## 1 Introduction  
Stay cables have low mass, low frequencies, and low damping ratio, so they are vulnerable to external excitation. The vibration of stay cable under ambient excitation (such as wind or earthquake loading) and live load can cause fatigue in cable attachment, which might reduce the bridge safety. There are mainly two methods to control cable vibration. The first method is to modify the cable surface to improve its aerodynamic performance [1; 2; 3] and the other one is to install the energy dissipating devices such as magnetorheological damper, friction damper; oil damper; or HDR damper [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. The installation of an external damper can increase the cable damping ratio. Consequently, its vibration was mitigated. Studies into the cable vibration with an external damper including an HRD damper have been investigated by many researchers [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. Single HDR damper device has widely been applied for stayed cables with short lengths as it can efficiently dissipated the vibration energy of the cable. [PERSON] et.al. [6] optimized the parameters of single HDR damper for obtaining the maximum damping of stay cable. In addition, [PERSON], and [PERSON]. [7] proposed the formulas for calculating the cable damping ratios of stay cable with a single HDR-damper. [PERSON] et al., further investigated on stay cable with HDR damper [11]. They found that the damping ratio has strong dependency on loss factor and damper location. However, their study did not consider the bending stiffness of taut cable for calculating the cable damping. Recently, [PERSON] et al. (2020) also proposed an approximated formula for estimating the damping ratio of a stay cable with single HDR-damper considering the uncertainties of boundaries conditions [12]. They concluded that the efficiency of the damper can be increased by adding more rotational restrain stiffness. According to previous studies [4; 9; 10; 11], the maximum modal damping of stay cable is directly proportional to the distance of the damper from its end. The effect of single HDR damper and other damper types on stay cable damping has attracted many further studies [4; 6; 8; 9; 11; 12; 13; 14; 15; 16; 17; 18]. However, due to the bridge aesthetic, the damper is often installed near the cable end within a few percentages of cable length that lessens the efficiency of the damper. Therefore, it is difficult to obtain an expected damping level using a single damper for very long-stay cables [4]. In this regard, the use of two dampers at different locations can provide a possible countermeasure to suppress the cable vibration by increasing the total damping [10]. [PERSON] et.al. [19] investigated the impact of two viscous dampers on taut cable damping. They figured out that the dampers located at opposite ends of taut cable can enhance the damping as compared to the case with dampers installed on one side near the bridge deck. [PERSON] and [PERSON]. [10] also confirmed that the total damping ratio of cable with two dampers is equal to the summation of two single damper effects. [PERSON] and [PERSON] [20] presented a numerical example of combining two oil dampers to increase the stay cable damping ratio. This study showed that damper stiffness, damper installation location, and oil-damper direction influent its effectiveness. Furthermore, according to [PERSON] and [PERSON] [21], bending stiffness affects noticeably the damping ratio of stay cable. These authors have developed a numerical framework to calculate vibration frequencies and damping ratios of sagging cables with non-negligible bending stiffness equipped with a viscous damper. [PERSON] and [PERSON] proposed the theoretical model for evaluating damping of stay cable with two friction dampers to reduce cable galloping considering cable bending stiffness [22].