## 1 Introduction  
Structural vibration control has been widely applied in civil engineering, such as wind-induced vibration reduction and seismic protection of high-rise buildings [1; 2; 3; 4; 5], seismic response mitigation of long span bridges [6; 7; 8], human-induced vibration control of footbridges and floor decks [9; 10], vibration control of stay cables [11; 12; 13], vibration control of wind turbines [14; 15; 16] and vibration control of marine structures [17; 16], etc. Energy dissipation devices, such as fluid viscous dampers (FVDs) [6; 18], magnetorheological (MR) dampers [7; 11; 12], bounding dampers [17; 19; 20; 21] and electromagnetic dampers [2; 9; 22; 23; 24; 25], can suppress structural vibration effectively. In recent years, an innovative control device named eddy current damper (ECD) has gained more and more attention in the field of structural vibration control. The eddy current damping is a kind of magnetic damping, which is contactless between the movable members. Therefore, wear and fatigue are minimized in the ECDs, minimum maintenance is required and lifetime is maximized, gently outperforming that of contact-type dampers. Compared to conventional FVDs, there is no working fluid and no fluid degradation and leakage problems exist in the ECDs. The ECDs can be made tube-like and produce an axial force [26], or coupled with a ball screw to form a rotary axial eddy current damper [27]. The eddy current damping unit can be also integrated with a mass and a spring to form an eddy current tuned mass damper (EC-TMD) [2; 22]. In its simplest form, an ECD device consists of a conductive plate and a pair of magnets. When the conductive plate moves through a magnetic field, the movement induces an eddy current in the conductive plate. The interaction of the eddy currents and the applied magnetic field induces a drag force that resists the relative motion between the conductive plate and the magnets. The drag force is also called the eddy current damping force. However, the energy dissipation density of such an eddy current damping system is usually very low, and this disadvantage does restrict its industrial applications [26]. In recent years, significant improvements have been made by either enhancing magnetic strength or increasing the speed of relative movement. For example, the use of back irons in the ECDs is shown to increase the damping coefficients by a factor of up to five [22]. The relative speed between the conductive plate and magnets can be amplified using a ball screw [27] or a linear magnetic gear [28]. A ball screw can transform low speed linear motion into high speed rotary motion and a linear magnetic gear can amplify the linear speed by several times. For the ECDs without velocity amplification, the relative velocity between the conductive plate and magnets is not very high if the external excitations are in the low frequency band. In such cases, the constitutive behavior of eddy current damping is linear for the relationship between the damping force and relative velocity [22]. When the external excitations are in very high frequency band such as earthquakes and impact loads, or the ECDs are enhanced with amplification mechanisms such as the ball screw or magnetic gear, the relative velocity between the conductive plate and magnets becomes much higher. In such cases, the constitutive behavior of eddy current damping is nonlinear due to the influence of the reaction field of the eddy current in the conductive plate [29]. It is common in both design and retrofit of structures to approximate the real multi-degree-of-freedom (MDOF) system with an equivalent single-degree-of-freedom (SDOF) system such that the latter represents the most relevant characteristics of the former for a particular motion mode. In this way, any preliminary attempt to design a supplemental damping system for a structure can be simplified allowing the designer to use an equivalent SDOF model instead.