Shock and Vibration

Volume 2018 (2018), Article ID 1247523, 35 pages

https://doi.org/10.1155/2018/1247523

## Nonlinear Dynamic Analysis of High-Voltage Overhead Transmission Lines

^{1}School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China^{2}College of Civil Engineering, Tongji University, Shanghai 200092, China^{3}State 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 [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 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.