# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 5 Conclusions  
This paper examined the damping estimation of an eight-story steel building structure equipped with oil dampers, located at Tohoku Institute of Technology. It was carried out within a proposed framework that consists of an enhanced strain-energy method and an improved Berman-Nagy method for model updating. The former method was first employed to quantify the additional damping and stiffness provided by oil dampers in the form of equivalent damping ratios and natural frequencies.  
Figure 8: Comparison of the predicted and measured floor acceleration: (**a**) updated model I; (**b**) updated model II.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Longitudinal Direction**} & \multicolumn{3}{c}{**Transversal Direction**} \\
**Updated** & \multicolumn{2}{c}{**Acceleration**} & \multicolumn{2}{c}{**Damper Force**} & \multicolumn{2}{c}{**Acceleration**} & \multicolumn{2}{c}{**Damper Force**} \\
**Models** & **4F** & **8F** & **1F** & **8F** & **4F** & **8F** & **1F** & **8F** \\ \hline Model I & 0.468 & 0.466 & 0.597 & 0.709 & 0.461 & 0.308 & 0.536 & 0.641 \\ Model II & 0.278 & 0.225 & 0.267 & 0.392 & 0.236 & 0.118 & 0.268 & 0.444 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Normalized root mean squared errors (NRMSEs) of the response predictions of updated models.  
Figure 9: Comparison of the predicted and measured damper force: (**a**) updated model I; (**b**) updated model II.  
Subsequently, the improved Berman-Nagy was implemented to update an initial model of the building by using its estimated modal parameters, whose accuracy were further verified through prediction validations. The case study demonstrated the effectiveness of the proposed framework. The main conclusions and findings obtained from this study are the following: (1) The updated models show a high precision in prediction for both floor acceleration and damper force, showing the estimated natural frequencies and damping ratios are accurate. Besides, the damper stiffness, though not large, can significantly worsen the model prediction precision and, therefore, should be taken into consideration in model updating. (2) The updated model by the improved Berman-Nagy method reproduces exactly the measured modal data, including mode participation factors. However, the non-diagonal elements of updated mass matrix are not negligible and become large when more modal data utilized. (3) The supplemental damping ratios for the first modes are about 0.02, while the damping contributions for second modes are negligible, indicating that most energy dissipated by oil dampers concentrates in the seismic motion of fundamental modes. (4) The proposed framework and methods are computationally efficient and have good potential in the post-earthquake health and safety assessment for passively controlled building structures.  
Conceptualization, P.Y. and S.X.; methodology, P.Y.; software, L.X. and M.C.; validation, P.Y.; formal analysis, P.Y.; writing--original draft preparation, P.Y.; writing--review and editing, P.Y. and L.X. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant No. 51778490 and 51978525), the Natural Science Foundation of Shanghai (Grant No. 20 ZR1461800), the Japan Society for Promotion of Science(Grant No. 18K04438 and 19F19777), and the Tohoku Institute of Technology research Grant.  
The authors declare no conflict of interest.