# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 1 Introduction  
The diagnosis of failures through the analysis of vibrations in rotating machinery is a technique that has proven to be very useful and reliable in the industry because it helps improve the performance and service life of machines. It is possible to describe the position, intensity, and force of the impact of a vibration in a mechanical system through their magnitudes of position, velocity, and acceleration, respectively. Accelerometers are the most widely used instruments to measure vibration in industrial machinery. The dynamic behavior of a body is characterized by integrating acceleration to obtain its speed and position. The main methods to integrate acceleration are based on the time domain and the frequency domain. The integration in the time domain is carried out by integrating the acceleration signal directly. During this integration, several errors are accumulated due to integration method, noise, phase shift from the signal processing, and unknown initial conditions of speed and position. Frequency domain integration converts the original signal to the frequency domain using the Fourier transform, then it performs the integral in the frequency domain, and finally it returns the signal to the time domain using the inverse Fourier transform. The Fourier transformations applied to the signal generate truncation errors, additional to those already mentioned above in the time domain. In any case, working with these errors is a challenging task for researchers in many areas of study in the fields of engineering and, specifically, predictive maintenance. In order to simplify the analysis, most fault diagnosis and prediction techniques eliminate noise by filtering but this practice leads to baseline errors from signal processing. Several literature methods, such as that described in [1], report power prediction and noise reduction techniques in the field of underwater acoustics [2]. However, eliminating noise means losing crucial information for the fault diagnosis, especially for nonlinear faults. This research focuses on phase shift analysis by phase plot and recurrence plot (RP) to understand the importance of signal processing. Phase shift errors can affect the accuracy of the results, especially in complex mechanical systems. This work could be applied in all engineering and research fields when transforming an acceleration signal into velocity and displacement, or when processing vibration spectra to eliminate noise. The research demonstrated that one small phase shift error becomes more critical in mechanical systems subjected to several vibration sources. The smoothing of a Gaussian noisy peak does not eliminate noise entirely and indeed it generates additional phase shifts. In 1890 [PERSON] [3] introduced the concept of RP as a graphical technique to study dynamical systems within the chaos theory. Then [PERSON], [PERSON], and [PERSON] presented the RP method formally in 1987 [4]. The RP helps the analysis of complex mechanical systems in phase space and the quantification of recurrence [5]. Another characteristic of the RP is that it can identify nonlinear vibration in nonstationary regimens. Cross recurrence plot (CRP) and recurrence quantification analysis (RQA) are RP extensions. Both CRP and RQA can analyze and quantify the time dependence behavior between two processes in a single time series [6]. The potential of the RP method has been demonstrated in many areas of science, such as medicine, physics, chemistry, biology, engineering, earth sciences, finance, and economics [7]. RP and empirical mode decomposition (EMD) were implemented to study the dynamics of traffic flow on a bridge [8]. Complex fractionated arterial electrograms were dynamically analyzed and quantified with RP and recurrence quantification analysis [9]. Recently, some researchers started to apply the RP method to identify and diagnose the condition of rotating machinery. The machine defect severity was assessed by estimating entropy from RP [10].