## 5 TMD Damping Analysis  
The TMD system is a commonly used energy dissipation and vibration reduction device in engineering structures, which consists of the solid mass, the spring, and the damper. When the main structure is subjected to vibration, the self-vibration frequency of the damper subsystem will be adjusted by changing the mass or stiffness close to the basic frequency of the main structure. Therefore, an inertial force acting in the opposite direction to the structure vibration of the structure will be produced on the sub-structure, from which the vibration response of the main structure is attenuated. After setting the tuned mass damper TMD, the total modal damping ratio of the controlled structure can be expressed as below:  
\[\xi_{r}=\xi_{sr}+\xi_{dr} \tag{11}\]  
where \(\xi_{sr}\) is the \(r\)-th order modal damping ratio of the original structure; \(\xi_{dr}\) is the additional \(r\)-th order modal damping ratio of the TMD. The response under the action of the pedestrian load is controlled by the first-order mode, so the TMD is arranged at the maximum displacement of the first-order vertical and first-order lateral vibration respectively, that is, transversely in the middle of the span and vertically near 1/4 L and 3/4 L.  
The process of determining the optimal parameters of the TMD according to [PERSON] [43] for different excitation and optimization objectives is as follows:  
1. The ratio \(u\) of the TMD mass to the generalized mass of the damping mode is selected. 2. According to the type of external load and the damping response, the optimal frequency ratio and the optimal damping ratio are determined, and the frequency of the tuning device calculated as below: \[f_{d}=\alpha_{\rm opt}\times f_{s}\] (12) \begin{table}
\begin{tabular}{c c c c} \hline
**Level** & **Description** & \
\begin{tabular}{c} **Vertical Acceleration** \\ **Limit (m/s\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Lateral Acceleration** \\ **Limit (m/s\({}^{2}\))** \\ \end{tabular} \\ \hline CL1 & Best & \textless{}0.5 & \textless{}0.1 \\ CL2 & Moderate & 0.5–1 & 0.1–0.3 \\ CL3 & Poor & 1–2.5 & 0.3–0.8 \\ CL4 & Unacceptable & \textgreater{}2.5 & \textgreater{}0.8 \\ \hline \end{tabular}
\end{table} 
Table 2: Criteria for comfort evaluation of pedestrian bridges. The equations for the optimum frequency ratio and the optimum damping ratio are as follows:  
\[\alpha_{\text{opt}}=1/(1+u) \tag{13}\]  
\[\xi_{\text{opt}}=(\frac{3u}{8(1+u)})^{0.5} \tag{14}\]  
(3) Physical parameters of TMD can be calculated as:  
\[ \begin{array}{l}m_{\text{TMD}}=um^{*}\\ k_{\text{TMD}}=\omega_{\text{TMD}}{}^{2}\times m_{\text{TMD}}\\ c=2\times m_{\text{TMD}}\times\omega_{\text{TMD}}\times\xi_{\text{opt}}\end{array}  \tag{15}\]  
where \(m^{*}\) is the modal generalized mass. Taking \(u=0.01\)-\(0.08\), the calculation results of the TMD physical parameters are shown in Tables 3 and 4. The maximum acceleration response analysis of the structure was carried out, and the acceleration time curve at \(u=  
The result shows that the setting of the TMD has a significant effect on the damping of the structure, and the larger the mass of the TMD, the better the damping effect; however there is no linear relationship between the mass of the TMD and the damping rate. According to Figure 9, the damping efficiency was higher with the generalized mass ratio \(u\) taken as 0.01-0.03 in the vertical direction. When \(u\) exceeds 0.02, the maximum vertical acceleration is less than 0.5 m/s\({}^{2}\), and the comfort level reaches the best level. When \(u\) continues to increase, the maximum vertical acceleration reduction rate slows down significantly. In Figure 10, with the increase of \(u\), the maximum lateral acceleration  
Figure 8: Time course of vertical acceleration of section at \(1/4\) L span. (\(u=0.02\)). Figure 6: Time course of vertical acceleration of section at \(1/4\) L span. (\(u=0.01\)). Figure 7: Time course of lateral acceleration of middle section. (\(u=0.02\)). reduction rate slowed down significantly.