LI Min*， YANG Jianhong， and WANG Xiaojing

School of Mechanical Engineering， University of Science and Technology Beijing， Beijing 100083， China

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Fault Feature Extraction of Rolling Bearing Based on an Improved Cyclical Spectrum Density Method

LI Min*， YANG Jianhong， and WANG Xiaojing

School of Mechanical Engineering， University of Science and Technology Beijing， Beijing 100083， China

The traditional cyclical spectrum density（CSD） method is widely used to analyze the fault signals of rolling bearing. All modulation frequencies are demodulated in the cyclic frequency spectrum. Consequently， recognizing bearing fault type is difficult. Therefore， a new CSD method based on kurtosis（CSDK） is proposed. The kurtosis value of each cyclic frequency is used to measure the modulation capability of cyclic frequency. When the kurtosis value is large， the modulation capability is strong. Thus， the kurtosis value is regarded as the weight coefficient to accumulate all cyclic frequencies to extract fault features. Compared with the traditional method，CSDK can reduce the interference of harmonic frequency in fault frequency， which makes fault characteristics distinct from background noise. To validate the effectiveness of the method， experiments are performed on the simulation signal， the fault signal of the bearing outer race in the test bed， and the signal gathered from the bearing of the blast furnace belt cylinder. Experimental results show that the CSDK is better than the resonance demodulation method and the CSD in extracting fault features and recognizing degradation trends. The proposed method provides a new solution to fault diagnosis in bearings.

cyclostationary， cyclical spectrum density， rolling bearing， fault diagnosis

1 Introduction1

Rolling bearing is one of the most widely used parts of mechanical equipment. The fault diagnosis of rolling bearing has an important role in equipment condition monitoring. Fault in a rolling bearing leads to a cyclical effect and significantly affects the normal operation of mechanical equipment because of the periodicity characteristic of mechanical rotary motion.

Resonance demodulation is one of the most conventional methods used in rolling bearing fault diagnosis. Demodulation is mainly achieved by Hilbert transformation，in which the selected of resonance band directly affects the results of the method if it isn‘t appropriate. What's more，this method is more suitable for diagnosis during the middle and later stages of a rolling bearing fault. However，Extracting weak or incipient defects is difficult using this method because the energy generated by the fault is frequently buried in noise duriing this stage.

Based on the non-stationary characteristics of a rolling bearing with fault， the wavelet transform method is gradually applied in rolling bearing fault diagnosis［1］. The key to wavelet transform is a type of Fourier transform with adjustable windows. Using the wavelet scale factor and the time shift factor， the method decomposes a signal into different frequency band observations. The wavelet packet method that evolved from the wavelet can divide the signal into an entire frequency band. However， the application of both methods is subject to the selection of the wavelet basis function. Once the wavelet basis is set， decomposition and reconstruction will no longer change and will no longer be adaptive to signal analysis.

To solve the problem of selecting wavelet basis functions，HUANG， et al［2］， conducted an extensive research on instantaneous frequency and proposed the experience modal decomposition（EMD） method in 1998. The core of this method is to decompose a signal into a series of intrinsic modes， and then obtain the instantaneous spectrum of each single intrinsic mode function after Hilbert transformation. Finally， the signal spectrum can be obtained by combining all instantaneous frequency spectrums. Unlike the certainty of the wavelet basis function， the basis function of EMD mainly depends on the original signal，and thus， the method is adaptive for signal analysis. Consequently， EMD is generally applied in fault diagnosis［3-5］. Nevertheless， EMD is still a type of“experience” algorithm without an appropriate mathematical model， rigorous theoretical foundation， and theoretical framework. Moreover， current literature on EMD is frequently confined to experiments and applied research.

Wavelet transform and EMD are powerful tools for non-stationary signal analysis. However， increased attention should be given to the direct correlation between the periodic time-varying characteristic of the signal of a rolling bearing and its working conditions. Given the symmetry and rotation of a rolling bearing， the signal of the fault rolling bearing includes a large amount of periodic impacts； moreover， the two-order statistics of the signal are cyclical［6-7］. Traditional analysis methods always ignore this important information. However， the cyclostationary signal processing method can extract the cycle component from the two-order statistics of the signal and simultaneously reduce background noise［8-9］.

Cyclostationary theory was proposed during the 1950s；meanwhile， the cyclostationary characteristic according to the non-stationary process was first proposed by BENNETT［10］. Furthermore， spectral correlation theory was developed and widely applied in equipment health monitoring［11-12］. At present， the cyclical spectrum density（CSD） method is typically applied in two aspects:（1） combined with existing methods to extract fault features，and （2） to introduce new cyclostationary indicators to determine the severity of a fault. On the one hand， wavelet［13-14］， EMD［15-16］， local mean decomposition［17］， and other methods are used for pretreatment to reduce background noise in the signal， and then CSD is employed to extract the fault characteristic frequency of a rolling bearing. On the other hand， the concept of the indicators of cyclostationarity（ICS） is used to evaluate the conditions of simple machines such as gears and bearings. ICS was applied for the first time in spall diagnosis in gears by RAAD， et al［18］. Bearing defects were detected using a similar ICS tool called integrated spectral correlation that is applied in acoustic emission（AE） signals［19］. In addition，ANTONIADIS， et al［20］， proposed using new cyclostationary indicators to determine the type of bearing faults. ANTONI［21］used cyclic frequency that was set to identify the severity of the inner or outer ring fault of a rolling bearing. DELVECHIO， et al［22］， addressed using first- and second-order cyclostationary（CS1and CS2） tools to process vibration signals from internal combustion engines during cold tests. In summary， CSD is essentially a demodulation method. Using the conventional CSD method will modulate all high-frequency modulations in cyclic frequency. The amplitude of each frequency is similar， and thus， the results of fault feature extraction are not ideal.

To solve existing problems in traditional CSD， this study proposes a spectral correlation analysis method based on kurtosis energy（CSDK）. The kurtosis of each slice that corresponds to cycle frequency is used to reflect the modulation capability of such frequency. The kurtosis values are used as the weighting coefficient to strengthen the main modulation frequency. Therefore， the essence of the CSDK method is to highlight failure frequency and weaken the influence of multiple harmonic frequencies.

This study selects low-speed and heavy-load equipment as the research object and uses the AE technique to pick up the signal. Comparing CSDK with the CSD and resonance demodulation methods， the experimental results proved the effectiveness of the proposed method in three kinds of signals: the simulation signal， the signals of rolling bearings acquired from the low-speed and heavy-load test bed， and the industrial signal of the rolling bearing of the blast furnace belt cylinder.

2 Principle of Traditional CSD

The statistical feature of a signal plays an important role in signal processing. Common statistical features include one-order statistic， two-order statistics， and higher-order statistics. The statistics of the stationary signal of each order are independent from the time domain. A signal with statistics of a certain order that change over time is called a non-stationary signal. An important subclass of the non-stationary signal， i.e.， the cyclostationary signal， has periodical or multi-periodical statistics. The signal to be analyzed is assumed to be x（t）. The Nth-order statistics with a time variation can be expressed as Eq. （1）:

When the rolling bearing is under a fault condition， the rotation motion produces the periodical modulating feature of the signal. Thus， the 2nd-order statistics of the signal appear to be periodical.

Therefore， this study selects the 2nd-order cyclostationary signal as the object of analysis.

x（t） is assumed to be a cyclostationary signal with a period of T0. Its 2nd-order cyclostationary statistic Rx（ t，τ）can be expressed as Eq. （4）:

Given that Rx（ t，τ） can be expanded in Fourier form because it is cyclical，

Rx（α，τ） is the Fourier coefficient in Eq. （5）. Through inverse Fourier transformation， the expression of Rx（α ，τ）can be presented as Eq. （6）:

then

In general， Rx（α，τ） is defined as the cyclic autocorrelation function （CAF）. Notably， if the sampling number N in practical applications is high， then T will be large. In this case， the result of the calculation is more accurate， whereas the high sampling number increases the calculation amount. Therefore， determining sampling number N in practical applications should consider both frequency resolution and computational cost.

For the stationary signals， the autocorrelation function and power spectral density function belong to a Fourier transform pair. The spectrum feature of 2nd-order statistics can be described as the power spectral density function. Similarly， for the cyclostationary signal， the CAF and CSD function belong to a Fourier transform pair. The CSD function can be expressed as Eq. （8）:

where α is the cyclic frequency and f is the natural frequency. CSD is a function of both αandf. When α=0， CSD degrades into a power spectral density function to represent the stationary component of the signal. When α≠0， the non-zero part represents the cyclostationary component of the signal.

It can be seen from the above description， the CSD function is a 3D spectrum α～f～A ， where A is an amplitude value of the spectrum. Considering cyclic frequency α as a fixed value， the vertical slices of the α axis express the natural frequency spectrum that is modulated or half-modulated by the fixed cyclic frequency. If natural frequency f is a fixed value， then the vertical slices of the f axis represent the natural frequency that is modulated or half-modulated by various cyclic frequencies. In general，the space slice method is used to focus on α orf frequency characteristics for convenient CSD application in fault diagnosis.

3 Principle of CSDK

Given the cyclical impact characteristic of the signal generated by a rolling bearing fault， multiple harmonic components are found in the frequency domain. Therefore，the demodulation result is easily disturbed by cyclic frequency α in traditional CSD analysis.

For multiple harmonic frequency interference， the concept of the weighted coefficient based on kurtosis is introduced to CSD analysis to extract modulated information effectively. The CSD slices that are perpendicular to the α axis show the modulation frequencies of the corresponding cyclic frequency α， which indicates that the energy of each slice can reflect its modulation capability. Meanwhile， as a 4th-order statistic，kurtosis is more sensitive to higher amplitude signals and can represent the energy of the data. Therefore， the kurtosis of each slice describes the modulation capability of its corresponding cyclic frequency. The weight coefficients are obtained by normalizing all the kurtosis. If the kurtosis is high， then the weight coefficient will also be high. The modulation capability of cyclic frequency α can then be effectively highlighted. Thus， the CSD method based on kurtosis energy is proposed to extract the modulation frequency component that is submerged in the traditional CSD method. Accordingly， the energy accumulation of CSDK along the α-axis direction can obtain the demodulation spectrum of the signal and achieve the objective of fault diagnosis.

The implementation procedure of the CSDK method is described as follows.

Step 1: Suppose f ranges from1f tomf. Calculate the CSD of the analyzed signal. Slice perpendicular to the α axis， and then， calculate the kurtosis of each cyclic frequency slice（）:KAα

Step 2: Normalize all the kurtosis values. Calculate the coefficients that reflect the weight of the modulation capability of the frequency. Normalization is realized by Eq.（10）:

Step 3: Obtain CSDK （，）Sfα′in the weighted process using the aforementioned coefficients， as shown in Eq. （11）. So the multiple cyclic frequencies can then be eliminated orreduced.

Step 4: Calculate the energy accumulation of CSDK in the α axis. The result used for fault diagnosis is （）:SAα′

4 Experimental Validation

4.1 Simulation signal analysis

A set of outer race fault signals of the rolling bearing is simulated to verify the validity of the method. The fault frequency of the original signal is 8 Hz， whereas its natural frequency is 200 Hz. Based on these frequencies， Gaussian white noise is added to form a new mixed signal. The signal in the time and frequency domains is shown in Fig. 1. The results of the CSD and CSDK analyses are provided in Figs. 2 and 3， respectively.

Fig. 1 Simulation signal in the time and frequency domains

Fig. 2 CSD result of the simulation signal

As shown in Figs. 2 and 3， the CSDK method can effectively reduce multi-harmonic interference compared with the traditional CSD method. The CSDK method also clears the 3D spectrum and effectively extracts the fault frequency of 8 Hz and the natural frequency of 200 Hz. The results of the α axis of CSD and CSDK energy accumulation are shown in Fig. 4.

Fig. 3 CSDK result of the simulation signal

Fig. 4 Energy accumulation of CSD

As shown in Fig. 4， all the multi-harmonic components of the original spectrum are demodulated by CSD after energy accumulation， and the amplitude of each frequency is similar. Hence， accurately determining fault type is difficult. As shown in Fig. 5， CSDK energy accumulation can effectively reduce most of the multi-harmonic frequencies and easily extract the modulation frequency of 8 Hz.

Fig. 5 Energy accumulation of CSDK

4.2 Signal analysis of a low-speed and heavy-load test bed

4.2.1 Introduction to the experimental equipment

The experimental equipment is a low-speed and heavy-load test bed（Fig. 6）. This test bed is driven by a 370 W inverter motor reducer box. The heavy-load condition is achieved through a magnetic powder brake.

Electric discharge machining is performed to produce varying degrees of pitting corrosion on the replaceablebearing outer race， which presents a fault degradation trend with the increase in the number and the depth of corrosion pits. As shown in Table 1， No. 0 represents the bearing under normal condition. Meanwhile， Nos. 2 to 8 indicates that the bearings tend to degrade gradually.

Fig. 6 Low-speed and heavy-load test bed

Table 1. Machining scheme of the outer race fault bearings

Motor speed is set to 1450 r/min and the reducer box transmission ratio is 21.81. Hence， the rotating frequency of the output shaft is 1.1 Hz. The sensor is installed in the vertical direction of the bearing house. The PCI-II equipment is used to acquire AE signals of the bearings. The sampling frequency is set to 500 kHz and the sampling duration is 16 s. The AE signals in the time domain are shown in Fig. 7.

As shown in Fig. 7， the AE signals， along with the increasing the number and the depth of corrosion pits，appear to increase in significance. Nevertheless， the AE signals cannot be used directly to evaluate the fault degree because these signals are complex； therefore， further analysis is necessary.

4.2.2 Results and comparison

The resonance demodulation， CSD， and CSDK methods are used to analyze the AE signals.

Based on the signal spectrum， the analyzed results of the resonance demodulation method are presented in Fig. 8.

The rotating frequency of the output shaft is 1.1 Hz， so the fault frequency of the outer race is calculated as 3.4 Hz. As shown in Fig. 8， although the resonance demodulation method can distinguish the outer race faults for each fault level signal， identifying the severity of bearing failure is difficult because the fault frequency amplitude does not gradually increase with the fault degradation trend.

The energy accumulation results in the α-axis direction using the CSD and CSDK methods are as follows.

As shown in Figs. 9 and 10， the CSD method can reflect the existence of fault frequencies in the outer race with the increasing depth of pitting corrosion. However， the spectrum diagram is strongly influenced by other frequency components； thus， classifying fault type directly is difficult. Meanwhile， the spectrum obtained by the CSDK method is clearer than that obtained by the CSD method. The CSDK method cannot only effectively extract the fault frequency characteristics of the bearing outer race， i.e.， 3.359 Hz， but can also reflect the degradation trend of failure.

Fig. 7 AE signals of the outer race in the rolling bearing

In addition， 2.2 Hz is twice the rotating frequency， which indicates that an installation problem exists between the output shaft of the reducer box and the main shaft. This problem is attributed to the reduction ratio of the reducer box being exceedingly great， such that coupling breaks down under low-speed and heavy-load conditions.

4.3 Industrial signal analysis

The proposed method is experimentally validated by analyzing the signal gathered from the rolling bearing of the blast furnace belt cylinder. The type of rolling bearing is 3524. The belt operates at a rate of 2 m/s， and the speed ofthe bearing is 57.72 r/min. The bearing exhibits the characteristic of low-speed and heavy-load， and thus，fatigue wear and gluing faults always appear and directly affect blast furnace production. The fault frequency of the outer race is calculated as 8.17 Hz according to the above parameters.

Fig. 8 Resonance demodulation result of the AE signals in the test bed

The acoustic emission acquisition system made by USA PAC Company is used to collect AE signals. The sensor is placed at the vertical direction of the bearing house， as shown in Fig. 11， and the sample frequency is set to be 1 MHz. The collected signals of incipient and serious defects are shown in Fig. 12.

The two types of signals are analyzed using the resonance demodulation method， the traditional CSD method， and the proposed CSDK method. The results of the resonance demodulation method are provided in Fig. 13. Energy accumulation using the CSD and CSDK methods is shown in Figs. 14 and 15， respectively.

Fig. 9 Energy accumulation of signals using CSD

As shown in Fig. 13， the resonance demodulation method fails to identify the outer race fault， regardless of whether it is an incipient or a serious defect. As shown in Fig. 14， CSD is unable to identify the type of fault correctly because the fault frequencies are submerged in strong background noise. Compared with Fig. 15（a）， Figs. 13（a）and 14（a） show that CSDK can demodulate the fault frequency （i.e.， 8.4 Hz） when the bearing exhibits early defects. The result indicates that the proposed method has more advantages in recognizing incipient defects than conventional methods. In addition， as shown in Fig. 15（b），the energy accumulation of CSDK denotes the fault frequency of the outer race（i.e.， 8.345 Hz） and its harmonic frequencies， which can obviously diagnose the health condition of the bearing.

Fig. 10 Energy accumulation of signals using CSDK

Fig. 11 Placement of sensor

5 Conclusions

（1） A new method based on kurtosis energy， i.e.， CSDK，is proposed. Analyses of the simulation signal， the outer race fault signals of the bearing in the test bed， and the AE signal of the bearing in industrial environments are performed.

（2） The CSDK method can avoid the complexity and uncertainty of the resonance demodulation method in selecting the resonance band.

Fig. 12 AE signal in the time domain

Fig. 13 Resonance demodulation result

Fig. 14 Energy accumulation using CSD

Fig. 15 Energy accumulation using CSDK

（3） The proposed method overcomes the problem of the traditional CSD method regarding interference in multiple harmonic frequencies. This method can also effectively reflect the fault of the bearing and its degradation trend under the conditions of a strong background noise and complex fault mode.

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Biographical notes

LI Min， female， born in 1980， is currently an associate professor at School of Mechanical Engineering， University of Science and Technology Beijing， China. Her research interests include equipment fault diagnosis， signal processing， and pattern recognition. She has published over 20 articles， received 1 provincial award， and granted 5 patents.

Tel: +86-10-62332329； E-mail: limin@ustb.edu.cn

YANG Jianhong， male， born in 1978， is currently an associate professor at School of Mechanical Engineering， University of Science and Technology Beijing， China. His research interests include equipment fault diagnosis and advanced detection technology. He has published over 30 articles， received 1 provincial award， and granted 3 patents.

Tel: +86-10-62332329； E-mail: yangjianhong@me.ustb.edu.cn

WANG Xiaojing， female， born in 1989， is currently a postgraduate student at School of Mechanical Engineering， University of Science and Technology Beijing， China. Her research interests include equipment fault diagnosis.

Tel: +86-10-62334126； E-mail: xiaojingwang@aliyun.com

10.3901/CJME.2015.0522.074， available online at www.springerlink.com； www.cjmenet.com； www.cjme.com.cn

* Corresponding author. E-mail: limin@ustb.edu.cn

Supported by Beijing Higher Education Young Elite Teacher Project（Grant No. YETP0373）， and National Natural Science Foundation of China（Grant Nos. 51004013， 50905013）

? Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015

July 14， 2014； revised May 7， 2015； accepted May 22， 2015