## 2 Materials and Methods
### Indices to Measure Damping Non-Proportionality  
The free-vibration problem of an undamped system is described by Equation (1) with \(\mathbf{c}=\mathbf{0}\) and \(a_{g}(t)=0\) resulting in an eigenproblem with real-valued solutions \(\dot{\phi}_{j}\) and \(\omega_{j}^{2}\). For a damped system, the free-vibration problem is solved transforming the general equation of motion into \(2N\) state-space; if the damping matrix \(\mathbf{c}\) is non-proportional, the corresponding eigenproblem produces complex-valued modes \(\psi_{j}\) and frequencies \(\lambda_{j}\)[26] (as customary, a different notation, is used to differentiate undamped real-valued vibration properties \(\mathbf{\Phi}\), \(\mathbf{\Omega}\), from damped complex-valued vibration properties \(\mathbf{\Psi}\), \(\mathbf{\Lambda}\), where \(\mathbf{\Lambda}\) is the complex-valued spectral matrix equivalent to \(\mathbf{\Omega}\) in damped systems, see Appendix A). The complex nature of modes indicates out-of-phase free vibration of the different degrees-of-freedom [27]. The complex modes can be graphically represented in the complex plane, each mode consisting of \(N\) vectors or points that, properly ordered, form a \(N\)-sided modal polygon. At the same time, as shown in the previous section, the corresponding modal damping matrix \(\hat{\mathbf{C}}\) is fully populated, rather than diagonal, and can be expressed as \(\hat{\mathbf{C}}=\hat{\mathbf{C}}_{d}+\hat{\mathbf{C}}_{o}\), where \(\hat{\mathbf{C}}_{d}\) is diagonal and \(\hat{\mathbf{C}}_{o}\) is populated with \(\hat{\mathbf{C}}_{o,jj}=0\). The differences between Equations (5) and (6) are due to the relative weight of \(\hat{\mathbf{C}}_{o}\) with respect to \(\hat{\mathbf{C}}_{d}\). Several authors have proposed indices to measure the degree of non-proportionality of damped systems. Using these two features (phase properties of the complex modes or modal polygons and relative importance of off-diagonal components of \(\hat{\mathbf{C}}\)). Thus, these non-proportionality indices (hereinafter referred to as NPIs) can be broadly divided in two families: those obtained from modal damping matrix components and those derived from properties of damped vibration modes. The first set is probably more adequate for design, as complex modes are not a usual output of conventional analysis programs, whereas the modal damping matrix of systems with added damping can be easily assembled from first principles. In all cases, a zero value of the index is indicative of proportional damping. Indices upper-bounded to unity are referred to as normal, and are of greater practical value, as they allow direct comparison between different systems. Some of the indices are mode-dependent; for this case, a practical global index can be based in all modes, a single mode, or a subset of the most relevant modes that can be selected by some specific criterion. In this study, the average across modes is used. An interesting question is the type of correlation (if any) between index values and the level of non-proportionality. This topic is addressed by the study described in the next section. The indices considered in this study (ordered first by their nature and then chronologically) are:  
Indices based on components or properties of the modal damping matrix:  
* \(I_{\text{WS1}}\), \(I_{\text{WS2}}\) by [PERSON] and [PERSON] (1977) [28]. * \(I_{\text{PS5}}\), \(I_{\text{PS6}}\) by [PERSON] and [PERSON] (1986) [29]. * \(I_{\text{CV1}}\) by [PERSON] and [PERSON] (1991) [30]. * \(I_{TL11}\) by [PERSON], [PERSON] and [PERSON] (1994) [31]. * \(I_{B1}\) by [PERSON] (1995) [32]. * Indices based on properties of the complex modes or modal polygons:
* \(I_{\text{PS1}}\), \(I_{\text{PS2}}\), \(I_{\text{PS3}}\), \(I_{\text{PS4}}\) by [PERSON] and [PERSON] (1986) [29]. * \(I_{\text{B2}}\), \(I_{\text{B3}}\) by [PERSON] (1999) [33]. * \(I_{LKZ1}\), \(I_{LKZ2}\), \(I_{LKZ3}\), \(I_{LKZ4}\), \(I_{LKZ5}\), \(I_{LKZ6}\) by [PERSON], [PERSON] and [PERSON] (2000; 2001) [34; 35]. * \(I_{\text{PF1}}\) by [PERSON] and [PERSON] (2000) [36].