Estimation of the Defect Width on the Outer Race of a Rolling Element Bearing under Time-Varying Speed Conditions

Fault diagnosis and failure prognostics for rolling element bearing are helpful for preventing equipment failure and predicting the remaining useful life (RUL) to avoid catastrophic failure. Spall size is an important fault feature for RUL prediction, and most research work has focused on estimating the fault size under constant speed conditions. However, estimation of the defect width under time-varying speed conditions is still a challenge. In this paper, a method is proposed to solve this problem. To enhance the entry and exit events, the edited cepstrum is used to remove the determined components.-e preprocessed signal is resampled from the time domain to the angular domain to eliminate the effect of speed variation and measure the defect size of a rolling element bearing on outer race. Next, the transient impulse components are extracted by local mean decomposition.-e entry and exit points when the roller passes over the defect width on the outer race were identified by further processing the extracted signal with timefrequency analysis based on the continuous wavelet transform. -e defect size can be calculated with the angle duration, which is measured from the identified entry and exit points. -e proposed method was validated experimentally.


Introduction
Rolling element bearings are one of the critical components in rotating machines, where faults such as the spalling and the pitting are commonly observed in the course of normal operation [1].erefore, the diagnosis of rotating machinery faults and failure prognostics is necessary to prevent equipment failure or predict the remaining useful life (RUL) to avoid catastrophic failure.e spall size is very small in the early stages of the fault and gradually deteriorated until the bearing breaks.erefore, spall size can be used as a good fault feature for RUL prediction.However, fault size estimation for rolling element bearings can be a challenging task due to the harsh and variable working conditions.
To estimate the defect size of rolling element bearings, kinds of signal processing methods have been proposed.Epps [2] identified two events caused by a bearing fault: the point of entry and the point of impact.e defect size of a bearing was estimated by the time to impact (TTI).Sawalhi and Randall [3] illustrated more detailed explanation of the two events by observing the vibration signatures of seeded faults.To improve the estimation performance, Sawalhi et al. [4] and Ismail and Sawalhi [5] proposed other methods to process the vibration signal, such as autoregressive inverse filtration, synchronous averaging, energy envelopes, and numerical differentiation.
Other signal processing methods have also been proposed to measure the defect size of a bearing under different conditions.Jena and Panigrahi [6] measured different defect sizes in the inner and outer race of a bearing under constant speed.
e ridge spectrum was derived from the continuous wavelet transform (CWT) to obtain an obvious indication of the time duration between the entry point and the exit point.Moustafa et al. [7] estimated the different seeded fault widths under low speed with an instantaneous angular speed (IAS) technique.
e IAS could effectively reveal the shaft speed variation of a bearing with a fault in the outer race when the rolling element passed through the defect area.Khanam et al. [8] detected different fault sizes in the outer race of a ball bearing by using the discrete wavelet transform analysis.e entry and exit events were pointed out clearly in the decomposed signal, and a good estimation of the defect size was obtained.Wang et al. [9] proposed a vibration signal processing methodology for extracting the fault size of a naturally generated fault and observing the propagation of a bearing fault under high-speed conditions.e entry and exit events in the vibration signal were enhanced by tacholess synchronous signal averaging (SSA) and the wavelet transform.Previous research has focused on estimating the defect size of bearings under constant speed conditions.e signal enhancement processing methods have been applied based on the assumption of constant speed conditions [10].However, rotating machinery sometimes works under timevarying speed conditions.In such conditions, the amplitude and fault characteristic frequency (FCF) of the rolling element bearing vibration signal are influenced by the timevarying speed [11].Hence, envelope analysis and other enhancement techniques based on constant speed conditions cannot be applied directly.
e main idea of OT is to transform the time-varying speed vibration signal in the time domain to the angular domain with constant angle interval sampling.
us, the effect caused by the timevarying speed is removed by resampling the signal into the angular domain.Provided that the defect width is constant, the shaft rotating angle is constant when the roller passes the defect width.erefore, the defect width can be calculated from the resampled signal.
Although OT can eliminate the effect of speed fluctuation, it is still difficult to estimate the defect size of the bearing on the outer race because the impulse component of the signal is smeared by the noise signal.To extract the impact component of the signal, signal processing techniques can be used to extract fault information, such as wavelet transform (WT), empirical mode decomposition (EMD), and local mean decomposition (LMD) for a nonstationary signal.LMD is a novel signal processing method that was proposed by Smith [17].It is an adaptive time-frequency signal processing method and has been successfully used for bearing fault diagnosis and has similar properties to EMD.In contrast to EMD, the LMD method uses the moving average to obtain the amplitude envelope instead of cubic spline interpolation.e overshooting and undershooting effect caused by cubic interpolation can be eliminated [18].A series of product functions (PFs) can be obtained by the LMD processing method, and each PF represents a monocomponent of the original signal, which contains multiple components.us, LMD method is used in this study to extract the impact component from the multicomponent signal.
Energy is lost when a roller enters the leading edge of a defect, and high energy is generated when a roller hits the ending edge of the defect.Considering this, the position of the entry and exit points can be identified with timefrequency analysis, which can represent the energy distribution of the vibration signal.To obtain the exact position of the entry and exit points, the continuous wavelet transform can be used based on the characteristics of high timefrequency resolution [19].
e Morlet wavelet has been widely used to diagnose rolling bearing faults because its shape characteristic is similar to the impulse component of a rolling bearing signal [20].To obtain the optimal wavelet that matches the vibration signal, it is very important to select appropriate parameters for the Morlet wavelet (the bandwidth and center frequency).Researchers have proposed methods to optimize the bandwidth and center frequency [21][22][23].Once the optimized bandwidth and center frequency are obtained, the denoised signal can be processed with the CWT based on the Morlet wavelet.
In this paper, a new signal processing scheme is proposed to detect the fault size of a rolling element bearing under time-varying speed conditions.e whole procedure of the proposed method is shown in Figure 1.To enhance the entry and exit events, the edited cepstrum technique is used to remove the determined component of the signal, and the preprocessed signal is resampled by constant angle interval sampling with a tachometer signal to eliminate the effect of speed variation and measure the defect width.e impact component of the fault bearing signal is obtained with the LMD method for further time-frequency analysis.e extracted component is then processed using the CWT, for which the Morlet wavelet is selected as the mother wavelet.
e entry and exit points, when the roller passes over the defect on the outer race, can be identified from the scalogram.e defect width is calculated with the angle duration, which is measured from the identified entry and exit points.
e proposed method was verified experimentally.

Cepstrum Prewhitening.
A bearing fault signal based on time-varying speed conditions consists of an impulse fault component, determined component, and random noise component.e cepstrum is the inverse Fourier transform of the log spectrum [23,24].e main purpose of using the cepstrum prewhitening (CPW) technique is to separate the determined component from the vibration signal [24,25].
Once the constant component signal has been removed, the entry and exit events are enhanced.For a given vibration signal x(t), the cepstrum is defined as follows: where q is the quefrency and X(f) is the Fourier transform of x(t): e real cepstrum can be obtained by the real part in equation (1).
Cepstrum analysis can concentrate the harmonic components into a series of peaks.e cepstrum peaks in the quefrency domain indicate periodic harmonic components 2 Shock and Vibration in the spectrum.It is a simple way to separate to deterministic components from the vibration signal by editing the amplitude of the real cepstrum.e cepstrum editing procedure is shown in Figure 2. ere are two ways to edit the real cepstrum signal.e first one is setting a zero value for the whole real cepstrum (except possibly at zero quefrency), so that the discrete harmonics and resonances are eliminated in the frequency domain.e prewhitened signal can be obtained by recombining the edited cepstrum signal with the phase information of the original signal and inverse transforming to the time domain.e other one is eliminating the deterministic excitations by removing the peaks with filtering operations, which is a cepstrum editing procedure.e edited cepstrum signal is then transformed to the frequency domain.e edited real cepstrum signal can be obtained by recombining the edited cepstrum signal with the phase information of the original signal and inverse transforming to time domain.
e cepstrum editing method was used to remove the constant components from the signal.By using the CPW technique, the entry and exit events of rolling elements passing the defect area were enhanced.However, it is still hard to determine the obvious entry and exit points in the time domain due to the effect of random noise and speed variation.To identify the entry and exit points, other techniques are used to analyze the nonstationary signal.

Resampling Method.
When the entry and exit points are identified, the duration of the rollers rolling over a defect is obtained.e fault size can then be estimated by the following equation (assuming the contact angle is zero): where f shaft is the speed of shaft, f s is the sampling rate, T impact is the time to impact, D roller is the roller diameter, and D p is the pitch diameter.However, for the time-varying speed conditions, the equation is not suitable because the shaft speed varies.If the angle of the shaft when passing the defect is known, the defect size can be calculated with the geometric parameters of the rolling bearing.erefore, to measure the defect size under time-varying conditions, the signal needs to be resampled from the time domain to the angular domain.e conventional resampling method uses a constant angle interval instead of a constant time interval.In the resampling procedure, the speed information is obtained by a tachometer.
e main steps of the resampling process include the following [13,26,27]: (1) Synchronous acquisition of the vibration and keyphasor signal (constant-time increment) (2) Obtaining the speed of the shaft from the keyphasor signal and the total phase can be calculated (3) Setting the resampling rate based on the maximum value of shaft speed, and then obtaining the evenangle increment and corresponding sampling time (4) Interpolating the vibration signal according to the even-angle increment

Local Mean Decomposition.
A series of product functions (PFs) can be obtained by the LMD processing method, and each PF is a monocomponent of the original signal [28].e LMD method has been widely used to extract fault features for diagnosing rolling element bearing faults [29][30][31].It can also be used as a signal denoising method by selecting an appropriate PF which contains the fault component signal.For a given signal u(t), the LMD decomposition procedure consists of the following steps [32]: Step 1. Extracting the extrema z i (i � 1, . . ., M) of the original signal u(t).
Step 2. Obtaining the local mean value m i and the amplitude envelope estimate a i from the two successive extrema.
ese local means are linked by straight lines with extending between successive extrema.
e moving averaging can be done with averaging the right endpoint of each local mean with the left endpoint of the next local mean.e local mean function was repeatedly smoothed until there was no same value between two successive points.By this way, the continuous local mean function m 11 (t) is obtained.e continuous local magnitude envelope function a 11 (t) can be obtained by the same way [17].
e continuous local mean function is subtracted from the original signal u(t):  5) and ( 6). e process will stop until s 1n (t) is the pure frequency modulated signal.e iteration is shown as follows: where s 1n (t) is Step 4. e envelop signal can be obtained as us, the envelope function a 1 (t) is expressed as the instantaneous amplitude.e instantaneous phase is θ 1 (t) � arc cos s 1n (t) .
e instantaneous frequency can be defined as Step 5. e first PF 1 (t) can be obtained from the product envelope function a 1 (t) and frequency modulated signal s 1n (t): Step 6.New data u(t) can be obtained by subtracting PF 1 (t) from the original data.en, steps 1-5 are repeated ktimes until u k (t) is a constant or does not contain oscillations.
Finally, the original signal can be reconstructed:

Continuous Wavelet Transform Analysis
e defect signature of a faulty bearing is similar to an impulse signal and is nonstationary.Energy is lost when the roller enters the leading edge of the defect area.e maximum energy is generated when the roller hits the ending edge of the defect area.Energy loss then occurs again when the roller departs the ending edge of defect area [6].Considering this, the position of the entry and exit points can be identified with time-frequency analysis, which can represent the energy distribution of the vibration signal.To obtain a high resolution of the signal, the CWT is a suitable method.A scalogram can be obtained with the coefficients of the CWT.e coefficient matrix of CWT is defined as where x(t) is the signal, a is the scale, b is the translation, and ψ a,b (t)is the mother wavelet: It is very important to select a proper mother wavelet function for wavelet analysis.e Morlet wavelet is selected as the mother wavelet to analyze the signal because the shape of the wavelet is similar to the impulse component.e function of the Morlet wavelet is defined as Its Fourier transform is where f b is the bandwidth parameter and f c is the center frequency.Equations ( 16) and (17) show that the time-frequency resolution of the wavelet depends on the bandwidth f b and center frequency f c [21].e bandwidth parameter controls the oscillation attenuation of the Morlet wavelet.A larger f b value results in a better frequency resolution, at the cost of a To obtain the best resolution, the parameters need to be optimized.Shannon wavelet entropy is a good indicator for measuring sparsity.Jiang et al. [21] proposed an optimal method based on the modified Shannon wavelet entropy.e modified wavelet entropy is defined as where p k i is calculated by us, the parameters of Morlet wavelet can be optimized in separate steps.An initial bandwidth f b ∈ [N, M] and center frequency f c ∈ [A, B] are chosen.To set the initial center frequency, a large initial bandwidth value is chosen.
e Shannon entropy is then calculated by increasing the center frequency from A to B. e initial center frequency can be set based on the minimum wavelet entropy.en, the bandwidth is increased from N to M, and the wavelet entropy E n (f b ) is calculated based on the initial center frequency.
e optimized f ob is selected based on the corresponding minimum wavelet entropy.By calculating the wavelet entropy by increasing the center frequency f c from A to Bwith f ob , the optimal f c value is obtained based on the corresponding minimum wavelet entropy.

Fault Size Estimation
Once the entry and exit points were identified based on the Morlet wavelet analysis, the shaft's angular distance when roller passes over the defect area can be obtained.From Figure 3, the defect size can be calculated with angular duration as follows:     Shock and Vibration 5 where R is the outer race's inner radius, L is the defect width, and θ cage and θ shaft are the angular distances of the cage element and the shaft, respectively, when a roller passes over the defect area.f cage and f shaft are the cage and shaft speeds, respectively.
e relation between f cage and f shaft is as follows: where d is the roller diameter, α is the contact angle, and D p is the pitch diameter.is equation can be rewritten as   Shock and Vibration us, once the angular distance is obtained, the fault size can be estimated.

Experiment Setup
Laboratory experiments were conducted with different fault defect sizes on the outer race to validate the proposed method.e experimental setup is shown in Figure 4. e test system consists of a 3-phase AC motor, a shaft supported by two rolling bearings, and an AC controller.
e vibration signal is collected by an acceleration sensor that is mounted on the housing of the faulty bearing.e speed signal is by a tachometer mounted on the end of the shaft.All the data are obtained by an acquisition card (National Instrument) with a sampling frequency of 12000 Hz. ree sets of rolling element bearings (NTN 30206) were used to estimate the defect size on the outer races.ree different defect sizes on the outer race (0.75, 1.5, and 2.5 mm) were made by electric discharge machining, as shown in Figure 5. e parameters of the rolling element bearing are listed in Table 1.

Results and Discussion
e original vibration signal and corresponding speed signal are shown in Figure 6. e amplitude of vibration signal increased as the shaft speed increased.e impact effect is more obvious when the defect size increases.e vibration signals were processed by the edited cepstrum to remove the determined component, and the results are shown in Figure 7.
e amplitude of the prewhitened signal decreased after removing the determined component.However, the entry and exit points still could not be identified in the impulse component signal as random noise effect.
To estimate the defect size on the outer race, the timedomain signal needs to be resampled in the angular domain.us, the signal processed by the edited cepstrum method needs to be resampled in the angular domain before using the LMD method to extract the impact fault component.
During the resampling procedure, the signal is first upsampled by a factor of 10 to avoid aliasing.e resampled signal is obtained by recombining it with the speed signal, which was obtained from keyphasor data.e LMD decomposition result of the resampled signal with different fault sizes on outer race is shown in Figure 8. Several PFs can be obtained with the LMD process.e first PF is selected for further analysis because it has the biggest correlation coefficient value and keeps the most of information from original signal.By utilizing the LMD signal denoising method, the response of entry event and the impulse response of exit event were enhanced when rolling elements passed over the defect area on the outer race.
When using the continuous wavelet transform to process the denoised signal, the bandwidth and center frequency first need to be selected.
e optimal parameters are selected based on the minimum Shannon wavelet entropy.e maximum order of this study is 400.
e bandwidth is initialized as 400 to set the initial center frequency value.e initial bandwidth range is 1 to 30 with interval increment of 0.1, and the center frequency range is 0.1 to 5 with interval increment of 0.05.e initial center frequency and optimal parameters of each defect size are shown in Figures 9-11.
en, the signal was processed with CWT.At low speed (below 300 rpm), it is difficult to estimate the defect size due to the low signal energy and small response of the sensor in the low frequency range [7].us, the defect size can be measured above 300 rpm.
To better demonstrate the processed result, a portion of the signal was selected to measure the defect size.Figures 12(a e high energy zone is generated when a roller hits the end edge of the defect.
e portion data are selected around this high energy zone.When a roller enters into the start edge of defect area, the energy decreased as the destressing effect.When a roller leaves the end edge of the   Shock and Vibration identified, the angle duration was extracted, and the defect width can be measured.e angle duration is estimated from the CWT scalogram for 10 portions of the signal that contain an impact component.e estimation result is shown in Table 2. e mean estimation value and deviation are obtained from the scalogram.e deviation values for the different defect sizes are 0.078, 0.066, and 0.05.e results show that the defect size on the outer race can be measured with the proposed method.

Conclusions
In this paper, a method was proposed to estimate the defect size on the outer race of a rolling bearing under time-varying speed conditions.To enhance the

Shock and Vibration
entry and exit events, the edited cepstrum was used to remove the determined components.To eliminate the speed variation effect and estimate the defect size, the edited signal was resampled with a constant angle interval.
e LMD method was used to effectively extract the transient impulse component from the resampled signal.
e CWT can provide full information about the energy distribution.
e entry and exit events of a roller passing over the defect on the outer race were identified clearly from the CWT spectrum.
e average deviation of the estimated defect size was 6.5%.e estimation results show that the proposed method can effectively estimate the defect size on the outer race under time-varying speed conditions.

Figure 2 :
Figure 2: A flow chart of the cepstrum editing procedure.

Figure 5 :
Figure 5: Different fault sizes on the outer races.

Figure 3 :
Figure 3: Model of roller passing over a defect.

Figure 6 :
Figure 6: Vibration and speed signal of a bearing with different fault sizes on the outer race: (a), (c), (e) original vibration signal; (b), (d), (f ) original speed signal.

6
)-12(c) show the scalogram analysis results of the CWT with different defect widths on the outer race.To show the processed result more clearly, the details of the time-frequency analysis are shown in Figures12(d) and 12(f ).e impact response of the fault rolling bearing is like a sharp spike.Jena and Panigrahi[6] illustrated how to measure the defect width by using the CWT scalogram.

Figure 7 :Figure 8 :
Figure 7: Prewhitening of vibration signal by edited cepstrum: (a) prewhitening signal of 0.75 mm defect size on the outer race.(b) Prewhitening signal of 1.5 mm defect size on the outer race.(c) Prewhitening signal of 2.5 mm defect size on the outer race.

Figure 10 :
Figure 10: Relation between wavelet entropy and bandwidth parameters with the center frequency setting: (a) 0.75 mm, (b) 1.5 mm, and 2.5 mm.
Ideally, if s 11 (t) was a pure frequency modulated signal, its envelope function a 12 (t) should satisfy the conditions: a 12 (t) � 1.If a 12 (t) ≠ 1, then s 11 (t) is taken as the new signal, and the mean values and amplitude envelope values are calculated by the extrema of s 11 (t).m 12 and a 12 (t) are calculated by the same way of m 11 (t), and a 11 (t)•s 12 (t) also can be calculated via m 12 and a 12 (t) based on equations ( e demodulated amplitude s 11 (t) is obtained with the envelope function a 11 (t):

Table 1 :
Geometrical parameters of the test bearing.