## 1 Introduction  
In recent years, with the construction of a large number of high-rise buildings and long-span bridges, the civil engineering structure has become more flexible, and the structural vibration problem has become more prominent [1; 2; 3]. Cable-stayed bridges and other cable-supported bridges use a large number of cables as force transmission components [1]. However, cables, as lightweight, high-transverse-flexibility, and low-intrinsic-damping structural components [3], are prone to vibrations induced by environmental disturbances, such as wind, rain, live load, etc. [4; 5; 6; 7]. Long-term large-amplitude vibration may lead to local fatigue damage of cables and can even affect the safety of the whole bridge. To reduce cable vibrations, there are many countermeasures that have been used in practical engineering, such as attaching dampers between cables and the bridge deck or tower, or installing cross-ties between adjacent cables, among others [8; 9; 10; 11; 12; 13]. In terms of current practical engineering applications, installing dampers is the most widely used measure worldwide [10; 11]. Dampers installed on the cable can enhance the cable system's modal damping and, thus, suppress various types of vibration of the cable. Several different kinds of mechanical dampers have been successfully used in practice--for example, oil dampers were used in the Sutong Bridge [11] and the Fred Hartman Bridge [14]; viscous-shear dampers were used in the Sutong Bridge [11]; magnetorheological dampers were successfully installed on Dongting Lake Bridge [15] and Binzhou Yellow River Bridge [16]; friction dampers were also used in Uddevalla Bridge [17]; and high-damping rubber (HDR) dampers were applied in the Tatara Bridge [18]. Among them, viscous dampers and viscous-shear dampers are the most commonly used cable dampers. However, in practice, these two types of dampers also have oil leakage problems, especially for the viscous dampers [19]. Moreover, the cable modal damping obtained from viscous-shear dampers is heavily dependent on the ambient temperature and vibration frequency; for this reason, this type of damper is mostly used for suppressing vibrations of short cables. The friction damper is insensitive to the micro-amplitude vibration of the cable owing to the large static friction force [20], and the MR damper requires long-term power supply and regular maintenance [21]. Today, with the development of material science, HDR dampers have become a kind of damper with bright application prospects, because the HDR damper has no liquid leakage problem and has a long service life. Cables with HDR damper systems have been studied in many papers. [PERSON] et al. [22] explored the damping characteristics of HDR dampers and verified their good damping performance via testing. To maximize the cable's modal damping, [PERSON] et al. [23] devised an empirical formula for optimizing the HDR damper parameters. [PERSON] et al. [22] proposed a numerical method to study the cable-HDR damper system. [PERSON] and [PERSON] established a sagged cable-HDR damper system model by taking the flexibility of damper support into account and derived asymptotic formulae for the cable's modal damping [24]. In previous studies, the damping contributed to the cable by the damper was related to its installation height. With the increase in cable length, it is difficult for a single damper to provide sufficient damping, because the damper can only be installed near the cable's end. [PERSON] and [PERSON] [25] studied the combined damping effect of a cable equipped with two dampers--i.e., a viscous damper and a HDR damper--installed near the two ends of the cable. A taut cable with two HDR dampers was investigated by [PERSON] et al. [26]. [PERSON] et al. [27] studied the combined damping effect of a viscous damper and an HDR damper installed on a shallow cable through theoretical and experimental methods. To mitigate the cable's high-mode vortex-induced vibration, [PERSON] et al.