Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions

Yong Ren; Wei Li; Bo Zhang; Zhencai Zhu; Fang Jiang

doi:10.3390/app9061157

Ren, Yong;Li, Wei;Zhang, Bo;Zhu, Zhencai;Jiang, Fang

2019-03-19 00:00:00

applied sciences Article Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions Yong Ren , Wei Li *, Bo Zhang , Zhencai Zhu and Fang Jiang School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, China; reny_cumt@163.com (Y.R.); zbcumt@163.com (B.Z.); zzc_cmee@163.com (Z.Z.); jiangfan25709@163.com (F.J.) * Correspondence: liwei_cmee@163.com Received: 12 February 2019; Accepted: 12 March 2019; Published: 19 March 2019 Abstract: Envelope analysis is a widely used method in fault diagnoses of rolling bearings. An optimal narrowband chosen for the envelope demodulation is critical to obtain high detection accuracy. To select the narrowband, the fast kurtogram (FK), which computes the kurtosis of a set of ﬁltered signals, is introduced to detect cyclic transients in a signal, and the zone with the maximum kurtosis is the optimal frequency band. However, the kurtosis value is affected by rotating frequencies and is sensitive to large random impulses which normally occur in industrial applications. These factors weaken the performance of the FK for extracting weak fault features. To overcome these limitations, a novel feature named Order Spectrum Correlated Kurtosis (OSCK) is proposed, replacing the kurtosis index in the FK, to construct an improved kurtogram called Fast Order Spectrum Correlated Kurtogram (FOSCK). A band-pass ﬁlter is used to extract the optimal frequency band signal corresponding to the maximum OSCK. The envelope of the ﬁltered signal is calculated using the Hilbert transform, and a low-pass ﬁlter is employed to eliminate the trend terms of the envelope. Then, the non-stationary ﬁltered envelope is converted in the time domain into the stationary envelope in the angular domain via Computed Order Tracking (COT) to remove the effects of the speed ﬂuctuation. The order structure of the angular domain envelope signal can then be used to determine the type of fault by identifying its characteristic order. This method offers several merits, such as ﬁne order spectrum resolution and robustness to both random shock and heavy noise. Additionally, it can accurately locate the bearing fault resonance band within a relatively large speed ﬂuctuation. The effectiveness of the proposed method is veriﬁed by a number of simulations and experimental bearing fault signals. The results are compared with several existing methods; the proposed method outperforms others in accurate bearing fault feature extraction under varying speed conditions. Keywords: fault diagnosis; fast kurtogram; order spectrum correlated kurtosis; rolling bearing; non-stationary 1. Introduction Rolling bearings are among the most commonly used support elements in rotating machinery. They are prone to faults under harsh working conditions. When a fault occurs on the inner or outer race of a bearing, a series of impulses is generated in the vibration signal as the bearing defect interacts with another surface, and the impacts excite high-frequency resonances where the signal-to-noise ratio (SNR) is higher than the other frequency regions in the bearing system, thereby inducing a modulating phenomenon [1,2]. However, many other sources of bearing vibration such as the waviness of rolling elements etc. always result in the emergence of side bands around the principal bearing frequencies, which are more pronounced at higher frequencies [3]. Therefore, accurately Appl. Sci. 2019, 9, 1157; doi:10.3390/app9061157 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1157 2 of 31 determining the high-frequency resonance band where the impulse occurs is key to successfully detecting bearing faults. In earlier research, the resonance band is often determined by experimental tests, which are time-consuming [4]. As a statistical index, kurtosis is sensitive to the peaks caused by abnormal vibrations, and it is usually used as a direct measure of the transient impulses of the signal. Nevertheless, kurtosis is easily affected by noise. To overcome this limitation, Frequency Domain Kurtosis (FDK) was proposed by Dwyer [5] to complement the ﬂaw of classical power spectral density (PSD), i.e., not being sensitive to the statistical nature of the signal. Inspired by this proposal, Spectral Kurtosis (SK) was presented by Antoni based on the Wold-Cramer theorem for non-stationary feature extraction in [6]. The basic idea of this approach is that the kurtosis at each frequency line of a signal is calculated to discover the presence of transients, and to indicate in which frequency bands these occur. For the convenience of industrial applications, Antoni further proposed the Fast Kurtogram (FK) using short-time Fourier transform (STFT) combined with 1/3 binary tree algorithms to split frequency bands to reduce computing time, as described in [7]. The fault impulses are extracted after a raw signal is processed by a band-ﬁlter for which the center frequency and bandwidth are optimized by FK. Since then, many studies have been conducted to enhance these theories [8–11]. Considering the problem whereby the parameters of the band-ﬁlter cannot be determined adaptively, Zhang et al. [12] combined genetic algorithms and FK to optimize the parameters. To extract transient impulsive signals under a low SNR condition, Wang et al. [13] proposed a time-frequency analysis method which combines the merits of ensemble local mean decomposition and FK to detect bearing faults. In [14], Lei proposed replacing STFT with wavelet package transform (WPT) to improve the kurtogram (WPTK). Recently, Wang proposed an enhanced kurtogram, in which the kurtosis values are calculated based on the envelope power spectrum of WPT nodes at different depths [15]. It is worth mentioning that each kurtosis value of the ﬁltered signal is calculated without source identiﬁcation in these methods, which is sometimes incorrect, especially when the vibration signal contains random knocks which usually have higher amplitudes, as well as kurtosis values which are far larger than those of real faults [4,16]. This effect means that the optimal frequency band corresponding to the maximum kurtosis is the resonance band containing random knocks, while the real fault signature is missing. To solve this problem, Barszcz et al. [17] proposed a higher resolution kurtosis index, the Protrugram, which is obtained by calculating the kurtosis of the envelope spectrum amplitudes of a narrow band ﬁltered signal along the frequency axis. However, the optimal ﬁlter bandwidth depends on a certain knowledge of the sought fault. In [18], McDonald took advantage of the periodicity of the faults and proposed Correlated Kurtosis (CK) to detect cyclic transients. To make the extracted fault characteristic clear, they proposed an iterative selection process for the ﬁrst and M-shift to maximize the CK. Combining CK and Redundant Second-Generation Wavelet Package Transform (RSGWPT), Chen proposed an improved kurtogram in [19]. In addition to these high-frequency resonance techniques, a non-resonance-based approach is desirable in an industrial environment, such as the Auto-regression moving average [20] and higher-order energy operator fusion methods [21]. The effectiveness of these methods has been veriﬁed when a shaft rotates at constant speeds. However, bearings usually operate at variable speed conditions in practice, which leads the fault features to no longer be discrete frequency lines, but rather, frequency bands related to the shaft rotating frequency [22–24]. During speed up and speed down processes, impulses induced by the faults are non-periodic in the time domain, which means that the method based on the indexes derived from the kurtosis index will be weakened in non-stationary feature extraction. Therefore, the question of how to recover the fault impulses from the signal collected in the varying speed conditions must be solved. In this paper, a new index, Order Spectrum Correlation Kurtosis (OSCK) is proposed. By replacing the OSCK with the kurtosis in the FK, an improved kurtogram, Fast Order Spectrum Correlation Kurtogram (FOSCK) is constructed. In this method, the original non-stationary vibration signal is ﬁltered by a 1/3 binary tree strategy, the envelope of each ﬁltered signal is calculated by using the Appl. Sci. 2019, 9, 1157 3 of 31 Hilbert transform, and the trend term of each envelope is eliminated by a low-pass ﬁlter whose cutoff frequency is lower than the minimum of the rotating frequency. Then each envelope signal is resampled into a stationary one in the angular domain using the Computed Order Tracking (COT) technique to remove the effects of speed ﬂuctuation. The OSCK of each resampling envelope signal is calculated and utilized to generate a diagram in which the frequency band corresponding to the highest value can then be considered for further analysis. A band-pass ﬁlter is set to maintain the desired band, and is used to extract the optimal frequency band signal. After that, the envelope of the ﬁltered signal is resampled into angular domain by using the COT. The order structure of the angular domain envelope signal can be used to determine the type of fault by identifying its characteristic order. Compared with the FK, the WPTK and the Protrugram, the proposed method can extract bearing fault characteristic information more exactly under relatively large speed ﬂuctuations and heavy interference environments. 2. Theoretical Background 2.1. Overview of Spectral Kurtosis and Fast Kurtogram To overcome the shortcomings of the power spectral density (PSD), which is not sensitive to the statistical nature of the signals, the frequency domain kurtosis (FDK) was ﬁrst introduced by Dwyer. It can highlight the frequency harmonic that is smeared because of random variation in the periodicity [5]. Inspired by this development, Antoni proposed spectral kurtosis (SK) in [6]. Different from FDK, which calculates the kurtosis of a particular frequency’s amplitude, SK calculates the kurtosis of the complex envelope of ﬁltered signals [17]. According to the Wold-Cramér decomposition theorem, a zero-mean non-stationary signal x(n) can be expressed as [7,25]: +l/2 j2p f n x(n) = H(n, f )e dX( f ) (1) l/2 where dX( f ) is a spectral increment and H(n, f ) is the complex envelope of x(n) at frequency f . The SK can be deﬁned as the fourth-order normalized cumulant [7]: D E j H(n, f )j K ( f ) = 2 (2) x D E j H(n, f )j where the symbol h i denotes the temporal average operator. The constant 2 is used here because H(n, f ) is complex. Considering the presence of added noise, the SK of the non-stationary process x(n) is described by: K ( f ) K ( f ) = (3) [1 + r( f )] where r( f ) is the noise-to-signal ratio at frequency f . The transients in signals increase the spectral kurtosis value. Therefore, SK possesses the ability to detect and localize the presence of transients from a signal. However, to detect a narrow-band transient buried in noise, SK depends both on frequency and frequency resolution. Although this can be performed by computing the SK of each combination of different central frequencies f and bandwidths (D f ) , this process is time consuming. To improve the calculation efﬁciency, Antoni utilized 1/3 binary tree algorithms to split the frequency band; then, the SK of each level m and bandwidth (D f ) were calculated to construct a 3D map-fast kurtogram, as shown in Figure 1. The horizontal axis represents the frequency, the vertical axis represents the level of the split frequency band, and the third dimension represents the color band, which is the kurtosis value of the ﬁltered signal’s envelope spectrum for each frequency bandwidth. The node with the largest kurtosis is chosen as the optimal band. However, the fast kurtogram calculates each kurtosis value of the ﬁltered signal without source identiﬁcation, which is sometimes incorrect. Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 35 However, the fast kurtogram calculates each kurtosis value of the filtered signal without source Appl. Sci. 2019, 9, 1157 4 of 31 identification, which is sometimes incorrect. Levels (∆ƒ) K(0,1) 0 1/2 K(1,1) K(1,2) 1 1/4 K(1.6,1) K(1.6,2) K(1.6,3) 1.6 1/6 K(2,1) K(2,2) K(2,3) K(2,4) 2 1/8 K(2.6,1) K(2.6,2) K(2.6,3) K(2.6,4) K(2.6,5) K(2.6,6) 2.6 1/12 K(3,1) K(3,2) K(3,3) K(3,4) K(3,5) K(3,6) K(3,7) K(3,8) 3 1/16 … … m+1 K(m,1) K(m,n) m 1/2 … … … … 1/81/4 3/81/2 Figure 1. The paving of the Fast Kurtogram. Figure 1. The paving of the Fast Kurtogram. 2.2. Order Spectrum Correlated Kurtosis 2.2. Order Spectrum Correlated Kurtosis In [18], McDonald found that the kurtosis value of a signal with a single impulse is always higher In [18], McDonald found that the kurtosis value of a signal with a single impulse is always than a signal containing consecutive periodicity of impulses. To solve the effect of random transients higher than a signal containing consecutive periodicity of impulses. To solve the effect of random on kurtosis, McDonald proposed Correlated Kurtosis (CK) to detect signal cyclic transients by using transients on kurtosis, McDonald proposed Correlated Kurtosis (CK) to detect signal cyclic the periodicity of the fault nature, and veriﬁed that CK can decrease the inference of single transient transients by using the periodicity of the fault nature, and verified that CK can decrease the impulses. The CK of a vibration signal x is deﬁned as [14]: inference of single transient impulses. The CK of a vibration signal x is defined as [14]: N M N M  å Õ x nmT  ∏ nm − T n=1 m=0 n=1 m=0 CK(x) = (4) CK() x = M+1 (4) M +1  å x 2  n n=1 n=1  where N is the length of x, T is the period of interest impulses, and M is the CK shift. where N is the length of x , T is the period of interest impulses, and M is the CK shift. In [26], it was also observed that the kurtosis value is considerably affected by shaft rotational In [26], it was also observed that the kurtosis value is considerably affected by shaft rotational frequency. To eliminate the inﬂuence of speed on kurtosis and the frequency smear, COT is employed frequency. To eliminate the influence of speed on kurtosis and the frequency smear, COT is to convert the non-stationary ﬁltered envelope time signal into the stationary vibration in the angular employed to convert the non-stationary filtered envelope time signal into the stationary vibration in domain. Based on the key-phase signal, which is used to obtain the sampling-time marks of the the angular domain. Based on the key-phase signal, which is used to obtain the sampling-time marks even-angle sampling, an interpolation scheme is employed for resampling the original time-domain of the even-angle sampling, an interpolation scheme is employed for resampling the original signal into the angular domain. Here, we use cubic spline interpolation. After that, the envelope order time-domain signal into the angular domain. Here, we use cubic spline interpolation. After that, the spectrum is utilized to expose the order structure in the signal, and the fault characteristic order (FCO) envelope order spectrum is utilized to expose the order structure in the signal, and the fault can be indicated clearly. characteristic order (FCO) can be indicated clearly. Previous research found that kurtosis is sensitive to external interference, especially in varying Previous research found that kurtosis is sensitive to external interference, especially in varying speed conditions, and the components contained in the signal have a single transient characteristic. speed conditions, and the components contained in the signal have a single transient characteristic. Thus, it is difficult to detect fault sensitive components. To address the above problem, the order Thus, it is difﬁcult to detect fault sensitive components. To address the above problem, the order spectrum analysis is combined with correlated kurtosis to form a new feature, OSCK, to detect the spectrum analysis is combined with correlated kurtosis to form a new feature, OSCK, to detect the fault-sensitive frequency band under varying speed conditions. fault-sensitive frequency band under varying speed conditions. The OSCK can be defined as follows: The OSCK can be deﬁned as follows: N M  N M  ∏Anm − T å Õ nmT o n=1 m=0 n=1 m=0 OSCK(, A T) = OSCK( A, T) = (5) (5) M +1 M+1  å AA  n  n=1  n=1 where A is the envelope order spectrum amplitudes, and T is the period of impulses. o Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 35 Appl. Sci. 2019, 9, 1157 5 of 31 where A is the envelope order spectrum amplitudes, and T is the period of impulses. 2.3. Procedure of the Proposed Method 2.3. Procedure of the Proposed Method Based on the discussion above, an improved kurtogram is proposed for rolling bearing fault Based on the discussion above, an improved kurtogram is proposed for rolling bearing fault diagnosis under varying speed conditions. The optimal frequency band corresponding to the maximum diagnosis under varying speed conditions. The optimal frequency band corresponding to the OSCK in the kurtogram is ﬁltered; then, the fault can be identiﬁed by envelope order spectrum analysis maximum OSCK in the kurtogram is filtered; then, the fault can be identified by envelope order of the ﬁltered signal. The scheme of the proposed method is shown in Figure 2, and the details are spectrum analysis of the filtered signal. The scheme of the proposed method is shown in Figure 2, described as follows: and the details are described as follows: Original vibration signal x(n) 1/3 binary tree algorithm to split frequency band Hilbert transform The envelope signal The envelope signal The envelope signal The envelope signal …… 1 2 2M-1 2M C (n) C (n) C (n) C (n) 1 1 M M 1 2 2M-1 2M The COT of C (n) The COT of C (n) The COT of C (n) The COT of C (n) 1 1 M M Key-phase 1 2 2M-1 2M and y (n) is and y (n) is …… and y (n) is and y (n) is 1 1 M M signal v(n) obtained obtained obtained obtained Autocorrelation Autocorrelation Autocorrelation Autocorrelation 1 2 2M-1 2M analysis of y (n) analysis of y (n) analysis of y (n) analysis of y (n) 1 1 M M 1 2 …… and the period T (n) and the period T (n) and the period and the period 1 1 2M-1 2M is obtained is obtained T (n) is obtained T (n) is obtained M M The order spectrum The order spectrum The order spectrum The order spectrum 1 2 2M-1 2M amplitude A (n) is amplitude A (n) is amplitude A (n) amplitude A (n) 1 1 2M M obtained by FFT of obtained by FFT of …… is obtained by FFT of is obtained by FFT 1 2 2M-1 2M y (n) y (n) y (n) of y (n) 1 1 M M 1 2 2M-1 2M-1 OSCK OSCK …… OSCK OSCK 1 1 M M The optimal frequency band corresponding to the largest OSCK The envelope order spectrum analysis by using the COT and FFT Figure 2. The ﬂowchart of proposed method. Figure 2. The flowchart of proposed method. Step 1. The original vibration signal x(n) and synchronous sampling key-phase signal v(n) Step 1. The original vibration signal x() n and synchronous sampling key-phase signal vn () measured by different accelerometers are loaded. measured by different accelerometers are loaded. Step 2. The signal x(n) is ﬁltered with a 1/3 binary tree strategy. Let h(n) be a low-pass prototype Step 2. The signal x() n is filtered with a 1/3 binary tree strategy. Let hn () be a low-pass ﬁlter, and two quasi-analytic low-pass and high-pass analysis ﬁlters h (n) and h (n) are constructed, l h prototype filter, and two quasi-analytic low-pass and high-pass analysis filters hn () and hn () which have the frequency bands [0; 1/4] and [1/4; 1/2], respectively: l h are constructed, which have the frequency bands [0; 1/4] and [1/4; 1/2], respectively: jpn/4 h (n) = h(n)e (6) jπ n/4 hn () =h(n)e (6) j3pn/4 2 h (n) = h(n)e , j = 1 (7) h Appl. Sci. 2019, 9, 1157 6 of 31 Different central frequency f and bandwidth (D f ) corresponding signals are iteratively ci obtained by using these ﬁlters in a pyramidal manner, which has tree-structured ﬁlter-banks and i m denote as x (n), where i = 1, 2, , 2 , m = 0, 1, , M 1, and M is the largest decomposition i i i level [7]. The envelope of each ﬁltered signal C (n) = x (n) + j Hil[x (n)] can be created, where m m m Hil is the Hilbert transform and the symbol represent the absolute value. The trend term of each j j envelope is eliminated by a low-pass ﬁlter whose cutoff frequency is lower than the minimum of the rotating frequency. Step 3. Each ﬁltered envelope signal C (n) is resampled in the angular domain. Each ﬁltered envelope signal is non-stationary in the time domain due to the variable speed operations that cause spectrum smearing and low autocorrelation. To solve this problem, COT is employed to convert the non-stationary envelope signal into the stationary envelope signal in the angular domain by using the key-phase signal v(n), whose length is the same as the original signal x(n). The resampling envelope signal denote as y (n). Step 4. The OSCK of each resampling envelope signal is calculated. First, the autocorrelation analysis of y (n) is performed to enhance the involved periodic impulsive feature and the autocorrelation coefﬁcient can be calculated by the following formula: h i h i i i i i å y y y y m,j m m,j+t m j=1 R = s (8) h i h i n n 2 2 i i i i y y y y å å m,j m m,j+t m j=1 j=1 i i where R is the autocorrelation coefﬁcient, y is the average value of signal y (n), and t denotes the length of the delay. Through autocorrelation operation, the periodic impulsive signal component related to the bearing fault is strengthened. The period T of impulses of interest is denoted as: T = argmax(R ) (9) o t Second, the order spectrum of y (n) is obtained by Fourier transform and the order spectrum amplitude denote as A (n). Last, the OSCK values are calculated using Equation (5). The OSCK values of all nodes are represented in the kurtogram. Step 5. The frequency band corresponding to the maximum OSCK value are ﬁltered by a band-pass ﬁlter, and the envelope of the ﬁltered signal is transformed into angular domain by using the COT, and the envelope order spectrum is used to map the angle domain signal to the order-dependent signal to identify the bearing fault characteristic order, which is usually calculated by Equations (10)–(12). The outer race fault characteristic order FCO , the inner race fault characteristic order FCO and the rolling element fault characteristic order FCO are formulated as follows: Z d FCO = 1 cos a (10) 2 D Z d FCO = 1 + cos a (11) 2 D Z d FCO = 1 ( ) cos a (12) 2d D where Z is the number of rolling elements, a is the contact angle, and d and D are the diameter of the rolling element and pitch diameter, respectively. Appl. Sci. 2019, 9, 1157 7 of 31 3. Simulations In this section, several simulations are used to demonstrate the effectiveness of the proposed method. Considering the complexity of the rotating system, the synthetic signals usually include three terms: deterministic components, including the fundamental frequency and harmonics of the shaft, which are caused by factors such as misalignment, eccentricity or imbalance. Random components, which represent a series of impulses excited by a fault, and measurement noise. The simulated signal is deﬁned as: x(t) = A cos(2pm f (t)t + f ) + [1 + l M(t)] B s(t t t ) + n(t) (13) å m m å n n n |{z} m n | {z } | {z } the noise the deterministic components the random components components where A and f are the amplitude and initial phase of the mth harmonic frequency of the shaft, m m respectively; f (t) is the instantaneous rotating frequency of the shaft; 1 + l M(t) denotes the amplitude modulation term, 1, l = 0 i f bearing outer race f ault 1 + l cos(2p f t), l 6= 0 i f bearing inner race f ault 1 + l M(t) = (14) 1 + l cos(2p f t), l 6= 0 i f bearing rolling element f ault cage 0 normal where f is the cage speed; B and t are the amplitude and occurrence time of the nth impulse, and cage n n the occurrence time t is determined according to the instantaneous rotating frequency f (t) and the fault order frequency f ; t is the coefﬁcient used to calculate slippage time, which varies from 1% to o n 2% of the time period of the fault impulse; s(t) is the impulse response function of the system; and n(t) is the Gaussian white noise that is uncorrelated with other components. The impulse response function can be written as b(tt t ) n n e sinf2p f (t t t )g, i f t t > 0 r n n n s(t) = (15) 0 otherwise where b is the structural damping coefﬁcient, and f is the resonance frequency. 3.1. Simple Simulation for the Study of COT Analysis after Time-Domain Filtering In the FOSCK, the COT must be used after the envelope demodulation analysis. In order to explain this and determine the inﬂuence of COT on impulse feature extraction, a simple outer race simulated signal that has a single resonant frequency and consists of a series of pure impulses is shown in Figure 3. The simulation signal parameters in the model are given in Table 1, where f is sampling frequency Table 1. Parameters of the simulation model. N (s) B ' f (Hz) f f (kHz) f (kHz) (kHz) n m o s r n 3 1 0 5–15 4.5 20 5 1.2 0.01 The rotating frequency is given by f (t) = 10 + 5 sin(10pt) (16) and illustrated in Figure 3a. The corresponding time-domain signal and angle-domain resampling signal are shown in Figure 3b,c, respectively. It is clear that the intervals of adjacent impulse responses Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 35 The rotating frequency is given by f ()tt =+ 10 5×sin(10π ) (16) The rotating frequency is given by and illustrated in Figure 3a. The corresponding time-domain signal and angle-domain resampling f ()tt =+ 10 5×sin(10π ) (16) signal are shown in Figure 3b,c, respectively. It is clear that the intervals of adjacent impulse Appl. Sci. 2019, 9, 1157 8 of 31 and illustrated in Figure 3a. The corresponding time-domain signal and angle-domain resampling responses change synchronously with the rotating frequency in the time-domain signal; the rotating signal are shown in Figure 3b,c, respectively. It is clear that the intervals of adjacent impulse frequency is larger, and the adjacent impulse intervals are smaller. Unlike that of the time-domain responses change synchronously with the rotating frequency in the time-domain signal; the rotating change signal synchr , the pu onously lse intewith rval is unchan the rotating ged in frequency the ang in le- the domain si time-domain gnal. signal; the rotating frequency is frequency is larger, and the adjacent impulse intervals are smaller. Unlike that of the time-domain larger, and the adjacent impulse intervals are smaller. Unlike that of the time-domain signal, the pulse signal, the pulse interva (a)l is unchanged in the angle-domain signal. interval is unchanged in the angle-domain signal. (a) 0 0.5 1 1.5 2 2.5 3 Time [s] (b) 0 0.5 1 1.5 2 2.5 3 Time [s] (b) -1 0 0.5 1 1.5 2 2.5 3 (c) Time [s] -1 0 0.5 1 1.5 2 2.5 3 (c) Time [s] -1 0 20 40 60 80 100 120 140 160 180 Angle[rad] -1 0 20 40 60 80 100 120 140 160 180 Angle[rad] Figure 3. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time domain and (c) the resampling signal in the angle domain. Figure 3. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time domain and (c) the resampling signal in the angle domain. Figure 3. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time The short-time Fourier transforms (STFTs) of the time domain and angle domain impulse domain and (c) the resampling signal in the angle domain. responses are shown in Figure 4a,b, respectively. The carrier frequencies of the time-domain signal The short-time Fourier transforms (STFTs) of the time domain and angle domain impulse are concentrated around the resonant frequency, while the carrier orders of the angle-domain signal responses are shown in Figure 4a,b, respectively. The carrier frequencies of the time-domain signal The short-time Fourier transforms (STFTs) of the time domain and angle domain impulse spread to a wider order scope. Therefore, it can be concluded that the COT procedure causes are concentrated around the resonant frequency, while the carrier orders of the angle-domain signal responses are shown in Figure 4a,b, respectively. The carrier frequencies of the time-domain signal distortion of the signal resonance band, which is very important in the resonance demodulation spread to a wider order scope. Therefore, it can be concluded that the COT procedure causes distortion are concentrated around the resonant frequency, while the carrier orders of the angle-domain signal analysis. of the signal resonance band, which is very important in the resonance demodulation analysis. spread to a wider order scope. Therefore, it can be concluded that the COT procedure causes distortion of the signal resonance band, which is very important in the resonance demodulation (a) (b) analysis. (a) (b) Figure 4. STFTs of the simulation: (a) STFT of the time-domain signal and (b) STFT of the Figure 4. STFTs of the simulation: (a) STFT of the time-domain signal and (b) STFT of the angular-domain signal. angular-domain signal. InFigure 4. additionSTF to the Ts of the simulation: ( above defects, as stated a) STFT of the time-domain signal and ( by A.B. Ming in [27], a simple ﬁlter b with ) STFT of the a ﬁxed cutof f angular-domain signal. order cannot deal with the angle-domain signal, whose carrier orders vary over time. The envelopes obtained by a low pass ﬁlter with a ﬁxed cutoff frequency in the time domain and a ﬁxed cutoff order in the angle domain, as shown in Figure 5a,b, respectively. The amplitude of the ﬁltered signal in the angle domain is distorted. Therefore, an envelope demodulation analysis must ﬁrst be carried out in the time domain, then the COT method is performed on the envelope. Velocity[Hz] Amplitude Amplitude Velocity[Hz] Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 35 In addition to the above defects, as stated by A.B. Ming in [27], a simple filter with a fixed cutoff order cannot deal with the angle-domain signal, whose carrier orders vary over time. The envelopes obtained by a low pass filter with a fixed cutoff frequency in the time domain and a fixed cutoff order in the angle domain, as shown in Figure 5a,b, respectively. The amplitude of the filtered signal Appl. in the ang Sci. 2019,le d 9, 1157 omain is distorted. Therefore, an envelope demodulation analysis must first be carr 9 of 31 ied out in the time domain, then the COT method is performed on the envelope. (a) 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 Time[s] (b) 0.8 0.6 0.4 0.2 0 20 40 60 80 100 120 140 160 180 Angle[rad] Figure 5. Envelopes of the ﬁltered signals: (a) envelope of the time-domain ﬁltered signal and (bFigure 5. ) envelope Env of the elopes of angular the f -domain iltered ﬁlter signals: ( ed signal. a) envelope of the time-domain filtered signal and (b) envelope of the angular-domain filtered signal. 3.2. Simple Simulation of the Inﬂuence of Rotational Speed Indication 3.2. Simple Simulation of the Influence of Rotational Speed Indication In this simulation, to illustrate the effect of shaft rotational frequency on the different index values intuitively, the rotating frequency is given by In this simulation, to illustrate the effect of shaft rotational frequency on the different index values intuitively, the rotating frequency is given by f (t) = 10, t = 0 3s f (t) = 10 + 5 (t 3), t = 3 6s (17) f (tt )== 10, 0∼ 3s f (t) = 25, t = 6 9s f ()tt =+ 10 5×( −3), t= 3∼ 6s (17) f (tt )== 25, 6∼ 9s The other parameters are the same as those mentioned above. The simulation signal that only contains Appl. Sci. the 2018, pur 8, x FO e impulsive R PEER REVIEW signal is shown in Figure 6. 10 of 35 The other parameters are the same as those mentioned above. The simulation signal that only contains the pure impulsive signal is shown in Figure 6. (a) 0 1 2 3 4 5 6 7 8 9 Time[s] (b) -1 0 1 2 3 4 5 6 7 8 9 Time[s] (c) -1 0 100 200 300 400 500 600 700 800 900 Angle[rad] Figure 6. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time domain and (c) the resampling signal in angle domain. Figure 6. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time domain and (c) the resampling signal in angle domain. It is generally assumed that a high kurtosis value is treated as a sign of the presence of faults in a rotating mechanical system. However, this assumption has no application to the varying speed case. It is generally assumed that a high kurtosis value is treated as a sign of the presence of faults in To demonstrate this special condition, both of the time domain signal and angular domain signal are a rotating mechanical system. However, this assumption has no application to the varying speed case. To demonstrate this special condition, both of the time domain signal and angular domain signal are equally divided into 45 signal segments. A different index value of every signal segment is calculated respectively, and normalized to construct a vector, as shown in Figure 7, where the blue and red lines are denote the normalized kurtosis value and the normalized CK value of each time domain signal segment. The green and black lines are CK and OSCK values correspond to the angle-domain signal segments. Overall, the CK and kurtosis of the signal segment decrease as the speed increases, both in the time domain and the angle domain. It is worth noting that the OSCK of the angle domain signal segments is sensitive to the speed fluctuation while not being affected by the size of the speed. Therefore, the OSCK index is more suitable for the extraction of resonance bands under varying speed conditions. (a) 0 1 2 3 4 5 6 7 8 9 Time [s] time domain kurtosis (b) time domain CK angle domain kurtosis angle domain CK angle domain OSCK 0.5 0 5 10 15 20 25 30 35 40 45 Number Figure 7. (a) The shaft rotational frequency, (b) the normalized kurtosis of different indexes. 3.3. Simple Simulation of the Influence of Random Shocks Indication Normalized kurtosis Velocity[Hz] Velocity[Hz] Amplitude Amplitude Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 35 (a) 0 1 2 3 4 5 6 7 8 9 Time[s] (b) -1 0 1 2 3 4 5 6 7 8 9 Time[s] (c) -1 0 100 200 300 400 500 600 700 800 900 Angle[rad] Figure 6. Simple simulation: (a) the shaft rotational frequency, (b) the impulse signal in the time domain and (c) the resampling signal in angle domain. It is generally assumed that a high kurtosis value is treated as a sign of the presence of faults in Appl. a rotating Sci. 2019 ,me 9, 1157 chanical system. However, this assumption has no application to the varying 10speed of 31 case. To demonstrate this special condition, both of the time domain signal and angular domain signal are equally divided into 45 signal segments. A different index value of every signal segment is equally divided into 45 signal segments. A different index value of every signal segment is calculated calculated respectively, and normalized to construct a vector, as shown in Figure 7, where the blue respectively, and normalized to construct a vector, as shown in Figure 7, where the blue and red lines and red lines are denote the normalized kurtosis value and the normalized CK value of each time are denote the normalized kurtosis value and the normalized CK value of each time domain signal domain signal segment. The green and black lines are CK and OSCK values correspond to the segment. The green and black lines are CK and OSCK values correspond to the angle-domain signal angle-domain signal segments. Overall, the CK and kurtosis of the signal segment decrease as the segments. Overall, the CK and kurtosis of the signal segment decrease as the speed increases, both speed increases, both in the time domain and the angle domain. It is worth noting that the OSCK of in the time domain and the angle domain. It is worth noting that the OSCK of the angle domain the angle domain signal segments is sensitive to the speed fluctuation while not being affected by signal segments is sensitive to the speed ﬂuctuation while not being affected by the size of the speed. the size of the speed. Therefore, the OSCK index is more suitable for the extraction of resonance Therefore, the OSCK index is more suitable for the extraction of resonance bands under varying bands under varying speed conditions. speed conditions. (a) 0 1 2 3 4 5 6 7 8 9 Time [s] time domain kurtosis (b) time domain CK angle domain kurtosis angle domain CK angle domain OSCK 0.5 0 5 10 15 20 25 30 35 40 45 Number Figure 7. (a) The shaft rotational frequency, (b) the normalized kurtosis of different indexes. Figure 7. (a) The shaft rotational frequency, (b) the normalized kurtosis of different indexes. 3.3. Simple Simulation of the Inﬂuence of Random Shocks Indication 3.3. Simple Simulation of the Influence of Random Shocks Indication A high kurtosis value is often treated as a sign of the presence of faults in bearing fault diagnosis. However, the kurtosis value of a signal with a single impulse is always higher than a signal containing consecutive periodicity of impulses. To illustrate the inﬂuence of random shocks on a chosen resonance band, two cases are considered here. The parameters of the simulated signal are shown in Table 2. Table 2. Parameters of the simulation model. f f f SNR s r1 r2 1 2 N (s) B ' f (Hz) f n m o n (kHz) (kHz) (kHz) (kHz) (kHz) (dB) Case 1 1 1 0 10–12 4.5 20 5 n 1.2 3 0.01 5 Case 2 1 1 0 10–12 4.5 20 5 7.3 1.2 3 0.01 5 3.3.1. Case 1: The Random Shocks Have the Same Resonant Frequency as the Fault Impulses The noise-free simulated mixed signal, which contains fault impulses and a random shock, its noise-added signal and their frequency spectrums are shown in Figure 8. The FOSCK of the simulated signal is paved in Figure 9a; the maximum OSCK is calculated at the 5.5th decomposition level and its corresponding frequency band is (4792, 5000) Hz. The corresponding envelope of the ﬁltered signal and its resampling envelope signal are shown in Figure 9b,c, respectively. The envelope order spectrum is shown in Figure 9d. It can be observed that the fault characteristic order and its harmonics are quite efﬁciently extracted. Normalized kurtosis Velocity[Hz] Velocity[Hz] Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 35 A high kurtosis value is often treated as a sign of the presence of faults in bearing fault diagnosis. However, the kurtosis value of a signal with a single impulse is always higher than a signal containing consecutive periodicity of impulses. To illustrate the influence of random shocks on a chosen resonance band, two cases are considered here. The parameters of the simulated signal are shown in Table 2. Table 2. Parameters of the simulation model. N (s) Bn φm ƒ (Hz) ƒo ƒs (kHz) ƒr1 (kHz) ƒr2 (kHz) β1 (kHz) β2 (kHz) τn SNR (dB) Case 1 1 1 0 10–12 4.5 20 5 \ 1.2 3 0.01 −5 Case 2 1 1 0 10–12 4.5 20 5 7.3 1.2 3 0.01 −5 3.3.1. Case 1: The Random Shocks Have the Same Resonant Frequency as the Fault Impulses The noise-free simulated mixed signal, which contains fault impulses and a random shock, its Appl. Sci. 2019, 9, 1157 11 of 31 noise-added signal and their frequency spectrums are shown in Figure 8. -3 x 10 (a) (b) random shock fr=5000 Hz -2 0 0.5 1 0 5000 10000 Time [s] Frequency [Hz] -3 x 10 (c) (d) 4 8 2 6 0 4 -2 2 -4 0 0 0.5 1 0 5000 10000 Time [s] Frequency [Hz] Figure 8. Simple simulation: (a) the simulated signal, (b) the frequency spectra of (a), (c) the Appl.Figure 8. Sci. 2018, 8,Si x FO mpl R P e s EER imul REaVIEW tion: ( a) the simulated signal, (b) the frequency spectra of (a), (c) th12 of e 35 noise-added signal with SNR = 5 dB and (d) the frequency spectra of (c). noise-added signal with SNR = −5 dB and (d) the frequency spectra of (c). (a) OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz (b) max c 0.5 The FOSCK of the simulated signal is paved in Figure 9a; the maximum OSCK is calculated at the 5.5th decomposition level and its corresponding frequency band is (4792, 5000) Hz. The fr=5000Hz 1.6 corresponding envelope of the filtered signal and its resampl 0ing en0. velope 2 0. sign 4 al are 0.6 shown 0.8 in Figure (c) Time [s] 9b,c, respectively. The envelope order spectrum is shown in Figure 9d. It can be observed that the 2.6 fault characteristic order and its harmonics are quite efficient 0.5 ly extracted. 3.6 10 20 30 40 50 60 Angle [rad] (d) 4.6 0.05 FCO 5.6 6 0 10 20 30 40 50 0 2,000 4000 6000 8,000 10,000 Order Frequency [Hz] Figure Figure 9. 9. The The results results o obtained btainedby bythe the FO FOSCK SCKfor for pro processing cessing the mi the mixed xed sig sinal gnalwith with same same r resonant esonant frfrequency: ( equency: (a) a) FOSCK, ( FOSCK, (bb ) ) the the envelope envelope of th of the e band-pass f band-pass ﬁlter iltered signal, ( ed signal, (c)cthe ) the resam resampling pling envelope envelope signal of (b,d) the envelope order spectrum of (c). signal of (b,d) the envelope order spectrum of (c). 3.3.2. Case 2: The Random Shocks Have Different Resonant Frequencies from the Fault Impulses 3.3.2. Case 2: The Random Shocks Have Different Resonant Frequencies from the Fault Impulses In this simulation, a random shock with a different resonance frequency from the fault impulses In this simulation, a random shock with a different resonance frequency from the fault impulses is added to the pure signal; its noise-added signal is shown in Figure 10. is added to the pure signal; its noise-added signal is shown in Figure 10. The paving of the FOSCK is shown in Figure 11a. The same to case 1, the optimal frequency band -3 corresponding to the maximum CK is calculated at the 5.5th decomposition level and its frequency x 10 (a) (b) band is (4792, 5000) Hz. The envelope of the ﬁltered signal and its resampling envelope signal are fr1=5000 Hz random shock shown in Figure 11b,c, respectively. The envelope order spectrum is shown in Figure 11d, in which it the fault characteristic order is obvious. fr2=7300 Hz Therefore, when dealing with a vibration signal with random shock interference, whether the -2 random shock has the same resonance frequency band as the fault impulses or not, the proposed index OSCK can locate the fault resonant frequency band exactly. -4 0 0.5 1 0 2,000 5000 7300 10,000 Time [s] Frequency [Hz] -3 x 10 (c) (d) 4 8 2 6 0 4 -2 2 -4 0 0 0.5 1 0 2,000 5000 7300 10,000 Time [s] Frequency [Hz] Figure 10. Simple simulation: (a) the simulated signal, (b) the frequency spectra of (a), (c) the noise-added signal with SNR= -5 dB and (d) the frequency spectra of (c). The paving of the FOSCK is shown in Figure 11a. The same to case 1, the optimal frequency band corresponding to the maximum CK is calculated at the 5.5th decomposition level and its frequency band is (4792, 5000) Hz. The envelope of the filtered signal and its resampling envelope signal are shown in Figure 11b,c, respectively. The envelope order spectrum is shown in Figure 11d, in which it the fault characteristic order is obvious. Level k Amplitude Amplitude Amplitude Amplitude Magnitude Magnitude Magnitude Magnitude Magnitude Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 35 (a) OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz (b) max c 0.5 fr=5000Hz 1.6 0 0 0.2 0.4 0.6 0.8 (c) Time [s] 2.6 0.5 3.6 4 10 20 30 40 50 60 Angle [rad] (d) 4.6 0.05 FCO 5.6 6 0 10 20 30 40 50 0 2,000 4000 6000 8,000 10,000 Order Frequency [Hz] Figure 9. The results obtained by the FOSCK for processing the mixed signal with same resonant frequency: (a) FOSCK, (b) the envelope of the band-pass filtered signal, (c) the resampling envelope signal of (b,d) the envelope order spectrum of (c). 3.3.2. Case 2: The Random Shocks Have Different Resonant Frequencies from the Fault Impulses In this simulation, a random shock with a different resonance frequency from the fault impulses Appl. Sci. 2019, 9, 1157 12 of 31 is added to the pure signal; its noise-added signal is shown in Figure 10. -3 x 10 (a) (b) fr1=5000 Hz random shock fr2=7300 Hz -2 -4 0 0.5 1 0 2,000 5000 7300 10,000 Time [s] Frequency [Hz] -3 (c) (d) x 10 4 8 2 6 0 4 -2 2 -4 0 0 0.5 1 0 2,000 5000 7300 10,000 Time [s] Frequency [Hz] Figure 10. Simple simulation: (a) the simulated signal, (b) the frequency spectra of (a), (c) the Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 35 Figure 10. Simple simulation: (a) the simulated signal, (b) the frequency spectra of (a), (c) the noise-added signal with SNR= 5 dB and (d) the frequency spectra of (c). noise-added signal with SNR= -5 dB and (d) the frequency spectra of (c). (a) (b) OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz max c 0.4 The paving of the FOSCK is shown in Figure 11a. The same to case 1, the optimal frequency 0.2 band corresponding to the maximum CK is calculated at the 5.5th decomposition level and its 1.6 frequency band is (4792, 5000) Hz. The envelope of the filtered 0 signal and 0.2 0.4 its re0. sampling en 6 0.8 velope (c) fr1=5000 Hz fr2=7300 Hz Time [s] signal are shown in Figure 11b,c, respectively. The envelope 0.4 order spectrum is shown in Figure 11d, 2.6 in which it the fault characteristic order is obvious. 0.2 3.6 4 10 20 30 40 50 60 (d) Angle [rad] 4.6 0.05 FCO 5.6 10 20 30 40 50 0 2,000 4,000 6,000 8000 10,000 Order Frequency [Hz] Figure 11. The results obtained by the FOSCK for processing the mixed signal with different resonant Figure 11. The results obtained by the FOSCK for processing the mixed signal with different resonant frequencies: (a) FOSCK, (b) the envelope of the band-pass ﬁltered signal, (c) the resampling envelope frequencies: (a) FOSCK, (b) the envelope of the band-pass filtered signal, (c) the resampling envelope signal of (b,d) the envelope order spectrum of (c). signal of (b,d) the envelope order spectrum of (c). 3.4. Simple Simulation of the Inﬂuence of Multiple Impact Sources Therefore, when dealing with a vibration signal with random shock interference, whether the random To match shock has the the simulation same reson closer to anc the e frequency real situation, band a deterministic as the fault im component pulses or not, the p and two random roposed index OSCK can locate the fault resonant frequency band exactly. shocks are added to a fault impulse signal, and a considerable amount of Gaussian noise is added too. The simulation signal parameters are given in Table 3. The different components and their frequency 3.4. Simple Simulation of the Influence of Multiple Impact Sources spectra are shown in Figure 12. To match the simulation closer to the real situation, a deterministic component and two random Table 3. Parameters of the simulation model. shocks are added to a fault impulse signal, and a considerable amount of Gaussian noise is added too. The simulation signal parameters are given in Table 3. The different components and their f f f SNR s r1 r2 1 2 N (s) B ' f (Hz) f n m o n (kHz) (kHz) (kHz) (kHz) (kHz) (dB) frequency spectra are shown in Figure 12. 1 1 0 10–15 4.5 20 5 7.3 1.2 3 0.01 5 Table 3. Parameters of the simulation model. N (s) Bn φm ƒ (Hz) ƒo ƒs (kHz) ƒr1 (kHz) ƒr2 (kHz) β1 (kHz) β2 (kHz) τn SNR (dB) The FK, WPTK and Protrugram are applied to analyze the mixed signal, and the results are shown 1 1 0 10–15 4.5 20 5 7.3 1.2 3 0.01 −5 in Figures 13–15, respectively. It is worth noting that in the following paragraphs, a wavelet packet basis (a) (f) 0.1 0.04 0 0.02 -0.1 0 0 0.5 1 0 50 100 150 200 (b) (g) Time [s] Frequency [Hz] 5 0.04 0 0.02 -5 0 0 0.5 1 0 5000 10000 -3 (c) (h) Time [s] Frequency [Hz] x 10 20 4 0 2 -20 0 0 0.5 1 0 5000 10000 (d) (i) Time [s] Frequency [Hz] 20 0.04 0 0.02 -20 0 0 0.5 1 0 5000 10000 (e) (j) Time [s] Frequency [Hz] 0.04 0.02 -20 0 0.5 1 0 5000 10000 Time [s] Frequency [Hz] Figure 12. Simulated signals: (a) a deterministic component signal, (b) the fault impulse signal, (c) the random shocks, (d) the synthetic signal without noise added, (e) the synthetic signal with noise-added and SNR = −5 and (f–j) the frequency spectra of the different components. Level k Level k Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Amplitude Amplitude Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 35 (a)OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz (b) max c 0.4 0.2 1.6 0 0.2 0.4 0.6 0.8 (c) fr1=5000 Hz fr2=7300 Hz Time [s] 0.4 2.6 0.2 3.6 4 10 20 30 40 50 60 (d) Angle [rad] 4.6 0.05 FCO 5.6 6 0 10 20 30 40 50 0 2,000 4,000 6,000 8000 10,000 Order Frequency [Hz] Appl. Sci. 2019, 9, 1157 13 of 31 Figure 11. The results obtained by the FOSCK for processing the mixed signal with different resonant frequencies: (a) FOSCK, (b) the envelope of the band-pass filtered signal, (c) the resampling envelope signal of (b,d) the envelope order spectrum of (c). db10 is used in the WPTK as that given in [14], and a bandwidth (BW) that includes the 3rd harmonic of the characteristic frequency is selected in the Protrugram according to the rules mentioned in [28]. Therefore, when dealing with a vibration signal with random shock interference, whether the The analysis results show that both the FK and the WPTK failed to detect the fault-sensitive resonance random shock has the same resonance frequency band as the fault impulses or not, the proposed frequency, and their optimal frequency bands were located near 7300 Hz, which corresponded to the index OSCK can locate the fault resonant frequency band exactly. random shock. From the envelopes of the ﬁltered signal shown in Figures 13 and 14b,c, the amplitudes of the 3.4. fault Simple Si impulses mulation of are rth elatively e Influence of small, Multi while ple Imp thearandom ct Sources shocks are obviously. The FCO and its harmonics are difﬁcult to identify from the envelope order spectra, as shown in Figures 13 and 14d. To match the simulation closer to the real situation, a deterministic component and two random Differing from the FK and WPTK, the Protrugram is shown in Figure 15a with BW equals to 300 Hz and shocks are added to a fault impulse signal, and a considerable amount of Gaussian noise is added the step is 50 Hz. The optimal frequency band relevant to the maximum kurtosis is the fault-sensitive too. The simulation signal parameters are given in Table 3. The different components and their frequency band. The corresponding envelope of the ﬁltered signal and its resampling envelope signal frequency spectra are shown in Figure 12. are shown in Figure 15b,c, respectively. The FCO and its harmonics are clearly shown in Figure 15d. The FOSCK is shown in Figure 16a, the optimal frequency band corresponding to the maximum OSCK Table 3. Parameters of the simulation model. is calculated at the 5.5th decomposition level, and its frequency band is (4792, 5000) Hz. The fault N (s) Bn φm ƒ (Hz) ƒo ƒs (kHz) ƒr1 (kHz) ƒr2 (kHz) β1 (kHz) β2 (kHz) τn SNR (dB) impulses are clear in the envelopes shown in Figure 16b,c. The FCO and its harmonics are clearly 1 1 0 10–15 4.5 20 5 7.3 1.2 3 0.01 −5 visible in Figure 16d, which means the FOSCK is robust to the varying speed and random shocks. (a) (f) Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 35 0.1 0.04 0 0.02 -0.1 0 The FK, WPTK and Protrugram are applied to analyze the mixed signal, and the results are 0 0.5 1 0 50 100 150 200 (b) (g) shown in Figures 13–15, respect Ti im vely e [s]. It is worth noting that in Fr t eq h ue e fo ncyllow [Hz] ing paragraphs, a wavelet 5 0.04 packet basis db10 is used in the WPTK as that given in [14], and a bandwidth (BW) that includes the 0 0.02 -5 0 3rd harmonic of the characteristic frequency is selected in the Protrugram according to the rules 0 0.5 1 0 5000 10000 -3 (c) (h) mentioned in [28]. The analysis results show that both the FK and the WPTK failed to detect the Time [s] Frequency [Hz] x 10 20 4 fault-sensitive resonance frequency, and their optimal frequency bands were located near 7300 Hz, 0 2 which corresponded to the random shock. From the envelopes of the filtered signal shown in -20 0 0 0.5 1 0 5000 10000 Figures 13 and 14b,c, the amplitudes of the fault impulses are relatively small, while the random (d) (i) Time [s] Frequency [Hz] 20 0.04 shocks are obviously. The FCO and its harmonics are difficult to identify from the envelope order 0 0.02 spectra, as shown in Figures 13 and 14d. Differing from the FK and WPTK, the Protrugram is shown -20 0 0 0.5 1 0 5000 10000 in Figure 15a with BW equals to 300 Hz and the step is 50 Hz. The optimal frequency band relevant (e) (j) Time [s] Frequency [Hz] to the maximum kurtosis is the fault-sensitive frequency band. The corresponding envelope of the 0.04 filtered signal and its resampling envelope signal ar 0.e shown 02 in Figure 15b,c, respectively. The FCO -20 and its harmonics are clearly shown in Figure 15d. The FOSCK is shown in Figure 16a, the optimal 0 0.5 1 0 5000 10000 Time [s] Frequency [Hz] frequency band corresponding to the maximum OSCK is calculated at the 5.5th decomposition level, and its frequency band is (4792, 5000) Hz. The fault impulses are clear in the envelopes shown in Figure 12. Simulated signals: (a) a deterministic component signal, (b) the fault impulse signal, (c) the Figure 12. Simulated signals: (a) a deterministic component signal, (b) the fault impulse signal, (c) Figure 16b,c. The FCO and its harmonics are clearly visible in Figure 16d, which means the FOSCK is random shocks, (d) the synthetic signal without noise added, (e) the synthetic signal with noise-added the random shocks, (d) the synthetic signal without noise added, (e) the synthetic signal with robust to the varying speed and random shocks. and SNR = 5 and (f–j) the frequency spectra of the different components. noise-added and SNR = −5 and (f–j) the frequency spectra of the different components. (a) FK =13.1 @ level 2.5, Bw= 1666.6667Hz, f =7500Hz (b) max c 1.6 0 0 0.2 0.4 0.6 0.8 1 (c) Time [s] 2.6 3.6 fr1=5000 Hz fr2=7300 Hz 4 10 20 30 40 50 60 Angle [rad] (d) 4.6 0.05 FCO 5.6 6 0 10 20 30 40 50 0 2,000 4,000 6000 8000 10000 Order Frequency [Hz] Figure Figure 13. 13. The Th results e resulobtained ts obtained by bthe y thFK e FK for fopr r pro ocessing cessing t the he mi mixed xed si signal: gnal: ( (a a)) FK, ( FK, (b b) t ) the he env envelope elope o of f the the band-pass f band-pass ﬁlter iltered ed signal, signal (c , ( ) c the ) the re resampling sampling envelope si envelope signal gnal of ( of (b b,,d d)) the the envelope envelope ord order er spectrum spectrum of (c of ( ). c). WPT kurtogram：@level 4, Bw= 625Hz,f =7812.5Hz (a) (b) 1 0 0.2 0.4 0.6 0.8 1 Time [s] fr2=7300 Hz (c) fr1=5000 Hz 10 20 30 40 50 60 Angle [rad] (d) 0.05 FCO 4 10 20 30 40 50 0 2,000 4,000 6000 8000 10,000 Order Frequency [Hz] Figure 14. The results obtained by the WPTK for processing the mixed signal: (a) WPTK, (b) the envelope of the band-pass filtered signal, (c) the envelope of the resampling of (b,d) the envelope order spectrum of (c). Level k Level k Level k Amplitude Amplitude Amplitude Amplitude Amplitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Magnitude Amplitude Amplitude Amplitude Amplitude Amplitude Amplitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 35 The FK, WPTK and Protrugram are applied to analyze the mixed signal, and the results are shown in Figures 13–15, respectively. It is worth noting that in the following paragraphs, a wavelet packet basis db10 is used in the WPTK as that given in [14], and a bandwidth (BW) that includes the 3rd harmonic of the characteristic frequency is selected in the Protrugram according to the rules mentioned in [28]. The analysis results show that both the FK and the WPTK failed to detect the fault-sensitive resonance frequency, and their optimal frequency bands were located near 7300 Hz, which corresponded to the random shock. From the envelopes of the filtered signal shown in Figures 13 and 14b,c, the amplitudes of the fault impulses are relatively small, while the random shocks are obviously. The FCO and its harmonics are difficult to identify from the envelope order spectra, as shown in Figures 13 and 14d. Differing from the FK and WPTK, the Protrugram is shown in Figure 15a with BW equals to 300 Hz and the step is 50 Hz. The optimal frequency band relevant to the maximum kurtosis is the fault-sensitive frequency band. The corresponding envelope of the filtered signal and its resampling envelope signal are shown in Figure 15b,c, respectively. The FCO and its harmonics are clearly shown in Figure 15d. The FOSCK is shown in Figure 16a, the optimal frequency band corresponding to the maximum OSCK is calculated at the 5.5th decomposition level, and its frequency band is (4792, 5000) Hz. The fault impulses are clear in the envelopes shown in Figure 16b,c. The FCO and its harmonics are clearly visible in Figure 16d, which means the FOSCK is robust to the varying speed and random shocks. (a) FK =13.1 @ level 2.5, Bw= 1666.6667Hz, f =7500Hz (b) max c 1.6 0 0.2 0.4 0.6 0.8 1 (c) Time [s] 2.6 3.6 fr1=5000 Hz fr2=7300 Hz 4 10 20 30 40 50 60 Angle [rad] (d) 4.6 0.05 FCO 5.6 6 0 10 20 30 40 50 0 2,000 4,000 6000 8000 10000 Order Frequency [Hz] Figure 13. The results obtained by the FK for processing the mixed signal: (a) FK, (b) the envelope of the band-pass filtered signal, (c) the resampling envelope signal of (b,d) the envelope order spectrum Appl. Sci. 2019, 9, 1157 14 of 31 of (c). WPT kurtogram：@level 4, Bw= 625Hz,f =7812.5Hz (a) (b) 0 0.2 0.4 0.6 0.8 1 Time [s] (c) fr2=7300 Hz fr1=5000 Hz 10 20 30 40 50 60 Angle [rad] (d) 0.05 FCO 10 20 30 40 50 0 2,000 4,000 6000 8000 10,000 Order Frequency [Hz] Figure 14. The results obtained by the WPTK for processing the mixed signal: (a) WPTK, (b) the Figure 14. The results obtained by the WPTK for processing the mixed signal: (a) WPTK, (b) the envelope of the band-pass ﬁltered signal, (c) the envelope of the resampling of (b,d) the envelope order envelope of the band-pass filtered signal, (c) the envelope of the resampling of (b,d) the envelope Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 35 spectrum of (c). order spectrum of (c). (a) The Protrugram analysis: Bw=300Hz,f =5053Hz,,step=100Hz (b) (a) The Protrugram analysis: Bw=300Hz,f =5053Hz,,step=100Hz (b) X: 5053 160 2 Y: X: 5 10 45 53 .5 Y: 145.5 fr2=7300 Hz 00 0.2 0.4 0.6 0.8 1 fr1=5000 Hz fr2=7300 Hz 0 0.2 0.4 0.6 0.8 1 fr1=5000 Hz (c) Time [s] (c) 4 Time [s] 00 10 20 30 40 50 60 0 10 20 30 40 50 60 Angle [rad] (d) Angle [rad] 0.1 (d) 80 FCO 0.1 FCO 0.05 60 0.05 00 10 20 30 40 50 0 2,000 4,000 6000 8000 10,000 0 10 20 30 40 50 Order 0 2,000 4,000 6000 8000 10,000 Frequency [Hz] Order Frequency [Hz] Figure 15. The results obtained by the Protrugram for processing the mixed signal: (a) Protrugram, Figure Figure 15. 15. The The results results obtaine obtaineddby by the the Protru Protrugram gramfor for processing processing the the m mixied xed sig signal: nal:( ( aa ) ) Protrugra Protrugram, m, (b) the envelope of the band-pass filtered signal, (c) the envelope of the resampling of (b,d) the (b (b ) the ) the envelope envelope of of the the ban band-pass d-pass fi ﬁlteredltered signal, signal, ( (c) the envelope c) the envelope of the r of esampling the resam of (b p,ling d) the ofenvelope (b,d) the envelope order spectrum of (c). order spectrum of (c). envelope order spectrum of (c). (a) (b) OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz (a) max c (b) OSCK =1 @ level 5.5, Bw= 208.3333Hz, f =4895.8333Hz max c 0 2 1 2 1.6 00 0.2 0.4 0.6 0.8 1 1.6 fr1=5000 Hz fr2=7300 Hz 0 0.2 0.4 0.6 0.8 1 (c) fr1=5000 Hz Time[s] fr2=7300 Hz (c) 4 Time[s] 2.6 2.6 3 2 3.6 3.6 0 10 20 30 40 50 60 10 20 30 40 50 60 4 Angle[rad] (d) 4.6 0.1 Angle[rad] FCO (d) 4.6 0.1 FCO 0.05 0.05 5.6 5.6 0 10 20 30 40 50 10 20 30 40 50 0 2000 4000 6000 8000 10000 Order 0 2000 4000 6000 8000 10000 Order Frequency [Hz] Frequency [Hz] Figure Figure 16. 16. The Ther results obta esults obtained ined by the FOSCK by the FOSCK for for proc processing essing the mi the mixed xedsignal: signal:( ( aa ) ) FOS FOSCK, CK, ( (bb ) ) the the Figure 16. The results obtained by the FOSCK for processing the mixed signal: (a) FOSCK, (b) the envelope envelope of th of the band-pass e band-pass f ﬁlter iled tered signal signal, (c), ( the c) the envelope envelope of the of the resampl resampling ofi(ng of ( b,d) the b,d envelope ) the envelope order envelope of the band-pass filtered signal, (c) the envelope of the resampling of (b,d) the envelope spectr order spectrum of ( um of (c). c). order spectrum of (c). 4. Experimental Evaluation 4. Experimental Evaluation 4. Experimental Evaluation To further examine the effectiveness of the proposed method, an experiment is carried out on To further examine the effectiveness of the proposed method, an experiment is carried out on a To further examine the effectiveness of the proposed method, an experiment is carried out on a a Spectra Quest Machinery Fault Simulator. The bearing test rig consists of an AC motor, a ﬂexible Spectra Quest Machinery Fault Simulator. The bearing test rig consists of an AC motor, a flexible Spectra Quest Machinery Fault Simulator. The bearing test rig consists of an AC motor, a flexible coupling to connect the shaft to the motor, a tachometer mounted on the motor, a rotor disk coupling to connect the shaft to the motor, a tachometer mounted on the motor, a rotor disk mounted onto the shaft, the outboard bearing housing, and two rolling element bearings. One of the mounted onto the shaft, the outboard bearing housing, and two rolling element bearings. One of the bearings without defects is located in the bearing housing closer to the motor, and the other one is bearings without defects is located in the bearing housing closer to the motor, and the other one is located farther from the motor. The ICP acceleration sensors are fixed on the bearing housing to located farther from the motor. The ICP acceleration sensors are fixed on the bearing housing to collect vibration signals at a sampling frequency of 20 kHz. A data acquisition instrument and a collect vibration signals at a sampling frequency of 20 kHz. A data acquisition instrument and a computer are used for the analysis. The test rig is shown in Figure 17. computer are used for the analysis. The test rig is shown in Figure 17. Kur Kur tots ois sis Level k Level k Level k Level k MM agni agni tutde ude Ma Ma gn gin tu itd ue de Magnitude Magnitude Amplitude Amplitude Am Am plpl itude itude Am Am plpl itude itude Am Am plpl itud itud e e Am Am plpl itude itude Amplitude Amplitude Appl. Sci. 2019, 9, 1157 15 of 31 coupling to connect the shaft to the motor, a tachometer mounted on the motor, a rotor disk mounted onto the shaft, the outboard bearing housing, and two rolling element bearings. One of the bearings without defects is located in the bearing housing closer to the motor, and the other one is located farther from the motor. The ICP acceleration sensors are ﬁxed on the bearing housing to collect vibration signals at a sampling frequency of 20 kHz. A data acquisition instrument and a computer are used for Appl. Sci. 2018, 8, x FOR PEER REVIEW 16 of 35 the analysis. The test rig is shown in Figure 17. AC motor Coupling Normal Rotor Faulty Outboard bearing bearing disk bearing housing Data acquisition Computer instrument Tachometer Inboard bearing Sensor housing Figure 17. The test bench for bearing fault detection. Figure 17. The test bench for bearing fault detection. The parameters of the bearings are listed in Table 4. The parameters of the bearings are listed in Table 4. Table 4. Parameters of the bearings. Table 4. Parameters of the bearings. Bearing Number Contact Pitch Ball Fault Severity BPFO BPFI Type of Balls Angle Diameter Diameter Bearing Number of Contact Pitch Ball Fault Severity BPFO BPFI Type Balls Angle Diameter Diameter 3/4” Rotor bearing ER-12K 8 0 1.318 in 0.3125 in 3.048 4.95 3/4” Rotor ER-12K 8 0 1.318 in 0.3125 in 3.048 4.95 bearing 4.1. Normal Bearing 4.1. Normal Bearing To illustrate the effectiveness of the proposed bearing fault diagnosis method, a baseline case is ﬁrst studied, in which both bearings are healthy, as shown in Figure 17. Due to relative motion, bearing To illustrate the effectiveness of the proposed bearing fault diagnosis method, a baseline case is components generate vibrator signals in operation, as shown in Figure 18a. The shaft rotational speed is first studied, in which both bearings are healthy, as shown in Figure 17. Due to relative motion, shown in Figure 18b. Figure 18c shows the frequency spectra of a healthy bearing as the shaft accelerates bearing components generate vibrator signals in operation, as shown in Figure 18a. The shaft from 20 Hz to 25 Hz within 3.347 s, i.e., the acceleration a equals 3/2 Hz/s. The time-frequency rotational speed is shown in Figure 18b. Figure 18c shows the frequency spectra of a healthy bearing representation (TFR) of the signal is obtained via STFT, as shown in Figure 18d. In the TFR, several as the shaft accelerates from 20 Hz to 25 Hz within 3.347 s, i.e., the acceleration a equals 3/2 Hz/s. The suspected resonance frequency bands, in which the energy is most concentrated, are adaptively time-frequency representation (TFR) of the signal is obtained via STFT, as shown in Figure 18d. In removed using different indexes in Figure 19. Based on the maximum of each index, an optimal the TFR, several suspected resonance frequency bands, in which the energy is most concentrated, are frequency band is selected for further analysis. In Figure 19a, the optimal frequency band is (4375, adaptively removed using different indexes in Figure 19. Based on the maximum of each index, an 4687.5) Hz in the FK. The WPTK is shown in Figure 19b, and the corresponding optimal frequency optimal frequency band is selected for further analysis. In Figure 19a, the optimal frequency band is band is (9376, 10,000) Hz at the 4th decomposition level. The Protrugram with BW = 400 Hz and step (4375, 4687.5) Hz in the FK. The WPTK is shown in Figure 19b, and the corresponding optimal = 50 Hz is shown in Figure 19c, and the maximum kurtosis is calculated at 4974 Hz. In Figure 19d, frequency band is (9376, 10,000) Hz at the 4th decomposition level. The Protrugram with BW = 400 the FOSCK, the optimal frequency band corresponding to the maximum CK is calculated at the 5th Hz and step = 50 Hz is shown in Figure 19c, and the maximum kurtosis is calculated at 4974 Hz. In decomposition level, and its frequency band is (3125, 3437.5) Hz. The original signal is ﬁltered by Figure 19d, the FOSCK, the optimal frequency band corresponding to the maximum CK is calculated using different band-pass ﬁlters corresponding to these optimal frequency bands. The envelope of at the 5th decomposition level, and its frequency band is (3125, 3437.5) Hz. The original signal is each ﬁltered signal is calculated by using the Hilbert transform and resampled into the angular domain filtered by using different band-pass filters corresponding to these optimal frequency bands. The by using COT. The envelope order spectrum analysis result is shown in Figure 20. In all the ﬁgures, envelope of each filtered signal is calculated by using the Hilbert transform and resampled into the the dominant order components are related to the shaft rotational order (SRO) and its harmonics. These angular domain by using COT. The envelope order spectrum analysis result is shown in Figure 20. In all the figures, the dominant order components are related to the shaft rotational order (SRO) and its harmonics. These results imply that both bearings are healthy. It is worth mentioning that the SRO and its harmonics appear due to rotor disk machining error and installation error. Appl. Sci. 2019, 9, 1157 16 of 31 results imply that both bearings are healthy. It is worth mentioning that the SRO and its harmonics Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 35 appear due to rotor disk machining error and installation error. Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 35 (a) (b) (a) (b) 0. 0.2 2 (a) (b) 26 Real speed Real speed 0.2 Real speed Coded pulse Coded pulse -0.2 Coded pulse -0.2 T Th he e ffiittttiin ng g sp spee eed d -0.2 The fitting speed 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 (c) (c) (d) (d) 22 22 10000 10000 10000 10000 0 0.5 1 1.5 2 2.5 3 (c) (d) 10000 10000 8000 8000 8000 8000 20 20 8000 8000 6000 6000 6000 6000 6000 6000 4000 4000 4000 4000 4000 4000 2000 2000 2000 2000 2000 2000 0 0 0 0 14 14 200 200 100 100 0 0 0. 0.5 5 1 1 1. 1.5 5 2 2 2. 2.5 5 3 3 0 0 0. 0.5 5 1 1 1. 1.5 5 2 2 2. 2.5 5 3 3 0 0 14 200 A Am mp plit lit 100 u ud de e 0 0.5 1 T Tiim m 1.e e 5 [s [s]] 2 2.5 3 Time [s] 0 0.5 1 T1. im5e [s] 2 2.5 3 Amplitude Time [s] Time [s] Figure 18. Figure 18. The s The siignal measured gnal measured from from a a normal bearing: ( normal bearing: (a a) t ) tiime-do me-dom main signal, ( ain signal, (b b) the ) the shaft shaft Figure 18. The signal measured from a normal bearing: (a) time-domain signal, (b) the shaft rotational Figure 18. The signal measured from a normal bearing: (a) time-domain signal, (b) the shaft rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. frequency rotational freq from 20 uency from 20 Hz to 25 Hz to 25 Hz, (c) the fr Hz equency , (c) the frequency spectrum of spectrum of (a,d) TFR by (ausing ,d) TFR by us STFT. ing STFT. (a) FK FK =0 =0.9 .9 @ @ le lev ve ell 5 5, B , Bw w= =3 31 12 2.5 .5 H Hz z,, f f = =4 453 531. 1.25 25 H Hz z (b) W WP PT T k ku ur rttogr ogra am m：：@ @llev evel el 4 4,, n no ode( de(4 4,,15 15) ) (a) (b) max c max c FK =0.9 @ level 5, Bw=312.5 Hz, f =4531.25 Hz 0 WPT kurtogram：@level 4, node(4,15) (a) (b) max c 0 0 1 1 1.6 1.6 1.6 2. 2.6 6 2 2 2.6 3 2 3. 3.6 6 3.6 4 4 3 4.6 4.6 4.6 4 4 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 0 2 2000 000 4 4000 000 6 6000 000 8 8000 000 10 10000 000 Frequency [Hz] Frequency[Hz] 0 2000 F4 re 000 quency [H 60 z00 ] 8000 10000 0 2000 F4 re 000 quency[H 6000 z] 8000 10000 Frequency [Hz] Frequency[Hz] T Th he e Pr Prot otr ru ugr gra am m a an na ally ysi sis s:: B Bw w= =40 400H 0Hz z,, ff = = 49 4974 74H Hz z,, s stte ep= p=5 50 0H Hz z.. OS OSCK CK = =1 1 @ @ le lev ve ell 5 5, B , Bw w= =3 31 12 2.5 .5 H Hz z,, f f = =3 32 281. 81.25 25 H Hz z (c) (d) (c) (d) c max c c max c The Protrugram analysis: Bw=400Hz, f = 4974Hz, step=50Hz. OSCK =1 @ level 5, Bw=312.5 Hz, f =3281.25 Hz (c) (d) c max c 300 300 X: 4974 1 1 X: 4974 X X:: 7989 7989 Y: 297.1 Y: 297.1 X: 4974 Y Y: 290. : 290.9 9 1 X: 7989 1. 1.6 6 Y: 297.1 250 250 Y : 290.9 1.6 200 2.6 2.6 2.6 3.6 3.6 3.6 4. 4.6 6 4.6 50 5 5 0 0 200 2000 0 40 4000 00 60 6000 00 80 8000 00 10 10000 000 0 0 2 2000 000 4 4000 000 6 6000 000 8 8000 000 1 1000 0000 0 0 2000 F F40 r re eq q 00 ue uenc ncy [ y [60 H Hz] z] 00 8000 10000 0 2000 Fr Fr 4equ equ 000en ency cy [[6 H H 000 z z]] 8000 10000 Frequency [Hz] Frequency [Hz] Figure 19. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Figure 19. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Figure 19. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) (c) Protrugram and (d) FOSCK. Figure 19. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. Fr Frequ equency ency [[H Hz] z] Ku Kurrttos osiis s Le Leve vell k k Frequency [Hz] Kurtosis Level k Am Amp plliittud ude e Amplitude Le Level vel k k Velocity [Hz] Velocity [Hz] Level k Le Lev ve ell k k Velocity [Hz] Level k Appl. Sci. 2019, 9, 1157 17 of 31 Appl. Sci. 2018, 8, x FOR PEER REVIEW 18 of 35 -3 -5 FK, Bw=312.5 Hz, fc=4531.25 Hz (a) (b) x 10 x 10 WPT kurtogram, node(4,15) 2.5 FCO o FCO 2 FCO FCO 1.5 1X SRO 0.5 1 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order -3 -3 The Protrugram, Bw=400 Hz, fc=4974 Hz FOSCK, Bw=312.5 Hz, fc= 3281.25Hz (c) (d) x 10 x 10 2.5 FCO 3.5 FCO 2 FCO FCO i 2.5 1.5 1 X SRO 4 X SRO 6 X SRO 9 X SRO 1X SRO 2X SRO 1.5 0.5 0.5 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure Figure 20. 20. TheThe envelope envelope ord order spectra er spectra ofof the the signal signalobtained obtained by d by dif ifferent m ferent methods: ethods: (a() FK, a) FK, (b( ) WPTK b) WPTK, , (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. 4.2. Diagnosis of a Bearing with an Outer Race Fault 4.2. Diagnosis of a Bearing with an Outer Race Fault In the outer race fault case, the right bearing is replaced by an outer race defect, as shown in In the outer race fault case, the right bearing is replaced by an outer race defect, as shown in Figure 17. Aiming to demonstrate that the method is robust to rotational speed changes and random Figure 17. Aiming to demonstrate that the method is robust to rotational speed changes and random shock interference, the vibration signals under three different acceleration conditions are analyzed, shock interference, the vibration signals under three different acceleration conditions are analyzed, as illustrated in Table 5. In addition, white Gaussian noise (SNR=-3 dB) is added to the collected as illustrated in Table 5. In addition, white Gaussian noise (SNR = 3 dB) is added to the collected signal to make the detection more challenging. signal to make the detection more challenging. Table 5. Parameters of each experiment. Table 5. Parameters of each experiment. Acceleration (Hz/s) Experimental Study #1 Experimental Study #2 Experimental Study #3 Acceleration (Hz/s) Experimental Study #1 Experimental Study #2 Experimental Study #3 a 4/3 3/2 3 a 4/3 3/2 3 4.2.1. Experimental Study #1 4.2.1. Experimental Study #1 The collected vibration signal and the rotating speed are shown in Figure 21a,b. The frequency spectrum of the vibration signal is shown in Figure 21c, in which spectrum smearing could be The collected vibration signal and the rotating speed are shown in Figure 21a,b. The frequency observed due to the variable rotating speeds. In addition, the TFR of the signal is blurry and lacks spectrum of the vibration signal is shown in Figure 21c, in which spectrum smearing could be observed detail due to background noise interference, which may come from other coupled machine due to the variable rotating speeds. In addition, the TFR of the signal is blurry and lacks detail due to components and the working environment, making it more difficult to identify the fault type in background noise interference, which may come from other coupled machine components and the Figure 21d. working environment, making it more difﬁcult to identify the fault type in Figure 21d. Figure 22 shows the signal analysis results for the outer race fault case when a is equal to 4/3 Hz/s. The FK is paved in Figure 22a, in which the optimal frequency band is (2916.67, 3333.33) Hz. Figure 22b shows the WPTK, in which the maximum kurtosis is calculated at the 4th decomposition level, and its corresponding optimal frequency band is (3125, 3750) Hz. The Protrugram is shown in Figure 22c and the center frequency is 638.3 Hz. Figure 22d gives the FOSCK, in which the maximum OSCK is calculated at the 3.5th level, and the optimal frequency band is (3333.33, 4166.67) Hz. Different Magnitude Magnitude Magnitude Magnitude Appl. Sci. 2019, 9, 1157 18 of 31 band-pass ﬁlters are used to ﬁlter out the corresponding frequency band signals, and their envelopes are calculated using the Hilbert transform. Then, each envelope of these ﬁltered signals is resampled into the angular domain by using COT, and the envelope order spectrum analysis results are shown in Figure 23a–d, respectively. It is clear that all the ﬁltered signals contain fault components, which also veriﬁes that the fault impulse has broadband characteristics. In Figure 23, the FCO of the bearing outer race fault and its triple octaves are very clear. Therefore, all the methods mentioned above can effectively detect the bearing outer race fault, as in the case of acceleration a = 4/3 Hz/s. Appl. Sci. 2018, 8, x FOR PEER REVIEW 19 of 35 (a) (b) Real speed Coded pulse The fitting speed -5 0 1 2 3 4 5 6 (c) (d) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 15 1000 500 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Amplitude Time [s] Time [s] Figure Figure 21. 21. TheThe signal signal measured from an outer race fault bea measured from an outer race fault bearing: ring: ( (aa )) time-domain signal, ( time-domain signal, (b b) the shaft ) the shaft Appl. Sci. 2018, 8, x FOR PEER REVIEW 20 of 35 Appl. Sci. rotational freq 2018, 8, x FOR P uency from 20 Hz to 25 EER REVIEW Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. 20 of 35 rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. FK =3.3 @ level 4.5, Bw=416.6667 Hz, f =3125 Hz WPT kurtogram:@level 4, node(4,5) (a) (b) Fig mau xre 22 shows the signal analy c sis results for the outer race fault case when a is equal to 4/3 FK =3.3 @ level 4.5, Bw=416.6667 Hz, f =3125 Hz WPT kurtogram:@level 4, node(4,5) (a) (b) max c 0 Hz/0 s. The FK is paved in Figure 22a, in which the optimal frequency band is (2916.67, 3333.33) Hz. Figure 22b shows the WPTK, in which the maximum kurtosis is calculated at the 4th decomposition 1.6 level, and its corresponding optimal frequency band is (3125, 3750) Hz. The Protrugram is shown in 1.6 Figure 22c and the center frequency is 638.3 Hz. Figure 22d gives the FOSCK, in which the maximum 2.6 2.6 OSCK is calculated at the 3.5th level, and the optimal 2 frequency band is (3333.33, 4166.67) Hz. Different band-pass filters are used to filter out the corresponding frequency band signals, and their 3.6 3.6 envelopes are calculated using the Hilbert transform. Then, each envelope of these filtered signals is 4 3 resampled into the angular domain by using COT, and the envelope order spectrum analysis results 4.6 4.6 are shown in Figure 23a–d, respectively. It is clear that all the filtered signals contain fault components, which also verifies that the fault impulse ha 4 s broadband characteristics. In Figure 23, 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] the FCO of the beari Freque ng outer ra ncy [Hz] ce fault and its triple octaves are very Frequ c enclear y [Hz.] Therefore, all the (c) (d) methods mentioned above can effectively detect the bearing outer race fault, as in the case of (c) (d) The Protrugram analysis: Bw=300 Hz, f =638.3 Hz, step=100 Hz. OSCK =1 @ level 3.5, Bw= 833.3333 Hz, f =3750 Hz The Protrugram analysis: Bw=300 Hz, f =638.3 Hz, step=100 Hz. max c OSCK =1 @ level 3.5, Bw= 833.3333 Hz, f =3750 Hz acceleration a = 4/3 Hz/s. c max c X: 638.3 1 X: 638.3 Y : 141.5 1.6 Y : 141.5 1.6 2.6 2.6 3.6 3.6 4.6 4.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Frequency [Hz] Frequency [Hz] Figure 22. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Figure 22. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) (c) Protrugram and (d) FOSCK. Figure 22. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. Kurtosis Kurtosis Level k Level k Frequency [Hz] Amplitude Level k Level k Level k Level k Velocity [Hz] Appl. Sci. 2019, 9, 1157 19 of 31 Appl. Sci. 2018, 8, x FOR PEER REVIEW 21 of 35 FK, Bw=416.6667 Hz, fc=3125 Hz WPT kurtogram, node(4,5) (a) (b) 0.1 0.1 X: 3.048 X: 3.048 Y: 0.08264 Y: 0.08 1X FCO 2X FCO 3X FCO 4X FCO 1X FCO 2X FCO 3X FCO 4X FCO 0.08 0.08 o o o o o o o o FCO -SRO FCO +SRO FCO -SRO FCO +SRO o o o o SRO SRO 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order The Protrugram, Bw=300 Hz, fc= 638.3Hz FOSCK, Bw=833.3333 Hz, fc=3750 Hz (c) (d) 0.2 0.1 X: 3.048 X: 3.048 Y: 0.16 Y: 0.08 1X FCO 2X FCO 3X FCO 0.08 o o o 1X FCO 4X FCO 0.15 2X FCO 3X FCO o o o o FCO +SRO FCO -SRO 0.06 FCO +SRO FCO -SRO 0.1 SRO SRO 0.04 0.05 0.02 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure 23. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Figure 23. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. 4.2.2. Experimental Study #2 4.2.2. Experimental Study #2 To verify that the proposed method is still effective under the conditions of random shock interference, the test is randomly knocked during the changing of speed between 20 Hz and 25 Hz with To verify that the proposed method is still effective under the conditions of random shock a equal to 3/2 Hz/s in this experiment, and the collected vibration signal and rotating speed signal interference, the test is randomly knocked during the changing of speed between 20 Hz and 25 Hz are shown in Figure 24a,b, respectively. The acquisition process is disturbed by a transient impact with a equal to 3/2 Hz/s in this experiment, and the collected vibration signal and rotating speed leading to an impulse with a large amplitude in the time domain, which is shown in Figure 24a, and signal are shown in Figure 24a,b, respectively. The acquisition process is disturbed by a transient the corresponding frequency spectrum and TFR are shown in Figure 24c,d, respectively. The optimal impact leading to an impulse with a large amplitude in the time domain, which is shown in Figure frequency bands corresponding to different indexes are shown in Figure 25. In Figure 25a,b, the 24a, and the corresponding frequency spectrum and TFR are shown in Figure 24c,d, respectively. optimal frequency band corresponding to the FK and the WPTK is (4375, 5000) Hz. The Protrugram The optimal frequency bands corresponding to different indexes are shown in Figure 25. In Figure is paved in Figure 25c with BW = 400 Hz and step = 100 Hz and the center frequency is 744.7 Hz. 25a,b, the optimal frequency band corresponding to the FK and the WPTK is (4375, 5000) Hz. The Figure 25d shows the FOSCK, and its optimal frequency band is (5833.33, 6666.67) Hz. The envelope Protrugram is paved in Figure 25c with BW = 400 Hz and step = 100 Hz and the center frequency is of the ﬁltered signal from the selected band and its corresponding envelope order spectrum are shown 744.7 Hz. Figure 25d shows the FOSCK, and its optimal frequency band is (5833.33, 6666.67) Hz. The in Figure 26. In Figure 26a,b, they failed to provide any bearing fault related signatures, while in envelope of the filtered signal from the selected band and its corresponding envelope order Figure 26c,d, the FCO and its harmonics in the envelope order spectrum can be clearly observed. spectrum are shown in Figure 26. In Figure 26a,b, they failed to provide any bearing fault related Therefore, in the case of bearing fault diagnosis under random shock interference, the Protrugram and signatures, while in Figure 26c,d, the FCO and its harmonics in the envelope order spectrum can be the FOSCK was better than that of the FK and the WPTK. clearly observed. Therefore, in the case of bearing fault diagnosis under random shock interference, the Protrugram and the FOSCK was better than that of the FK and the WPTK. Magnitude Magnitude Magnitude Magnitude Appl. Sci. 2019, 9, 1157 20 of 31 Appl. Sci. 2018, 8, x FOR PEER REVIEW 22 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 22 of 35 (a) (b) 10 25 (a) (b) Real speed 10 25 Cod Rea ed l sp peed ulse Th Cod e fited ting p u slpeed se -10 The fitting speed 0 0.5 1 1.5 2 2.5 3 -10 (c) (d) 10000 10000 0 0.5 1 1.5 2 2.5 3 (c) (d) 10000 10000 8000 8000 8000 8000 6000 6000 6000 6000 4000 4000 4000 4000 2000 2000 2000 2000 0 0 1000 500 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0 Amplitude Time [s] 1000 500 0 0.5 1 1.5 2 2.5 3 Time [s] 0 0.5 1 1.5 2 2.5 3 Amplitude Time [s] Time [s] Figure 24. The signal measured from an outer race fault bearing: (a) time-domain signal, (b) the shaft Figure 24. The signal measured from an outer race fault bearing: (a) time-domain signal, (b) the shaft Figure 24. The signal measured from an outer race fault bearing: (a) time-domain signal, (b) the shaft rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. FK =35.4 @ level 4, Bw=625 Hz, f =4687.5 Hz WPT kurtogram:@level 4, node(4,7) (a) (b) max c (a) FK =35.4 @ level 4, Bw=625 Hz, f =4687.5 Hz (b) WPT kurtogram:@level 4, node(4,7) max c 1.6 1 1.6 2.6 2 2.6 3.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) The Protrugram analysis: Bw=400 Hz, f =744.7 Hz, step=100 Hz OSCK =1 @ level 3.5, Bw= 833.3333 Hz, f =6250 Hz (c) c (d) max c The Protrugram analysis: Bw=400 Hz, f =744.7 Hz, step=100 Hz OSCK =1 @ level 3.5, Bw= 833.3333 Hz, f =6250 Hz c max c X: 744.7 1 Y : 153.5 X: 744.7 1 Y : 153.5 1.6 120 1.6 2.6 2.6 80 3 80 3 3.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] 0 2000 4000 6000 8000 10000 Frequency [Hz] 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Figure 25. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Figure 25. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) (c) Protrugram and (d) FOSCK. Figure 25. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. Frequency [Hz] Kurtosis Level k Frequency [Hz] Kurtosis Level k Amplitude Amplitude Level k Velocity [Hz] Level k Level k Velocity [Hz] Level k Appl. Sci. 2019, 9, 1157 21 of 31 Appl. Sci. 2018, 8, x FOR PEER REVIEW 23 of 35 FK, Bw=625 Hz, f =4687.5 Hz FK, Bw=625 Hz, fc=4687.5 Hz (a) 0.1 X: 3.048 Y: 0.08 0.08 0.06 0.04 0.02 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 Time [s] Order (b) WPT kurtogram, node(4,7) WPT kurtogram, node(4,7) 0.02 X: 3.048 Y: 0.01625 0.015 FCO 0.01 0.005 0 0 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 Time [s] Order (c) The Protrugram, Bw=400 Hz, f =744.7 Hz The Protrugram, Bw=400 Hz, f =744.7 Hz c c 2 0.1 X: 3.048 Y: 0.08 0.08 1.5 1x FCO 2X FCO 3X FCO o o o 0.06 FCO +SRO FCO -SRO 0.04 0.5 0.02 0 0 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 Time [s] Order (d) FOSCK, Bw=833.3333 Hz, f = 6250 Hz FOSCK, Bw=833.3333 Hz, fc=6250 Hz 0.1 2.5 X: 3.048 Y: 0.08 0.08 1X FCO 2X FCO 3X FCO o o o 0.06 1.5 FCO -SRO FCO +SRO o o 1 0.04 SRO 0.5 0.02 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 Time [s] Order Figure 26. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Amplitude Amplitude Amplitude Amplitude Magnitude Magnitude Magnitude Magnitude Appl. Sci. 2018, 8, x FOR PEER REVIEW 24 of 35 Appl. Sci. 2019, 9, 1157 22 of 31 Figure 26. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. 4.2.3. Experimental Study #3 4.2.3. Experimental Study #3 In this experiment, to verify that the proposed method is still effective in the case of rapidly In this experiment, to verify that the proposed method is still effective in the case of rapidly changing speed and wild ﬂuctuations, the vibration signal and rotational speed are collected during changing speed and wild fluctuations, the vibration signal and rotational speed are collected during the speed increase from 18 Hz to 28 Hz with a equal to 3 Hz/s. Figure 27a,b show the raw signal and the speed increase from 18 Hz to 28 Hz with a equal to 3 Hz/s. Figure 27a,b show the raw signal and its rotational speed, respectively. The optimal frequency band corresponding to different indexes is its rotational speed, respectively. The optimal frequency band corresponding to different indexes is shown in Figure 28. The FK is paved in Figure 28a and the optimal frequency band is [5625, 6250] Hz. shown in Figure 28. The FK is paved in Figure 28a and the optimal frequency band is [5625, 6250] The optimal frequency band in the WPTK is [1875, 2500] Hz, as shown in Figure 28b. The Protrugram Hz. The optimal frequency band in the WPTK is [1875, 2500] Hz, as shown in Figure 28b. The is shown in Figure 28c, where the BW equals 400 Hz and step is 100 Hz and the center frequency Protrugram is shown in Figure 28c, where the BW equals 400 Hz and step is 100 Hz and the center is 2105 Hz. Figure 28d shows the FOSCK, in which the maximum OSCK is calculated at the 3.5th frequency is 2105 Hz. Figure 28d shows the FOSCK, in which the maximum OSCK is calculated at decomposition level, and its corresponding optimal frequency band is [1666.67, 2500] Hz. The envelope the 3.5th decomposition level, and its corresponding optimal frequency band is [1666.67, 2500] Hz. order spectrum analysis results are shown in Figure 29a–d. In Figure 29a,d, the FCO and its quadruple The envelope order spectrum analysis results are shown in Figure 29a–d. In Figure 29a,d, the FCO octaves are very clear, especially in Figure 29d, and more harmonic components of the FCO can and its quadruple octaves are very clear, especially in Figure 29d, and more harmonic components of be found. Although the FCO can be found in Figure 29b,c, only the ﬁrst two octaves are obvious. the FCO can be found. Although the FCO can be found in Figure 29b,c, only the first two octaves are Therefore, compared with the WPTK and the Protrugram, the FK and the FOSCK are more sensitive to obvious. Therefore, compared with the WPTK and the Protrugram, the FK and the FOSCK are more the fault impulse resonance frequency under these conditions. sensitive to the fault impulse resonance frequency under these conditions. (a) (b) Real speed 26 Coded pulse -5 The fitting speed 0 0.5 1 1.5 2 2.5 3 (c) (d) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 14 1000 500 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Amplitude Time [s] Time [s] Figure 27. The signal measured from an outer race fault bearing: (a) time-domain signal, (b) the shaft Figure 27. The signal measured from an outer race fault bearing: (a) time-domain signal, (b) the shaft Appl. Sci. 2018, 8, x FOR PEER REVIEW 25 of 35 rotational frequency from 18 Hz to 28 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 18 Hz to 28 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. FK =1.6 @ level 4, Bw=625 Hz, f =5937.5 Hz WPT kurtogram: @level4, node(4,3) (a) (b) max c 1.6 2.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) Figure 28. Cont. The Protrugram analysis: Bw=300 Hz, f =2105 Hz, step=100 Hz. OSCK =1 @ level 3.5, Bw=833.3333 Hz, f =2083.3333 Hz c max c X: 2105 Y: 126.4 1.6 2.6 3.6 4.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency[Hz] Frequency [Hz] Figure 28. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. Kurtosis Level k Frequency [Hz] Amplitude Level k Level k Velocity [Hz] Appl. Sci. 2018, 8, x FOR PEER REVIEW 25 of 35 FK =1.6 @ level 4, Bw=625 Hz, f =5937.5 Hz WPT kurtogram: @level4, node(4,3) (a) (b) max c 1.6 2.6 3.6 Appl. Sci. 2019, 9, 1157 23 of 31 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) The Protrugram analysis: Bw=300 Hz, f =2105 Hz, step=100 Hz. OSCK =1 @ level 3.5, Bw=833.3333 Hz, f =2083.3333 Hz c max c X: 2105 Y: 126.4 1.6 2.6 3.6 4.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency[Hz] Frequency [Hz] Figure 28. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Figure 28. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Appl. Sci. 2018, 8, x FOR PEER REVIEW 26 of 35 (c) Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. FK, Bw=625 Hz, fc=5937.5 Hz WPT kurtogram, node(4,3) (a) (b) 0.1 0.1 X: 3.048 X: 3.048 Y : 0.08432 Y: 0.08 0.08 0.08 2X FCO 3X FCO 4X FCO 1X FCO 2X FCO 1X FCO o o o o o o FCO +SRO FCO -SRO FCO -SRO o FCO +SRO 0.06 0.06 SRO SRO 0.04 0.04 0.02 0.02 0 5 10 15 20 25 0 5 10 15 20 25 Order Order (c) The Protrugram, Bw=300 Hz, fc=2105 Hz (d) FOSCK, Bw=833.3333 Hz, fc=2083.3333 Hz 0.1 0.1 X: 3.048 X: 3.048 Y: 0.08 Y: 0.08 0.08 0.08 1X FCO 2X FCO 3X FCO 4X FCO o o o 1X FCO 2X FCO FCO -SRO FCO +SRO 5X FCO o o FCO -SRO FCO +SRO SRO o o 0.06 0.06 SRO 6X FCO 0.04 0.04 0.02 0.02 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure 29. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Figure 29. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. 4.3. Diagnosis of a Bearing with an Inner Race Fault 4.3. Diagnosis of a Bearing with an Inner Race Fault In this test, the setup is the same as in the previous test, except that the left normal bearing is In this test, the setup is the same as in the previous test, except that the left normal bearing is replaced by one with an inner race fault, and the right bearing is normal. Three different experiments replaced by one with an inner race fault, and the right bearing is normal. Three different are carried out to prove the effectiveness of the proposed method for inner race fault diagnosis. White experiments are carried out to prove the effectiveness of the proposed method for inner race fault Gaussian noise (SNR=-3 dB) is also added to the collected signal in each experiment. The shaft diagnosis. White Gaussian noise (SNR=-3 dB) is also added to the collected signal in each rotational speed increases from 20 Hz to 25 Hz following a nearly linear pattern with three different experiment. The shaft rotational speed increases from 20 Hz to 25 Hz following a nearly linear accelerations, which are displayed in Table 6. pattern with three different accelerations, which are displayed in Table 6. Table 6. Parameters of each experiment. Acceleration (Hz/s) Experimental Study #4 Experimental Study #5 Experimental Study #6 a 4/3 3/2 3 4.3.1. Experimental Study #4 As shown in Figure 30a,b, the vibration signal and the rotating speed are collected at the same time during the speed increase from 20 Hz to 25 Hz, and the acceleration a equals 4/3 Hz/s. The frequency spectrum and TFR of the signal are blurred due to the variable rotating speed and the background noise, as shown in Figure 30c,d. Kurtosis Magnitude Magnitude Level k Level k Magnitude Magnitude Level k Appl. Sci. 2019, 9, 1157 24 of 31 Table 6. Parameters of each experiment. Acceleration (Hz/s) Experimental Study #4 Experimental Study #5 Experimental Study #6 a 4/3 3/2 3 4.3.1. Experimental Study #4 As shown in Figure 30a,b, the vibration signal and the rotating speed are collected at the same time during the speed increase from 20 Hz to 25 Hz, and the acceleration a equals 4/3 Hz/s. The frequency spectrum and TFR of the signal are blurred due to the variable rotating speed and the background noise, as shown in Figure 30c,d. Appl. Sci. 2018, 8, x FOR PEER REVIEW 27 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 27 of 35 (a) (b) (a) (b) Real speed Real speed Coded pulse Coded pulse The fitting speed -2 The fitting speed -2 0 1 2 3 4 5 6 (c) (d) 0 1 2 3 4 5 6 10000 10000 (c) (d) 10000 10000 8000 8000 8000 8000 6000 6000 6000 6000 4000 4000 4000 4000 2000 2000 2000 2000 0 0 400 200 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 0 15 400 200 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Amplitude Time [s] Time [s] Amplitude Time [s] Time [s] Figure Figure 30. 30. TheThe signal signal measure measured fr dom from an an inner inner race fault bearing: ( race fault bearing: (aa )) time-domain signal, time-domain signal,((b b ) the ) the shaft shaft Figure 30. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. TheThe optimal optimal frequency frequency bands bancorr ds corr esponding esponding to to di differ ffer entent indexe indexes ar s eare shown shown in in Figur Fige ure 31.31. The The FK The optimal frequency bands corresponding to different indexes are shown in Figure 31. The FK and the WPTK have the same optimal frequency band, which is [625, 1250] Hz as shown in and the WPTK have the same optimal frequency band, which is [625, 1250] Hz as shown in Figure 31a,b. FK and the WPTK have the same optimal frequency band, which is [625, 1250] Hz as shown in Figure 31a,b. The Protrugram is paved in Figure 31c with BW equal to 400 Hz and a step of 100 Hz The Protrugram is paved in Figure 31c with BW equal to 400 Hz and a step of 100 Hz and the center Figure 31a,b. The Protrugram is paved in Figure 31c with BW equal to 400 Hz and a step of 100 Hz and the center frequency is 425.5 Hz. In Figure 31d, the optimal frequency band of the FOSCK is frequency is 425.5 Hz. In Figure 31d, the optimal frequency band of the FOSCK is [3125, 3750] Hz. and the center frequency is 425.5 Hz. In Figure 31d, the optimal frequency band of the FOSCK is [3125, 3750] Hz. The envelope order spectra of the filtered signals corresponding to different indexes The envelope order spectra of the ﬁltered signals corresponding to different indexes are displayed [3125, 3750] Hz. The envelope order spectra of the filtered signals corresponding to different indexes are displayed in Figure 32. The envelope order spectra do not contain any noticeable FCO in Figure in Figure 32. The envelope order spectra do not contain any noticeable FCO in Figure 32a–c, which are displayed in Figure 32. The envelope order spectra do not contain any noticeable FCO in Figure 32a–c, which means that the FK, the WPTK and the Protrugram failed to identify the appropriate means that the FK, the WPTK and the Protrugram failed to identify the appropriate fault sensitive 32a–c, which means that the FK, the WPTK and the Protrugram failed to identify the appropriate fault sensitive resonance frequency band. In Figure 32d, the FCO and its third octaves can be resonance fault sensitiv frequency e reson band. ance frequenc In Figury band. In e 32d, the Figu FCOre 32 andd, the its thir FCO a d octaves nd its thi can r be d oct identiﬁed aves can be in the identified in the envelope order spectrum, although the third harmonics are masked by heavy envelope identified order in spectr the envelope um, although order spectr the thir um, d harmonics although the thi are masked rd haby rmoni heavy cs abackgr re masked by ound noise, heavand y background noise, and the fault component can still be identified. Therefore, the FOSCK has the best background noise, and the fault component can still be identified. Therefore, the FOSCK has the best the fault component can still be identiﬁed. Therefore, the FOSCK has the best ability to detect bearing ability to detect bearing inner race faults in this case. ability to detect bearing inner race faults in this case. inner race faults in this case. FK =3.1 @ level 4, Bw=625 Hz, f =937.5 Hz WPT kurtogram: @level4, node(4,1) (a) (b) max c FK =3.1 @ level 4, Bw=625 Hz, f =937.5 Hz WPT kurtogram: @level4, node(4,1) (a) (b) 0 max c 1.6 1.6 2.6 2.6 3.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Frequency [Hz] Frequency [Hz] (c) (d) (c) (d) The Protrugram analysis: Bw=400 Hz, f =425.5 Hz, step=100 Hz. OSCK =1 @ level 4, Bw=625 Hz, f =3437.5 Hz Figure 31. Cont. c max c The Protrugram analysis: Bw=400 Hz, f =425.5 Hz, step=100 Hz. OSCK =1 @ level 4, Bw=625 Hz, f =3437.5 Hz c max c 350 X: 425.5 X: 4 Y2 : 3 5.5 46.8 Y: 346.8 1 1.6 1.6 2.6 2.6 3.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Frequency [Hz] Kurtosis Frequency [Hz] Level k Kurtosis Frequency [Hz] Level k Amplitude Amplitude Level k Velocity [Hz] Level k Level k Velocity [Hz] Level k Appl. Sci. 2018, 8, x FOR PEER REVIEW 27 of 35 (a) (b) Real speed Coded pulse The fitting speed -2 0 1 2 3 4 5 6 (c) (d) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 15 400 200 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Amplitude Time [s] Time [s] Figure 30. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. The optimal frequency bands corresponding to different indexes are shown in Figure 31. The FK and the WPTK have the same optimal frequency band, which is [625, 1250] Hz as shown in Figure 31a,b. The Protrugram is paved in Figure 31c with BW equal to 400 Hz and a step of 100 Hz and the center frequency is 425.5 Hz. In Figure 31d, the optimal frequency band of the FOSCK is [3125, 3750] Hz. The envelope order spectra of the filtered signals corresponding to different indexes are displayed in Figure 32. The envelope order spectra do not contain any noticeable FCO in Figure 32a–c, which means that the FK, the WPTK and the Protrugram failed to identify the appropriate fault sensitive resonance frequency band. In Figure 32d, the FCO and its third octaves can be identified in the envelope order spectrum, although the third harmonics are masked by heavy background noise, and the fault component can still be identified. Therefore, the FOSCK has the best ability to detect bearing inner race faults in this case. FK =3.1 @ level 4, Bw=625 Hz, f =937.5 Hz WPT kurtogram: @level4, node(4,1) (a) (b) max c 1.6 2.6 3.6 Appl. Sci. 2019, 9, 1157 25 of 31 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) The Protrugram analysis: Bw=400 Hz, f =425.5 Hz, step=100 Hz. OSCK =1 @ level 4, Bw=625 Hz, f =3437.5 Hz c max c X: 425.5 Y: 346.8 1.6 2.6 3.6 Appl. Sci. 2018 0 , 8,2 x FO 000 R P4000 EER REVIEW 6000 8000 10000 28 of 35 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Figure 31. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Figure 31. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. FK, Bw=625 Hz, fc=937.5 Hz WPT kurtogram, node(4,1) (a) (b) 0.02 0.03 X: 4.95 X: 4.95 1X SRO Y: 0.018 Y: 0.02507 0.015 0.025 2X SRO 1X SRO 0.02 3X SRO 2X SRO 0.01 0.015 3X SRO 4X SRO 0.01 4X SRO 0.005 0.005 0 5 10 15 20 25 0 5 10 15 20 25 Order Order (c) The Protrugram, Bw=400 Hz, fc=425.5 Hz (d) FOSCK, Bw=625 Hz, fc=3437.5 Hz 0.01 X: 4.95 Y: 0.008 X: 4.95 0.04 Y: 0.03576 0.008 FCO -2XSRO 1X FCO FCO +2XSRO 2X FCO 3X FCO i i i i i FCO -SRO FCO +SRO i i 1X SRO 0.03 0.006 SRO 2X SRO 0.02 0.004 3X SRO 0.01 0.002 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure 32. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Figure 32. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. 4.3.2. Experimental Study #5 4.3.2. Experimental Study #5 In this experiment, the vibration and the rotational speed signal are collected during the speed In this experiment, the vibration and the rotational speed signal are collected during the speed increase from 20 Hz to 25 Hz with a equal to 3/2 Hz/s, as shown in Figure 33a,b, respectively. As can increase from 20 Hz to 25 Hz with a equal to 3/2 Hz/s, as shown in Figure 33a,b, respectively. As can be seen in Figure 33c,d, the faulty bearing signatures cannot be detected either through the frequency be seen in Figure 33c,d, the faulty bearing signatures cannot be detected either through the spectrum or the envelope order spectrum directly due to the frequency smearing caused by the varying frequency spectrum or the envelope order spectrum directly due to the frequency smearing caused speed and heavy noise. by the varying speed and heavy noise. The analysis of the vibration signal performed by different methods is shown in Figure 34. One can ﬁnd in Figure 34a that the maximum kurtosis occurs at level 4, and the optimal frequency band in (a) (b) the FK is [5000, 5625] Hz. The WPTK is shown in Figure 34b, and the optimal frequency band is Real speed Coded pulse -1 The fitting speed 0 0.5 1 1.5 2 2.5 3 (c) (d) 10000 10000 8000 8000 6000 6000 4000 4000 2000 2000 0 0 15 400 200 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Amplitude Time [s] Time [s] Figure 33. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft rotational frequency from 20 Hz to 25 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. Magnitude Magnitude Frequency [Hz] Frequency [Hz] Kurtosis Level k Amplitude Amplitude Level k Magnitude Magnitude Velocity [Hz] Velocity [Hz] Level k Appl. Sci. 2018, 8, x FOR PEER REVIEW 28 of 35 Figure 31. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. FK, Bw=625 Hz, fc=937.5 Hz WPT kurtogram, node(4,1) (a) (b) 0.02 0.03 X: 4.95 X: 4.95 Y: 0.018 1X SRO Y: 0.02507 0.015 0.025 2X SRO 1X SRO 0.02 3X SRO 2X SRO 0.01 0.015 3X SRO 4X SRO 0.01 4X SRO 0.005 0.005 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order (c) The Protrugram, Bw=400 Hz, fc=425.5 Hz (d) FOSCK, Bw=625 Hz, fc=3437.5 Hz 0.01 X: 4.95 Y: 0.008 X: 4.95 0.04 Y: 0.03576 0.008 FCO -2XSRO 1X FCO FCO +2XSRO 2X FCO 3X FCO i i i i i FCO -SRO FCO +SRO i i 0.03 1X SRO 0.006SRO 2X SRO 0.02 0.004 3X SRO 0.01 0.002 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure 32. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Appl. Sci. 2019, 9, 1157 26 of 31 (c) Protrugram and (d) FOSCK. 4.3.2. Experimental Study #5 [1875, 2500] Hz. Figure 34c shows the Protrugram with BW equal to 400 Hz and step equal to 100 Hz In this experiment, the vibration and the rotational speed signal are collected during the speed and the center frequency is 537.6 Hz. The optimal frequency band of the FOSCK is [2500, 3750] Hz, increase from 20 Hz to 25 Hz with a equal to 3/2 Hz/s, as shown in Figure 33a,b, respectively. As can as shown in Figure 34d. The results of the envelope order spectrum analysis are shown in Figure 35a–d. be seen in Figure 33c,d, the faulty bearing signatures cannot be detected either through the The FCO and its third octaves can be identiﬁed in the envelope order spectrum obtained by the FOSCK, frequency spectrum or the envelope order spectrum directly due to the frequency smearing caused as shown in Figure 35d. Therefore, the FOSCK has the best ability to detect bearing inner race faults in by the varying speed and heavy noise. this case. (a) (b) Real speed Coded pulse -1 The fitting speed Appl. Sci. 2018, 8, x FOR PEER REVIEW 29 of 35 0 0.5 1 1.5 2 2.5 3 (c) (d) 10000 10000 The analysis of the vibration signal performed by different methods is shown in Figure 34. One 8000 8000 can find in Figure 34a that the maximum kurtosis occurs at level 4, and the optimal frequency band 6000 6000 in the FK is [5000, 5625] Hz. The WPTK is shown in Figure 34b, and the optimal frequency band is 4000 4000 [1875, 2500] Hz. Figure 34c shows the Protrugram with BW equal to 400 Hz and step equal to 100 Hz 2000 2000 and the center frequency is 537.6 Hz. The optimal frequency band of the FOSCK is [2500, 3750] Hz, 0 0 15 as shown in Figure 34d. The results of the envelope order spectrum analysis are shown in Figure 400 200 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Amplitude Time [s] Time [s] 35a–d. The FCO and its third octaves can be identified in the envelope order spectrum obtained by the FOSCK, as shown in Figure 35d. Therefore, the FOSCK has the best ability to detect bearing inner Figure Figure 33. 33. The The signal signal measure measured d fr from om an an inner inner race fault bearing: ( race fault bearing: (a)atime-domain ) time-domain signal, signal, (b() bthe ) the shaft shaft race faults in this case. rotational rotational freq frequency uency from 20 Hz to 25 from 20 Hz to 25 Hz, Hz (c , ( )cthe ) the frequency spectrum of frequency spectrum of (a,(d a) ,d TFR ) TFR by us by using inSTFT g STFT. . FK =5 @ level 4, Bw=625 Hz, f =5312.5 Hz WPT kurtogram:@level 4, node(4,3) (a) (b) max c 1.6 2.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) The Protrugram analysis: Bw=400Hz, f =537.6 Hz, step=100Hz OSCK =1 @ level 3, Bw=1250 Hz, f =3125 Hz c max c X: 537.6 Y: 437 1.6 2.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Figure 34. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, Figure 34. The optimal frequency band obtained by different methods: (a) FK, (b) WPTK, (c) (c) Protrugram and (d) FOSCK. Protrugram and (d) FOSCK. Kurtosis Magnitude Magnitude Level k Frequency [Hz] Amplitude Level k Magnitude Magnitude Level k Velocity [Hz] Appl. Sci. 2019, 9, 1157 27 of 31 Appl. Sci. 2018, 8, x FOR PEER REVIEW 30 of 35 Appl. Sci. 2018, 8, x FOR PEER REVIEW 30 of 35 FK, Bw=625 Hz, fc=5312.5 Hz WPT kurtogram, node(4,3) (a) FK, Bw=625 Hz, fc=5312.5 Hz (b) WPT kurtogram, node(4,3) 0.01 (a) (b) 0.01 0.01 X: 4.95 0.01 X Y: 4 : 0 .9 .0 5 08 X: 4.95 Y: 0.008 X: 4.95 Y: 0.008088 0.008 Y: 0.008088 0.008 0.008 0.008 FCO -2XSRO 1X FCO 1X FCO FCO +2XSRO 2X FCO 2X FCO FCO -2XSRO SRO i i i i i i i FCO -2XSRO 1X FCO 2X FCO FCO -2XSRO 1X FCO FCO +2XSRO 2X FCO SRO i i 0.006 i i i i FCO -SRO FCO +SRO FCO +2XSRO 0.006 0.006 i FCO -SRO FCO +SRO i i i i FCO -SRO FCO +SRO FCO +2XSRO 0.006 i FCO -SRO FCO +SRO i i i i 0.004 0.004 0.004 SRO 0.004 SRO 0.002 0.002 0.002 0.002 0 0 0 0 5 10 15 20 25 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Order Order (c) The Protrugram, Bw=400 Hz, fc=537.6 Hz (d) FOSCK, Bw=1250 Hz, fc=3125 Hz (c) The Protrugram, Bw=400 Hz, fc=537.6 Hz (d) FOSCK, Bw=1250 Hz, fc=3125 Hz 0.01 0.01 0.016 0.016 X: 4.95 X Y: 4 : 0 .9 .0 5 08 X: 4.95 0.014 Y: 0.008 X Y: 4 : 0 .9 .0 5 1261 0.014 0.008 Y: 0.01261 0.008 0.012 0.012 1x SRO FCO -2XSRO 1X FCO FCO +2XSRO 2X FCO 3X FCO i i i i i 1x SRO FCO -2XSRO 1X FCO FCO +2XSRO 2X FCO 3X FCO 0.01 0.006 i i i i i FCO -SRO FCO +SRO 2X SRO 0.01 0.006 i i 2X SRO FCO -SRO FCO +SRO i i 3X SRO 0.008 3X SRO SRO 0.008 4X SRO SRO 0.004 4X SRO 0.006 0.004 0.006 0.004 0.004 0.002 0.002 0.002 0.002 0 5 10 15 20 25 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Order Order Figure 35. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Figure 35. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Figure 35. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. 4.3.3. Experimental Study #6 4.3.3. Experimental Study #6 4.3.3. Experimental Study #6 Similar to experiment three, the vibration signal and rotational speed are collected as the speed Similar to experiment three, the vibration signal and rotational speed are collected as the speed Similar to experiment three, the vibration signal and rotational speed are collected as the speed increases from 18 Hz to 28 Hz with a equal to 3 Hz/s to verify that the proposed method is still increases from 18 Hz to 28 Hz with a equal to 3 Hz/s to verify that the proposed method is still increases from 18 Hz to 28 Hz with a equal to 3 Hz/s to verify that the proposed method is still effective in the case of rapidly changing speed and wild ﬂuctuations. These signals are shown in effective in the case of rapidly changing speed and wild fluctuations. These signals are shown in effective in the case of rapidly changing speed and wild fluctuations. These signals are shown in Figure 36a,b, respectively. The frequency spectrum and the TFR of the vibration signal are shown in Figure 36a,b, respectively. The frequency spectrum and the TFR of the vibration signal are shown in Figure 36a,b, respectively. The frequency spectrum and the TFR of the vibration signal are shown in Figure 36c,d, respectively. Figure 36c,d, respectively. Figure 36c,d, respectively. (a) (b) (a) (b) 30 Real speed Real speed Coded pulse Coded pulse The fitting speed -2 The fitting speed -2 0 0.5 1 1.5 2 2.5 3 (c) (d) 0 0.5 1 1.5 2 2.5 3 25 (c) (d) 10000 10000 10000 10000 8000 8000 8000 8000 6000 6000 6000 6000 4000 4000 4000 4000 2000 2000 2000 2000 0 0 400 200 0 0.5 1 1.5 2 2.5 3 15 0 0 0 0.5 1 1.5 2 2.5 3 400 200 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Amplitude Time [s] Time [s] Amplitude Time [s] Time [s] Figure 36. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft Figure 36. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft Figure 36. The signal measured from an inner race fault bearing: (a) time-domain signal, (b) the shaft rotational frequency from 18 Hz to 28 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 18 Hz to 28 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. rotational frequency from 18 Hz to 28 Hz, (c) the frequency spectrum of (a,d) TFR by using STFT. Figure 37 shows the signal analysis results of the inner race fault case. The FK is paved in Figure Figure 37 shows the signal analysis results of the inner race fault case. The FK is paved in Figure 37a, in which the optimal frequency band is [625, 1250] Hz. Figure 37b shows the WPTK, in which 37a, in which the optimal frequency band is [625, 1250] Hz. Figure 37b shows the WPTK, in which Magnitude Magnitude Frequency [Hz] Magnitude Magnitude Frequency [Hz] Am Am plp itud litud e e Magnitude Magnitued Magnitude Magnitued Ve Vlo elo cic ty ity [H [H z]z] Appl. Sci. 2019, 9, 1157 28 of 31 Figure 37 shows the signal analysis results of the inner race fault case. The FK is paved in Appl. Sci. 2018, 8, x FOR PEER REVIEW 31 of 35 Figure 37a, in which the optimal frequency band is [625, 1250] Hz. Figure 37b shows the WPTK, in which the maximum kurtosis is calculated at the 4th decomposition level, and its corresponding the maximum kurtosis is calculated at the 4th decomposition level, and its corresponding optimal optimal frequency band is [1875, 2500] Hz. The Protrugram is shown in Figure 37c and the center frequency band is [1875, 2500] Hz. The Protrugram is shown in Figure 37c and the center frequency frequency is 543.5 Hz. Figure 37d shows the FOSCK, and its optimal frequency band is [0, 1250] Hz. is 543.5 Hz. Figure 37d shows the FOSCK, and its optimal frequency band is [0, 1250] Hz. The The envelope order spectra of the ﬁltered signals are shown in Figure 38a–d. It can be seen that only envelope order spectra of the filtered signals are shown in Figure 38a–d. It can be seen that only the the envelope order spectrum obtained by the FOSCK can extract the ﬁrst three octaves in Figure 38d. envelope order spectrum obtained by the FOSCK can extract the first three octaves in Figure 38d. Therefore, the FOSCK is better than other methods in bearing inner race fault diagnosis in this case. Therefore, the FOSCK is better than other methods in bearing inner race fault diagnosis in this case. FK =3.4 @ level 4, Bw=625 Hz, f =937.5 Hz WPT kurtogram: @level4, node(4,3) (a) (b) max c 1.6 2.6 3.6 4.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] (c) (d) The Protrugram analysis: Bw=500 Hz, f =543.5 Hz, step=100 Hz. OSCK =1 @ level 3, Bw=1250 Hz, f =625 Hz c max c X: 543.5 Y: 264.2 1.6 2.6 3.6 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Frequency [Hz] Frequency [Hz] Figure Figure 37. 37. The The optimal optimal freq frequency uency band o band obtained btained by bydif dif ferent method ferent methods: s: (a) FK (a), ( F b K, ) WPTK, (b) WPTK, (c) Appl. Sci. 2018, 8, x FOR PEER REVIEW 32 of 35 (c) Pr Protrugram and ( otrugram and ( d d ) FO ) FOSCK. SCK. FK, Bw=625 Hz, fc= 937.5Hz WPT kurtogram, node(4,3) (a) (b) 0.02 1X SRO X: 4.95 X: 4.95 Y: 0.018 0.015 Y: 0.01374 0.015 2X SRO SRO FCO -2XSRO FCO FCO +2XSRO 3X SRO i i i 0.01 FCO 0.01 0.005 0.005 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Order Order Figure 38. Cont. (c) The Protrugram, Bw=500 Hz, fc=543.5 Hz (d) FOSCK, Bw=1250 Hz, fc=625 Hz 0.03 0.03 SRO X: 4.95 X: 4.95 Y: 0.028 0.025 0.025 Y: 0.02263 2X FCO 1X FCO 0.02 0.02 i 1X SRO FCO -2XSRO FCO +2XSRO i i 2X SRO 0.015 0.015 3X FCO FCO -SRO FCO +SRO i i i 0.01 3X SRO 0.01 0.005 0.005 0 10 20 30 40 50 0 5 10 15 20 25 Order Order Figure 38. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. 5. Discussion Regarding the various bearing fault signals obtained under different conditions, the diagnoses results are shown in Table 7. Note that the fault is supposed to be diagnosed successfully only if the FCO and its first three harmonics or above are identified effectively, as mentioned in [28]. Table 7. Bearing fault diagnosis result. Fault Location Outer Race Fault Inner Race Fault Experimental Study #1 #2 #3 #4 #5 #6 FK Y N Y N N N WPTK Y N N N N N Protrugram Y Y N N N N FOSCK Y Y Y Y Y Y * Y means Yes and N means No. From the diagnosis results, it is clear that the FOSCK is capable of detecting bearing faults in all cases. It is also concluded that: (1) the fault impact has broadband characteristics and causes different resonances; (2) the Protrugram and FOSCK have better robustness against random impulse disturbance; (3) the FOSCK and FK can diagnose the outer race fault effectively under a large range of the speed fluctuations conditions, which verifies the OSCK index proposed in Section 2 is sensitive to the speed fluctuation while not being affected by the size of the speed; (4) the comparison between the results of experimental studies #1 and #2 shows that both of the Protrugram and FOSCK can suppress the influence of acceleration changes; (5) under the same fault severity Kurtosis Level k Magnitude Magnitude Magnitude Magnitude Level k Level k Appl. Sci. 2018, 8, x FOR PEER REVIEW 32 of 35 FK, Bw=625 Hz, fc= 937.5Hz WPT kurtogram, node(4,3) (a) (b) 0.02 1X SRO X: 4.95 X: 4.95 Y: 0.018 0.015 Y: 0.01374 0.015 2X SRO SRO FCO -2XSRO FCO FCO +2XSRO 3X SRO i i i 0.01 FCO 0.01 0.005 0.005 0 0 Appl. Sci. 2019, 9, 1157 29 of 31 0 5 10 15 20 25 0 5 10 15 20 25 Order Order (c) The Protrugram, Bw=500 Hz, fc=543.5 Hz (d) FOSCK, Bw=1250 Hz, fc=625 Hz 0.03 0.03 SRO X: 4.95 X: 4.95 Y: 0.028 0.025 0.025 Y: 0.02263 2X FCO 1X FCO 0.02 0.02 i 1X SRO FCO -2XSRO FCO +2XSRO i i 2X SRO 0.015 0.015 3X FCO FCO -SRO FCO +SRO i i i 0.01 0.01 3X SRO 0.005 0.005 0 10 20 30 40 50 0 5 10 15 20 25 Order Order Figure 38. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, Figure 38. The envelope order spectra of the signal obtained by different methods: (a) FK, (b) WPTK, (c) Protrugram and (d) FOSCK. (c) Protrugram and (d) FOSCK. 5. Discussion 5. Discussion Regarding the various bearing fault signals obtained under different conditions, the diagnoses Regarding the various bearing fault signals obtained under different conditions, the diagnoses results are shown in Table 7. Note that the fault is supposed to be diagnosed successfully only if the results are shown in Table 7. Note that the fault is supposed to be diagnosed successfully only if the FCO and its ﬁrst three harmonics or above are identiﬁed effectively, as mentioned in [28]. FCO and its first three harmonics or above are identified effectively, as mentioned in [28]. Table 7. Bearing fault diagnosis result. Table 7. Bearing fault diagnosis result. Fault Location Outer Race Fault Inner Race Fault Fault Location Outer Race Fault Inner Race Fault Experimental Study #1 #2 #3 #4 #5 #6 Experimental Study #1 #2 #3 #4 #5 #6 FK Y N Y N N N FK Y N Y N N N WPTK Y N N N N N Protrugram Y Y N N N N WPTK Y N N N N N FOSCK Y Y Y Y Y Y Protrugram Y Y N N N N * Y means Yes and N means No. FOSCK Y Y Y Y Y Y * Y means Yes and N means No. From the diagnosis results, it is clear that the FOSCK is capable of detecting bearing faults in From the diagnosis results, it is clear that the FOSCK is capable of detecting bearing faults in all all cases. It is also concluded that: (1) the fault impact has broadband characteristics and causes cases. It is also concluded that: (1) the fault impact has broadband characteristics and causes different resonances; (2) the Protrugram and FOSCK have better robustness against random impulse different resonances; (2) the Protrugram and FOSCK have better robustness against random impulse disturbance; (3) the FOSCK and FK can diagnose the outer race fault effectively under a large range of disturbance; (3) the FOSCK and FK can diagnose the outer race fault effectively under a large range the speed ﬂuctuations conditions, which veriﬁes the OSCK index proposed in Section 2 is sensitive to of the speed fluctuations conditions, which verifies the OSCK index proposed in Section 2 is the speed ﬂuctuation while not being affected by the size of the speed; (4) the comparison between sensitive to the speed fluctuation while not being affected by the size of the speed; (4) the the results of experimental studies #1 and #2 shows that both of the Protrugram and FOSCK can comparison between the results of experimental studies #1 and #2 shows that both of the Protrugram suppress the inﬂuence of acceleration changes; (5) under the same fault severity conditions, the energy and FOSCK can suppress the influence of acceleration changes; (5) under the same fault severity of the inner race fault is dispersed due to the modulation, the local SNR is lower, in which case the envelope order spectra obtained by the existing methods fails to provide any bearing fault related signature. However, the FOSCK is capable of detecting bearing inner race fault in all cases. The results of performance comparison of the FK, the WPTK, the Protrugram and the FOSCK are summarized in Table 8. Besides, considering the FOSCK is robust to the random shock and heavy noise, the method can be applied for exacting random impulses caused by earthquake, in which the random impulses have similar characteristics with bearing fault impulses [29]. Table 8. Method robustness. Interference Random Shock Large Speed Fluctuation Different Acceleration Heavy Noise FK N Y P N WPTK N N P N Protrugram Y N Y N FOSCK Y Y Y Y * Y means Yes, N means No and P means Pending. Magnitude Magnitude Magnitude Magnitude Appl. Sci. 2019, 9, 1157 30 of 31 6. Conclusions This paper proposes a new feature OSCK based on the COT and CK, and by replacing the OSCK with the kurtosis in the FK, an improved kurtogram the FOSCK is constructed. In the case of simulated signal analysis, the COT procedure may cause warp of the signal resonance band and distortion of the signal amplitude, which means that the COT method must be used after other signal enhancement methods. Compared with other indexes, the OSCK is sensitive to the speed ﬂuctuation while not affected by the size of the speed, so it is more suitable for locating fault-sensitive frequency bands under variable speed conditions. The results of the simulated and experimental bearing vibration signals analyses show that compared with the FK, the WPTK and the Protrugram, the proposed method in this paper can extract fault characteristic information more exactly under different operating conditions and interference environments. In the FOSCK, the COT is carried out many times, which will increases the computational cost. Our work will focus on solving this problem in the future. Author Contributions: Y.R. designed the experiments and analyzed the datasets; W.L., B.Z. and Z.Z. performed the experiments and analyzed part of the dataset; Y.R. and F.J. wrote the paper. All authors contributed to discussing and revising the manuscript. Funding: This work was supported by National Natural Science Foundation of China (No. 51605478), Natural Science Foundation of Jiangsu Province (Nos. BK20160276, BK20160251), China Postdoctoral Science Foundation (No. 2017M621862), Jiangsu Planned Projects for Postdoctoral Research Funds (No.1701193B) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Acknowledgments: The authors would like to thank all of the reviewers for their constructive comments. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Randall, R.B.; Antoni, J. Rolling element bearing diagnostics—A tutorial. Mech. Syst. Signal Process. 2011, 25, 485–520. [CrossRef] 2. 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Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions

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Publisher: Multidisciplinary Digital Publishing Institute
ISSN: 2076-3417
DOI: 10.3390/app9061157
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Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions

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Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions

Fault Diagnosis of Rolling Bearings Based on Improved Kurtogram in Varying Speed Conditions

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