# Mode Shape-Based Damage Detection Method (MSDI): Experimental Validation
## 2 Materials and Methods  
### The Mode Shape Damage Index (MSDI)  
The MAC criterion provides a degree of consistency between two states (undamaged and damaged), and it ranges between 0 and 1. The MAC matrix is defined as  
\[MAC_{(k,l)}=\frac{\left|\left\{\phi_{k}^{u}\right\}^{T}\left\{\phi_{l}^{d} \right\}\right|^{2}}{\left(\left\{\phi_{k}^{u}\right\}^{T}\left\{\phi_{k}^{u} \right\}\right)\left(\left\{\phi_{l}^{d}\right\}^{T}\left\{\phi_{l}^{d} \right\}\right)}, \tag{1}\]  
where \(\left\{\phi_{k}^{u}\right\}\) represents the \(k\)-th mass-normalized mode shape vector in the undamaged state and \(\left\{\phi_{l}^{d}\right\}\) represents the \(l\)-th mass-normalized mode shape vector in the damaged state. The values of the diagonal elements of the MAC matrix represent the correlation between corresponding mode shapes. If there is no damage and no noise in the signal, the diagonal elements of the MAC matrix are equal to 1, while values that are less than 1 reveal a weak correlation between the two mode shapes that can indicate the presence of damage (reduction of stiffness). The non-diagonal elements of the MAC matrix are usually equal to 0 because their values represent the correlation of inconsistent modes. Therefore, the MSDI damage localization algorithm uses only the diagonal elements (\(k=l\)) of the MAC matrix represented by the squared value of the trace of the MAC matrix defined as  
\[\gamma_{trMAC}=\left(trMAC\right)^{2}. \tag{2}\]  
This value ranges between zero and \(n^{2}\), where \(n\) represents the number of correlated mode shapes. If \(\gamma_{trMAC}=n^{2}\), then the compared mode shape vectors are highly correlated and almost identical, which means that there is no change in stiffness. When the value tends to 0, the vectors of the compared mode shapes are entirely inconsistent. The uncertainties such as noise in the signal, which can occur in the experimental determination of mode shapes, are ignored. Further, a modified matrix named \(\Delta\)MAC was developed, which is sensitive even to a small difference between correlated mode shapes in the undamaged and damaged state. The diagonal elements of the \(\Delta\)MAC matrix are defined as follows  
\[\Delta\alpha_{kl}=\alpha_{kl}{}^{\gamma_{trMAC}},\;k=l, \tag{3}\]  
where \(\alpha_{kl}\) and \(\Delta\alpha_{kl}\) are the diagonal elements of the original MAC matrix and the modified \(\Delta\)MAC matrix, respectively. Theoretically, if there is no change in stiffness and no noise in the signal, the mode shapes match entirely, and the modified \(\Delta\)MAC matrix equals the original MAC matrix. The mode shape damage index (MSDI) is a damage indicator based on the difference between modified modal displacements in the undamaged and damaged state in each \(i\)-th node of a structural element (i.e., plate) as follows  
\[\left|MSDI_{i}\right|=\frac{\left(\overline{\Phi}_{k}^{u}\right)_{i}-\left( \overline{\Phi}_{l}^{d}\right)_{i}}{\left(\overline{\[PERSON]}_{k}^{u}\right)_{i}}, \tag{4}\]  
where \(\left(\overline{\[PERSON]}_{k}^{u}\right)_{i}\) and \(\left(\overline{\[PERSON]}_{l}^{d}\right)_{i}\) are the modified modal displacements in each \(i\)-th node given by the following equations  
\[\left(\overline{\[PERSON]}_{k}^{u}\right)_{i}=\sum_{k=1}^{n}\left(\left(\phi_{k}^{u }\right)_{i}^{2}\cdot\Delta\alpha_{kl}\right), \tag{5}\]  
\[\left(\overline{\[PERSON]}_{l}^{d}\right)_{i}=\sum_{l=1}^{n}\left(\left(\phi_{l}^{d }\right)_{i}^{2}\cdot\Delta\alpha_{kl}\right), \tag{6}\]where \(\left(\phi_{k}^{u}\right)_{i}\) and \(\left(\phi_{l}^{d}\right)_{i}\) denote the values of modal displacements of the \(k\)-th mode shape in the undamaged state and the \(l\)-th mode shape in the damaged state in each \(i\)-th node of the plate. The normalized modal displacements are squared to highlight a local change of the mode shape in the damaged state compared to the undamaged state. Furthermore, modal displacements are multiplied by the corresponding value \(\Delta\alpha_{kl}\) calculated from the modified matrix \(\Delta\)MAC.