# Performance of Strengthened Accelerated Oscillator Damper for Vibration Control of Bridges
## 6 Concluding Remarks  
In order to solve the problem of the secondary system arrangement and improve the damping efficiency of the AOD with equivalent values of parameters (i.e., Improvement needed 1 mentioned in Introduction), this paper introduces the SAOD-LS, in which the secondary spring and damper are connected to the primary structure. We derive the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Seismic Wave Number** & **Seismic Wave Name** \\ \hline \multirow{3}{*}{Far-field} & 1 & El Centro \\  & 2 &  
SAOD-LS system's motion equation from the Lagrange Dynamic and verify it using the FE approach. The SAOD-LS demonstrates a significant capacity to reduce the peak amplitude of the response. Considering that in practical applications, \(r\) can reach about 10, raising \(r\) will further improve the damping performance [40]. As a result, the SAOD-LS offers a wide range of useful applications. SAOD-LS's rapid response to providing the damping effect is a significant mechanical characteristic that is strongly tied to its unique nature as a SDOF damper. Future anti-ice collisions on piers may utilize this mechanical characteristic. Additionally, an innovative nonlinear spring device (NSD) was proposed, wherein connecting rods and linear springs are arranged in a diamond pattern. Its constitutive relationship can be expressed as a piecewise function, which is checked by the FE analysis. In summary, the device exhibits the following mechanical characteristics: when the relative displacement between the NSD's two terminals is small, the force (\(F\)) is also small; however, if the relative displacement between the NSD's two terminals is large, the device will generate a considerably large force (\(F\)). The device lays the groundwork for future applications in fields such as civil engineering and machinery. For instance, in the context of controlling displacement and force along the longitudinal direction of the bridge, it enables the controlled release of small displacements caused by temperature changes and regular vehicle loads. However, in the event of abnormal excitations, such as earthquakes, the device exerts significant force to prevent excessive displacement of the main girder. Also, in order to realize the Improvement needed 2, mentioned in Introduction, the SAOD-NSD, which is an SDOF system made by combining the NSD and SAOD-LS, has a much better damping effect in the near-resonance area than the SAOD-LS. At exact resonance, the SAOD-NSD exhibits a resonance-like phenomenon, but the peak value of the response is much smaller. When the characteristic frequency of an earthquake closely matches the natural frequency of the structure, the utilization of NSD for shock absorption becomes highly significant. It is noteworthy that while Case 3 in Example 4 fully utilizes the nonlinear stage of the NSD, it also stays within the maximal applicability range of the NSD. In future research, on the one hand, it is crucial to ensure a sufficient safety margin for NSDs to prevent damage from excessive force. On the other hand, it is critical that we make great efforts to ensure that NSD's strong nonlinear stage plays an active role in the task. Otherwise, the NSD cannot fully utilize its distinctive features. Last but not least, this research studies the potential application of the proposed SAOD-LS and SAOD-NSD to a practical bridge project, demonstrating that both devices effectively absorb shocks in practical applications. By analyzing the response to various seismic waves, it is evident that both devices exhibit strong robustness. It is noteworthy that the SAOD-LS and SAOD-NSD also have certain restrictions on their applications. With the SAOD-LS as an example, engineers ought to calculate the values of \(M\), \(C\), and \(K\) in the Equation (7) for practical use while designing the device. Practical engineering generally establishes \(m_{1}\), \(k_{1}\), and \(c_{1}\), so the focus should solely be on determining the values of \(m_{2}r^{2}\), \(k_{2}\big{(}r+1\big{)}^{2}\), and \(c_{2}(r+1)^{2}\).