## 5 Conclusions
### [PERSON] and [PERSON] and [PERSON] [29] proposed several indices. After normalization of the modal vectors to the maximum of each mode (so that the modal displacements magnitudes are no larger than 1), the area of the polygon formed by the complex plane representation of the modal vectors for a given mode \(j\), \(A_{1,j}\), is taken as representative of the phase differences between modes. This area is compared with the maximum possible area \(A_{1,\text{max}}\) corresponding to a regular polygon inscribed in a circle of unit radius, given by \(N\cos(\pi/N)\sin(\pi/N)\). Thus:  
\[I_{PS1,j}=\frac{A_{1,j}}{A_{1,max}}. \tag{11}\]  
As discussed by [PERSON] [33], the polygon area is ambiguous in the sense that it is dependent on the order of DOFs, but this difficulty is avoided considering only the convex polygon. Another difficulty is that this index is not defined for systems with only 2 DOFs; this could be overridden by adding the point (0,0) to the set, giving rise to an alternative index, not discussed here (in this study \(N>2\) for all systems).  
Higher modes will always present a higher index because the presence of nodal points ensures that points representing modal vectors will lie in different quadrants of the complex plane. To override this problem, modal vectors in the 2 nd or 3 rd quadrant can be rotated 180\({}^{\circ}\), so that all polygons lie in the 1 st and 4 th quadrant. These are referred to normalized-rotated modal vectors. The polygon area in mode \(j\) is referred to as \(A_{2,j}\), and the maximum area \(A_{2,\text{max}}\) is given by \((N-1)\cos(\pi/2(N-1))\sin(\pi/(2N-1))\), that is half the area of a regular polygon of (2\(N\)-2) sides inscribed in a circle of unit radius; the index reads:  
\[I_{PS2,j}=\frac{A_{2,j}}{A_{2,max}}, \tag{12}\]  
where the area of the convex polygon should be considered, and the index is again undefined for 2 DOF systems.  
Additionally, considering the phase of the normalized-rotated complex modes, \(\theta_{nj}=\arg(\varphi_{nj})\) (where \(\arg(\cdot)\) indicates the argument function), these authors define the simple index:  
\[I_{PS3,j}=\frac{\sum_{n=1}^{N}2\abs{\theta_{nj}}}{N\pi}, \tag{13}\]  
normalized assuming the phase angle lies between \(-\pi/2\) and \(\pi/2\). This normal index is not independent of mode rotations; for instance, if two DOFs present the same maximum modal coordinate the value of the index will depend upon which one is chosen to normalize. To avoid this problem, a second, more meaningful normal index is given by:  
\[I_{PS4,j}=\frac{\max(\abs{\theta_{nj}})-\min(\abs{\theta_{nj}})}{\pi}, \tag{14}\]  
corresponding to the normalized angle spanned by the normalized-rotated modal vectors.  
Finally, two indices are derived based on the normalized modal damping matrix \(\hat{\mathbf{C}}\); the relative magnitude of the non-diagonal terms can be used as a measure of the degree of non-proportionality:  
\[I_{PS5}=\frac{\sum_{j=1}^{N}\sum_{n=1,n\
eq j}^{N}\abs{\hat{C}_{nj}}}{\sum_{j =1}^{N}\sum_{n=1}^{N}\abs{\hat{C}_{nj}}}, \tag{15}\]  
a normal index which is simple to compute; a mode-dependent index can be calculated ignoring the first summation sign in the numerator and denominator:  
\[I_{PS5,j}=\frac{\sum_{n=1,n\
eq j}^{N}\abs{\hat{C}_{nj}}}{\sum_{n=1}^{N}\abs{ \hat{C}_{nj}}}. \tag{16}\]In this case the global value obtained by Equation (A13) is not coincident with the average of the modal cases obtained by Equation (A14) \(I_{PS5,ap}\). An optional way to measure the importance of off-diagonal terms is given by the matrix determinant. If \(\hat{\mathbf{C}}_{o}\) is the matrix formed suppressing the diagonal terms in \(\hat{\mathbf{C}}\),  
\[I_{PS6}=\frac{\det(\hat{\mathbf{C}}_{o})}{\det(\hat{\mathbf{C}})},\] (A15)  
a normal index that cannot be applied to single modes. A third index is proposed by these authors based on the response of the system to harmonic motion. Because it involves full evaluation of the system's response, it is not a practical option for design purposes and is left out of this study.