# Low-Speed Bearing Fault Diagnosis Based on Permutation and Spectral Entropy Measures
## 4 Methodology  
According to the theory of rolling element bearings, once a bearing has incipient damage, several characteristic harmonics arise in the frequency spectrum [68]. The frequency and size of the harmonics are related to the damage size and type, respectively (the bigger the damage size, the greater the harmonic magnitude). This asseveration explains the application of the SE to the PE signal. When the original vibration signal is damaged, the PE signal can detect the damage, because of the changes in the fundamental harmonics of the signal. To quantify the trend and the harmonics, and hence the damage, SE is applied to the PE signal. To analyze the raw vibration signals \(\left\{x_{t}\right\}_{t=1,\ldots,T}\), the first step is calculating the PE signal \(\left\{p_{t}\right\}_{t=1,\ldots,T^{\prime}}\), with \(T^{\prime}=T-W_{p}\). To obtain \(p_{1}\), Equation (9) is applied to \(\left\{x_{1},x_{2}\ldots,x_{W_{p}}\right\}\). The same step is then applied \(\left\{x_{2},x_{3},\ldots,x_{W_{p}+1}\right\}\) for \(p_{2}\). This process is continuously performed until the set \(\left\{x_{T-W_{p}+1},\ldots,x_{T}\right\}\). This calculation has two main parameters: the window size \(W_{p}\) and the permutation order \(O_{p}\). For the \(W_{p}\) selection, Figure 2 shows several PE signals calculated for a 30 s vibration signal (approximately five complete rotations) from a D2 bearing at 10 rpm as example. \(W_{p}\) is different for each signal, with values from 128 to 16384 samples (using powers of two for calculation purposes) and \(O_{p}=3\). The results reveal that, for values lower than \(W_{p}=2048\), unlike values greater than \(W_{p}=2048\), a larger amount of noise and the loss of any representativeness of the original signal can be observed in the calculated signal. Figure 1: Test-rig for low speed bearings: (**a**) General view of the test-rig. (**b**) Position of sensor is highlighted with the red arrow. (**c**) Damage seeded. The selection of the \(O_{p}\) value is performed with several PE signals computed using a process similar to that used for \(W_{p}\) selection. The calculation uses \(O_{p}=\{3,4,5,6\}\), as \(W_{p}\) covers less than \(7!\) samples. The results have the same waveform, but the amplitude reduces for higher \(O_{p}\) values (Figure 3). Because a higher value requires more computational effort and amplitude has no influence on the SE, the signal \(O_{p}=3\) is employed to reduce computing time. Figure 4 shows graphically the steps followed for the analysis of a vibration signal; they are summarized as follows:  
Figure 3: Several PE signals calculated from a D2 bearing at 10 rpm, with \(W_{p}=2048\) and varying \(O_{p}\). Figure 2: Several permutation entropy (PE) signals calculated from a D2 bearing at 10 rpm, with \(O_{p}=3\) and varying \(W_{p}\). The raw vibration signal is used for the PE process. As a result, the PE signal is obtained from the PE process. 2. The PE signal is later used by the SE process to calculate the SE value according to the rotation time. The SE calculation of the PE signals is normalized to compare results [59]. For values close to 1, SE indicates uniform PSD of the analyzed signal. In contrast, values close to zero imply a concentration of the PSD in some harmonics of the frequency domain. Unlike that for PE, the SE calculation has no further parameters. One last parameter to take into account is the elapsed time. As mentioned, the calculation is to be used in a CBM system, where incipient damage can take weeks, months, or even years to emerge. Incipient damage can be spotted anywhere on the bearing; therefore, for the entire bearing, the minimum time analyzed is the time needed for one rotation. For the case of 5 and 10 rpm, the bearing takes 12 s and 6 s for a full rotation. As the signals from the data set are 20 min long, there are approximately 100 rotations for the case of 5 rpm and 200 rotations for 10 rpm. The computation is performed to analyze the SE values calculated for one or more rotations, to integrate the results in a diagnostic model.