# Nonlinear Dynamic Analysis of High-Voltage Overhead Transmission Lines

Authors: Meng Zhang; Guifeng Zhao; Jie Li
Journal: Shock and Vibration
 (2018)

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## Abstract

According to a generalized Hamilton’s principle, three-dimensional (3D) nonlinear vibration equations for overhead transmission lines that consider geometric nonlinearity are established. Based on the characteristics of an actual transmission line, the 3D equations are simplified to two-dimensional equations, and the nonlinear vibration behavior of transmission lines is investigated by combining theoretical analysis with numerical simulation. The results show that transmission lines have inherently nonlinear vibration characteristics. When in free vibration, a transmission line can undergo nonlinear internal resonance, even when its initial out-of-plane energy is relatively low; as its initial out-of-plane energy increases, the coupling of in-plane and out-of-plane vibration becomes stronger. When forced to vibrate by an external excitation, due to the combined action of internal and primary resonance, the vibration energy of a transmission line transfers from the out-of-plane direction to the in-plane direction that is not directly under the excitation, resulting in an increase in the dynamic tension and the displacement amplitude of the transmission line. Increasing damping can consume the vibration energy of a transmission line but cannot prevent the occurrence of internal resonance.


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## Body

## 1. Introduction

High-voltage overhead transmission lines (HVOTLs) are important carriers within power systems, and their safety directly affects the normal operation of the whole power system. Currently, aluminum conductor steel-reinforced (ACSR) cables are the most widely used transmission lines in engineering applications. ACSR cables, characterized by their light weight, small damping, and long spans, can only bear axial tension. Generally, the vibration of a transmission line in the plane where the transmission line is located under the action of gravity is referred to as in-plane vibration, and the vibration in the plane vertical to the aforementioned plane is referred to as out-of-plane vibration. A transmission line is often considered as a single-cable structure in calculation. According to the suspended cable theory [1], the in-plane stiffness of a transmission line is a result of its internal force, and the out-of-plane stiffness of a transmission line is relatively low. Therefore, the vibration of a transmission line exhibits conspicuous “large displacement, small-strain” geometrically nonlinear characteristics. In particular, when induced by wind and rain or strong wind, transmission lines are more prone to relatively strong nonlinear vibration. In severe cases, such vibration can cause damage to transmission lines. Therefore, an in-depth investigation of the nonlinear vibration of HVOTLs is of great significance.Research on cables has progressed from statics to dynamics and from the linear theory to the nonlinear theory [1–14]. In the static theory, cables are viewed as two-force bars with an aim to simplify calculation. As a result, the results of calculation based on the static theory differ relatively significantly from the actual situation [1, 2]. In comparison, the natural frequencies of the out-of-plane and in-plane vibration of a suspended cable obtained based on the linear dynamic theory of cables are relatively accurate. Moreover, it can also be derived from the linear dynamic theory of cables that out-of-plane and in-plane modes of vibration are completely independent of one another [3]. However, cables are flexible structures; consequently, their motion exhibits significantly nonlinear motion characteristics and very complex forms that are a result of the combined action of external and internal resonance. In addition, the geometric and physical parameters of cables also have an impact on their dynamic response [4–14]. Evidently, even the linear dynamic theory cannot accurately describe the forms of motion of cables. It is irrefutable that the wind-induced vibration response of long-span, lightweight, and small-damping transmission lines has inherently nonlinear characteristics [15–24]. However, the current design specification still requires the use of the linear static theory of suspended cables to design transmission lines, which clearly cannot meet the engineering demand. The abovementioned linear design theory is easily mastered and applied in engineering, but its shortcomings are also obvious. For example, it assumes that the relationship between the static load and the response of the whole line is a linear function in calculations, which obviously does not conform to the nonlinear mechanical properties of a transmission line. Furthermore, there is a significant difference between the in-plane and out-of-plane vibration of a transmission line, and the former has obvious nonlinearity stiffness that will induce additional tension of the transmission line. Therefore, it is not enough to calculate the response of a transmission line only by its first-order out-of-plane vibration. In fact, as a highly flexible line system, it is inevitable to consider the multiorder vibrations of a transmission line, and the dynamic tension generated by the vibration of the transmission line under the action of the pulsating wind is the main load that acts on the transmission tower.Based on the aforementioned analysis, according to a generalizedHamilton’s principle, three-dimensional (3D) nonlinear differential equations of motion for horizontally suspended transmission lines that consider the initial deflection are established in this work. Based on the characteristics of an actual transmission line, the 3D equations are simplified to two-dimensional (2D) equations. On this basis, the method of multiple scales is employed to theoretically study the nonlinear internal resonance behavior of transmission lines. Moreover, a higher-orderRunge-Kutta method is also used to perform a numerical analysis of the nonlinear vibration characteristics of transmission lines to reveal the characteristics of the coupled action of the out-of-plane and in-plane vibration of transmission lines when in nonlinear vibration, with the aim of providing a basis for the reasonable design of HVOTLs.


## 2. 3D Nonlinear Equations of Motion for Elastic Transmission Lines

### 2.1. Establishing 3D Equations of Motion

Figure1 shows an elastic transmission line with its two ends hinge-supported at the same height. The initial static equilibrium geometric configuration S0 of the transmission line in the plane Oxy is used as the reference location and is represented by function x(y). dc, Lc, and Tc(y,t) represent the sag and span of the dynamic tension in the transmission line, respectively. SD represents the 3D dynamic configuration of the transmission line under an external excitation P(t). P(x,t), P(y,t), and P(z,t) represent the components of the external excitation P(t) in the x-, y-, and z-axes, respectively. Ec and Ac represent the elastic modulus and the cross-sectional area of the transmission line, respectively. γc represents the dead weight of the transmission line per unit length (γc=mcg, where g represents the acceleration of gravity). uc(y,t), vc(y,t), and wc(y,t) represent the displacements of the transmission line along the longitudinal in-plane vibration (y-axis direction), the transverse in-plane vibration (x-axis direction), and the transverse out-of-plane vibration (z-axis direction), respectively.Figure 1
Spatial configuration of a transmission line.To simplify analysis, the following assumptions [12, 25] are made: (a) the flexural, torsional, and shear stiffness of the transmission line is sufficiently low and thus negligible; (b) the transmission line only bears tension in the axial direction, and its axial strain, when vibrating, is sufficiently small and thus negligible; (c) only geometric nonlinearity is considered, whereas material nonlinearity is not considered.When the tension in the transmission line is in the initial static equilibrium state, the lengthds0 of the differential element at any arbitrary location yc along the y-axis direction is(1)ds0=1+∂x∂y2dy.According to the definition of Lagrangian strain, initially, the lengthds of the differential element of the transmission line under the action of gravity is(2)ds=1+∂x/∂y21+εc0dy,where εc0 represents the initial static strain of the transmission line, which is calculated using the following equation:(3)εc0=Tcy,0EcAc,where Tc(y,0) represents the initial tension in the transmission line.The length of the differential element of the transmission line after it has undergone dynamic deformation under an external excitationP(t) is (4)dsD=1+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2dy.The total dynamic strain [12] of the transmission line in a motion state under the external excitation is(5)εc=dsD-dsds=1+εc01+∂x/∂y21+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2-1.The transient total tensionTc(y,t) in the differential element of the transmission line in a motion state is(6)Tcy,t=Tcy,0+EcAcεc=Tcy,0+11+∂x/∂y21+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2-1EcAc.According to a generalizedHamilton’s principle,(7)δ∫t1t2Kv-Udt+∫t1t2δWdt=0,where Kv, U, δW represent the kinetic energy and the strain energy of the transmission line and the virtual work done by nonconservative forces (including the virtual work δWc done by the damping force and the virtual work δWP done by the external excitation), respectively.The kinetic energy of the transmission line is(8)Kv=12∫0Lcγc1+∂x/∂y2g1+εc0u˙c2+v˙c2+w˙c2dy,where γc1+∂x/∂y2/g(1+εc0) represents the mass of the transmission line in the static equilibrium state per unit length and u˙c, v˙c, and w˙c represent the vibration velocities of the transmission line in the y-, x-, and z-axis directions, respectively.According to the aforementioned assumptions, the strain energy of the transmission line can only be generated by an axial tension. According to the principle of virtual work, the strain energy is(9)U=∫0LcTcy,0εcds+12∫0LcEcAcεc2ds.The virtual work done by the damping force is(10)δWc=-∫0Lcccu˙cδuc+v˙cδvc+w˙cδwcdy,where cc represents the structural damping of the transmission line.The virtual work done by the external excitationP(t) is(11)δWP=∫0LcPy,tδuc+Px,tδvc+Pz,tδwcdy.By substituting (8)–(11) into (7) and considering the boundary conditions, the following is satisfied at the locations yc=0 and yc=Lc:(12)δuc=δvc=δwc=0.Thus, theEuler equations for the transmission line along the virtual displacement δuc, δvc, and δwc directions are obtained:(13)∂∂yEcAc1+∂uc/∂y1+∂x/∂y2+Tcy,0-EcAc1+∂uc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0u¨c+ccu˙c-Py,t,∂∂yEcAc∂x/∂y+∂vc/∂y1+∂x/∂y2+Tcy,0-EcAc∂x/∂y+∂vc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,∂∂yEcAc∂wc/∂y1+∂x/∂y2+Tcy,0-EcAc∂wc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.Because the sag-to-span ratiodc/Lc of a transmission line is generally very small, ∂uc/∂y≪1,  ∂vc/∂y≪1,  ∂wc/∂y≪1, and ∂x/∂y≪1 under small-strain conditions. Thus, these small quantities can each be expanded in a Taylor series, and their higher-order terms are negligible. In other words, in the abovementioned equations,(14)11+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2≈1-∂uc∂y-∂uc∂y2+∂x∂y·∂vc∂y+12∂x∂y2+12∂vc∂y2+12∂wc∂y2.Let(15)η=∂uc∂y-∂uc∂y2+∂x∂y·∂vc∂y+12∂x∂y2+12∂vc∂y2+12∂wc∂y2.By substituting (15) into (13), nonlinear equations of motion for the elastic transmission line that is horizontally hinge-supported at the two ends are obtained:(16)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η1+∂uc∂y=γc1+∂x/∂y2g1+εc0u¨c+ccu˙c-Py,t,(17)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η∂x∂y+∂vc∂y=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,(18)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η∂wc∂y=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.In (16)–(18), u¨c,v¨c,w¨c represent the accelerations of vibration of the transmission line along the y-, x-, and z-axes, respectively.The above set of partial differential equations contains all the main vibration characteristics of the initial elastic transmission line continuum system, that is, the quadratic and cubic coupling terms associated with the initial deflection, the axial tension, and the spatial motion, which can be used to analyze the nonlinear dynamic behavior of the transmission line.


### 2.2. Equations of Coupled In-Plane and Out-of-Plane Motion for a Hinge-Supported Transmission Line

Considering that the axial stiffnessEcAc of a transmission line is generally relatively high and its axial elongation due to vibration is far smaller than its transverse vibration amplitude, the axial inertia force of a transmission line when in vibration is negligible. Under normal circumstances, the axial damping of a transmission line is relatively small; thus, the axial damping force of a transmission line when in motion is negligible. In addition, in Figure 1, since there is no external excitation along the axial direction of the transmission line (y-axis direction), P(y,t)=0. Thus, (16) can be rewritten as(19)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η1+∂uc∂y=0.The following approximation relation is then used:(20)11+∂x/∂y2≈1-12∂x∂y2.By substituting (20) into (19), integrating the two sides of (19) over the span [0,Lc] of the transmission line once, and ignoring the terms unrelated to time t, we have(21)Tcy,01-12∂x∂y2·∂uc∂y-Tcy,0-EcAcη-Tcy,0-EcAcη·∂uc∂y=ħt,where ħ(t) is an arbitrary function. By dividing the two sides of (21) by EcAc and considering that the initial tension Tc(y,0) in a transmission line is generally far smaller than its axial stiffness EcAc and that the axial deformation of a transmission line is a higher-order small quantity compared to its vertical in-plane deformation and out-of-plane deformation, we have(22)∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2≈ħtEcAc.To determineħ(t), the two sides of (22) are integrated over the interval [0,Lc] and the transmission line boundary conditions represented by (23) are used. After simplification, we have (24).(23)uc0,t=0,ucLc,t=0,(24)ħtEcAc·Lc=∫0Lc∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2dy.Let us set(25)λt=ħtEcAc=1Lc∫0Lc∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2dy.By substituting (22) and (25) into (17) and (18), respectively, we have(26)∂∂yTcy,0·1-12∂x∂y2·∂vc∂y-Tcy,0-EcAc·λt·∂x∂y-∂∂yTcy,0-EcAc·λt·∂vc∂y=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,(27)∂∂yTcy,0·1-12∂x∂y2·∂wc∂y-Tcy,0-EcAc·λt·∂wc∂y=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.For ease of analysis, by use ofGalerkin’s modal truncation method, the in-plane and out-of-plane response of the transmission line is represented by(28)vcy,t=∑i=1Nφiy·qit,wcy,t=∑i=1Nφiy·qit.In (28), φi(y)=sin⁡(iπy/Lc) is the ith-order vibration mode function (the two ends of the conductor line are considered to be hinge-connected), N represents the retained mode order, and qi(t) represents the generalized coordinates. By substituting it into (26) and (27), (28) can be transformed to ordinary differential equations of motion, which are subsequently solved. Equation (28) is subjected to first-order model truncation usingGalerkin’s modal truncation method; that is, set(29)vcy,t=φvy·qvt=sin⁡πyLc·qvt,wcy,t=φwy·qwt=sin⁡πyLc·qwt.By substituting (29) into (26) and (27), subsequently multiplying the two sides of (26) by φv(y), integrating the resulting equation over the interval [0,Lc], multiplying the two sides of (27) by φw(y), and integrating the resultant equation over the interval [0,Lc], discrete equations of coupled in-plane and out-of-plane motion for the transmission line are obtained:(30)a1q¨vt+a2qvt+a3qv2t+a4qw2t+a5qv3t+a6qw2tqvt+a7q˙vt=Pv,b1q¨wt+b2qwt+b3qvtqwt+b4qv2tqwt+b5qw3t+b6q˙wt=Pw.The factors in (30) are calculated as follows:(31)a1=∫0Lcγc1+∂x/∂y2g1+εc0φv2ydy,a2=-∫0Lc∂∂yTcy,0·1-12∂x∂y2·φv′y-Tcy,0-EcAcLc·∂x∂y·∫0Lc∂x∂y·φv′ydyφvydy,a3=∫0Lc∂∂yTcy,0-EcAc2Lc·∂x∂y·∫0Lcφv′2ydy+Tcy,0-EcAcLc·φv′y∫0Lc∂x∂y·φv′ydyφvydy,a4=∫0Lc∂∂yTcy,0-EcAc2Lc·∂x∂y·∫0Lcφw′2ydyφvydy,a5=∫0Lc∂∂yTcy,0-EcAc2Lc·φv′y·∫0Lcφv′2ydyφvydy,a6=∫0Lc∂∂yTcy,0-EcAc2Lc·φv′y·∫0Lcφw′2ydyφvydy,a7=∫0Lcccφv2ydy,Pv=∫0LcPx,tφvydy,b1=∫0Lcγc1+∂x/∂y2g1+εc0φw2ydy,b2=-∫0Lc∂∂yTcy,0·1-12∂x∂y2·φw′yφwydy,b3=∫0Lc∂∂yTcy,0-EcAcLc·φw′y·∫0Lc∂x∂y·φv′ydyφwydy,b4=∫0Lc∂∂yTcy,0-EcAc2Lc·φw′y·∫0Lcφv′2ydyφwydy,b5=∫0Lc∂∂yTcy,0-EcAc2Lc·φw′y·∫0Lcφw′2ydyφwydy,b6=∫0Lcccφw2ydy,Pw=∫0LcPz,tφwydy.As demonstrated in (30), the in-plane and out-of-plane motion of the transmission line exhibits cubic nonlinearity, whereas the coupled in-plane and out-of-plane motion exhibits linearity, quadratic nonlinearity, and cubic nonlinearity; that is, the nonlinear coupling is very strong.


## 2.1. Establishing 3D Equations of Motion

Figure1 shows an elastic transmission line with its two ends hinge-supported at the same height. The initial static equilibrium geometric configuration S0 of the transmission line in the plane Oxy is used as the reference location and is represented by function x(y). dc, Lc, and Tc(y,t) represent the sag and span of the dynamic tension in the transmission line, respectively. SD represents the 3D dynamic configuration of the transmission line under an external excitation P(t). P(x,t), P(y,t), and P(z,t) represent the components of the external excitation P(t) in the x-, y-, and z-axes, respectively. Ec and Ac represent the elastic modulus and the cross-sectional area of the transmission line, respectively. γc represents the dead weight of the transmission line per unit length (γc=mcg, where g represents the acceleration of gravity). uc(y,t), vc(y,t), and wc(y,t) represent the displacements of the transmission line along the longitudinal in-plane vibration (y-axis direction), the transverse in-plane vibration (x-axis direction), and the transverse out-of-plane vibration (z-axis direction), respectively.Figure 1
Spatial configuration of a transmission line.To simplify analysis, the following assumptions [12, 25] are made: (a) the flexural, torsional, and shear stiffness of the transmission line is sufficiently low and thus negligible; (b) the transmission line only bears tension in the axial direction, and its axial strain, when vibrating, is sufficiently small and thus negligible; (c) only geometric nonlinearity is considered, whereas material nonlinearity is not considered.When the tension in the transmission line is in the initial static equilibrium state, the lengthds0 of the differential element at any arbitrary location yc along the y-axis direction is(1)ds0=1+∂x∂y2dy.According to the definition of Lagrangian strain, initially, the lengthds of the differential element of the transmission line under the action of gravity is(2)ds=1+∂x/∂y21+εc0dy,where εc0 represents the initial static strain of the transmission line, which is calculated using the following equation:(3)εc0=Tcy,0EcAc,where Tc(y,0) represents the initial tension in the transmission line.The length of the differential element of the transmission line after it has undergone dynamic deformation under an external excitationP(t) is (4)dsD=1+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2dy.The total dynamic strain [12] of the transmission line in a motion state under the external excitation is(5)εc=dsD-dsds=1+εc01+∂x/∂y21+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2-1.The transient total tensionTc(y,t) in the differential element of the transmission line in a motion state is(6)Tcy,t=Tcy,0+EcAcεc=Tcy,0+11+∂x/∂y21+∂uc∂y2+∂x∂y+∂vc∂y2+∂wc∂y2-1EcAc.According to a generalizedHamilton’s principle,(7)δ∫t1t2Kv-Udt+∫t1t2δWdt=0,where Kv, U, δW represent the kinetic energy and the strain energy of the transmission line and the virtual work done by nonconservative forces (including the virtual work δWc done by the damping force and the virtual work δWP done by the external excitation), respectively.The kinetic energy of the transmission line is(8)Kv=12∫0Lcγc1+∂x/∂y2g1+εc0u˙c2+v˙c2+w˙c2dy,where γc1+∂x/∂y2/g(1+εc0) represents the mass of the transmission line in the static equilibrium state per unit length and u˙c, v˙c, and w˙c represent the vibration velocities of the transmission line in the y-, x-, and z-axis directions, respectively.According to the aforementioned assumptions, the strain energy of the transmission line can only be generated by an axial tension. According to the principle of virtual work, the strain energy is(9)U=∫0LcTcy,0εcds+12∫0LcEcAcεc2ds.The virtual work done by the damping force is(10)δWc=-∫0Lcccu˙cδuc+v˙cδvc+w˙cδwcdy,where cc represents the structural damping of the transmission line.The virtual work done by the external excitationP(t) is(11)δWP=∫0LcPy,tδuc+Px,tδvc+Pz,tδwcdy.By substituting (8)–(11) into (7) and considering the boundary conditions, the following is satisfied at the locations yc=0 and yc=Lc:(12)δuc=δvc=δwc=0.Thus, theEuler equations for the transmission line along the virtual displacement δuc, δvc, and δwc directions are obtained:(13)∂∂yEcAc1+∂uc/∂y1+∂x/∂y2+Tcy,0-EcAc1+∂uc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0u¨c+ccu˙c-Py,t,∂∂yEcAc∂x/∂y+∂vc/∂y1+∂x/∂y2+Tcy,0-EcAc∂x/∂y+∂vc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,∂∂yEcAc∂wc/∂y1+∂x/∂y2+Tcy,0-EcAc∂wc/∂y1+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.Because the sag-to-span ratiodc/Lc of a transmission line is generally very small, ∂uc/∂y≪1,  ∂vc/∂y≪1,  ∂wc/∂y≪1, and ∂x/∂y≪1 under small-strain conditions. Thus, these small quantities can each be expanded in a Taylor series, and their higher-order terms are negligible. In other words, in the abovementioned equations,(14)11+∂uc/∂y2+∂x/∂y+∂vc/∂y2+∂wc/∂y2≈1-∂uc∂y-∂uc∂y2+∂x∂y·∂vc∂y+12∂x∂y2+12∂vc∂y2+12∂wc∂y2.Let(15)η=∂uc∂y-∂uc∂y2+∂x∂y·∂vc∂y+12∂x∂y2+12∂vc∂y2+12∂wc∂y2.By substituting (15) into (13), nonlinear equations of motion for the elastic transmission line that is horizontally hinge-supported at the two ends are obtained:(16)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η1+∂uc∂y=γc1+∂x/∂y2g1+εc0u¨c+ccu˙c-Py,t,(17)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η∂x∂y+∂vc∂y=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,(18)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η∂wc∂y=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.In (16)–(18), u¨c,v¨c,w¨c represent the accelerations of vibration of the transmission line along the y-, x-, and z-axes, respectively.The above set of partial differential equations contains all the main vibration characteristics of the initial elastic transmission line continuum system, that is, the quadratic and cubic coupling terms associated with the initial deflection, the axial tension, and the spatial motion, which can be used to analyze the nonlinear dynamic behavior of the transmission line.


## 2.2. Equations of Coupled In-Plane and Out-of-Plane Motion for a Hinge-Supported Transmission Line

Considering that the axial stiffnessEcAc of a transmission line is generally relatively high and its axial elongation due to vibration is far smaller than its transverse vibration amplitude, the axial inertia force of a transmission line when in vibration is negligible. Under normal circumstances, the axial damping of a transmission line is relatively small; thus, the axial damping force of a transmission line when in motion is negligible. In addition, in Figure 1, since there is no external excitation along the axial direction of the transmission line (y-axis direction), P(y,t)=0. Thus, (16) can be rewritten as(19)∂∂yEcAc1+∂x/∂y2+Tcy,0-EcAc1-η1+∂uc∂y=0.The following approximation relation is then used:(20)11+∂x/∂y2≈1-12∂x∂y2.By substituting (20) into (19), integrating the two sides of (19) over the span [0,Lc] of the transmission line once, and ignoring the terms unrelated to time t, we have(21)Tcy,01-12∂x∂y2·∂uc∂y-Tcy,0-EcAcη-Tcy,0-EcAcη·∂uc∂y=ħt,where ħ(t) is an arbitrary function. By dividing the two sides of (21) by EcAc and considering that the initial tension Tc(y,0) in a transmission line is generally far smaller than its axial stiffness EcAc and that the axial deformation of a transmission line is a higher-order small quantity compared to its vertical in-plane deformation and out-of-plane deformation, we have(22)∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2≈ħtEcAc.To determineħ(t), the two sides of (22) are integrated over the interval [0,Lc] and the transmission line boundary conditions represented by (23) are used. After simplification, we have (24).(23)uc0,t=0,ucLc,t=0,(24)ħtEcAc·Lc=∫0Lc∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2dy.Let us set(25)λt=ħtEcAc=1Lc∫0Lc∂uc∂y+∂x∂y·∂vc∂y+12∂wc∂y2+12∂vc∂y2dy.By substituting (22) and (25) into (17) and (18), respectively, we have(26)∂∂yTcy,0·1-12∂x∂y2·∂vc∂y-Tcy,0-EcAc·λt·∂x∂y-∂∂yTcy,0-EcAc·λt·∂vc∂y=γc1+∂x/∂y2g1+εc0v¨c+ccv˙c-Px,t,(27)∂∂yTcy,0·1-12∂x∂y2·∂wc∂y-Tcy,0-EcAc·λt·∂wc∂y=γc1+∂x/∂y2g1+εc0w¨c+ccw˙c-Pz,t.For ease of analysis, by use ofGalerkin’s modal truncation method, the in-plane and out-of-plane response of the transmission line is represented by(28)vcy,t=∑i=1Nφiy·qit,wcy,t=∑i=1Nφiy·qit.In (28), φi(y)=sin⁡(iπy/Lc) is the ith-order vibration mode function (the two ends of the conductor line are considered to be hinge-connected), N represents the retained mode order, and qi(t) represents the generalized coordinates. By substituting it into (26) and (27), (28) can be transformed to ordinary differential equations of motion, which are subsequently solved. Equation (28) is subjected to first-order model truncation usingGalerkin’s modal truncation method; that is, set(29)vcy,t=φvy·qvt=sin⁡πyLc·qvt,wcy,t=φwy·qwt=sin⁡πyLc·qwt.By substituting (29) into (26) and (27), subsequently multiplying the two sides of (26) by φv(y), integrating the resulting equation over the interval [0,Lc], multiplying the two sides of (27) by φw(y), and integrating the resultant equation over the interval [0,Lc], discrete equations of coupled in-plane and out-of-plane motion for the transmission line are obtained:(30)a1q¨vt+a2qvt+a3qv2t+a4qw2t+a5qv3t+a6qw2tqvt+a7q˙vt=Pv,b1q¨wt+b2qwt+b3qvtqwt+b4qv2tqwt+b5qw3t+b6q˙wt=Pw.The factors in (30) are calculated as follows:(31)a1=∫0Lcγc1+∂x/∂y2g1+εc0φv2ydy,a2=-∫0Lc∂∂yTcy,0·1-12∂x∂y2·φv′y-Tcy,0-EcAcLc·∂x∂y·∫0Lc∂x∂y·φv′ydyφvydy,a3=∫0Lc∂∂yTcy,0-EcAc2Lc·∂x∂y·∫0Lcφv′2ydy+Tcy,0-EcAcLc·φv′y∫0Lc∂x∂y·φv′ydyφvydy,a4=∫0Lc∂∂yTcy,0-EcAc2Lc·∂x∂y·∫0Lcφw′2ydyφvydy,a5=∫0Lc∂∂yTcy,0-EcAc2Lc·φv′y·∫0Lcφv′2ydyφvydy,a6=∫0Lc∂∂yTcy,0-EcAc2Lc·φv′y·∫0Lcφw′2ydyφvydy,a7=∫0Lcccφv2ydy,Pv=∫0LcPx,tφvydy,b1=∫0Lcγc1+∂x/∂y2g1+εc0φw2ydy,b2=-∫0Lc∂∂yTcy,0·1-12∂x∂y2·φw′yφwydy,b3=∫0Lc∂∂yTcy,0-EcAcLc·φw′y·∫0Lc∂x∂y·φv′ydyφwydy,b4=∫0Lc∂∂yTcy,0-EcAc2Lc·φw′y·∫0Lcφv′2ydyφwydy,b5=∫0Lc∂∂yTcy,0-EcAc2Lc·φw′y·∫0Lcφw′2ydyφwydy,b6=∫0Lcccφw2ydy,Pw=∫0LcPz,tφwydy.As demonstrated in (30), the in-plane and out-of-plane motion of the transmission line exhibits cubic nonlinearity, whereas the coupled in-plane and out-of-plane motion exhibits linearity, quadratic nonlinearity, and cubic nonlinearity; that is, the nonlinear coupling is very strong.


## 3. Theoretical Analysis of the Characteristics of the Nonlinear Coupled In-Plane and Out-of-Plane Internal Resonance of Transmission Lines

To further study the nonlinear coupled in-plane and out-of-plane dynamic characteristics of an overhead transmission line, we conducted the following analysis with the help of the multiple-scale (MS) method [26–29], which is an effective method for the analysis of nonlinear dynamics.Assume that the external excitation is a harmonic excitation: that is,(32)Px,t=Avcos⁡Ωxt,Pz,t=Awcos⁡Ωzt,where Av and Aw represent the amplitudes of the in-plane excitation and the out-of-plane excitation, respectively, on the transmission line, and Ωx and Ωz represent the circular frequencies of the in-plane excitation and the out-of-plane excitation, respectively.Here, the in-plane excitation on the transmission line is not considered, that is, the generalized in-plane excitation loadPv is assumed to be zero. Now, the generalized out-of-plane excitation load on the transmission line is set to Pw=P0cos⁡Ωzt. In addition, the parameters ε, α1, α2, μ1, μ2, κ1, κ2, β1, β2, γ1, and f2 are introduced to transform (30) to dimensionless equations:(33)q¨vt+ω12qvt+εα1qv2t+εκ1qw2t+εβ1qv3t+εγ1qw2tqvt+2εμ1q˙vt=0,q¨wt+ω22qwt+εα2qvtqwt+εκ2qv2tqwt+εβ2qw3t+2εμ2q˙wt=2εf2cos⁡Ωzt,where ω1 and ω2 represent the circular frequencies of the in-plane and out-of-plane vibrations of the transmission line, respectively. The abovementioned parameters have the following conversion relationships:(34)ω12=a2a1,2εμ1=a7a1,εα1=a3a1,εκ1=a4a1,εβ1=a5a1,εγ1=a6a1,ω22=b2b1,2εμ2=b6b1,εα2=b3b1,εκ2=b4b1,εβ2=b5b1,2εf2=P0b1.The abovementioned set of nonlinear equations can be solved using the MS method, the basic idea of which is to regard the expansion of the response as a function with multiple independent variables (or multiple scales) [26].The following independent variables are introduced:(35)Tn=εntn=0,1,…,that is,(36)T0=t,T1=εt,….Thus, the derivative with respect tot is transformed to the expansion of a partial derivative with respect to Tn, that is,(37)ddt=dT0dt∂∂T0+dT1dt∂∂T1+⋯=D0+εD1+ε2D2+⋯,d2dt2=dT0dt∂2∂T02+2ε∂2∂T0∂T1+⋯=D02+2εD0D1+ε2D21+2D1D2+⋯.Hence, the solutions to (33) can be expressed as follows:(38)qvt,ε=qv0T0,T1,T2,…+εqv1T0,T1,T2,…+ε2qv2T0,T1,T2,…+⋯.qwt,ε=qw0T0,T1,T2,…+εqw1T0,T1,T2,…+ε2qw2T0,T1,T2,…+⋯,By substituting (37) and (38) into (33), we have(39)D12qv2+2μ1D1qv2+2μ1D2qv1+2D0D1qv3+2μ1D0qv3+β1qv13+2α1qv1qv2+6β1qv0qv1qv2+2α1qv0qv3+3β1q2v0qv3+γ1qw02qv3+2γ1qv2qw0qw1+γ1qw12qv1+2γ1qv1qw0qw2+2κ1qw1qw2+2γ1qv0qw1qw2+2κ1qw0qw3+2γ1qv0qw0qw3ε4+D02qv3+ω12qv3+2μ1D2qv0+D12qv1+2μ1D1qv1+2D0D1qv2+2μ1D0qv2+α1qv12+3β1qv0q2v1+2α1qv0qv2+3β1qv02qv2+γ1qv2qw02+2γ1qv1qw0qw1+κ1qw12+γ1qv0qw12+2κ1qw0qw2+2γ1qv0qw0qw2ε3+D02qv2+ω12qv2+D12qv0+2μ1D0qv0+2D0D1qv1+2μ1D0qv1+3β1qv02qv1+2κ1qw0qw1+γ1qw02qv1+2γ1qw0qw1qv0+2α1qv0qv1ε2+D02qv1+ω12qv1+2D0D1qv0+2μ1D0qv0+α1qv02+κ1qw02+β1qv03+γ1qw02qv0ε+D02qv0+ω12qv0=0,D12qw2+2μ2D1qw2+2μ2D0qw3+2D0D1qw3+2μ2D2qw1+β2qw13+α2qv3qw0+2κ2qv0qv3qw0+2κ2qv1qv2qw0+κ2qv12qw1+α2qv2qw1+2κ2qv0qv2qw1+α2qv1qw2+2κ2qv0qv1qw2+6β2qw0qw1qw2+α2qv0qw3+κ2qv02qw3+3β2q2w0qw3ε4+D02qw3+ω22qw3+2μ2D2qw0+D12qw1+2μ2D1qw1+2D0D1qw2+2μ2D0qw2+α2qv2qw0+κ2qv12qw0+3β2qw0q2w1+α2qv1qw1+3β2qw02qw2+2κ2qv0qv2qw0+2κ2qv0qv1qw1+κ2qv02qw0+α2qv0qw2ε3+D02qw2+ω22qw2+D12qw0+2μ2D1qw0+2D0D1qw2+2μ2D0qw2+D12qw1+2μ2D1qw1+3β2qw02qw2+3β2qw0qw12+2κ2qv0qv1qw1+2κ2qv0qv2qw0+α2qv2qw0+α2qv1qw1+α2qv0qw2+κ2qv02qw2ε2+D02qw1+ω22qw1+2D0D1qw0+2μ2D0qw0+α2qv0qw0+κ2qv02qw0+β2qw03ε+D02qw0+ω22qw0=2εf2cos⁡Ωzt.By setting the sum of the coefficients of the same power ofε to zero, we haveε 0: (40)D02qv0+ω12qv0=0,D02qw0+ω22qw0=0,ε 1:(41)D02qv1+2D0D1qv0+2μ1D0qv0+ω12qv1+β1qv03+a1qv02+γ1qw02qv0+κ1qw02=0,D02qw1+2D0D1qw0+β2qw03+ω22qw1+a2qv0qw0+2μ2D0qw0+κ2qv02qw0=2f2cos⁡Ωzt,ε 2:(42)D02qv2+ω12qv2+D12qv0+2μ1D0qv0+2D0D1qv1+2μ1D0qv1+3β1qv02qv1+2κ1qw0qw1+γ1qw02qv1+2γ1qw0qw1qv0+2α1qv0qv1=0,D02qw2+ω22qw2+D12qw0+2μ2D1qw0+2D0D1qw2+2μ2D0qw2+D12qw1+2μ2D1qw1+3β2qw02qw2+3β2qw0qw12+2κ2qv0qv1qw1+2κ2qv0qv2qw0+α2qv2qw0+α2qv1qw1+α2qv0qw2+κ2qv02qw2=0,ε 3:(43)D02qv3+ω12qv3+2μ1D2qv0+D12qv1+2μ1D1qv1+2D0D1qv2+2μ1D0qv2+α1qv12+3β1qv0q2v1+2α1qv0qv2+3β1qv02qv2+γ1qv2qw02+2γ1qv1qw0qw1+κ1qw12+γ1qv0qw12+2κ1qw0qw2+2γ1qv0qw0qw2=0,D02qw3+ω22qw3+2μ2D2qw0+D12qw1+2μ2D1qw1+2D0D1qw2+2μ2D0qw2+α2qv2qw0+κ2qv12qw0+3β2qw0q2w1+α2qv1qw1+3β2qw02qw2+2κ2qv0qv2qw0+2κ2qv0qv1qw1+κ2qv02qw0+α2qv0qw2=0.By solving the set of equations in (40), we have(44)qv0=AvT1,T2exp⁡iω1T0+A-vT1,T2exp⁡-iω1T0,qw0=AwT1,T2exp⁡iω2T0+A-wT1,T2exp⁡-iω2T0,where Av and Aw are unknown complex functions and A-v and A-w are the conjugates of Av and Aw, respectively. For ease of description, conjugate terms are denoted by cc. The governing equations for Av and Aw can be solved by requiring qv0, qv1, qw0, and qw1 to be functions with a period of T0.By substituting (44) into (41), we have(45)D02qv1+ω12qv1=-2iω1D1Avei2ω1T0-2iω1μ1Avei2ω1T0-γ1Aw2Avei2ω2+ω1T0-γ1A-w2Aveiω1-2ω2T0-3β1Av2A-veiω1T0-2γ1AwA-wAveiω1T0-β1Av3ei3ω1T0-α1Av2ei2ω1T0-2α1AvA-v-2κ1AwA-w-κ1Aw2ei2ω2T0+cc,D02qw1+ω12qw1=-2iω2D1Awei2ω2T0-2iω2μ2Awei2ω2T0-3β2Aw2A-weiω2T0-2κ2AvA-vAweiω2T0-β2Aw3e3iω2T0-α2AvAweiω1+ω2T0-α2AvA-weiω1-ω2T0-κ2Av2Awei2ω1+ω2T0-κ2A-v2Aweiω2-2ω1T0+f2eiΩzT0+cc.From (45), we can easily find that the terms that contain ei2ω1T0 and ei2ω2T0 can lead to secular terms in the solutions of the above equation. In addition, from (30), we know that the coupling of the in-plane and out-of-plane motion of the transmission line exhibits linearity, quadratic nonlinearity, and cubic nonlinearity. Thus, if parameters exist that allow the natural frequencies of the in-plane and out-of-plane vibration of the transmission line to satisfy any of the relations ω1≈ω2,  ω1≈2ω2, and ω2≈2ω1, the solutions to the abovementioned partial differential equations that describe the motion of the transmission line will contain an additional term that links qv1 and qw1, which is exactly the secular term caused by internal resonance. Therefore, when solving (45), we should consider the cases of ω1≈ω2, ω1≈2ω2, and ω2≈2ω1. In view of the complexity of the problem-solving and the purpose of this study, we merely take the case of ω1≈ω2 as an example to illustrate the analysis process.Assume that the natural frequencies of the in-plane and out-of-plane vibrations of the transmission line meet the following relationship:(46)ω2-ω1=εσ,σ=O1.Thus,(47)ω2+ω1T0=2ω1T0+εσT0=2ω1T0+σT1=2ω2T0+σT1,2ω2T0=2ω1+εσT0=2ω2T0+2σT1.By substituting (47) into (45), we can obtain the requirements to eliminate the secular terms as follows:(48)2iω1D1Av+μ1Av+α1Av2+κ1Aw2ei2σT1=0,2iω2D1Aw+μ2Aw+α2AvAweiσT1=0.The following formulas are introduced:(49)Ar=ar2ejθr,r=v,w.By substituting (49) into (48) and separating the real part from the imaginary part, we have(50)D1av+μ1av+α1av22ω1sin⁡θv+κ1aw24ω1sin⁡2θw+σT1sec⁡θv=0,avD1θv-csc⁡θv4ω1α1av2cos⁡2θv+κ1aw2cos⁡2θw+σT1=0,D1aw+μ2aw+α2avaw4ω2cos⁡θwsin⁡θv+θw+σT1=0,awD1θw+α2avaw4ω2cos⁡θwcos⁡θv+θw+σT1=0.Let us introduce the new phase angle as follows:(51)φ1=2θw+σT1,φ2=θv+θw+σT1.By substituting (51) into (50), we have(52)D1av+μ1av+α1av22ω1sin⁡θv+κ1aw24ω1sin⁡φ1sec⁡θv=0,avD1θv-csc⁡θv4ω1α1av2cos⁡2θv+κ1aw2cos⁡φ1=0,D1aw+μ2aw+α2avaw4ω2cos⁡θwsin⁡φ2=0,awD1θw+α2avaw4ω2cos⁡θwcos⁡φ2=0.When solving (52), we must discuss whether the damping of the system is considered. In addition, even if we can obtain the one-order approximate steady-state solution of the equation, we must investigate its asymptotic stability to determine whether it actually exists. Hence, we can see that the above issue is so complicated that it should be specifically studied. Due to the limitation of the abovementioned approximation method, it is difficult to obtain the theoretical solutions to the equations of coupled in-plane and out-of-plane motion for transmission lines. Therefore, in the subsequent section, we intend to perform the quantitative analysis of (30) by using numerical methods.Because the main purpose of this article is to reveal the inherently nonlinear vibration characteristics of transmission lines, the above MS analysis is merely used to determine the requirements for eliminating secular terms in the solutions of the partial differential equations of transmission lines. Because the second-order approximation solutions can provide enough information for most practical problems, it is usually necessary to give only the second-order approximation solutions of a nonlinear system by using the MS method [27–29]. For this reason, we consider only the power ofεup to 2 in this paper. From the requirements for eliminating secular terms, it is clear that internal resonance is considered to exist in the transmission line if parameters exist that allow the natural frequencies of the in-plane and out-of-plane vibration of the transmission line to satisfy any of the following relations: ω1≈ω2, ω1≈2ω2, and ω2≈2ω1.


## 4. Numerical Analysis of the Characteristics of the Nonlinear Coupled In-Plane and Out-of-Plane Internal Resonance of Transmission Lines

In numerical analysis, theRunge-Kutta method is widely used to obtain the approximate solutions of differential equations. This method is so accurate that most computer packages designed to find numerical solutions for differential equations will use it by default—the fourth-orderRunge-Kutta method. Therefore, we employ a higher-orderRunge-Kutta method to obtain the numerical solutions to these equations, with the aim of revealing the nonlinear vibration characteristics of transmission lines.The following state variables are introduced:(53)Y1=qv,Y2=q˙v,Y3=qw,Y˙4=q˙w.Thus, (30) can be transformed to state functions:(54)Y˙1=Y2,Y˙2=Pv-a2Y1-a3Y12-a4Y32-a5Y13-a6Y32Y1-a7Y2a1,Y˙3=Y4,Y˙4=Pw-b2Y3-b3Y1Y3-b4Y12Y3-b5Y33-b6Y4b1.


### 4.1. Calculation of the Parameters

A relatively large single-span transmission line in a high-voltage transmission line system in East China (Figure2) is selected as an example for analysis [30]. The conductors of this span of the transmission line are composed of LGJ-630/45 ASCR cables. Table 1 lists the design parameters of the selected transmission line. The maximum design wind speed at a height of 10 m above the ground for safe operation of the selected transmission line is 25.3 m/s. In the actual transmission line, the conductors of each phase are quad-bundled conductors. To simplify calculation, in the finite element (FE) modeling process, the quad-bundled conductors of each phase are equivalently simplified to one conductor based on the following principles: the bundled conductors have the same windward area, the same total operational tension, and the same linear density. Table 2 lists the parameters of the transmission line after simplification.Table 1
Original design parameters of the selected transmission line.
Items Diameterd  (mm) Cross-sectional areaAc    (mm2) Linear densitymc  (kg/m) Elastic modulusEc (MPa) Average operational tension (N) Tension at design wind speed(N) Breaking force(N) SpanLc  (m) Sagdc  (m) Conductors 33.6 666.55 2.06 63,000 35,316 43,688 141,265 480 16.6Table 2
Design parameters of the selected transmission line after simplification.
Items Diameterd  (mm) Densityρc  (kg/m3) Linear densitymc  (kg/m) Linear loadqc  (N/m) Initial strain  εc0 Conductors 134.4 3,090.5 8.24 80.752 0.00084Figure 2
A high-voltage transmission line.The sag-to-span ratiodc/Lc of the selected transmission line is calculated based on the parameters listed in Table 1 to be approximately 1/29, which is far smaller than 1/8. Thus, (30) can be used to analyze the nonlinear vibration characteristics of this transmission line.


### 4.2. Modal Analysis of the Transmission Line

Based on the actual characteristics of the selected transmission line in combination with the design parameters listed in Tables1 and 2, LINK10 elements in ANSYS are used to construct an FE model for the selected transmission line. First, a modal analysis is performed to determine the natural frequency of vibration of each order of the transmission line model; this provides a basis for the subsequent numerical simulation. To validate the FE model constructed for a single transmission line, in this section, the FE simulation results with respect to the natural vibration frequencies of the transmission line are also compared with the theoretical solutions.A transmission line is a typical type of suspended cable structure, and the analytical solutions for the natural frequencies of its vibration can be obtained based on the single-cable theory. A transmission line resists external loading primarily by self-stretching and consequently will undergo relatively large displacement under loading. Therefore, the geometric nonlinearity of a transmission line must be taken into consideration in the calculation. When constructing an FE model of a transmission line, one crucial point is to determine the initial configuration of the cable. Specifically, on the one hand, this means to determine the spatial location of the transmission line by determining the equilibrium and deformation compatibility equations for the transmission line based on the existing suspended cable theory; on the other hand, this means to transition the transmission line from its initial stress-free state to an initial loaded state based on a certain configuration-seeking approach.The catenary and parabolic methods are the main methods used to determine the equilibrium equation for a suspended cable. The former is an accurate method, whereas the latter is an approximate method. However, when the sag-to-span ratiodc/Lc of a cable is less than 1/8, the parabolic method can also yield a relatively accurate solution for the initial configuration of the transmission line. Here, assuming that the vertical load on a transmission line is evenly distributed along the span, as shown in Figure 3, the equation for the spatial configuration of the transmission line is(55)xy=qc2Tc0yLc-y+ΔcLcy,where qc represents the vertical load on the transmission line, which in this case is the dead weight of the transmission line; Tc0 represents the initial horizontal tension in the transmission line (Tc0=qcLc2/8dc); Lc represents the horizontal span of the transmission line; Δc represents the difference in height between the two ends of the transmission line; and dc represents the mid-span sag of the transmission line.Figure 3
A schematic diagram of the calculation for a single cable when the load is evenly distributed along the span.Considering that a transmission line generally has a small sag, the first-order natural frequency corresponding to the out-of-plane vibration of the transmission line is the smallest frequency of all the frequencies of its vibration. According to the linear vibration theory of suspended cables, for a horizontal suspended cable with two ends hinged at the same height and whose mass is evenly distributed, the natural frequencies of vibration and the corresponding modes under the action of gravity alone are as follows:(a) The out-of-plane swing and the in-plane vibration of the suspended cable are not coupled. Thus, the natural circular frequencies of the out-of-plane swing and the corresponding modes can be expressed as follows:(56)ωw=nπLcTc0mc,n=1,2,3,…,Φw=Ansin⁡nπyLc,n=1,2,3,…,where ωw represents the natural circular frequency of the out-of-plane vibration of the suspended cable, n represents the modal number, mc represents the mass of the suspended cable per unit length, Φw represents the out-of-plane mode of vibration of the suspended cable, and An represents the modal amplitude (each of the other symbols has the same meaning as previously described).(b) The in-plane antisymmetric vibration of the suspended cable will not generate an increment in its horizontal dynamic tension. The natural circular frequencies of its vibration and the corresponding modes can be expressed as follows:(57)ωv=2nπLcTc0mc,n=1,2,3,…,Φv=Bnsin2nπyLc,n=1,2,3,…,where ωv represents the natural circular frequency of the in-plane antisymmetric vibration of the suspended cable, n represents the modal number, Φv represents the mode of the in-plane antisymmetric vibration of the suspended cable, and Bn represents the modal amplitude (each of the other symbols has the same meaning as previously described).(c) The in-plane symmetric vibration of the suspended cable will generate an additional increment in its dynamic tension. The natural circular frequencies of its vibration and the corresponding modes can be determined by solving the following transcendental equations:(58)tanϖ2=ϖ2-4λ2ϖ23,ϖ=ωvantLcTc0/mc,λ2=mcgLc/Tc02LcTc0Lc/EcAc,Φvant=h~Tc0ϖ21-tan⁡ϖ2sin⁡ϖyLc-cos⁡ϖyLc,where ωvant represents the natural circular frequency of the in-plane symmetric vibration of the suspended cable, Φvant represents the in-plane symmetric mode of vibration of the suspended cable, Ec represents the elastic modulus of the suspended cable, Ac represents the cross-sectional area of the suspended cable, and h~ represents the additional tension in the suspended cable (each of the other symbols has the same meaning as previously described).By substituting the parameters listed in Tables1 and 2 into (55)–(58), the theoretical solutions for the natural vibration frequencies of the first eight orders of the transmission line in the three directions can be obtained (they are shown in Table 3). Table 4 summarizes the linear natural vibration frequencies of the transmission line obtained from FE simulation.Table 3
Theoretical linear natural vibration frequencies of the transmission line model.
Calculation equation First order Second order Third order Fourth order Fifth order Sixth order Seventh order Eighth order Out-of-plane vibration frequencyfw (Hz) f w = ω w ( 2 π ) 0.136 0.273 0.409 0.546 0.682 0.818 0.955 1.091 In-plane symmetric vibration frequencyfv (Hz) f v = ω v ( 2 π ) 0.344 0.46 – – – – – – In-plane antisymmetric vibration frequencyfvant (Hz) f v a n t = ω v a n t ( 2 π ) 0.273 0.546 0.818 1.091 1.364 1.637 1.91 2.182Table 4
Natural vibration frequencies of the transmission line model obtained from FE simulation.
Calculation equation First order Second order Third order Fourth order Fifth order Sixth order Seventh order Eighth order Out-of-plane vibration frequencyfw (Hz) f w = ω w ( 2 π ) 0.136 0.272 0.408 0.544 0.68 0.816 0.952 1.088 In-plane symmetric vibration frequencyfv (Hz) f v = ω v ( 2 π ) 0.346 0.46 0.686 0.954 1.225 1.497 1.769 2.042 In-plane antisymmetric vibration frequencyfvant (Hz) f v a n t = ω v a n t ( 2 π ) 0.27 0.544 0.816 1.088 1.36 1.633 1.91 2.18A comparison of Tables3 and 4 shows that the natural vibration frequencies of the transmission line model obtained from FE simulation are in relatively good agreement with the theoretical values, with the largest difference being approximately 1%. This indicates that the transmission line model constructed in this work is relatively reliable. Moreover, Tables 3 and 4 also demonstrate that the following relationships between the natural frequencies of the in-plane and out-of-plane modes of vibration of the transmission line always exist: ωv≈ωw, ωv≈2ωw, and ωv≈3ωw. In other words, within the scope of the linear theory, the natural vibration frequencies of the transmission line are always multiples of one another.Figure4 shows the modes of vibration of the first nine orders obtained from FE simulation. As demonstrated in Figure 4, of all the modes of vibration of the transmission line, the first-order out-of-plane vibration is the most easily excitable. The higher-order frequencies of in-plane vibration are approximately integer multiples of the fundamental frequency, and this relationship also exists between the frequencies of in-plane antisymmetric vibration and the fundamental frequency. In addition to the relationship to the fundamental frequency, the higher-order frequencies of the out-of-plane vibration are also approximately equal to those of the in-plane vibration, and the in-plane and out-of-plane modes of vibration corresponding to the same frequency are also similar to one another.Figure 4
Modes of vibration of the transmission line of the first nine orders obtained from FE simulation.
(a)
First order, 0.136 Hz (b)
Second order, 0.27 Hz (c)
Third order, 0.272 Hz (d)
Fourth order, 0.346 Hz (e)
Fifth order, 0.408 Hz (f)
Sixth order, 0.46 Hz (g)
Seventh order, 0.544 Hz (h)
Eighth order, 0.544 Hz (i)
Ninth order, 0.68 Hz


### 4.3. Numerical Analysis of the Nonlinear Internal Resonance of the Transmission Line When in Free Vibration

When only the free vibration of the transmission line under the action of the initial displacement is considered, in (54), Pv=0 and Pw=0.The structural damping of a transmission line is often relatively small, and it can generally be set to 0.0005. In addition, the effects of the structural damping of a transmission line on its vibration, particularly under strong wind conditions, are relatively insignificant, whereas aerodynamic damping relatively significantly affects the vibration of a transmission line. By comprehensively considering the aforementioned factors, in this study, when analyzing the free vibration of the transmission line, the dampingcc of its in-plane and out-of-plane vibration is set to 0.001. In the following sections, the effects of the initial potential energy of the transmission line on its internal resonance are analyzed in two scenarios.


#### 4.3.1. When the Initial Potential Energy of the Transmission Line Is Relatively Low (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Small)

Let us set the initial out-of-plane mid-span displacement and the initial velocity of out-of-plane motion of the transmission line to 1 m and 0.00005 m/s, respectively, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=1,  q˙w=0.00005. The displacements of the transmission line obtained from numerical simulation are then divided by the horizontal span Lc to obtain dimensionless displacements. The dimensionless velocities of in-plane and out-of-plane can also be obtained from the numerical simulation results of the in-plane and out-of-plane velocities multiplied by the first-order out-of-plane vibration period T0 and then divided by the horizontal span Lc. The results are presented in Figures 5–7.Figure 5
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively low.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time historyFigure 6
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively low.
(a)
In-plane (b)
Out-of-planeFigure 7
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively low.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement responses in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure5, when the initial energy of the transmission line is relatively low, its in-plane vibration displacement is relatively small, with the maximum amplitude being approximately 1/10 that of its out-of-plane vibration displacement; in addition, the “beat vibration” phenomenon is inconspicuous in the response time history. As demonstrated in the Fourier spectra of the displacement response of the transmission line (Figure 6), the first-order mode of vibration is the primary out-of-plane mode of vibration. The out-of-plane vibration of the transmission line gives rise to the first-order in-plane antisymmetric mode of vibration and the first-order symmetric mode of vibration. In addition, the first-order in-plane symmetric vibration energy is also greater than the first-order in-plane antisymmetric vibration energy, and the in-plane vibration energy amplitude is smaller than the out-of-plane vibration energy amplitude by one order of magnitude. This indicates that the transmission line is inherently capable of generating nonlinear internal resonance and can undergo internal resonance even when its initial energy is relatively low, but energy transfer and mode coupling are quite weak under this condition. As demonstrated in Figure 7, the out-of-plane and in-plane modes of vibration of the transmission line both exhibit periodic motion characteristics.


#### 4.3.2. When the Initial Potential Energy of the Transmission Line Is Relatively High (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Large)

Let us set the initial out-of-plane mid-span displacement of the transmission line to a relatively large value (10 m), the initial velocity of its out-of-plane mid-span motion to 0.00005 m/s, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=10,  q˙w=0.00005. Figures 8–10 show the numerical simulation results.Figure 8
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively high.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacementFigure 9
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively high.
(a)
In-plane (b)
Out-of-planeFigure 10
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively high.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement response in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure8, when the initial out-of-plane displacement of the transmission line is relatively large, as a result of the internal resonance the system energy constantly transfers between the modes that are in the ωv≈ωw, ωv≈2ωw, and ωv≈3ωw relationships; in addition, the system energy also transfers between degrees of freedom that are similarly related to multiples of the frequency. On the one hand, the out-of-plane vibration energy transfers to the in-plane direction, resulting in vibration with a relatively large amplitude. Figure 8 also demonstrates that the order of magnitude of the in-plane vibration displacement amplitude is commensurate with the out-of-plane vibration displacement amplitude; a phenomenon similar to “beat vibration” is also observed in the displacement response time history; that is, as the out-of-plane vibration response gradually decreases, the in-plane vibration response at the corresponding time gradually increases and vice versa. Furthermore, from Figures 5 and 8, we can also see that both of the maximum in-plane and out-of-plane vibration amplitudes decrease with time. This is principally because the damping dissipates the free vibration energy. On the other hand, whenGalerkin’s method is employed in numerical simulation to perform first-order modal truncation on the vibration response of the transmission line to simplify the analytical process, the simulation results show that the second-order out-of-plane mode of vibration of the transmission line, as well as its first-order symmetric mode of vibration and its first- to fourth-order antisymmetric modes of vibration, are still excited. As a result of the internal resonance, the first-order out-of-plane mode of vibration, which has relatively high energy, gives rise to the second-order out-of-plane mode of vibration, whose frequency is twice its own frequency. Because the natural frequencies of the second-order out-of-plane mode of vibration and the first-order in-plane antisymmetric mode of vibration are closely spaced, the out-of-plane vibration energy gradually transfers to the in-plane direction. In addition, because the natural frequency of the in-plane vibration of each order is two or three times the in-plane fundamental frequency, the energy also transfers between different in-plane modes of vibration. The Fourier spectra of the displacement response shown in Figure 9 clearly demonstrate that the in-plane vibration also contains the first-order out-of-plane mode of vibration. Similarly, as demonstrated in Figure 10, the in-plane motion of the transmission line is strongly coupled to its out-of-plane motion.The aforementioned analysis shows that the out-of-plane vibration energy of the transmission line can give rise not only to in-plane antisymmetric vibration but also to first-order in-plane symmetric vibration, whose energy is the highest of all the in-plane modes of vibration. According to the suspended cable theory, in-plane symmetric vibration is the primary factor that causes changes in the dynamic tension in a suspended cable. To further illustrate the effects of internal resonance on the dynamic tension in the transmission line, the dynamic tension in the transmission line is calculated under the aforementioned two initial potential energy conditions (see Figure11 for the results). In Figure 11, the tension is a dimensionless value obtained by dividing the dynamic tension Tc obtained from simulation by the initial tension Tc0. As demonstrated in Figure 11, when the transmission line vibrates with relatively low initial potential energy, the dynamic tension in the line is always positive, indicating that the transmission line is under tension. Meanwhile, when the line is in vibration, the dynamic tension in the transmission line is approximately the same as the initial tension. This suggests that when the transmission line vibrates freely with relatively low initial potential energy, the effects of internal resonance on the tension in the transmission line are negligible. The spectral distribution of the dynamic tension shown in Figure 11(c) also demonstrates that when the initial potential energy of the transmission line is relatively low, the tension in the transmission line is almost unaffected by its vibration. When the initial potential energy of the transmission line is relatively high, the dynamic tension in the transmission line may reach twice the initial tension. The spectral distribution of the dynamic tension shown in Figure 11(d) also demonstrates that the increase in the tension in the line is related to its in-plane vibration. This indicates that when the transmission line is in motion with a large amplitude, its in-plane vibration is coupled to its out-of-plane vibration; that is, when the initial out-of-plane energy of the transmission line is relatively high, significant internal resonance occurs in the transmission line when it starts vibrating, resulting in the transfer of out-of-plane vibration energy to the in-plane direction. This in turn not only gives rise to in-plane antisymmetric vibration but also causes in-plane symmetric vibration, ultimately leading to a significant increase in the tension in the transmission line.Figure 11
Dynamic tension in the transmission line and its Fourier spectra under two initial energy conditions.
(a)
Dynamic tension time history of the transmission line when its initial energy is relatively low (b)
Dynamic tension time history of the transmission line when its initial energy is relatively high (c)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively low (d)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively high


#### 4.3.3. Effects of Damping on the Nonlinear Internal Resonance of the Transmission Line

It was noted in the previous sections that the structural damping of a transmission line is very small. To effectively reduce the vibration responses of transmission lines, control devices (e.g., dampers, spiral dampers, damping spacers, and damper lines) are often installed on transmission lines in engineering practice; these control devices can generate relatively large damping. Thus, in this section, the effects of damping on the internal resonance of the transmission line are analyzed.Here, the same initial potential energy conditions as those used in Section4.3.2 are used, but the initial damping is varied. Figures 12 and 13 show the free vibration response of the transmission line obtained in the simulation under various damping conditions. A comparison of Figures 12 and 13 with Figures 8–10 shows that, under the same conditions, increasing the damping of the transmission line results in a significant decrease in the displacement of and the tension in the transmission line as well as a decrease in the duration of vibration of the transmission line. This indicates that increasing the damping of the transmission line can effectively consume its vibration energy, control its vibration response amplitude, and prevent an increase in its dynamic tension but cannot hamper the occurrence of its nonlinear internal resonance.Figure 12
Free vibration response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 13
Free vibration response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


### 4.4. Numerical Analysis of the Nonlinear Internal Resonance of the Transmission Line When in Forced Vibration

Based on the aforementioned analysis, we know that the natural frequencies of the lower-order modes of vibration of the overhead transmission line are relatively low. However, in an actual environment, excitation energy (primarily wind load energy) is mainly distributed in the 0-1 Hz band, particularly in the 0–0.2 Hz band. Therefore, under an external excitation (e.g., a wind load), there is a relatively high probability that the transmission line will resonate with it. To further study the effects of the internal resonance of the transmission line on its wind-induced vibration, in this section the effects of the internal resonance of the transmission line under a harmonic excitation on the primary resonance are first analyzed; based on the results, the forced vibration characteristics of the transmission line under a wind load are investigated.


#### 4.4.1. Analysis of the Internal Resonance of the Transmission Line under a Harmonic Excitation

Here, it is assumed that the transmission line is only subjected to an out-of-plane harmonic excitationPw=P0cos⁡Ωzt. The relationship between the circular frequency Ωz of the excitation and the natural circular frequency ωw of the out-of-plane vibration of the transmission line is characterized by introducing a detuning parameter σ1, a nonnegative real number k, and a small parameter ε:(59)Ωz=kωw+εσ1.To study the primary resonance response of the system,k, ε, and σ1 are set to 1, 0.001, and 10, respectively. By substituting these values into (51), the frequency fz of the out-of-plane excitation is determined: fz=Ωz/(2π)=0.146Hz. The natural frequency fw of the first-order out-of-plane vibration of the transmission line that is directly induced by the excitation is as follows: fw=ωw/(2π)=0.136Hz. fz approximately equals fw. Thus, it can be considered that Ωz≈ωw.Let us set the amplitudeP0 of the generalized excitation load to 3305 N and the initial conditions for simulation to qv=0,  q˙v=0,  qw=0.1,  q˙w=0.00005. By substituting these values into (54), the nonlinear forced vibration response of the transmission line under various damping conditions is determined. Figures 14–17 show the numerical simulation results obtained when the damping cc is set to 0.001, 0.01, 0.05, and 0.1, respectively.Figure 14
Primary resonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 15
Primary resonance response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 16
Primary resonance response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 17
Primary resonance response of the transmission line when its dampingcc is set to 0.1.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure14, the out-of-plane vibration of the transmission line directly induced by the excitation has a very large amplitude, suggesting that the transmission line is forced to resonate when Ωz≈ωw. Due to the internal resonance, the vibration energy of the transmission line transfers from the out-of-plane direction to the in-plane direction, which is not directly under excitation. In addition, the in-plane vibration of the transmission line also exhibits resonance characteristics. In other words, due to the internal resonance, the forced resonance of the transmission line can involve several modes of resonance. When only considering the small structural damping, the resonance amplitude of the transmission line will jump; that is, within a certain period of time, the response occurs at a relatively small amplitude, and vibration with a relatively large amplitude occurs at a certain time and can continue steadily for an extended period of time. This differs from the resonance of a linear system. Due to the nonlinear coupling and internal resonance, the forced resonance modulates the vibration frequency of the system, and the frequencies of the out-of-plane and in-plane vibration of the transmission line are no longer linear natural frequencies and are also not the same as the excitation frequency but instead are manifested as a relatively wide distribution of vibration energy (Figures 14(c) and 14(d)). As demonstrated in the phase diagrams of the in-plane and out-of-plane response (Figures 14(e) and 14(f)), the in-plane and out-of-plane response is aperiodic with an amplitude that jumps. Considering that the external excitation has a very large amplitude due to the nonlinear internal resonance, the forced resonance excites the transmission line to resonate internally, resulting in a sharp increase in the dynamic tension in the transmission line. In addition, because of the in-plane vibration amplitude jumps, a corresponding jumping phenomenon is also observed in the dynamic tension time history curve (Figure 14(g)). As shown in the Fourier spectrum of the dynamic tension (Figure 14(h)), the dynamic tension in the transmission line is mainly significantly affected by the in-plane vibration. Because its amplitude is far greater than the design-breaking load, the dynamic tension may cause the transmission to break.Figure15 shows the simulation results obtained by increasing the damping cc to 0.01, while retaining the other conditions. As demonstrated in Figure 15, amplitude jumps are eliminated due to the damping effects, but internal resonance still exists. The in-plane vibration of the transmission line not directly induced by the excitation has a large amplitude, which are of the same order of magnitude of the out-of-plane vibration. In addition, the frequency modulation by the forced resonance becomes more complicated compared to the situation in which the damping is small (Figures 15(c) and 15(d)). As demonstrated in the phase trajectory of the steady-state response (Figures 15(e) and 15(f)), there are still jumps in the in-plane vibration amplitude of the transmission line and the corresponding dynamic tension amplitude, and the maximum value of these jumps can be as high as 3 times the initial tension. This indicates that it is still possible for the transmission line to be broken under these conditions.Figures16 and 17 show the simulation results that are obtained when the damping cc is further increased to 0.05 and 0.1, respectively, while retaining the other conditions. As demonstrated in Figures 16 and 17, due to the large damping effects, the vibration amplitude of the transmission line reaches a stable state within a short period of time; however, the large damping is unable to prevent the occurrence of internal resonance, and the in-plane vibration still has a very large amplitude, the numerical value of which is commensurate with that of the out-of-plane vibration. The dynamic tension in the transmission line when it is in steady-state vibration can still reach approximately three times the initial tension. This suggests that even under relatively large damping effects, the forced resonance of the transmission line cannot be maintained as the steady-state mode of motion directly induced by the excitation but is instead manifested as the nonlinear vibration of coupled modes. Due to the frequency-modulating effect of the forced resonance, the natural frequencies of the modes of vibration of the transmission line directly induced by the excitation more accurately approach the excitation frequency, and, correspondingly, the modulated frequencies inherit the relationships between the original linear natural frequencies: they are either closely spaced with or multiples of one another. In addition, the vibration still contains higher-order modes of vibration.As demonstrated in the phase trajectories of the response (Figures16(e), 16(f), 17(e), and 17(f)), when its damping is relatively high, the resonance response of the transmission line reaches saturation. In other words, because the nonlinear vibration equations for the transmission line exhibit quadratic nonlinearity and the linear natural vibration frequencies of the transmission line satisfy the ωw≈ωv and ωv≈2ωw conditions, when the frequency Ωz of the external excitation is approximately equal to ωw and the excitation amplitude meets a certain condition, the response energy of the mode of vibration directly induced by the excitation reaches saturation, and all the input excitation energy enters another mode. Thus, while its damping can be increased by installing energy dissipating equipment in engineering practice, the motion of a transmission line exhibits coupled in-plane and out-of-plane vibration characteristics due to the nonlinear internal resonance; in addition, a transmission line can still continue to vibrate with a relatively large amplitude under an external excitation with a relatively large amplitude, and the vibration energy continuously transfers between relevant modes without being attenuated. Due to the presence of this response saturation phenomenon, the dynamic tension in a transmission line when in vibration can still possibly reach a higher level even when its damping is set to a relatively large value, which is disadvantageous to practical engineering.To compare the vibration response of the transmission line when not in resonance with the vibration response of the transmission line when in resonance, the following set of parameters is selected: frequencyfz of the external excitation: 0.68 Hz; damping cc: 0.001; all the other parameters are the same as those used to simulate the transmission line when in resonance. Figure 18 shows the simulation results. A comparison of Figures 14 and 18 shows that when the external excitation amplitude and the damping of the transmission line remain unchanged, the vibration amplitude of the transmission line is far smaller when not in resonance than when in resonance. While an internal resonance phenomenon can still be observed numerically when the transmission line is not in resonance, the internal resonance-induced in-plane vibration amplitude is very small and is negligible. When the transmission line is not in resonance, the vibration of the system is primarily composed of the out-of-plane forced vibration induced by the out-of-plane excitation (Figure 18(d)); in addition, the dynamic tension in the transmission line changes to a relatively small extent (Figure 18(g)). It can be seen that internal resonance has a great influence on the main resonance of transmission line, so the design of transmission line needs to consider its influence reasonably.Figure 18
Nonresonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


#### 4.4.2. Analysis of the Internal Resonance of the Transmission Line When in Wind-Induced Vibration

Under actual operating conditions, overhead transmission lines will inevitably undergo wind-induced vibration. Therefore, in this section, the nonlinear vibration characteristics of the transmission line under a wind load are studied. Considering that the vertical in-plane wind load on a transmission line is generally very small, to simplify the analytical process, it is assumed that the wind load acts in the out-of-plane direction of the transmission line. The quasi-steady hypothesis is used in calculation: when calculating the wind load on the transmission line, only the effects of the vibration of the transmission line on the wind speed are considered, whereas the aerodynamic coupling effects are not considered: that is, the load termsP(x,t) and P(z,t) in (26) and (27) can be expressed as follows:(60)Px,t=0.5ρaDcCDva-x-v˙c2,Pz,t=0.5ρaDcCDva-z-w˙c2,where ρa represents the air density; Dc represents the windward width of the transmission line; CD represents the shape factor of the transmission line; va-x and va-z represent the wind speeds along the x- and z-axis directions, respectively; v˙c,w˙c represent the in-plane and out-of-plane vibration velocities of the transmission line, respectively.By substituting (60) into (26) and (27) and finding the first-order modal solutions for the in-plane and out-of-plane vibration usingGalerkin’s modal truncation method, the continuous partial differential equations can be transformed to discrete ordinary partial equations, as shown in (30), in which the factors a7, b6, Pv, and Pw are calculated using the following equations (all the other factors are calculated using the same equations as introduced in Section 2.2):(61)a7=∫0Lccc+ρaDcCDva-xφv2ydy,b6=∫0Lccc+ρaDcCDva-zφw2ydy,Pv=∫0Lc0.5ρaDcCDva-x2φvydy,Pw=∫0Lc0.5ρaDcCDva-z2φwydy.Wind speeds used in this section are selected from the database of wind speeds recorded by the field wind measurement platform installed on the transmission line (Figure3). Three wind speed time histories with different average winds are selected, which are three wind speed records with an average value U0 of 5.2, 9.56, and 20.1 m/s, respectively. These three wind speeds are equivalent to the Beaufort Scales 3, 5, and 8, respectively. The in-plane vertical wind load on the transmission line is converted from the vertical fluctuating wind speed recorded by the wind speed measurement device and its value is generally small and negligible. Figure 19 shows the frequency spectra corresponding to the aforementioned three wind speed time histories. As demonstrated in Figure 19, the wind speed energy is mainly distributed in the 0-1 Hz band.Figure 19
Fourier spectra of the wind speed.
(a)
Fourier spectrum of the wind speed (U0=5.2 m/s) (b)
Fourier spectrum of the wind speed (U0=9.56 m/s) (c)
Fourier spectrum of the wind speed (U0=20.1 m/s)Equation (30) is solved using a higher-orderRunge-Kutta method. The damping cc of the transmission line is set to 0.01, and the initial conditions for calculation are set as follows:(62)qv=0,q˙v=0,qw=0.5,q˙w=0.00005.Based on Tables3 and 4, we know that the natural frequencies of the first-order in-plane and out-of-plane vibration of the transmission satisfy the following condition: ωv≈3ωw. Figures 20–22 show the numerical simulation results with respect to the vibration response of the transmission line under a wind load.Figure 20
Wind-induced vibration response of the transmission line whenU0=5.2 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 21
Wind-induced vibration response of the transmission line whenU0=9.56 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 22
Wind-induced vibration response of the transmission line whenU0=20.1 m/s.
(a)
Dimensionless in-plane mid-span displacement (b)
Dimensionless out-of-plane mid-span displacement (c)
In-plane mid-span phase-plane response (d)
Out-of-plane mid-span phase-plane response (e)
Fourier spectrum of the in-plane mid-span displacement response (f)
Fourier spectrum of the out-of-plane mid-span displacement response (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure20, when U0=5.2 m/s, the out-of-plane mid-span vibration amplitude of the transmission line can exceed 1 m, and its in-plane mid-span vibration amplitude is far greater than its cross-sectional diameter; in addition, the in-plane vibration is associated with a noticeable jumping phenomenon (Figure 20(a)). This indicates that the transmission line also undergoes internal resonance when in wind-induced vibration and that the vibration energy transfers between the out-of-plane and the in-plane modes of vibration. The overall vibration of the transmission line is manifested as coupled in-plane and out-of-plane vibration (Figure 20(b)). In-plane vibration amplitude jumps can also be found by observing the phase trajectories of the transmission line (Figures 20(c) and 20(d)). The out-of-plane vibration is a complex aperiodic motion. Due to the excitation from the wind load and the frequency-modulating effect of the internal resonance, the natural characteristics of the in-plane and out-of-plane vibration undergo relatively significant changes, and its energy is distributed in a relatively wide frequency band (Figures 20(e) and 20(f)). In addition, due to the in-plane vibration, there are significant jumps in the dynamic tension amplitude of the transmission line (Figures 20(g) and 20(h)).Figure21 shows the simulation results obtained when the average wind speed U0 is increased to 9.56 m/s. As demonstrated in Figure 21, the transmission line vibrates with a large amplitude; in addition, the out-of-plane mid-span vibration amplitude of the transmission line can reach half of its maximum sag, and its in-plane mid-span vibration amplitude can also reach half of its maximum out-of-plane vibration amplitude (Figure 21(a)); moreover, there are also jumps in the in-plane and out-of-plane vibration amplitudes of the transmission line. However, due to the inherent randomness of a wind load, the internal resonance of the transmission line when in wind-induced vibration is not as conspicuous as the internal resonance that occurs when the line is in free vibration or under a harmonic excitation. However, as demonstrated by the response frequency spectra, changes in the natural vibration characteristics of the system are more significant than those that occur in a linear system. Higher-order in-plane and out-of-plane modes of vibration of the transmission line can be excited by a wind load with a relatively high wind speed (Figures 21(e) and 21(f)); under this condition, the phase trajectory of the response has an oval shape, and there are jumps in the displacement amplitude (the outer and inner trajectories represent lower-order and higher-order modes of vibration, resp.) (Figures 21(c) and 21(d)). A comparison of Figures 20 and 21 shows that as the wind speed increases, the dynamic tension in the transmission line increases. Because the dynamic tension is mainly affected by the in-plane vibration, the jumps in the dynamic tension amplitude are quite noticeable. If the external wind excitation lasts for a sufficiently long period of time, the dynamic tension in the transmission line can even surpass its breaking load limit at some moments during the vibration process (Figures 21(g) and 21(h)), resulting in damage to the transmission line. This should warrant attention in practical engineering.Figure22 shows the simulation results obtained when the average wind speed U0 is increased to 20.1 m/s. As demonstrated in Figure 22, the vibration amplitude of the transmission line further increases under a strong wind; in addition, the in-plane vibration amplitude of the transmission line is even greater than its out-of-plane vibration amplitude, and the out-of-plane and in-plane vibration amplitudes of the transmission line are of the same order of magnitude as its maximum arc sag (Figures 22(a) and 22(b)) and can even surpass its maximum arc sag, which will result in damage to the transmission line under practical conditions. As demonstrated in the in-plane and out-of-plane displacement response time histories and phase-plane response (Figures 22(c) and 22(d)), the wind-induced vibration of the transmission line exhibits significant nonlinear vibration characteristics; in addition, the out-of-plane vibration is also coupled to the in-plane vibration, and the vibration energy of the structure transfers from the out-of-plane modes that are under the wind load to the in-plane modes.A comparison of Figures22(e) and 22(f) shows that the energy-containing range of the out-of-plane mid-span displacement response of the transmission line is 0.09–0.294 Hz and the energy-containing range of its in-plane mid-span displacement response is 0.12–0.25 Hz, whereas the main energy-containing range of the wind load is 0.01–1 Hz. Thus, under a strong wind, the wind-induced vibration of the transmission line modulates not only the line’s vibration amplitude but also its vibration frequency and can cause the transmission line to undergo high-frequency vibration; in addition, the dynamic tension in the transmission line increases significantly and can even reach or surpass its breaking load in a short time, which is very disadvantageous to the transmission line under practical conditions. A comparison of Figures 14 and 22 shows that the motion of the transmission line under a strong wind is similar to that of a primary resonance under harmonic excitation; that is, the amplitude of vibration is very large, and the in-plane vibration is strongly coupled to its out-of-plane vibration.


## 4.1. Calculation of the Parameters

A relatively large single-span transmission line in a high-voltage transmission line system in East China (Figure2) is selected as an example for analysis [30]. The conductors of this span of the transmission line are composed of LGJ-630/45 ASCR cables. Table 1 lists the design parameters of the selected transmission line. The maximum design wind speed at a height of 10 m above the ground for safe operation of the selected transmission line is 25.3 m/s. In the actual transmission line, the conductors of each phase are quad-bundled conductors. To simplify calculation, in the finite element (FE) modeling process, the quad-bundled conductors of each phase are equivalently simplified to one conductor based on the following principles: the bundled conductors have the same windward area, the same total operational tension, and the same linear density. Table 2 lists the parameters of the transmission line after simplification.Table 1
Original design parameters of the selected transmission line.
Items Diameterd  (mm) Cross-sectional areaAc    (mm2) Linear densitymc  (kg/m) Elastic modulusEc (MPa) Average operational tension (N) Tension at design wind speed(N) Breaking force(N) SpanLc  (m) Sagdc  (m) Conductors 33.6 666.55 2.06 63,000 35,316 43,688 141,265 480 16.6Table 2
Design parameters of the selected transmission line after simplification.
Items Diameterd  (mm) Densityρc  (kg/m3) Linear densitymc  (kg/m) Linear loadqc  (N/m) Initial strain  εc0 Conductors 134.4 3,090.5 8.24 80.752 0.00084Figure 2
A high-voltage transmission line.The sag-to-span ratiodc/Lc of the selected transmission line is calculated based on the parameters listed in Table 1 to be approximately 1/29, which is far smaller than 1/8. Thus, (30) can be used to analyze the nonlinear vibration characteristics of this transmission line.


## 4.2. Modal Analysis of the Transmission Line

Based on the actual characteristics of the selected transmission line in combination with the design parameters listed in Tables1 and 2, LINK10 elements in ANSYS are used to construct an FE model for the selected transmission line. First, a modal analysis is performed to determine the natural frequency of vibration of each order of the transmission line model; this provides a basis for the subsequent numerical simulation. To validate the FE model constructed for a single transmission line, in this section, the FE simulation results with respect to the natural vibration frequencies of the transmission line are also compared with the theoretical solutions.A transmission line is a typical type of suspended cable structure, and the analytical solutions for the natural frequencies of its vibration can be obtained based on the single-cable theory. A transmission line resists external loading primarily by self-stretching and consequently will undergo relatively large displacement under loading. Therefore, the geometric nonlinearity of a transmission line must be taken into consideration in the calculation. When constructing an FE model of a transmission line, one crucial point is to determine the initial configuration of the cable. Specifically, on the one hand, this means to determine the spatial location of the transmission line by determining the equilibrium and deformation compatibility equations for the transmission line based on the existing suspended cable theory; on the other hand, this means to transition the transmission line from its initial stress-free state to an initial loaded state based on a certain configuration-seeking approach.The catenary and parabolic methods are the main methods used to determine the equilibrium equation for a suspended cable. The former is an accurate method, whereas the latter is an approximate method. However, when the sag-to-span ratiodc/Lc of a cable is less than 1/8, the parabolic method can also yield a relatively accurate solution for the initial configuration of the transmission line. Here, assuming that the vertical load on a transmission line is evenly distributed along the span, as shown in Figure 3, the equation for the spatial configuration of the transmission line is(55)xy=qc2Tc0yLc-y+ΔcLcy,where qc represents the vertical load on the transmission line, which in this case is the dead weight of the transmission line; Tc0 represents the initial horizontal tension in the transmission line (Tc0=qcLc2/8dc); Lc represents the horizontal span of the transmission line; Δc represents the difference in height between the two ends of the transmission line; and dc represents the mid-span sag of the transmission line.Figure 3
A schematic diagram of the calculation for a single cable when the load is evenly distributed along the span.Considering that a transmission line generally has a small sag, the first-order natural frequency corresponding to the out-of-plane vibration of the transmission line is the smallest frequency of all the frequencies of its vibration. According to the linear vibration theory of suspended cables, for a horizontal suspended cable with two ends hinged at the same height and whose mass is evenly distributed, the natural frequencies of vibration and the corresponding modes under the action of gravity alone are as follows:(a) The out-of-plane swing and the in-plane vibration of the suspended cable are not coupled. Thus, the natural circular frequencies of the out-of-plane swing and the corresponding modes can be expressed as follows:(56)ωw=nπLcTc0mc,n=1,2,3,…,Φw=Ansin⁡nπyLc,n=1,2,3,…,where ωw represents the natural circular frequency of the out-of-plane vibration of the suspended cable, n represents the modal number, mc represents the mass of the suspended cable per unit length, Φw represents the out-of-plane mode of vibration of the suspended cable, and An represents the modal amplitude (each of the other symbols has the same meaning as previously described).(b) The in-plane antisymmetric vibration of the suspended cable will not generate an increment in its horizontal dynamic tension. The natural circular frequencies of its vibration and the corresponding modes can be expressed as follows:(57)ωv=2nπLcTc0mc,n=1,2,3,…,Φv=Bnsin2nπyLc,n=1,2,3,…,where ωv represents the natural circular frequency of the in-plane antisymmetric vibration of the suspended cable, n represents the modal number, Φv represents the mode of the in-plane antisymmetric vibration of the suspended cable, and Bn represents the modal amplitude (each of the other symbols has the same meaning as previously described).(c) The in-plane symmetric vibration of the suspended cable will generate an additional increment in its dynamic tension. The natural circular frequencies of its vibration and the corresponding modes can be determined by solving the following transcendental equations:(58)tanϖ2=ϖ2-4λ2ϖ23,ϖ=ωvantLcTc0/mc,λ2=mcgLc/Tc02LcTc0Lc/EcAc,Φvant=h~Tc0ϖ21-tan⁡ϖ2sin⁡ϖyLc-cos⁡ϖyLc,where ωvant represents the natural circular frequency of the in-plane symmetric vibration of the suspended cable, Φvant represents the in-plane symmetric mode of vibration of the suspended cable, Ec represents the elastic modulus of the suspended cable, Ac represents the cross-sectional area of the suspended cable, and h~ represents the additional tension in the suspended cable (each of the other symbols has the same meaning as previously described).By substituting the parameters listed in Tables1 and 2 into (55)–(58), the theoretical solutions for the natural vibration frequencies of the first eight orders of the transmission line in the three directions can be obtained (they are shown in Table 3). Table 4 summarizes the linear natural vibration frequencies of the transmission line obtained from FE simulation.Table 3
Theoretical linear natural vibration frequencies of the transmission line model.
Calculation equation First order Second order Third order Fourth order Fifth order Sixth order Seventh order Eighth order Out-of-plane vibration frequencyfw (Hz) f w = ω w ( 2 π ) 0.136 0.273 0.409 0.546 0.682 0.818 0.955 1.091 In-plane symmetric vibration frequencyfv (Hz) f v = ω v ( 2 π ) 0.344 0.46 – – – – – – In-plane antisymmetric vibration frequencyfvant (Hz) f v a n t = ω v a n t ( 2 π ) 0.273 0.546 0.818 1.091 1.364 1.637 1.91 2.182Table 4
Natural vibration frequencies of the transmission line model obtained from FE simulation.
Calculation equation First order Second order Third order Fourth order Fifth order Sixth order Seventh order Eighth order Out-of-plane vibration frequencyfw (Hz) f w = ω w ( 2 π ) 0.136 0.272 0.408 0.544 0.68 0.816 0.952 1.088 In-plane symmetric vibration frequencyfv (Hz) f v = ω v ( 2 π ) 0.346 0.46 0.686 0.954 1.225 1.497 1.769 2.042 In-plane antisymmetric vibration frequencyfvant (Hz) f v a n t = ω v a n t ( 2 π ) 0.27 0.544 0.816 1.088 1.36 1.633 1.91 2.18A comparison of Tables3 and 4 shows that the natural vibration frequencies of the transmission line model obtained from FE simulation are in relatively good agreement with the theoretical values, with the largest difference being approximately 1%. This indicates that the transmission line model constructed in this work is relatively reliable. Moreover, Tables 3 and 4 also demonstrate that the following relationships between the natural frequencies of the in-plane and out-of-plane modes of vibration of the transmission line always exist: ωv≈ωw, ωv≈2ωw, and ωv≈3ωw. In other words, within the scope of the linear theory, the natural vibration frequencies of the transmission line are always multiples of one another.Figure4 shows the modes of vibration of the first nine orders obtained from FE simulation. As demonstrated in Figure 4, of all the modes of vibration of the transmission line, the first-order out-of-plane vibration is the most easily excitable. The higher-order frequencies of in-plane vibration are approximately integer multiples of the fundamental frequency, and this relationship also exists between the frequencies of in-plane antisymmetric vibration and the fundamental frequency. In addition to the relationship to the fundamental frequency, the higher-order frequencies of the out-of-plane vibration are also approximately equal to those of the in-plane vibration, and the in-plane and out-of-plane modes of vibration corresponding to the same frequency are also similar to one another.Figure 4
Modes of vibration of the transmission line of the first nine orders obtained from FE simulation.
(a)
First order, 0.136 Hz (b)
Second order, 0.27 Hz (c)
Third order, 0.272 Hz (d)
Fourth order, 0.346 Hz (e)
Fifth order, 0.408 Hz (f)
Sixth order, 0.46 Hz (g)
Seventh order, 0.544 Hz (h)
Eighth order, 0.544 Hz (i)
Ninth order, 0.68 Hz


## 4.3. Numerical Analysis of the Nonlinear Internal Resonance of the Transmission Line When in Free Vibration

When only the free vibration of the transmission line under the action of the initial displacement is considered, in (54), Pv=0 and Pw=0.The structural damping of a transmission line is often relatively small, and it can generally be set to 0.0005. In addition, the effects of the structural damping of a transmission line on its vibration, particularly under strong wind conditions, are relatively insignificant, whereas aerodynamic damping relatively significantly affects the vibration of a transmission line. By comprehensively considering the aforementioned factors, in this study, when analyzing the free vibration of the transmission line, the dampingcc of its in-plane and out-of-plane vibration is set to 0.001. In the following sections, the effects of the initial potential energy of the transmission line on its internal resonance are analyzed in two scenarios.


### 4.3.1. When the Initial Potential Energy of the Transmission Line Is Relatively Low (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Small)

Let us set the initial out-of-plane mid-span displacement and the initial velocity of out-of-plane motion of the transmission line to 1 m and 0.00005 m/s, respectively, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=1,  q˙w=0.00005. The displacements of the transmission line obtained from numerical simulation are then divided by the horizontal span Lc to obtain dimensionless displacements. The dimensionless velocities of in-plane and out-of-plane can also be obtained from the numerical simulation results of the in-plane and out-of-plane velocities multiplied by the first-order out-of-plane vibration period T0 and then divided by the horizontal span Lc. The results are presented in Figures 5–7.Figure 5
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively low.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time historyFigure 6
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively low.
(a)
In-plane (b)
Out-of-planeFigure 7
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively low.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement responses in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure5, when the initial energy of the transmission line is relatively low, its in-plane vibration displacement is relatively small, with the maximum amplitude being approximately 1/10 that of its out-of-plane vibration displacement; in addition, the “beat vibration” phenomenon is inconspicuous in the response time history. As demonstrated in the Fourier spectra of the displacement response of the transmission line (Figure 6), the first-order mode of vibration is the primary out-of-plane mode of vibration. The out-of-plane vibration of the transmission line gives rise to the first-order in-plane antisymmetric mode of vibration and the first-order symmetric mode of vibration. In addition, the first-order in-plane symmetric vibration energy is also greater than the first-order in-plane antisymmetric vibration energy, and the in-plane vibration energy amplitude is smaller than the out-of-plane vibration energy amplitude by one order of magnitude. This indicates that the transmission line is inherently capable of generating nonlinear internal resonance and can undergo internal resonance even when its initial energy is relatively low, but energy transfer and mode coupling are quite weak under this condition. As demonstrated in Figure 7, the out-of-plane and in-plane modes of vibration of the transmission line both exhibit periodic motion characteristics.


### 4.3.2. When the Initial Potential Energy of the Transmission Line Is Relatively High (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Large)

Let us set the initial out-of-plane mid-span displacement of the transmission line to a relatively large value (10 m), the initial velocity of its out-of-plane mid-span motion to 0.00005 m/s, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=10,  q˙w=0.00005. Figures 8–10 show the numerical simulation results.Figure 8
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively high.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacementFigure 9
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively high.
(a)
In-plane (b)
Out-of-planeFigure 10
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively high.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement response in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure8, when the initial out-of-plane displacement of the transmission line is relatively large, as a result of the internal resonance the system energy constantly transfers between the modes that are in the ωv≈ωw, ωv≈2ωw, and ωv≈3ωw relationships; in addition, the system energy also transfers between degrees of freedom that are similarly related to multiples of the frequency. On the one hand, the out-of-plane vibration energy transfers to the in-plane direction, resulting in vibration with a relatively large amplitude. Figure 8 also demonstrates that the order of magnitude of the in-plane vibration displacement amplitude is commensurate with the out-of-plane vibration displacement amplitude; a phenomenon similar to “beat vibration” is also observed in the displacement response time history; that is, as the out-of-plane vibration response gradually decreases, the in-plane vibration response at the corresponding time gradually increases and vice versa. Furthermore, from Figures 5 and 8, we can also see that both of the maximum in-plane and out-of-plane vibration amplitudes decrease with time. This is principally because the damping dissipates the free vibration energy. On the other hand, whenGalerkin’s method is employed in numerical simulation to perform first-order modal truncation on the vibration response of the transmission line to simplify the analytical process, the simulation results show that the second-order out-of-plane mode of vibration of the transmission line, as well as its first-order symmetric mode of vibration and its first- to fourth-order antisymmetric modes of vibration, are still excited. As a result of the internal resonance, the first-order out-of-plane mode of vibration, which has relatively high energy, gives rise to the second-order out-of-plane mode of vibration, whose frequency is twice its own frequency. Because the natural frequencies of the second-order out-of-plane mode of vibration and the first-order in-plane antisymmetric mode of vibration are closely spaced, the out-of-plane vibration energy gradually transfers to the in-plane direction. In addition, because the natural frequency of the in-plane vibration of each order is two or three times the in-plane fundamental frequency, the energy also transfers between different in-plane modes of vibration. The Fourier spectra of the displacement response shown in Figure 9 clearly demonstrate that the in-plane vibration also contains the first-order out-of-plane mode of vibration. Similarly, as demonstrated in Figure 10, the in-plane motion of the transmission line is strongly coupled to its out-of-plane motion.The aforementioned analysis shows that the out-of-plane vibration energy of the transmission line can give rise not only to in-plane antisymmetric vibration but also to first-order in-plane symmetric vibration, whose energy is the highest of all the in-plane modes of vibration. According to the suspended cable theory, in-plane symmetric vibration is the primary factor that causes changes in the dynamic tension in a suspended cable. To further illustrate the effects of internal resonance on the dynamic tension in the transmission line, the dynamic tension in the transmission line is calculated under the aforementioned two initial potential energy conditions (see Figure11 for the results). In Figure 11, the tension is a dimensionless value obtained by dividing the dynamic tension Tc obtained from simulation by the initial tension Tc0. As demonstrated in Figure 11, when the transmission line vibrates with relatively low initial potential energy, the dynamic tension in the line is always positive, indicating that the transmission line is under tension. Meanwhile, when the line is in vibration, the dynamic tension in the transmission line is approximately the same as the initial tension. This suggests that when the transmission line vibrates freely with relatively low initial potential energy, the effects of internal resonance on the tension in the transmission line are negligible. The spectral distribution of the dynamic tension shown in Figure 11(c) also demonstrates that when the initial potential energy of the transmission line is relatively low, the tension in the transmission line is almost unaffected by its vibration. When the initial potential energy of the transmission line is relatively high, the dynamic tension in the transmission line may reach twice the initial tension. The spectral distribution of the dynamic tension shown in Figure 11(d) also demonstrates that the increase in the tension in the line is related to its in-plane vibration. This indicates that when the transmission line is in motion with a large amplitude, its in-plane vibration is coupled to its out-of-plane vibration; that is, when the initial out-of-plane energy of the transmission line is relatively high, significant internal resonance occurs in the transmission line when it starts vibrating, resulting in the transfer of out-of-plane vibration energy to the in-plane direction. This in turn not only gives rise to in-plane antisymmetric vibration but also causes in-plane symmetric vibration, ultimately leading to a significant increase in the tension in the transmission line.Figure 11
Dynamic tension in the transmission line and its Fourier spectra under two initial energy conditions.
(a)
Dynamic tension time history of the transmission line when its initial energy is relatively low (b)
Dynamic tension time history of the transmission line when its initial energy is relatively high (c)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively low (d)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively high


### 4.3.3. Effects of Damping on the Nonlinear Internal Resonance of the Transmission Line

It was noted in the previous sections that the structural damping of a transmission line is very small. To effectively reduce the vibration responses of transmission lines, control devices (e.g., dampers, spiral dampers, damping spacers, and damper lines) are often installed on transmission lines in engineering practice; these control devices can generate relatively large damping. Thus, in this section, the effects of damping on the internal resonance of the transmission line are analyzed.Here, the same initial potential energy conditions as those used in Section4.3.2 are used, but the initial damping is varied. Figures 12 and 13 show the free vibration response of the transmission line obtained in the simulation under various damping conditions. A comparison of Figures 12 and 13 with Figures 8–10 shows that, under the same conditions, increasing the damping of the transmission line results in a significant decrease in the displacement of and the tension in the transmission line as well as a decrease in the duration of vibration of the transmission line. This indicates that increasing the damping of the transmission line can effectively consume its vibration energy, control its vibration response amplitude, and prevent an increase in its dynamic tension but cannot hamper the occurrence of its nonlinear internal resonance.Figure 12
Free vibration response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 13
Free vibration response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


## 4.3.1. When the Initial Potential Energy of the Transmission Line Is Relatively Low (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Small)

Let us set the initial out-of-plane mid-span displacement and the initial velocity of out-of-plane motion of the transmission line to 1 m and 0.00005 m/s, respectively, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=1,  q˙w=0.00005. The displacements of the transmission line obtained from numerical simulation are then divided by the horizontal span Lc to obtain dimensionless displacements. The dimensionless velocities of in-plane and out-of-plane can also be obtained from the numerical simulation results of the in-plane and out-of-plane velocities multiplied by the first-order out-of-plane vibration period T0 and then divided by the horizontal span Lc. The results are presented in Figures 5–7.Figure 5
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively low.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time historyFigure 6
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively low.
(a)
In-plane (b)
Out-of-planeFigure 7
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively low.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement responses in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure5, when the initial energy of the transmission line is relatively low, its in-plane vibration displacement is relatively small, with the maximum amplitude being approximately 1/10 that of its out-of-plane vibration displacement; in addition, the “beat vibration” phenomenon is inconspicuous in the response time history. As demonstrated in the Fourier spectra of the displacement response of the transmission line (Figure 6), the first-order mode of vibration is the primary out-of-plane mode of vibration. The out-of-plane vibration of the transmission line gives rise to the first-order in-plane antisymmetric mode of vibration and the first-order symmetric mode of vibration. In addition, the first-order in-plane symmetric vibration energy is also greater than the first-order in-plane antisymmetric vibration energy, and the in-plane vibration energy amplitude is smaller than the out-of-plane vibration energy amplitude by one order of magnitude. This indicates that the transmission line is inherently capable of generating nonlinear internal resonance and can undergo internal resonance even when its initial energy is relatively low, but energy transfer and mode coupling are quite weak under this condition. As demonstrated in Figure 7, the out-of-plane and in-plane modes of vibration of the transmission line both exhibit periodic motion characteristics.


## 4.3.2. When the Initial Potential Energy of the Transmission Line Is Relatively High (i.e., When the Initial Mid-Span Displacement of the Transmission Line Is Relatively Large)

Let us set the initial out-of-plane mid-span displacement of the transmission line to a relatively large value (10 m), the initial velocity of its out-of-plane mid-span motion to 0.00005 m/s, and both its initial in-plane mid-span displacement and the initial velocity of its in-plane motion to 0; that is, the initial conditions described in (53) are set as follows: qv=0,  q˙v=0,  qw=10,  q˙w=0.00005. Figures 8–10 show the numerical simulation results.Figure 8
In-plane and out-of-plane mid-span displacement time histories of the transmission line when its initial potential energy is relatively high.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacementFigure 9
Fourier spectra of the in-plane and out-of-plane mid-span displacements of the transmission line when its initial potential energy is relatively high.
(a)
In-plane (b)
Out-of-planeFigure 10
Configuration of the out-of-plane motion and the phase-space response of the transmission line when its initial potential energy is relatively high.
(a)
Configuration of out-of-plane motion at various times (b)
In-plane and out-of-plane displacement response in the phase space (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectoryAs demonstrated in Figure8, when the initial out-of-plane displacement of the transmission line is relatively large, as a result of the internal resonance the system energy constantly transfers between the modes that are in the ωv≈ωw, ωv≈2ωw, and ωv≈3ωw relationships; in addition, the system energy also transfers between degrees of freedom that are similarly related to multiples of the frequency. On the one hand, the out-of-plane vibration energy transfers to the in-plane direction, resulting in vibration with a relatively large amplitude. Figure 8 also demonstrates that the order of magnitude of the in-plane vibration displacement amplitude is commensurate with the out-of-plane vibration displacement amplitude; a phenomenon similar to “beat vibration” is also observed in the displacement response time history; that is, as the out-of-plane vibration response gradually decreases, the in-plane vibration response at the corresponding time gradually increases and vice versa. Furthermore, from Figures 5 and 8, we can also see that both of the maximum in-plane and out-of-plane vibration amplitudes decrease with time. This is principally because the damping dissipates the free vibration energy. On the other hand, whenGalerkin’s method is employed in numerical simulation to perform first-order modal truncation on the vibration response of the transmission line to simplify the analytical process, the simulation results show that the second-order out-of-plane mode of vibration of the transmission line, as well as its first-order symmetric mode of vibration and its first- to fourth-order antisymmetric modes of vibration, are still excited. As a result of the internal resonance, the first-order out-of-plane mode of vibration, which has relatively high energy, gives rise to the second-order out-of-plane mode of vibration, whose frequency is twice its own frequency. Because the natural frequencies of the second-order out-of-plane mode of vibration and the first-order in-plane antisymmetric mode of vibration are closely spaced, the out-of-plane vibration energy gradually transfers to the in-plane direction. In addition, because the natural frequency of the in-plane vibration of each order is two or three times the in-plane fundamental frequency, the energy also transfers between different in-plane modes of vibration. The Fourier spectra of the displacement response shown in Figure 9 clearly demonstrate that the in-plane vibration also contains the first-order out-of-plane mode of vibration. Similarly, as demonstrated in Figure 10, the in-plane motion of the transmission line is strongly coupled to its out-of-plane motion.The aforementioned analysis shows that the out-of-plane vibration energy of the transmission line can give rise not only to in-plane antisymmetric vibration but also to first-order in-plane symmetric vibration, whose energy is the highest of all the in-plane modes of vibration. According to the suspended cable theory, in-plane symmetric vibration is the primary factor that causes changes in the dynamic tension in a suspended cable. To further illustrate the effects of internal resonance on the dynamic tension in the transmission line, the dynamic tension in the transmission line is calculated under the aforementioned two initial potential energy conditions (see Figure11 for the results). In Figure 11, the tension is a dimensionless value obtained by dividing the dynamic tension Tc obtained from simulation by the initial tension Tc0. As demonstrated in Figure 11, when the transmission line vibrates with relatively low initial potential energy, the dynamic tension in the line is always positive, indicating that the transmission line is under tension. Meanwhile, when the line is in vibration, the dynamic tension in the transmission line is approximately the same as the initial tension. This suggests that when the transmission line vibrates freely with relatively low initial potential energy, the effects of internal resonance on the tension in the transmission line are negligible. The spectral distribution of the dynamic tension shown in Figure 11(c) also demonstrates that when the initial potential energy of the transmission line is relatively low, the tension in the transmission line is almost unaffected by its vibration. When the initial potential energy of the transmission line is relatively high, the dynamic tension in the transmission line may reach twice the initial tension. The spectral distribution of the dynamic tension shown in Figure 11(d) also demonstrates that the increase in the tension in the line is related to its in-plane vibration. This indicates that when the transmission line is in motion with a large amplitude, its in-plane vibration is coupled to its out-of-plane vibration; that is, when the initial out-of-plane energy of the transmission line is relatively high, significant internal resonance occurs in the transmission line when it starts vibrating, resulting in the transfer of out-of-plane vibration energy to the in-plane direction. This in turn not only gives rise to in-plane antisymmetric vibration but also causes in-plane symmetric vibration, ultimately leading to a significant increase in the tension in the transmission line.Figure 11
Dynamic tension in the transmission line and its Fourier spectra under two initial energy conditions.
(a)
Dynamic tension time history of the transmission line when its initial energy is relatively low (b)
Dynamic tension time history of the transmission line when its initial energy is relatively high (c)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively low (d)
Fourier spectrum of the dynamic tension in the transmission line when its initial energy is relatively high


## 4.3.3. Effects of Damping on the Nonlinear Internal Resonance of the Transmission Line

It was noted in the previous sections that the structural damping of a transmission line is very small. To effectively reduce the vibration responses of transmission lines, control devices (e.g., dampers, spiral dampers, damping spacers, and damper lines) are often installed on transmission lines in engineering practice; these control devices can generate relatively large damping. Thus, in this section, the effects of damping on the internal resonance of the transmission line are analyzed.Here, the same initial potential energy conditions as those used in Section4.3.2 are used, but the initial damping is varied. Figures 12 and 13 show the free vibration response of the transmission line obtained in the simulation under various damping conditions. A comparison of Figures 12 and 13 with Figures 8–10 shows that, under the same conditions, increasing the damping of the transmission line results in a significant decrease in the displacement of and the tension in the transmission line as well as a decrease in the duration of vibration of the transmission line. This indicates that increasing the damping of the transmission line can effectively consume its vibration energy, control its vibration response amplitude, and prevent an increase in its dynamic tension but cannot hamper the occurrence of its nonlinear internal resonance.Figure 12
Free vibration response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 13
Free vibration response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


## 4.4. Numerical Analysis of the Nonlinear Internal Resonance of the Transmission Line When in Forced Vibration

Based on the aforementioned analysis, we know that the natural frequencies of the lower-order modes of vibration of the overhead transmission line are relatively low. However, in an actual environment, excitation energy (primarily wind load energy) is mainly distributed in the 0-1 Hz band, particularly in the 0–0.2 Hz band. Therefore, under an external excitation (e.g., a wind load), there is a relatively high probability that the transmission line will resonate with it. To further study the effects of the internal resonance of the transmission line on its wind-induced vibration, in this section the effects of the internal resonance of the transmission line under a harmonic excitation on the primary resonance are first analyzed; based on the results, the forced vibration characteristics of the transmission line under a wind load are investigated.


### 4.4.1. Analysis of the Internal Resonance of the Transmission Line under a Harmonic Excitation

Here, it is assumed that the transmission line is only subjected to an out-of-plane harmonic excitationPw=P0cos⁡Ωzt. The relationship between the circular frequency Ωz of the excitation and the natural circular frequency ωw of the out-of-plane vibration of the transmission line is characterized by introducing a detuning parameter σ1, a nonnegative real number k, and a small parameter ε:(59)Ωz=kωw+εσ1.To study the primary resonance response of the system,k, ε, and σ1 are set to 1, 0.001, and 10, respectively. By substituting these values into (51), the frequency fz of the out-of-plane excitation is determined: fz=Ωz/(2π)=0.146Hz. The natural frequency fw of the first-order out-of-plane vibration of the transmission line that is directly induced by the excitation is as follows: fw=ωw/(2π)=0.136Hz. fz approximately equals fw. Thus, it can be considered that Ωz≈ωw.Let us set the amplitudeP0 of the generalized excitation load to 3305 N and the initial conditions for simulation to qv=0,  q˙v=0,  qw=0.1,  q˙w=0.00005. By substituting these values into (54), the nonlinear forced vibration response of the transmission line under various damping conditions is determined. Figures 14–17 show the numerical simulation results obtained when the damping cc is set to 0.001, 0.01, 0.05, and 0.1, respectively.Figure 14
Primary resonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 15
Primary resonance response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 16
Primary resonance response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 17
Primary resonance response of the transmission line when its dampingcc is set to 0.1.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure14, the out-of-plane vibration of the transmission line directly induced by the excitation has a very large amplitude, suggesting that the transmission line is forced to resonate when Ωz≈ωw. Due to the internal resonance, the vibration energy of the transmission line transfers from the out-of-plane direction to the in-plane direction, which is not directly under excitation. In addition, the in-plane vibration of the transmission line also exhibits resonance characteristics. In other words, due to the internal resonance, the forced resonance of the transmission line can involve several modes of resonance. When only considering the small structural damping, the resonance amplitude of the transmission line will jump; that is, within a certain period of time, the response occurs at a relatively small amplitude, and vibration with a relatively large amplitude occurs at a certain time and can continue steadily for an extended period of time. This differs from the resonance of a linear system. Due to the nonlinear coupling and internal resonance, the forced resonance modulates the vibration frequency of the system, and the frequencies of the out-of-plane and in-plane vibration of the transmission line are no longer linear natural frequencies and are also not the same as the excitation frequency but instead are manifested as a relatively wide distribution of vibration energy (Figures 14(c) and 14(d)). As demonstrated in the phase diagrams of the in-plane and out-of-plane response (Figures 14(e) and 14(f)), the in-plane and out-of-plane response is aperiodic with an amplitude that jumps. Considering that the external excitation has a very large amplitude due to the nonlinear internal resonance, the forced resonance excites the transmission line to resonate internally, resulting in a sharp increase in the dynamic tension in the transmission line. In addition, because of the in-plane vibration amplitude jumps, a corresponding jumping phenomenon is also observed in the dynamic tension time history curve (Figure 14(g)). As shown in the Fourier spectrum of the dynamic tension (Figure 14(h)), the dynamic tension in the transmission line is mainly significantly affected by the in-plane vibration. Because its amplitude is far greater than the design-breaking load, the dynamic tension may cause the transmission to break.Figure15 shows the simulation results obtained by increasing the damping cc to 0.01, while retaining the other conditions. As demonstrated in Figure 15, amplitude jumps are eliminated due to the damping effects, but internal resonance still exists. The in-plane vibration of the transmission line not directly induced by the excitation has a large amplitude, which are of the same order of magnitude of the out-of-plane vibration. In addition, the frequency modulation by the forced resonance becomes more complicated compared to the situation in which the damping is small (Figures 15(c) and 15(d)). As demonstrated in the phase trajectory of the steady-state response (Figures 15(e) and 15(f)), there are still jumps in the in-plane vibration amplitude of the transmission line and the corresponding dynamic tension amplitude, and the maximum value of these jumps can be as high as 3 times the initial tension. This indicates that it is still possible for the transmission line to be broken under these conditions.Figures16 and 17 show the simulation results that are obtained when the damping cc is further increased to 0.05 and 0.1, respectively, while retaining the other conditions. As demonstrated in Figures 16 and 17, due to the large damping effects, the vibration amplitude of the transmission line reaches a stable state within a short period of time; however, the large damping is unable to prevent the occurrence of internal resonance, and the in-plane vibration still has a very large amplitude, the numerical value of which is commensurate with that of the out-of-plane vibration. The dynamic tension in the transmission line when it is in steady-state vibration can still reach approximately three times the initial tension. This suggests that even under relatively large damping effects, the forced resonance of the transmission line cannot be maintained as the steady-state mode of motion directly induced by the excitation but is instead manifested as the nonlinear vibration of coupled modes. Due to the frequency-modulating effect of the forced resonance, the natural frequencies of the modes of vibration of the transmission line directly induced by the excitation more accurately approach the excitation frequency, and, correspondingly, the modulated frequencies inherit the relationships between the original linear natural frequencies: they are either closely spaced with or multiples of one another. In addition, the vibration still contains higher-order modes of vibration.As demonstrated in the phase trajectories of the response (Figures16(e), 16(f), 17(e), and 17(f)), when its damping is relatively high, the resonance response of the transmission line reaches saturation. In other words, because the nonlinear vibration equations for the transmission line exhibit quadratic nonlinearity and the linear natural vibration frequencies of the transmission line satisfy the ωw≈ωv and ωv≈2ωw conditions, when the frequency Ωz of the external excitation is approximately equal to ωw and the excitation amplitude meets a certain condition, the response energy of the mode of vibration directly induced by the excitation reaches saturation, and all the input excitation energy enters another mode. Thus, while its damping can be increased by installing energy dissipating equipment in engineering practice, the motion of a transmission line exhibits coupled in-plane and out-of-plane vibration characteristics due to the nonlinear internal resonance; in addition, a transmission line can still continue to vibrate with a relatively large amplitude under an external excitation with a relatively large amplitude, and the vibration energy continuously transfers between relevant modes without being attenuated. Due to the presence of this response saturation phenomenon, the dynamic tension in a transmission line when in vibration can still possibly reach a higher level even when its damping is set to a relatively large value, which is disadvantageous to practical engineering.To compare the vibration response of the transmission line when not in resonance with the vibration response of the transmission line when in resonance, the following set of parameters is selected: frequencyfz of the external excitation: 0.68 Hz; damping cc: 0.001; all the other parameters are the same as those used to simulate the transmission line when in resonance. Figure 18 shows the simulation results. A comparison of Figures 14 and 18 shows that when the external excitation amplitude and the damping of the transmission line remain unchanged, the vibration amplitude of the transmission line is far smaller when not in resonance than when in resonance. While an internal resonance phenomenon can still be observed numerically when the transmission line is not in resonance, the internal resonance-induced in-plane vibration amplitude is very small and is negligible. When the transmission line is not in resonance, the vibration of the system is primarily composed of the out-of-plane forced vibration induced by the out-of-plane excitation (Figure 18(d)); in addition, the dynamic tension in the transmission line changes to a relatively small extent (Figure 18(g)). It can be seen that internal resonance has a great influence on the main resonance of transmission line, so the design of transmission line needs to consider its influence reasonably.Figure 18
Nonresonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


### 4.4.2. Analysis of the Internal Resonance of the Transmission Line When in Wind-Induced Vibration

Under actual operating conditions, overhead transmission lines will inevitably undergo wind-induced vibration. Therefore, in this section, the nonlinear vibration characteristics of the transmission line under a wind load are studied. Considering that the vertical in-plane wind load on a transmission line is generally very small, to simplify the analytical process, it is assumed that the wind load acts in the out-of-plane direction of the transmission line. The quasi-steady hypothesis is used in calculation: when calculating the wind load on the transmission line, only the effects of the vibration of the transmission line on the wind speed are considered, whereas the aerodynamic coupling effects are not considered: that is, the load termsP(x,t) and P(z,t) in (26) and (27) can be expressed as follows:(60)Px,t=0.5ρaDcCDva-x-v˙c2,Pz,t=0.5ρaDcCDva-z-w˙c2,where ρa represents the air density; Dc represents the windward width of the transmission line; CD represents the shape factor of the transmission line; va-x and va-z represent the wind speeds along the x- and z-axis directions, respectively; v˙c,w˙c represent the in-plane and out-of-plane vibration velocities of the transmission line, respectively.By substituting (60) into (26) and (27) and finding the first-order modal solutions for the in-plane and out-of-plane vibration usingGalerkin’s modal truncation method, the continuous partial differential equations can be transformed to discrete ordinary partial equations, as shown in (30), in which the factors a7, b6, Pv, and Pw are calculated using the following equations (all the other factors are calculated using the same equations as introduced in Section 2.2):(61)a7=∫0Lccc+ρaDcCDva-xφv2ydy,b6=∫0Lccc+ρaDcCDva-zφw2ydy,Pv=∫0Lc0.5ρaDcCDva-x2φvydy,Pw=∫0Lc0.5ρaDcCDva-z2φwydy.Wind speeds used in this section are selected from the database of wind speeds recorded by the field wind measurement platform installed on the transmission line (Figure3). Three wind speed time histories with different average winds are selected, which are three wind speed records with an average value U0 of 5.2, 9.56, and 20.1 m/s, respectively. These three wind speeds are equivalent to the Beaufort Scales 3, 5, and 8, respectively. The in-plane vertical wind load on the transmission line is converted from the vertical fluctuating wind speed recorded by the wind speed measurement device and its value is generally small and negligible. Figure 19 shows the frequency spectra corresponding to the aforementioned three wind speed time histories. As demonstrated in Figure 19, the wind speed energy is mainly distributed in the 0-1 Hz band.Figure 19
Fourier spectra of the wind speed.
(a)
Fourier spectrum of the wind speed (U0=5.2 m/s) (b)
Fourier spectrum of the wind speed (U0=9.56 m/s) (c)
Fourier spectrum of the wind speed (U0=20.1 m/s)Equation (30) is solved using a higher-orderRunge-Kutta method. The damping cc of the transmission line is set to 0.01, and the initial conditions for calculation are set as follows:(62)qv=0,q˙v=0,qw=0.5,q˙w=0.00005.Based on Tables3 and 4, we know that the natural frequencies of the first-order in-plane and out-of-plane vibration of the transmission satisfy the following condition: ωv≈3ωw. Figures 20–22 show the numerical simulation results with respect to the vibration response of the transmission line under a wind load.Figure 20
Wind-induced vibration response of the transmission line whenU0=5.2 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 21
Wind-induced vibration response of the transmission line whenU0=9.56 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 22
Wind-induced vibration response of the transmission line whenU0=20.1 m/s.
(a)
Dimensionless in-plane mid-span displacement (b)
Dimensionless out-of-plane mid-span displacement (c)
In-plane mid-span phase-plane response (d)
Out-of-plane mid-span phase-plane response (e)
Fourier spectrum of the in-plane mid-span displacement response (f)
Fourier spectrum of the out-of-plane mid-span displacement response (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure20, when U0=5.2 m/s, the out-of-plane mid-span vibration amplitude of the transmission line can exceed 1 m, and its in-plane mid-span vibration amplitude is far greater than its cross-sectional diameter; in addition, the in-plane vibration is associated with a noticeable jumping phenomenon (Figure 20(a)). This indicates that the transmission line also undergoes internal resonance when in wind-induced vibration and that the vibration energy transfers between the out-of-plane and the in-plane modes of vibration. The overall vibration of the transmission line is manifested as coupled in-plane and out-of-plane vibration (Figure 20(b)). In-plane vibration amplitude jumps can also be found by observing the phase trajectories of the transmission line (Figures 20(c) and 20(d)). The out-of-plane vibration is a complex aperiodic motion. Due to the excitation from the wind load and the frequency-modulating effect of the internal resonance, the natural characteristics of the in-plane and out-of-plane vibration undergo relatively significant changes, and its energy is distributed in a relatively wide frequency band (Figures 20(e) and 20(f)). In addition, due to the in-plane vibration, there are significant jumps in the dynamic tension amplitude of the transmission line (Figures 20(g) and 20(h)).Figure21 shows the simulation results obtained when the average wind speed U0 is increased to 9.56 m/s. As demonstrated in Figure 21, the transmission line vibrates with a large amplitude; in addition, the out-of-plane mid-span vibration amplitude of the transmission line can reach half of its maximum sag, and its in-plane mid-span vibration amplitude can also reach half of its maximum out-of-plane vibration amplitude (Figure 21(a)); moreover, there are also jumps in the in-plane and out-of-plane vibration amplitudes of the transmission line. However, due to the inherent randomness of a wind load, the internal resonance of the transmission line when in wind-induced vibration is not as conspicuous as the internal resonance that occurs when the line is in free vibration or under a harmonic excitation. However, as demonstrated by the response frequency spectra, changes in the natural vibration characteristics of the system are more significant than those that occur in a linear system. Higher-order in-plane and out-of-plane modes of vibration of the transmission line can be excited by a wind load with a relatively high wind speed (Figures 21(e) and 21(f)); under this condition, the phase trajectory of the response has an oval shape, and there are jumps in the displacement amplitude (the outer and inner trajectories represent lower-order and higher-order modes of vibration, resp.) (Figures 21(c) and 21(d)). A comparison of Figures 20 and 21 shows that as the wind speed increases, the dynamic tension in the transmission line increases. Because the dynamic tension is mainly affected by the in-plane vibration, the jumps in the dynamic tension amplitude are quite noticeable. If the external wind excitation lasts for a sufficiently long period of time, the dynamic tension in the transmission line can even surpass its breaking load limit at some moments during the vibration process (Figures 21(g) and 21(h)), resulting in damage to the transmission line. This should warrant attention in practical engineering.Figure22 shows the simulation results obtained when the average wind speed U0 is increased to 20.1 m/s. As demonstrated in Figure 22, the vibration amplitude of the transmission line further increases under a strong wind; in addition, the in-plane vibration amplitude of the transmission line is even greater than its out-of-plane vibration amplitude, and the out-of-plane and in-plane vibration amplitudes of the transmission line are of the same order of magnitude as its maximum arc sag (Figures 22(a) and 22(b)) and can even surpass its maximum arc sag, which will result in damage to the transmission line under practical conditions. As demonstrated in the in-plane and out-of-plane displacement response time histories and phase-plane response (Figures 22(c) and 22(d)), the wind-induced vibration of the transmission line exhibits significant nonlinear vibration characteristics; in addition, the out-of-plane vibration is also coupled to the in-plane vibration, and the vibration energy of the structure transfers from the out-of-plane modes that are under the wind load to the in-plane modes.A comparison of Figures22(e) and 22(f) shows that the energy-containing range of the out-of-plane mid-span displacement response of the transmission line is 0.09–0.294 Hz and the energy-containing range of its in-plane mid-span displacement response is 0.12–0.25 Hz, whereas the main energy-containing range of the wind load is 0.01–1 Hz. Thus, under a strong wind, the wind-induced vibration of the transmission line modulates not only the line’s vibration amplitude but also its vibration frequency and can cause the transmission line to undergo high-frequency vibration; in addition, the dynamic tension in the transmission line increases significantly and can even reach or surpass its breaking load in a short time, which is very disadvantageous to the transmission line under practical conditions. A comparison of Figures 14 and 22 shows that the motion of the transmission line under a strong wind is similar to that of a primary resonance under harmonic excitation; that is, the amplitude of vibration is very large, and the in-plane vibration is strongly coupled to its out-of-plane vibration.


## 4.4.1. Analysis of the Internal Resonance of the Transmission Line under a Harmonic Excitation

Here, it is assumed that the transmission line is only subjected to an out-of-plane harmonic excitationPw=P0cos⁡Ωzt. The relationship between the circular frequency Ωz of the excitation and the natural circular frequency ωw of the out-of-plane vibration of the transmission line is characterized by introducing a detuning parameter σ1, a nonnegative real number k, and a small parameter ε:(59)Ωz=kωw+εσ1.To study the primary resonance response of the system,k, ε, and σ1 are set to 1, 0.001, and 10, respectively. By substituting these values into (51), the frequency fz of the out-of-plane excitation is determined: fz=Ωz/(2π)=0.146Hz. The natural frequency fw of the first-order out-of-plane vibration of the transmission line that is directly induced by the excitation is as follows: fw=ωw/(2π)=0.136Hz. fz approximately equals fw. Thus, it can be considered that Ωz≈ωw.Let us set the amplitudeP0 of the generalized excitation load to 3305 N and the initial conditions for simulation to qv=0,  q˙v=0,  qw=0.1,  q˙w=0.00005. By substituting these values into (54), the nonlinear forced vibration response of the transmission line under various damping conditions is determined. Figures 14–17 show the numerical simulation results obtained when the damping cc is set to 0.001, 0.01, 0.05, and 0.1, respectively.Figure 14
Primary resonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement time history (b)
Out-of-plane mid-span displacement time history (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 15
Primary resonance response of the transmission line when its dampingcc is set to 0.01.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 16
Primary resonance response of the transmission line when its dampingcc is set to 0.05.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 17
Primary resonance response of the transmission line when its dampingcc is set to 0.1.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure14, the out-of-plane vibration of the transmission line directly induced by the excitation has a very large amplitude, suggesting that the transmission line is forced to resonate when Ωz≈ωw. Due to the internal resonance, the vibration energy of the transmission line transfers from the out-of-plane direction to the in-plane direction, which is not directly under excitation. In addition, the in-plane vibration of the transmission line also exhibits resonance characteristics. In other words, due to the internal resonance, the forced resonance of the transmission line can involve several modes of resonance. When only considering the small structural damping, the resonance amplitude of the transmission line will jump; that is, within a certain period of time, the response occurs at a relatively small amplitude, and vibration with a relatively large amplitude occurs at a certain time and can continue steadily for an extended period of time. This differs from the resonance of a linear system. Due to the nonlinear coupling and internal resonance, the forced resonance modulates the vibration frequency of the system, and the frequencies of the out-of-plane and in-plane vibration of the transmission line are no longer linear natural frequencies and are also not the same as the excitation frequency but instead are manifested as a relatively wide distribution of vibration energy (Figures 14(c) and 14(d)). As demonstrated in the phase diagrams of the in-plane and out-of-plane response (Figures 14(e) and 14(f)), the in-plane and out-of-plane response is aperiodic with an amplitude that jumps. Considering that the external excitation has a very large amplitude due to the nonlinear internal resonance, the forced resonance excites the transmission line to resonate internally, resulting in a sharp increase in the dynamic tension in the transmission line. In addition, because of the in-plane vibration amplitude jumps, a corresponding jumping phenomenon is also observed in the dynamic tension time history curve (Figure 14(g)). As shown in the Fourier spectrum of the dynamic tension (Figure 14(h)), the dynamic tension in the transmission line is mainly significantly affected by the in-plane vibration. Because its amplitude is far greater than the design-breaking load, the dynamic tension may cause the transmission to break.Figure15 shows the simulation results obtained by increasing the damping cc to 0.01, while retaining the other conditions. As demonstrated in Figure 15, amplitude jumps are eliminated due to the damping effects, but internal resonance still exists. The in-plane vibration of the transmission line not directly induced by the excitation has a large amplitude, which are of the same order of magnitude of the out-of-plane vibration. In addition, the frequency modulation by the forced resonance becomes more complicated compared to the situation in which the damping is small (Figures 15(c) and 15(d)). As demonstrated in the phase trajectory of the steady-state response (Figures 15(e) and 15(f)), there are still jumps in the in-plane vibration amplitude of the transmission line and the corresponding dynamic tension amplitude, and the maximum value of these jumps can be as high as 3 times the initial tension. This indicates that it is still possible for the transmission line to be broken under these conditions.Figures16 and 17 show the simulation results that are obtained when the damping cc is further increased to 0.05 and 0.1, respectively, while retaining the other conditions. As demonstrated in Figures 16 and 17, due to the large damping effects, the vibration amplitude of the transmission line reaches a stable state within a short period of time; however, the large damping is unable to prevent the occurrence of internal resonance, and the in-plane vibration still has a very large amplitude, the numerical value of which is commensurate with that of the out-of-plane vibration. The dynamic tension in the transmission line when it is in steady-state vibration can still reach approximately three times the initial tension. This suggests that even under relatively large damping effects, the forced resonance of the transmission line cannot be maintained as the steady-state mode of motion directly induced by the excitation but is instead manifested as the nonlinear vibration of coupled modes. Due to the frequency-modulating effect of the forced resonance, the natural frequencies of the modes of vibration of the transmission line directly induced by the excitation more accurately approach the excitation frequency, and, correspondingly, the modulated frequencies inherit the relationships between the original linear natural frequencies: they are either closely spaced with or multiples of one another. In addition, the vibration still contains higher-order modes of vibration.As demonstrated in the phase trajectories of the response (Figures16(e), 16(f), 17(e), and 17(f)), when its damping is relatively high, the resonance response of the transmission line reaches saturation. In other words, because the nonlinear vibration equations for the transmission line exhibit quadratic nonlinearity and the linear natural vibration frequencies of the transmission line satisfy the ωw≈ωv and ωv≈2ωw conditions, when the frequency Ωz of the external excitation is approximately equal to ωw and the excitation amplitude meets a certain condition, the response energy of the mode of vibration directly induced by the excitation reaches saturation, and all the input excitation energy enters another mode. Thus, while its damping can be increased by installing energy dissipating equipment in engineering practice, the motion of a transmission line exhibits coupled in-plane and out-of-plane vibration characteristics due to the nonlinear internal resonance; in addition, a transmission line can still continue to vibrate with a relatively large amplitude under an external excitation with a relatively large amplitude, and the vibration energy continuously transfers between relevant modes without being attenuated. Due to the presence of this response saturation phenomenon, the dynamic tension in a transmission line when in vibration can still possibly reach a higher level even when its damping is set to a relatively large value, which is disadvantageous to practical engineering.To compare the vibration response of the transmission line when not in resonance with the vibration response of the transmission line when in resonance, the following set of parameters is selected: frequencyfz of the external excitation: 0.68 Hz; damping cc: 0.001; all the other parameters are the same as those used to simulate the transmission line when in resonance. Figure 18 shows the simulation results. A comparison of Figures 14 and 18 shows that when the external excitation amplitude and the damping of the transmission line remain unchanged, the vibration amplitude of the transmission line is far smaller when not in resonance than when in resonance. While an internal resonance phenomenon can still be observed numerically when the transmission line is not in resonance, the internal resonance-induced in-plane vibration amplitude is very small and is negligible. When the transmission line is not in resonance, the vibration of the system is primarily composed of the out-of-plane forced vibration induced by the out-of-plane excitation (Figure 18(d)); in addition, the dynamic tension in the transmission line changes to a relatively small extent (Figure 18(g)). It can be seen that internal resonance has a great influence on the main resonance of transmission line, so the design of transmission line needs to consider its influence reasonably.Figure 18
Nonresonance response of the transmission line when its dampingcc is set to 0.001.
(a)
In-plane mid-span displacement (b)
Out-of-plane mid-span displacement (c)
Fourier spectrum of the in-plane mid-span displacement (d)
Fourier spectrum of the out-of-plane mid-span displacement (e)
In-plane mid-span phase trajectory (f)
Out-of-plane mid-span phase trajectory (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tension


## 4.4.2. Analysis of the Internal Resonance of the Transmission Line When in Wind-Induced Vibration

Under actual operating conditions, overhead transmission lines will inevitably undergo wind-induced vibration. Therefore, in this section, the nonlinear vibration characteristics of the transmission line under a wind load are studied. Considering that the vertical in-plane wind load on a transmission line is generally very small, to simplify the analytical process, it is assumed that the wind load acts in the out-of-plane direction of the transmission line. The quasi-steady hypothesis is used in calculation: when calculating the wind load on the transmission line, only the effects of the vibration of the transmission line on the wind speed are considered, whereas the aerodynamic coupling effects are not considered: that is, the load termsP(x,t) and P(z,t) in (26) and (27) can be expressed as follows:(60)Px,t=0.5ρaDcCDva-x-v˙c2,Pz,t=0.5ρaDcCDva-z-w˙c2,where ρa represents the air density; Dc represents the windward width of the transmission line; CD represents the shape factor of the transmission line; va-x and va-z represent the wind speeds along the x- and z-axis directions, respectively; v˙c,w˙c represent the in-plane and out-of-plane vibration velocities of the transmission line, respectively.By substituting (60) into (26) and (27) and finding the first-order modal solutions for the in-plane and out-of-plane vibration usingGalerkin’s modal truncation method, the continuous partial differential equations can be transformed to discrete ordinary partial equations, as shown in (30), in which the factors a7, b6, Pv, and Pw are calculated using the following equations (all the other factors are calculated using the same equations as introduced in Section 2.2):(61)a7=∫0Lccc+ρaDcCDva-xφv2ydy,b6=∫0Lccc+ρaDcCDva-zφw2ydy,Pv=∫0Lc0.5ρaDcCDva-x2φvydy,Pw=∫0Lc0.5ρaDcCDva-z2φwydy.Wind speeds used in this section are selected from the database of wind speeds recorded by the field wind measurement platform installed on the transmission line (Figure3). Three wind speed time histories with different average winds are selected, which are three wind speed records with an average value U0 of 5.2, 9.56, and 20.1 m/s, respectively. These three wind speeds are equivalent to the Beaufort Scales 3, 5, and 8, respectively. The in-plane vertical wind load on the transmission line is converted from the vertical fluctuating wind speed recorded by the wind speed measurement device and its value is generally small and negligible. Figure 19 shows the frequency spectra corresponding to the aforementioned three wind speed time histories. As demonstrated in Figure 19, the wind speed energy is mainly distributed in the 0-1 Hz band.Figure 19
Fourier spectra of the wind speed.
(a)
Fourier spectrum of the wind speed (U0=5.2 m/s) (b)
Fourier spectrum of the wind speed (U0=9.56 m/s) (c)
Fourier spectrum of the wind speed (U0=20.1 m/s)Equation (30) is solved using a higher-orderRunge-Kutta method. The damping cc of the transmission line is set to 0.01, and the initial conditions for calculation are set as follows:(62)qv=0,q˙v=0,qw=0.5,q˙w=0.00005.Based on Tables3 and 4, we know that the natural frequencies of the first-order in-plane and out-of-plane vibration of the transmission satisfy the following condition: ωv≈3ωw. Figures 20–22 show the numerical simulation results with respect to the vibration response of the transmission line under a wind load.Figure 20
Wind-induced vibration response of the transmission line whenU0=5.2 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 21
Wind-induced vibration response of the transmission line whenU0=9.56 m/s.
(a)
Dimensionless mid-span displacement (b)
Relationship between the out-of-plane and in-plane mid-span displacements (c)
In-plane mid-span phase trajectory (d)
Out-of-plane mid-span phase trajectory (e)
Fourier spectrum of the in-plane mid-span displacement (f)
Fourier spectrum of the out-of-plane mid-span displacement (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionFigure 22
Wind-induced vibration response of the transmission line whenU0=20.1 m/s.
(a)
Dimensionless in-plane mid-span displacement (b)
Dimensionless out-of-plane mid-span displacement (c)
In-plane mid-span phase-plane response (d)
Out-of-plane mid-span phase-plane response (e)
Fourier spectrum of the in-plane mid-span displacement response (f)
Fourier spectrum of the out-of-plane mid-span displacement response (g)
Dynamic tension time history (h)
Fourier spectrum of the dynamic tensionAs demonstrated in Figure20, when U0=5.2 m/s, the out-of-plane mid-span vibration amplitude of the transmission line can exceed 1 m, and its in-plane mid-span vibration amplitude is far greater than its cross-sectional diameter; in addition, the in-plane vibration is associated with a noticeable jumping phenomenon (Figure 20(a)). This indicates that the transmission line also undergoes internal resonance when in wind-induced vibration and that the vibration energy transfers between the out-of-plane and the in-plane modes of vibration. The overall vibration of the transmission line is manifested as coupled in-plane and out-of-plane vibration (Figure 20(b)). In-plane vibration amplitude jumps can also be found by observing the phase trajectories of the transmission line (Figures 20(c) and 20(d)). The out-of-plane vibration is a complex aperiodic motion. Due to the excitation from the wind load and the frequency-modulating effect of the internal resonance, the natural characteristics of the in-plane and out-of-plane vibration undergo relatively significant changes, and its energy is distributed in a relatively wide frequency band (Figures 20(e) and 20(f)). In addition, due to the in-plane vibration, there are significant jumps in the dynamic tension amplitude of the transmission line (Figures 20(g) and 20(h)).Figure21 shows the simulation results obtained when the average wind speed U0 is increased to 9.56 m/s. As demonstrated in Figure 21, the transmission line vibrates with a large amplitude; in addition, the out-of-plane mid-span vibration amplitude of the transmission line can reach half of its maximum sag, and its in-plane mid-span vibration amplitude can also reach half of its maximum out-of-plane vibration amplitude (Figure 21(a)); moreover, there are also jumps in the in-plane and out-of-plane vibration amplitudes of the transmission line. However, due to the inherent randomness of a wind load, the internal resonance of the transmission line when in wind-induced vibration is not as conspicuous as the internal resonance that occurs when the line is in free vibration or under a harmonic excitation. However, as demonstrated by the response frequency spectra, changes in the natural vibration characteristics of the system are more significant than those that occur in a linear system. Higher-order in-plane and out-of-plane modes of vibration of the transmission line can be excited by a wind load with a relatively high wind speed (Figures 21(e) and 21(f)); under this condition, the phase trajectory of the response has an oval shape, and there are jumps in the displacement amplitude (the outer and inner trajectories represent lower-order and higher-order modes of vibration, resp.) (Figures 21(c) and 21(d)). A comparison of Figures 20 and 21 shows that as the wind speed increases, the dynamic tension in the transmission line increases. Because the dynamic tension is mainly affected by the in-plane vibration, the jumps in the dynamic tension amplitude are quite noticeable. If the external wind excitation lasts for a sufficiently long period of time, the dynamic tension in the transmission line can even surpass its breaking load limit at some moments during the vibration process (Figures 21(g) and 21(h)), resulting in damage to the transmission line. This should warrant attention in practical engineering.Figure22 shows the simulation results obtained when the average wind speed U0 is increased to 20.1 m/s. As demonstrated in Figure 22, the vibration amplitude of the transmission line further increases under a strong wind; in addition, the in-plane vibration amplitude of the transmission line is even greater than its out-of-plane vibration amplitude, and the out-of-plane and in-plane vibration amplitudes of the transmission line are of the same order of magnitude as its maximum arc sag (Figures 22(a) and 22(b)) and can even surpass its maximum arc sag, which will result in damage to the transmission line under practical conditions. As demonstrated in the in-plane and out-of-plane displacement response time histories and phase-plane response (Figures 22(c) and 22(d)), the wind-induced vibration of the transmission line exhibits significant nonlinear vibration characteristics; in addition, the out-of-plane vibration is also coupled to the in-plane vibration, and the vibration energy of the structure transfers from the out-of-plane modes that are under the wind load to the in-plane modes.A comparison of Figures22(e) and 22(f) shows that the energy-containing range of the out-of-plane mid-span displacement response of the transmission line is 0.09–0.294 Hz and the energy-containing range of its in-plane mid-span displacement response is 0.12–0.25 Hz, whereas the main energy-containing range of the wind load is 0.01–1 Hz. Thus, under a strong wind, the wind-induced vibration of the transmission line modulates not only the line’s vibration amplitude but also its vibration frequency and can cause the transmission line to undergo high-frequency vibration; in addition, the dynamic tension in the transmission line increases significantly and can even reach or surpass its breaking load in a short time, which is very disadvantageous to the transmission line under practical conditions. A comparison of Figures 14 and 22 shows that the motion of the transmission line under a strong wind is similar to that of a primary resonance under harmonic excitation; that is, the amplitude of vibration is very large, and the in-plane vibration is strongly coupled to its out-of-plane vibration.


## 5. Conclusions

In this study, 3D nonlinear vibration equations for transmission lines that consider geometric nonlinearity are established. Based on the boundary conditions and the assumptions used for modeling, the 3D equations are simplified to 2D equations.Galerkin’s modal truncation method is employed to simplify the continuous partial differential equations to discrete partial differential equations that are analyzed using the method of multiple scales and solved by a higher-orderRunge-Kutta method. On this basis, the nonlinear dynamic characteristics of transmission lines when in free and forced vibration, respectively, are analyzed. The main conclusions drawn from this study are summarized as follows:( 1 ) A transmission line has inherent nonlinear vibration characteristics. When in vibration, the in-plane motion of a transmission line is coupled to its out-of-plane motion, and linear, quadratic nonlinear, and cubic nonlinear modes of internal resonance can easily occur.( 2 ) When a transmission line vibrates freely, nonlinear internal resonance can occur even when the initial out-of-plane energy of the transmission line is relatively low; however, under this condition, energy transfer and mode coupling phenomena are not noticeable. As its initial out-of-plane energy increases, the internal resonance of a transmission line becomes prominent and the coupling of its in-plane and out-of-plane vibration becomes stronger, resulting in a transfer of out-of-plane vibration energy to the in-plane direction and an increase in the in-plane vibration amplitude and the dynamic tension in the transmission line. Due to the transmission line’s mode coupling and nonlinear internal resonance, the out-of-plane and in-plane vibration frequencies of the line differ relatively significantly from the natural vibration frequencies of a linear system.( 3 ) Under a harmonic excitation, a transmission line undergoes forced resonance. Under the combined action of internal and primary resonance, the in-plane vibration of the transmission line is coupled to its out-of-plane vibration and the vibration energy transfers from the out-of-plane direction to the in-plane direction that is not directly under the excitation, resulting in a significant increase in the dynamic tension and the displacement amplitude of the transmission line. In addition, as the external excitation amplitude increases, the forced resonance effects become more significant, and the forced resonance has a modulating effect on the vibration frequency of the transmission line. As a result of the forced resonance, the in-plane and out-of-plane vibration energy of the transmission line is distributed over a relatively wide range.( 4 ) When a transmission line is in wind-induced vibration, as the wind speed increases, the nonlinear internal resonance of the transmission line increases; the frequency-modulating effect of its nonlinear internal resonance also increases, which may cause the transmission line to undergo high-frequency vibration. Because the in-plane vibration amplitude of a transmission line and its dynamic tension increase significantly as the wind speed increases, a noticeable amplitude jumping phenomenon occurs; consequently, when in vibration for a long period of time, the dynamic tension amplitude of the transmission line may exceed the breaking load limit, resulting in the breaking of the transmission line.( 5 ) Regardless of whether a transmission line is in free or forced vibration, increasing its damping can effectively consume its vibration energy, thereby allowing the vibration amplitude of the transmission line to reach a stable state within a short period of time. However, increasing the damping cannot prevent the occurrence of nonlinear internal resonance in a transmission line. When a transmission line is in forced vibration and the external excitation has a relatively large amplitude, even if the damping of the transmission line is increased to a relatively large value, the transmission line can still maintain its nonlinear vibration of coupled modes with a large amplitude due to the nonlinear internal resonance; in addition, the vibration energy also constantly transfers between the coupled modes and does not decrease significantly. When a transmission line is in vibration for an extended period of time, this response saturation phenomenon can still cause the dynamic tension in the transmission line to surpass the design-breaking load, thereby causing damage to the transmission line.



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*Source: 1247523-2018-04-19.xml*