## 3 Results
### The Detection of PCRRf Patterns
#### 3.2.1 The Extraction of the PCRRf1 Patterns
In Figure 18 curve 1 shows the PCRRf1a pattern (already described above, without DUM), curve 2 shows the PCRRf1c pattern (the DUM placed at 60 degrees before the marker 2, relative to the direction of rotation), curve 3 shows the PCRRf1d pattern (the DUM placed at 60 degrees after the marker 2), and curve 4 shows the PCRRf1e pattern (the DUM placed at 180 degrees). It would be expected that the centrifugal force generated by the dynamic imbalance (acting as a rotational force pushing radially against the bearings) would alter the PCRRf1 patterns. As can be clearly seen, there are no major differences between the patterns, indicating that these mechanical imbalances have an insignificant influence (mainly due to the low rotor speed). It should be noted that this rotating centrifugal force has another effect: it excites the motor to vibrate on its support. The mechanical power delivered by the motor to supply this vibration is obviously reflected in the IAEPf. This effect has not been highlighted here (e.g., by phase-shifting of patterns). Figure 17: A magnified view of Figure 16 in area A, with PCRRf1a pattern (1) and PCRRf1b pattern (3). Some resources provided by the analysis of the PCRRf1a pattern are presented below. The FASS procedure was used to obtain the synthetic description of the PCRRf1a pattern (by the coordinates of the \(n\) points on the curve). An approximate analytical description of this pattern can be carried out as the sum of \(r\) sinusoidal components, with a sample formally described as  
\[P_{pTpla}[k\!\cdot\!\Delta t]\approx\sum_{i=1}^{r}A_{fai}\cdot\sin(2\!\cdot\! \pi\!\cdot\!B_{fai}\!\cdot\!k\!\cdot\!\Delta t+C_{fai})\text{ with }k=1\div n \tag{6}\]  
A simple procedure (already introduced in [45]) is also available here to find the constants \(A_{fai}\) (as amplitudes), \(B_{fai}\) (as frequencies) and \(C_{fai}\) (as phases at the origin of time). Similarly to Figure 16 (which contains two periods of the PCRRf1s pattern), the synthetic definition of the PCRRf1a pattern is extended to more than one period (e.g., 10 periods). Thus, the description of a generic sample \(P_{pTpla}[k\!\cdot\!\Delta t]\) from the synthetic definition of the PCRRf1a pattern is available for \(k=1\div 10\!\cdot\!n\). Using the _Curve Fitting Tool_ from MATLAB (R2019b), applied to the extended PCRRf1a pattern, it is possible to quite accurately find the values of the constants from Equation (6) up to a reasonable upper limit \(r\) (here \(r=15\)). The extended PCRRf1a pattern (with ten periods) rather than the normal pattern (with one period) is used to increase the accuracy of determining the values of the constants \(A_{fai}\), \(B_{fai}\) and \(C_{fai}\). A model with a sum of eight sinusoidal components was used in the curve-fitting procedure. A first curve-fitting procedure produces the values of the first eight sets of constants (for \(i=1\div 8\)). The eight harmonic components defined by these sets are mathematically removed from the extended PCRRf1a pattern. The curve-fitting procedure is then reapplied to the remainder of the pattern, finding another eight sets of constants (for \(i=9\div 16\)). One set is ignored, corresponding to a component with insignificant amplitude. The values of the constants thus determined are given in Table 2 in ascending order of frequencies \(B_{fai}\). It is quite obvious that, as predicted, FASS has produced two notable results: first, it has removed all signal components that are not harmonically correlated with the \(T_{p1a}\) period (and the signal noise, too), and second, it has retained only the pattern sinusoidal components that are harmonically correlated with this period. In other words, there is a harmonic correlation between the \(B_{fai}\) frequencies (\(B_{fai}\approx i\!\cdot\!B_{fai1}\)). Some harmonics are missing.