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Shock and Vibration
Volume 2018 (2018), Article ID 1247523, 35 pages
https://doi.org/10.1155/2018/1247523
Research Article

Nonlinear Dynamic Analysis of High-Voltage Overhead Transmission Lines

1School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China
2College of Civil Engineering, Tongji University, Shanghai 200092, China
3State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Guifeng Zhao; nc.ude.uzz@oahzfg

Received 24 July 2017; Revised 15 February 2018; Accepted 8 March 2018; Published 19 April 2018

Academic Editor: Songye Zhu

Copyright © 2018 Meng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [114]. 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 [414]. 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 [1524]. 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 generalized Hamilton’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-order Runge-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

Figure 1 shows an elastic transmission line with its two ends hinge-supported at the same height. The initial static equilibrium geometric configuration of the transmission line in the plane is used as the reference location and is represented by function . , , and represent the sag and span of the dynamic tension in the transmission line, respectively. represents the 3D dynamic configuration of the transmission line under an external excitation . , , and represent the components of the external excitation in the -, -, and -axes, respectively. and represent the elastic modulus and the cross-sectional area of the transmission line, respectively. represents the dead weight of the transmission line per unit length (, where represents the acceleration of gravity). , , and represent the displacements of the transmission line along the longitudinal in-plane vibration (-axis direction), the transverse in-plane vibration (-axis direction), and the transverse out-of-plane vibration (-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 length of the differential element at any arbitrary location along the -axis direction is

According to the definition of Lagrangian strain, initially, the length of the differential element of the transmission line under the action of gravity iswhere represents the initial static strain of the transmission line, which is calculated using the following equation:where 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 excitation is

The total dynamic strain [12] of the transmission line in a motion state under the external excitation is

The transient total tension in the differential element of the transmission line in a motion state is

According to a generalized Hamilton’s principle,where , , represent the kinetic energy and the strain energy of the transmission line and the virtual work done by nonconservative forces (including the virtual work done by the damping force and the virtual work done by the external excitation), respectively.

The kinetic energy of the transmission line iswhere represents the mass of the transmission line in the static equilibrium state per unit length and , , and represent the vibration velocities of the transmission line in the -, -, and -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

The virtual work done by the damping force iswhere represents the structural damping of the transmission line.

The virtual work done by the external excitation isBy substituting (8)–(11) into (7) and considering the boundary conditions, the following is satisfied at the locations and :

Thus, the Euler equations for the transmission line along the virtual displacement , , and directions are obtained:

Because the sag-to-span ratio of a transmission line is generally very small, ,  ,  , and 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,

LetBy substituting (15) into (13), nonlinear equations of motion for the elastic transmission line that is horizontally hinge-supported at the two ends are obtained:

In (16)–(18), represent the accelerations of vibration of the transmission line along the -, -, and -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 stiffness 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 (-axis direction), . Thus, (16) can be rewritten as

The following approximation relation is then used:

By substituting (20) into (19), integrating the two sides of (19) over the span of the transmission line once, and ignoring the terms unrelated to time , we havewhere is an arbitrary function. By dividing the two sides of (21) by and considering that the initial tension in a transmission line is generally far smaller than its axial stiffness 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

To determine , the two sides of (22) are integrated over the interval and the transmission line boundary conditions represented by (23) are used. After simplification, we have (24).

Let us set

By substituting (22) and (25) into (17) and (18), respectively, we have

For ease of analysis, by use of Galerkin’s modal truncation method, the in-plane and out-of-plane response of the transmission line is represented by

In (28), is the th-order vibration mode function (the two ends of the conductor line are considered to be hinge-connected), represents the retained mode order, and 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 using Galerkin’s modal truncation method; that is, set

By substituting (29) into (26) and (27), subsequently multiplying the two sides of (26) by , integrating the resulting equation over the interval , multiplying the two sides of (27) by , and integrating the resultant equation over the interval , discrete equations of coupled in-plane and out-of-plane motion for the transmission line are obtained:The factors in (30) are calculated as follows:

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 [2629], which is an effective method for the analysis of nonlinear dynamics.

Assume that the external excitation is a harmonic excitation: that is,where and represent the amplitudes of the in-plane excitation and the out-of-plane excitation, respectively, on the transmission line, and and 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 load is assumed to be zero. Now, the generalized out-of-plane excitation load on the transmission line is set to . In addition, the parameters , , , , , , , , , , and are introduced to transform (30) to dimensionless equations:where and 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:

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:that is,

Thus, the derivative with respect to is transformed to the expansion of a partial derivative with respect to , that is,

Hence, the solutions to (33) can be expressed as follows:

By substituting (37) and (38) into (33), we have

By setting the sum of the coefficients of the same power of to zero, we have

:

:

:

:

By solving the set of equations in (40), we havewhere and are unknown complex functions and and are the conjugates of and , respectively. For ease of description, conjugate terms are denoted by . The governing equations for and can be solved by requiring , , , and to be functions with a period of .

By substituting (44) into (41), we have

From (45), we can easily find that the terms that contain and 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 ,  , and , the solutions to the abovementioned partial differential equations that describe the motion of the transmission line will contain an additional term that links and , which is exactly the secular term caused by internal resonance. Therefore, when solving (45), we should consider the cases of , , and . In view of the complexity of the problem-solving and the purpose of this study, we merely take the case of 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:

Thus,

By substituting (47) into (45), we can obtain the requirements to eliminate the secular terms as follows:

The following formulas are introduced:

By substituting (49) into (48) and separating the real part from the imaginary part, we have

Let us introduce the new phase angle as follows:

By substituting (51) into (50), we have

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 [2729]. For this reason, we consider only the power of ε up 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: , , and .

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

In numerical analysis, the Runge-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-order Runge-Kutta method. Therefore, we employ a higher-order Runge-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:

Thus, (30) can be transformed to state functions:

4.1. Calculation of the Parameters

A relatively large single-span transmission line in a high-voltage transmission line system in East China (Figure 2) 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.
Table 2: Design parameters of the selected transmission line after simplification.
Figure 2: A high-voltage transmission line.

The sag-to-span ratio 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 Tables 1 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 ratio 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 iswhere represents the vertical load on the transmission line, which in this case is the dead weight of the transmission line; represents the initial horizontal tension in the transmission line (); represents the horizontal span of the transmission line; represents the difference in height between the two ends of the transmission line; and 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:where represents the natural circular frequency of the out-of-plane vibration of the suspended cable, represents the modal number, represents the mass of the suspended cable per unit length, represents the out-of-plane mode of vibration of the suspended cable, and 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:where represents the natural circular frequency of the in-plane antisymmetric vibration of the suspended cable, represents the modal number, represents the mode of the in-plane antisymmetric vibration of the suspended cable, and 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:where represents the natural circular frequency of the in-plane symmetric vibration of the suspended cable, represents the in-plane symmetric mode of vibration of the suspended cable, represents the elastic modulus of the suspended cable, represents the cross-sectional area of the suspended cable, and 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 Tables 1 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.
Table 4: Natural vibration frequencies of the transmission line model obtained from FE simulation.

A comparison of Tables 3 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: , , and . In other words, within the scope of the linear theory, the natural vibration frequencies of the transmission line are always multiples of one another.

Figure 4 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.
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), and .

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 damping 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: ,  ,  ,  . The displacements of the transmission line obtained from numerical simulation are then divided by the horizontal span 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 and then divided by the horizontal span . The results are presented in Figures 57.

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.
Figure 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.
Figure 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.

As demonstrated in Figure 5, 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: ,  ,  ,  . Figures 810 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.
Figure 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.
Figure 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.

As demonstrated in Figure 8, 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 , , and 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, when Galerkin’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 Figure 11 for the results). In Figure 11, the tension is a dimensionless value obtained by dividing the dynamic tension obtained from simulation by the initial tension . 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.
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 Section 4.3.2 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 810 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 damping is set to 0.01.
Figure 13: Free vibration response of the transmission line when its damping is set to 0.05.
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 excitation . The relationship between the circular frequency of the excitation and the natural circular frequency of the out-of-plane vibration of the transmission line is characterized by introducing a detuning parameter , a nonnegative real number , and a small parameter :

To study the primary resonance response of the system, , , and are set to 1, 0.001, and 10, respectively. By substituting these values into (51), the frequency of the out-of-plane excitation is determined: Hz. The natural frequency of the first-order out-of-plane vibration of the transmission line that is directly induced by the excitation is as follows: Hz. approximately equals . Thus, it can be considered that .

Let us set the amplitude of the generalized excitation load to 3305 N and the initial conditions for simulation to ,  ,  ,  . By substituting these values into (54), the nonlinear forced vibration response of the transmission line under various damping conditions is determined. Figures 1417 show the numerical simulation results obtained when the damping is set to 0.001, 0.01, 0.05, and 0.1, respectively.

Figure 14: Primary resonance response of the transmission line when its damping is set to 0.001.
Figure 15: Primary resonance response of the transmission line when its damping is set to 0.01.
Figure 16: Primary resonance response of the transmission line when its damping is set to 0.05.
Figure 17: Primary resonance response of the transmission line when its damping is set to 0.1.

As demonstrated in Figure 14, 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 . 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.

Figure 15 shows the simulation results obtained by increasing the damping 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.

Figures 16 and 17 show the simulation results that are obtained when the damping 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 (Figures 16(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 and conditions, when the frequency of the external excitation is approximately equal to 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: frequency of the external excitation: 0.68 Hz; damping : 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 a