## 1 Introduction  
A cross-rope suspension (CRS) system is a widely used type of transmission line structure that was developed from a guyed V-tower system. CRS was firstly proposed in 1974 [1]. In the CRS system, the transmission line is supported by transversally crossing suspension cables connected to the towers on both ends. Thus, the conventional steel towers were removed, and the CRS system was always considered to be optimized compared with the line-tower system. Moreover, the CRS system was more suitable in mountainous areas, which largely improved the construction difficulty and efficiency. The CRS structural system built in Cape Town, South Africa, was called the \"invisible tower line\" [2]. Its engineering experience indicated that CRS could significantly reduce the visual impact on the surrounding environment and protect the original natural landscape. Generally, the span of the suspension cable is limited to 200 m. For the application in a complex environment, the span of CRS was expanded beyond 1000 m while insulators were installed between each pair of conductors, which became a novel long-span cross-rope suspension structural system, as shown in Figure 1[3]. The finite element method (FEM) was widely adopted for analyzing complex flexible structures with large geometric deformation so far, as well as CRS [4]. The analysis of the initial shape (mechanical sag) of suspension cable and conductors was the basis of further dynamic analysis of long-span CRS. [PERSON] and [PERSON] [5] established the initial form-finding method of overhead transmission lines, which is based on the form-finding of cable structures. Based on the FE model of CRS, dynamic properties and structural responses induced by wind load were studied. CRS transmission line structures were designed to withstand several different load cases. For mountainous environments, extreme wind and combined wind and ice in cold climates governed the design load for CRS. [PERSON] presented a new method to determine wind loading on transmission line conductors based on fluid-structure interaction analysis [6]. Based on the historical weather data in Ontario, [PERSON] analyzed the wind load on bare and ice-covered conductors [7]. [PERSON] used the finite element strategy and developed the study of aluminum conductor steel reinforced (ACSR) by wind-induced load and compared it with experimental data [8]. Wind-induced vibration and its vibration controlling measurement were the major consideration of the structural design of long-span CRS. Flexible structures, such as long cable and cable-supported structures, were susceptible to wind-induced vibration due to their strong geometric nonlinear [9; 10]. Long-span CRS was considered to be a typical wind-sensitive structure, and its wind-induced vibration was always focused on by researchers. Vortex-induced vibration (VIV) was recognized as one of the most common types of wind-include vibration that occurred in overhead lines, which was attributed to the process of vortex shedding [11]. VIV significantly induced dynamic bending stress on the ends of conductors and enhanced the risk of structural failure. Models of CRS with different element types based on finite element analysis were established, and the influence of various models on the overall response of structure was compared [12]. Dampers, such as tuned mass damper, magnetorheological damper and Stockbridge damper, were adopted to provide additional energy dissipation for reducing wind-induced structural vibration, also applicable for VIV of transmission line [13]. [PERSON] presented an analytical model of a novel VIV damper with an increased number of resonant frequencies [14]. Generally, the maximum steady-state amplitude of the conductor during the VIV process was solved by energy balance method [15; 16]. Therefore, the conductors' dynamic bending stress cycle results in independent elongated deformation of each rope of conductors and further surface erosion and fatigue [17].