Bearing failure prediction using Wigner-Ville distribution, modified Poincare mapping and fast Fourier transform

2.1. Fast Fourier Transform (FFT)

An appropriate signal processing method can easily detect changes in the vibration signal caused by the fault in the bearing components. Traditional analysis has relied on spectrum analysis like Fast Fourier Transform (FFT). FFT transforms a vibration signal $x\left(t\right)$ from time based domain to frequency domain. [14] Henceforth, giving us the frequency spectrum that includes all the signal’s fundamental frequency and its harmonics:

where $x\left(t\right)$ is time domain response of any system, $F\left(\omega \right)$ is Fourier Transform of $x\left(t\right)$.

In this algorithm, the assumption is, within a single time interval, the frequency change is small, such that there is no violation of the necessity of a stationary signal for frequency transformation. If the frequency change is significant within this time interval, then the FFT will yield an error in the actual value of the signal.

FFT is easy to implement and the most common vibration signal processing tool. However, this method does not give us information about the time dependence of the vibration signal being analysed, as the FFT results are averaged over the signal’s time duration. This becomes a problem when analysing non-stationary signals. It is often beneficial to acquire a correlation between the time and frequency contents of the signal in such cases. FFT analysis is thus not suitable for the detection of the bearing characteristics, which are an important part of the vibration signature. Hence, more powerful signal processing methods are needed to unambiguously detect all faults [15].

2.2. Wigner-Ville distribution (WVD)

The limitation of the FFT mentioned in Section 2.1 has led to the introduction of joint time-frequency signal processing methods, such as the Wigner-Ville Distribution (WVD), Short-Time Fourier Transform (STFT) [16] and others. These methods map a signal into a two- dimensional (2D) function of time and frequency.

WVD is a good analysis method because of its good time and frequency resolution. WVD allows an accurate description of spectral events associated with fast changes in the bearing vibration signature. However, the WVD present difficulties due to the presence of interference terms when dealing with multi-component signals [17, 18]. Taking into consideration the characteristics of bearing signature in the selection of the analysis and time smoothing windows, smoothing the WVD, alleviates this problem and allows the extraction of important cues on the time-frequency plane [19]. The WVD in discrete form can be defined as:

where $W_x(t,\omega )$ is the Wigner-Ville distribution in both the time domain $t$ and frequency domain $f$, $x\left(t\right)$ is the time signal, $T$ is the sampling period, and $L$ is the length of time data used in the transform. The quadratic operation on the signal causes the WVD to be a bilinear transformation. For a composite signal $x=x_1+x_2$:

The third term in the above equation is known as a cross-term. It is due to the interference between the two components. A cross-term appears midway between the two components and oscillates in time, at a rate equal to the frequency separation between them. Its amplitude is proportional to the product of the two components’ amplitudes. To avoid this aliasing problem arising in the computation of the WVD, the original real signal is transformed into a complex analytic signal. It also reduces the bandwidth to one half, thus avoiding double Nyquist rate sampling which is required for a real signal. The analytic signal is obtained by adding the signal’s Hilbert transform $H\left[x\left(t\right)\right]$, as it’s imaginary part or by using its definition in the frequency domain. The WVDs of the analytic and real signal are related by:

where, $xa\left(t\right)=x\left(t\right)+jH\left[x\left(t\right)\right]$.

The window length limits the accuracy of extracting frequency information relative to the duration of the signal. The area (time bandwidth product) of the window function in the time-frequency plane remains fixed once the window function is defined, which means that the time and frequency resolutions cannot be increased simultaneously.

This method can be used to examine a time domain signal and provide an instantaneous frequency spectrum at each instant whereas a frequency domain analysis like Fourier transform gives us a frequency spectrum averaged over the entire signal period. The WVD can be viewed as a 3-dimensional plot of real-valued WVD function, time and frequency or as a contour plot. The 3D and contour WVD plot of a simple continuous sinusoidal signal is shown in Fig. 1. The plot shows that the energy is evenly distributed over the entire duration of the signal.

Fig. 1a) Sinusoidal signal; b) sinusoidal signal with -3 D white noise, c) 3D WVD plot, d) contour plot of WVD

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Similarly, Fig. 2 shows the WVD plots for a discontinuous sinusoidal signal of the same frequency. The plot shows how the energy in the signal is discontinuous and unevenly distributed at that frequency.

Due to the windowing, there are non-zero values around the signal frequency component in both plots. The WVD plot is an excellent tool to detect the change in signal energy and henceforth, the fault that is causing it.

Fig. 2a) Discontinuous sinusoidal signal, b) discontinuous sinusoidal signal with –3 Db white noise, c) 3D WVD plot, d) contour plot of WVD

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2.3. Modified Poincare mapping

The theory of chaos and nonlinear dynamics have spread across almost every field of contemporary science over the period of last ten decades. Chaotic method has provided new conceptual and theoretical methods to capture and understand the complex behaviours of nonlinear dynamic systems [20]. It is completely different from other methods used to analyse vibration signal.

In fluid mechanics chaos phenomena had been known, but only now chaotic vibrations have been observed in low-order mechanical systems. This has prompted the development of new ways of looking at dynamical solutions, such as Lyapunov exponents, fractal dimensions and Poincare Maps [20, 21]. Ehrich [1, 7], Myers et al., Abarbanel et al., Oppenheim et al., and Singer et al. were the first few to use chaotic vibration analysis to process signals [22-26].

By the principles of the Poincare Mapping, a time sequenced data of $\left\{x\left(t_1\right),x\left(t_2\right),\dots ,x\left(t_k\right),\dots ,x\left(t_N\right)\right\}$was recorded and by representing $x\left(t_k\right)$ by $x_k$, data point $x_{n+1}$ can be determined by the values of $x_k$ such that:

If a moving particle is displayed in the phase plane as, (${x(t}_k)$, ${\dot{x}(t}_k)$), one can represent the motion discretely in a plot using $x_k=x\left(t_k\right)$ and $y_k=\dot{x}\left(t_k\right)$. This sequence of points forms a two-dimensional map of:

When the sampling times $t_k$ are chosen according to a specific position, this map is called a Poincare Map.

The primary importance of this study is to find a method to detect the location and severity of the bearing component failure in the rotor system. Hence, we are interested in every position of bearing and rotor running from 0° to 360°. This map is called the Poincare Map for a specific angle ($\theta$) position when the sampling rate is equal to the rotor speed. For a series of vibration data collected over $N$ cycles of revolutions, $N$ points of data in the Poincare Map are obtained.

The distance of each point $(x_k,y_k)$ to the origin, $d_k^{\theta }$ in a Poincare map can be calculated as shown in Eq. (7) and averaged as shown in Eq. (8):

The averaged points connected from 0° to 360° in polar coordinates gives us the average modified Poincare map [27-29] associated with the rotor speed. If the sampling rate is taken equal to the cage/ball carrier speed, then the modified Poincare map associated with the cage speed is obtained.

When the sampling rate is different and less than the rotor speed, multiple data points for each cycle of revolution are obtained. Let the rotor frequency to be $f_r$, sampling frequency to be $f_s$, the angle to be, $\varphi$, that the bearing rotates in the time between two data acquisition and is given by Eq. (9):

Plotting the vibration signal, $d_{avg}^{k\varnothing }$ with the angle, $k\varphi$ in polar coordinates gives the modified Poincare mapping.

4.1. Failure prediction from FFT graphs

The FFT graphs can be only used to detect major defects in the bearing. The remaining faults are not detected by the FFT method. Hence, more powerful signal processing methods are therefore needed to unambiguously detect all faults.

Fig. 6 show the FFT on the vibration signal acquired from good and two bad bearings [30-32]. From these figures it can be inferred that for a good bearing the amplitudes of the peaks in the FFT graph are small, as seen in Fig. 6(a), and for a bad bearing frequency corresponding to the failure and its harmonics are excited – the peaks seen in Fig. 6(b) are BPFO and its integral multiples and the peaks in Fig. 6(c) are BPF and its integral multiples [33]. The amplitude of vibration is measured and compared with vibration severity chart. If the amplitude reaches rough zone, preventive maintenance should be done. The artificial neural network and evolutionary algorithm are used to flag the user that the bearing needs to be changed [34, 35].

Fig. 6FFT graph for: a) a good bearing, b) a bearing with race failure, c) a bearing with ball failure

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4.2. Failure prediction from WVD plots

A signal acquired from a good bearing would consists mainly of vibration signature of the shaft rotation and the motor frequency. As the change in signal energy would be zero or negligible over the entire frequency spectrum, the WVD contour plot for such a signal would have vertical lines. Hence, it can be inferred that a bearing is in good condition if the WVD contour plot has vertical lines (lines parallel to time axis) as seen in Fig. 7(b).

Whereas, a signal acquired from a bad bearing would consists mainly of vibration signature of the bearing defect. As there would be continuous change in signal energy with a frequency corresponding to the failure, the WVD contour plot for such a signal would have horizontal lines (lines parallel to frequency axis), showing the discontinuity in the energy over the entire frequency spectrum. Fig. 8 shows the WVD plots for a bearing with a race failure and Fig. 9 shows the plots for a bearing with ball failure.

Fig. 7a) 3D plot and b) contour plot, of the WVD for a good bearing

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Fig. 8a) 3D plot and b) contour plot, of the WVD for a bearing with race failure

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Fig. 9a) 3D plot and b) contour plot, of the WVD for a bearing with ball failure

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4.3. Failure prediction from modified Poincare mapping

The modified Poincare Mapping is plotted as mentioned in Section 2.3, taking the vibration signature (displacement) as $x\left(t_k\right)$. Poincare mapping of vibration signature for a good bearing without defects, the plot will be concentric with the amplitude very less in comparison and few data points as outliers, as seen in Fig. 10(a). The bearing with a defect in the race, has the plot with skewed data points protruding in the direction the defect as seen in Fig. 10(b). The position of the protrusion gives us the location of the failure and the extent of protrusion gives us the severity of the failure. The modified Poincare map for bearing with defect in the ball is shown in Fig. 10(c). The amplitude is more in comparison to the plot of a good bearing and there are a lot of outliers at fixed angular intervals.

Fig. 10Modified Poincare mapping: a) a good bearing, b) a bearing with race failure, and c) a bearing with ball failure

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The data is acquired for three types of bearing (i) good bearing (ii) bearing with defect in race (iii) bearing ball failure. This data is acquired over a period, by running the setup shown in Fig. 3 and then these three methods are used for analysis to understand the type of failure.

The methods discussed in this section gives the notion of failed bearing based on the change in signature of FFT with respect to its amplitude and presence of side bands near different bearing failures. ANN and evolutionary algorithms are used to predict failure based of the amplitude reaching the severe zone of vibration severity chart. The presence of horizontal lines showing the discontinuity in the energy over the entire frequency spectrum in case of WVD vis-à-vis the discrete vertical lines in the contour plot of good bearing, gives a qualitative measure of bearing failure. In case of Poincare map, a polar plot of amplitude, the change in nature of plot, over a period, is taken and the observation of changes are corroborated with the maps of good bearing. The skewed nature of the graph in Fig. 10(b) with increased amplitude, gives a clear picture of bearing race failure in a direction. In case of Fig. 10(c), the amplitude increases in all directions and presence of outliers gives an indication of ball failure [6, 36].