# Performance of Strengthened Accelerated Oscillator Damper for Vibration Control of Bridges
## 4 Comparison of Shock Absorption Performance between SAOD-NSD and SAOD-LS
\begin{table}
\begin{tabular}{c c c c c} \hline \(k_{A}\) (kN/m) & \(k_{B}\) (kN/m) & \(I_{1}\) (m) & \(I_{2}\) (m) & \(I_{3}\) (m) \\ \hline
47.25 & 2425.85 & 0.80 & 0.35 & 0.40 \\ \hline \end{tabular}
\end{table} 
Table 3: Values of NSD parameters in Example 3 and Case 6 of Example 4. Figure 6: \(x_{1 max}\) comparison of SAOD-NSD and SAOD-LS. In this modeling, the exact resonance frequency (4.53 Hz) is shown by the red dash line, and the values of SAOD-NSD and SAOD-LS’s \(x_{1 max}\) at exact resonance are represented by the symbol \(\odot\). In Case 5, the values of \(k_{2}\) and \(k_{3}\) are set at 1389.4 kN/m. As for Case 6, the NSD parameters \(k_{2}\) and \(k_{3}\) are consistent with the values specified in Table 3. Additionally, the external excitation in Example 4 is represented by the formula \(\tilde{x}_{g}=-38\)sin(2\(\pi\)\(\cdot\)1.81t). For further details on the other parameters of both Case 5 and Case 6, please refer to Table 4.  
The natural frequencies of Case 4 and Case 5 are both 1.81 Hz, thereby satisfying resonance. The main objective of Case 6's structure design is to minimize the impact of the force change of NSD during both the tension and compression stages. The organization of Case 5 facilitates the comparison of Case 6. The NSD's equivalent stiffness in Case 6 is identical to that of Case 5, making both cases comparable. Figure 8 displays the time-history displacements of \(m_{1}\) in Case 4, Case 5, and Case 6, specifically for the first 12 s. The FE technique is used to calculate the results. We use NSD for a nonlinear-spring element. The following modeling steps are identical to those in Example 1. Figure 8 demonstrates that in Case 5, the SAOD-LS exhibits a notable damping effect at resonance, resulting in a reduction in the displacement peak at resonance. In comparison, the peak resonance of the SAOD-NSD in Case 6 appears lower. Meanwhile, the time-history \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(m_{1}\) (kg) & \(m_{2}\) (kg) & \(m_{3}\) (kg) & \(k_{1}\) (kN/m) & \(c_{1}\) (Ns/m) & \(c_{2}\) (Ns/m) & \(c_{3}\) (Ns/m) & \(r_{2}\) & \(r_{3}\) \\ \hline
29,485 & 19,165 & 19,165 & 3800 & 3740 & 237 & 237 & 3 & 3 \\ \hline \end{tabular}
\end{table} 
Table 4: Values of other parameters of Case 5 and Case 6 in Example 4. Figure 8: Example 4 result comparison: (**a**) time-history displacement response of \(m_{1}\). \(\otimes\) reflects the \(x_{1 max}\) (i.e., the maximum time-history displacement of primary mass \(m_{1}\)) within the initial 12 s of this case; (**b**) amplitude spectrum result. Figure 7: Schematic diagram of Case 5 or Case 6.  
response indicates that the SAOD-NSD exhibits a resonance-like phenomenon, but with a significant reduction in its response peak. Subsequently, aiming to facilitate a more in-depth investigation of the damping mechanism of the NSD in comparison to LS, the amplitude spectrum is employed for analyzing both the frequency components and the amplitude of the response. The \(m_{1}\) displacement responses of Case 4, Case 5, and Case 6 during the first 300 s are subjected to Fourier transformation. We analyze the amplitude spectra of the filtered data and present the results in Figure 8b. Upon examination of Figure 8b, it is evident that the frequency of 1.81 Hz corresponds to both the natural frequency of the structure and the frequency of external excitation in Case 4 and Case 5. Notably, the peak value at Point B (Case 5) is considerably lower than that at Point A (Case 4), corroborating the results obtained through time-history analysis. Given that the equivalent stiffness of the NSD in Case 6 is equal to the stiffness of the LS in Case 5, it can be inferred that the frequency at Point D in Case 6 is also 1.81 Hz. Additionally, the NSD contributes to Point C, which displays a greater amplitude than Point D. When comparing Case 5 and Case 6, it becomes evident that the NSD not only consumes low-frequency energy but also demonstrates remarkable energy dissipation capacity.