# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 3 Results
### Phase Shift Effects on Multi-Source Functions
Quantification analysis [7,39] is presented in Table 1, which reports the RQA quantifications of parameters from CRP for the single function analysis with and without phase shift. The table shows that the entropy of diagonal length and the length of the longest vertical line had a variation of 1% and 2%, respectively. The other parameters remained constant. According to the results, it is clear that the RP patterns were not affected by the considered phase shift. In the case of signals with multiple excitations, the solution is not the same with phase shift and acceleration integration. Considering a function with three different signals sources, then:  
\[x=A\,\sin(wt)+B\,\sin(nwt)+C\,\sin(mwt) \tag{9}\]  
\[\dot{x}=A\,\sin[w(t+\phi_{1}/w)]+B\,n\,\sin[nw(t+\phi_{2}/nw)]+C\,m\,\sin[mwt+ \phi_{3}/mw] \tag{10}\]  
where \(n,m\in\mathbb{Z}\) for synchronous functions and \(n,m\in\mathbb{Q}\) for asynchronous functions. Then, if \(n\
eq m\) and \(\Delta sp\) is the shift phase error, the shift phases for each function are different:  
\[\Delta sp=\frac{\phi_{1}}{w}=\frac{\phi_{2}}{nw}=\frac{\phi_{3}}{mw} \tag{11}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **For \(\mathbf{\phi=0}\)** & **For \(\mathbf{\phi=\pi/8}\)** & **Variation (\%)** \\ \hline \(RR\) & 0.1 & 0.1 & 0 \\ \hline \(DET\) & 1.0 & 1.0 & 0 \\ \hline \(ENTR\) & 11.3 & 11.4 & 1 \\ \hline \(LAM\) & 499.0 & 499.0 & 0 \\ \hline \(L\) & 2.6 & 2.6 & 0 \\ \hline \(L_{max}\) & 1.0 & 1.0 & 0 \\ \hline \(TT\) & 9.8 & 9.8 & 0 \\ \hline \(V_{max}\) & 45.0 & 46.0 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Recurrence quantification analysis (RQA) quantification of the single function with different phase shifts. Figure 6: Recurrence plot for \(x=\sin(2t+\phi)\): (**a**) without phase shift (\(\phi=0\)); (**b**) with phase shift (\(\phi=\pi/8\)). The behavior of a function with many different vibration sources is not the same when it is affected by a phase shift because \(\phi_{1}\
eq\phi_{2}\
eq\phi_{3}\). Thus, \(\hat{x}(t)\
eq\hat{x}(t+\phi)\). This means that a phase shift error changes the original signal because it produces a different displacement for each source of vibration. The following examples are presented to illustrate this hypothesis. When considering two functions with three different vibrations sources, their displacement and velocity are described by the following equations:  
\[x=A\,\sin(nwt+\phi)+B\,\cos(mwt+\phi)+C\,\sin(pwt+\phi) \tag{12}\]  
The values of the variables were: amplitude \(A=B=C=1\) mm, \(n=3\pi/2\), \(m=43\pi/3\), \(p=131\pi/7\), and frequency \(w=2\) rad/s. Each function was calculated for \(\phi_{1}=0\) and \(\phi_{2}=\pi/8\) rad and the normalized results were compared graphically. The evolution of the state variable for both cases is plotted in Figure 7a. The graph shows that the trajectories do not match because of phase shift effects. Those differences can be visualized better in the phase plane in Figure 7b. According to the results, it can be observed clearly that the phase shift affects the original function. The influence of the phase shift in the RP can be clearly observed in Figure 8. The threshold distance was \(\epsilon=0.10\) and there is a significant variation of the pattern due the phase shift, in addition to the displacement along the main diagonal. Figure 7: Evolution of the state variables and phase plane for the multiple asynchronous function of Equation (12): (**a**) without phase shift (\(\phi=0\)); (**b**) with phase shift (\(\phi=\pi/8\)). The quantification analysis is presented in Table 2, which reports the RQA quantifications of parameters from the analysis of the multiple asynchronous function with and without phase shift. The biggest change was found for the laminarity with 17%. For the length of longest vertical line the change was \(-4\%\) and for the trapping time it was 3%. The other parameters remained constant. According to the results, the RP patterns were affected by the phase shift, as seen for the laminarity.