## 2 Theory of the Inverse Analysis of Structural Damage
### Proof of the Principle of Damage Identification (Sensitivity Analysis)
Therefore, it is assumed that no mass change has occurred in the structure, and Equation (7) can be written as:  
\[\frac{\partial MKEn_{ij}}{\partial p}=\Big{[}2\omega_{i}[M]_{j}\quad\frac{1}{ 2}\{\phi_{i}\}^{Tr}[M]_{j}\{\phi_{i}\}\Big{]}\bigg{[}\frac{\frac{\partial\{ \phi_{i}\}}{\partial p}\{\phi_{i}\}}{\frac{\partial\omega_{i}}{\partial p}\bigg{\}}. \tag{8}\]The second vector, which is the first order derivative of the modal parameters, is derived and can be further studied in references [24; 30] as:  
\[\left\{ \begin{array}{c}\frac{\partial\{\phi_{i}\}}{\partial p}\\ \frac{\partial\omega_{i}}{\partial p}\end{array} \right\}=\left[ \begin{matrix}[K]- \omega_{i}[M]&-[M]\{\phi_{i}\}\\ -\{\phi_{i}\}^{Tr}[M]&0\end{matrix} \right]\left\{ \begin{array}{c}-\left( \frac{\partial[K]}{\partial p_{i}}-\omega_{i}\frac{\partial[M]}{\partial p} \right)\{\phi_{i}\}\\ \frac{1}{2}\{\phi_{i}\}^{Tr}\frac{\partial[M]}{\partial p}\{\phi_{i}\}\end{array}  \right\}. \tag{9}\]  
By substitution of Equation (9) into Equation (10), the MKE sensitivity of _jth_ element for _ith_ mode of vibration will be as:  
\[\frac{\partial MKEn_{ij}}{\partial p}=\{\phi_{i}\}^{Tr}[\omega_{i}[M]_{j}\quad \frac{1}{2}[M]_{j}\{\phi_{i}\}][K_{U/D}]^{-1}[F_{p}]\{\phi_{i}\}. \tag{10}\]  
In which  
\[K_{U/D}= \begin{bmatrix}[K]-\omega_{i}[M]&-[M]\{\phi_{i}\}\\ -\{\phi_{i}\}^{Tr}[M]&0\end{bmatrix} , \tag{11}\]  
and  
\[F_{p}= \begin{bmatrix}-\left(\frac{\partial[K]}{\partial p_{j}}-\omega_{i}\frac {\partial[M]}{\partial p}\right)\\ \frac{1}{2}\{\phi_{i}\}^{Tr}\frac{\partial[M]}{\partial p}\end{bmatrix} , \tag{12}\]  
where the subscript \(U/D\) refers to the undamped system with distinct and well-spaced natural frequency. The parameter \(p_{i}\) is the physical parameter that is used to calculate the elemental mass and stiffness matrices. By assuming that for the _ith_ element there is a corresponding parameter \(p_{i}\) which can be considered to be the [PERSON]'s modulus of elasticity or mass density, or any other parameter of interest. In most of the vibration-based damage identification approaches, it is assumed that the mass matrix of the structure will not change due to the occurrence of damage. Hence, the parameter \(p_{i}\) is assumed to be the [PERSON]'s modulus of elasticity. This assumption involves only the stiffness changes of the system and limits the mass change of the structure to zero. Therefore, the sensitivity matrix of MKEn with respect to parameter \(E_{i}\) is written as:  
\[\frac{\partial MKEn_{ij}}{\partial E_{j}}=\{\phi_{i}\}^{Tr}[\omega_{i}[M]_{j} \quad\frac{1}{2}[M]_{j}\{\phi_{i}\}][K_{U/D}]^{-1}\Big{[}F_{E_{j}}\Big{]}\{ \phi_{i}\}, \tag{13}\]  
where \(F_{p}= \begin{bmatrix}-\frac{\partial[K]}{\partial p}\\ 0\end{bmatrix} \). Using the first order Taylor series expansion, the ratio of change of MKEn can be written as:  
\[\Delta MKEn_{ij}=\sum_{j=1}^{n}\frac{\partial MKEn_{ij}}{\partial E_{i}} \Delta E_{j}, \tag{14}\]  
and the ratio of change of \([K]\) can be written as:  
\[[\Delta K]=\sum_{j=1}^{n}\frac{\partial[K]}{\partial E_{j}}\Delta E_{i}, \tag{15}\]  
To calculate the matrix variation [\(\Delta\)K], which describes the global stiffness matrix change between the intact and healthy structure, the global stiffness matrix should be calculated using the assembly of the elemental stiffness matrix that indicates damage in each element. Hence, the [\(\Delta\)K] can be calculated as:  
\[[\Delta K]=[K]-[K]^{D}=\sum_{j=1}^{n}\alpha_{j}[K]_{j},\ \big{(}0\leq\alpha_{j} \leq 1\big{)}. \tag{16}\]The alpha parameter in the above equation indicates the severity of damage. The zero value indicates no damage, and the value of one indicates a full damage corresponding to the \(jth\) element. By multiplying Equation (10) by the \(\Delta E_{j}\) and summing up for all \(E_{j}\), it yields:  
\[\Delta MKEn_{ij}=\{\phi_{i}\}^{T\!\!T}[\omega_{i}[M]_{j}\quad\tfrac{1}{2}[M]_{j }\{\phi_{i}\}][K_{U/D}]^{-1} \begin{bmatrix}-\sum\limits_{j=1}^{n}\alpha_{j}[K] _{j}\\ 0\end{bmatrix}