# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 3 Robust Damage Identification Scheme
This means that \(\alpha_{0}\) can satisfy the CMSE equation corresponding to residual sub-function \(Y_{1}\) better than that corresponding to \(Y_{2}\). It can be further envisaged that \(\alpha_{0}\) has a larger chance of being predicted when the linear system is completely composed of CMSE equations corresponding to \(Y_{1}\) like residual sub-functions. The main task hereafter is discriminating and eliminating the redundant equations that correspond to relatively large \(\|\mathbf{R}_{m}^{\prime}(\hat{\mathbf{\alpha}})\|_{2}\), which can be obtained by calculating the residual sensitivity of the CMSE to damage:  
\[\mathbf{R}_{m}^{\prime}(\mathbf{\alpha})=\frac{\partial\mathbf{C}_{m}^{*}}{\partial\mathbf{ \alpha}^{T}}\mathbf{\alpha}+\mathbf{C}_{m}^{*}\frac{\partial\mathbf{\alpha}}{\partial\mathbf{ \alpha}^{T}}-\frac{\partial b_{m}^{*}}{\partial\mathbf{\alpha}^{T}} \tag{23}\]  
It is assumed that only a few elements are slightly damaged; hence, \(\mathbf{\alpha}\approx 0\). Then Equation (23) is reduced to  
\[\mathbf{R}_{m}^{\prime}(\mathbf{\alpha})\approx\mathbf{C}_{m}-\frac{\partial b_{m}^{*}}{ \partial\mathbf{\alpha}^{T}} \tag{24}\]  
By considering \(\partial b_{m}^{*}/\partial\mathbf{\alpha}^{T}=\left[\partial b_{m}^{*}/\partial \alpha_{1},\cdots,\partial b_{m}^{*}/\partial\alpha_{n},\cdots,\partial b_{m} ^{*}/\partial\alpha_{Ne}\right]\), \(\|\mathbf{R}_{m}^{\prime}(\hat{\mathbf{\alpha}})\|_{2}\) can be calculated and defined as a sensitivity index to ascertain whether the \(m\)-th equation is redundant:  
\[S_{m}=\|\mathbf{R}_{m}^{\prime}(\hat{\mathbf{\alpha}})\|_{2}=\text{sqrt}\left(\sum_{n =1}^{Ne}\bigl{|}C_{m,n}^{*}-\partial b_{m}^{*}/\partial\alpha_{n}\bigr{|}^{2}\right) \tag{25}\]  
Substituting \(C_{m,n}^{*}=\left(\phi_{i}\right)^{T}\!\mathbf{K}_{n}\phi_{j}^{*}\) and \(b_{m}^{*}=\left(1-\lambda_{j}^{*}/\lambda_{i}\right)\!\left(\phi_{i}\right)^{ T}\!\mathbf{K}\phi_{j}^{*}\) into Equation (26), it can be shown that  
\[S_{m}=\text{sqrt}\Bigl{(}\sum_{n=1}^{Ne}\Bigl{|}\left(\phi_{i}\right)^{T}\! \mathbf{K}_{n}\phi_{j}^{*}-\partial\Bigl{[}\left(1-\lambda_{j}^{*}/\lambda_{i }\right)\!\left(\phi_{i}\right)^{T}\!\mathbf{K}\phi_{j}^{*}\Bigr{]}/\partial \alpha_{n}\Bigr{|}^{2}\Bigr{)} \tag{26}\]  
Substituting Equation (13) into Equation (27) yields  
\[S_{m}=\text{sqrt}\Bigl{(}\sum_{n=1}^{Ne}\Bigl{|}\left(\phi_{i}\right)^{T}\! \mathbf{K}_{n}\phi_{j}^{*}+\partial\Bigl{[}\left(1-\lambda_{j}^{*}/\lambda_{i }\right)\!\left(\phi_{i}\right)^{T}\!\mathbf{K}\phi_{j}^{*}\Bigr{]}E_{n}/ \partial E_{n}^{*}\Bigr{|}^{2}\Bigr{)} \tag{27}\]  
In order to compute Equation (28), one first considers computing \(\partial\lambda_{j}^{*}/\partial E_{n}^{*}\). The derivative of Equation (7) with respect to \(E_{n}^{*}\) can be expressed  
\[\left(\mathbf{K}^{*}-\lambda_{j}^{*}\mathbf{M}\right)\!\frac{\partial\phi_{j} ^{*}}{\partial E_{n}^{*}}=-\left(\frac{\partial\mathbf{K}^{*}}{\partial E_{n} ^{*}}-\lambda_{j}^{*}\frac{\partial\mathbf{M}}{\partial E_{n}^{*}}\right)\! \phi_{j}^{*}+\frac{\partial\lambda_{j}^{*}}{\partial E_{n}^{*}}\mathbf{M}\phi_ {j}^{*} \tag{28}\]  
Assuming that the eigenvector \(\mathbf{\phi}_{j}^{*}\) is mass-normalized yields  
\[\left(\mathbf{\phi}_{j}^{*}\right)^{T}\!\mathbf{M}\phi_{j}^{*}=1 \tag{29}\]  
Figure 1: A comparison of two residual sub-functions with different residual-damage sensitivity. Pre-multiplying Equation (29) by \((\mathbf{\phi}_{j}^{*})^{T}\) and substituting Equations (7) and (30) into Equation (29) yields  
Figure 1.0  
Here, it is always \(\partial\mathbf{M}/\partial E_{n}^{*}=0\) because the mass distributions of the structure are independent of the modulus of elasticity.