# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 4 Application to the Investigated Building
### Damping Estimation of the Building  
Implementations of the enhanced strain-energy method require the building modal data and the oil damper parameters. These characteristics are identified by adopting a standard time-domain identification technique [7; 35] and using the selected earthquake measurements. The identification is separately performed in each horizontal direction, which facilitates a better consideration of the impact of the excitation/response amplitude for oil dampers. Precisely, for oil damper identification, the input-output earthquake data corresponds to the input displacement and output force data, with the governing equation given by Equation (33), while, for modal identification, the data is the acceleration of the ground floor (input) and two monitored floors (output), with modal-superposition models as governing equation [35]. Table 1 lists the identification results for the four monitored oil dampers, in which I and II represent the damper types. Recall that Type I oil dampers were installed on the ground floor, while Type II oil dampers were placed on other upper floors. It is evident from the results that a discrepancy exists between the identified parameters of same-type dampers, verifying their anticipated amplitude-dependent characteristics. Table 2 shows the identified (translational) modal parameters of the first two modes of the building. The remaining higher modes are of small importance, and identification of their properties suffers to a much larger degree by measurement noise [35]. Notably, the identified mode shapes and mode participation factors in Table 2 are determined by using the so-called mode scale factor, which is commonly adopted to (mass) normalize measured mode shapes. Furthermore, to obtain complete mode shapes, an expansion technique suggested in Reference [27] is adopted, though note that other alternatives exist for resolving this issue [19]. The implementations of this expansion approach are omitted here since it is a common practice. For clarity of explanation, the expanded mode shape is denoted hereinafter as \(\phi_{i}\), for the \(i\)th mode of the building. Table 3 shows the estimated equivalent damping ratios (\(\xi_{eq,i}\)) and natural frequencies (\(\omega_{eq,i}\)) by the enhanced strain-energy method. Results of the inherent damping ratios (\(\xi_{s,i}\)) and natural frequencies (\(\omega_{s,i}\)) for the building itself are also reported in this table. They are obtained by subtracting those equivalent estimates from the corresponding identified modal parameters in Table 2. It can be seen from Table 3 that the estimated equivalent damping ratios have a sharp decrease when mode order increases and the second equivalent damping ratios are negligible. It indicates that the energy dissipated by oil dampers concentrates in the seismic motion of the fundamental modes. On the other side, for equivalent natural frequencies, an increasing trend is observed. Overall, they are relatively small compared with the identified natural frequencies of the building. However, it does not indicate that the estimated equivalent natural frequencies are unimportant. Instead, they can significantly affect the prediction accuracy of updated models, as will be shown in the next section. Besides, the enhanced strain-energy method also allows quantifying the contribution from a specific damper to the equivalent damping ratios and natural frequencies, as evident from Equations (3) and (8). Figure 5 plots the damping and stiffness contribution in percentages from the oil dampers of different floors along with the two horizontal directions. The dampers on the first floor dominate the contribution, up to 70% for damping and up to 50% for stiffness, which is attributed to its larger story drift. Because the damping and stiffness contribution from oil dampers is proportional to the square values of \(\phi_{i,dij}\), as indicated in Equations (3) and (8).