## 4 Slit Damper Systems with Recentering SMA Bars  
In order to overcome the disadvantage of existing steel damper models, slit damper systems with recentering SMA bars, which are intended to improve self-centering capability, are proposed in this study. The detail drawings for the proposed damper systems are shown in Figure 5. Four superelastic SMA bars are equipped with the steel slit damper, and connected to both flanges by clamping themselves on the nuts. These superelastic SMA bars, round in shape, can achieve weld-free design for the purpose of avoiding any heat treatment effect. The welding heat applied to superelastic SMA material at room temperature tends to cause the change of the material property, and results in the loss of recentering capability. Furthermore, round section SMA bars offer both stiffness and strength for equally resisting external forces in any direction. The diameters of all SMA bars are a uniform 30 mm so that the same recentering forces are offered to various slit dampers with their different capacities. The steel slit dampers that are identical to experimental specimens presented in Section 3 are used to construct the proposed damper system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Model ID**} & \multicolumn{4}{c}{**Measured Dimensions**} & \multirow{2}{*}{\(\mathbf{b/l_{0}}\)} \\ \cline{2-2} \cline{4-5}  & \(\mathbf{t}\) & \(\mathbf{b}\) & \(\mathbf{l_{0}}\) \\ \hline SL1 (SL1-SMA) & 8.0 & 14.9 & 97.0 & 0.155 \\ SL2 (SL2-SMA) & 8.0 & 15.0 & 87.1 & 0.172 \\ SL3 (SL3-SMA) & 8.0 & 15.1 & 77.0 & 0.195 \\ SL4 (SL4-SMA) & 8.0 & 16.9 & 99.2 & 0.172 \\ SL5 (SL5-SMA) & 8.0 & 16.8 & 88.3 & 0.191 \\ SL6 (SL6-SMA) & 8.0 & 16.5 & 79.0 & 0.215 \\ \hline \end{tabular}
\end{table}
Table 1: Geometric design parameters for experimental slip damper models.  
Figure 4: Response mechanism of slit damper models, (**a**): Bending Moment Diagram, (**b**): Deformation Shape.  
The response mechanism of the recentering slit damper system, which is achieved by the behavior of the steel slit damper and the superelastic SMA bending bars, is presented in Figure 6. The reverse phase transformation changing from martensite to austenite upon unloading leads in a major way to self-centering behavior, which offers core competence to SMA materials without residual strain at the room temperature. Accordingly, the proposed damper system possesses excellent recentering capability resulting from this unique characteristic referred to as super-elasticity or pseudo-elasticity. The parallelogram hysteresis loop represents stable energy dissipation formed by the yielding of the slender strips. The displacement (or strain) hardening affected by the base material's property is also involved in this loop. In spite of providing vibration damping to the structure, energy dissipation induced by metallic yielding has a tendency to produce considerable permanent deformation (\(\Delta_{\mathrm{slit},p}\)) during cyclic loading. Finally, the total behavior of the recentering slit damper system proposed herein can be simulated by superposing superelastic SMA behavior and energy dissipation behavior. As can be seen in Figure 6, these component behaviors are assembled in parallel depending on how they interact each other. Owing to such parallel assemblage, the total force of the recentering slit damper system (\(P\)) can be determined by adding together component forces under the corresponding displacement (\(\Delta\)) as follows:  
\[P=P_{\mathrm{SMA}}+P_{\mathrm{slit}} \tag{1}\]  
Figure 5: Design of the slit damper system with recentering SMA bars (**a**): Front View; (**b**): Plan View and (**c**): 3D View.  
\[\Delta=\Delta_{\rm SMA}=\Delta_{\rm slit} \tag{2}\]  
The amount of permanent deformation in the recentering slit damper system is entirely dependent on recentering force simply regulated by the size of the superelastic SMA bending bars.