[17], and Zhang [18, 19] analyzed global bifurcation and chaos by the techniques of phase portrait, waveform, and Poincaré map. The global perturbation method is employed to analyze the global bifurcations and chaotic dynamics of the string-beam coupled system in [20]. Yao [21] introduced how to employ the extended Melnikov method to analyze the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Based on the averaging equation obtained by the perturbation method, the nonlinear dynamics of the string-beam coupled system is studied in [17–21]. In addition, Luongo [22] used the first-order multiple scales method to analyze the amplitude-frequency function of cable under the external excitation and obtained a bifurcation diagram. Luongo [23, 24] established a model of cable with two DOFs considering the effects of static swing and dynamic torsion and used second-order multiple scales method to solve the corresponding nonlinear ordinary differential vibration equation of the model. Benedettini [25] and Srinil [26] used second-order multiple scales method to analyze the linear stability of stay-cable under main external resonance and also studied the stability of stay-cable under 1 : 1 internal resonance. Based on second-order multiple scales method, Yao [27] studied the vibration characteristic of nonlinear oscillations system with internal resonance of 1 : 1 and 2 : 1. Benedettini [28] also used the fourth-order multiple scales method to solve the nonlinear partial differential equation of shallow cable under weak periodic excitation and analyzed the multivalue effect of the amplitude-frequency function of shallow cable.The above research papers mainly used the averaging method and the first-order, second-order, third-order, and fourth-order multiple scales methods to solve the nonlinear ordinary differential equation. These methods all fall under the category of perturbation method, which is also a commonly used method for solving galloping equation of iced transmission lines. However, few scholars have studied the influences of perturbation method on the accuracy of approximate solutions of galloping equation of transmission lines. Based on this concept, this paper systematically studied the accuracy of the approximate solutions obtained by perturbation method. Firstly, the galloping governing equation of iced transmission line under wind excitation is obtained by physical modeling. The quadratic and cubic nonlinear restoring force and the Rayleigh damping in the governing equation have constituted the hysteretic nonlinear restoring force, which is widely representative. Next, the approximate analytical solution of the governing equation is obtained by using the averaging method and the different-order’s multiple scales method. Finally, the accuracy of the analytical solution is obtained by comparing the analytical solution with the numerical solution. Through the above analysis, this paper studied the influence of quadratic restoring force (q2) on the galloping system of transmission lines for the first time and found that the quadratic restoring force makes the vibration center of the system shift and the magnitude of the restoring force is directly related to drift of the system. It is also found in this paper that the existence of quadratic restoring force also has affected the use of the perturbation method. The approximate analytical solution of the averaging method cannot reflect the drift of the vibration center of the system, but the approximate analytical solution of the multiple scales method (first-order, second-order, third-order, and fourth-order) can reflect the drift. In addition, this paper also analyzed the accuracy of different-order’s multiple scales method for solving weakly nonlinear systems. As the condition of the system nonlinearity (damping force and restoring force) increasing with the changing of physical and aerodynamic parameters, it is found that the accuracy of the fourth-order multiple scales method is higher than that of the lower-order multiple scales method. From the theoretical point of view, the conclusion of this paper can give a reasonable explanation for the phenomenon of the drift of vibration center of the system. In practical application, the conclusion of this paper can guide researchers and engineers to use reasonable perturbation method to study weak nonlinear system. More importantly, the conclusions obtained in this paper would be helpful to the solutions of galloping equation of iced transmission lines and could also give some references to practical engineering.


## 2. Dynamic Model of Iced Transmission Lines

Most of high-voltage transmission lines in plain areas can be equivalent to equal-height transmission lines. Therefore, single-span equal-height transmission lines with two fixed supports are considered in this paper. The mechanical model of single-span iced transmission lines is established as shown in Figure1, and it is considered that the wind is along z-axis direction as shown in Figure 2, which is the most dangerous condition. The transmission lines are assumed to be homogenous and linearly elastic with negligible torsion, bending, and shear stiffness. The initial static equilibrium configuration of the transmission lines under gravity is ξ1, and when the static equilibrium configuration of the transmission lines changes under the action of other external loads, the corresponding equilibrium configuration becomes ξ2 (dynamic equilibrium configuration).Figure 1
The model of single-span equal-height transmission lines.Figure 2
The model of cross section of iced transmission lines. (a) Physical model. (b) Force analysis model.
(a)(b)Figure1 is the model of single-span equal-height transmission lines, and, in Figure 1, u (s, t) and vs,t are the displacements measured from the dynamic equilibrium configuration in x-axis and y-axis directions, respectively; and the associated static equilibrium configuration of transmission lines can be described through the parabola y = 4d[s/L-(s/L)2] [29], in which s is the curvilinear abscissa, l is the span length, and d is the sag.In order to facilitate the study of the aerodynamic loads of iced transmission lines, it is assumed that the ice is changeless along the transmission lines and the iced shape is a common crescent iced shape.Figure2 is the model of cross section of iced transmission lines, Figure (a) is the physical model, and Figure (b) is the force analysis model. In Figure 2, O1z2 is the axis of symmetry of the model of cross section, O1z1 is the direction in which the horizontal wind acts on, O1z is the horizontal axis, α is wind attack angle, α0 is initial wind attack angle, αt is relative wind attack angle, U is horizontal wind velocity, U0 is relatively wind velocity, and v˙ is the vertical galloping velocity.From Figure2, the following can be obtained: (1)tanα=v˙U≈α.The relative wind acts on the iced transmission lines, resulting in an air drag (FD) along the relative wind direction and an upward air lift (FL) perpendicular to the relative wind direction; therefore, the following can be obtained: (2)Fy=FLcosα−FDsinα.Considering small deformation, that is, sin (α)≈α, cos (α)≈1, (2) can be simplified as follows:(3)Fy=FL−αFD.According to fluid-induced vibration theory, the expressions ofFL and FD can be listed [30]:(4a)FL=12ρU2DCL,(4b)FD=12ρU2DCD,where CL is the coefficient of aerodynamic lift, CD is the coefficient of aerodynamic drag, ρ is the air density, and D is the diameter of transmission lines.Substituting (4a) and (4b) into (3), based on Taylor’ law, the aerodynamic coefficients in y-axis direction can be fitted with a cubic curve:(5)Cy=A′α+B′α3+C′α2+C0,whereA′, B′, C′, and C0 are aerodynamic coefficients related to the aerodynamic loads.According to (2)–(4a) and (4b), the following can be obtained: (6)Fy=12ρU2DCy.According to (5)–(6), the following can be obtained: (7)Fy=a˜v˙+b˜v˙3+c˜v˙2+d˜,where(8)a˜=ρUDA′2,b˜=ρDB′2U,c˜=ρDC′2,d˜=ρU2DC02.The initial wind attack angle and icing shape of transmission lines remain unchanged, and the quasistatic assumption is adopted; then,A′, B′, C′, and C0 are constants. Since the density ρ and the diameter D are also constants, the coefficient (c˜) is a constant. The fitting coefficient (c˜) does not change with the wind velocity (U), so coefficient (c˜) is also a constant term. Therefore, the coefficient (c˜) of aerodynamic load is not considered in this paper. In addition, the fitting coefficient (d˜) does not change with the vertical wind velocity (v˙), and the force of aerodynamic coefficient (d˜) is a constant. Then, the coefficient (d˜) has nothing to do with the generation of transmission line galloping. Similarly, the coefficient (d˜) of aerodynamic load is not considered in this paper.Den Hartog [9] found that the horizontal vibration amplitude of iced transmission lines was much smaller than the vertical amplitude, indicating that galloping mainly occurred in the vertical direction. Based on this concept, only the galloping in the vertical direction for iced transmission lines is considered in this paper. And according to [28–30], the governing equation in the vertical direction of iced transmission lines is(9)Hv′+ESy′+v′∫0ly′v′+v′22dx′+Fy−μv˙=mv¨,where H is the tension of the transmission lines, E is Young’s modulus of the transmission lines, and S is the cross-sectional area of transmission lines. v′is the first derivative of the vertical motion function with respect to x, y′is the first derivative of the parabolic equation with respect to x, v˙ and v¨ are the first derivative and second derivative of the vertical motion function with respect to time t, respectively, μ represents the structural damping, and m represents the self-weight per unit unstretched length.


## 3. Numerical Solutions

The displacementvx,t in (9) can be written as(10)vx,t=fxqt,where f (x) is the mode shapes and q (t) is the time function.Based on Galerkin method [31], substituting (10) into (9) can obtain the nonlinear ordinary differential equation:(11)q¨+ω2q+c1q2+c2q3+μ∗−c3q˙+c4q˙3=0.Equation (11) represents the galloping equation of transmission lines under the action of horizontal wind, and the parameters in (11) are(12)ω2=1mI0Im,c1=12dESmlI1Im,c2=ES2mlI2Imμ∗=μm,c3=a˜m,c4=b˜mIbIm,where(13)I0=H∫0lf′·f′dx+64d2ESl5∫0lfdx2,I1=∫0lf′2dx∫0lfdx,I2=∫0lf′2dx2,Im=∫0lf2dx,Ib=∫0lf4dx.Equation (11) contains the first-order and third-order nonlinear damping terms, the second-order and the third-order nonlinear restoring force term of q. The coefficient of the aerodynamic loads in the system includes the first-order and third-order terms, which is consistent with damping term of the Rayleigh equation (Rayleigh damping). The aerodynamic loads are presented in the form of the Rayleigh damping in the nonlinear galloping equation. Moreover, the nonlinear restoring force term is consistent with the Duffing equation. Therefore, (11) can be regarded as the combined form of the Duffing equation and the Rayleigh equation.For the physical parameters and aerodynamic parameters of iced transmission lines, refer to [32] shown in Table 1.Table 1
The physical parameters of transmission line.
ParameterSymbolUnitsValueStrainHN30 000SpanLm125.88Young's modulusEN/mm247 803.3DiameterDm0.0286Mass per unit lengthmKg/m2.379Traverse areaSmm2423.24Air mass densityρKg/m31.2929Wind velocityUm/s10.0Sagdm1.5432Vertical dumpingμ/0.0005Aerodynamic parametersA′/−0.1667Aerodynamic parametersB′/8.3581Substituting the physical parameters and aerodynamic parameters in Table1 into (11) and using the Runge-Kutta function [33–36] in MATLAB, we can obtain numerical solutions of the galloping equation. Figure 3 is the displacement response curve obtained by numerical method, Figure 3(a) is the time history of displacement curve obtained by numerical method, Figure 3(b) is the time history of velocity curve obtained by numerical method, and Figure 3(c) is the phase diagram obtained by numerical method.Figure 3
Displacement response curve obtained by numerical method. (a) Time history of displacement curve. (b) Time history of velocity curve. (c) Phase diagram.
(a)(b)(c)It can be seen from Figure3 that vibration amplitude and vibration velocity would increase with time gradually and vibration amplitude and vibration velocity tend to be stable after 850 s. The vibration amplitude and vibration velocity generate a stable limit cycle in Figure 3(c). The maximal vibration amplitude q = -0.57 m (0.51 m) in the vertical direction, which indicates that the vibration center of system is not symmetric about the equilibrium point. And the nonlinearity of the transmission lines leads to the drift of vibration center in the process of galloping.


## 4. The Approximate Solutions of Averaging Method

The approximate solutions can be easily got by averaging method and it is more convenient to calculate the partial differential galloping equation. Quoting solution form of small parameter in [16], (11) can be written as(14)q¨+ω2q=−εc1q2+c3q˙+c2q3+μq˙+c4q˙3,where ε is a small but finite parameter. Because (14) can be seen as a weakly nonlinear autonomous system, the vibration solutions of (14) are as follows:(15a)q=a^cosωt+θ,(15b)q˙=−a^ωsinωt+θ,where a^t is an amplitude function and θ (t) is a phase function.The amplitude function can be obtained by averaging method, which is(16)a^=11/a02+3ω2c4/4c3+μeεc3+μt−3ω2c4/4c3+μ.In the same way, the phase function can also be obtained by averaging method(17)θ=θ0+3c2εt81/a02+3ω2c4/4c3+μeεc3+μt−3ω2c4/4c3+μ.According to (15a) and (15b)-(17), the approximate solutions of (14) are(18)q=11/a02+3ω2c4/4c3+μeεc3+μt−3ω2c4/4c3+μcosωt+3c2εt81/a02+3ω2c4/4c3+μeεc3+μt−3ω2c4/4c3+μ.In (18), a0 represents the initial disturbance of the transmission lines. Taking the derivative of (18) with respect to t, the following can be obtained: (19a)q˙=p0+n0a^2a^cosωt+θ−ω+g0a^2a^sinωt+θ,where(19b)p0=−εc3+μ2,n0=−3εω2c48,g0=3εc28ω.In order to more intuitively reflect the accuracy of the approximate solutions of averaging method, Figure4 is the comparison curves of numerical solutions and approximate solutions of averaging method.Figure 4
Comparison curves of numerical solutions and approximate solutions of averaging method. (a) Time history of displacement curve. (b) Phase diagram.
(a)(b)As shown in Figure4, there are some differences in value of amplitudes (q) between two curves. For example, the phase diagram in Figure 4 does not overlap, and the vibration centers of the two curves do not overlap either. The point with coordinates (0, 0) in Figure 4 represents the vibration center of the approximate solutions of the averaging method, and the point with coordinates (−0.02, 0) represents the vibration center of the numerical solutions. To this end, the curve of the approximate solutions of the averaging method is symmetrical about the coordinate center, and the phase diagram of the numerical method is not symmetrical about the coordinate center. In other words, the approximate solutions of the averaging method do not reflect the drift of the vibration center of the transmission lines.


## 5. Approximate Solutions of Multiple Scales Method

### 5.1. Fourth-Order Multiple Scales Method

The multiple scales method is a commonly used to solve galloping equation of iced transmission lines. According to (1)–(10), the nonlinear galloping equation of the transmission lines under steady wind velocity is established, that is, (11):(20)q¨+ω2q+c1q2+c2q3+μ∗−c3q˙+c4q˙3=0,where, due to the inclusion of quadratic and cubic nonlinear terms, the aerodynamic coefficient of the aerodynamic loads in the system also includes the first-order and third-order terms. It is generally considered that the damping and nonlinear terms have little effect on the system. To this end, the solution form of small parameter is(21)q=εq¯,q˙=ε2q.¯,μ∗q˙=ε3μ¯q.¯,q˙3=ε5q.¯3,where q¯ ,μ¯, and q˙¯ are new variables of the system.In order to make the equation easy to express,q,μ, and q˙ are still used to represent q¯, μ¯, and q˙¯. Substituting (21) into (20), we can obtain (22)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε4c4q˙3=0.The fourth-order approximate solutions have been discussed, and setq as(23)q=q0+εq1+ε2q2+ε3q3+ε4q4.A new independent time variableTn=εnt(n = 0, 1, 2, 3, 4) was obtained by multiple scales method, in which n represents the order of dividing the time variable. The larger the value of n, the greater the amount of calculation and the complexity. Then, let the partial differential operator symbol for deriving t be(24)DT=D0T0+εD1T1+ε2D2T2+ε3D3T3+ε4D4T4Substituting (23) and (24) into (22) and equating coefficients of like powers of εn (n = 0, 1, 2, 3, and 4) lead to the following linear ordinary equations, respectively:(25a)D02q0+ω2q0=0,(25b)D02q1+ω2q1=−2D0D1q0−c1q02,(25c)D02q2+ω2q2=−D12q0+2D0D2q0+2D0D1q1+2c1q0q1+c2q03+μ−c3D0q0,(25d)D02q3+ω2q3=−D12q1+2D0D3q0+D1D2q0+D0D2q1+D0D1q2+c1q12+2q0q2+3c2q02q1+μ+c3D0q1+D1q0,(25e)D02q4+ω2q4=−D12q2+D22q0+2D0D4q0+D1D3q0+D0D3q1+D1D2q1+D0D2q2+D0D1q3+2c1q1q2+q0q3+3c2q02q2+q0q12+μ+c3D1q1+D2q0+D0q2+c4D03q0,where Dk represents the partial derivative of Tk; the solutions of (25a) can be written in this form(26)q0=AT1,T2,T3,T4eiwT0+A¯T1,T2,T3,T4e−iwT0,where A can be set as(27)AT1,T2,T3,T4=12a4T1,T2,T3,T4expiβ4T1,T2,T3,T4,where ai (t) and βi (t), respectively, represent the amplitude function and phase function of the system (i = 1, 2, 3, 4). Substituting (26) into (25a)-(25b), the following can be obtained: (28a)D1A=0,(28b)q1=c13ω2A2e2iwT0−c1ω2AA¯+cc.Substituting (26), (28a), and (28b) into (25c), we obtain(29a)D2A=3c22ω−5c123ω3iA2A¯−12μ+c3A+cc,(29b)q2=c1212ω4+c28ω2A3e3iwT0+cc.Substituting (26) and (28a)–(29b) into (25d), we obtain(30a)D3A=0,(30b)q3=c19ω32c3−2u−4c32A2e2iωt+59c1354ω6−31c1c212ω4A3A¯e2iωt+c1354ω6+c1c212ω4A4e4iωt+5c1c2ω4−19c139ω6A2A¯2+cc.Substituting (26) and (28a)-(30b) into (25e), we obtain(31)D4A=−485c14108ω7+15c2216ω3−173c12c212ω5iA3A¯2+4c12c329ω4−2c12c39ω4+2μc129ω4−3ω2c42A2A¯−μ28ω+c34ω+c328ωiA+cc.Combining (24) and (27)-(31) to obtain the averaging equation about amplitude and phase of the system,(32a)a˙4=−ε22u+c3a+c12c329ω4−c12c318ω4+μc1218ω4−3ω2c48ε4a3,(32b)β˙4=ε23c28ω−5c1212ω3a2−ε4485c141728ω7+15c22256w3−173c12c2192ω5a4−ε4μ28w+c34ω+c328ω.After integrating (32a) and (32b) over time, the following can be obtained:(33a)a4=11/a02+n/me−2mt−n/m.(33b)β4=β0−jhm22n3lnma02e−2mt+ne−2mt−n+jhm24nma02e−2mt+ne−2mt−n−2−jhm3n3t−kt−jhm22n1+mna02ma02+nma02e−2mt+ne−2mt−n−1.In (33a)-(33b), β0 represents the integral constant, and other parameters are(33c)m=−ε22μ+c3,k=ε4μ28ω+c34ω+c328ω,n=c12c329ω4−c12c318ω4+μc1218ω4−3ω2c48ε4,j=ε23c28ω−5c1212ω3,h=485c141728ω7+15c22256ω3−173c12c2192ω5ε4.Combining (23), (26), (27), and (33a)–(33c) to obtain the expression of zero-order amplitude,(34)qε0=a4·cosψ4.Combining (23), (26), (27), (28b), and (33a)–(33c) to obtain the expressions of first-order amplitude,(35)qε1=a4·cosψ+ε6w2c1a42cos2ψ−3.Combining (23), (26), (27), (28b), (29b), and (33a)–(33c) to obtain the expressions of second-order amplitude,(36)qε2=a4·cosψ+ε6w2c1a42cos2ψ−3+ε2c1248w4+c232w2a43·cos3ψ.Combining (23), (26), (27), (28b), (29b), (30b), and (33a)–(33c) to obtain the expressions of third-order amplitude,(37)qε3=a4·cosψ4+ε6ω2c1a42cos2ψ4−3+ε2c1248ω4+c232ω2a43·cos3ψ4+ε3−c19ω3u+c3a42·cos2ψ4+59c13432ω6−31c1c296ω4a44·cos2ψ4+c13432ω6+c1c296ω4a44·cos4ψ4+5c1c28ω4−19c1372ω6a44,where ψ4=ωt+β4.Equations (34), (36), and (37), respectively, correspond to approximate solutions of zero-order (qε0), second-order (qε2), and third-order (qε3) amplitude of multiple scales method. In order to obtain the accuracy of (34), (36), and (37), the time history displacement curves and phase diagram about the numerical solutions and the approximate solutions of multiple scales method are compared. Figure 5 is the comparison curves about numerical solutions and the approximate solutions of multiple scales method.Figure 5
Comparison curves of numerical solutions and approximate solutions of multiple scales method. (a) Zero-order expression of amplitude (qε0). (b) Second-order expression of amplitude (qε2). (c) Third-order expression of amplitude (qε3).
(a)(b)(c)As shown in Figure5, the time history displacement curve obtained by multiple scales method has the same phenomenon as that obtained by averaging method, which has shown that the expressions of zero-order amplitude cannot accurately describe the vibration amplitude of the nonlinear galloping equation. However, the expressions of second-order amplitude and the expressions of third-order amplitude are very accurate in describing the vibration amplitude of nonlinear galloping equation.


### 5.2. Third-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of third-order small parameter, letε4c4q˙3in (22) beε3c4q˙3; then(38)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε3c4q˙3=0.Similarly, using the multiple scales method to solve (38), the following can be obtained:(39a)a3=11/a02+n3/m3e−2m3t−n3/m3,(39b)β3=β0−m3kε2n3t−ε28ωc32+ε3μc34ωt−kε22n3lnm3a02e−2m3t+n3e−2m3t−n3,where k3=3c2/8ω−5c12/12ω3;m3=−εc3/2+ε2μ/2+ε3c32/32ω2;n3=ε31/18ω4c02c3−3/8c4ω2,The approximate solutions of multiple scales method of the solution form of third-order small parameter (T0, T1, T2, and T3) are(40)qε0,ε1,ε2,ε3=a⋅cosψ3+εa2c16ω2⋅cos2ψ3−c12ω2+ε2c1248ω4+c232ω2a3⋅cos3ψ3+19ω3a2c1c3⋅sin2ψ3+ε3c19ω3sin2ψ3−c1c3254ω4cos2ψ3a2+59c13432ω6−31c1c296ω4a4cos2ψ3+ωc432+3c2c3128ω3+17c12c3576ω5a3sin3ψ3+c1c296ω4+c13432ω6a4cos4ψ3+5c1c28ω4−19c1372ω6a4,where ψ3=ωt+β3.


### 5.3. Second-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of second-order small parameter, letε4c4q˙3in (22) beε2c4q˙3; then,(41)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε2c4q˙3=0.Similarly, using the multiple scales method to solve (46), the following can be obtained:(42a)a2=11/a02+n2/m2e−2m2t−n2/m2,(42b)β2=β0−m2k2ε2n2t−k2ε22n2lnm2a02e−2m2t+n2e−2m2t−n2,where m2=−ε2μ+εc3/2;n2=−3/8ε2ω2c4;k2=3c2/8ω−5c12/12ω3,The approximate solutions of multiple scales method of the solution form of second-order small parameter (T0, T1, and T2) are(43)qε0,ε1,ε2=a2·cosψ2+ε16ω2c1a22·cos2ψ2−12ω2c1a22+ε2c232ω2+c1248ω4a23·cos3ψ2+132ωc4a23·sin3ψ2+c1c39ω3a22sin2ψ2,where ψ2=ωt+β2.


### 5.4. First-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of first-order small parameter, quoting the solution form of small parameter in [16, 37–40], (11) can be written as(44)q¨+ω2q+εc1q2+c3q˙+c2q3+μq˙+c4q˙3=0.Similarly, using the multiple scales method to solve (44), the following can be obtained:(45a)a1=11/a02+n1/m1e−2m1t−n1/m1,(45b)β1=β0−3εm1c28ωn1t−3εc216ωn1lnm1a02e−2m1t+n1e−2m1t−n1,where m1=−εc3+μ/2;n1=−3εω2c4/8;The approximate solutions of multiple scales method of the solution form of first-order small parameter (T0 and T1) are(46)qε0,ε1=a1·cosψ1+εc16ω2a12·cos2ψ1+3c232ω2a13·cos3ψ1−c12ω2a2+c4ω32a13·sin3ψ1,where ψ1=ωt+β1.


## 5.1. Fourth-Order Multiple Scales Method

The multiple scales method is a commonly used to solve galloping equation of iced transmission lines. According to (1)–(10), the nonlinear galloping equation of the transmission lines under steady wind velocity is established, that is, (11):(20)q¨+ω2q+c1q2+c2q3+μ∗−c3q˙+c4q˙3=0,where, due to the inclusion of quadratic and cubic nonlinear terms, the aerodynamic coefficient of the aerodynamic loads in the system also includes the first-order and third-order terms. It is generally considered that the damping and nonlinear terms have little effect on the system. To this end, the solution form of small parameter is(21)q=εq¯,q˙=ε2q.¯,μ∗q˙=ε3μ¯q.¯,q˙3=ε5q.¯3,where q¯ ,μ¯, and q˙¯ are new variables of the system.In order to make the equation easy to express,q,μ, and q˙ are still used to represent q¯, μ¯, and q˙¯. Substituting (21) into (20), we can obtain (22)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε4c4q˙3=0.The fourth-order approximate solutions have been discussed, and setq as(23)q=q0+εq1+ε2q2+ε3q3+ε4q4.A new independent time variableTn=εnt(n = 0, 1, 2, 3, 4) was obtained by multiple scales method, in which n represents the order of dividing the time variable. The larger the value of n, the greater the amount of calculation and the complexity. Then, let the partial differential operator symbol for deriving t be(24)DT=D0T0+εD1T1+ε2D2T2+ε3D3T3+ε4D4T4Substituting (23) and (24) into (22) and equating coefficients of like powers of εn (n = 0, 1, 2, 3, and 4) lead to the following linear ordinary equations, respectively:(25a)D02q0+ω2q0=0,(25b)D02q1+ω2q1=−2D0D1q0−c1q02,(25c)D02q2+ω2q2=−D12q0+2D0D2q0+2D0D1q1+2c1q0q1+c2q03+μ−c3D0q0,(25d)D02q3+ω2q3=−D12q1+2D0D3q0+D1D2q0+D0D2q1+D0D1q2+c1q12+2q0q2+3c2q02q1+μ+c3D0q1+D1q0,(25e)D02q4+ω2q4=−D12q2+D22q0+2D0D4q0+D1D3q0+D0D3q1+D1D2q1+D0D2q2+D0D1q3+2c1q1q2+q0q3+3c2q02q2+q0q12+μ+c3D1q1+D2q0+D0q2+c4D03q0,where Dk represents the partial derivative of Tk; the solutions of (25a) can be written in this form(26)q0=AT1,T2,T3,T4eiwT0+A¯T1,T2,T3,T4e−iwT0,where A can be set as(27)AT1,T2,T3,T4=12a4T1,T2,T3,T4expiβ4T1,T2,T3,T4,where ai (t) and βi (t), respectively, represent the amplitude function and phase function of the system (i = 1, 2, 3, 4). Substituting (26) into (25a)-(25b), the following can be obtained: (28a)D1A=0,(28b)q1=c13ω2A2e2iwT0−c1ω2AA¯+cc.Substituting (26), (28a), and (28b) into (25c), we obtain(29a)D2A=3c22ω−5c123ω3iA2A¯−12μ+c3A+cc,(29b)q2=c1212ω4+c28ω2A3e3iwT0+cc.Substituting (26) and (28a)–(29b) into (25d), we obtain(30a)D3A=0,(30b)q3=c19ω32c3−2u−4c32A2e2iωt+59c1354ω6−31c1c212ω4A3A¯e2iωt+c1354ω6+c1c212ω4A4e4iωt+5c1c2ω4−19c139ω6A2A¯2+cc.Substituting (26) and (28a)-(30b) into (25e), we obtain(31)D4A=−485c14108ω7+15c2216ω3−173c12c212ω5iA3A¯2+4c12c329ω4−2c12c39ω4+2μc129ω4−3ω2c42A2A¯−μ28ω+c34ω+c328ωiA+cc.Combining (24) and (27)-(31) to obtain the averaging equation about amplitude and phase of the system,(32a)a˙4=−ε22u+c3a+c12c329ω4−c12c318ω4+μc1218ω4−3ω2c48ε4a3,(32b)β˙4=ε23c28ω−5c1212ω3a2−ε4485c141728ω7+15c22256w3−173c12c2192ω5a4−ε4μ28w+c34ω+c328ω.After integrating (32a) and (32b) over time, the following can be obtained:(33a)a4=11/a02+n/me−2mt−n/m.(33b)β4=β0−jhm22n3lnma02e−2mt+ne−2mt−n+jhm24nma02e−2mt+ne−2mt−n−2−jhm3n3t−kt−jhm22n1+mna02ma02+nma02e−2mt+ne−2mt−n−1.In (33a)-(33b), β0 represents the integral constant, and other parameters are(33c)m=−ε22μ+c3,k=ε4μ28ω+c34ω+c328ω,n=c12c329ω4−c12c318ω4+μc1218ω4−3ω2c48ε4,j=ε23c28ω−5c1212ω3,h=485c141728ω7+15c22256ω3−173c12c2192ω5ε4.Combining (23), (26), (27), and (33a)–(33c) to obtain the expression of zero-order amplitude,(34)qε0=a4·cosψ4.Combining (23), (26), (27), (28b), and (33a)–(33c) to obtain the expressions of first-order amplitude,(35)qε1=a4·cosψ+ε6w2c1a42cos2ψ−3.Combining (23), (26), (27), (28b), (29b), and (33a)–(33c) to obtain the expressions of second-order amplitude,(36)qε2=a4·cosψ+ε6w2c1a42cos2ψ−3+ε2c1248w4+c232w2a43·cos3ψ.Combining (23), (26), (27), (28b), (29b), (30b), and (33a)–(33c) to obtain the expressions of third-order amplitude,(37)qε3=a4·cosψ4+ε6ω2c1a42cos2ψ4−3+ε2c1248ω4+c232ω2a43·cos3ψ4+ε3−c19ω3u+c3a42·cos2ψ4+59c13432ω6−31c1c296ω4a44·cos2ψ4+c13432ω6+c1c296ω4a44·cos4ψ4+5c1c28ω4−19c1372ω6a44,where ψ4=ωt+β4.Equations (34), (36), and (37), respectively, correspond to approximate solutions of zero-order (qε0), second-order (qε2), and third-order (qε3) amplitude of multiple scales method. In order to obtain the accuracy of (34), (36), and (37), the time history displacement curves and phase diagram about the numerical solutions and the approximate solutions of multiple scales method are compared. Figure 5 is the comparison curves about numerical solutions and the approximate solutions of multiple scales method.Figure 5
Comparison curves of numerical solutions and approximate solutions of multiple scales method. (a) Zero-order expression of amplitude (qε0). (b) Second-order expression of amplitude (qε2). (c) Third-order expression of amplitude (qε3).
(a)(b)(c)As shown in Figure5, the time history displacement curve obtained by multiple scales method has the same phenomenon as that obtained by averaging method, which has shown that the expressions of zero-order amplitude cannot accurately describe the vibration amplitude of the nonlinear galloping equation. However, the expressions of second-order amplitude and the expressions of third-order amplitude are very accurate in describing the vibration amplitude of nonlinear galloping equation.


## 5.2. Third-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of third-order small parameter, letε4c4q˙3in (22) beε3c4q˙3; then(38)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε3c4q˙3=0.Similarly, using the multiple scales method to solve (38), the following can be obtained:(39a)a3=11/a02+n3/m3e−2m3t−n3/m3,(39b)β3=β0−m3kε2n3t−ε28ωc32+ε3μc34ωt−kε22n3lnm3a02e−2m3t+n3e−2m3t−n3,where k3=3c2/8ω−5c12/12ω3;m3=−εc3/2+ε2μ/2+ε3c32/32ω2;n3=ε31/18ω4c02c3−3/8c4ω2,The approximate solutions of multiple scales method of the solution form of third-order small parameter (T0, T1, T2, and T3) are(40)qε0,ε1,ε2,ε3=a⋅cosψ3+εa2c16ω2⋅cos2ψ3−c12ω2+ε2c1248ω4+c232ω2a3⋅cos3ψ3+19ω3a2c1c3⋅sin2ψ3+ε3c19ω3sin2ψ3−c1c3254ω4cos2ψ3a2+59c13432ω6−31c1c296ω4a4cos2ψ3+ωc432+3c2c3128ω3+17c12c3576ω5a3sin3ψ3+c1c296ω4+c13432ω6a4cos4ψ3+5c1c28ω4−19c1372ω6a4,where ψ3=ωt+β3.


## 5.3. Second-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of second-order small parameter, letε4c4q˙3in (22) beε2c4q˙3; then,(41)q¨+ω2q+εc1q2+c3q˙+ε2c2q3+μq˙+ε2c4q˙3=0.Similarly, using the multiple scales method to solve (46), the following can be obtained:(42a)a2=11/a02+n2/m2e−2m2t−n2/m2,(42b)β2=β0−m2k2ε2n2t−k2ε22n2lnm2a02e−2m2t+n2e−2m2t−n2,where m2=−ε2μ+εc3/2;n2=−3/8ε2ω2c4;k2=3c2/8ω−5c12/12ω3,The approximate solutions of multiple scales method of the solution form of second-order small parameter (T0, T1, and T2) are(43)qε0,ε1,ε2=a2·cosψ2+ε16ω2c1a22·cos2ψ2−12ω2c1a22+ε2c232ω2+c1248ω4a23·cos3ψ2+132ωc4a23·sin3ψ2+c1c39ω3a22sin2ψ2,where ψ2=ωt+β2.


## 5.4. First-Order Multiple Scales Method

In order to analyze the accuracy of approximate solutions obtained by multiple scales method of the solution form of first-order small parameter, quoting the solution form of small parameter in [16, 37–40], (11) can be written as(44)q¨+ω2q+εc1q2+c3q˙+c2q3+μq˙+c4q˙3=0.Similarly, using the multiple scales method to solve (44), the following can be obtained:(45a)a1=11/a02+n1/m1e−2m1t−n1/m1,(45b)β1=β0−3εm1c28ωn1t−3εc216ωn1lnm1a02e−2m1t+n1e−2m1t−n1,where m1=−εc3+μ/2;n1=−3εω2c4/8;The approximate solutions of multiple scales method of the solution form of first-order small parameter (T0 and T1) are(46)qε0,ε1=a1·cosψ1+εc16ω2a12·cos2ψ1+3c232ω2a13·cos3ψ1−c12ω2a2+c4ω32a13·sin3ψ1,where ψ1=ωt+β1.


## 6. Influences of Parameter on the Approximate Solutions

Under the condition that the physical parameters and aerodynamic parameters remain constant, Sections4 and 5.1 discuss the accuracy of the approximate solutions of the nonlinear galloping equation by the perturbation method. This section studies whether the changes of physical parameters and aerodynamic coefficients would cause the nonlinear changing (damping and restoring force) of the system, so as to affect the accuracy of the analytical solution of the perturbation method. And the major contents of this section are mainly the following:(1)
The change of tension and sag will cause the change of parameters in the nonlinear galloping equation.(2)
The change of Young's modulus will cause the change of parameters in the nonlinear galloping equation.(3)
The change of wind velocity will cause the change of parameters in the nonlinear galloping equation.


### 6.1. The Influences of Tension on the Approximate Solutions

For flexible cable structures, the relationship between tension and sag isd = mgL2/8H [41], where tension (H) is inversely proportional to sag (d) and tension (H) is proportional to span (L). The iced transmission lines are a type of flexible cable structure. The greater the tension is, the greater the structural stiffness will be. As the tension of iced transmission lines changing, the natural frequency term ω2 and the coefficient c1 of quadratic term will also change. Table 2 compares the accuracy of approximate solutions of perturbation method under different tension.Table 2
The accuracy comparison of perturbation methods under different tension.
Tension sagNumerical solutions amplitude (-/+)/driftAveraging method amplitude (-/+)/errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Fourth-order multiple scalesq (ε3)20 000 N 2.309 m(−0.415 m, 0.372 m) 0.022 m(−0.390 m, 0.387 m) 4.925%(−0.410 m, 0.368 m) 1.141%(−0.410 m, 0.367 m) 1.260%(−0.412 m, 0.366 m) 1.230%30 000 N 1.539 m(−0.452 m, 0.411 m) 0.021 m(−0.428 m, 0.429 m) 4.881%(−0.450 m, 0.409 m) 0.551%(−0.449 m, 0.409 m) 0.690%(−0.449 m, 0.410 m) 0.534%40 000 N 1.154 m(−0.427 m, 0.400 m) 0.014 m( 0.415 m, 0.415 m) 3.180%(−0.429 m, 0.402 m) 0.433%(−0.429 m, 0.402 m) 0.417%(−0.429 m, 0.403 m) 0.592%50 000 N 0.924 m(−0.395 m, 0.377 m) 0.009 m(−0.387 m, 0.390 m) 2.813%(−0.401 m, 0.377 m) 0.768%(−0.398 m, 0.379 m) 0.675%(−0.396 m, 0.382 m) 0.786%60 000 N 0.770 m(−0.365 m, 0.354 m) 0.006 m(−0.363 m, 0.362 m) 1.303%(−0.368 m, 0.a357 m) 0.802%(−0.368 m, 0.357 m) 0.884%(−0.368 m, 0.358 m) 0.891%In Table2, firstly, substituting the tension H = 20 000, 30 000, 40 000, 50 000, and 60 000 N into (12)-(13), the corresponding parameters (&) are obtained; secondly, substituting these parameters (&) into (11), then the numerical solution in Table 2 is obtained by solving (11) with Runge-Kutta function in MATLAB; finally, substituting the parameters (&) into (18), (46), (43), and (37), the approximate analytical solution in Table 2 is obtained. It is worth noting that, in Table 2, the amplitude represents the mean value of the largest absolute value of the vibration amplitude (±) after the system is stable; the drift represents the difference-value between the vibration center of the system and the equilibrium point of the system; the error represents the difference between the approximate analytical solution and the numerical solution. And the definition of the amplitude, drift, and error in Tables 3-6 is consistent with Table 2, respectively.Table 3
The accuracy comparison of perturbation methods under different Young’s modulus.
Young’s modulus (N/m2)Numerical solutions driftAveraging method errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Fourth-order multiple scalesq (ε3)9 560.7(−0.509 m, 0.494 m) 0.008 m(−0.503 m, 0.504 m) 1.549%(−0.503 m, 0.496 m) 0.698%(−0.513 m, 0.495 m) 0.516%(−0.511 m, 0.494 m) 0.247%47 803.3(−0.452 m, 0.412 m) 0.021 m(−0.429 m, 0.428 m) 4.573%(−0.450 m, 0.409 m) 0.553%(−0.449 m, 0.409 m) 0.655%(−0.449 m, 0.410 m) 0.567%239 017(−0.300 m, 0.264 m) 0.019 m(−0.276 m, 0.276 m) 6.187%(−0.296 m, 0.261 m) 1.542%(−0.296 m, 0.260 m) 1.529%(−0.295 m, 0.260 m) 1.696%478 033(−0.226 m, 0.203 m) 0.011 m(−0.210 m, 0.212 m) 5.651%(−0.222 m, 0.2000 m) 1.624%(−0.223 m, 0.198 m) 1.924%(−0.222 m, 0.198 m) 2.033%Table 4
The vibration center drift of the system under different Young’s modulus.
Young’s modulus9 561 N/m232 507 N/m278 399 N/m2170 183 N/m2239 021 N/m2Amplitude drift(−0.509 m, 0.494 m) 0.008 m(−0.473 m, 0.438 m) 0.017 m(−0.418 m, 0.371 m) 0.023 m(−0.339 m, 0.295 m) 0.022 m(−0.301 m, 0.263 m) 0.019 mYoung’s modulus284 913 N/m2330 805 N/m2376697 N/m2468 481 N/m2478 033 N/m2Amplitude drift(−0.282 m, 0.247 m) 0.018 m(−0.267 m, 0.236 m) 0.016 m(−0.253 m, 0.223 m) 0.015 m(−0.229 m, 0.205 m) 0.012 m(−0.226 m, 0.203 m) 0.011 mTable 5
Comparison of the accuracy ofq (ε0), q (ε1), q (ε2), and q (ε3) under different wind velocity.
Wind velocity (m/s)Numerical solutions driftApproximate solutionsq (ε0) errorApproximate solutionsq (ε1) errorApproximate solutionsq (ε2) errorApproximate solutionsq (ε3) error4(−0.214 m, 0.205 m) 0.005(−0.212 m, 0.212 m) 2.293%(−0.217 m, 0.207 m) 1.456%(−0.217 m, 0.207 m) 1.477%(−0.217 m, 0.208 m) 1.595%8(−0.452 m, 0.412 m) 0.020(−0.429 m, 0.430 m) 4.693%(−0.450 m, 0.409 m) 0.514%(−0.450 m, 0.410 m) 0.435%(-0.451 m, 0.411 m) 0.201%12(−0.700 m, 0.608 m) 0.046(−0.647 m, 0.647 m) 6.942%(−0.693 m, 0.601 m) 1.077%(−0.695 m, 0.602 m) 0.898%(−0.697 m, 0.604 m) 0.541%16(−0.960 m, 0.796 m) 0.082(−0.866 m, 0.867 m) 9.328%(−0.949 m, 0.784 m) 1.329%(−0.952 m, 0.787 m) 1.007%(−0.956 m, 0.791 m) 0.539%20(−1.237 m, 0.978 m) 0.130(−1.087 m, 1.088 m) 11.695%(−1.217 m, 0.959 m) 1.824%(−1.222 m, 0.961 m) 1.472%(−1.229 m, 0.973 m) 0.608%24(−1.535 m, 1.163 m) 0.186(−1.321 m, 1.321 m) 13.742%(−1.512 m, 1.132 m) 2.093%(−1.521 m, 1.139 m) 1.483%(−1.526 m, 1.156 m) 0.621%28(−1.834 m, 1.333 m) 0.251(−1.551 m, 1.557 m) 16.152%(−1.822 m, 1.290 m) 1.914%(−1.834 m, 1.309 m) 1.790%(−1.839 m, 1.328 m) 0.285%30(−1.977 m, 1.409 m) 0.284(−1.664 m, 1.664 m) 16.971%(−1.968 m, 1.360 m) 1.918%(−1.988 m, 1.380 m) 0.902%(−1.987 m, 1.411 m) 0.351%32(−2.115 m, 1.487 m) 0.314(−1.776 m, 1.779 m) 17.822%(−2.126 m, 1.431 m) 2.145%(−2.149 m, 1.460 m) 1.730%(−2.154 m, 1.490 m) 1.012%Mean error010.299%1.586%1.243%0.639%Table 6
The accuracy comparison of perturbation methods under different wind velocity.
Wind velocity (m/s)Averaging method errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Third-order multiple scalesq (ε0, ε1, ε2, and ε3)Fourth-order multiple scalesq (ε3)4(−0.210 m, 0.210 m) 2.228%(−0.217 m, 0.207 m) 1.428%(−0.217 m, 0.207 m) 1.439%(−0.218 m, 0.207 m) 1.405%(−0.217 m, 0.208 m) 1.595%8(−0.429 m, 0.428 m) 4.404%(−0.449 m, 0.410 m) 0.499%(−0.449 m, 0.409 m) 0.552%(−0.450 m, 0.407 m) 0.668%(−0.451 m, 0.411 m) 0.201%12(−0.648 m, 0.646 m) 6.851%(−0.695 m, 0.599 m) 1.169%(−0.693 m, 0.599 m) 1.223%(−0.693 m, 0.597 m) 1.425%(−0.697 m, 0.604 m) 0.541%16(−0.863 m, 0.862 m) 9.126%(−0.949 m, 0.786 m) 1.169%(−0.948 m, 0.785 m) 1.300%(−0.946 m, 0.779 m) 1.800%(−0.956 m, 0.791 m) 0.539%20(−1.081 m, 1.080 m) 11.529%(−1.216 m, 0.961 m) 1.762%(−1.213 m, 0.957 m) 2.078%(−1.207 m, 0.951 m) 2.608%(−1.229 m, 0.973 m) 0.608%24(−1.305 m, 1.305 m) 13.586%(−1.507 m, 1.139 m) 1.955%(−1.502 m, 1.129 m) 2.563%(−1.481 m, 1.116 m) 3.780%(−1.526 m, 1.156 m) 0.621%28(−1.529 m, 1.533 m) 15.844%(−1.818 m, 1.303 m) 1.534%(−1.805 m, 1.289 m) 2.430%(−1.765 m, 1.284 m) 3.702%(−1.839 m, 1.328 m) 0.285%30(−1.634 m, 1.636 m) 16.724%(−1.960 m, 1.365 m) 1.966%(−1.947 m, 1.355 m) 2.641%(−1.897 m, 1.357 m) 3.849%(−1.987 m, 1.411 m) 0.351%32(−1.745 m, 1.743 m) 17.321%(−2.117 m, 1.447 m) 1.412%(−2.098 m, 1.437 m) 2.105%(−2.035 m, 1.429 m) 3.837%(−2.154 m, 1.490 m) 1.012%Mean error10.846%1.303%1.815%2.563%0.639%As shown in Table2, with the increasing in tension, the coefficient c1 of the quadratic term in (11) would decrease. With the decreasing in the quadratic term coefficient c1, the drift of vibration center of the transmission lines would continue to decrease. Therefore, the quadratic restoring force term has a direct effect on the drift of vibration center of the system. In addition, the error of the approximate solutions of the averaging method also continues to decrease, which shows that the existence of the quadratic restoring force term is directly related to the accuracy of approximate solutions of the averaging method.It can be seen in Table2 that the approximate solutions of the first-order, second-order, and fourth-order expression of amplitude of multiple scales method have basically the same accuracy for the nonlinear galloping equation. Under the same parameter conditions, increasing the tension of the transmission lines would reduce the vibration amplitude of the transmission lines.


### 6.2. The Influence of Young's Modulus on the Approximate Solutions

Young's modulus is a structural parameter of iced transmission lines. The greater Young modulus is, the greater the stiffness of the transmission lines will be. The natural frequency termω2, quadratic term coefficient c1, and cubic term coefficient c2 in (11) also change with the changing of Young's modulus of iced transmission lines. Table 3 compares the accuracy of approximate solutions of perturbation method under different Young’s modulus.In Table3, Young’s modulus E = 9 560.7, 47 803.3, 239 017, and 478 033 N/m2. With the increase in Young’s modulus, it can be seen from Table 3 that the drift of vibration center of the system increases first and then decreases. And the error of the averaging method increases first and then decreases, which also shows that the drift of vibration center of the system is directly related to the accuracy of the averaging method.It can be seen from Table3 that when Young's modulus increases from E = 9 560.7 N/m2 to E = 478 033 N/m2, the error of different-order multiple scales method increases from 0.516% to 2.033%. Therefore, when only the elastic modulus of the system is changed, the accuracy of the first-order, second-order, and fourth-order multiscale methods is basically the same. As Young’s modulus increases, the vibration amplitude of the iced transmission lines would decrease.In order to verify the influence of Young’s modulus on the drift of vibration center of the system, Table4 compares the drift of vibration center under different Young’s modulus. The natural frequency term ω2, the quadratic term coefficient c1, and the cubic term coefficient c2 of Young’s modulus E = 9 561–478 033 N/m2 are introduced into (11); then the drift of vibration center is obtained by Runge-Kutta function in MATLAB.With the increase in Young's modulus fromE = 9 561 N/m2 to E = 78 399 N/m2, the vibration center drift first increases from 0.008 m to 0.023 m. When Young's modulus increases from E = 78 399 N/m2 to E = 478 033 N/m2, the drift decreases by 0.011 m. In addition, when Young's modulus E = 9 560.7 N/m2 increases to E = 478 033 N/m2, however, the amplitude of iced transmission line decreases from (−0.509 m, 0.494 m) to (−0.226 m, 0.203 m).As Young’s modulus gradually increases, the structural stiffness also increases. Under the combined contribution of the quadratic coefficientc1 and the cubic coefficient c2, the vibration amplitude of the iced transmission lines decreases gradually, and the vibration center drift of the system increases first and decreases later. Under the same parameter conditions, increasing Young’s modulus of the transmission lines is beneficial to reducing the vibration amplitude of transmission lines.


### 6.3. The Relationship between the Coefficient and the Drift

The galloping of iced transmission lines is affected by the transmission line's own structure, the shape of the transmission line's ice-cover shape, and wind loads. In general, the vibration amplitude and vibration velocity gradually increase with the increase in wind velocity.In order to determine whether the increase in wind velocity affects the accuracy of the approximate solutions of multiple scales methods, the following sections would discuss the influence of different wind velocity on the accuracy of the multiple scales method and averaging method. Figure6(a) shows the time history displacement diagram when the wind velocity is 8 m/s. The vibration amplitude of the transmission line increases slowly with time, until the amplitude gradually stabilizes at 1 050 s. And when the transmission lines are galloping, the vibration amplitude is 0.412 m and −0.452 m. The wind velocity in Figure 6(b) is 10 m/s, the galloping time of transmission lines is reduced to 650 s, and the galloping amplitude is increased to 0.796 m and −0.960 m. As the wind velocity increases, the time for the transmission line to reach galloping is shortened, and the amplitude of the galloping is increased.Figure 6
The time history displacement diagram under different wind velocity ((c)1 = 0, (c)1 ≠ 0). (a) The coefficient (c)1≠0 (8 m/s). (b) The coefficient (c)1≠0 (10 m/s). (c) The coefficient (c)1 = 0 (8 m/s). (d) The coefficient (c)1 = 0 (10 m/s).
(a)(b)(c)(d)Through the comparison of different tensions in Table2 and different Young’s modulus in Table 3, it is found that the value of the quadratic coefficient is related to the drift of the vibration center. Then, in order to clarify the relationship between the coefficient of the quadratic term and the drift of the vibration center, let the coefficient of the quadratic term in (11) be equal to 0. And Figures 6(c) and 6(d) correspond to wind velocities U = 8 m/s and U = 10 m/s, respectively. After comparing Figures 6(a) and 6(c), it can be found that the existence of the quadratic coefficient makes the vibration amplitude of the negative axis larger in Figure 6(a); that is, the phenomenon of vibration center drift occurs. Figures 6(b) and 6(d) also reveal the same phenomenon. To this end, it is easy to derive that the existence of quadratic restoring force makes the vibration center of the system drift.


### 6.4. The Effect of Wind Velocity on the Accuracy ofq (ε0), q (ε1), q (ε2), and q (ε3)

The Rayleigh damping coefficientsc3 and c4 in (11) would change with the wind velocity. Under the condition of different wind velocity, the accuracy of q (ε0), q (ε1), q (ε2), and q (ε3) is studied in Table 5. q (ε0), q (ε1), q (ε2), and q (ε3) represent zero-order equation (34), first-order equation (35), second-order equation (36), and third-order equation (37), and all of them are the approximate analytical solutions of multiple scales method. And the wind velocity U = 4–32 m/s is shown in Table 5.It can be seen from Table5 that the drift of the vibration center changes from 0.005 m of 4 m/s to 0.314 m of 32 m/s. Although the changing of wind velocity does not lead to a changing in value of quadratic term coefficient c1, the drift of vibration center would increase with the increase in wind velocity. It is shown that when there is quadratic restoring force in the system, the drift of vibration center would also change with wind velocity change.In addition, the error of the expressions of zero-order amplitude increases from 2.293% of wind velocityU = 4 m/s to 17.822% of wind velocity U = 32 m/s; the error of the expressions of first-order amplitude is less than 2.145%; the error of the expressions of second-order amplitude is less than 1.790%; the error of the expressions of third-order amplitude is generally less than 1.012%. As the wind velocity increases, the nonlinearity and Rayleigh damping of the system gradually strengthen, and the accuracy of the approximate solutions of multiple scales method tends to decrease overall. In the interval of 4–32 m/s wind velocity, the expressions of first-order amplitude correct the drift of the vibration center of the transmission lines. And the mean error of the expressions of zero-order amplitude is 10.299%, which has been corrected to 1.586% by the expressions of first-order amplitude.From the above analysis, it can be seen that, with the increase in wind velocity, the vibration center drift of iced transmission lines would increase gradually in Table5. The greater the drift is, the greater the error of the approximate solutions will be. The approximate solutions of expressions of zero-order amplitude of multiple scales method are not suitable to describe the galloping process with high wind velocity. However, the expressions of first-order amplitude of multiple scales method can correct the error of expressions of zero-order amplitude.The expressions of second-order amplitude reduce the mean error of the expressions of first-order amplitude from 1.586% to 1.243%. The expressions of third-order amplitude reduce the mean error of the expressions of second-order amplitude from 1.243% to 0.639%. Therefore, with the increase in the order of expression of amplitude, the error of approximate solutions by multiple scales method has been corrected; that is, the high-order approximate solutions of multiple scales method can correct the error of low-order approximate solutions.


### 6.5. The Effect of Wind Velocity on the Accuracy ofq, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3)

Section5.4 has discussed the effect of wind velocity on the accuracy of q (ε0), q (ε1), q (ε2), and q (ε3) with the same solution form of small parameter. Therefore, this section will discuss the effect of wind velocity on the accuracy of the approximate solutions obtained by the perturbation method with different solution form of small parameter. And q, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3), respectively, represent the approximate solutions of different solution form of small parameter. Equations (18), (46), (43), (40), and (37) are the averaging method (q) and first-order multiple scales method (q (ε0, ε1)), second-order multiple scales method (q (ε0, ε1 , ε2)), third-order multiple scales method (q (ε0, ε1, ε2, ε3)), and fourth-order multiple scales method (q (ε3)) approximate solutions of different solution form of small parameter, respectively.Table6 compares the accuracy of q, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3) under different wind velocity. The wind velocity U = 4–32 m/s is shown in Table 6. The error in Table 6 was obtained by comparing the approximate solutions in Table 6 with the numerical solutions in Table 5.As the wind velocity increases, the nonlinearity and Rayleigh damping of the iced transmission line structure gradually increase. The error of averaging method becomes larger (from 2.228% of 4 m/s to 17.321% of 32 m/s). The error ofq (ε0 and ε1) is less than 1.966%; the error of q (ε0, ε1, and ε2) is gradually increased from 1.439% of the wind velocity U = 4 m/s to 2.641% of the wind velocity U = 30 m/s. The error of q (ε0, ε1, ε2, and ε3) is gradually increased from 1.405% of the wind velocity U = 4 m/s to 3.837% of the wind velocity U = 32 m/s. The error of q (ε3) is only less than 1.600%, and the error is only less than 0.630% when the wind velocity is 8–30 m/s, which indicates that the accuracy of the fourth-order multiple scales perturbation method is better than that of the low-order multiple scales perturbation method when the range of wind velocity is 8–30 m/s. By comparing the mean error of multiple scales method, it can also be known that the accuracy of the first-order and fourth-order of multiple scales method of the solutions form of small parameter is more precise for galloping equation. By comparing the approximate solutions of the first-order multiple scales method and fourth-order multiple scales method, it is found that the higher the order is, the more difficult the solutions and the more complex the expression will be. Therefore, when the wind velocity ranges from 4 to 32 m/s and the accuracy requirements are not high, the multiple scales method of first-order solutions form is more suitable for calculation and analysis; when the accuracy requirements are higher, the multiple scales method of fourth-order solutions form is better than the multiple scales method of first-order solutions form.


## 6.1. The Influences of Tension on the Approximate Solutions

For flexible cable structures, the relationship between tension and sag isd = mgL2/8H [41], where tension (H) is inversely proportional to sag (d) and tension (H) is proportional to span (L). The iced transmission lines are a type of flexible cable structure. The greater the tension is, the greater the structural stiffness will be. As the tension of iced transmission lines changing, the natural frequency term ω2 and the coefficient c1 of quadratic term will also change. Table 2 compares the accuracy of approximate solutions of perturbation method under different tension.Table 2
The accuracy comparison of perturbation methods under different tension.
Tension sagNumerical solutions amplitude (-/+)/driftAveraging method amplitude (-/+)/errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Fourth-order multiple scalesq (ε3)20 000 N 2.309 m(−0.415 m, 0.372 m) 0.022 m(−0.390 m, 0.387 m) 4.925%(−0.410 m, 0.368 m) 1.141%(−0.410 m, 0.367 m) 1.260%(−0.412 m, 0.366 m) 1.230%30 000 N 1.539 m(−0.452 m, 0.411 m) 0.021 m(−0.428 m, 0.429 m) 4.881%(−0.450 m, 0.409 m) 0.551%(−0.449 m, 0.409 m) 0.690%(−0.449 m, 0.410 m) 0.534%40 000 N 1.154 m(−0.427 m, 0.400 m) 0.014 m( 0.415 m, 0.415 m) 3.180%(−0.429 m, 0.402 m) 0.433%(−0.429 m, 0.402 m) 0.417%(−0.429 m, 0.403 m) 0.592%50 000 N 0.924 m(−0.395 m, 0.377 m) 0.009 m(−0.387 m, 0.390 m) 2.813%(−0.401 m, 0.377 m) 0.768%(−0.398 m, 0.379 m) 0.675%(−0.396 m, 0.382 m) 0.786%60 000 N 0.770 m(−0.365 m, 0.354 m) 0.006 m(−0.363 m, 0.362 m) 1.303%(−0.368 m, 0.a357 m) 0.802%(−0.368 m, 0.357 m) 0.884%(−0.368 m, 0.358 m) 0.891%In Table2, firstly, substituting the tension H = 20 000, 30 000, 40 000, 50 000, and 60 000 N into (12)-(13), the corresponding parameters (&) are obtained; secondly, substituting these parameters (&) into (11), then the numerical solution in Table 2 is obtained by solving (11) with Runge-Kutta function in MATLAB; finally, substituting the parameters (&) into (18), (46), (43), and (37), the approximate analytical solution in Table 2 is obtained. It is worth noting that, in Table 2, the amplitude represents the mean value of the largest absolute value of the vibration amplitude (±) after the system is stable; the drift represents the difference-value between the vibration center of the system and the equilibrium point of the system; the error represents the difference between the approximate analytical solution and the numerical solution. And the definition of the amplitude, drift, and error in Tables 3-6 is consistent with Table 2, respectively.Table 3
The accuracy comparison of perturbation methods under different Young’s modulus.
Young’s modulus (N/m2)Numerical solutions driftAveraging method errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Fourth-order multiple scalesq (ε3)9 560.7(−0.509 m, 0.494 m) 0.008 m(−0.503 m, 0.504 m) 1.549%(−0.503 m, 0.496 m) 0.698%(−0.513 m, 0.495 m) 0.516%(−0.511 m, 0.494 m) 0.247%47 803.3(−0.452 m, 0.412 m) 0.021 m(−0.429 m, 0.428 m) 4.573%(−0.450 m, 0.409 m) 0.553%(−0.449 m, 0.409 m) 0.655%(−0.449 m, 0.410 m) 0.567%239 017(−0.300 m, 0.264 m) 0.019 m(−0.276 m, 0.276 m) 6.187%(−0.296 m, 0.261 m) 1.542%(−0.296 m, 0.260 m) 1.529%(−0.295 m, 0.260 m) 1.696%478 033(−0.226 m, 0.203 m) 0.011 m(−0.210 m, 0.212 m) 5.651%(−0.222 m, 0.2000 m) 1.624%(−0.223 m, 0.198 m) 1.924%(−0.222 m, 0.198 m) 2.033%Table 4
The vibration center drift of the system under different Young’s modulus.
Young’s modulus9 561 N/m232 507 N/m278 399 N/m2170 183 N/m2239 021 N/m2Amplitude drift(−0.509 m, 0.494 m) 0.008 m(−0.473 m, 0.438 m) 0.017 m(−0.418 m, 0.371 m) 0.023 m(−0.339 m, 0.295 m) 0.022 m(−0.301 m, 0.263 m) 0.019 mYoung’s modulus284 913 N/m2330 805 N/m2376697 N/m2468 481 N/m2478 033 N/m2Amplitude drift(−0.282 m, 0.247 m) 0.018 m(−0.267 m, 0.236 m) 0.016 m(−0.253 m, 0.223 m) 0.015 m(−0.229 m, 0.205 m) 0.012 m(−0.226 m, 0.203 m) 0.011 mTable 5
Comparison of the accuracy ofq (ε0), q (ε1), q (ε2), and q (ε3) under different wind velocity.
Wind velocity (m/s)Numerical solutions driftApproximate solutionsq (ε0) errorApproximate solutionsq (ε1) errorApproximate solutionsq (ε2) errorApproximate solutionsq (ε3) error4(−0.214 m, 0.205 m) 0.005(−0.212 m, 0.212 m) 2.293%(−0.217 m, 0.207 m) 1.456%(−0.217 m, 0.207 m) 1.477%(−0.217 m, 0.208 m) 1.595%8(−0.452 m, 0.412 m) 0.020(−0.429 m, 0.430 m) 4.693%(−0.450 m, 0.409 m) 0.514%(−0.450 m, 0.410 m) 0.435%(-0.451 m, 0.411 m) 0.201%12(−0.700 m, 0.608 m) 0.046(−0.647 m, 0.647 m) 6.942%(−0.693 m, 0.601 m) 1.077%(−0.695 m, 0.602 m) 0.898%(−0.697 m, 0.604 m) 0.541%16(−0.960 m, 0.796 m) 0.082(−0.866 m, 0.867 m) 9.328%(−0.949 m, 0.784 m) 1.329%(−0.952 m, 0.787 m) 1.007%(−0.956 m, 0.791 m) 0.539%20(−1.237 m, 0.978 m) 0.130(−1.087 m, 1.088 m) 11.695%(−1.217 m, 0.959 m) 1.824%(−1.222 m, 0.961 m) 1.472%(−1.229 m, 0.973 m) 0.608%24(−1.535 m, 1.163 m) 0.186(−1.321 m, 1.321 m) 13.742%(−1.512 m, 1.132 m) 2.093%(−1.521 m, 1.139 m) 1.483%(−1.526 m, 1.156 m) 0.621%28(−1.834 m, 1.333 m) 0.251(−1.551 m, 1.557 m) 16.152%(−1.822 m, 1.290 m) 1.914%(−1.834 m, 1.309 m) 1.790%(−1.839 m, 1.328 m) 0.285%30(−1.977 m, 1.409 m) 0.284(−1.664 m, 1.664 m) 16.971%(−1.968 m, 1.360 m) 1.918%(−1.988 m, 1.380 m) 0.902%(−1.987 m, 1.411 m) 0.351%32(−2.115 m, 1.487 m) 0.314(−1.776 m, 1.779 m) 17.822%(−2.126 m, 1.431 m) 2.145%(−2.149 m, 1.460 m) 1.730%(−2.154 m, 1.490 m) 1.012%Mean error010.299%1.586%1.243%0.639%Table 6
The accuracy comparison of perturbation methods under different wind velocity.
Wind velocity (m/s)Averaging method errorFirst-order multiple scalesq (ε0 and ε1)Second-order multiple scalesq (ε0, ε1, and ε2)Third-order multiple scalesq (ε0, ε1, ε2, and ε3)Fourth-order multiple scalesq (ε3)4(−0.210 m, 0.210 m) 2.228%(−0.217 m, 0.207 m) 1.428%(−0.217 m, 0.207 m) 1.439%(−0.218 m, 0.207 m) 1.405%(−0.217 m, 0.208 m) 1.595%8(−0.429 m, 0.428 m) 4.404%(−0.449 m, 0.410 m) 0.499%(−0.449 m, 0.409 m) 0.552%(−0.450 m, 0.407 m) 0.668%(−0.451 m, 0.411 m) 0.201%12(−0.648 m, 0.646 m) 6.851%(−0.695 m, 0.599 m) 1.169%(−0.693 m, 0.599 m) 1.223%(−0.693 m, 0.597 m) 1.425%(−0.697 m, 0.604 m) 0.541%16(−0.863 m, 0.862 m) 9.126%(−0.949 m, 0.786 m) 1.169%(−0.948 m, 0.785 m) 1.300%(−0.946 m, 0.779 m) 1.800%(−0.956 m, 0.791 m) 0.539%20(−1.081 m, 1.080 m) 11.529%(−1.216 m, 0.961 m) 1.762%(−1.213 m, 0.957 m) 2.078%(−1.207 m, 0.951 m) 2.608%(−1.229 m, 0.973 m) 0.608%24(−1.305 m, 1.305 m) 13.586%(−1.507 m, 1.139 m) 1.955%(−1.502 m, 1.129 m) 2.563%(−1.481 m, 1.116 m) 3.780%(−1.526 m, 1.156 m) 0.621%28(−1.529 m, 1.533 m) 15.844%(−1.818 m, 1.303 m) 1.534%(−1.805 m, 1.289 m) 2.430%(−1.765 m, 1.284 m) 3.702%(−1.839 m, 1.328 m) 0.285%30(−1.634 m, 1.636 m) 16.724%(−1.960 m, 1.365 m) 1.966%(−1.947 m, 1.355 m) 2.641%(−1.897 m, 1.357 m) 3.849%(−1.987 m, 1.411 m) 0.351%32(−1.745 m, 1.743 m) 17.321%(−2.117 m, 1.447 m) 1.412%(−2.098 m, 1.437 m) 2.105%(−2.035 m, 1.429 m) 3.837%(−2.154 m, 1.490 m) 1.012%Mean error10.846%1.303%1.815%2.563%0.639%As shown in Table2, with the increasing in tension, the coefficient c1 of the quadratic term in (11) would decrease. With the decreasing in the quadratic term coefficient c1, the drift of vibration center of the transmission lines would continue to decrease. Therefore, the quadratic restoring force term has a direct effect on the drift of vibration center of the system. In addition, the error of the approximate solutions of the averaging method also continues to decrease, which shows that the existence of the quadratic restoring force term is directly related to the accuracy of approximate solutions of the averaging method.It can be seen in Table2 that the approximate solutions of the first-order, second-order, and fourth-order expression of amplitude of multiple scales method have basically the same accuracy for the nonlinear galloping equation. Under the same parameter conditions, increasing the tension of the transmission lines would reduce the vibration amplitude of the transmission lines.


## 6.2. The Influence of Young's Modulus on the Approximate Solutions

Young's modulus is a structural parameter of iced transmission lines. The greater Young modulus is, the greater the stiffness of the transmission lines will be. The natural frequency termω2, quadratic term coefficient c1, and cubic term coefficient c2 in (11) also change with the changing of Young's modulus of iced transmission lines. Table 3 compares the accuracy of approximate solutions of perturbation method under different Young’s modulus.In Table3, Young’s modulus E = 9 560.7, 47 803.3, 239 017, and 478 033 N/m2. With the increase in Young’s modulus, it can be seen from Table 3 that the drift of vibration center of the system increases first and then decreases. And the error of the averaging method increases first and then decreases, which also shows that the drift of vibration center of the system is directly related to the accuracy of the averaging method.It can be seen from Table3 that when Young's modulus increases from E = 9 560.7 N/m2 to E = 478 033 N/m2, the error of different-order multiple scales method increases from 0.516% to 2.033%. Therefore, when only the elastic modulus of the system is changed, the accuracy of the first-order, second-order, and fourth-order multiscale methods is basically the same. As Young’s modulus increases, the vibration amplitude of the iced transmission lines would decrease.In order to verify the influence of Young’s modulus on the drift of vibration center of the system, Table4 compares the drift of vibration center under different Young’s modulus. The natural frequency term ω2, the quadratic term coefficient c1, and the cubic term coefficient c2 of Young’s modulus E = 9 561–478 033 N/m2 are introduced into (11); then the drift of vibration center is obtained by Runge-Kutta function in MATLAB.With the increase in Young's modulus fromE = 9 561 N/m2 to E = 78 399 N/m2, the vibration center drift first increases from 0.008 m to 0.023 m. When Young's modulus increases from E = 78 399 N/m2 to E = 478 033 N/m2, the drift decreases by 0.011 m. In addition, when Young's modulus E = 9 560.7 N/m2 increases to E = 478 033 N/m2, however, the amplitude of iced transmission line decreases from (−0.509 m, 0.494 m) to (−0.226 m, 0.203 m).As Young’s modulus gradually increases, the structural stiffness also increases. Under the combined contribution of the quadratic coefficientc1 and the cubic coefficient c2, the vibration amplitude of the iced transmission lines decreases gradually, and the vibration center drift of the system increases first and decreases later. Under the same parameter conditions, increasing Young’s modulus of the transmission lines is beneficial to reducing the vibration amplitude of transmission lines.


## 6.3. The Relationship between the Coefficient and the Drift

The galloping of iced transmission lines is affected by the transmission line's own structure, the shape of the transmission line's ice-cover shape, and wind loads. In general, the vibration amplitude and vibration velocity gradually increase with the increase in wind velocity.In order to determine whether the increase in wind velocity affects the accuracy of the approximate solutions of multiple scales methods, the following sections would discuss the influence of different wind velocity on the accuracy of the multiple scales method and averaging method. Figure6(a) shows the time history displacement diagram when the wind velocity is 8 m/s. The vibration amplitude of the transmission line increases slowly with time, until the amplitude gradually stabilizes at 1 050 s. And when the transmission lines are galloping, the vibration amplitude is 0.412 m and −0.452 m. The wind velocity in Figure 6(b) is 10 m/s, the galloping time of transmission lines is reduced to 650 s, and the galloping amplitude is increased to 0.796 m and −0.960 m. As the wind velocity increases, the time for the transmission line to reach galloping is shortened, and the amplitude of the galloping is increased.Figure 6
The time history displacement diagram under different wind velocity ((c)1 = 0, (c)1 ≠ 0). (a) The coefficient (c)1≠0 (8 m/s). (b) The coefficient (c)1≠0 (10 m/s). (c) The coefficient (c)1 = 0 (8 m/s). (d) The coefficient (c)1 = 0 (10 m/s).
(a)(b)(c)(d)Through the comparison of different tensions in Table2 and different Young’s modulus in Table 3, it is found that the value of the quadratic coefficient is related to the drift of the vibration center. Then, in order to clarify the relationship between the coefficient of the quadratic term and the drift of the vibration center, let the coefficient of the quadratic term in (11) be equal to 0. And Figures 6(c) and 6(d) correspond to wind velocities U = 8 m/s and U = 10 m/s, respectively. After comparing Figures 6(a) and 6(c), it can be found that the existence of the quadratic coefficient makes the vibration amplitude of the negative axis larger in Figure 6(a); that is, the phenomenon of vibration center drift occurs. Figures 6(b) and 6(d) also reveal the same phenomenon. To this end, it is easy to derive that the existence of quadratic restoring force makes the vibration center of the system drift.


## 6.4. The Effect of Wind Velocity on the Accuracy ofq (ε0), q (ε1), q (ε2), and q (ε3)

The Rayleigh damping coefficientsc3 and c4 in (11) would change with the wind velocity. Under the condition of different wind velocity, the accuracy of q (ε0), q (ε1), q (ε2), and q (ε3) is studied in Table 5. q (ε0), q (ε1), q (ε2), and q (ε3) represent zero-order equation (34), first-order equation (35), second-order equation (36), and third-order equation (37), and all of them are the approximate analytical solutions of multiple scales method. And the wind velocity U = 4–32 m/s is shown in Table 5.It can be seen from Table5 that the drift of the vibration center changes from 0.005 m of 4 m/s to 0.314 m of 32 m/s. Although the changing of wind velocity does not lead to a changing in value of quadratic term coefficient c1, the drift of vibration center would increase with the increase in wind velocity. It is shown that when there is quadratic restoring force in the system, the drift of vibration center would also change with wind velocity change.In addition, the error of the expressions of zero-order amplitude increases from 2.293% of wind velocityU = 4 m/s to 17.822% of wind velocity U = 32 m/s; the error of the expressions of first-order amplitude is less than 2.145%; the error of the expressions of second-order amplitude is less than 1.790%; the error of the expressions of third-order amplitude is generally less than 1.012%. As the wind velocity increases, the nonlinearity and Rayleigh damping of the system gradually strengthen, and the accuracy of the approximate solutions of multiple scales method tends to decrease overall. In the interval of 4–32 m/s wind velocity, the expressions of first-order amplitude correct the drift of the vibration center of the transmission lines. And the mean error of the expressions of zero-order amplitude is 10.299%, which has been corrected to 1.586% by the expressions of first-order amplitude.From the above analysis, it can be seen that, with the increase in wind velocity, the vibration center drift of iced transmission lines would increase gradually in Table5. The greater the drift is, the greater the error of the approximate solutions will be. The approximate solutions of expressions of zero-order amplitude of multiple scales method are not suitable to describe the galloping process with high wind velocity. However, the expressions of first-order amplitude of multiple scales method can correct the error of expressions of zero-order amplitude.The expressions of second-order amplitude reduce the mean error of the expressions of first-order amplitude from 1.586% to 1.243%. The expressions of third-order amplitude reduce the mean error of the expressions of second-order amplitude from 1.243% to 0.639%. Therefore, with the increase in the order of expression of amplitude, the error of approximate solutions by multiple scales method has been corrected; that is, the high-order approximate solutions of multiple scales method can correct the error of low-order approximate solutions.


## 6.5. The Effect of Wind Velocity on the Accuracy ofq, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3)

Section5.4 has discussed the effect of wind velocity on the accuracy of q (ε0), q (ε1), q (ε2), and q (ε3) with the same solution form of small parameter. Therefore, this section will discuss the effect of wind velocity on the accuracy of the approximate solutions obtained by the perturbation method with different solution form of small parameter. And q, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3), respectively, represent the approximate solutions of different solution form of small parameter. Equations (18), (46), (43), (40), and (37) are the averaging method (q) and first-order multiple scales method (q (ε0, ε1)), second-order multiple scales method (q (ε0, ε1 , ε2)), third-order multiple scales method (q (ε0, ε1, ε2, ε3)), and fourth-order multiple scales method (q (ε3)) approximate solutions of different solution form of small parameter, respectively.Table6 compares the accuracy of q, q (ε0 and ε1), q (ε0, ε1, and ε2), q (ε0, ε1, ε2, and ε3), and q (ε3) under different wind velocity. The wind velocity U = 4–32 m/s is shown in Table 6. The error in Table 6 was obtained by comparing the approximate solutions in Table 6 with the numerical solutions in Table 5.As the wind velocity increases, the nonlinearity and Rayleigh damping of the iced transmission line structure gradually increase. The error of averaging method becomes larger (from 2.228% of 4 m/s to 17.321% of 32 m/s). The error ofq (ε0 and ε1) is less than 1.966%; the error of q (ε0, ε1, and ε2) is gradually increased from 1.439% of the wind velocity U = 4 m/s to 2.641% of the wind velocity U = 30 m/s. The error of q (ε0, ε1, ε2, and ε3) is gradually increased from 1.405% of the wind velocity U = 4 m/s to 3.837% of the wind velocity U = 32 m/s. The error of q (ε3) is only less than 1.600%, and the error is only less than 0.630% when the wind velocity is 8–30 m/s, which indicates that the accuracy of the fourth-order multiple scales perturbation method is better than that of the low-order multiple scales perturbation method when the range of wind velocity is 8–30 m/s. By comparing the mean error of multiple scales method, it can also be known that the accuracy of the first-order and fourth-order of multiple scales method of the solutions form of small parameter is more precise for galloping equation. By comparing the approximate solutions of the first-order multiple scales method and fourth-order multiple scales method, it is found that the higher the order is, the more difficult the solutions and the more complex the expression will be. Therefore, when the wind velocity ranges from 4 to 32 m/s and the accuracy requirements are not high, the multiple scales method of first-order solutions form is more suitable for calculation and analysis; when the accuracy requirements are higher, the multiple scales method of fourth-order solutions form is better than the multiple scales method of first-order solutions form.


## 7. Conclusions

The accuracy of approximate solutions of perturbation method for nonlinear galloping equation of iced transmission lines with small sag is studied in this paper. Then, the approximate solutions obtained by perturbation method are compared with the numerical method. The following conclusions can be listed:(1)
For the iced transmission lines excited by wind loads, the existence of the quadratic nonlinear restoring force term in the ordinary differential galloping equation would cause the drift of vibration center as the transmission line gallops. The greater the coefficient of the quadratic nonlinear restoring force term is, the greater the drift of the vibration center will be.(2)
For galloping equations with quadratic nonlinear restoring force terms, the error between approximate solutions of the averaging method and numerical solutions is 17.321% when the wind velocityU = 32 m/s. Therefore, it is not suitable to use the averaging method to solve the vibration equation where the system contains the quadratic term.(3)
When the wind range is 4–32 m/s, the mean error between the expressions of zero-order amplitude and numerical solutions is 10.299%, which has been corrected to 1.586% by the expressions of first-order amplitude. With the same solution form of small parameter, the approximate solutions of high-order multiple scales method can correct the error of the approximate solutions of low-order multiple scales method.(4)
With the enhancement of structural nonlinearity and Rayleigh damping, the error between the approximate solutions of multiple scales method and numerical solutions would increase. Using the multiple scales method of solution form of first-order small parameter, the mean error between approximate solutions and numerical solutions is less than 1.303% when the range of wind velocity is 4–32 m/s; and the mean error between approximate solutions of fourth-order multiple scales method and numerical solutions is less than 0.639%.



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*Source: 6651629-2021-03-27.xml*