# Performance of Strengthened Accelerated Oscillator Damper for Vibration Control of Bridges
## 4 Comparison of Shock Absorption Performance between SAOD-NSD and SAOD-LS  
Figure 5 presents the mechanical model diagram of the SAOD-NSD, which is created by replacing the NSD with the constant value \(k_{2}\) of SAOD-LS. The only distinction between SAOD-NSD and SAOD-LS would be the secondary spring \(k_{2}\). In the SAOD-LS, \(k_{2}\) is an ideal spring with a certain stiffness coefficient, but in the SAOD-NSD, \(k_{2}\) refers to the NSD presented in Section 3. To provide a clearer understanding of the exceptional damping abilities of the SAOD-NSD, we have selected Example 3 for illustration in this section. The parameters for the SAOD-LS and external excitation are set based on the values listed in Table 2.  
By differentiating Equation (20) with respect to \(z\), we can derive a function that describes the NSD's equivalent stiffness, which corresponds to a specific \(z\). This enables us to ascertain the equivalent stiffness for a particular set of geometric parameters (specifically, \(l_{1}\), \(l_{2}\), and \(l_{3}\)) and spring constants (\(k_{A}\) and \(k_{B}\)). Referring to Example 3, Table 3 outlines the NSD parameters as mentioned. The NSD equivalent stiffness is calculated to be 1389.4 kN/m, which coincides with the SAOD-LS \(k_{2}\) in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(m_{1}\) (kg) & \(m_{2}\) (kg) & \(k_{1}\) (kN/m) & \(k_{2}\) (kN/m) & \(c_{1}\) (Ns/m) & \(c_{2}\) (Ns/m) & \(x_{1}\) (0) (m) & \(v_{1}\) (0) (m/s) & \(a\) (m/s\({}^{2}\)) & \(r\) \\ \hline
29,485 & 300 & 3800 & 1389.4 & 3740 & 237 & 0 & 0 & \(-5\) & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Values of parameters in Example 3. Figure 4: The relationship between the equivalent external force, \(F\), and the tension (or compression) of NSD, \(z\), obtained through FE simulations and theoretical derivations. Figure 5: Schematic diagram of SAOD-NSD. Using Table 2 and Equation (7), it can be determined that the natural frequency of SAOD-LS (\(f_{SAOD-LS}\)) is 4.53 Hz. In Example 3, the value of \(f_{load}\) falls within the range of \(0.95f_{SAOD-LS}\) to \(1.05f_{SAOD-LS}\), which is classified as the near-resonance region according to reference [45]. The value \(x_{1 max}\) is defined as the peak value of the \(m_{1}\)'s time-history displacement response to external excitation. Figure 6 compares the \(x_{1 max}\) curve between the SAOD-NSD and the SAOD-LS under 50 s harmonic loads at different frequencies (i.e., \(f_{load}\)). In this instance, \(x_{1 max}\) is calculated by the FE method. During the FE modeling process, the NSD is integrated into a spring element that exhibits nonlinear behavior depicted in Equation (20). The other modeling methods remain consistent with those used in Example 1. Figure 6 clearly illustrates that at the exact resonance point, the \(x_{1 max}\) of the SAOD-NSD is 22 mm, notably less than the 162 mm observed in the SAOD-LS. Throughout the entire near-resonance region, the \(x_{1 max}\) of the SAOD-NSD remains below 40% of that of SAOD-LS. This indicates that the NSD exhibits exceptional damping effectiveness in the near-resonance region. Similar to many SDOF dampers, time-history analysis reveals a distinct resonance phenomenon when the parameters of the SAOD-LS and external stimulation remain constant. In addition, despite the SAOD-NSD demonstrating resonance-like characteristics, its resonance peak can be significantly reduced. This study utilizes Example 4, which involves three distinct cases [40]. Case 4 is the classic SDOF mass-spring-damper system, as depicted in Figure 6.1a in [46]. In Case 4, the mass (\(m_{1}\)) is set at 29,485 kg, the spring constant (\(k_{1}\)) at 3800 kN/m, and the damping coefficient (\(c_{1}\)) at 3740 Ns/m. Please refer to Figure 7 for schematic sketches depicting Case 5 and Case 6. The main structural forms of Case 5 and Case 6 are exactly the same, except for the fact that \(k_{2}\) and \(k_{3}\) in Case 5 are LSs (i.e., linear springs), whereas in Case 6, they are NSDs (i.e., the nonlinear spring devices).