## 1 Introduction  
Stay cables are susceptible to vibrations because of their low inherent damping [1, 2]. The transverse cable amplitude may become unacceptably large when the excitation frequency due to vertex-shedding matches one of the cable eigenfrequencies whereby the entire excitation energy is transmitted to one modal component [2, 3, 4, 5, 6, 7, 8, 9, 10]. Such cable vibrations with amplitudes in the range of 0.3 m to 0.6 m have been observed on several stay-cable bridges, e.g., the Dongting Lake Bridge in China [11], the Franjo Tudjman Bridge in Croatia [12] or the Alamillo Bridge in Spain [13]. The common countermeasure is to install transverse cable dampers close to the deck anchor [14, 15, 16, 17, 18]. The selection of the damper position relative to the cable length, which is often between 2% and 3%, is a critical issue. On the one hand, the damper position must be great enough to guarantee at least the required damping ratios in the targeted modes. On the other hand, the damper must not be positioned too far away from the deck anchor because of aesthetic reasons, additional costs of tall damper supports and geometrical conflicts of the damper support with the anchor of the neighbour cable. The common procedure is to optimize the viscous damper coefficient to the targeted modes in order to guarantee at the required damping in lowest and highest targeted modes whereby the damper position is automatically minimized [19]. This analytical method is based on optimum modal damping, assuming that the cable can be modelled as a taut string, i.e., a cable without bending rigidity and with simply supported ends [14; 15; 16]. However, real cables show the effects due to bending rigidity and fixed support end conditions which necessitates a different design of the damper viscous coefficient and reduces the attainable cable damping ratio significantly as presented in [20; 21; 22]. The studies [20; 21; 22] estimate that the bending rigidity and fixed support end conditions reduce the attainable cable damping ratio in one target mode by approximately 20% compared to the attainable damping ratio assuming a taut string behaviour. However, the experimental investigations on a single strand with fixed support end conditions and bending rigidity described in [23; 24] demonstrate that the cable motion at typical damper positions is reduced by 20% to 40% compared to the taut string model prediction. This means that the cable damping ratio due to a transverse damper is at least 20% to 40% smaller than expected from the closed-form solution 1/2 a/L (a: damper position, L: cable length, valid for a/L \(\approx\) 1 %) being valid for a taut string behaviour. This estimation agrees well with the experiments reported in [25; 26; 27; 28; 29; 30; 31] which show that the measured additional cable damping ratio due to the transverse damper, i.e., the total cable damping ratio minus the inherent cable damping ratio, is often around 50% to 70% of the theoretical value 1/2 a/L being valid for a taut string behaviour. Besides the effect of the reduced damper motion, other effects may also lead to reduced cable damping ratios such as insufficient damper activation due to manpower excitation of real stay cables with great modal mass [32], friction damping [33; 34; 35], excitation of higher order modes necessitating low pass filtering during post-processing of the measurement data [29; 35], force tracking errors in case of controllable dampers [24; 25; 26; 27; 36; 37; 38; 39; 40] and the fact that the damping estimation 1/2 a/L is valid for a/L \(\approx\) 1 % but transverse dampers are commonly positioned at 2% to 3% of the cable length. This testimony of cable damper tests and even the development of other damper concepts [41; 42; 43] show that efficient and reliable stay cable dampers is a need to ensure the serviceability of cable-stayed bridges [44].