# Damping Estimation of an Eight-Story Steel Building Equipped with Oil Dampers
## 3 A Framework for Damping Estimation of Passively Controlled Structures
### An Improved Berman-Nagy Method
#### 3.2.2 A New Procedure for Mass Matrix Updating  
This new procedure considers the constraints of symmetry, modal orthogonality, as well as mode participation factors in mass matrix updating. Its formulation process follows the guidance in reference [23]. Specifically, the cost function to be minimized is defined as  
\[\varepsilon=\|\mathbf{M}_{a}^{-1/2}(\mathbf{M}-\mathbf{M}_{a})\mathbf{M}_{a}^{ -1/2}\| \tag{16}\]  
subjected to constraints as follows:  
\[\mathbf{M}=\mathbf{M}^{\mathrm{T}} \tag{17}\]  
\[\mathbf{\Phi}^{\mathrm{T}}\mathbf{M}\mathbf{\Phi}=\mathbf{I} \tag{18}\]  
\[\mathbf{\Phi}^{\mathrm{T}}\mathbf{M}\mathbf{I}_{n}=\mathbf{P} \tag{19}\]  
where \(\mathbf{P}\) is the vector of measured mode participation factors. Using the Lagrange multiplier method, the Lagrangian function may be formulated as  
\[J=\varepsilon+\sum_{ij}\Psi_{ij}\big{(}\mathbf{M}-\mathbf{M}^{\mathrm{T}} \big{)}_{ij}+\sum_{ij}\lambda_{ij}\big{(}\mathbf{\Phi}^{\mathrm{T}}\mathbf{M} \mathbf{\Phi}-\mathbf{I}\big{)}_{ij}+\sum_{i}\gamma_{i}\big{(}\mathbf{\Phi}^{ \mathrm{T}}\mathbf{M}\mathbf{I}_{n}-\mathbf{P}\big{)}_{i} \tag{20}\]  
where \(\Psi_{ij}\), \(\lambda_{ij}\), and \(\gamma_{i}\) are the undetermined multiplier coefficients, corresponding to the constraint equations above, and their matrix forms are denoted as \(\mathbf{\Psi}\), \(\mathbf{\Lambda}\), and \(\mathbf{\Gamma}\), respectively. Differentiating Equation (20) with respect to each element in \(\mathbf{M}\) and setting these results equal to zero, it leads to  
\[\mathbf{M}=\mathbf{M}_{a}-\frac{1}{2}\mathbf{M}_{a}\Big{(}\mathbf{\Phi} \mathbf{\Lambda}\mathbf{\Phi}^{\mathrm{T}}+\mathbf{\Phi}\mathbf{\Pi}_{n}^{ \mathrm{T}}+\mathbf{\Psi}-\mathbf{\Psi}^{\mathrm{T}}\Big{)}\mathbf{M}_{a} \tag{21}\]  
Substituting Equation (21) into Equation (17) and arranging the result, the solution for \(\mathbf{\Psi}-\mathbf{\Psi}^{\mathrm{T}}\) is  
\[\mathbf{\Psi}-\mathbf{\Psi}^{\mathrm{T}}=\frac{1}{2}\Big{(}\mathbf{\Phi} \mathbf{\Lambda}^{\mathrm{T}}\mathbf{\Phi}^{\mathrm{T}}+\mathbf{I}_{n} \mathbf{\Gamma}^{\mathrm{T}}\mathbf{\Phi}^{\mathrm{T}}-\mathbf{\Phi} \mathbf{\Lambda}\mathbf{\Phi}^{\mathrm{T}}-\mathbf{\Phi}\mathbf{\Pi}_{n}^{ \mathrm{T}}\Big{)} \tag{22}\]  
Inserting Equation (22) into Equation (21) to update mass matrix, it follows that  
\[\mathbf{M}=\mathbf{M}_{a}-\frac{1}{4}\mathbf{M}_{a}\mathbf{\Phi}\Big{(} \mathbf{\Lambda}+\mathbf{\Lambda}^{\mathrm{T}}\Big{)}\mathbf{\Phi}^{ \mathrm{T}}\mathbf{M}_{a}-\frac{1}{4}\mathbf{M}_{a}\Big{(}\mathbf{\Phi} \mathbf{\Pi}_{n}^{\mathrm{T}}+\mathbf{I}_{n}\mathbf{\Gamma}^{\mathrm{T}} \mathbf{\Phi}^{\mathrm{T}}\Big{)}\mathbf{M}_{a} \tag{23}\]  
Substitute Equation (23) into Equation (18), we can get  
\[\mathbf{\Lambda}+\mathbf{\Lambda}^{\mathrm{T}}=-4\mathbf{m}_{a}^{-1}(\mathbf{ I}-\mathbf{m}_{a})\mathbf{m}_{a}^{-1}-\mathbf{\Gamma}\mathbf{P}_{a}^{\mathrm{T}} \mathbf{m}_{a}^{-1}-\mathbf{m}_{a}^{-1}\mathbf{P}_{a}\mathbf{\Gamma}^{\mathrm{T}} \tag{24}\]where \(\mathbf{m}_{a}=\mathbf{\Phi}^{\mathrm{T}}\mathbf{M}_{a}\mathbf{\Phi}\), \(\mathbf{P}_{a}=\mathbf{\Phi}^{\mathrm{T}}\mathbf{M}_{a}\mathbf{I}_{n}\). Eliminating the item \(\mathbf{\Lambda}+\mathbf{\Lambda}^{\mathrm{T}}\) in Equation (23), mass matrix is rewritten as  
\[\mathbf{M}=\mathbf{M}_{B}+\frac{1}{4}\mathbf{M}_{a}\mathbf{\Phi}\mathbf{\Gamma}\Big{(} \mathbf{P}_{a}^{\mathrm{T}}\mathbf{m}_{a}^{-1}\mathbf{\Phi}^{\mathrm{T}}-\mathbf{I }_{n}^{\mathrm{T}}\Big{)}\mathbf{M}_{a}+\frac{1}{4}\mathbf{M}_{a}\Big{(}\mathbf{ \Phi}\mathbf{m}_{a}^{-1}\mathbf{P}_{a}+\mathbf{I}_{n}\Big{)}\mathbf{\Gamma}^{ \mathrm{T}}\mathbf{\Phi}^{\mathrm{T}}\mathbf{M}_{a} \tag{25}\]  
Substitute Equation (25) into the last constraint, Equation (19), the solution for \(\mathbf{\Gamma}\) is  
\[\mathbf{M}=\mathbf{M}^{\mathrm{T}} \tag{17}\]0  
where \(c=\Big{(}\mathbf{P}_{a}^{\mathrm{T}}\mathbf{m}_{a}^{-1}\mathbf{\Phi}^{\mathrm{T}} -\mathbf{I}_{n}^{\mathrm{T}}\Big{)}\mathbf{M}_{a}\mathbf{I}_{n}\) is a constant.