# Refined Study on Free Vibration of a Cable with an Inertial Mass Damper
## 1 Introduction  
Long steel stay cables, commonly used in cable-stayed bridges, are highly susceptible to dynamic excitations due to their high flexibility and low inherent damping [1; 2; 3; 4; 5]. Large oscillations may reduce life span of cables and have detrimental effects on public confidence in the safety of the bridge [6; 7]. Structural vibration control methods, including passive control [8], semi-active control [9; 10], and active control [11], have been well developed and used for protecting structures against dynamic loading. For stay cables, modifying aerodynamic cable surface [12], connecting multiple cables together [13], installing external dampers [14; 15], and combining external dampers with cross-ties [16; 17], have been proposed to eliminate such vibrations. Among them, transversely attaching a passive viscous damper to the stay cable has been widely implemented in practical applications [18; 19]. However, the supplemental damping induced by passive viscous dampers would be insufficient to eliminate vibrations effects of super long stay-cables since the damper location is typically restricted to the vicinity of the bridge deck for aesthetic and practical reasons [20; 21]. As a more efficient solution, active or semi-active control, especially MR-based smart damping technology has attracted extensive attention from the community [22; 23]. The MR damper can produce a damper force-deformation relationship with the negative stiffness behavior that benefits damper efficiency when the linear quadratic regulator (LQR) algorithm is employed [24; 25; 26; 27]. Recently, an innovative mobile tuned mass damper (TMD) system, consisting of a mobile TMD device and a semi-active MR damper, has shown a better control performance than its passive counterparts when the fuzzy control algorithms were employed [28]. However, active or semi-active control requires an external stable power supply, a sensing system, a controller, etc. [29; 30; 31; 32], which is complicated and costly. To solve this problem, great efforts have been devoted to seeking innovative passive dampers with similar negative stiffness mechanism. Passive negative stiffness dampers (NSDs), the superposition of negative stiffness devices with passive viscous dampers, have been theoretically and experimentally proved to be able to provide superior achievable modal damping to the cable over that of traditional passive viscous dampers [33; 34; 35; 36; 37; 38]. Recent study has also shown that the combined effects of a concentrated mass and a viscous damper can also behave like that of the passive NSD [39]. However, its control performance seems to be sensitive to the extra mass attachment on the cable. As an alternative, many inerter-based devices have been developed, and their control performance advantages have been well demonstrated for civil engineering structures [40; 41; 42; 43; 44; 45; 46; 47; 48]. In particular, the behaviors of typical inertial mass dampers (IMDs) [49; 50; 51] or tuned inerter dampers (TIDs) [52; 53] to suppress cable vibrations have been theoretically investigated and identified through an ideal taut-cable model. The results indicated that similar to the NSD, the IMD can also show negative stiffness that benefits damper efficiency. Moreover, there is no instability issue for the IMD, while it would be observed when passive negative stiffness of the NSD is extremely larger. A TID system, where the traditional tuned mass damper (TMD) mass is replaced by an inerter, can offer the potential for much higher mass ratios than that of the TMD. To facilitate efficient damper design and accurately predict the dynamic behavior of a cable with a passive or semi-active damper, the influences of the cable sag and the cable flexural rigidity on the damper efficiency have been investigated. It was found that nearly symmetric vibrations of the cable, especially in the first mode, were significantly affected by the cable sag, while the antisymmetric vibrations were hardly affected by the cable sag [54; 55; 56].