## 4 Case Study  
In this section, an engineering project of a pedestrian bridge, for which TMDs need to be installed, will be introduced in detail as a case study. The pedestrian bridge is 45 m long and 6 m wide. It is a simply supported pedestrian bridge. Due to the excessive vibration under human-induced excitations, it has a serviceability problem. The finite element model was established in Midas Gen. The structural modes are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Mode** & **Frequency/Hz** & **UX/\%** & **UV/\%** & **UZ/\%** & **RX/\%** & **RY/\%** & **RZ/\%** \\ \hline
1 & 1.006 & 94.445 & 3.788 & 0.058 & 0.033 & 0.001 & 0.588 \\
2 & 1.488 & 3.564 & 84.251 & 0.000 & 6.749 & 0.011 & 0.007 \\
3 & 1.946 & 0.063 & 0.026 & 75.499 & 0.208 & 0.014 & 0.000 \\
4 & 2.098 & 0.055 & 2.405 & 0.097 & 41.152 & 0.225 & 0.019 \\
5 & 2.482 & 0.133 & 8.134 & 0.035 & 28.389 & 0.040 & 0.000 \\
6 & 2.726 & 0.594 & 0.006 & 0.001 & 0.029 & 0.000 & 94.733 \\
7 & 4.276 & 0.000 & 0.000 & 0.003 & 0.000 & 0.000 & 0.000 \\
8 & 4.278 & 0.000 & 0.000 & 0.000 & 0.000 & 0.002 & 0.000 \\
9 & 4.425 & 0.026 & 0.002 & 0.000 & 0.002 & 0.390 & 0.003 \\
10 & 4.681 & 0.000 & 0.006 & 0.000 & 0.005 & 43.800 & 0.109 \\ \hline \hline \end{tabular} Notes: UX, UY and UZ are modal mass participation coefficient in transverse, longitudinal and vertical direction. RX, RY and RZ are rotation modal mass participation coefficient in transverse, longitudinal and vertical direction.  
\end{table} 
Table 4: Modes of the pedestrian bridge. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mu\) & \(\xi_{s}\) & 0.010 & 0.020 & 0.030 & 0.040 & 0.050 & 0.060 & 0.070 & 0.080 & 0.090 & 0.100 \\ \hline
0.001 & 3.286 & 1.543 & 1.968 & 1.834 & 1.783 & 1.779 & 1.657 & 1.554 & 1.492 & 1.308 \\
0.002 & 0.964 & 1.531 & 1.914 & 1.812 & 1.721 & 1.746 & 1.699 & 1.542 & 1.434 & 1.567 \\
0.003 & 0.989 & 1.293 & 1.232 & 1.391 & 1.619 & 1.712 & 1.715 & 1.778 & 1.693 & 1.580 \\
0.004 & 0.126 & 0.817 & 1.277 & 1.518 & 1.552 & 1.524 & 1.531 & 1.531 & 1.591 & 1.680 \\
0.005 & 0.971 & 1.131 & 1.546 & 1.553 & 1.550 & 1.671 & 1.626 & 1.559 & 1.553 & 1.467 \\
0.006 & 0.651 & 1.286 & 1.275 & 1.365 & 1.568 & 1.549 & 1.575 & 1.638 & 1.573 & 1.529 \\
0.007 & 0.895 & 0.893 & 0.884 & 1 Because the transverse and longitudinal loads of pedestrians are relatively smaller than the vertical load [21], the effect of transverse and longitudinal loads can usually be ignored. From Table 4, it is clear that the third vertical mode is dominant. The third order frequency is 1.946 Hz, and the vertical modal mass participation coefficient is 75.499%. This pedestrian bridge is about 357,500 kg in total. Therefore, the third vertical modal participation mass is 269,909 kg. The damping ratio of this pedestrian bridge is set to be 2%. The vertical mode shape of the pedestrian bridge is shown in Figure 4.  
In order to solve the serviceability problem, a TMD is designed. The TMD mass ratio was chosen to be 1%. According to Table 1, the TMD frequency ratio is 0.997 and the TMD damping ratio is 0.063. According to the Den Hartog method, the TMD frequency ratio is 0.995 and the TMD damping ratio is 0.061. According to the Ioi and Ikeda method, the TMD frequency ratio is 0.998 and the TMD damping ratio is 0.064. The dynamic amplification factors of the three TMDs under different frequency harmonic excitations are compared in Figure 5.  
It is shown in Figure 5 that the maximum dynamic amplification factor without a TMD is 25.00, while with the TMDs designed based on the AFSA, Den Hartog and Ioi and Ikeda methods it is 9.41, 9.56 and 9.48, respectively. Compared to the situation without a TMD, the TMD designed based on the AFSA, Den Hartog and Ioi and Ikeda methods has a reduction of 62.36%, 61.76% and 62.08%, respectively. It can be seen that the vibration control effect of the TMD designed based on the AFSA method is slightly better than that of the TMD designed based on the remaining two methods.