Abstract
The autocorrelation function is combined with wavelet transform and cyclostationary theory (WTAFCT) in place of threshold denoising, and meanwhile the mean power ratio (MPR) is calculated by the proposed method. Furthermore, extracted characteristic as well as calculated MPR is used to identify compound faults of rolling bearings in aeroengine based on casing vibration acceleration signalincluding the ones of common rolling bearing (inner race rotates and outer race is constant) and intershaft bearing (corotates with outer and inner race). A comparative analysis was carried out between conventional researches (cyclostationary theory (CT) or wavelet transform combined with threshold value denoising (WTTD)) and proposed WTAFCT method. Additionally, the effect of sensors installation direction for feature separation and extraction of compound faults is considered. The results indicate that the proposed WTAFCT method can separate and extract characteristics of compound faults exactly and identify fault types of bearings precisely, no matter sensors are installed horizontally or vertically, while CT or WT cannot.
1. Introduction
The fault modes of rolling bearing are usually compound in real aeroengines due to the intercoupling among faults [1, 2]. On the contrary, more researches have been mainly concentrated on single fault rather than compound ones at present [3, 4]. The faults of rolling bearing would result in increased vibration and even lead to a flameout during flight because the rotor is locked, which would cause tremendous threat and damage to aeroengine [5, 6]. Therefore, it is crucial and essential to monitor the working conditions and identify the fault types of rolling bearings. The intershaft bearing, which corotates with outer race and inner race, is widely used in aeroengine at present [5]. The faults of intershaft bearing are probably frequent due to high operating temperature, insufficient lubrication and large dynamic load, etc. In 2007, Hu, et al. analyzed the corotating and counter rotating between inner and outer race of cylindrical intershaft bearing in aeroengine by quasi static method [7]. Liao, et al. designed an intershaft bearing experiment rig according to kinetic and structural features of dualrotor turbofan engine [8].
Rotate mode and symmetric or closely symmetric physical structure provides vibration signal of aeroengine with cyclostationary feature. Therefore, the condition monitoring and fault diagnosis in bearings of aeroengine can be studied in cyclostationary theoretical frame [9, 10]. On the one hand, cyclostationary theory reflects signal statistics varying with time and offsets the deficiency in the processing of cyclostationary signals; on the other hand, periodic changes of signal statistics are taken to simplify the treatment of normal nonstationary signals. The most representative study of cyclostationary theory is from J. Antoni [9, 11, 12]. Luo et al. used secondorder directional cyclostationarity to analyze complex signals from journal bearing which supports a rotor system operating with oil whip [13]. Based on chirplet path pursuit and order cyclostationary demodulation, Xu., et al. proposed a diagnosis method of roller bearing in 2013 [14]. In considering that fact that the effect of bearing fault diagnosis with maximum second order cyclostationary blind deconvolution (CYCBD) depends on the precision of characteristic frequency of choice and the length of filter, Huang Baoyu, et al. implemented fault diagnosis of bearings by proposing the method which optimizes CYCBD with Cuckoo search algorithm (CSA) and treats Improved maximum harmonic significance index (IHSI) as the basis of optimization [16]. Wu Bin, et al. proposed the calculation method for order ratio cyclostationary autocorrelation function, combined the demodulation function of cyclostationary analysis to vibration signals and continuous hidden Markov model (CHMM), and thereby proposed a fault diagnosis method for rolling bearings applicable to the operation with variable speed, and then verified the feasibility of method [17].
Wavelet transform is widely applied to identify fault weak and compound of bearings due to its timefrequency localization characteristics [15, 1820]. For example, Shen Zhengwei, et al. proposed a wavelet construction method based on fractional order and carried out the extraction of characteristics of bearing faults [21]. Liao Chuanjun, et al. analyzed the signal characteristics of acoustic emission (AE) and firstly proposed the analysis method of wavelet redistribution scale spectrum based on AE signals according to the extraction principles of fault characteristics of AE signals and features as a result of mechanical faults or injury, and then applied the method to the identification of injury type and parts of rolling bearings in acoustic emission inspection [22].
The applied studies of bearings have been improved by scholar studies, domestic and overseas. However, these studies, including CT and WTTD, are mostly concentrated on the signals from bearing block rather than casing signals, and a single fault than compound faults [36, 20]. Signal distortion, energy attenuation and noise effect make casing signal much more complex and weaker than the signal of bearing block, which is a kind of comprehensive and complicated response from many parts of aeroengine. Additionally, feature extraction of compound faults is far more difficult than that of single fault [3, 4]. That is, no matter traditional studies focus on the signals from bearing block or single fault, they will result in an extremely large gap between theoretical study and practical engineering application, which adds to the difficulty of bearing fault diagnosis in aeroengine.
Considering approximately symmetric physical structure and rotation way of aeroengine, the analysis of vibration signals of casing within the framework of cyclostationary theory can produce more precise status monitoring and fault identification of bearings. Autocorrelation function can maintain the periodicity of signal and reduce noise, and convert frequency modulation (FM) signal which usually exists in fault signal of bearing to amplitude modulation (AM) signal [10]. From the above, the autocorrelation function is introduced to wavelet transform and combined with cyclostationary theory (WTAFCT) for separating and extracting the characteristic of compound faults of rolling bearingincluding the ones of intershaft bearings and normal rolling bearings. Meanwhile, to identify the fault types of rolling bearing in more visual and simpler manners, the MPR is calculated by proposed WTAFCT method. The comparison analysis is implemented among CT, WTTD and WTAFCT. Finally, the effect of installation direction of sensors on feature separation and extraction is considered.
2. WTAFCT method
2.1. Wavelet transforms (WT)
Due to good local nature of wavelet transform, the frequency resolution is higher in low frequency and time resolution higher in high frequency. This just fits the nature of low frequency signals (slow change) and high frequency signals (fast change). Therefore, it is feasible to combine wavelet transform and cyclostationary theory to extract characteristics of compound faults and identify a fault type.
Suppose $y\left(t\right)$ is a continuous signal, and then wavelet transform is defined as the integral of product of signal $y\left(t\right)$ and basic wavelet function $\psi \left(t\right)$:
The scaling function is as follow:
where $j$ is the scale factor, and $k$ is the translation factor.
Then by the DWT (discrete wavelet transform) the wavelet coefficients of the signal can be defined by the following equations:
where ${a}_{{2}^{i}}\left(k\right)$ is a scale twolevel decomposition function of discrete wavelet transform; ${b}_{{2}^{i}}\left(k\right)$ is the position decomposition function of discrete wavelet transform.
2.2. Autocorrelation function (AF)
As the vibration signals of aeroengine are typical cyclostationary, they usually include modulation components when a bearing fault happens. For that, we can go through cyclic autocorrelation function analysis and take the slices of function under some cyclic frequencies for absolute value demodulation to solve modulation frequency of signals, and then implement the characteristic extraction and fault identification of compound faults.
Assuming one signal decomposed by wavelet transform is $Z\left(t\right)$, and $\tau $ is introduced to obtain, and t represents time, $N$ is length of$\mathrm{}Z\left(t\right)$:
Defined autocorrelation function ${R}_{z}\left(\tau \right)=x\left(t\right)$ and then ${R}_{x}(\tau ,\alpha )={R}_{x}^{f}\left(\tau \right)$ represents the cycleautocorrelation function of ${R}_{z}\left(\tau \right)$ when cycle frequency is equal to $\alpha $.
Given one amplitudemodulated signal ${R}_{z}\left(\tau \right)$:
$A$/$B$ respectively keeps the amplitude of carrier signal and modulating signal. ${f}_{n}$, ${f}_{z}$ remains respectively the frequency of carrier and modulation signal, so we can also conclude:
We can have Eq. (9) according to Eqs. (78):
It is very clear that cyclicautocorrelation function is not equal to zero when and only when cycle frequency $f=0$, $f=\pm {f}_{n}$, $f=\pm 2{f}_{n}$, $f=\pm 2{f}_{z}$, $f=\pm ({f}_{n}\pm 2{f}_{z})$ and $f=\pm (2{f}_{n}\pm 2{f}_{z})$. Suppose $f$ is selected and not equal to zero ($f=0\mathrm{}$represents stationary signal), the slice signal of cycleautocorrelation function is analyzed by given $f$ value. Additionally, the fault characteristics can be separated and extracted by obtaining frequency spectrum of slice signal.
2.3. Characteristic frequency of bearings
Considering the onetoone correspondence relationship between a fault type and characteristic frequency of bearings, the extracted feature frequency can be used to monitor running status and identify the fault type of bearings [5,6]. Supposing pitch diameter is$D$, rolling elements diameter and number is respectively $d$ and $Z$, and contact angle $\alpha $ is equal to zero. Rotating speed of outer race and inner race is respectively ${n}_{e}$ and ${n}_{i}$. If outer and inner race rotates in the same direction, the sign of ${n}_{e}$ and ${n}_{i}$ is the same, otherwise, opposite. If outer or inner race is constant, then corresponding ${n}_{e}$ or${n}_{i}$ is equal to zero. ${f}_{d}$ is the absolute difference of rotational speed between inner and outer race of bearings. The characteristic frequency of bearings can be expressed as follows:
Characteristic frequency ${f}_{i}$ of inner ring:
Outer ring ${f}_{o}$:
Rolling body ${f}_{b}$:
Characteristic frequency ${f}_{oc}$ of holder and outer ring in rubbing:
Characteristic frequency ${f}_{ic}$ of holder and inner ring in rubbing:
2.4. Mean power ratio (MPR)
A vibration signal, $g\left(n\right)$, and its Fourier transform, $g\left(k\right)$, is defined, then the mean square value which is mean power $E\left({g}^{2}\right)$ of $g\left(k\right)$ such that:
where $N$ represents the length of $g\left(k\right)$. Supposing a frequency band is given, then the mean power of the selected frequency band can be calculated according to Eq. (17). Meanwhile, the MPR $P\left({g}^{2}\right)$ of selected different frequency bands can be shown as follows:
where ${m}_{1}$, ${m}_{2}$, ${n}_{1}$, ${n}_{2}$ respectively represents the starting point and end point of selected different frequency band of $g\left(k\right)$, and $P\left({g}^{2}\right)$ is MPR of selected different frequency bands.
3. Compound faults characteristic separation and extraction
When a fault occurs in bearings, the amplitude of vibration signal will be modulated by characteristic frequency which matches the failure type of bearings. That is, to monitor the running state of bearing and identify the type of fault, the $f$ in Eq. (9) can be valued according to the characteristic frequency obtained by Eqs. (1014) or the double. Considering that the fault characteristic of bearings, in casing response, is weak and complex, cyclic statistic alone probably cannot identify the compound faults. Based on above consideration, a new method of WTAFCT is proposed and the corresponding MPR is calculated according to Eqs. (916) in order to separate and extract the characteristics of compound faults of bearings. Meanwhile the comparison analysis is carried out in conventional method (CT and WTTD) and WTAFCT. Specifically, the conventional research is shown in Fig. 1(a), and proposed WTAFCT method is shown in Fig. 1(b).
Specific process of each scheme is as follow:
(1) In Fig. 1(a), the concrete steps of characteristic extraction based on CT and WTTD are as follow:
1) CT method: for vibration signals from bearing block, directly make slick analysis with cyclic autocorrelation function within the framework of cyclostationary theory and extract fault characteristics based on the frequency spectrum of slice signals.
2) WTTD method: combine wavelet transform and threshold value denoising to carry out characteristic extraction and fault identification. Firstly, separate vibration signals of bearing block with wavelet transform; secondly, denoise approximate signals and detail signals as a result of wavelet decomposition based on wavelet threshold value denoising; finally, choose the frequency spectrum of approximate signals or detail signals of good extraction effect for characteristic extraction and identification of fault.
(2) In Fig. 1(b), specific steps of characteristic extraction with the proposed WTAFCT method are as follow:
Firstly, subject vibration signals of casing to timefrequency local decomposition based on wavelet transform; secondly, characteristic enhancement and denoising of detail signals and approximate signals after wavelet transform with autocorrelation function; thirdly, detail signals or approximate signals of better characteristic extraction effect are given slice analysis based on cyclic autocorrelation function; fourthly, calculate the frequency spectrum of slice signals. Meanwhile, for more intuitive analysis and judgment, we have calculated the average power of each characteristic frequency spot according to Eqs. (1516). At last, based on average power and frequency spectrum of slice signals, extract characteristics of compound faults of bearings and identify a fault type.
Fig. 1researches scheme for a) traditional research and b) proposed WTAFCT method
a)
b)
The special corresponding relationship among different schemes and methods is shown in Table 1.
Table 1the corresponding relationship among different scheme and method
Traditional research  Proposed new method  
Scheme A  Scheme B  Scheme C  Scheme D 
CT alone  WTTD  WTAFCT  MPR basis for WTAFCT 
4. Compound faults of intershaft bearing
Limited by the space, the compound faults of intershaft bearing are analyzed. The dynamical model of one birotor aeroengine is shown in Fig. 2. The vibration acceleration sensors are installed vertically and horizontally on the rear casing in aeroengine. Inner and outer race of intershaft bearing rotates in the same direction and the sample frequency is 16384 Hz. In the stage of test running of the aeroengine, the standard value of the slide oil spectrum overstepped restricted value, and meanwhile the monitored vibration value of highpressurerotor was increasing with rotational speedincluding horizontal and vertical direction of rear casing, which proves to be inconsistent with the previous test running of the model aeroengine. To seek for a fault cause, casing vibration acceleration signal is analyzed. The specific geometry parameter of intershaft bearing is shown in Table 2. The ${f}_{r}$, ${f}_{oc}$, ${f}_{ic}$, ${f}_{o}$, ${f}_{i}$ and ${f}_{b}$.
Fig. 2Rotor supporting dynamical model of one birotor aeroengine
Table 2the geometric parameter of intershaft bearing
Rolling elements diameter  Rolling elements numbers  Pitch diameter  Inner race radius  Outer race radius 
18.06 mm  28  212.76 mm  97.35 mm  115.41 mm 
4.1. Case study
4.1.1. Cyclostationary theoryscheme A
To separate and extract the compound fault characteristics of intershaft bearing, the slice signal of cyclic autocorrelation function, on different cyclic frequency positions, is analyzed. The rotational speed of highpressure and lowpressure rotor is respectively 13260 r/min and 3810 r/min, and the corresponding rotational frequency proves respectively 221 Hz (13260/60) and 63.5 Hz (3810/60), respectively. The feature frequency of inner race, outer race and rolling element of intershaft bearing is respectively equal to 2562.2 Hz, 2162.8 Hz and 924.8 Hz by calculation according to table.2 and Eqs. (1012). The feature frequency of retainer rubbing outer race and retainer rubbing inner race is respectively 72.1 Hz, 85.4 Hz by calculation according to Table 2 and Eqs. (1314). Limited by space, the vibration acceleration signal of horizontal direction is analyzed firstly. The results are displayed in Fig. 3. The Fig. 3(a) is the timedomain of original casing vibration acceleration signal. The Fig. 3(b)(d) is frequency spectrum and its local amplification. Fig. 3(e) is the slice signal of cyclic autocorrelation when timedelay is equal to zero. Fig. 3(f) represents threedimensional slice graph of cyclic autocorrelation, and $\mathrm{x}$\$\mathrm{y}$\$\mathrm{z}$ axis is respectively cyclic frequency\time delay\vibration amplitude.
Analyzing Fig. 3, the following conclusions can be drawn:
(1) There exists obvious rotational frequency (221.0 Hz) and its double frequency (442.0 Hz) of highpressure rotor in Fig. 3(b)(d); meanwhile, there is weak rotational frequency of lowpressure rotor (63.5 Hz).
(2) By further analysis, the weak frequency component 71.3 Hz and 142.3 Hz can be observed in Fig. 3(d). These 2 frequency components are approximately equal to the feature frequency of outer race rubbing against retainer of intershaft bearing (72.1 Hz) and its double. However, compared with other frequency components in the spectrum, these characteristics are very weak.
(3) In the analysis of Fig. 3(e)(f), it can be observed that no matter in slice signal or threedimensional slice graph of cyclic autocorrelation function, there is no obvious feature frequency of intershaft bearing.
Namely, cyclic autocorrelation function alone cannot separate and extract the characteristic frequency and identify the compound fault types of intershaft bearing because of the weakness and complexity of fault characteristic in casing response of aeroengine.
Fig. 3a) timedomain signal, b)d) frequency spectrum, e) slice signal, f) threedimensional slice graph
4.1.2. Wavelet transform is combined with threshold denoisingscheme B
Wavelet transform is used to extract the features of compound faults of intershaft bearing, and the data selected is the same with section 4.1.1 and the results are shown in Fig. 4. Considering sym Nseries of wavelets are featured by orthogonality, compactness and approximate symmetry, sym6 wavelets are selected to actualize signal separation and the number of decomposition layers is 5. The unifiedthreshold denoising method is selected. Fig. 4(a) is the approximate signal a5 (the optimal is selected) after denoising and Fig. 4(b) is the frequency spectrum of Fig. 4(a).
In Fig. 4(b), there exists obvious frequency 221.0 Hz and 63.5 Hz, which respectively matches the rotational frequency of highpressure and lowpressure rotor. Meanwhile, there exists the frequency component 291 Hz, and it is exactly equal to the sum of rotational frequency (221 Hz) of highpressure rotor and feature frequency (72.1 Hz) of retainer rubbing outer race (221+72.1 = 293.1 Hz $\approx $ 291). It can identify the fault of outer race rubbing retainer in intershaft bearing, while the feature frequency of inner race rubbing retainer cannot be observed.
Fig. 4Approximate signal a5 and its frequency spectrum
4.2. WTAFCY: proposed new methodshown in Fig. 1(b)
To separate and extract the characteristic frequency of compound faults, and meanwhile identify and judge the fault types of intershaft bearing precisely, the autocorrelation function is introduced to wavelet transform and combined with cyclostationary theory. For a comparison, the data, wavelet function (sym6) and selected decomposing level (5) is the same with section 3.1. The specific steps of proposed method are shown in Fig. 1(b) marked by blue. The results are displayed on Fig. 5.
Fig. 5a)b) timedelay slice signal, c)f) threedimensional slice graphs: approximate signal a5 based on WTAFCY method
Fig. 5(a)(b) is the slice signal of cyclic autocorrelation function which corresponds with autocorrelation function of reconstructed approximate signal a5. Fig. 5(c)(f) represents threedimensional slice graphs and its local amplification by proposed WTAFCY method, and $\mathrm{x}$\$\mathrm{y}$\$\mathrm{z}$ axis respectively represents cyclic frequency, time delay signal and vibration amplitude. Fig. 5(a)(b) is respectively corresponding with cyclic frequency –2000 Hz~2000 Hz and 0 Hz~800 Hz. Fig. 6(c)(f) is respectively corresponding with cyclic frequency 0 Hz~3000 Hz, 0 Hz~800 Hz, 0 Hz ~100 Hz and 600 Hz~800 Hz.
By precisely analyzing Fig. 5, the following conclusions can be drawn:
In Fig. 5(a)(f), no matter in the timedelay slice signals or in threedimensional slice graphs based on proposed WTAFCY method, it can be observed that there is outstanding frequency 70.1 Hz, which is approximately equal to the feature frequency of retainer rubbing against outer race. After further analysis of Fig. 5(a)(f), the outstanding frequency 512.2 Hz can be found, which just matches the 6multiplication frequency of retainer rubbing inner race (85.4 Hz) of intershaft bearing.
It can be concluded that the intershaft bearing of aeroengine has faults, and the fault types are supposed to include retainer rubbing inner race and retainer rubbing outer race. Further studies are implemented in order to verify the correctness of analysis based on proposed WTAFCY method, and the frequency spectrum of slice signal is shown in Fig. 6. Fig. 6(a)(d) is respectively matching with timedelay slice signal and their frequency spectrum, and the slice positions are corresponding with cyclic frequency equal to retainer rubbing outer race feature frequency and its double (the result is similar if we select cyclic frequency equals to retainer rubbing inner race feature frequency or twice).
Fig. 6a)d) Timedelay slice signal and their frequency spectrum – approximate signal a5: for a), c) cyclic frequency equal to foc, for b), d) cyclic frequency equal to 2foc
Analyzing Fig. 6(a) and (c), there exists outstanding frequency 255.6 Hz which is just corresponding with triple frequency of feature frequency of retainer rubbing inner race (85.4 Hz). Meanwhile, in Fig. 6(b) and (d), there is the frequency 143.0 Hz that matches with double frequency of feature frequency of retainer rubbing outer race (72.1 Hz).
Namely, the intershaft bearing may have the compound faults of inner race rubbing against retainer and outer race rubbing against retainer. To further prove the accuracy of analysis, we disassembled the aeroengine and the picture of intershaft bearing is as shown in Fig. 7.
By carefully observing Fig. 7, it can be found that there is abrasion trace founded in the retainer with inner race and outer race of intershaft bearing. It is consistent with the results according to singlechannel casing vibration acceleration signal based on proposed WTAFCY method. That is, the compound faults characteristic can be separated and extracted; moreover, compound fault types of intershaft bearing of aeroengine can be identified correctly and effectively based on proposed WTAFCY method, while the traditional researchincluding CT and WTTD cannot.
Fig. 7Abrasion diagram of intershaft bearing
a)
b)
4.3. The effect of sensor installed direction
In order to verify the sensitivity of WTAFCY method to sensor installation direction when separating and extracting compound faults of intershaft bearing, the data selected was collected vertically at the same moment with section 3.1. The same wavelet function (sym6) and decomposition level (5 levels) is chosen, and the results are shown in Fig. 8. Fig. 8(a) is original time domain signal. Fig. 8(b)(c) is the timedelay slice signal of cyclic autocorrelation function. Fig. 8(d)(g) is the timedelay slice signal and frequency spectrum when cyclic frequency is equal to retainer rubbing outer ring and its double frequency (the result is similar in the case that the feature frequency of retainer and inner race or its double is selected)
Analyze Fig. 8 that is corresponding with sensor installed in vertical direction of rear casing of aeroengine and compare it with section 3.2 (sensor installed in horizontal direction). The following conclusions can be drawn:
(1) In Fig. 8 (b), (c), (e) and (g), when the data from the sensor in vertical direction is selected, there exists highlighted frequency components, 70 Hz and 145 Hz, in the slice signals of cyclic autocorrelation function. It is consistent with sensor installed in horizontal direction.
(2) In Fig. 8(bd) and (f), there exists outstanding frequency components 256 Hz and 512 Hz. They correspond to the 3 time and 6time of feature frequency (85.4 Hz) of retainer rubbing against inner race.
Namely, no matter with the signals collected by sensors in horizontal or vertical direction, the proposed WTAFCY method can equally identify the compound faults of intershaft bearing and the results of horizontal and vertical directions are similar (except the difference in amplitude).
4.4. MPR: a more visual way to identify
In order to identify fault types of bearings in simpler and more visual ways, the MPR is calculated by WTAFCY method according to Eqs. (1516) and Fig. 6, Fig. 8. The results are exhibited on Fig. 9 and Table 3.
The Fig. 9 (a)(b) respectively represents sensor installation in horizontal and vertical direction. The abscissa 1\2\3\4\5\6 respectively represents feature frequency band of rotational speed, retainer rubbing outer race, retainer rubbing inner race, outer race, inner race and rolling element. The feature frequencies bands are taken from subtract 5 Hz to add 5 Hz on the basis of calculated frequency according to Eqs. (1016), in the consideration that there is a deviation between theoretical calculation and practice obtaining. For example, suppose that certain characteristic frequency is equal to MHz by calculation, and then the corresponding frequency band is selected from ($\mathrm{M}$5) Hz to ($\mathrm{M}$+5) Hz, and double characteristic frequency band is selected from (2×$\mathrm{M}$5) Hz to (2×$\mathrm{M}$+5) Hz and so on. The most representative characteristic frequency band of every feature frequency is selected in this paper. The ordinate represents calculated MPR between the selected characteristics frequency band and total frequency band (0 Hz4000 Hz).
Fig. 8a) time domain, b)c) slice signals, d)g) time delay slice signalsapproximate signal a5 for d), e) cyclic frequency equal to foc for e), g) cyclic frequency equal to 2foc – vertical direction
After analyzing Fig. 9 and Table 3, the following conclusions can be drawn:
It is observed from the Fig. 9(a)(b) and Table 3, the calculated MPR is outstanding only in the positions where the feature frequency bands of retainer rubbing against outer and inner race are located, which is corresponding with 2 and 3 of $x$axis. The other MPRs obtained (the feature frequency bands of rotate frequency, inner race and rolling element) are not obvious.
Fig. 9MPRproposed WTAFCY method
a) Different characteristic frequency band Sensor installation on horizontal direction
b) Different characteristic frequency band Sensor installation on vertical direction
Table 3Calculated MPR based on WTAFCY method
Direction  Feature frequency  
${f}_{r}$  ${f}_{oc}$  ${f}_{bc}$  ${f}_{o}$  ${f}_{i}$  ${f}_{b}$  
Horizontal  2.42  37.7  45.4  0.06  0.06  0.09 
Vertical  2.43  33.4  45.6  0.07  0.06  0.10 
4.5. Common rolling bearing: proposed method
different compound faults (inner race and rolling elements) is selected for verifying the accuracy of WTAFCY method in common rolling bearings. The faulted rolling bearing is displayed on Fig. 10 and its geometric parameters are: the rolling elements diameter and numbers is respectively 9.6 mm and 7, and the pitch diameter is 36 mm. The rotational speed is 1492.5 r/min, and the rotational frequency is equal to 21.22 Hz. The feature frequency of retainer, rolling element, outer race and inner race of rolling bearing is respectively equal to 9.1 Hz, 43.3 Hz and 110.3 Hz by calculation according to Eqs. (1012). The same wavelet function (sym6) and decomposition level (5 levels) is chosen, and the results are shown in Fig. 11 and Table 4. Fig. 11(a) is time domain signal. Fig. 11(b)(c) is frequency spectrum of the timedelay slice signal when cyclic frequency is respectively equal to the feature characteristic of rolling elements and inner race. The Table 4 is the function value of calculated MPR by proposed method.
Fig. 10Faulted rolling bearing
Table 4Calculated MPR based on WTAFCY method
Frequency horizontal  ${f}_{r}$  ${f}_{oc}$  ${f}_{bc}$  ${f}_{o}$  ${f}_{i}$  ${f}_{b}$ 
Vertical  21.2  –  –  3.03  131.2  61.44 
After analyzing Fig. 11(b)(c), it can be found that the outstanding fault feature frequency is 112 Hz which is corresponding with feature frequency (110.3) of inner race and 178.9 Hz which matches 4times of rolling elements feature frequency (43.3 Hz). Therefore, compound faults of rolling bearing can be identified.
The results are similar in Fig. 11(d) and Fig. 59, the calculated MPR is outstanding only in those positions of the rotational speed, inner race and rolling element characteristic frequency bands, which is corresponding with 1, 5 and 6 of $\mathrm{x}$axis.
That is to say, no matter for common rolling bearings or intershaft bearings of aeroengine, the results of characteristic extraction of compound faults remains the same based on proposed method.
Fig. 11a) Time domain of common bearing, b)c) slice signals, cyclic frequency equal to fi and fb, d) MPR – vertical direction
5. Conclusions
The autocorrelation function is combined with wavelet transform and cyclostationary theory to identify the compound faults including the ones of intershaft bearings and common rolling bearings. Meanwhile, the MPR is calculated in order to identify the fault type of bearings in simpler and more visual ways. The following conclusions can be drawn:
1) The cyclic autocorrelation alone and wavelet transform combined with threshold value denoising cannot extract the fault characteristic frequency of intershaft bearings or common rollings.
2) The proposed method can not only extract and separate the compound faults feature precisely, but also can identify the compound fault type of intershaft bearing as well as common rolling bearings correctly and effectively.
3) The proposed method is insensitivity to installation direction of sensors. No matter acceleration sensors are installed vertically or horizontally, the proposed method can extract the characteristic frequency of bearing correctly.
4) The calculated MPR can also identify the compound faults type of intershaft bearings and common rolling bearings in more visual and simpler way no matter acceleration sensors are installed vertically or horizontally.
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 51605309), Natural Science Foundation of Liaoning Province (Grant No. 2019ZD0219), Aeronautical Science Foundation of China (Grant No. 201933054002) and Department of Education of Liaoning Province (Grant No. JYT19042).
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