## 2 Theory of the Inverse Analysis of Structural Damage
### Proof of the Principle of Damage Identification (Sensitivity Analysis)  
As has been shown in the literature review, there are very few studies that implement MKEn for inverse analysis of structural damage; MSEn is more widely used as a main structural damage detection indicator, due to its direct link to the stiffness matrix of the structure. Nevertheless, due to the concept of energy conversion, both MKEn and MSEn mutually transform between each other, which makes them both highly sensitive to the occurrence of structural damage, particularly when stiffness alterations due to damage exist. In this regard, the derivation of MKEn and MSEn change ratios are presented in order to formulate the objective function necessary for the inverse problem of structural damage identification [9; 10; 11; 24; 30]. The sensitivity analysis is first conducted on the MKEn to proof its sensitivity for damage when the change is made in [PERSON]'s modulus of elasticity to explain the rate of change in MKEn when implementing a minor perturbation in the system. In structural dynamics, the free vibration of the FE model of the structure with \(N_{m}\) degrees of freedom, the following equation can be written as:  
\[[K]\{\phi_{i}\}=\omega_{i}[M]\{\phi_{i}\}, \tag{1}\]  
where \([K]\) and \([M]\) are the \(N_{m}\times N_{m}\) global stiffness and mass matrices of the structure, respectively; \(\{\phi_{i}\}\) and \(\omega_{i}\) are mode shape and modal frequencies of the \(ith\) mode of vibration, respectively. According to continuous damage mechanics, and due to various internal and external conditions surrounding the structure, damage occurs and in turn alters the free vibration equation, which can be expressed by a scalar variable \(\alpha_{j}=(\alpha_{1},\alpha_{2},\dots,\ \alpha_{n})\) corresponding to \(n\) number of elements and \(\alpha_{j}\in[0,1]\). As the damage might occur in some elements, then, by discretizing the model, the global stiffness and mass matrices can be formulated by the scalar multiplication of the elemental matrices and damage parameters; hence, the global stiffness and mass matrices after damage can be written as:  
\[[K]^{\rm D}=\sum_{j=1}^{n}\alpha_{j}[K]_{j}, \tag{2}\]  
and  
\[[M]^{\rm D}=\sum_{j=1}^{n}\alpha_{j}[M]_{j}, \tag{3}\]  
where the superscript \(D\) refers to the damage of the structure; \([K]_{j}\) and \([M]_{j}\) are the element stiffness and mass matrices of the \(j^{th}\) element of the structure; n is the total number of elements. From the theory of modal energy, the global \(MKEn\) of the intact structure and the \(MKEn^{\rm D}\) of the damaged structure can be derived from the mode shapes and the stiffness matrix of the structure as:  
\[MKEn=\frac{1}{2}\omega_{i}\{\phi_{i}\}^{Tr}[M]_{j}\{\phi_{i}\}, \tag{4}\]  
and  
\[MKEn^{\rm D}=\frac{1}{2}\omega_{i}^{D}\Big{\{}\phi_{i}^{D}\Big{\}}^{Tr}\Big{[} M^{\rm D}\Big{]}_{j}\Big{\{}\phi_{i}^{D}\Big{\}}. \tag{5}\]  
where the superscript \(Tr\) refers to the transpose. As noted in [24], when damage occurs in a structure, the ratio of change of MKEn in each element can serve as a damage indicator, as:  
\[MKEn_{ij}=\frac{\Delta MKEn_{ij}}{MKEn_{ij}}=\frac{MKEn_{ij}^{D}-MKEn_{ij}}{ MKEn_{ij}}. \tag{6}\]  
The sensitivity of MKEn, with respect to a damage parameter \(p\), which can be any physical parameter such as Young's modulus, mass density, etc. [24; 30], can be written as:  
\[\frac{\partial MKEn_{ij}}{\partial p}=\frac{1}{2}\bigg{(}\frac{\partial\omega _{i}}{\partial p}\{\phi_{i}\}^{Tr}[M]_{j}\{\phi_{i}\}+\omega_{i}\{\phi_{i}\}^ {Tr}\frac{\partial[M]_{j}}{\partial p}\{\phi_{i}\}+2\omega_{i}[M]_{j}\frac{ \partial\{\phi_{i}\}}{\partial p}\bigg{)}. \tag{7}\]  
It is assumed that damage has only occurred due to the reduction of [PERSON]'s modulus (i.e., reduction in stiffness).