# TMD Damping for Structures with Uncertain Modal Parameters
## 6 Conclusions  
This research investigates if increased TMD damping helps to reduce the maximum structural response when the actual eigenfrequency and modal mass deviate from their nominal values that are used for TMD design. The relevant computations are made for  
\begin{table}
\begin{tabular}{c c c c} \hline \(\mu\) & \(m_{1}/m_{1,nom}\) & \(\gamma=1.125\) Beneficial & \(\gamma=1.25\) Beneficial & \(\gamma=1\) Beneficial \\ \hline
10\% & 0.8 & all frequency ratios & \(f_{1}/f_{1,nom}<0.99\) & no frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.03\) & no frequency ratios \\ \hline
10\% & 0.9 & \(f_{1}/f_{1,nom}<0.97\) & \(f_{1}/f_{1,nom}<0.92\) & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.01\) & \(f_{1}/f_{1,nom}>1.03\) & all other frequency ratios \\ \hline
10\% & 1 & \(f_{1}/f_{1,nom}<0.89\) & \(f_{1}/f_{1,nom}<0.85\) & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.04\) & \(f_{1}/f_{1,nom}>1.08\) & all other frequency ratios \\ \hline
10\% & 1/0.9 & \(f_{1}/f_{1,nom}<0.84\) & no lower bound & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.08\) & \(f_{1}/f_{1,nom}>1.15\) & all other frequency ratios \\ \hline
10\% & 1/0.8 & no lower bound & no lower bound & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.16\) & no upper bound & all other frequency ratios \\ \hline \end{tabular}
\end{table}
Table 4: Beneficial TMD damping increase factors depending on actual modal properties and for TMD mass ratio \(\mu=10\%\).  
\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
4\% & 0.8 & all frequency ratios & all frequency ratios & no frequency ratios \\ \hline
4\% & 0.9 & \(f_{1}/f_{1,nom}<0.99\) & \(f_{1}/f_{1,nom}<0.94\) & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.01\) & \(f_{1}/f_{1,nom}>1.04\) & all other frequency ratios \\ \hline
4\% & 1 & \(f_{1}/f_{1,nom}<0.93\) & \(f_{1}/f_{1,nom}<0.88\) & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.04\) & \(f_{1}/f_{1,nom}>1.09\) & all other frequency ratios \\ \hline
4\% & 1/0.9 & \(f_{1}/f_{1,nom}<0.90\) & \(f_{1}/f_{1,nom}<0.85\) & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.07\) & \(f_{1}/f_{1,nom}>1.12\) & all other frequency ratios \\ \hline
4\% & 1/0.8 & \(f_{1}/f_{1,nom}<0.83\) & no lower bound & all other frequency ratios \\  & & \(f_{1}/f_{1,nom}>1.14\) & no upper bound & all other frequency ratios \\ \hline \end{tabular}
\end{table}
Table 3: Beneficial TMD damping increase factors depending on actual modal properties and for TMD mass ratio \(\mu=4\%\).  
four typical TMD mass ratios, reasonable deviations between actual and nominal modal properties, and harmonic excitation. The general picture is that increased TMD damping is beneficial when the actual eigenfrequency significantly deviates from its nominal value, while the difference between actual and nominal modal mass is of minor relevance. However, the problem is too complex to be able to deduce a simple rule of thumb when increased TMD damping is beneficial. Therefore, the beneficial TMD damping increase factors are given as functions of TMD mass ratio and deviations between actual and nominal modal parameters. These data can be used by engineers for TMD projects.  
This research received no external funding.  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The author gratefully acknowledges the support of Maurer SE, Munich, Germany, Maurer Engineering GmbH, Munich, Germany, and Maurer Switzerland GmbH, Pfaffhausen, Switzerland.  
Author [PERSON] was employed by the company Maurer Switzerland GmbH.