# Free Vibration of a Taut Cable with Two Discrete Inertial Mass Dampers
## 1 Introduction  
With the flourishing development of materials and construction technologies, civil engineering structures are becoming larger, lighter, and more flexible, especially for long-span bridges. Cable-stayed is a common option for bridges in the medium to long-span ranges due to its unique structural formation, economic advantage, and esthetic value [1]. However, as important load-bearing components of cable-stayed bridges, stay cables are highly susceptible to dynamic excitations due to their high flexibility and low intrinsic damping [2; 3]. Frequent and excessive amplitude cable vibrations may lead to fatigue failure of cables. These problems may inevitably shorten the service life and cause the risk of losing public confidence in cable-stayed bridges. To guarantee structural safety, several solutions have been proposed to dampen cable vibrations, which include modifying aerodynamic surface of cables [4], connecting cables together via cross tie [5], and attaching external dampers on cables [6; 7; 8; 9]. Though these practical measures have been well applied in the field, each has its own shortcomings. Changing the surface of the cable is difficult to implement for retrofit and may increase drag forces athigh wind velocities [10]. Cross-ties are incapable of direct energy dissipation and make the aesthetics of cable-stayed bridges deteriorate [11]. Compared to the two methods above, attaching external dampers on the cable seems to be more promising. Nevertheless, the installation location of a passive viscous damper is typically restricted to within a few percentage points of the cable length from the cable anchorage [12]. As expected, passive viscous dampers cannot provide sufficient damping to eliminate vibrations for super-long cables, such as the Sutong Bridge, with cables nearly 600 m long. Moreover, the results based on both theoretical and experimental studies indicated that the existence of the cable sag [13; 14], the cable flexural rigidity [15; 16], the damper stiffness [17], and the damper support stiffness [18] or their coexistence [19; 20; 21; 22; 23] would have adverse impacts on the efficiency of passive viscous dampers. An active damper can produce a force-deformation relationship with the negative-stiffness behavior that benefits damper efficiency when the linear quadratic regulator (LQR) algorithm is employed [24; 25]. However, active dampers often require high power sources beyond practical limits and are thus rarely used for cable vibration mitigation in real bridges. Alternatively, semiactive dampers, which can produce similar hysteresis and achieve control performance comparable to that of active dampers, were proposed [26; 27; 28; 29]. For instance, the semiactive control based on magnetorheological dampers has been successfully applied on the Dongting Lake Bridge [30], Binzhou Bridge [31], and Sutong Bridge [32]. Compared to active dampers, semiactive dampers require less power. Nevertheless, possible implementations of semiactive dampers on site still require an external stable power supply, a sensing system, and a controller, which seems to be complicated and costly. This fact has inspired researchers to introduce a negative stiffness mechanism into passive dampers to mitigate cable vibrations. Recently, several representative passive dampers with negative stiffness mechanisms, including pre-spring negative stiffness dampers (pre-spring NSDs) [33; 34] and magnetic negative stiffness dampers (magnetic NSDs) [35; 36], have been successfully developed. Negative stiffness dampers have well demonstrated to be capable of providing superior damping over that of traditional passive viscous dampers [37; 38; 39]. However, extremely large passive negative stiffness may make the NSD lose its stability. Alternatively, an inerter has the potential to provide similar negative stiffness without a stability problem [40].