# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 1 Introduction
Bearing degradation was evaluated by RP quantification and Kalman filter [11]. Vibration analysis was used in combination with EMD to detect bearing degradation by introducing a technique called autocorrelation function [12]. RP and phase space analysis were applied to identify the parameters in cutting processes [13]. Cutting stability in the milling process of nickel-based alloy Inconel 625 was analyzed by [PERSON] et al. [14]. [PERSON] et al. [15] studied the effect of friction on the dynamic response of a rotor rubbing housing by wavelet, phase plane and RP, and they concluded that their models could represent the friction. However, the dynamic response model was unable to predict the behavior observed in the field. Nonlinearities in a gearbox and other mechanical components were studied with RP by [PERSON] [16]. His final observations confirm that RP is a powerful technique to analyze the dynamic behavior of nonlinear mechanical systems. Nevertheless, RP has some limitations: the first one is the correlation between the topology and the characteristics of the system; the second one relates to the phase plane construction when measurement data come from an accelerometer. The basis of the RP is the phase plane that is based on the Hamilton's principle using the relationship between kinetic energy (velocity) and potential energy (displacement). These two quantities are calculated from an accelerometer by performing a double integral of the time series obtained from the instrument. The double integral of data in the time domain in complex systems can lose valuable information if it is not solved correctly. In the fault diagnosis of rotating machinery, vibration is measured by accelerometers that can pick up high-frequency with high sensitivity. The double integration of acceleration is applied to estimate displacement or position in some practical applications. [PERSON] et al. [17] demonstrated that the double integration of acceleration did not provide accurate information about the displacement due to noise and measurement errors, and they evaluated a correction method based on the Kalman filter to increase the accuracy of the results. Some authors apply several filters to make the double integration, but unfortunately, valuable information in the high-frequencies can be lost in this way. Other researchers integrated velocity and displacement from an accelerometer in the frequency domain [18] and proposed some techniques to manage errors arising during the process,which are mainly due to unknown initial conditions for velocity and displacement, and to the noise at low and high frequencies. Other researchers [19] estimated the drift by using the mean of the upper and lower envelopes of signals after integration from acceleration to velocity and displacement. [PERSON] et al. [20] introduced a way of measuring vibration displacement using a three-axes accelerometer. At first, they worked in the frequency domain to remove the trend items and noises. Then, the resulting acceleration was converted into the acceleration spectrum by Fourier transform and, finally, the acceleration spectrum was transformed into displacement spectrum. [PERSON] et al. proposed a Kernel regression approach to obtain displacement in the time domain from acceleration data [21]. [PERSON] et al. [22] proposed to correct the integration of acceleration with the baseline of the shift or drift phenomenon. In the vibration tests of autopilot missiles, baseline drift and heavy pollution are still a problem. [PERSON] and [PERSON] [23] suggested a morphological high filtering method to filter out the drift noise. One explicit time integration technique was developed by [PERSON] and [PERSON] [24] to analyze linear and nonlinear problems in structural dynamics. This method controls the amount of numerical dissipation in the high-frequency range if the matrix is diagonal, and it does not require evaluation of the initial acceleration vector. A compound fault diagnosis of bearings using improved fast spectral kurtosis with VMD was presented by [PERSON] et al.