## 3 Results
### The Detection of PCRRf Patterns
#### 3.2.1 The Extraction of the PCRRf1 Patterns
Based on Equation (6)--where \(A_{fai}\), \(B_{fai}\) and \(C_{fai}\) have been replaced by \(A_{ai}\), \(B_{ai}\) and \(C_{ai}\) from Table 3, the analytical pattern PCRR1a was built (with two periods) as shown in Figure 22 (curve 1), superimposed on the analytical pattern PCRRf1a (curve 2, also with two periods). It is  
Figure 21: the partial FFT spectrum of the analytical PCRRf1a extended pattern (10 periods); 2—the lower part of the curve of the inverse transmissibility of the BMAF filter (1/_Tr(f)_) used to obtain IAEPf from IEP. obvious that the PCRR1a pattern filtering using the BMAF (with \(h\) = 2001) produces the PCRRf1a pattern. Of course, a more complete description of the analytical PCRR1a pattern can be obtained if the component identification on synthetic extended PCRRf1a is performed by curve fitting for \(r>15\). As can be seen from the FFT spectrum in Figure 20, there are many other small amplitude sinusoidal components that can be considered in this description. It is obvious that the analytical PCRR1a pattern can only be deduced by knowing the analytical approximation of the PCRR1a pattern (from Equation (6), here, based on the data in Table 2). In other words, it is not possible to directly use the synthetically defined PCRRf1a pattern for this purpose. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(i\) & \(A_{fit}\) & **1/Tr(B\({}_{fit}\))** & \(A_{ni}\) = \(A_{fit}\):** 1/Tr(B\({}_{fit}\)) & \(B_{ai}\) & \(C_{ai}\) \\  & **[W]** & **[W]** & **[Hz]** & **[rad]** \\ \hline
1 & 3.945 & 1.103 & 4.351 & 12.4857 & Fundamental & 1.725 \\ \hline
2 & 1.913 & 1.568 & 2.999 & 24.9714 & 1 st harmonic & \(-\)1.747 \\ \hline
3 & 0.02436 & 3.311 & 0.0807 & 37.4491 & 2 nd harmonic & 1.059 \\ \hline
4 & 0.02832 & 5.578 & 0.1580 & 62.4364 & 4 th harmonic & \(-\)2.031 + \(\pi\) = 1.105 \\ \hline
5 & 0.1168 & 4.704 & 0.5494 & 74.9142 & 5 th harmonic & 2.07 + \(\pi\) = 5.2115 \\ \hline
6 & 0.09591 & 7.697 & 0.7382 & 87.4078 & 6 th harmonic & \(-\)1.45 + \(\pi\) = 1.6915 \\ \hline
7 & 0.1688 & 608 & 102.6304 & 99.8856 & 7 th harmonic & 0.9121 \\ \hline
8 & 0.06114 & 10.111 & 0.6182 & 112.3474 & 8 th harmonic & 1.169 \\ \hline
9 & 0.04019 & 7.841 & 0.3151 & 124.8570 & 9 th harmonic & 0.5538 \\ \hline
10 & 0.02553 & 12.035 & 0.3073 & 137.3507 & 10 th harmonic & 1.596 \\ \hline
11 & 0.02038 & 20.398 & 0.4157 & 324.6760 & 25 th harmonic & 2.634 \\ \hline
12 & 0.02024 & 28.90 & 0.5849 & 337.0901 & 26 th harmonic & 1.23 \\ \hline
13 & 0.06175 & 26.684 & 1.6477 & 424.6253 & 33 rd harmonic & \(-\)2.366 \\ \hline
14 & 0.05657 & 37.25 & 2.1072 & 437.0394 & 34 th harmonic & 1.26 \\ \hline
15 & 0.01367 & 45.09 & 0.6164 & 536.8296 & 42 nd harmonic & \(-\)0.5324 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The values of the constants \(A_{ai}\), \(B_{ai}\) and \(C_{ai}\) involved in the analytical description of the PCRR1a pattern using Equation (6). Figure 22: Graphical descriptions of two periods from 1, the analytical PCRR1a pattern, and 2, the analytical PCRRf1a pattern. It is interesting to look at the similarities between the PCRR1a and PCRR1b patterns (PCRR1b being similarly obtained from PCRRf1b). Figure 23 shows these two patterns superimposed (curve 1 as PCRR1a and curve 2 as PCRR1b, both with two periods). As can be clearly seen, there are some similarities but also some relatively large differences (the patterns do not fit perfectly). The first plausible reason for these differences is the accuracy of the curve-fitting procedure used to find the mathematical description of the synthetic patterns, as it is less accurate in describing the low-amplitude sinusoidal components (which also have high frequencies) in PCRR1 fa and b. An approach to a more accurate curve-fitting procedure is needed as a future challenge. We believe that, for the time being, the use of analytical PCRRf1a and b patterns or their FFT spectra is more reliable for motor condition monitoring.