# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 2 Theoretical Background  
### CMSE Method  
If \(\mathbf{M}\), \(\mathbf{K}\) and \(\mathbf{M^{*}}\), \(\mathbf{K^{*}}\) are used to denote the mass and stiffness matrices for baseline FEM of the healthy structure and the measured damaged structure, respectively, the following can be written:  
\[\mathbf{K}\phi_{i}=\lambda_{i}\mathbf{M}\phi_{i} \tag{6}\]  
\[\mathbf{K^{*}}\phi^{*}_{j}=\lambda^{*}_{j}\mathbf{M^{*}}\phi^{*}_{j} \tag{7}\]  
where \(\lambda_{i}\) and \(\lambda^{*}_{j}\) are the \(i\)-th and the \(j\)-th eigenvalue of the baseline FEM of the healthy structure and the measured damaged structure, respectively. The CMSE method is developed under the assumption that the mass distributions of the healthy and damaged structures are not known, but do not change, that is \(\mathbf{M^{*}}=\mathbf{M}\). In the following derivation, as \(\lambda_{i}\), \(\phi_{i}\), \(\mathbf{K}\), \(\lambda^{*}_{j}\), and \(\phi^{*}_{j}\) are presumably known, the unknowns are \(\mathbf{K^{*}}\) and \(\mathbf{M}\). Pre-multiplying Equation (6) by \(\left(\mathbf{\phi}_{j}^{*}\right)^{T}\) and Equation (7) by \(\left(\mathbf{\phi}_{i}\right)^{T}\) yields  
\[\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{K}\mathbf{\phi}_{i}=\lambda_{i}(\mathbf{\phi}_ {j}^{*})^{T}\mathbf{M}\mathbf{\phi}_{i} \tag{8}\]  
\[\left(\mathbf{\phi}_{i}\right)^{T}\mathbf{K}^{\prime}\mathbf{\phi}_{j}^{*}=\lambda_{j} ^{*}(\mathbf{\phi}_{i})^{T}\mathbf{M}\mathbf{\phi}_{j}^{*} \tag{9}\]  
Since \(\mathbf{M}\) and \(\mathbf{K}\) are both symmetric matrices, also noting the transpose of a scalar equals to itself, one thus has  
\[(\mathbf{\phi}_{j}^{*})^{T}\mathbf{M}\mathbf{\phi}_{i}=(\mathbf{\phi}_{i})^{T}\mathbf{M} \mathbf{\phi}_{j}^{*} \tag{10}\]  
\[(\mathbf{\phi}_{j}^{*})^{T}\mathbf{K}\mathbf{\phi}_{i}=(\mathbf{\phi}_{i})^{T}\mathbf{K} \mathbf{\phi}_{j}^{*} \tag{11}\]  
Assuming that all structural elements are damaged, the stiffness matrix of the damaged structure can be written as  
\[\mathbf{K}^{*}=\mathbf{K}-\sum\
olimits_{n=1}^{N_{e}}\alpha_{n}\mathbf{K}_{n} \tag{12}\]  
where \(\alpha_{n}\) (\(0\leq\alpha_{n}\leq 1\)) is the extent of damage of the \(n\)-th element. It is difficult to model the damage in sufficient detail for an unknown type of damage. Here, it is assumed that the damage to a structure can be represented by a decrease in the modulus of elasticity of each structural element:  
\[E_{n}^{*}=(1-\alpha_{n})E_{n} \tag{13}\]  
where \(E_{n}\) and \(E_{n}^{*}\) are the moduli of elasticity of the \(n\)-th element of the baseline FEM and the damaged structure, respectively. Herein, \(E_{n}\) is regarded as a known constant but \(E_{n}^{*}\) is regarded as an unknown variable. The objective herein is to evaluate \(\alpha_{n}\) corresponding to each structural element. Dividing Equation (9) by Equation (8), and using the scalar identities of Equations (10) and (11) yields  
\[\frac{(\mathbf{\phi}_{i})^{T}\mathbf{K}^{\prime}\mathbf{\phi}_{j}^{*}}{(\mathbf{\phi}_{j}^ {*})^{T}\mathbf{K}\mathbf{\phi}_{i}}=\frac{\lambda_{j}^{*}}{\lambda_{i}} \tag{14}\]  
Finally, substituting Equation (12) into Equation (14), the following is obtained:  
\[\sum\
olimits_{n=1}^{N_{e}}\alpha_{n}\mathbf{C}_{m,n}^{*}=b_{m}^{*} \tag{15}\]  
where \(\mathbf{C}_{m,n}^{*}=\left(\mathbf{\phi}_{i}\right)^{T}\mathbf{K}_{n}\mathbf{\phi}_{j} ^{*}\), as given by Equation (5), is the CMSE of the \(n\)-th structural element associated with the \(m\)-th MC, and the right-hand side \(b_{m}^{*}=(1-\lambda_{j}^{*}/\lambda_{i})C_{m}\), where \(\mathbf{C}_{m}^{*}=\sum\
olimits_{n=1}^{N_{e}}\mathbf{C}_{m,n}^{*}=\left(\bm {\phi}_{i}\right)^{T}\mathbf{K}\mathbf{\phi}_{j}^{*}\) is the CMSE of the overall structural system. If the first \(N_{i}\) analytical modes and \(N_{j}\) measured modes are obtained, then \(N_{q}=N_{i}\times N_{j}\) MCs are available and \(N_{q}\) linear equations can be constructed, which can be assembled in a matrix of the form as given by Equation (1), where  
\[\mathbf{C}^{*}=\left[