## 3 Proposed Health Indicators  
The leading hypothesis behind the following argument is that the informative content on the damage severity is hidden beneath the energy related to the fault harmonics in the enveloped signals. To this purpose, the authors introduced the normalized power spectral density (NPSD) [42] obtained from the ratio between the PSD and the mean power of the signal, which is \(RMS^{2}\) (Equations (14)-(16)).  
\[E_{x}=\int_{-\infty}^{+\infty}x^{2}(t)dt \tag{14}\]  
\[RMS^{2}=\frac{1}{T}\int_{-\infty}^{+\infty}x^{2}(t)dt=\frac{E_{x}}{T} \tag{15}\]  
\[NPSD(f)=\frac{PSD(f)}{RMS^{2}} \tag{16}\]  
where \(E_{x}\) is the signal energy.  
One of the properties of the NPSD is that its integration in the frequency domain is unitary (Equation (17)).  
\[\int_{-\infty}^{+\infty}NPSD(f)df=1 \tag{17}\]  
Essentially, this is due to the fact that the integration of NPSD in a given frequency band results in the fraction of the signal power contained in such a band. Therefore, the health indicator (HI) proposed in Equation (18) gives an insight into the portion of power related to a frequency range with central frequency \(f_{c}\) and bandwidth \(\Delta f\).  
\[HI(f_{c},\Delta f)=\int_{f_{c}-\frac{\Delta f}{2}}^{f_{c}+\frac{\Delta f}{2}} NPSD(f)df \tag{18}\]  
Established that the conjunction between the SK and envelope analysis provided proper damage detectability into processed spectra, NPSD and HI were expected to give energetic information purely related to the fault, as long as they are evaluated in x_BPFI_ harmonics. Indeed, thanks to NPSD and HI, the contribution given to the signal power solely imputable to the damage was investigated. This provided the remarkable possibility of isolating the contribution given to a global signal feature (mean power) by a very specific event, which is damage. Additionally, the physical significance of these features made the authors assume two interesting aspects. First, it was reasonable to hypothesize that the greater the defect size was, the more accentuated the relative energetic contribution given by the fault to the extracted signal. Hence, NPSD and HI were expected to be sensitive to damage severity. Secondly, the physical units of these quantities permitted a direct comparison between signals of different natures. Actually, the indicators referred to energetic quantities with respect to the total signal energy, which is different for each type of signal. Consequently, relative energetic features rather than absolute were adopted to compare different physical information in the pursuit of a correlation.  
The described features were insofar evaluated in the x_BPFI_ harmonics (Table 4) of the accelerometric and pressure signals, and the sensitivity to the load and damage level was explored.  
Eventually, the correlation coefficient \(r\) (Equation (19)) was considered as a measure of the linear dependence between the vibration and pressure features.  
\[r=\frac{1}{P-1}\sum_{i=1}^{P}\left(\frac{FP_{i}-\mu_{FP}}{\sigma_{FP}}\right) \left(\frac{FV_{i}-\mu_{FV}}{\sigma_{FV}}\right) \tag{19}\]  
where \(P\) is the number of observations, \(FP\) is the feature extracted from the pressure signal, \(\mu_{FP}\) and \(\sigma_{FP}\) are the mean and the standard deviation of \(FP\), \(FV\) is the feature extracted from the vibration signal and \(\mu_{FV}\) and \(\sigma_{FV}\) are the mean and the standard deviation of \(FV\).