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Meshing frequency modulation (MFM) index-based kurtogram for planet bearing fault detection
Journal of Sound and Vibration 432 (2018) 437e453
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Meshing frequency modulation (MFM) index-based kurtogram for planet bearing fault detection Tianyang Wang a, Fulei Chu a, *, Zhipeng Feng b a b
Department of Mechanical Engineering, Tsinghua University, Beijing 100084 China School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 November 2017 Received in revised form 20 June 2018 Accepted 20 June 2018
Identifying the bearing fault-induced impulsive components in the frequency domain is a key step in the corresponding fault detection. However, gearbox vibration signals often significant interrupt the rolling bearing fault diagnosis, particularly in the detection of planet bearing faults under the background noise of the planetary gearbox. Except in the case of a high amplitude, a gear meshing-related vibration may also affect the identification of the planet bearing fault-induced resonance frequency band. To solve this problem, a meshing frequency modulation index (indexMFM)-based kurtogram utilizing a particular gearbox related phenomenon is proposed. The underlying mechanism is such that although the gear meshing-related spectral components are always more prominent in relatively higher-frequency band than the planet bearing-induced resonance frequency band in impulsiveness, the gear meshing-related impulsive components modulate the gear meshing frequency, yet the faulty bearing-induced one does not. Exploiting this difference, the planet bearing fault-induced impulsive components can be directly identified from the strong gear vibration interruption by determining the bearing fault-related resonance frequency band in the indexMFM-based kurtogram. The effectiveness of the proposed method is separately verified using simulated and experimental data. © 2018 Elsevier Ltd. All rights reserved.
Handling Editor: Z Su Keywords: Planet bearing Fault diagnosis Meshing frequency modulation Meshing frequency modulation index-based kurtogram
1. Introduction In contrast to general rolling bearings, the planet bearing has several unique features due to the complex structure of the planetary gearbox [1]. This intricate structure, which provides the advantages of a large transmission ratio, a strong loadbearing capacity, and lightness, affects the corresponding planet bearing fault detection [2]. First, the unique structure of the planetary gearbox makes the operating mode of the planet bearing different from the traditional one [3]. On one hand, the planet bearing not only rotates with its shaft as the center but also orbits around the axis of the sun gear shaft. On the other hand, the inner race of the planet bearing stands still with its shaft, and the outer race rotates together with the planetary gear. In this way, the formula for the planet bearing FCF changes at the same time. To solve this problem, Jain [4] and Feng [3] deduced the corresponding planet bearing FCF through the dynamic model and numeral model separately and predicted the corresponding spectrum pattern with a faulty planet bearing. Wang et al. [2] analyzed the sideband patterns around the FCF in the envelope spectrum with different fault positions.
* Corresponding author. E-mail addresses: [email protected] (T. Wang), chufl@mail.tsinghua.edu.cn (F. Chu). https://doi.org/10.1016/j.jsv.2018.06.051 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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Nomenclature indexMFM Meshing frequency modulation index fs Sampling frequency fmeshing Meshing frequency l Decomposition level fbandwidth Frequency bandwidth in determining indexMFM N Length of filtered result fplanetarygear Planetary gear rotational frequency fcarrier Carrier rotational frequency b Structural damping characteristic ti Random slippage Tp Time interval between fault-induced impulses Tp,outer Time interval between outer race fault-induced impulses Tp,inner Time interval between inner race fault-induced impulses A m,bearing Amplitude of mth impulse-induced by localized planet bearing fault ur,bearing Resonance frequency excited by faulty bearing fcfp,outer Fault characteristic frequency(FCF) of planet bearing outer race fault fcfp,inner FCF of planet bearing inner race fault Z Number of rolling elements d Diameter of rolling element D Pitch diameter 4 Contact angle Ak,gear kth impulse-induced by gear meshing ur,gear Higher frequency that modulates meshing frequency Tp,gear Time interval between two adjacent gear-induced impulses
In addition to changing the operating mode, the planetary gearbox meshing vibration usually interrupts the extraction of the planet bearing fault features in both the time and frequency domains. In the time domain, the planet bearing fault-induced impulses are always buried in the gear meshing vibration owing to the long and time-varying distance between the fault position and the sensor mounted on the outer case. The gear noise can be considered as one of the common interruptions of the bearing fault detection [5]. Algorithms designed to eliminate the timeinvariant components, such as linear prediction [6], self-adaptive noise cancellation [7], discrete random separation [8] and time synchronous averaging [9], have been used to eliminate the gear meshing noise. However, in the case of the planetary gearboxes the effectiveness of these elimination algorithms is unsatisfactory. First, the significantly rich sidebands around the meshing frequency and its harmonics can be readily identified even in the absence of faults, which affects the effectiveness of the traditional gear noise elimination algorithms. Furthermore, the impulses caused by the planet bearing fault are very weak compared with the gear interruption. Thus, the traditional algorithms may interrupt the planet bearing part when they are applied for gear noise elimination. To solve this problem, Fan [10] directly mounted an internal sensor on the carrier, which can measure the planet bearing vibration with less gear interruption. Although this strategy can partly highlight weak planet bearing fault-related vibration, the corresponding internal sensor cannot be installed on every planetary gearbox in real engineering owing to cost or design limitations. In the frequency domain, the planetary gear meshing noise can influence the planet bearing fault detection by interrupting the identification of the planet bearing fault-induced resonance (PBFR) frequency band. In general, the recognition of the optimal bearing fault-induced resonance frequency band is essential for the final diagnosis, because it allows the prominent FCF and its harmonics to be identified in the corresponding envelope spectrum of the bandpass filtering result. To determine the resonance frequency band, several algorithms [11e14] have been proposed, among which the spectral kurtosis (SK) [13] based methods are the most widely used and effective. However, as in the case of the planetary gearbox, the gear meshing vibration also leads to impulsive components in the higher frequency band, and these frequency components are always more evident than the planet bearing fault-related resonance. Hence, it is difficult to identify the planet bearing-induced resonance frequency bands directly using traditional algorithms. In response to this problem, Wang [2] proposed an SKRgram based demodulation technique, in which the kurtogram of the monitored planetary gearbox in a healthy condition is used as a baseline, and the planet bearing fault-induced resonance vibration can be located by calculating the ratio of the kurtogram of the faulty signal to the baseline. Nevertheless, the health baseline is not always available, and it is challenging to obtain an optimal health baseline for different operating conditions. According to the above analysis, it is difficult to realize planet bearing fault detection without auxiliary devices, such as an internal sensor or the health baseline. That is, it is challenging to highlight the planet bearing fault component from the strong
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gear noise without any prior knowledge. As such, a new algorithm is proposed in this paper for detecting planet bearing faults without auxiliary equipment or a health baseline. In this algorithm, a particular characteristic of the gear meshing vibration is utilized to select the bearing fault-induced impulsive vibration in the frequency domain. The principle is that the impulsive frequency band induced by the gear meshing modulates the meshing frequency, and the resonance frequency band resulting from the planet bearing fault does not modulate the meshing frequency at all. In real applications, the kurtogram of the raw signal is first calculated to determine how the impulsiveness is distributed in different frequency bands and levels. It should be mentioned that some of these impulsive frequency bands are caused by the planet bearing fault, and some of them are resulted from the gear meshing. A meshing frequency modulation (MFM) index called indexMFM is then constructed to determine which impulsive bands in the corresponding kurtogram are related to the meshing vibration. Furthermore, the potential frequency bands related to the planet bearing fault, which is called potential PBFR frequency bands, are selected from the indexMFM-based kurtogram, which is obtained by setting the SK values of particular frequency bands to zero if the corresponding indexMFM is lower than a selected threshold. Finally, the raw signal is filtered with the PBFR frequency bands, and the envelope spectra of the filtering results are used to perform final fault detection of the planet bearing. The remainder of the paper is structured as follows: Section 2 specifies the proposed algorithm, which includes obtaining the raw kurtogram, calculating the meshing index, determining the potential planet bearing fault-related band according to the IndexMFM-based kurtogram, and detecting the planet bearing fault with the envelope spectrum. Sections 3 and 4 testify the effectiveness of the proposed algorithm with simulated and experimental cases, respectively. The conclusion is drawn in Section 5. 2. Proposed planet bearing fault diagnosis method As previously explained, one of the major difficulties of planet bearing fault detection without any prior knowledge lies in distinguishing the planet bearing fault-related resonance frequency band from the strong gear meshing vibration noise. This is because the resonance component induced by the planet bearing fault is too weak to be recognized from more prominent gear meshing-related components. First, an example is presented to verify this statement with figures in Ref. [2]. Fig. 1 (a) and (b) show kurtograms of healthy planetary gearbox vibration and vibration with a planet bearing outer race fault under the same operating environment, respectively. It can be seen that the kurtograms obtained under different working conditions are almost the same. The frequency bands with prominent SK values are mostly induced by gear meshing vibration. Hence, it is challenging to select PBFR frequency band directly according to the kurtogram shown in Fig. 1(b). To solve this problem, a distinctive characteristic of the gear meshing vibration-induced impulsive frequency bands is utilized. Although the gear meshing frequency can lead to concentrated frequency bands with higher SK values in the kurtogram, these gear-related frequency bands modulate the meshing frequency. In contrast, the BPFR frequency bands do not modulate the gear meshing frequency. Exploiting this difference, an indexMFM-based kurtogram is proposed to select the planet bearing fault-induced frequency band directly. The corresponding algorithm is introduced in the following three parts. 2.1. Obtaining distribution of impulsiveness based on raw kurtogram In this section, a traditional fast kurtogram algorithm [15] is used to analyze the raw vibration. The raw signal is first decomposed with a binary tree structure and then extended to a 1/3-binary tree structure. Next, the SK values of each filtered results are calculated to form a kurtogram. During the formulation of the kurtogram, the decomposition level should be set with the certain value with the limitation given by the following equation:
k j . l ¼ fs =2 fmeshing
Fig. 1. Kurtograms of planetary gearbox vibrations: (a) healthy condition, (b) faulty condition with the planet bearing outer race fault [2].
(1)
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where, P:R is the rounding-down operator, fs is the sampling frequency, and fmeshing is the meshing frequency of the monitored planetary gearbox. The rationale for setting the decomposition level according to Eq. (1) is that this equation can ensure that the frequency spans of the envelope spectrums of all the filtered results contain the planetary gear meshing frequency. Only under this condition, the indexMFM proposed in Section 2.2.1 can be calculated. In detail, the IndexMFM is constructed based on the magnitudes of the spectral lines at the meshing frequency and its harmonics in the envelope spectrum of all the filtered results. This strategy will not be processed if one of the frequency spans of all the filtered results is smaller than the meshing frequency. In other words, the smallest frequency span of all the filtered results should be larger than or equal to the meshing frequency. If the decomposition level is set as l, for instance, the smallest frequency span of the filtered results equals to fs/2 l. Hence, the fs/2 l should be larger than the meshing frequency, which gives the decomposition level an upper limitation of fs/2/fmeshing. 2.2. Construction of indexMFM-based kurtogram In the raw kurtogram of the planetary gearbox with a planet bearing fault, most of the frequency bands with a relatively high SK are induced by the gear meshing vibration rather than the bearing fault. Therefore, it is difficult to determine the optimal PBFR frequency band according to the traditional kurtogram. In this section, an indexMFM is proposed for determining which frequency bands with relatively high SK values are related to the planetary gearbox meshing vibration. An indexMFMbased kurtogram is then constructed to determine the potential frequency bands induced by the planet bearing fault. 2.2.1. Construction of indexMFM As previously mentioned, if a frequency band in the kurtogram with a relatively high SK value is induced by the gear meshing vibration, it modulates the meshing frequency. Consequently, the meshing frequency and its harmonics are prominent in the corresponding envelope spectrum. However, the frequency bands induced by the planet bearing fault do not modulate the meshing frequency at all. Exploiting this difference, a new criterion called the MFM index is proposed in this subsection for evaluating the prominence of the meshing frequency and its harmonics in an envelope spectrum. Supposing that x(t) of length N is the filtered result of the raw signal with a certain frequency band, the envelope spectrum of x(t) is denoted by ES(f). The discrete version of x(t) and ES(f) are x[n] and ES[k] respectively, where n, k ¼ 1,2,3, …, N. The MFM index is defined as, NH X indexMFM x½n; fmeshing ¼ i¼1
1 RankES½k
!
(2)
ES½k max jkDf ifmeshing jdf
where NH represents the harmonic order whose value could equal to 1, 2, or 3 according to the frequency range of ES[k]. In detail, if the upper limit of the frequency range of ES[k] is larger than 3 fmeshing, NH equals to 3. If the upper limit of the frequency range of ES[k] is larger than 2 fmeshing and smaller than 3 fmeshing, NH equals to 2. If the upper limit of the frequency range of ES[k] is larger than fmeshing and smaller than 2 fmeshing NH equals to 1. Df is the frequency resoES½k denotes the largest envelope spectral magnitude of ES[k] in the frequency range of [i fmeshing - df, lution. max jkDf ifmeshing jdf i fmeshing þ df ]. In fact, max ES½k represents the magnitude of the peak at the meshing frequency. Here, a frequency jkDf ifmeshing jdf interval, [i fmeshing - df, i fmeshing þ df ], is used for extracting the real meshing frequency from the envelope spectrum because the rotational speed cannot be a constant value in real engineering which may result in a difference between the theoretical and real meshing frequencies. The allowable frequency deviation df is set as 5 Hz in this paper. RankB[k](b) is a newly defined function, where B[k] is a data series and b is a member of B[k]. The RankB[k](b) represents the ranking number of b of B[k] after all the data in B[k] are sorted in a descending order by magnitude. Supposing that B[k] equals to [20, 56, 2, 8, 30, 100] and b equals to 30 which is a member of B[k]. Just because 30 is the third largest value of B[k], RankB[k](b) equals to 3. ! ES½k represents the ranking number of max ES½k in the ES[k] after all the data in max jkDf ifmeshing jdf jkDf ifmeshing jdf ES[k] are sorted in a descending order by magnitude. In real applications, if the meshing frequency and its harmonics are prominent in the envelope spectrum (ES[k]) of the filtered result (x[n]), the corresponding ranking numbers are relatively low, leading to a higher indexMFM. Thus, a higher indexMFM indicates that the corresponding frequency band is closely related to the gear meshing frequency.
Hence, RankES½k
2.2.2. IndexMFM-based kurtogram construction As indicated by the above analysis, the gear meshing vibration always interrupts the extraction of the PBFR frequency band because the gear meshing vibration-induced frequency bands in the raw kurtogram are more prominent than those caused by the planet bearing fault. In light of this problem, a unique characteristic is explored, the impulsive frequency bands related to the gear meshing vibration modulates the meshing frequency, while the planet bearing fault-induced frequency bands do not
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modulate the meshing frequency. Exploiting this characteristic, an indexMFM is proposed and used as a criterion to determine which frequency bands in the raw kurtogram are related to the gear meshing vibration. By eliminating these gear meshing vibration related frequency bands, the resonance frequency bands induced by the planet bearing fault can be selected from the rest of the frequency bands easily, without the gear noise interruption. In detail, an indexMFM-based kurtogram is constructed to determine which frequency bands are related to the planet bearing faults directly, via the following four-step strategy: STEP1: The indexMFM values of all the filtered results obtained in Section 2.1 are calculated. In this step, the raw signal is first filtered with the 1/3 binary tree filter banks, and then the indexMFM values of all the filtered results are calculated using the algorithm proposed in Section 2.2.1. STEP2: The median of all the indexMFM values is set as the threshold. As mentioned previously, the indexMFM is proposed to determine which frequency bands in the raw kurtogram are related to the gear meshing vibration. Hence, if an optimal indexMFM threshold can be properly selected, the frequency bands whose indexMFM is larger than the selected threshold are related to the gear meshing vibration and should be eliminated as interruptions. The PBFR frequency bands can be selected from the rest of the frequency bands easily without the gear vibration interruption. However, it is challenging to select the best threshold to perfectly divide the two types of frequency bands, because there is no clear boundary between them. With a threshold that is too high, the algorithm cannot eliminate the gear noise, and with a threshold that is too low, the algorithm may eliminate the PBFR frequency bands. To address this issue, a statistical value that can reflect the distribution of the indexMFM values may be effective. Considering that the distribution of the indexMFM values can reflect how the meshing frequency is modulated to different frequency bands, the characteristics of the MFM must be introduced as a background before the threshold setting. Specifically, the meshing frequency is modulated to a relatively concentrated area rather than all of the frequency bands as observed in Fig. 1(a). In fact, the higher indexMFM values are concentrated in a few frequency bands, and most of the indexMFM values are relatively small. Therefore, a competent threshold should guarantee the elimination of the gear interruption and reflect the average distribution of all the indexMFM values simultaneously, which makes the median a suitable candidate. First, the median of all the indexMFM values is definitely smaller than the prominent indexMFM values in the raw kurtogram, which can ensure the elimination of the gear interruption. Second, the median can reflect the average distribution of all the indexMFM values in some level. Hence, the median of all the indexMFM values is selected as the threshold in this section to guarantee that most of the gear meshing vibration related frequency bands are separated from the ones caused by the planet bearing fault. It should be mentioned that the median of all the indexMFM values is not always the best threshold. It is a relatively conservative threshold that can guarantee that most of the gear meshing vibration related frequency bands with prominent SK values in the raw kurtogram can be identified. STEP3: All of the SK values of the raw kurtogram are edited according to following rule: With the algorithms described above, all of the filtered results have their own SK value and indexMFM value. If the indexMFM value of a filtered result is larger than the threshold determined in STEP 2, the edited SK value is set as zero, because a high indexMFM value indicates that the corresponding frequency band is closely related to the gear meshing vibration. In contrast, if the index value is smaller than the threshold, the edited SK values should be equal to the original SK value of the raw kurtogram. STEP 4: The kurtogram with edited SK values is considered as the indexMFM-based kurtogram. In the aforementioned four-step strategy, most of the frequency bands related to the gear meshing vibration are located, and the corresponding SK values are set as zeros. Hence, the remaining frequency bands may contain the PBFR frequency bands. A flowchart of the four-step strategy is presented in Fig. 2. 2.3. Determination of planet bearing fault-induced optimal resonance frequency band and accomplishment of fault detection In the indexMFM-based kurtogram, most of the SK values of frequency bands related to the gear meshing vibration are set as zero. In this way, the PBFR frequency band remains. Hence, it is easier to identify the optimal resonance frequency bands induced by the planet bearing fault from the indexMFM-based kurtogram than from the raw kurtogram. The particular steps are listed as follows: First, the areas with concentrated high SK values are identified from the indexMFM-based kurtogram and considered as the potential optimal PBFR frequency areas. Then, the frequency bands with the highest SK values are selected from each potential optimal PBFR frequency area and considered as the potential filter bands. Furthermore, the raw signal is filtered with these potential filter bands, whereby several envelope spectra are calculated. Finally, the obtained envelope spectra are compared with the planet bearing envelope fault pattern constructed in Ref. [2] to perform the final planet bearing fault detection. A flowchart of the proposed method for planet bearing fault detection without any auxiliary equipment or prior information is presented in Fig. 3, showing the aforementioned algorithm elements. It should be mentioned that the algorithm proposed in this section cannot handle the case where the frequency bands induced by the planet bearing fault and those that modulate the meshing frequency overlap exactly with each other.
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Fig. 2. Flowchart of the proposed strategy for editing the SK values of the raw kurtogram.
3. Simulation analyses In this section, the proposed algorithm is tested using two simulated planetary gearbox vibration signals with a planet bearing inner race fault and outer race fault. The models are derived from those proposed in Ref. [2] without the healthy planet bearing part. In detail, the synthetic signals of these two fault conditions can be written as follows:
xouter ðtÞ ¼ xbearing;outer ðtÞ þ xgear ðtÞ þ nðtÞ
(3)
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Fig. 3. Flowchart of the proposed method for planet bearing fault detection.
xinner ðtÞ ¼ xbearing;inner ðtÞ þ xgear ðtÞ þ nðtÞ
(4)
where, xouter(t) and xinner(t) are the simulated vibration signals of the planetary gearbox with planet bearing outer race and inner race faults, respectively. xbearing, outer(t) and xbearing, inner(t) are the vibrations induced by planet bearing faults. xgear(t) represents the gear interruption, and n(t) represents the background noise. The bearing fault-induced vibration part xbearing, outer(t) and xbearing, inner(t) can be expressed by Eqs. (5) and (6), respectively
m P b tmTp;outer ti i¼M xbearing;outer ðtÞ ¼ Am;bearing sin 2pfplanetarygear t sinð2pfcarrier tÞe … m¼M !! ! m m X X ti u t mTp;outer ti sin ur;bearing t mTp;outer M X
i¼M
i¼M
xbearing;inner ðtÞ ¼
M X
P
m
b tmTp;inner
ti
i¼M Am;bearing sinð2pfcarrier tÞe … m¼M !! ! m m X X ti u t mTp;inner ti sin ur;bearing t mTp;inner
i¼M
(5)
(6)
i¼M
Most of the parameters in these two equations are the same. fplanetarygear represents the planetary gear rotational frequency (PRF), and fcarrier represents the carrier rotational frequency (CRF). b represents the structural damping characteristic. ti represents the effect of random slippage of the rolling elements. In this case, the slippage is uniformly distributed with a zero mean and a standard deviation of 0.01Tp~0.02Tp, where Tp is the time interval between the fault-induced impulses. If the planet bearing has an outer race fault, Tp equals Tp,outer, and if the planet bearing has an inner race fault, Tp should be set as Tp,inner. u(t) is a unit step function. Am,bearing is the amplitude of the mth impulse-induced by the localized planet bearing fault. ur,bearing is resonance frequency excited by the faulty bearing. Here, the impulse amplitude and resonance frequency remain the same under the two conditions of the inner race and out race faults.
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T. Wang et al. / Journal of Sound and Vibration 432 (2018) 437e453 Table 1 Details of the parameters of simulation model. Parameters
Corresponding values
Length of signal Planet bearing rotational frequency Sampling rate FCF for outer race fault FCF for inner race fault Damping characteristic (b) PBFR frequency (ur,bearing) Gear resonance frequency (ur,gear) Amplitude of planet bearing-induced impulse Amplitude of gear-induced impulse Magnitude coefficient (l) Planetary gear rotational frequency (PRF) Carrier rotational frequency (CRF) Repeat frequency of gear-induced impulse
8s 20 Hz 20480 Hz 3.7 fr 4.5 fr 800 6000 Hz 2000 Hz 1 3 0.1 10 Hz 7 Hz 40 Hz
Apart from these parameters, there are two differences between the two equations. First, sin(2pfplantgeart) is present only in Eq. (5), because the localized fault on the planet bearing outer race not only orbits around the center of the carrier but also rotates with the planet gear. Hence, the amplitude of the bearing outer race fault impulses is modulated by the PRF. Second, the time periods of the outer race and inner race fault-induced impulses differ, as indicated by Eqs (7) and (8). (Tp,outer and Tp,inner)
Fig. 4. Simulation case with the outer race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram, (e) envelope spectrum of the filtered result obtained using the selected optimal filter band.
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. nfplanetarygear d 1 cos 4 Tp;outer ¼ 1 fcfp;outer ¼ 1 D 2
(7)
. nfplanetarygear d Tp;inner ¼ 1 fcfp;inner ¼ 1 1 cos 4 D 2
(8)
where fcfp, outer and fcfp, inner are the fault characteristic frequencies of the planet bearing outer and inner race fault, respectively. fplanetarygear is the rotational frequency of the planetary gear. Z, d, D, and 4 are the number of rolling elements, the diameter of rolling element, the pitch diameter, and the contact angle, respectively. Considering that the gear meshing vibration also leads to impulsive frequency bands in the raw kurtogram, the simulated gear part is constructed according to the following equation.
xgear ðtÞ ¼
K X
Ak;gear ebðtkTp;gear Þ sin ur;gear t kTp;gear u t kTp;gear
(9)
k¼K
where Ak,gear is the amplitude of the kth impulse-induced by the gear meshing, ur,gear is the higher frequency that modulates the meshing frequency, and Tp,gear is the time interval between the two adjacent gear-induced impulses. Because the gear meshing-induced higher frequency band always significant interrupts the extraction of the PBFR frequency band, Ak,gear is set to be higher than Am,bearing. With the equations listed above, two signals that separately simulate the planetary gearbox vibration with planet bearing outer and inner race fault are constructed. Table 1 shows the corresponding parameter values.
Fig. 5. Simulation case with the inner race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram, (e) envelope spectrum of the filtered result using the selected optimal filter band.
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Fig. 4(a) shows the simulated planetary signal with the planet bearing outer race fault. Fig. 4(b) shows the corresponding kurtogram obtained using the traditional SK algorithm, where the frequency band with the highest SK value is located around 2000 Hz. The corresponding center frequency (2133 Hz) is close to the preset gear-related resonance frequency(2000 Hz). Hence, it is difficult to identify PBFR frequency using the traditional SK algorithm. As such, the algorithms proposed in Section 2 are employed to analyze the simulation signals. First, the filter results acquired in calculating the kurtogram are obtained, along with the indexMFM values of all these filter signals. Fig. 4(c) presents these index values in a manner similar to a kurtogram. Here, only the frequency bands around 2000 Hz possess relatively high indexMFM values, the planet bearing-induced resonance frequency band cannot be identified at all. It is proven that the indexMFM can be used to determine which frequency bands are related to the gear meshing and which are not. Then, the median of these index values is set as the threshold. Here, the threshold equals 0.0034. With the indexMFM-based threshold, some SK values of the original raw kurtogram are set as zero according to the rule introduced in Section 2.2. Fig. 4(d) shows the indexMFM-based kurtogram, where the gear meshing-related areas are removed, and the planet bearing-related area around 6000 Hz can be identified. In the area around 6000 Hz, the frequency band (center frequency: 5920 Hz, level: 5) with the highest SK value is selected as the potential filter band. For the final fault detection, the raw signal is filtered with the selected potential filter band, and the envelope spectrum of the filter result is displayed in Fig. 4(e). Here, the colored vertical lines with different types show the planet bearing envelope fault pattern (outer race fault). In detail, the blue solid lines represent the 1st and 2nd FCFs; the red dashed lines represent the PRF sidebands (including the left RPF (LPRF) and right RPF (RPRF)) around the FCF; the black dotted lines represent the CRF sidebands (including the LCRF and RCRF) around the PRF sidebands; and the green dash-dotted lines represent the CRF sidebands (including LCRF and RCRF) around the FCF. The obtained envelope spectrum matches the envelope fault pattern. Thus, the effectiveness of the proposed algorithm is partly confirmed.
Fig. 6. Analysis results obtained with different SNRs. (a) indexMFM-based kurtogram under -5 dB, (b) indexMFM-based kurtogram under 10 dB, (c) indexMFMbased kurtogram under 20 dB, (d) optimal envelope spectrum under -5 dB, (e) optimal envelope spectrum under 10 dB, (f) optimal envelope spectrum under 20 dB.
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Fig. 7. Experimental set-up for the planet bearing fault diagnosis.
Furthermore, simulated planetary gearbox vibration with a planet bearing inner race fault is used to testify the robustness of the proposed algorithm. Fig. 5 (a) ~ (e) are the counterparts of Fig. 4(a) ~ (e), respectively. In detail, Fig. 5(a) and (b) show the raw vibration and kurtogram, respectively. Similar to Fig. 4(b), the frequency bands with prominent SK values are related to gear meshing vibration, and the planet bearing-induced resonance frequency bands around 6000 Hz are less prominent and difficult to identify. Fig. 5(c) ~ (e) show the analysis results obtained using the proposed algorithm. Fig. 5(c) shows all the indexMFM values, from which only the gear meshing-related frequency bands (around 2000 Hz) can be identified. On the basis of all the indexMFM values, the threshold in inner race fault detection is set as 0.0033. According to the rules described in Section 2.2, Fig. 5(d) shows the indexMFM-based kurtogram, from which concentrated frequency bands with high SK values are observed around 6000 Hz. Hence, the planet bearing inner race faultinduced resonance frequency band can also be identified, and the center frequency and the level of the optimal resonance frequency band are 6080 Hz and 4, respectively. Fig. 5(e) compares the envelope spectrum of the filter result obtained using the optimal filter band with the envelope fault pattern for the planet bearing inner race fault, revealing that the envelope spectrum matches the pattern very well. In addition, three other simulated signals with different SNRs are added to investigate how the algorithm proposed in Section 2 performs under different noise conditions. Here, it is worth mentioning that the gear interruption is considered as the main noise rather than the signal toward the planet bearing fault-induced vibration. Hence, the SNR of the simulated signal should be reset by changing the amplitude of the gear magnitude, rather than the background white noise. Fig. 6(a)e(c) show the indexMFM-based kurtograms under different SNRs: 5, 10, and 20 dB. Fig. 6(d)e(f) show the corresponding envelope spectrum with selected optimal frequency bands chosen from Fig. 6(a)e(c). In the case of the SNR of 5 dB, the preset PBFR frequency (6000 Hz) can be also located in the indexMFM-based kurtogram shown in Fig. 6(a) with the frequency-band parameters (center frequency: 6080 Hz; level: 4). The corresponding envelope spectrum of the filtered result is shown in Fig. 6(d), where the FCF and the carrier sidebands are clearly observed. In the case of the SNR of 10 dB, the preset bearing fault-related frequency band is not as prominent as that shown in Fig. 6(a). With the given frequency-band parameters (center frequency: 6240 Hz; level: 2), the envelope spectrum of the filtered result is shown in Fig. 6(e). In contrast to Fig. 6(d), although the planet bearing FCF and the sidebands can also be located, other spectral interruptions are much higher. The SNR of 20 dB is the worst case. On one hand, the planet bearing-related frequency band almost cannot be located from the indexMFM-based kurtogram shown in Fig. 6(c). The selected frequency band with the parameters (center frequency: 7333 Hz; level: 2) is far from the preset one. On the other hand, the planet bearing FCF and the sidebands are too weak to be detected in the corresponding envelope spectrum of the filtered result.
Fig. 8. Planet bearing damage: (a) outer race fault, (b) inner race fault.
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Table 2 Parameters and the corresponding characteristic frequencies of the planetary gearbox and its planet bearing.
Planetary gearbox parameters
Planetary rolling bearing parameters
Characteristic frequencies of the planetary gearbox and its planet bearing
Parameter
Specific value
Sun gear teeth Planetary gear teeth Ring gear teeth Bearing type Number of balls Fitch diameter (mm) Ball diameter (mm) Contact angle (º) Rotational frequency of sun gear (Hz) Rotational frequency of carrier (Hz) Rotational frequency of planetary gear (Hz) FCFouter of planet bearing (Hz) FCFinner of planet bearing (Hz)
13 38(3) 92 NJ304 10 36 9 0 24.97 3.09 4.39 16.47 27.45
Table 3 Theoretical values of characteristic frequencies in the fault envelope pattern. Characteristic frequency
Theoretical value of characteristic frequencies Fault envelope pattern for outer race fault
Fault envelope pattern for inner race fault
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF RCRF sideband of LPRF RPRF sideband of 1st FCF LCRF sideband of RPRF RCRF sideband of RPRF LCRF sideband of 1st FCF RCRF sideband of 1st FCF 2nd FCF
16.47 12.08 8.99 15.17 20.86 17.77 23.95 19.56 13.38 32.92
27.45 N/A N/A N/A N/A N/A N/A 24.36 30.54 54.41
Fig. 9. Planet bearing with an outer race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram.
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4. Experimental tests For this section, both the test rig (at the University of Science and Technology, Beijing lab) and the experimental data are the same as those used in Ref. [2]. In addition to confirming the effectiveness of the proposed indexMFM-based kurtogram algorithm for planet bearing fault detection, the corresponding analysis results can be compared with the results obtaining using the algorithm proposed in Ref. [2]. The layout of the test rig and the faulty planet bearings are shown in Figs. 7 and 8, respectively. As shown in Fig. 7, the planetary gearbox is driven by a motor, and the corresponding rotational speed is measured by the tachometer placed between the motor and the monitored planetary gearbox. An electromagnetic brake is mounted with the output shaft of the planetary gearbox to provide additional torque. During the measurement, the accelerometer, which is mounted on the top of the planetary gearbox (indicated by a red arrow), is used to obtain the vibration signal. Fig. 8 shows the artificially added localized faults on the outer race and the inner race of a planet bearing. As shown, two cutting defects are separately added on these two places. The corresponding parameters and the theoretical values of the characteristic frequencies of the fault envelope pattern are shown in Tables 2 and 3, respectively. 4.1. Planet bearing with outer race fault For the case with a planet bearing outer race fault, the raw signal and the corresponding raw kurtogram are shown in Fig. 9(a) and (b), respectively.
Fig. 10. Planet bearing with an outer race fault: (a) envelope spectrum of the filtered result with parameters of (3680, 5), (b) envelope spectrum of the filtered result with parameters of (7200, 5), (c) envelope spectrum of the filtered result with parameters of (6400, 2).
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Table 4 Errors with different filtering parameters in the case where the outer race of the planet bearing has a fault.. Characteristic frequencies
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF RCRF sideband of LPRF RPRF sideband of 1st FCF LCRF sideband of RPRF RCRF sideband of RPRF LCRF sideband of 1st FCF RCRF sideband of 1st FCF 2nd FCF Average error
Errors with different filtering parameters fc ¼ 3680 k¼5
fc ¼ 7200 k¼5
fc ¼ 6400 k¼2
Optimal parameters in Ref. [2]
0.07 0.04 0.013 0.04 0.02 0.02 0.03 0.03 0.08 0.09 0.0433
0.04 0.03 0.034 0.04 0 0.04 0.01 0.04 ✕ 0.07 0.0337
0.04 0.01 0.066 0 0.11 ✕ 0.05 0.03 0.01 0.09 0.0451
0.04 0.02 0.034 0.05 0 0.02 0.02 ✕ ✕ 0.08 0.033
In Fig. 9(b), the frequency bands with higher SK values are concentrated around 1000 Hz. It is stated in Ref. [2] that the area around 1000 Hz is not related to the planet bearing fault. Hence, the traditional SK algorithm cannot identify the weak PBFR frequency band. Here, the level for the kurtogram calculation is set as 6 according to Eq. (1). Using the algorithms proposed in Section 2, the indexMFM values of all the decomposition results are calculated, as shown in Fig. 9(c), in which the frequency bands with relatively high indexMFM values are concentrated below 2000 Hz, and the center frequency of the frequency band with the highest indexMFM value is 640 Hz. It can be seen that the area of higher SK values in Fig. 9(b) almost coincides with the area of higher indexMFM values in Fig. 9(c). This phenomenon confirms that the indexMFM values can be used to identify the gear meshing-related frequency band in the raw kurtogram. With the help of the indexMFM values shown in Fig. 9(c), the threshold can be determined as the corresponding median. This, together with the rules introduced Section 2.2.2 yields the indexMFM-based kurtogram shown in Fig. 9(d). Comparing the raw kurtogram in Fig. 9(b) and the indexMFM-based kurtogram in Fig. 9(d) reveals that the area with prominent SK values below 2000 Hz in the raw kurtogram is not observed in Fig. 9(d). This confirms that the proposed algorithm can remove the gear meshing-related frequency bands with prominent SK values from the raw kurtogram.
Fig. 11. Planet bearing with an inner race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram.
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Furthermore, the potential optimal filter bands can be selected from the indexMFM-based kurtogram. Specifically, there are three areas (marked by the red dashed rectangles) with relatively high SK values in Fig. 9(d). According to the analysis in Section 2, these three areas may contain the planet bearing fault-induced frequency bands, and the three frequency bands with the highest SK values in each area are considered as the potential PBFR frequency bands. The parameters of the potential PBFR frequency bands are (3680, 5), (7200, 5), and (6400, 2), where the first value inside the parentheses is the center frequency in hertz and the second one is the level. With these three frequency bands, the raw signal is filtered. Fig. 10 shows the corresponding envelope spectra of the filtered results. In these three envelope spectra, the colored straight lines represent the fault envelope pattern for the planet bearing outer race fault constructed according to Table 2. If prominent peaks can be located at the specific theoretical values, the corresponding x-axis value of the prominent peak is shown at the top left of the straight lines, and the theoretical value is shown inside the bracket, such as “x-axis value (theoretical value)”. Otherwise, the corresponding expression follows the form of “(theoretical value)”. In Fig. 10(a), spectral peaks are observed at all of the theoretical values. In Fig. 10(b), there are no peaks at the RCRF sideband of the 1st FCF, and in Fig. 10(c), no peak is observed at the LCRF sideband of the RPRF. It should be mentioned that, the meanings of different types of straight line are as same as the ones introduced in the simulation cases. The differences between the theoretical values and the x-axis values of the detected peaks at different characteristic frequencies are defined as the errors and shown in Table 4. If no peak exists at the theoretical characteristic frequency, “ ” is shown instead of the error value. A small error means that the FCF has been located. Hence, the frequency band with the parameters of (3680, 5) is the optimal planet PBFR frequency band, and the monitored planet bearing can be considered as having an outer race fault.
Fig. 12. Planet bearing with an inner race fault: (a) envelope spectrum of the filtered result with parameters of (3040, 5), (b) envelope spectrum of the filtered result with parameters of (7040, 4.5), (c) envelope spectrum of the filtered result with parameters of (6400, 2).
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Table 5 Errors with different filtering parameters in the case where the inner race of the planet bearing has a fault.. Characteristic frequencies
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF 2nd FCF Average error
Errors with different filtering parameters fc ¼ 3040 k¼5
fc ¼ 7040 k ¼ 4.5
fc ¼ 6400 k¼2
Optimal parameters in Ref. [2]
0.11 0.06 0.06 0 0.0575
0.02 ✗ 0.13 0.06 0.07
0.01 0.01 0.09 0.06 0.0425
0.01 0.03 0.05 0.01 0.033
The errors of the traditional algorithm proposed in Ref. [2] are also listed in Table 4(the last column). Although the average error of the proposed algorithm is higher than that (0.033) of the traditional algorithm, the average error of 0.0433 is low enough for a convincing diagnosis result. Moreover, the proposed indexMFM-based method does not require a health baseline signal. 4.2. Planet bearing with inner race fault In this subsection, the proposed indexMFM-based algorithm is validated using experimental data for a planet bearing with an inner race fault. The test rig and the parameters are the same as those introduced in Section 4.1. Figs. 11 and 12 are the counterparts of Figs. 9 and 10, respectively. Fig. 11(a) shows the raw waveform, and Fig. 11 (b) and (c) shows the raw kurtogram and the distribution of the indexMFM values, respectively. According to these two figures, the areas with prominent SK and indexMFM values are similar and are induced by the gear meshing-related frequency band. This similarity suggests that the traditional SK based algorithm cannot identify the PBFR frequency band, and the indexMFM can detect the gear meshing-related area. Using indexMFM values shown in Fig. 11(c), the threshold is set as 4.18 107. Fig. 11(d) shows the indexMFM-based kurtogram, based on which, three PBFR frequency bands can be determined. The corresponding parameters are (3040,5), (7040,4.5), and (6400,2), respectively. Fig. 12(a)e(c) show the envelope spectra obtained using the selected frequency bands together with the fault envelope pattern of the planet bearing inner race fault. The errors between the theoretical values and the x-axis values of the selected peaks are listed in Table 5, among which the frequency band with the parameters of (6400,2) represents the optimal resonance frequency band. With this frequency band, spectral peaks are observed in all of the theoretical values, indicating that the monitored planet bearing has an inner race fault. For comparison, the errors obtained using the traditional algorithm are listed in Table 5. The average error obtained using the proposed algorithm is 0.0425. Although this is larger than the average error for the traditional algorithm, 0.0425 is a small enough error for a credible fault detection result. 5. Conclusion An indexMFM-based kurtogram algorithm was proposed to solve the problem of planet bearing fault detection without using any auxiliary equipment or a health baseline. To eliminate the strong gear interruption from the frequency domain, a new indexMFM is formulated to identify which frequency bands result from the gear meshing vibration in the raw kurtogram. The plant bearing fault-induced resonance frequency bands can be identified from the indexMFM-based kurtogram. Simulation and experimental tests demonstrate the accuracy of the fault envelope pattern and the effectiveness of the planet bearing fault detection algorithm, respectively. Acknowledgment The research work described in this paper was supported by National Natural Science Foundation of China under Grant Nos. 51605244, 51335006. The valuable comments and suggestions from the editor and two reviewers are very much appreciated. References [1] [2] [3] [4]
Y. Lei, J. Lin, M.J. Zuo, Z. He, Condition monitoring and fault diagnosis of planetary gearboxes: a review, Measurement 48 (2014) 292e305. T. Wang, Q. Han, F. Chu, Z. Feng, A new SKRgram based demodulation technique for planet bearing fault detection, J. Sound Vib. 385 (2016) 330e349. Z. Feng, H. Ma, M.J. Zuo, Amplitude and frequency demodulation analysis for fault diagnosis of planet bearings, J. Sound Vib. 382 (2016) 395e412. S. Jain, H. Hunt, Vibration response of a wind-turbine planetary gear set in the presence of a localized planet bearing defect, in: ASME 2011 International Mechanical Engineering Congress and Exposition (IMECE2011), Denver, Colorado, USA, 2011, pp. 943e952. [5] R.B. Randall, J. Antoni, Rolling element bearing diagnostics-A tutorial, Mech. Syst. Signal Process. 25 (2011) 485e520. [6] J. Antoni, R.B. Randall, Unsupervised noise cancellation for vibration signals: part Idevaluation of adaptive algorithms, Mech. Syst. Signal Process. 18 (2004) 89e101.
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[7] T. Wang, M. Liang, J. Li, W. Cheng, C. Li, Bearing fault diagnosis under unknown variable speed via gear noise cancellation and rotational order sideband identification, Mech. Syst. Signal Process. 62e63 (2015) 30e53. [8] J. Antoni, R.B. Randall, Unsupervised noise cancellation for vibration signals: part IIda novel frequency-domain algorithm, Mech. Syst. Signal Process. 18 (2004) 103e117. [9] P.D. Mcfadden, M.M. Toozhy, Application of synchronous averaging to vibration monitoring of rolling element bearings, Mech. Syst. Signal Process. 14 (2000) 891e906. [10] Z. Fan, H. Li, A hybrid approach for fault diagnosis of planetary bearings using an internal vibration sensor, Measurement (2015) 71e80. [11] T. Wang, M. Liang, J. Li, W. Cheng, Rolling element bearing fault diagnosis via fault characteristic order (FCO) analysis, Mech. Syst. Signal Process. 45 (2014) 139e153. [12] M. Liang, I. Soltani Bozchalooi, An energy operator approach to joint application of amplitude and frequency-demodulations for bearing fault detection, Mech. Syst. Signal Process. 24 (2010) 1473e1494. [13] J. Antoni, The spectral kurtosis: a useful tool for characterising non-stationary signals, Mech. Syst. Signal Process. 20 (2006) 282e307. [14] Y. Wang, J. Xiang, R. Markert, M. Liang, Spectral kurtosis for fault detection, diagnosis and prognostics of rotating machines: a review with applications, Mech. Syst. Signal Process. 66e67 (2016) 679e698. [15] J. Antoni, Fast computation of the kurtogram for the detection of transient faults, Mech. Syst. Signal Process. 21 (2007) 108e124.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Meshing frequency modulation (MFM) index-based kurtogram for planet bearing fault detection Tianyang Wang a, Fulei Chu a, *, Zhipeng Feng b a b
Department of Mechanical Engineering, Tsinghua University, Beijing 100084 China School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 November 2017 Received in revised form 20 June 2018 Accepted 20 June 2018
Identifying the bearing fault-induced impulsive components in the frequency domain is a key step in the corresponding fault detection. However, gearbox vibration signals often significant interrupt the rolling bearing fault diagnosis, particularly in the detection of planet bearing faults under the background noise of the planetary gearbox. Except in the case of a high amplitude, a gear meshing-related vibration may also affect the identification of the planet bearing fault-induced resonance frequency band. To solve this problem, a meshing frequency modulation index (indexMFM)-based kurtogram utilizing a particular gearbox related phenomenon is proposed. The underlying mechanism is such that although the gear meshing-related spectral components are always more prominent in relatively higher-frequency band than the planet bearing-induced resonance frequency band in impulsiveness, the gear meshing-related impulsive components modulate the gear meshing frequency, yet the faulty bearing-induced one does not. Exploiting this difference, the planet bearing fault-induced impulsive components can be directly identified from the strong gear vibration interruption by determining the bearing fault-related resonance frequency band in the indexMFM-based kurtogram. The effectiveness of the proposed method is separately verified using simulated and experimental data. © 2018 Elsevier Ltd. All rights reserved.
Handling Editor: Z Su Keywords: Planet bearing Fault diagnosis Meshing frequency modulation Meshing frequency modulation index-based kurtogram
1. Introduction In contrast to general rolling bearings, the planet bearing has several unique features due to the complex structure of the planetary gearbox [1]. This intricate structure, which provides the advantages of a large transmission ratio, a strong loadbearing capacity, and lightness, affects the corresponding planet bearing fault detection [2]. First, the unique structure of the planetary gearbox makes the operating mode of the planet bearing different from the traditional one [3]. On one hand, the planet bearing not only rotates with its shaft as the center but also orbits around the axis of the sun gear shaft. On the other hand, the inner race of the planet bearing stands still with its shaft, and the outer race rotates together with the planetary gear. In this way, the formula for the planet bearing FCF changes at the same time. To solve this problem, Jain [4] and Feng [3] deduced the corresponding planet bearing FCF through the dynamic model and numeral model separately and predicted the corresponding spectrum pattern with a faulty planet bearing. Wang et al. [2] analyzed the sideband patterns around the FCF in the envelope spectrum with different fault positions.
* Corresponding author. E-mail addresses: [email protected] (T. Wang), chufl@mail.tsinghua.edu.cn (F. Chu). https://doi.org/10.1016/j.jsv.2018.06.051 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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Nomenclature indexMFM Meshing frequency modulation index fs Sampling frequency fmeshing Meshing frequency l Decomposition level fbandwidth Frequency bandwidth in determining indexMFM N Length of filtered result fplanetarygear Planetary gear rotational frequency fcarrier Carrier rotational frequency b Structural damping characteristic ti Random slippage Tp Time interval between fault-induced impulses Tp,outer Time interval between outer race fault-induced impulses Tp,inner Time interval between inner race fault-induced impulses A m,bearing Amplitude of mth impulse-induced by localized planet bearing fault ur,bearing Resonance frequency excited by faulty bearing fcfp,outer Fault characteristic frequency(FCF) of planet bearing outer race fault fcfp,inner FCF of planet bearing inner race fault Z Number of rolling elements d Diameter of rolling element D Pitch diameter 4 Contact angle Ak,gear kth impulse-induced by gear meshing ur,gear Higher frequency that modulates meshing frequency Tp,gear Time interval between two adjacent gear-induced impulses
In addition to changing the operating mode, the planetary gearbox meshing vibration usually interrupts the extraction of the planet bearing fault features in both the time and frequency domains. In the time domain, the planet bearing fault-induced impulses are always buried in the gear meshing vibration owing to the long and time-varying distance between the fault position and the sensor mounted on the outer case. The gear noise can be considered as one of the common interruptions of the bearing fault detection [5]. Algorithms designed to eliminate the timeinvariant components, such as linear prediction [6], self-adaptive noise cancellation [7], discrete random separation [8] and time synchronous averaging [9], have been used to eliminate the gear meshing noise. However, in the case of the planetary gearboxes the effectiveness of these elimination algorithms is unsatisfactory. First, the significantly rich sidebands around the meshing frequency and its harmonics can be readily identified even in the absence of faults, which affects the effectiveness of the traditional gear noise elimination algorithms. Furthermore, the impulses caused by the planet bearing fault are very weak compared with the gear interruption. Thus, the traditional algorithms may interrupt the planet bearing part when they are applied for gear noise elimination. To solve this problem, Fan [10] directly mounted an internal sensor on the carrier, which can measure the planet bearing vibration with less gear interruption. Although this strategy can partly highlight weak planet bearing fault-related vibration, the corresponding internal sensor cannot be installed on every planetary gearbox in real engineering owing to cost or design limitations. In the frequency domain, the planetary gear meshing noise can influence the planet bearing fault detection by interrupting the identification of the planet bearing fault-induced resonance (PBFR) frequency band. In general, the recognition of the optimal bearing fault-induced resonance frequency band is essential for the final diagnosis, because it allows the prominent FCF and its harmonics to be identified in the corresponding envelope spectrum of the bandpass filtering result. To determine the resonance frequency band, several algorithms [11e14] have been proposed, among which the spectral kurtosis (SK) [13] based methods are the most widely used and effective. However, as in the case of the planetary gearbox, the gear meshing vibration also leads to impulsive components in the higher frequency band, and these frequency components are always more evident than the planet bearing fault-related resonance. Hence, it is difficult to identify the planet bearing-induced resonance frequency bands directly using traditional algorithms. In response to this problem, Wang [2] proposed an SKRgram based demodulation technique, in which the kurtogram of the monitored planetary gearbox in a healthy condition is used as a baseline, and the planet bearing fault-induced resonance vibration can be located by calculating the ratio of the kurtogram of the faulty signal to the baseline. Nevertheless, the health baseline is not always available, and it is challenging to obtain an optimal health baseline for different operating conditions. According to the above analysis, it is difficult to realize planet bearing fault detection without auxiliary devices, such as an internal sensor or the health baseline. That is, it is challenging to highlight the planet bearing fault component from the strong
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gear noise without any prior knowledge. As such, a new algorithm is proposed in this paper for detecting planet bearing faults without auxiliary equipment or a health baseline. In this algorithm, a particular characteristic of the gear meshing vibration is utilized to select the bearing fault-induced impulsive vibration in the frequency domain. The principle is that the impulsive frequency band induced by the gear meshing modulates the meshing frequency, and the resonance frequency band resulting from the planet bearing fault does not modulate the meshing frequency at all. In real applications, the kurtogram of the raw signal is first calculated to determine how the impulsiveness is distributed in different frequency bands and levels. It should be mentioned that some of these impulsive frequency bands are caused by the planet bearing fault, and some of them are resulted from the gear meshing. A meshing frequency modulation (MFM) index called indexMFM is then constructed to determine which impulsive bands in the corresponding kurtogram are related to the meshing vibration. Furthermore, the potential frequency bands related to the planet bearing fault, which is called potential PBFR frequency bands, are selected from the indexMFM-based kurtogram, which is obtained by setting the SK values of particular frequency bands to zero if the corresponding indexMFM is lower than a selected threshold. Finally, the raw signal is filtered with the PBFR frequency bands, and the envelope spectra of the filtering results are used to perform final fault detection of the planet bearing. The remainder of the paper is structured as follows: Section 2 specifies the proposed algorithm, which includes obtaining the raw kurtogram, calculating the meshing index, determining the potential planet bearing fault-related band according to the IndexMFM-based kurtogram, and detecting the planet bearing fault with the envelope spectrum. Sections 3 and 4 testify the effectiveness of the proposed algorithm with simulated and experimental cases, respectively. The conclusion is drawn in Section 5. 2. Proposed planet bearing fault diagnosis method As previously explained, one of the major difficulties of planet bearing fault detection without any prior knowledge lies in distinguishing the planet bearing fault-related resonance frequency band from the strong gear meshing vibration noise. This is because the resonance component induced by the planet bearing fault is too weak to be recognized from more prominent gear meshing-related components. First, an example is presented to verify this statement with figures in Ref. [2]. Fig. 1 (a) and (b) show kurtograms of healthy planetary gearbox vibration and vibration with a planet bearing outer race fault under the same operating environment, respectively. It can be seen that the kurtograms obtained under different working conditions are almost the same. The frequency bands with prominent SK values are mostly induced by gear meshing vibration. Hence, it is challenging to select PBFR frequency band directly according to the kurtogram shown in Fig. 1(b). To solve this problem, a distinctive characteristic of the gear meshing vibration-induced impulsive frequency bands is utilized. Although the gear meshing frequency can lead to concentrated frequency bands with higher SK values in the kurtogram, these gear-related frequency bands modulate the meshing frequency. In contrast, the BPFR frequency bands do not modulate the gear meshing frequency. Exploiting this difference, an indexMFM-based kurtogram is proposed to select the planet bearing fault-induced frequency band directly. The corresponding algorithm is introduced in the following three parts. 2.1. Obtaining distribution of impulsiveness based on raw kurtogram In this section, a traditional fast kurtogram algorithm [15] is used to analyze the raw vibration. The raw signal is first decomposed with a binary tree structure and then extended to a 1/3-binary tree structure. Next, the SK values of each filtered results are calculated to form a kurtogram. During the formulation of the kurtogram, the decomposition level should be set with the certain value with the limitation given by the following equation:
k j . l ¼ fs =2 fmeshing
Fig. 1. Kurtograms of planetary gearbox vibrations: (a) healthy condition, (b) faulty condition with the planet bearing outer race fault [2].
(1)
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where, P:R is the rounding-down operator, fs is the sampling frequency, and fmeshing is the meshing frequency of the monitored planetary gearbox. The rationale for setting the decomposition level according to Eq. (1) is that this equation can ensure that the frequency spans of the envelope spectrums of all the filtered results contain the planetary gear meshing frequency. Only under this condition, the indexMFM proposed in Section 2.2.1 can be calculated. In detail, the IndexMFM is constructed based on the magnitudes of the spectral lines at the meshing frequency and its harmonics in the envelope spectrum of all the filtered results. This strategy will not be processed if one of the frequency spans of all the filtered results is smaller than the meshing frequency. In other words, the smallest frequency span of all the filtered results should be larger than or equal to the meshing frequency. If the decomposition level is set as l, for instance, the smallest frequency span of the filtered results equals to fs/2 l. Hence, the fs/2 l should be larger than the meshing frequency, which gives the decomposition level an upper limitation of fs/2/fmeshing. 2.2. Construction of indexMFM-based kurtogram In the raw kurtogram of the planetary gearbox with a planet bearing fault, most of the frequency bands with a relatively high SK are induced by the gear meshing vibration rather than the bearing fault. Therefore, it is difficult to determine the optimal PBFR frequency band according to the traditional kurtogram. In this section, an indexMFM is proposed for determining which frequency bands with relatively high SK values are related to the planetary gearbox meshing vibration. An indexMFMbased kurtogram is then constructed to determine the potential frequency bands induced by the planet bearing fault. 2.2.1. Construction of indexMFM As previously mentioned, if a frequency band in the kurtogram with a relatively high SK value is induced by the gear meshing vibration, it modulates the meshing frequency. Consequently, the meshing frequency and its harmonics are prominent in the corresponding envelope spectrum. However, the frequency bands induced by the planet bearing fault do not modulate the meshing frequency at all. Exploiting this difference, a new criterion called the MFM index is proposed in this subsection for evaluating the prominence of the meshing frequency and its harmonics in an envelope spectrum. Supposing that x(t) of length N is the filtered result of the raw signal with a certain frequency band, the envelope spectrum of x(t) is denoted by ES(f). The discrete version of x(t) and ES(f) are x[n] and ES[k] respectively, where n, k ¼ 1,2,3, …, N. The MFM index is defined as, NH X indexMFM x½n; fmeshing ¼ i¼1
1 RankES½k
!
(2)
ES½k max jkDf ifmeshing jdf
where NH represents the harmonic order whose value could equal to 1, 2, or 3 according to the frequency range of ES[k]. In detail, if the upper limit of the frequency range of ES[k] is larger than 3 fmeshing, NH equals to 3. If the upper limit of the frequency range of ES[k] is larger than 2 fmeshing and smaller than 3 fmeshing, NH equals to 2. If the upper limit of the frequency range of ES[k] is larger than fmeshing and smaller than 2 fmeshing NH equals to 1. Df is the frequency resoES½k denotes the largest envelope spectral magnitude of ES[k] in the frequency range of [i fmeshing - df, lution. max jkDf ifmeshing jdf i fmeshing þ df ]. In fact, max ES½k represents the magnitude of the peak at the meshing frequency. Here, a frequency jkDf ifmeshing jdf interval, [i fmeshing - df, i fmeshing þ df ], is used for extracting the real meshing frequency from the envelope spectrum because the rotational speed cannot be a constant value in real engineering which may result in a difference between the theoretical and real meshing frequencies. The allowable frequency deviation df is set as 5 Hz in this paper. RankB[k](b) is a newly defined function, where B[k] is a data series and b is a member of B[k]. The RankB[k](b) represents the ranking number of b of B[k] after all the data in B[k] are sorted in a descending order by magnitude. Supposing that B[k] equals to [20, 56, 2, 8, 30, 100] and b equals to 30 which is a member of B[k]. Just because 30 is the third largest value of B[k], RankB[k](b) equals to 3. ! ES½k represents the ranking number of max ES½k in the ES[k] after all the data in max jkDf ifmeshing jdf jkDf ifmeshing jdf ES[k] are sorted in a descending order by magnitude. In real applications, if the meshing frequency and its harmonics are prominent in the envelope spectrum (ES[k]) of the filtered result (x[n]), the corresponding ranking numbers are relatively low, leading to a higher indexMFM. Thus, a higher indexMFM indicates that the corresponding frequency band is closely related to the gear meshing frequency.
Hence, RankES½k
2.2.2. IndexMFM-based kurtogram construction As indicated by the above analysis, the gear meshing vibration always interrupts the extraction of the PBFR frequency band because the gear meshing vibration-induced frequency bands in the raw kurtogram are more prominent than those caused by the planet bearing fault. In light of this problem, a unique characteristic is explored, the impulsive frequency bands related to the gear meshing vibration modulates the meshing frequency, while the planet bearing fault-induced frequency bands do not
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modulate the meshing frequency. Exploiting this characteristic, an indexMFM is proposed and used as a criterion to determine which frequency bands in the raw kurtogram are related to the gear meshing vibration. By eliminating these gear meshing vibration related frequency bands, the resonance frequency bands induced by the planet bearing fault can be selected from the rest of the frequency bands easily, without the gear noise interruption. In detail, an indexMFM-based kurtogram is constructed to determine which frequency bands are related to the planet bearing faults directly, via the following four-step strategy: STEP1: The indexMFM values of all the filtered results obtained in Section 2.1 are calculated. In this step, the raw signal is first filtered with the 1/3 binary tree filter banks, and then the indexMFM values of all the filtered results are calculated using the algorithm proposed in Section 2.2.1. STEP2: The median of all the indexMFM values is set as the threshold. As mentioned previously, the indexMFM is proposed to determine which frequency bands in the raw kurtogram are related to the gear meshing vibration. Hence, if an optimal indexMFM threshold can be properly selected, the frequency bands whose indexMFM is larger than the selected threshold are related to the gear meshing vibration and should be eliminated as interruptions. The PBFR frequency bands can be selected from the rest of the frequency bands easily without the gear vibration interruption. However, it is challenging to select the best threshold to perfectly divide the two types of frequency bands, because there is no clear boundary between them. With a threshold that is too high, the algorithm cannot eliminate the gear noise, and with a threshold that is too low, the algorithm may eliminate the PBFR frequency bands. To address this issue, a statistical value that can reflect the distribution of the indexMFM values may be effective. Considering that the distribution of the indexMFM values can reflect how the meshing frequency is modulated to different frequency bands, the characteristics of the MFM must be introduced as a background before the threshold setting. Specifically, the meshing frequency is modulated to a relatively concentrated area rather than all of the frequency bands as observed in Fig. 1(a). In fact, the higher indexMFM values are concentrated in a few frequency bands, and most of the indexMFM values are relatively small. Therefore, a competent threshold should guarantee the elimination of the gear interruption and reflect the average distribution of all the indexMFM values simultaneously, which makes the median a suitable candidate. First, the median of all the indexMFM values is definitely smaller than the prominent indexMFM values in the raw kurtogram, which can ensure the elimination of the gear interruption. Second, the median can reflect the average distribution of all the indexMFM values in some level. Hence, the median of all the indexMFM values is selected as the threshold in this section to guarantee that most of the gear meshing vibration related frequency bands are separated from the ones caused by the planet bearing fault. It should be mentioned that the median of all the indexMFM values is not always the best threshold. It is a relatively conservative threshold that can guarantee that most of the gear meshing vibration related frequency bands with prominent SK values in the raw kurtogram can be identified. STEP3: All of the SK values of the raw kurtogram are edited according to following rule: With the algorithms described above, all of the filtered results have their own SK value and indexMFM value. If the indexMFM value of a filtered result is larger than the threshold determined in STEP 2, the edited SK value is set as zero, because a high indexMFM value indicates that the corresponding frequency band is closely related to the gear meshing vibration. In contrast, if the index value is smaller than the threshold, the edited SK values should be equal to the original SK value of the raw kurtogram. STEP 4: The kurtogram with edited SK values is considered as the indexMFM-based kurtogram. In the aforementioned four-step strategy, most of the frequency bands related to the gear meshing vibration are located, and the corresponding SK values are set as zeros. Hence, the remaining frequency bands may contain the PBFR frequency bands. A flowchart of the four-step strategy is presented in Fig. 2. 2.3. Determination of planet bearing fault-induced optimal resonance frequency band and accomplishment of fault detection In the indexMFM-based kurtogram, most of the SK values of frequency bands related to the gear meshing vibration are set as zero. In this way, the PBFR frequency band remains. Hence, it is easier to identify the optimal resonance frequency bands induced by the planet bearing fault from the indexMFM-based kurtogram than from the raw kurtogram. The particular steps are listed as follows: First, the areas with concentrated high SK values are identified from the indexMFM-based kurtogram and considered as the potential optimal PBFR frequency areas. Then, the frequency bands with the highest SK values are selected from each potential optimal PBFR frequency area and considered as the potential filter bands. Furthermore, the raw signal is filtered with these potential filter bands, whereby several envelope spectra are calculated. Finally, the obtained envelope spectra are compared with the planet bearing envelope fault pattern constructed in Ref. [2] to perform the final planet bearing fault detection. A flowchart of the proposed method for planet bearing fault detection without any auxiliary equipment or prior information is presented in Fig. 3, showing the aforementioned algorithm elements. It should be mentioned that the algorithm proposed in this section cannot handle the case where the frequency bands induced by the planet bearing fault and those that modulate the meshing frequency overlap exactly with each other.
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Fig. 2. Flowchart of the proposed strategy for editing the SK values of the raw kurtogram.
3. Simulation analyses In this section, the proposed algorithm is tested using two simulated planetary gearbox vibration signals with a planet bearing inner race fault and outer race fault. The models are derived from those proposed in Ref. [2] without the healthy planet bearing part. In detail, the synthetic signals of these two fault conditions can be written as follows:
xouter ðtÞ ¼ xbearing;outer ðtÞ þ xgear ðtÞ þ nðtÞ
(3)
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Fig. 3. Flowchart of the proposed method for planet bearing fault detection.
xinner ðtÞ ¼ xbearing;inner ðtÞ þ xgear ðtÞ þ nðtÞ
(4)
where, xouter(t) and xinner(t) are the simulated vibration signals of the planetary gearbox with planet bearing outer race and inner race faults, respectively. xbearing, outer(t) and xbearing, inner(t) are the vibrations induced by planet bearing faults. xgear(t) represents the gear interruption, and n(t) represents the background noise. The bearing fault-induced vibration part xbearing, outer(t) and xbearing, inner(t) can be expressed by Eqs. (5) and (6), respectively
m P b tmTp;outer ti i¼M xbearing;outer ðtÞ ¼ Am;bearing sin 2pfplanetarygear t sinð2pfcarrier tÞe … m¼M !! ! m m X X ti u t mTp;outer ti sin ur;bearing t mTp;outer M X
i¼M
i¼M
xbearing;inner ðtÞ ¼
M X
P
m
b tmTp;inner
ti
i¼M Am;bearing sinð2pfcarrier tÞe … m¼M !! ! m m X X ti u t mTp;inner ti sin ur;bearing t mTp;inner
i¼M
(5)
(6)
i¼M
Most of the parameters in these two equations are the same. fplanetarygear represents the planetary gear rotational frequency (PRF), and fcarrier represents the carrier rotational frequency (CRF). b represents the structural damping characteristic. ti represents the effect of random slippage of the rolling elements. In this case, the slippage is uniformly distributed with a zero mean and a standard deviation of 0.01Tp~0.02Tp, where Tp is the time interval between the fault-induced impulses. If the planet bearing has an outer race fault, Tp equals Tp,outer, and if the planet bearing has an inner race fault, Tp should be set as Tp,inner. u(t) is a unit step function. Am,bearing is the amplitude of the mth impulse-induced by the localized planet bearing fault. ur,bearing is resonance frequency excited by the faulty bearing. Here, the impulse amplitude and resonance frequency remain the same under the two conditions of the inner race and out race faults.
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T. Wang et al. / Journal of Sound and Vibration 432 (2018) 437e453 Table 1 Details of the parameters of simulation model. Parameters
Corresponding values
Length of signal Planet bearing rotational frequency Sampling rate FCF for outer race fault FCF for inner race fault Damping characteristic (b) PBFR frequency (ur,bearing) Gear resonance frequency (ur,gear) Amplitude of planet bearing-induced impulse Amplitude of gear-induced impulse Magnitude coefficient (l) Planetary gear rotational frequency (PRF) Carrier rotational frequency (CRF) Repeat frequency of gear-induced impulse
8s 20 Hz 20480 Hz 3.7 fr 4.5 fr 800 6000 Hz 2000 Hz 1 3 0.1 10 Hz 7 Hz 40 Hz
Apart from these parameters, there are two differences between the two equations. First, sin(2pfplantgeart) is present only in Eq. (5), because the localized fault on the planet bearing outer race not only orbits around the center of the carrier but also rotates with the planet gear. Hence, the amplitude of the bearing outer race fault impulses is modulated by the PRF. Second, the time periods of the outer race and inner race fault-induced impulses differ, as indicated by Eqs (7) and (8). (Tp,outer and Tp,inner)
Fig. 4. Simulation case with the outer race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram, (e) envelope spectrum of the filtered result obtained using the selected optimal filter band.
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. nfplanetarygear d 1 cos 4 Tp;outer ¼ 1 fcfp;outer ¼ 1 D 2
(7)
. nfplanetarygear d Tp;inner ¼ 1 fcfp;inner ¼ 1 1 cos 4 D 2
(8)
where fcfp, outer and fcfp, inner are the fault characteristic frequencies of the planet bearing outer and inner race fault, respectively. fplanetarygear is the rotational frequency of the planetary gear. Z, d, D, and 4 are the number of rolling elements, the diameter of rolling element, the pitch diameter, and the contact angle, respectively. Considering that the gear meshing vibration also leads to impulsive frequency bands in the raw kurtogram, the simulated gear part is constructed according to the following equation.
xgear ðtÞ ¼
K X
Ak;gear ebðtkTp;gear Þ sin ur;gear t kTp;gear u t kTp;gear
(9)
k¼K
where Ak,gear is the amplitude of the kth impulse-induced by the gear meshing, ur,gear is the higher frequency that modulates the meshing frequency, and Tp,gear is the time interval between the two adjacent gear-induced impulses. Because the gear meshing-induced higher frequency band always significant interrupts the extraction of the PBFR frequency band, Ak,gear is set to be higher than Am,bearing. With the equations listed above, two signals that separately simulate the planetary gearbox vibration with planet bearing outer and inner race fault are constructed. Table 1 shows the corresponding parameter values.
Fig. 5. Simulation case with the inner race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram, (e) envelope spectrum of the filtered result using the selected optimal filter band.
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Fig. 4(a) shows the simulated planetary signal with the planet bearing outer race fault. Fig. 4(b) shows the corresponding kurtogram obtained using the traditional SK algorithm, where the frequency band with the highest SK value is located around 2000 Hz. The corresponding center frequency (2133 Hz) is close to the preset gear-related resonance frequency(2000 Hz). Hence, it is difficult to identify PBFR frequency using the traditional SK algorithm. As such, the algorithms proposed in Section 2 are employed to analyze the simulation signals. First, the filter results acquired in calculating the kurtogram are obtained, along with the indexMFM values of all these filter signals. Fig. 4(c) presents these index values in a manner similar to a kurtogram. Here, only the frequency bands around 2000 Hz possess relatively high indexMFM values, the planet bearing-induced resonance frequency band cannot be identified at all. It is proven that the indexMFM can be used to determine which frequency bands are related to the gear meshing and which are not. Then, the median of these index values is set as the threshold. Here, the threshold equals 0.0034. With the indexMFM-based threshold, some SK values of the original raw kurtogram are set as zero according to the rule introduced in Section 2.2. Fig. 4(d) shows the indexMFM-based kurtogram, where the gear meshing-related areas are removed, and the planet bearing-related area around 6000 Hz can be identified. In the area around 6000 Hz, the frequency band (center frequency: 5920 Hz, level: 5) with the highest SK value is selected as the potential filter band. For the final fault detection, the raw signal is filtered with the selected potential filter band, and the envelope spectrum of the filter result is displayed in Fig. 4(e). Here, the colored vertical lines with different types show the planet bearing envelope fault pattern (outer race fault). In detail, the blue solid lines represent the 1st and 2nd FCFs; the red dashed lines represent the PRF sidebands (including the left RPF (LPRF) and right RPF (RPRF)) around the FCF; the black dotted lines represent the CRF sidebands (including the LCRF and RCRF) around the PRF sidebands; and the green dash-dotted lines represent the CRF sidebands (including LCRF and RCRF) around the FCF. The obtained envelope spectrum matches the envelope fault pattern. Thus, the effectiveness of the proposed algorithm is partly confirmed.
Fig. 6. Analysis results obtained with different SNRs. (a) indexMFM-based kurtogram under -5 dB, (b) indexMFM-based kurtogram under 10 dB, (c) indexMFMbased kurtogram under 20 dB, (d) optimal envelope spectrum under -5 dB, (e) optimal envelope spectrum under 10 dB, (f) optimal envelope spectrum under 20 dB.
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Fig. 7. Experimental set-up for the planet bearing fault diagnosis.
Furthermore, simulated planetary gearbox vibration with a planet bearing inner race fault is used to testify the robustness of the proposed algorithm. Fig. 5 (a) ~ (e) are the counterparts of Fig. 4(a) ~ (e), respectively. In detail, Fig. 5(a) and (b) show the raw vibration and kurtogram, respectively. Similar to Fig. 4(b), the frequency bands with prominent SK values are related to gear meshing vibration, and the planet bearing-induced resonance frequency bands around 6000 Hz are less prominent and difficult to identify. Fig. 5(c) ~ (e) show the analysis results obtained using the proposed algorithm. Fig. 5(c) shows all the indexMFM values, from which only the gear meshing-related frequency bands (around 2000 Hz) can be identified. On the basis of all the indexMFM values, the threshold in inner race fault detection is set as 0.0033. According to the rules described in Section 2.2, Fig. 5(d) shows the indexMFM-based kurtogram, from which concentrated frequency bands with high SK values are observed around 6000 Hz. Hence, the planet bearing inner race faultinduced resonance frequency band can also be identified, and the center frequency and the level of the optimal resonance frequency band are 6080 Hz and 4, respectively. Fig. 5(e) compares the envelope spectrum of the filter result obtained using the optimal filter band with the envelope fault pattern for the planet bearing inner race fault, revealing that the envelope spectrum matches the pattern very well. In addition, three other simulated signals with different SNRs are added to investigate how the algorithm proposed in Section 2 performs under different noise conditions. Here, it is worth mentioning that the gear interruption is considered as the main noise rather than the signal toward the planet bearing fault-induced vibration. Hence, the SNR of the simulated signal should be reset by changing the amplitude of the gear magnitude, rather than the background white noise. Fig. 6(a)e(c) show the indexMFM-based kurtograms under different SNRs: 5, 10, and 20 dB. Fig. 6(d)e(f) show the corresponding envelope spectrum with selected optimal frequency bands chosen from Fig. 6(a)e(c). In the case of the SNR of 5 dB, the preset PBFR frequency (6000 Hz) can be also located in the indexMFM-based kurtogram shown in Fig. 6(a) with the frequency-band parameters (center frequency: 6080 Hz; level: 4). The corresponding envelope spectrum of the filtered result is shown in Fig. 6(d), where the FCF and the carrier sidebands are clearly observed. In the case of the SNR of 10 dB, the preset bearing fault-related frequency band is not as prominent as that shown in Fig. 6(a). With the given frequency-band parameters (center frequency: 6240 Hz; level: 2), the envelope spectrum of the filtered result is shown in Fig. 6(e). In contrast to Fig. 6(d), although the planet bearing FCF and the sidebands can also be located, other spectral interruptions are much higher. The SNR of 20 dB is the worst case. On one hand, the planet bearing-related frequency band almost cannot be located from the indexMFM-based kurtogram shown in Fig. 6(c). The selected frequency band with the parameters (center frequency: 7333 Hz; level: 2) is far from the preset one. On the other hand, the planet bearing FCF and the sidebands are too weak to be detected in the corresponding envelope spectrum of the filtered result.
Fig. 8. Planet bearing damage: (a) outer race fault, (b) inner race fault.
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Table 2 Parameters and the corresponding characteristic frequencies of the planetary gearbox and its planet bearing.
Planetary gearbox parameters
Planetary rolling bearing parameters
Characteristic frequencies of the planetary gearbox and its planet bearing
Parameter
Specific value
Sun gear teeth Planetary gear teeth Ring gear teeth Bearing type Number of balls Fitch diameter (mm) Ball diameter (mm) Contact angle (º) Rotational frequency of sun gear (Hz) Rotational frequency of carrier (Hz) Rotational frequency of planetary gear (Hz) FCFouter of planet bearing (Hz) FCFinner of planet bearing (Hz)
13 38(3) 92 NJ304 10 36 9 0 24.97 3.09 4.39 16.47 27.45
Table 3 Theoretical values of characteristic frequencies in the fault envelope pattern. Characteristic frequency
Theoretical value of characteristic frequencies Fault envelope pattern for outer race fault
Fault envelope pattern for inner race fault
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF RCRF sideband of LPRF RPRF sideband of 1st FCF LCRF sideband of RPRF RCRF sideband of RPRF LCRF sideband of 1st FCF RCRF sideband of 1st FCF 2nd FCF
16.47 12.08 8.99 15.17 20.86 17.77 23.95 19.56 13.38 32.92
27.45 N/A N/A N/A N/A N/A N/A 24.36 30.54 54.41
Fig. 9. Planet bearing with an outer race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in the frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram.
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4. Experimental tests For this section, both the test rig (at the University of Science and Technology, Beijing lab) and the experimental data are the same as those used in Ref. [2]. In addition to confirming the effectiveness of the proposed indexMFM-based kurtogram algorithm for planet bearing fault detection, the corresponding analysis results can be compared with the results obtaining using the algorithm proposed in Ref. [2]. The layout of the test rig and the faulty planet bearings are shown in Figs. 7 and 8, respectively. As shown in Fig. 7, the planetary gearbox is driven by a motor, and the corresponding rotational speed is measured by the tachometer placed between the motor and the monitored planetary gearbox. An electromagnetic brake is mounted with the output shaft of the planetary gearbox to provide additional torque. During the measurement, the accelerometer, which is mounted on the top of the planetary gearbox (indicated by a red arrow), is used to obtain the vibration signal. Fig. 8 shows the artificially added localized faults on the outer race and the inner race of a planet bearing. As shown, two cutting defects are separately added on these two places. The corresponding parameters and the theoretical values of the characteristic frequencies of the fault envelope pattern are shown in Tables 2 and 3, respectively. 4.1. Planet bearing with outer race fault For the case with a planet bearing outer race fault, the raw signal and the corresponding raw kurtogram are shown in Fig. 9(a) and (b), respectively.
Fig. 10. Planet bearing with an outer race fault: (a) envelope spectrum of the filtered result with parameters of (3680, 5), (b) envelope spectrum of the filtered result with parameters of (7200, 5), (c) envelope spectrum of the filtered result with parameters of (6400, 2).
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Table 4 Errors with different filtering parameters in the case where the outer race of the planet bearing has a fault.. Characteristic frequencies
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF RCRF sideband of LPRF RPRF sideband of 1st FCF LCRF sideband of RPRF RCRF sideband of RPRF LCRF sideband of 1st FCF RCRF sideband of 1st FCF 2nd FCF Average error
Errors with different filtering parameters fc ¼ 3680 k¼5
fc ¼ 7200 k¼5
fc ¼ 6400 k¼2
Optimal parameters in Ref. [2]
0.07 0.04 0.013 0.04 0.02 0.02 0.03 0.03 0.08 0.09 0.0433
0.04 0.03 0.034 0.04 0 0.04 0.01 0.04 ✕ 0.07 0.0337
0.04 0.01 0.066 0 0.11 ✕ 0.05 0.03 0.01 0.09 0.0451
0.04 0.02 0.034 0.05 0 0.02 0.02 ✕ ✕ 0.08 0.033
In Fig. 9(b), the frequency bands with higher SK values are concentrated around 1000 Hz. It is stated in Ref. [2] that the area around 1000 Hz is not related to the planet bearing fault. Hence, the traditional SK algorithm cannot identify the weak PBFR frequency band. Here, the level for the kurtogram calculation is set as 6 according to Eq. (1). Using the algorithms proposed in Section 2, the indexMFM values of all the decomposition results are calculated, as shown in Fig. 9(c), in which the frequency bands with relatively high indexMFM values are concentrated below 2000 Hz, and the center frequency of the frequency band with the highest indexMFM value is 640 Hz. It can be seen that the area of higher SK values in Fig. 9(b) almost coincides with the area of higher indexMFM values in Fig. 9(c). This phenomenon confirms that the indexMFM values can be used to identify the gear meshing-related frequency band in the raw kurtogram. With the help of the indexMFM values shown in Fig. 9(c), the threshold can be determined as the corresponding median. This, together with the rules introduced Section 2.2.2 yields the indexMFM-based kurtogram shown in Fig. 9(d). Comparing the raw kurtogram in Fig. 9(b) and the indexMFM-based kurtogram in Fig. 9(d) reveals that the area with prominent SK values below 2000 Hz in the raw kurtogram is not observed in Fig. 9(d). This confirms that the proposed algorithm can remove the gear meshing-related frequency bands with prominent SK values from the raw kurtogram.
Fig. 11. Planet bearing with an inner race fault: (a) raw waveform, (b) raw kurtogram, (c) distribution of the indexMFM in frequency and level domains, (d) potential filter band in the indexMFM-based kurtogram.
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Furthermore, the potential optimal filter bands can be selected from the indexMFM-based kurtogram. Specifically, there are three areas (marked by the red dashed rectangles) with relatively high SK values in Fig. 9(d). According to the analysis in Section 2, these three areas may contain the planet bearing fault-induced frequency bands, and the three frequency bands with the highest SK values in each area are considered as the potential PBFR frequency bands. The parameters of the potential PBFR frequency bands are (3680, 5), (7200, 5), and (6400, 2), where the first value inside the parentheses is the center frequency in hertz and the second one is the level. With these three frequency bands, the raw signal is filtered. Fig. 10 shows the corresponding envelope spectra of the filtered results. In these three envelope spectra, the colored straight lines represent the fault envelope pattern for the planet bearing outer race fault constructed according to Table 2. If prominent peaks can be located at the specific theoretical values, the corresponding x-axis value of the prominent peak is shown at the top left of the straight lines, and the theoretical value is shown inside the bracket, such as “x-axis value (theoretical value)”. Otherwise, the corresponding expression follows the form of “(theoretical value)”. In Fig. 10(a), spectral peaks are observed at all of the theoretical values. In Fig. 10(b), there are no peaks at the RCRF sideband of the 1st FCF, and in Fig. 10(c), no peak is observed at the LCRF sideband of the RPRF. It should be mentioned that, the meanings of different types of straight line are as same as the ones introduced in the simulation cases. The differences between the theoretical values and the x-axis values of the detected peaks at different characteristic frequencies are defined as the errors and shown in Table 4. If no peak exists at the theoretical characteristic frequency, “ ” is shown instead of the error value. A small error means that the FCF has been located. Hence, the frequency band with the parameters of (3680, 5) is the optimal planet PBFR frequency band, and the monitored planet bearing can be considered as having an outer race fault.
Fig. 12. Planet bearing with an inner race fault: (a) envelope spectrum of the filtered result with parameters of (3040, 5), (b) envelope spectrum of the filtered result with parameters of (7040, 4.5), (c) envelope spectrum of the filtered result with parameters of (6400, 2).
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Table 5 Errors with different filtering parameters in the case where the inner race of the planet bearing has a fault.. Characteristic frequencies
1st FCF LPRF sideband of 1st FCF LCRF sideband of LPRF 2nd FCF Average error
Errors with different filtering parameters fc ¼ 3040 k¼5
fc ¼ 7040 k ¼ 4.5
fc ¼ 6400 k¼2
Optimal parameters in Ref. [2]
0.11 0.06 0.06 0 0.0575
0.02 ✗ 0.13 0.06 0.07
0.01 0.01 0.09 0.06 0.0425
0.01 0.03 0.05 0.01 0.033
The errors of the traditional algorithm proposed in Ref. [2] are also listed in Table 4(the last column). Although the average error of the proposed algorithm is higher than that (0.033) of the traditional algorithm, the average error of 0.0433 is low enough for a convincing diagnosis result. Moreover, the proposed indexMFM-based method does not require a health baseline signal. 4.2. Planet bearing with inner race fault In this subsection, the proposed indexMFM-based algorithm is validated using experimental data for a planet bearing with an inner race fault. The test rig and the parameters are the same as those introduced in Section 4.1. Figs. 11 and 12 are the counterparts of Figs. 9 and 10, respectively. Fig. 11(a) shows the raw waveform, and Fig. 11 (b) and (c) shows the raw kurtogram and the distribution of the indexMFM values, respectively. According to these two figures, the areas with prominent SK and indexMFM values are similar and are induced by the gear meshing-related frequency band. This similarity suggests that the traditional SK based algorithm cannot identify the PBFR frequency band, and the indexMFM can detect the gear meshing-related area. Using indexMFM values shown in Fig. 11(c), the threshold is set as 4.18 107. Fig. 11(d) shows the indexMFM-based kurtogram, based on which, three PBFR frequency bands can be determined. The corresponding parameters are (3040,5), (7040,4.5), and (6400,2), respectively. Fig. 12(a)e(c) show the envelope spectra obtained using the selected frequency bands together with the fault envelope pattern of the planet bearing inner race fault. The errors between the theoretical values and the x-axis values of the selected peaks are listed in Table 5, among which the frequency band with the parameters of (6400,2) represents the optimal resonance frequency band. With this frequency band, spectral peaks are observed in all of the theoretical values, indicating that the monitored planet bearing has an inner race fault. For comparison, the errors obtained using the traditional algorithm are listed in Table 5. The average error obtained using the proposed algorithm is 0.0425. Although this is larger than the average error for the traditional algorithm, 0.0425 is a small enough error for a credible fault detection result. 5. Conclusion An indexMFM-based kurtogram algorithm was proposed to solve the problem of planet bearing fault detection without using any auxiliary equipment or a health baseline. To eliminate the strong gear interruption from the frequency domain, a new indexMFM is formulated to identify which frequency bands result from the gear meshing vibration in the raw kurtogram. The plant bearing fault-induced resonance frequency bands can be identified from the indexMFM-based kurtogram. Simulation and experimental tests demonstrate the accuracy of the fault envelope pattern and the effectiveness of the planet bearing fault detection algorithm, respectively. Acknowledgment The research work described in this paper was supported by National Natural Science Foundation of China under Grant Nos. 51605244, 51335006. The valuable comments and suggestions from the editor and two reviewers are very much appreciated. References [1] [2] [3] [4]
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