# Order-Based Identification of Bearing Defects under Variable Speed Condition
## 2 Update of SPRO SPRI and SPRR Features  
Under stationary operating conditions, the impulses characterizing a bearing defect occur with a constant frequency. The characteristic frequency of an outer race defect (f\({}_{\text{o}}\)), an inner race defect (f\({}_{\text{i}}\)), and a ball defect (f\({}_{\text{b}}\)) are given respectively by:  
\[\text{f}_{\text{o}} =\frac{\text{N}_{\text{b}}}{2}\text{f}_{\text{r}}\left(1+\frac{ \text{d}_{\text{ball}}}{\text{D}_{\text{m}}}\text{.}\cos(\alpha)\right) \tag{1}\] \[\text{f}_{\text{i}} =\frac{\text{N}_{\text{b}}}{2}\text{f}_{\text{r}}\left(1-\frac{ \text{d}_{\text{ball}}}{\text{D}_{\text{m}}}\text{.}\cos(\alpha)\right)\] (2) \[\text{f}_{\text{b}} =\frac{\text{D}_{\text{m}}}{2\text{d}_{\text{ball}}}\text{f}_{ \text{r}}\left(1-\left(\frac{\text{d}_{\text{ball}}}{\text{D}_{\text{m}}} \text{.}\cos(\alpha)\right)^{2}\right) \tag{3}\]  
where f\({}_{\text{r}}\) is the rotation frequency of the shaft, N\({}_{\text{b}}\), d\({}_{\text{ball}}\), and, D\({}_{\text{m}}\) present respectively the number of balls, the ball diameter, and the pitch diameter of the bearing. \(\alpha\) is the contact angle of the bearing, (\(\alpha=0\) for the bearing considered in this work). The frequency spectrum of an outer race defect is dominated by f\({}_{\text{o}}\) and its harmonics. Similarly, an inner race defect (resp. ball defect) spectrum includes f\({}_{\text{i}}\) (resp. f\({}_{\text{b}}\)), its harmonics, but also sidebands corresponding to the rotation frequency of the inner race (resp. the cage) distributed around the defect frequency. SPRO, SPRI, and SPRR identify defects in the outer race, the inner race, and the ball, respectively. They are obtained by dividing the sum of peaks corresponding to the defect (fundamental and harmonics) by the sum of all peaks in the frequency spectrum. The returned value will therefore reflect the probability of the presence of defects. They are given by:  
\[\text{SPRO} =\frac{\text{K}\sum_{i=1}^{\text{H}}\text{P}_{\text{O}}(\text{h})}{ \sum_{k=1}^{\text{K}}\text{s}(\text{k})} \tag{4}\] \[\text{SPRI} =\frac{\text{K}\sum_{i=1}^{\text{H}}\text{P}_{\text{I}}(\text{h})}{ \sum_{k=1}^{\text{K}}\text{s}(\text{k})}\] (5) \[\text{SPRR} =\frac{\text{K}\sum_{i=1}^{\text{H}}\text{P}_{\text{R}}(\text{h})}{ \sum_{k=1}^{\text{K}}\text{s}(\text{k})} \tag{6}\]  
s(k) is the frequency spectrum with \(\text{k}\in(1,2\,\dots\text{K})\). K is the number of lines in the frequency spectrum. \(\text{P}_{\text{O}}(\text{h})\), \(\text{P}_{\text{I}}(\text{h})\), and \(\text{P}_{\text{R}}(\text{h})\) are, the peak values of the h\({}^{\text{th}}\) harmonics of \(\text{f}_{\text{o}}\), \(\text{f}_{\text{i}}\), and \(\text{f}_{\text{b}}\), respectively, with \(\text{h}\in(1,2\,\dots\text{H})\). H is the number of harmonics corresponding to the defect. SPRO, SPRI, and SPRR are derived from the frequency-domain feature SPR (Spectrum Peak Ratio) proposed by [PERSON] et al. [3], which has proven its effectiveness in the characterization of bearing localized defects under stationary operating conditions [3]. Referring to [12], under variable speed, bearing defect frequencies are time-varying. The conventional EA is therefore no longer suitable for the identification of bearing defects (problem of frequency smearing). Envelope order tracking is used in this work as an alternative to the conventional EA method. Based on a constant angular increment resampling, the non-stationary signal in the frequency domain transforms into a stationary signal in the order domain. Each defect is therefore now characterized by a constant order value whatever the rotation speed. The order is defined as the frequency normalized by the shaft speed [13], given by:  
\[1=\frac{60\,\text{f}_{\text{r}}}{\text{n}} \tag{7}\]  
where I is the order, \(\text{f}_{\text{r}}\) (Hz) represents the frequency of the shaft, and n (rev/min) denotes the shaft rotating speed.