# TMD Damping for Structures with Uncertain Modal Parameters
## 5 Beneficial TMD Damping Increase Factors  
In the following, beneficial TMD damping increase factors are given, i.e., factors that reduce the maximum displacement response of the primary structure compared to nominal TMD damping. As already mentioned in Section 4, the criterion that \(\max(X_{1}(\gamma\
eq 1))\) is smaller than \(\max(X_{1}(\gamma=1))\) cannot take into consideration the duration of deviated modal properties of the primary structure and the time instant of a possible vibration event. Due to the complexity of the investigated problem, the beneficial TMD increase factors depend on TMD mass ratio and the deviations between actual and nominal modal parameters. Therefore, for each TMD mass ratio and for the selected values of \(m_{1}/m_{1,nom}\) (0.8, 0.9, 1, 1/0.9, 1/0.8), three beneficial TMD increase factors with associated frequency ratios \(f_{1}/f_{1,nom}\) are given in Tables 1-4. The selected three beneficial TMD increase factors are the following:  
* 1 (no increased TMD damping);
* 1.25 (maximum value considered); and
* 1.125 (half of maximum value considered). \begin{table}
\begin{tabular}{c c c c c} \hline \(\mu\) & \(m_{1}/m_{1,nom}\) & \(\gamma\) = 1.125 Beneficial & \(\gamma\) = 1.25 Beneficial & \(\gamma\) = 1 Beneficial \\ \hline
0.5\% & 0.8 & all frequency ratios & \(f_{1}/f_{1,nom}<0.99\) & no frequency ratios \\ \hline
0.5\% & 0.9 & \(f_{1}/f_{1,nom}<0.99\) & \(f_{1}/f_{1,nom}<0.97\) & all other frequency ratios \\ \hline
0.5\% & 1 & \(f_{1}/f_{1,nom}<0.97\) & \(f_{1}/f_{1,nom}<0.95\) & all other frequency ratios \\ \hline
0.5\% & 1/0.9 & \(f_{1}/f_{1,nom}<0.96\) & \(f_{1}/f_{1,nom}<0.92\) & all other frequency ratios \\ \hline
0.5\% & 1/0.8 & \(f_{1}/f_{1,nom}<0.93\) & \(f_{1}/f_{1,nom}<0.85\) & all other frequency ratios \\ \hline \end{tabular}
\end{table} 
Table 1: Beneficial TMD damping increase factors depending on actual modal properties and for TMD mass ratio \(\mu\) = 0.5%. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mu\) & \(m_{1}/m_{1,nom}\) & \(\gamma\) = 1.125 Beneficial & \(\gamma\) = 1.25 Beneficial & \(\gamma\) = 1 Beneficial \\ \hline
1\% & 0.8 & all frequency ratios & \(f_{1}/f_{1,nom}<0.99\) & no frequency ratios \\ \hline
1\% & 0.9 & \(f_{1}/f_{1,nom}>1.01\) & \(f_{1}/f_{1,nom}<0.96\) & all other frequency ratios \\ \hline
1\% & 1 & \(f_{1}/f_{1,nom}<0.96\) & \(f_{1}/f_{1,nom}<0.93\) & all other frequency ratios \\ \hline
1\% & 1/0.9 & \(f_{1}/f_{1,nom}>1.03\) & \(f_{1}/f_{1,nom}>1.07\) & all other frequency ratios \\ \hline
1\% & 1/0.9 & \(f_{1}/f_{1,nom}<0.94\) & \(f_{1}/f_{1,nom}<0.89\) & all other frequency ratios \\ \hline
1\% & 1/0.8 & \(f_{1}/f_{1,nom}>1.05\) & \(f_{1}/f_{1,nom}>1.11\) & all other frequency ratios \\ \hline \end{tabular}
\end{table} 
Table 2: Beneficial TMD damping increase factors depending on actual modal properties and for TMD mass ratio \(\mu\) = 1%. The information provided in these four tables may be used by engineers to decide which TMD damping increase factor may be appropriate for their TMD projects. To demonstrate how the engineer can use the provided table data, one example is given in the following. A TMD for a footbridge with a typical mass ratio of 4% is considered. Hence, the data of Table 3 must be considered. Commonly, the modal mass of the footbridge is estimated from the finite element model and is not measured using accelerometer numbers to ensure high spatial mode shape resolution. As a result, the actual modal mass is not known. Therefore, \(m_{1}/m_{1,nom}=1\) is considered. The uncertainty in the eigenfrequency due to temperature effects and life loads is assumed to be \(\pm 10\%\), i.e., the actual eigenfrequency of the first vertical bending mode may vary between 2.2 Hz and 1.8 Hz when the nominal eigenfrequency is 2.0 Hz. For this case, Table 3 shows that the TMD dash pot damper shall be designed for the damping ratio \(\gamma\)\(c_{2}\) with damping increase factor \(\gamma=1.125\) because \(f_{1}\) may be lower than 0.93 \(f_{1,nom}\), and \(f_{1}\) may be higher than 1.04 \(f_{1,nom}\).