## 2 Nonlinear Constitutive Behavior of an Eddy Current Damper  
Generally speaking, the mathematical models for supplemental damping devices are quite complicated. The most widely used energy dispassion devices are FVDs. Experimental evidence [30] reveals that the constitutive behavior of FVDs for the force-velocity relationship can be described by a fractional velocity power law as follows\[F_{FVD}=c_{d}v^{\alpha} \tag{1}\]  
where \(c_{d}\) is a damping coefficient, \(\alpha\) is a velocity exponent, and \(v\) is the relative velocity between two ends of the damper. The exponent \(\alpha\) is responsible for the nonlinear damping of FVDs and depends upon the hydraulic circuit employed. Typically, \(\alpha\) ranges from 0.35 to 1.00 for seismic mitigation applications [18]. For eddy current dampers, the constitutive behavior models for the force-velocity relationship are quite different from those of FVDs. When the relative speed is low, the force-velocity relationship is linear, and it becomes nonlinear when the relative speed goes higher. In 2014, [PERSON] proposed a rotary axial eddy current damper [27], in which a ball screw is employed to amplify the relative speed by transforming the linear motion to a rotary motion. The longitudinal and transverse cross sections of the rotary axial ECD are shown in Figure 1. The damper comprises two main components, i.e., a ball screw assembly and an eddy current damping generator. The ball screw assembly comprises a ball screw drive pair, a stator and a rotor respectively made of magnetic conductive materials. The eddy current damping generators comprise permanent magnets (PMs) and conductive plate, which are arranged on the stator and the outer rotor, respectively. In order to characterize the constitutive behavior for the force-velocity relationship of such a new kind of rotary axial ECDs, both finite element simulations and full scale laboratory experiments are conducted. It is found that the nonlinear constitutive behavior of the force-velocity relationship can be well characterized by the [PERSON]'s model [29] as follows  
\[F_{ECD}=F_{max}\frac{2}{\frac{v}{v_{\sigma}}\ +\ \frac{v_{\sigma}}{v}} \tag{2}\]  
where \(F_{max}\) is the maximum damping force, and \(v_{\sigma}\) is the critical relative velocity when the damping force reaches the maximum value. It should be noted that \(F_{max}\) and \(v_{\sigma}\) are dependent on the design parameters of the damper, such as the remanence of the PMs, the thickness of the conductive plate and the air gap, etc.. The measured data points from full scale laboratory experiments, the finite element method (FEM) simulation curve and the fitting curve by the [PERSON]'s model are plotted in Figure 2. The designed \(v_{\sigma}\) for the tested damper is 0.259 m/s. It can be seen that the [PERSON]'s model can depict the nonlinear constitutive behavior of such kind of ECDs very well. In the following, the energy dissipation and response mitigation analysis are all based on this model. From Figure 2, it can be seen that the eddy current damping force is approximately linearly proportional to the relative velocity in the low speed region, gradually increasing with decreasing slope when the relative velocity becomes higher, reaching a maximum value at the critical speed, and then decreasing for much higher speeds. Although the experimental data do not cover the decreasing stage due to the limitations of the testing machine, the finite element simulation curve has revealed the characteristic of damping force decreasing at much higher speeds. This characteristic of eddy current damping has also been  
Figure 1: Cross sections of the rotary axial eddy current damper (ECD): (**a**) Longitudinal view; (**b**) transverse view. reported in other literatures [29; 31; 32; 33]. This unique characteristic can protect the damper and the connection part between the damper and structure from damage when an over-load is exerted on the damper.