## 2 Materials and Methods
### Dynamic Materials Testing  
As shown in Figure 5, a pair of WMDs was used to design an isolator to evaluate the symmetric tension-compression behavior under sinusoidal excitations. The setup of the dynamic mechanical test is illustrated in Figure 4 wherein is featured the SDS-200 hydraulic universal testing machine equipped with a digital data acquisition and control system, with a load capacity of 200 kN and a test stroke of \(\pm 50\) mm. Fictures connecting the isolator and the machine gripper were designed to satisfy the need for symmetric tension-compression excitation, and a sine test protocol with a peak amplitude of 0.1 mm and a constant frequency of 1 Hz was executed. A pre-test was carried out for more than 10 cycles until the hysteresis curves overlapped, to stabilize the hysteresis behavior of the NiTi WMD. The sampling procedure consisted of 30 loading-unloading cycles repeated three times to reduce error. The design of the one-factor test is shown in Table 4. The entire test involves variables such as relative density, heat treatment temperature, and heat treatment time to determine their impact on the WMDs' final damping capacity and stiffness. Dynamic stiffness can be viewed as the ability of the WMD to resist an external oscillatory load. It differs from static stiffness concerning the calculations and can be expressed by the load that is required to produce unit displacement on the damper, i.e.,  
\[K_{\text{d}}=\frac{F_{\text{max}}-F_{\text{min}}}{2X_{0}}, \tag{2}\]  
where the \(F_{\text{max}}\) and \(F_{\text{min}}\) are the maximum and minimum restoring force sampled from the hysteresis curve, and \(X_{0}\) is the peak motion amplitude of the test. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Material**} & **Relative Density** & \
\begin{tabular}{c} **Heat Treatment** \\ **Temperature (\({}^{\text{c}}\)C)** \\ \end{tabular} &
\begin{tabular}{c} **Heat Treatment** \\ **Time (min)** \\ \end{tabular} \\ \hline
1 & 304, NiTi & 0.31 & – & – \\
2 & NiTi & 0.25, 0.31, 0.38 & – & – \\
3 & NiTi & 0.31 & 150, 200, 250 & 30 \\
4 & NiTi & 0.31 & 200 & \(15,30,45\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Test groups with various factors of material, relative density, heat treatment temperature, and heat treatment time. Figure 5: SDS-200 dynamic testing machine and the setup details of the NiTi WMDs. The loss factor can be written in the form of energy dissipation, \(\Delta W\), divided by the elastic potential, \(W\), within a complete cycle of vibration, which gives  
\[\eta_{\mathrm{d}}=\frac{\Delta W}{2\pi W}, \tag{3}\]  
where the energy dissipation can be obtained by  
\[\Delta W=\int F\mathrm{d}x=-\omega X_{0}\int_{0}^{T}F\sin(\omega t+\alpha) \mathrm{d}t=-\frac{2\pi X_{0}}{N}\sum_{i=1}^{N}F_{i}\sin(\frac{2\pi i}{N}+ \alpha), \tag{4}\]  
where the \(F\) is the restoring force, \(\omega\) the circular frequency, \(\alpha\) the phase angle, and \(N\) the dynamic load cycles. The elastic potential is given by  
\[W=\frac{1}{2}KX_{0}^{2}, \tag{5}\]  
Substituting Equations (4) and (5) into Equation (3), the dynamic loss factor can be written as  
\[\eta_{\mathrm{d}}=-\frac{4f}{f_{0}(F_{\mathrm{max}}-F_{\mathrm{min}})}\sum_{i =1}^{N}F_{i}\sin(\frac{2\pi i}{N}+\alpha) \tag{6}\]
As illustrated in Figure 7, the sine sweep test was performed with input signals of which the acceleration ranges from 0.25 g to 2.0 g and frequency ranges from 10 Hz to 200 Hz. For all of the tests, the input acceleration gradually increased at lower frequencies until it reached the designated values and was kept constant for the rest of the test. A pre-test feasibility analysis was performed to see if the system was well configured and to stabilize the damping performance of the WMDs. In the present work, the damping capacity of the NiTi WMDs is mainly evaluated by the transmissibility and the amplification factor.