## 5 Conclusions
### [PERSON], [PERSON] and [PERSON]  
These authors proposed six different indices [34; 35]; three of them are based on modal displacements and three on properties of the modal damping matrix. To obtain the former three indices, it is first necessary to manipulate the modes with a procedure similar to the one proposed by [PERSON]. The modal vector of mode \(j\), \(\mathbf{\Psi}_{j}\) is normalized to its maximum component \(\mathbf{\Psi}_{j,\max}\), and rotated counterclockwise by \(45^{\circ}\):  
\[\widetilde{\mathbf{\Psi}}_{j}=\frac{\mathbf{\Psi}_{j}}{\overline{\mathbf{\Psi }}_{j,\max}}\exp(\frac{i\pi}{4}); \tag{16}\]  
the first index proposed by the authors is based on the increasing correlation (due to modal displacement alignment) between imaginary and real parts of \(\widetilde{\mathbf{\Psi}}_{j}\) as proportionality increases. Thus:  
\[I_{LKZ1,j}=1-\frac{\left|\widetilde{\mathbf{\Psi}}_{R,j}^{\top}\widetilde{ \mathbf{\Psi}}_{\text{Ij}}\right|}{\sqrt{\left[\widetilde{\mathbf{\Psi}}_{R,j }^{\top}\widetilde{\mathbf{\Psi}}_{\text{Rj}}\right]\left[\widetilde{\mathbf{ \Psi}}_{\text{Ij}}^{\top}\widetilde{\mathbf{\Psi}}_{\text{Ij}}\right]}}, \tag{17}\]where the second part of the right term measures the correlation between real and imaginary parts of the modal vector. The best fit angle of the rotated displacements is found with a least square method:  
\[\theta_{j}=\arctan\left( \begin{array}{c}\overset{\overset{\sim}{\mathbf{V}}}{ \mathbf{F}_{R,j}}\overset{\overset{\sim}{\mathbf{V}}}{\mathbf{V}}_{I,j}\\ \overset{\sim}{\mathbf{V}}\overset{\sim}{\mathbf{F}}_{R,j}\overset{\overset{ \sim}{\mathbf{V}}}{\mathbf{F}}_{R,j}\end{array} \right), \tag{10}\]  
where \(\arctan\left(\cdot\right)\) is the arctangent function, \(\overset{\sim}{\mathbf{F}}_{R,j}\) is the real part of \(\overset{\sim}{\mathbf{V}}_{j}\) and \(\overset{\sim}{\mathbf{V}}_{I,j}\) is its imaginary part. Modal vectors are then rotated clockwise by the best fit angle:  
\[\overset{\sim}{\mathbf{V}}_{j}=\overset{\sim}{\mathbf{V}}_{j}\exp\left(-i \theta_{j}\right). \tag{11}\]  
The imaginary part of \(\overset{\sim}{\mathbf{V}}_{j}\) is minimal (in the least square sense). The second index is simply a measure of the scatter of the imaginary parts of \(\overset{\sim}{\mathbf{V}}_{j}\):  
\[I_{LKZ2,j}=\frac{\sum_{n=1}^{N}\left|\overset{\sim}{\mathbf{V}}_{I,nj}\right| }{N}. \tag{12}\]  
To find the third index, the normalized modal matrix is first decomposed as:  
\[\overset{\sim}{\mathbf{V}}=\overset{\sim}{\mathbf{V}}_{R}\overset{\sim}{ \mathbf{T}}=\overset{\sim}{\mathbf{V}}_{R}\left(\mathbf{I}+i\overset{\sim}{ \mathbf{T}}_{1}\right), \tag{13}\]  
where \(\overset{\sim}{\mathbf{V}}_{R}\) is the real part of \(\overset{\sim}{\mathbf{V}}_{R}\), \(\overset{\sim}{\mathbf{T}}\) is complex with real part \(\overset{\sim}{\mathbf{T}}_{R}=\mathbf{I}\), and imaginary part \(\overset{\sim}{\mathbf{T}}_{1}=\overset{\sim}{\mathbf{V}}_{R}^{-1}\overset{ \sim}{\mathbf{V}}_{1}\). The magnitude of \(\overset{\sim}{\mathbf{T}}_{1}\) is linked to modal coupling and therefore the following index is proposed:  
\[I_{LKZ3,j}=\frac{\sum_{n=1}^{N}\left|\overset{\sim}{T}_{I,nj}\right|}{N}. \tag{14}\]  
The next three indices proposed by these authors are based on modal information. Using the expansion of the normalized modal matrix:  
\[\hat{\mathbf{\Psi}}=\hat{\mathbf{\Phi}}\hat{\mathbf{T}}\Rightarrow\hat{ \mathbf{T}}=\hat{\mathbf{\Phi}}^{-1}\hat{\mathbf{\Psi}}. \tag{15}\]  
The fourth index is a counterpart of \(I_{PS5}\) based on \(\hat{\mathbf{T}}\) instead of \(\hat{\mathbf{C}}\):  
\[I_{LKZ4}=\frac{\sum_{j=1}^{N}\sum_{n=1,n\
eq j}^{N}\left|\hat{T}_{nj}\right|} {\sum_{j=1}^{N}\sum_{n=1}^{N}\left|\hat{T}_{nj}\right|}, \tag{16}\]  
it must be noticed that \(I_{LKZ4}\) is dependent from the normalization used for both the damped modes \(\hat{\mathbf{\Psi}}\) and undamped modes \(\hat{\mathbf{\Phi}}\).