# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 4 Application to the Investigated Building
### Model Updating of the Building  
The model updating of the building itself is separately performed for its two horizontal directions. Specifically, for each updating case, the natural frequencies (\(\omega_{s,i}\)), expanded mode shapes (\(\phi_{i}\)), and mode participation factors (\(p_{i}\)) of the first two modes are regrouped and utilized to correct mass and stiffness matrices by the improved Berman-Nagy method, i.e., Equation (27) for mass matrix updating and Equation (29) for stiffness matrix updating. Besides, for inherent structural damping, the damping matrix is modeled through a common modal damping assumption and is adjusted to match the estimated damping ratios (\(\xi_{s,i}\)), as listed in Table 3. Damping ratios for the remaining higher modes are generally large and are constrained to a predefined value (taken in the case study later as 10%). Note that using a large damping ratio for these modes, though not critical, can efficiently suppress higher-frequency components in the building response predictions. First, mass and stiffness matrices are updated to demonstrate the effectiveness of the improved Berman-Nagy method. Two updating cases of using different modal data are considered to illustrate the impact of additional matrix \(\Delta\mathbf{M}\) on updated mass matrix. Specifically, the regrouped modal parameters of the first mode and the first two modes are separately utilized to update the nominal mass matrix \(\mathbf{M}_{d}\). Figure 6 presents the new, updated mass matrix \(\mathbf{M}\) by Equation (27). For comparisons, the updated mass matrix \(\mathbf{M}_{B}\) by the traditional Berman-Nagy method is also reported. The results in this figure correspond to the transversal direction FE model. It is evident from the results in Figure 6 that \(\Delta\mathbf{M}\), the difference between \(\mathbf{M}\) and \(\mathbf{M}_{B}\), has an increasing impact on the updated mass matrix when more modal data is utilized. In the first updating case, the predicted mode participation factor by \(\mathbf{M}\) exactly matches the measured one (48.952), while the prediction by \(\mathbf{M}_{B}\) is 46.884. This prediction error is large but causes small changes in the two updated mass matrices, as shown in Figure 6a. When the regrouped modal data of the first two modes is  
Figure 6: Updated mass matrices by Equations (13) and (27): **(a)** updating results by using the first mode; **(b)** updating results by using the first two modes. utilized, **M** and **M\({}_{B}\)** differ significantly in both diagonal and off-diagonal elements. The predicted mode participation factors are listed in Table 4, where the improved method results are denoted as \(p_{i,i}\). Again, **M** reproduces exactly the measured mode participation factors, while **M\({}_{B}\)** fails in the first two modes cases. Figure 7 shows the new, updated stiffness matrices by the improved method and using the regrouped modal data of the first two modes. Unlike updated mass matrices, changes in updated stiffness matrices are small. Notably, the updated mass and stiffness matrices by the improved method exactly possess the modal data, including natural frequencies, mode shapes, and mode participation factors. Then, the precision of the estimated inherent damping ratios (\(\xi_{s,i}\)) and natural frequencies (\(\omega_{s,i}\)) is considered. The updated models with new, updated mass and stiffness matrices are utilized for the building response prediction, according to Equation (34). The linear Maxwell models with identified parameters shown in Table 1 are assigned to the same-type oil dampers, i.e., the parameter configures of Type I for the ground-floor oil dampers and those of Type II for the other floors oil dampers.