Sage Journals: Discover world-class research

Abstract

On the basis of empirical mode decomposition and teager energy operator, the new approach for rolling bearing fault diagnosis based on phase difference correction method is presented. It focuses on improving the identification precision of characteristic defect frequency of rolling bearing. Firstly, fault vibration signal is decomposed into finite number of intrinsic mode functions by empirical mode decomposition method. Secondly, computing the kurtosis value and correlation coefficient between intrinsic mode functions and original signal, then the reasonable intrinsic mode function is selected to demodulate by teager energy operator through the correlation coefficient and kurtosis value. Finally, the accurate characteristic defect frequency can be estimated by using phase difference correction method to correct the demodulation spectrum. The proposed method is applied in actual fault signals of rolling bearing with inner race damage and outer race damage. The results showed that, compared with the method without spectrum correction, the improved method receives better effect to identify characteristic defect frequency.

Keywords

Rolling bearing phase difference correct method teager energy operator fault diagnosis empirical mode decomposition

Introduction

Rolling bearing is the critical and vulnerable component for most rotating machinery. A number of conventional faults in rotating machinery are closely associated with bearing; bearing fault may lead entire system to catastrophic failure. Hence, bearing fault diagnosis (BFD) is very important and the early estimation of bearing faults can significantly prevent the system from damage. Many approaches have been studied throughout these last years, of which a typical research subject is to extract and identify the characteristic defect frequencies (CDF) of rolling bearing. CDF is the low-frequency defects and it is one important feature for BFD. It is well known that the fault types of bearing can be easily determined by CDF. However, CDF is usually difficult to be extracted due to the modulation in fault vibration signal of bearing. Therefore, the key step of BFD is to extract and identify the CDF with high precision.

As a matter of fact, the characteristics of fault vibration signals of rolling bearing are non-linear and non-stationary, thus it is difficult to obtain CDF. In order to effectively extract the CDF of roller bearings, a variety of signal processing techniques have been developed such as wavelet transform (WT), empirical mode decomposition (EMD), etc. These two famous techniques demonstrate the ability to decompose the original vibration signal of bearing into different frequency bands.¹ Many research have successfully applied the wavelet analysis for BFD,^2–4 however, WT suffers drawbacks, i.e. border distortion and energy leakage,^5,6 more importantly, in the strict sense, WT is not adaptive due to its fixed-scale frequency resolution.⁷ Aiming at these drawbacks, Huang et al. proposed the self-adaptive signal decomposition technique for non-linear and non-stationary signal, which is called EMD.⁸ The complicated vibration signal can be decomposed into a finite number intrinsic mode functions (IMFs) (intrinsic mode functions) efficiently and adaptively by EMD, the frequency component of each IMF from high to low and changed with the original signal itself. Therefore, EMD is an effective tool for treating vibration signals of rolling bearing.⁹ Moreover, some researches have combined the EMD with other techniques applied to BFD, such as Hilbert transform (HT),¹⁰ support vector machine (SVM),¹¹ wavelet packet transform (WPT).⁶ Autoregressive (AR) model,¹² and energy operator (EO).¹³

We know that the fault vibration signal of bearing always present the characteristic of modulation. Therefore, fault vibration signal demodulation analysis is necessary for BFD. At present, HT and EO are two common methods for signal demodulation. The performance of these two methods are analyzed and compared in Potamianos and Maragos.¹⁴ From Potamianos and Maragos,¹⁴ the demodulation effect of HT is inferior to EO demodulation and the computation of EO is also simple to perform. The non-linear EO (teager energy operator) valid for mono-component amplitude modulated (AM) signal and the frequency modulated (FM) signal is proposed in Kaiser,¹⁵ which is a fast and simple algorithm for BFD.^1,16 However, a majority of bearing fault vibration signals in real cases are all multi-component modulated signals. Therefore, EMD method is usually combined with teager energy operator (TEO) to decompose the multi-component signal into the mono-component signal. Many articles have applied this method to bearing fault detection with good effects.^17–19 It shows that the EMD–TEO method is an ideal option for the CDF extraction.

Most of existing researches are only concerning about how to effectively extract the CDF, thus far, few works have been reported on the precision of CDF. So far, the precision of CDF is not good as we expected, especially for shorter data length. Recently, discrete spectrum correction (DSC) methods have been widely applied to signal processing for improving estimation accuracy of signal parameters. Therefore, the goal of this paper is to improve the precision of CDF by using DSC methods. The phase difference correction (PDC) method is a kind of DSC which has the good anti-noise property and is easy to be implemented.²⁰ It was first proposed by McMahon and Barrett,²¹ hitherto, three types of PDC methods can be described as: time shifting (TS),^22,23 window length changing(WLC),^24,25 and correction method based on asymmetrical windows (AW).^26,27 TS method needs to choose an appropriate parameter L of time delay. And the AW method has presented very satisfactory computational accuracy, however, it needs the extra iterations. The WLC PDC method has the advantages of low computational effort and relatively high accuracy.

It is generally known that the fault vibration signal of bearing with low signal-to-noise ratios (SNR) and CDF is always overwhelmed by heavy noise. Based on the characteristics of vibration signals of rolling bearing, taking advantage of WLC, this study use WLC method for vibration signal analysis after being preprocessed by EMD-TEO and combine with kurtosis, correlation coefficients to select an appropriate IMF. The remainder of the paper is organized as follows: the principle of method is presented in “Basic principle” sections. Application of the improved method in the vibration analysis of rolling bearing with inner race (IR) damage and outer race (OR) damage are presented in “Simulation and experimental results” section. Finally, the conclusions are drawn in “Conclusion” section.

Basic principle

EMD method

The vibration signal can be decomposed into a finite number IMFs by EMD method, which is developed for non-stationary and non-linear signal analysis. Each IMF must satisfy the definition is mentioned in Huang et al.⁸ According to the definition, any signal can be decomposed as follows

s (t) = \sum_{i = 1}^{n} c_{i} (t) + r_{n} (t)

(1)

where c_i(t) represents the IMFs, which includes different frequencies from high to low, and r_n(t) is the final residual, which denotes the ideal tendency of signal s(t). The EMD process can be summarized as follows:

Step 1: Confirm all the local extreme points of s(t), then getting the upper envelop e_u(t) by connecting all the local maxima and repeat the procedure to connect all the local minima to obtain lower envelop e_l(t).

Step 2: Compute the mean of the envelopes m(t) = (e_u(t) + e_l(t))/2.

Step 3: Let h₁(t) = s(t) − m(t), if h₁(t) meet the condition of IMF, then h₁(t) is the first component of s(t), that means h₁(t) is c₁(t).

Step 4: If h₁(t) does not satisfies the condition of IMF, let h₁(t) replace s(t), and repeat steps 1–3 until h₁(t) is c₁(t).

Step 5: The remaining part of the signal r₁(t) are obtained by r₁(t) = s(t) − h₁(t), and repeat the above process, r₂(t) = r₁(t) − h₂(t),… r_n(t) = r_n_− 1(t) − h_n(t).

Step 6: After repeated iterative process, the signal s(t) can be expressed as equation (1).

The above discussion shows that the signal can be decomposed into different IMFs by EMD method, however, EMD method faces the problem of how to select a reasonable IMF for fault diagnosis of rolling bearing. Therefore, kurtosis metric is introduced to assist with making reasonable analysis, and select the optimum IMF. Kurtosis is a statistical parameter, which is sensitive to the shock signal and the value is described as^28,29

K = \frac{(1 / N) \sum_{l = 1}^{L} (x_{l} - μ)^{4}}{[(1 / N) \sum_{l = 1}^{L} (x_{l} - μ)^{2}]^{2}} = \frac{E (x - μ)^{4}}{σ^{4}}

(2)

in which μ is the average value of the amplitudes, σ is the standard deviation of signal. The kurtosis value is directly proportional to the magnitude of impulse component in IMF, the vibration signal bears more impulse component when the kurtosis value K > 3, which means this IMF contains more fault feature information and IMF is useful for extracting CDF.

Meanwhile, in the case of the kurtosis value K > 3, correlation coefficient is introduced as another metric to remove pseudocomponents of IMFs. As is known from the property of autocorrelation function (ACF), the ACF of periodic function is the periodic function with the same period of itself. An important aspect of vibration signals of rolling bearing is its periodical, therefore, ACF can be fully used to highlight the periodical of vibration signals and IMFs. As well as the pseudocomponents of IMFs can be effectively removed. The ACF of each IMF and the original signal defined as

C (m) = \sum_{j = 0}^{M - 1} x (j) x (j + m) / M

(3)

Then, the ACF is normalized to get the correlation coefficients between IMF ACF and original signal ACF

c_{j} = \sum_{j = 1}^{2 M - 1} C_{IMFj} (j) C_{x} (j) / \sqrt{\sum_{j = 1}^{2 M - 1} C_{IMFj}^{2} (j) \sum_{j = 1}^{2 M - 1} C_{x}^{2} (j)}

(4)

where C_x(j) and C_IMFi(j) represent original signal autocorrelation function and IMF autocorrelation function, respectively. The IMF has a good correlation with original signal if the correlation coefficients c_j > 0.5, and that means this IMF should be retained.³⁰

Teager energy operator

In order to effectively extract the CDF, the optimum IMF of vibration signal need to be demodulated. The TEO was proposed by Kaiser to measure the energy of the mechanical process that generated a single time-varying signal, which offers valuable signal processing tools for demodulating amplitude and frequency from AM–FM signal. The TEO for discrete time signal x(n) is¹⁵

ψ [x (n)] = [x (n)]^{2} - x (n - 1) x (n + 1) = A^{2} {sin}^{2} (ω) \approx A^{2} ω^{2}

(5)

From equation (5), we can see the output of TEO is in direct proportion to the square of amplitude A and frequency ω. Thus TEO is able to enhance the transient characteristic of vibration signal very effectively, moreover, since the output of TEO is obtained by only calculating the three adjacent sample points, the entire computation can be simplified. Therefore, the TEO implementation is simple and efficient for signal demodulating.

WLC PDC method

The WLC is a kind of PDC method, which is nearly applicable for all classical window functions and has relatively high accuracy, good anti-noise property. Without loss of generality, consider a single-frequency signal in the following formula

y (t) = A cos (2 π f_{0} + θ)

(6)

Make discrete Fourier transform (DFT) on weighted signal y_w(t) = y(t)w_T(t) with the different length of time-shifted Hanning window (N and N/2). In DFT, we assume that the number of acquired samples is N and the sampling frequency is f_s, then the phase of the two spectral peaks (k₁,k₂) can be written in the following form

\begin{matrix} ϕ_{1} = θ - π T (f_{1} - f_{0}) = θ - π \frac{N}{f_{s}} (\frac{f_{s} k_{1}}{N} - \frac{f_{s} k_{0}}{N}) \\ = θ - π (k_{1} - k_{0}) \end{matrix}

(7)

and

\begin{matrix} ϕ_{2} = θ - \frac{π T}{2} (f_{2} - f_{0}) = θ - \frac{π}{2} \frac{N}{f_{s}} (\frac{2 f_{s} k_{2}}{N} - \frac{f_{s} k_{0}}{N}) \\ = θ - \frac{π}{2} (2 k_{2} - k_{0}) \end{matrix}

(8)

where T = N/fs, the frequency resolution is f_Δ = f_s/N, k is the number of the spectrum lines, and k₀ = f₀/f_Δ. Consider the frequency error Δf = f₁ − f₀, and normalized frequency correction value Δk = k₁ − k₀ (Δk ∈ [−0.5, 0.5]). The phase difference value Δϕ can be written as

Δ ϕ = \frac{π (2 k_{2} - k_{1})}{2} - \frac{π Δ k}{2}

(9)

From equation (9) we can obtain

Δ k = 2 [\frac{π (2 k_{2} - k_{1})}{2} - Δ ϕ] / π

(10)

The corrected frequency is

f_{0} = f_{1} - Δ f = (k_{1} - Δ k) f_{s} / N

(11)

Then, the accurate CDF can be worked out by equation (11)

Diagnosis approach for rolling bearings based on WLC PDC method

The proposal can be summarized as follows

Perform the EMD to decompose the vibration signals into different IMF components as c₁…c_n.

Obtain the kurtosis value and correlation coefficient according to equation (2) and equation (4), then choose the optimum IMF component from above IMF components according to kurtosis value and correlation coefficient.

Compute the output of optimum IMF component by TEO.

Apply WLC phase difference method to the output of IMF component based on Hanning-windowed.

Finally, we get accurate corrected CDF.

Simulation and experimental results

In this section, we apply the proposed method (PM) to vibration data of real deep groove ball bearing, the IR fault and OR fault are discussed, respectively.

Database

The bearing vibration data from the Case Western Reserve University bearing data center³¹ are used in the experiment. The data were collected from accelerometer, which was placed at the drive-end bearing of the motor housing and the sample frequency is f_s = 12 kHz. The IR fault and OR fault were recorded with motor loads of 0–3 horsepower and test bearing with fault diameters of 0.007 in. and 0.021 in. The test bearing is 6205-2RS JEM SKF deep groove ball bearing, pitch diameter P_d = 1.537 in., ball diameter B_d = 0.3126 in., the number of balls Z = 9, and contact angle a = 0°.

Inner race

Figure 1 illustrates the time domain waveform of IR with fault diameter 0.007 in. and the motor speed is 1797 r/min; the data length 1–2048. The CDF of the IR can be calculated by

f_{ir} = \frac{{nf}_{r}}{2} (1 + \frac{B_{d}}{P_{d}} cos α)

(12)

and f_ir = 162.185973 Hz. Table 1 shows the kurtosis value and correlation coefficients of c₁, c₂, … , c₅. We can clearly see that c₁ with maximum kurtosis and correlation coefficient. Therefore, c₁ is used for demodulation by TEO.

Figure 1.

Time domain waveform of the inner race fault signal.

Table 1.

The kurtosis and correlation coefficients.

IMFs	1	2	3	4	5
Kurtosis	4.9624	4.5973	3.1499	3.1253	2.7249
Correlation coefficients	0.9580	0.2076	0.1101	0.0084	0.0163

IMFs: intrinsic mode functions.

Figure 2 shows the demodulated CDF and corrected CDF of c₁. The CDF is obtained by WLE method (f_C = 162.033267 Hz and double 2f_C = 324.127852 Hz) which is close to the calculated value by equation (12) compared with TEO demodulation (f_T = 164.0625 Hz and double 2f_T = 322.265625 Hz). According to the value of CDF, we can determine that the roller bearing has the IR fault.

Figure 2.

Demodulation spectrum of c₁ and corrected value.

It is worth noting that the kurtosis value of c₂ is also greater than 3. It is considered the consequences of ignoring some fault information if only just c₁ is selected for demodulation, Figure 3 shows the demodulation spectrum and corrected value of c₂. As shown in the figure, although the peak at the frequency f_c = 161.994473 Hz, which is similar to the CDF of IR: 162.185973 Hz. Nevertheless, the amplitude of CDF is very small, therefore, c₂ lays little effect on faults diagnosis, which can be ignored.

Figure 3.

Demodulation spectrum of c₂ and corrected value.

In order to evaluate applicability of the proposed algorithm, we investigate two fault diameters (0.007 in. and 0.021 in.) with different motor speeds. We use the root mean square error (RMSE) as a quantitative indicator of 80 random samples from bearing vibration data, and the data length of each sample is 2048. We define the corrected frequency error and demodulated frequency error as: fe_c = f_C − f_ir and fe_t = f_T − f_ir, respectively. The RMSE of fe_c and fe_t is given by

{RMSE}_{{fe}_{c}} = \sqrt{\frac{\sum_{i = 1}^{N} (f_{Cl} - f_{ir})^{2}}{N}} = \sqrt{\frac{\sum_{i = 1}^{N} {fe}_{c}^{2}}{N}}

(13)

and

{RMSE}_{{fe}_{t}} = \sqrt{\frac{\sum_{l = 1}^{N} (f_{Tl} - f_{ir})^{2}}{N}} = \sqrt{\frac{\sum_{i = 1}^{N} {fe}_{t}^{2}}{N}}

(14)

where N denotes the the number of samples, CDF _c is the corrected CDF of each sample and f_ir is the theoretical value by equation (12). Figures 4 and 5 show the RMSE of fe_c for each sample. In general, the proposed algorithm exhibits a good performance with high accuracy in most cases. However, in some cases, such as Figure 4 (1748 r/min) and Figure 5 (1752 r/min), fe_t has achieved higher accuracy, particularly in cases of Figure 5, this is because the demodulated CDF is pretty close to f_ir in this case (f_T_{(1752 r/min)} = 158.203125 Hz and f_ir_{(1752 r/min)} = 158.124554 Hz).

Figure 4.

RMSE for 80 random samples (0.007 in., N = 2048). RMSE: root mean square error.

Figure 5.

RMSE for 80 random samples (0.021 in., N = 2048). RMSE: root mean square error.

The RMSE of fe_c for 50 random samples are shown in Figures 6 and 7, and the data length of each sample is 4096. Firstly, the accuracy of both fe_c and fe_t increase with the growth of data length. Secondly, Figure 6 (1721 r/min) and Figure 7 (1752 r/min) illustrate the same situation as Figure 5. In general, by observing the simulation results, it is concluded that in most cases the proposed algorithm has higher precision than TEO demodulation.

Figure 6.

RMSE for 50 random samples (0.007 in., N = 4096). RMSE: root mean square error.

Figure 7.

RMSE for 50 random samples (0.021 in., N = 4096). RMSE: root mean square error.

Outer race

For localized defects of OR, we mainly consider the defects are located at different positions with same fault diameter. The localized defects of OR are located at 3:00 o'clock (directly in the load zone) and at 6:00 o'clock (orthogonal to the load zone), respectively, then the CDF of the OR can be calculated by

f_{or} = \frac{{nf}_{r}}{2} (1 - \frac{B_{d}}{P_{d}} cos α)

(15)

The RMSE of fe_c for 80 random samples with fault diameters (0.007 in.) at different location (3:00 o'clock, 6:00 o'clock) are shown in Figures 8 and 9, respectively, the data length of each sample is 2048.

Figure 8.

RMSE for 80 random samples (0.007 in., N = 2048, 3:00 o'clock). RMSE: root mean square error.

Figure 9.

RMSE for 80 random samples (0.007 in., N = 2048, 6:00 o'clock). RMSE: root mean square error.

As shown in Figures 8 and 9, we can clearly see that the maximum RMSE of fe_t is nearly equal to 2.4, and fe_t are still so large in most cases compared with the fe_c except in Figure 8 (RMSEfe_c_(1774r/min) = 0.521114 and RMSEfe_t_(1774r/min) = 0.451514) and Figure 9 (RMSEfe_c_(1773r/min) = 0.376766 and RMSEfe_t_(1773r/min) = 0.461368) due to the same reason as Figure 5.

Figures 10 and 11 show the simulation results for each sample with 4096 sample points. Generally, RMSE decrease as the data length increase and the trend of the fe_c is relatively stable with different motor speeds. In general, the proposal gives satisfactory results for fault detection with different locations of localized defects and different motor speeds, which can completely meet the requirement of the fault diagnosis.

Figure 10.

RMSE for 50 random samples (0.007 in., N = 4096, 3:00 o'clock). RMSE: root mean square error.

Figure 11.

RMSE for 50 random samples (0.007 in., N = 4096, 6:00 o'clock). RMSE: root mean square error.

Compare and influence of window

In addition to PDC method, windowed interpolated DFT (WIpDFT) algorithm is another popular method of DSC. It was first proposed by Jain et al.³² and has been widely used in harmonic analysis. Figure 12 shows the RMSE of CDF for PM and WIpDFT³³ based on Hanning window with different data length and different defects location.

Figure 12.

RMSE of PM and WIpDFT. (a) RMSE for 80 random samples IR007 N = 2048; (b) RMSE for 80 random samples IR021 N = 2048; (c) RMSE for 50 random samples OR007 at 3:00 o'clock N = 4096; (d) RMSE for 50 random samples OR007 at 6:00 o'clock N = 4096. RMSE: root mean square error; WIpDFT: windowed interpolated discrete Fourier transform.

From Figure 12 it can be observed that the accuracy of both method is quite high to identify the CDF of bearing and WIpDFT is even slightly better than PM in some cases. Both of these methods can meet the precision expectations of BFD. However, the process of WIpDFT always needs polynomial approximation if there is no analytical formulas for estimating the parameters of signal. In other words, WIpDFT therefore relies heavily on window with poor applicability. Being different from WIpDFT, WLC can be applied to all classical windows and without iteration, that is the reason why the WLC is chosen for PM. Although the accuracy of WLC is inferior to WIpDFT in some cases, it is good enough to precisely identify the CDF. In order to illustrate the accuracy of CDF influenced by window selection and to highlight the applicability of WLC, the proposed algorithm is used in four classic windows: Blackman, Blackman–Harris (B–H) window and Rife–Vincent (I)-5 (5-RV(I)), there are classical 3–5 term cosine window.

As shown in Figure 13(a), firstly, although shorter data for analysis (N = 1024), PM with different windows still presents satisfactory results, which shows a good generality of WLC. Secondly, the precision level decreases as term of window increases, and the precision of CDF based Hanning window (2-term window) is greatest because it gives relatively narrow main-lobe width. From Figure 13(b), it is clear that the Hanning window with the narrowest main-lobe width, which brings good frequency resolution of Hanning window.

Figure 13.

Performance of different windows. (a) RMSE for 100 samples (N = 1024,IR007); (b) normalized logarithm spectrum of windows. RMSE: root mean square error.

Conclusion

According to the characteristics of TEO and vibration signal of rolling bearing, TEO can effectively identify the CDF by demodulating the vibration signal. However, this method receives good effects only when the data length of sample with more data point, so the accuracy of demodulating is dependent largely on the number of sampling points.

The improved algorithm based on the WLC PDC is proposed in this paper, which effectively enhances the identification accuracy of the CDF and is easy to be implemented, therefore it is very suitable for the application of embedded system.

The simulation results from vibration signals with IR and OR faults indicate that combining the TEO demodulating with the WLC PDC technique can effectively improve the precision of CDF identification with fewer sampling points in most cases.

The identification accuracy of CDF is influenced by the main-lobe width, and narrow main-lobe width may obtain better effects.

Footnotes

Acknowledgements

The authors want to thank the anonymous reviewers for their helpful comments, which significantly improved the quality of the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: the Chongqing Municipal Education Commission (Grant No. KJ1600428).

References

Rodríguez

Alonso

Ferrer

et al.

Application of the Teager–Kaiser energy operator in bearing fault diagnosis. ISA Trans 2013; 52: 278–284.

Rubini

Meneghetti

. Application of the envelope and wavelet transform analyses for the diagnosis of incipient faults in ball bearings. Mech Syst Signal Processing 2001; 15: 287–302.

Fan

Zuo

. Gearbox fault detection using hilbert and wavelet packet transform. Mech Syst Signal Processing 2006; 20: 966–982.

Lou

Loparo

. Bearing fault diagnosis based on wavelet transform and fuzzy inference. Mech Syst Signal Processing 2004; 18: 1077–1095.

Zhu

Wang

et al.

Study on two application problems in wavelet transform. J Vib Eng 2002; 15: 228–232.

Peng

Tse

Chu

. A comparison study of improved Hilbert–Huang transform and wavelet transform: application to fault diagnosis for rolling bearing. Mech Syst Signal Processing 2005; 19: 974–988.

Rai

Mohanty

. Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert–Huang transform. Mech Syst Signal Processing 2007; 21: 2607–2615.

Huang

Shen

Long

et al.

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc London A 1998; 454: 903–995.

Yang

. Application of the EMD method in the vibration analysis of ball bearings. Mech Syst Signal Processing 2007; 21: 2634–2644.

10.

Cheng

Yang

. Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings. Mech Syst Signal Processing 2005; 19: 259–270.

11.

Yang

Cheng

. A fault diagnosis approach for roller bearing based on IMF envelope spectrum and SVM. Measurement 2007; 40: 943–950.

12.

Cheng

Yang

. A fault diagnosis approach for roller bearings based on EMD method and AR model. Mech Syst Signal Processing 2006; 20: 350–362.

13.

Cheng

. The application of energy operator demodulation approach based on EMD in machinery fault diagnosis. Mech Syst Signal Processing 2007; 21: 668–677.

14.

Potamianos

Maragos

. A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation. Signal Processing 1994; 37: 95–120.

15.

Kaiser JF. On a simple algorithm to calculate the energy of a signal. In: International conference on acoustics, speech, and signal processing, Albuquerque, NM, USA, April 1990, pp.381–384.

16.

Wang

Feng

Hao

. Fault diagnosis of rolling element bearings based on Teager energy operator. J Vib Shock 2012; 31: 1–5, 85.

17.

Li H and Zheng H. Bearing fault detection using envelope spectrum based on EMD and TKEO, In: Proceedings of 5th international conference on fuzzy systems and knowledge discovery, October 2008, pp.142–146.

18.

Zheng

Yang

. Bearing fault diagnosis based on EMD and Teager kaiser energy operator. J Vib Shock 2008; 27: 15–17, 22.

19.

Kwak

Lee

Ahn

et al.

Fault detection of roller-bearings using signal processing and optimization algorithms. Sensors 2014; 14: 283–298.

20.

Zhu

Ding

et al.

Noise influence on estimation of signal parameter from the phase difference of discretefourier transforms. Mech Syst Signal Processing 2002; 16: 991–1004.

21.

McMahon

Barrett

. An efficient method for the estimation of the frequency of a single tone in noise from the phases of discrete fourier transforms. Signal Processing 1986; 11: 169–177.

22.

Ding

Xie

. Phase difference correcting method for phase and frequency in spectral analysis. Mech Syst Signal Processing 2000; 14: 835–843.

23.

Zhu

Xiong

. An approach for amplitude correction in line spectrum. Mech Syst Signal Processing 2003; 17: 551–560.

24.

Ding

Zhu

. A phase difference correcting method on spectrum adapting to any window function. Acta Electron Sin 2001; 29: 1–3.

25.

Ding

Yang

. Improvement and anti-noise performance analysis of window-length changing phase difference correction. J South China Univ Technol 2007; 35: 210–213.

26.

Luo

Xie

. Phase difference methods based on asymmetric windows. Mech Syst Signal Processing 2015; 54: 52–67.

27.

Luo

Xie

. Asymmetric windows and their application in frequency estimation. J Algorithms Comput Technol 2015; 9: 389–412.

28.

Pachaud

Salvetatand

Fray

. Crest factor and kurtosis contributions to identify defects inducing periodical impulsive forces. Mech Syst Signal Processing 1997; 11: 903–916.

29.

Drona

Bolaersa

Rasolofondraibe

. Improvement of the sensitivityof the scalar indicators (crest factor, kurtosis) using a de-noising method by spectral subtraction: application to the detection of defects in ball bearings. J Sound Vib 2004; 270: 61–73.

30.

Chen

Tang

. Ensemble empirical mode decomposition de-noising method based on correlation coefficients for vibration signal of rotor system. J Vib Meas Diagn 2012; 32: 542–546.

31.

Case Western Reserve University Bearing Data Center Website, http://csegroups.case.edu/bearingdatacenter/home (2016).

32.

Jain

Collins

Davis

. High-accuracy analog measurements via interpolated FFT. IEEE Trans Instrum Meas 1979; 28: 113–122.

33.

Belega

Dallet

. Multifrequency signal analysis by interpolated DFT method with maximum sidelobe decay. Windows 2009; 42: 420–426.

Application of the phase difference correct method in rolling bearing fault diagnosis

Abstract

Keywords

Introduction

Basic principle

EMD method

Teager energy operator

WLC PDC method

Diagnosis approach for rolling bearings based on WLC PDC method

Simulation and experimental results

Database

Inner race

Outer race

Compare and influence of window

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References