# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 3 Robust Damage Identification Scheme
Further, defining a purely geometric stiffness submatrix \(\mathbf{K}_{n}^{0}=\mathbf{K}_{n}^{*}/E_{n}^{*}\), also noting \(\mathbf{K}^{*}=\sum_{t=1}^{Ne}\mathbf{K}_{t}^{*}\) and \(\partial\mathbf{K}_{t}^{*}/\partial E_{n}^{*}=0\) when \(t\
eq n\), Equation (31) can be simplified as  
Figure 1.1  
where \(P_{j,n}^{0}=(\mathbf{\phi}_{j}^{*})^{T}\mathbf{K}_{n}^{0}\mathbf{\phi}_{j}^{*}\) is the purely geometric MSE of the \(n\)-th structural element associated with the \(j\)-th mode of the damaged structure. Moreover, as for \(\partial\bigg{[}(\mathbf{\phi}_{i})^{T}\mathbf{K}\mathbf{\phi}_{j}^{*}\bigg{]}/ \partial E_{n}^{*}\), a compact matrix form can be written [32]:  
\[\frac{\partial\bigg{[}(\mathbf{\phi}_{i})^{T}\mathbf{K}\mathbf{\phi}_{j}^{*}\bigg{]}} {\partial E_{n}^{*}}=\Big{[} \begin{array}{cc}(\mathbf{\phi}_{i})^{T}\mathbf{K}& 0\end{array} \Big{]}\left\{ \begin{array}{c}\frac{\partial\mathbf{\phi}_{j}^{*} }{\partial E_{n}^{*}}\\ \frac{\partial\lambda_{j}^{*}}{\partial E_{n}^{*}}\end{array} \right\} \tag{32}\]  
According to [PERSON] and [PERSON] [33], the eigenpairs derivations for the damaged structure with respect to \(E_{n}^{*}\) can be obtained by  
\[\left\{ \begin{array}{c}\frac{\partial\mathbf{\phi}_{j}^{*}}{\partial E_{n}^{*}} \\ \frac{\partial\lambda_{j}^{*}}{\partial E_{n}^{*}}\end{array} \right\}=\left[ \begin{array}{cc}\mathbf{K}^{*}-\lambda_{j}^{*}\mathbf{M}&-\mathbf{M}\mathbf{ \phi}_{j}^{*}\\ -\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{M}&0\end{array} \right]\left\{ \begin{array}{c}-\left(\frac{\partial\mathbf{K}^{*}}{\partial E_{n}^{*}}- \lambda_{j}^{*}\frac{\partial\mathbf{M}}{\partial E_{n}^{*}}\right)\mathbf{\phi}_{ j}^{*}\\ \frac{1}{2}\left(\mathbf{\phi}_{j}^{*}\right)^{T}\frac{\partial\mathbf{M}}{ \partial E_{n}^{*}}\mathbf{\phi}_{j}^{*}\end{array} \right\} \tag{33}\]  
As it is still assumed that only a few elements are slightly damaged, the overall stiffness matrix of the damaged structure can be replaced by the healthy one. Equation (34) can thus be reduced to  
\[\left\{ \begin{array}{c}\frac{\partial\mathbf{\phi}_{j}^{*}}{\partial E_{n}^{*}} \\ \frac{\partial\lambda_{j}^{*}}{\partial E_{n}^{*}}\end{array} \right\}\approx- \left[ \begin{array}{cc}\mathbf{K}-\lambda_{j}^{*}\mathbf{M}&-\mathbf{M}\mathbf{ \phi}_{j}^{*}\\ -\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{M}&0\end{array} \right]^{-1}\left\{ \begin{array}{c}\mathbf{K}_{n}^{0}\mathbf{\phi}_{j}^{*}\\ 0\end{array} \right\} \tag{34}\]  
Substituting Equation (35) into Equation (33), the derivative of \((\mathbf{\phi}_{i})^{T}\mathbf{K}\mathbf{\phi}_{j}^{*}\) with respect to \(E_{n}^{*}\) can be obtained:  
\[\frac{\partial\bigg{[}(\mathbf{\phi}_{i})^{T}\mathbf{K}\mathbf{\phi}_{j}^{*}\bigg{]}}{ \partial E_{n}^{*}}=-\Big{[} \begin{array}{cc}(\mathbf{\phi}_{i})^{T}\mathbf{K}& 0\end{array} \Big{]}\left[ \begin{array}{cc}\mathbf{K}-\lambda_{j}^{*}\mathbf{M }&-\mathbf{M}\mathbf{\phi}_{j}^{*}\\ -\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{M}&0\end{array} \right]^{-1}\left\{ \begin{array}{c}\mathbf{K}_{n}^{0}\mathbf{\phi}_{j}^{*}\\ 0\end{array} \right\} \tag{35}\]  
Then, substituting Equations (32) and (36) into Equation (21) and simplifying yields the final expression of the sensitivity index as follows:  
\[S_{m}^{*}=\text{sqrt}\left(\sum_{n=1}^{Ne}\right)\hskip-2.845276 pt\left[C_{m,n }^{*}-\frac{\overline{P}_{j,n}^{*}C_{m}^{*}}{\lambda_{i}}-\left(1-\frac{ \lambda_{i}^{*}}{\lambda_{i}}\right)\hskip-2.845276 pt\left[ \begin{array}{cc}( \mathbf{\phi}_{i})^{T}\mathbf{K}&0\end{array} \right]\left[ \begin{array}{cc} \mathbf{K}-\lambda_{i}^{*}\mathbf{M}&-\mathbf{M}\mathbf{\phi}_{j}^{*}\\ -\left(\mathbf{\phi}_{j}^{*}\right)^{T}\mathbf{M}&0\end{array} \right]^{-1}\left\{ \begin{array}{c}\mathbf{K}_{n}\mathbf{\phi}_{j}^{*}\\ 0\end{array} \right\}\right]^{2}\right) \tag{36}\]  
where \(\overline{P}_{j,n}^{*}=P_{j,n}^{*0}E_{n}=(\mathbf{\phi}_{j}^{*})^{T}\mathbf{K}_{n} \mathbf{\phi}_{j}^{*}\) is an approximation for \(P_{j,n}^{*}\).