# Order-Based Identification of Bearing Defects under Variable Speed Condition
## 3 Validation of the Proposed Features
### Results and Discussion
#### 3.3.2 Features Extraction  
The proposed features expressed in Equations (11)-(13) are extracted directly from the order domain. A transformation of the signal from time to order domain is therefore required. The order tracking method used in this work is based on the availability of an encoder speed measurement. The signals acquired with a uniform \(\Delta\)t are resampled in the angular domain based on a digital adaptive resampling algorithm. The \(\Delta\)t uniform data are therefore transformed into uniformly angle spaced data. To carry out the transformation from the time domain to the angular domain, a reference signal is required to define the uniform angular intervals. Generally, this reference signal is taken as the tachometer signal measured on the bearing drive shaft. The signal resampling process carried out in this work can be summarized as follows: (i) a tachometer signal is derived from an available encoder signal and used as a phase increment reference to obtain the even-angle sampling  
Figure 3: Raw vibration signal of a faulty bearing with an outer race defect, (Rotation speed decreasing from 25.4 Hz to 10.3 Hz).  
time marks. (ii) An interpolation scheme is used to achieve an even-angle sampling of the considered time series according to the tachometer time marks. (iii) Order spectrum analysis is used to extract features from the even-angle data.  
Figure 4: Kurtogram of the faulty bearing vibration signal.  
Figure 5: Filtered vibration signal of the faulty bearing.  
As explained before, an even-angle sampling of the considered bearing vibration signal is carried out according to the tachometer time marks. The resampled signal is shown in Figure 6c, indicating equidistant defect impulses (in contrast to those in Figure 3). This confirms the stationarity of the re-sampled signal in the angular domain. It should be pointed out that the abscissa of the resampled signal is no longer the time, but the rotation angle of the shaft in revolutions.  
An ordinary envelop spectrogram of the raw time signal in focus is calculated referring to [12] and displayed in Figure 7a. The latter indicates the presence of outer race defect frequency \(\mathrm{f_{o}}=3.57\times\mathrm{f_{r}}\) and its harmonics. This frequency varies according to the shaft rotational speed. Figure 7b, on the other hand, shows the envelope order spectrogram obtained based on the resampled envelope of the time signal in the angular domain. It is clear that the defect order values (\(\mathrm{O_{o}}=3.57\) and its harmonics) are constant regardless of the operation speed, confirming the robustness of the expressions of the updated features under variable speed conditions. As explained in Section 2, \(\mathrm{SPRO_{a}}\), \(\mathrm{SPRIA_{a}}\), and \(\mathrm{SPRR_{a}}\) are directly extracted from the envelope order spectrum of the signal. The envelope order spectrum of the signal in focus is given in Figure 8. It is dominated by the order of the outer race defect and its harmonics, confirming the efficiency of the proposed features.  
Figure 6: Resampling of the bearing vibration signal; (**a**) Tacho pulses; (**b**) Speed profile; (**c**) Resampled signal.