## 3 Results
### The Detection of PCRRf Patterns
#### 3.2.1 The Extraction of the PCRRf1 Patterns
For example, the third harmonic (with a frequency of 49.9428 Hz) is missing because it is very close to the 50 Hz frequency and is removed by the BMAF filter used to  
Figure 18: A representation of two periods from PCRR1 patterns obtained without DUM (curve 1) and with three different angular positions of DUM on the jaw coupling hub (curves 2, 3 and 4). define IAEPf1a from IEP. The absence of the other harmonics is due to the peculiarities of the shape of the PCRRf1a pattern. The quality of the analytical description of the PCRRf1a pattern, based on the definition from Equation (6) and the values in Table 2, can be illustrated simply by plotting the two patterns, as shown in Figure 19.  
Figure 19: Representation of PCRRf1a patterns (on two periods): 1—with synthetic definition (by FASS); 2—with analytical definition (using Equation (6) with \(r=15\), found by curve fitting); 3—the difference between these patterns (as residual). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & \(A_{fai}\) & \(B_{fai}\) & & \(C_{fai}\) \\  & [**W**] & & [**Hz**] & & [**rad**] \\ \hline
1 & 3.945 & \(12.4857=1/T_{p1a}=B_{\#1}\) & Fundamental & 1.725 \\ \hline
2 & 1.913 & \(24.9714=2\cdot B_{\#1}\) & 1 st harmonic & \(-1.747\) \\ \hline
3 & 0.02436 & \(37.4491=2.9993\cdot B_{\#1}\) & 2 nd harmonic & 1.059 \\ \hline
4 & 0.02832 & \(62.4364=5.0006\cdot B_{\#1}\) & 4 th harmonic & \(-2.031\) \\ \hline
5 & 0.1168 & \(74.9142=6\cdot B_{\#1}\) & 5 th harmonic & 2.07 \\ \hline
6 & 0.09591 & \(87.4078=7\cdot B_{\#1}\) & 6 th harmonic & \(-1.45\) \\ \hline
7 & 0.1688 & \(99.8856=8\cdot B_{\#1}\) & 7 th harmonic & 0.9121 \\ \hline
8 & 0.06114 & \(112.3474=8.9980\cdot B_{\#1}\) & 8 th harmonic & 1.169 \\ \hline
9 & 0.04019 & \(124.8570=10\cdot B_{\#1}\) & 9 th harmonic & 0.5538 \\ \hline
10 & 0.02553 & \(137.3507=11\cdot B_{\#1}\) & 10 th harmonic & 1.596 \\ \hline
11 & 0.02038 & \(324.6760=26\cdot B_{\#1}\) & 25 th harmonic & 2.634 \\ \hline
12 & 0.02024 & \(337.0901=26.9980\cdot B_{\#1}\) & 26 th harmonic & 1.23 \\ \hline
13 & 0.06175 & \(424.6253=34.0089\cdot B_{\#1}\) & 33 rd harmonic & \(-2.366\) \\ \hline
14 & 0.05657 & \(437.0394=35.0032\cdot B_{\#1}\) & 34 th harmonic & 1.26 \\ \hline
15 & 0.01367 & \(536.8296=42.9955\cdot B_{\#1}\) & 42 nd harmonic & \(-0.5324\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The values of the constants \(A_{fai}\), \(B_{fai}\) and \(C_{fai}\) involved in the analytical description of the PCRRf1a pattern using Equation (6). This figure shows two periods of the synthetic pattern (in blue, curve 1, visible when zooming into area A, where the overlap is worst) and two periods of the analytical pattern (in red, curve 2). A very good analytical description makes the two patterns overlap very well, and the (insignificant) differences are shown (as an example) in zoom-in area A.  
Curve 3 (in black) shows the variation over time of the difference (sample by sample) between the two patterns (as a residual, what remains after the mathematical subtraction of the analytical model from the synthetic one). Obtaining analytical descriptions of the patterns facilitates the application of more advanced motor condition monitoring strategies (e.g., related to a phenomenon described by the evolution of a particular harmonic). There is another way to describe the content of the PCRRf1a pattern based on the FFT (Fast Fourier Transform) spectrum. In order to obtain a high resolution of the spectrum (in frequency), it is essential to use the FFT of the extended PCRRf1a pattern (e.g., for 10 periods, as before). The FFT spectrum of the extended synthetic PCRRf1a pattern is shown in Figure 20, window 1 (in the frequency range of \(0\div 550\) Hz). A zoomed-in portion of this spectrum from region A (the same frequency range, \(0\div 0.2\) W range on the \(y\)-axis) is shown in window 2. A zoomed-in portion of this spectrum from region B (the same frequency range, \(0\div 0.01\) W range on the \(y\)-axis) is shown in window 3.