## 4 Discussion  
### Comparative Analysis of Damping Mechanism under Dynamic Load: NiTi WMD versus SS 304 WMD  
A comparative analysis of system dynamics is conducted here to provide a clearer insight into the mechanism that underlies the hysteresis behavior of the two different types of WMD. According to reference [3], the SS 304 WMDs under dynamic load can be described by the single-degree-of-freedom (SDOF) system depicted in Figure 24a and expressed by the following first-order differential equation,  
\[F(t)=c\frac{\mathrm{d}y}{\mathrm{d}t}+k\cdot y(t)+z(t), \tag{9}\]  
where \(F(t)\) is the restoring force, \(c\) is the damper constant, \(k\) is the stiffness of the WMD, and z(t) is the hysteresis damping force generated by the dry friction between individual wires of the WMD. In addition to the friction dissipation between individual wire coils, the phase-transformation damping mechanism also plays a significant role in the dynamic mechanical behavior of NiTi WMD. By adding an exclusive term \(\zeta\) (\(\Gamma\), \(\tau\), \(t\)) describing the phase transformation effect to Equation (9), the SDOF system of NiTi WMD shown in Figure 24b now becomes  
\[F^{\prime}(t)=c^{\prime}\frac{\mathrm{d}y}{\mathrm{d}t}+k^{\prime}\cdot y(t)+ z^{\prime}(t)+\zeta(\Gamma,\tau,t) \tag{10}\]  
where \(c^{\prime},k^{\prime}\), and \(z^{\prime}(t)\) are the damper constant, stiffness, and frictional damping force of the NiTi WMD. The term \(\zeta\) (\(\Gamma\), \(\tau\), \(t\)) denotes the nonlinear stiffness and damping force exclusively contributed by the stress-induced martensite transformation and can be written as  
\[\zeta(\Gamma,\tau,t)=c_{\mathrm{niti}}y_{\mathrm{m}}\omega\cos(\omega t+\varphi _{y})+k_{\mathrm{niti}}y_{\mathrm{m}}\sin(\omega t+\varphi_{y}), \tag{11}\]  
where \(\omega\) the circular frequency of the dynamic load, and \(k_{\mathrm{niti}}\) is the stiffness component contributed by the NiTi material, which is given by [21]  
\[k_{\mathrm{niti}}(\Gamma,\tau)=k_{\mathrm{A}}+\xi(\Gamma,\tau)(k_{\mathrm{M}} -k_{\mathrm{A}}), \tag{12}\]  
where \(\xi\) is the martensite volume fraction that is supposed to be a function of heat treatment temperature and heat treatment time. \(k_{\mathrm{A}}\) and \(k_{\mathrm{M}}\) are the stiffness of NiTi wire in an austenite state and a martensite state when \(\xi\) is equal to 0 and 1, respectively. The damping coefficient of the NiTi material, \(c_{\mathrm{niti}}\), can be written as  
\[c_{\mathrm{niti}}=\frac{W_{\mathrm{d}}}{2\pi k_{\mathrm{niti}}\varepsilon_{ \mathrm{m}}^{2}}, \tag{13}\]  
where \(W_{\mathrm{d}}\) is the energy dissipation caused by the phase transformation of the NiTi material, and \(\varepsilon_{m}\) is the maximum strain the material undergoes. Figure 24: First-order SDOF system of WMD isolator under dynamic load. (**a**) SS 304 WMD; (**b**) SE NiTi WMD. Based on the theoretical analysis above, the SS 304 WMD's hysteresis behavior is solely contributed by the friction dissipation of individual wire coils. As for NiTi WMD, however, the phase transformation damping works in parallel with the dry friction damping, which renders the NiTi WMD a greater loss factor under different displacement amplitudes, as mentioned in Section 3.2.1. Both types of WMDs demonstrate reduced loss factors with the increasing displacement amplitude. On the one hand, the elastic potential increases even faster than the dissipation energy as displacement amplitude increases, resulting in a decrease in both types of WMD's loss factors. On the other hand, the loss modulus of NiTi wire generally exhibits a mild decrease with the increasing strain, which also causes a decline in the damping capacity of the NiTi WMD. However, the NiTi WMD's loss factor shows a slower declining trend than the SS 304 WMDs, implying that compensation is provided by the phase transformation mechanism of the NiTi wire. The NiTi wire softens because of the stress-induced martensite transformation, as mentioned in Section 3.2.1, leading to a slower increase in the elastic potential than in the SS 304 WMD.