# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 3 A Framework for Damping Estimation of Passively Controlled Structures
### An Enhanced Strain-Energy Method  
#### 3.1.1 Estimation of Equivalent Damping Ratios  
The traditional strain-energy method assumes passively controlled structure vibrates in harmonic motion with a frequency consistent with one of their natural frequencies and deflects proportionally to the corresponding mode shape. Based on these assumptions, the equivalent damping ratio provided by added dampers can be estimated by  
\[\xi_{eq,i}=\frac{1}{4\pi}\frac{E_{D,i}}{E_{S,i}} \tag{1}\]  
where \(\xi_{eq,i}\) is the equivalent damping ratio for the \(i\)th mode, \(E_{D,i}\) is the energy dissipated by dampers in one cycle of motion, \(E_{S,i}\) is the corresponding maximum strain energy stored in the entire structure system, which is estimated by  
\[E_{S,i}=\frac{1}{2}\Delta_{i}^{2}\Phi_{i}^{\mathrm{T}}\mathbf{K}\Phi_{i}=\frac {1}{2}\Delta_{i}^{2}\omega_{i}^{2} \tag{2}\]  
Figure 3: A typical set of earthquake measurements of the building: (**a**) Accelerations of monitored floors; (**b**) Force and displacement of monitored oil dampers.  
where \(\mathbf{K}\) is the stiffness matrix of the entire structure, \(\omega_{i}\) and \(\Phi_{i}\) are, respectively, the \(i\)th circular frequency and mass-normalized mode shape, \(\Delta_{i}\Phi_{i}\) denotes the maximum structural deformation vector in one cycle of motion, \(\Delta_{i}\) is an unknown factor determining the \(i\)th mode vibration amplitude.  
Formulation of \(E_{D,i}\) depends on damper types. For the case added dampers modeled by linear Maxwell models, \(E_{D,i}\) is given by [15]  
\[E_{D,i}=\pi\sum_{j}\frac{\omega_{i}}{1+\tau_{d,j}^{2}\omega_{i}^{2}}c_{d,j} \Delta_{i}^{2}\phi_{i,dj}^{2} \tag{3}\]  
where \(\tau_{d,j}=c_{d,j}/k_{d,j}\) is the relax time for the \(j\)th damper, \(c_{d,j}\) and \(k_{d,j}\) are the damping and stiffness coefficients, respectively. \(\phi_{i,dj}\) is the axial deformation of the \(j\)th damper in the \(i\)th mass-normalized mode shape.  
Substitute Equations (2) and (3) into Equation (1) yields the \(i\)th equivalent damping ratio. For simplifying this formula, a coordinate transformation, \(\Delta_{i}\Phi_{i,d}=\mathbf{T}_{D}\Delta_{i}\Phi_{i}\), is introduced between damper deformation \(\Delta_{i}\Phi_{i,d}\) (local damper coordinate system) and structure deformation \(\Delta_{i}\Phi_{i}\) (global coordinate system). \(\mathbf{T}_{D}\) is a predefined transformation matrix determined by damper configure details. By substituting such transformation into Equation (3), we can get  
\[E_{D,i}=\pi\omega_{i}\Delta_{i}^{2}\Phi_{i}^{T}\widetilde{\mathbf{C}}_{D,i} \Phi_{i} \tag{4}\]  
where the matrix \(\widetilde{\mathbf{C}}_{D,i}\) is given by  
\[\widetilde{\mathbf{C}}_{D,i}=\mathbf{T}_{D}^{T}diag\Bigg{\{}\frac{c_{d,j}}{1+ \tau_{d,j}^{2}\omega_{i}^{2}}\Bigg{\}}\mathbf{T}_{D} \tag{5}\]  
It can be regarded as the equivalent damping matrix for the \(i\)th mode provided by added dampers. The equivalent damping ratio is ultimately expressed by  
\[\xi_{eq,i}=\frac{\Phi_{i}^{T}\widetilde{\mathbf{C}}_{D,i}\Phi_{i}}{2\omega_{i}} \tag{6}\]  
Notably, such an expression follows the classical definition of modal damping ratios and assign strong physically meanings to the calculations of equivalent damping ratios.