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Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information
Journal Pre-proof Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information Yonghao Miao, Ming Zhao, Yinggang Yi, Jing Lin
PII: DOI: Reference:
S0019-0578(19)30459-8 https://doi.org/10.1016/j.isatra.2019.10.005 ISATRA 3378
To appear in:
ISA Transactions
Received date : 30 April 2019 Revised date : 8 October 2019 Accepted date : 8 October 2019 Please cite this article as: Y. Miao, M. Zhao, Y. Yi et al., Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information. ISA Transactions (2019), doi: https://doi.org/10.1016/j.isatra.2019.10.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd on behalf of ISA.
*Title page showing Author Details
Journal Pre-proof Yonghao Miao, PhD 1. School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China 2. Science & Technology on Reliability and Environmental Engineering Laboratory, Beihang
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University, Xueyuan Road No. 37, Haidian District, Beijing, China Email: [email protected]
Ming Zhao, PhD, Associate Professor
1. Shaanxi Key Laboratory of Quality Assurance and Diagnosis of Mechanical Products, School of
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Mechanical Engineering, Xi’an Jiaotong University, Xi'an 710049, China
2. School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37,
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Haidian District, Beijing 100191, China Email: [email protected]
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Yinggang Yi, PhD candidate
School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China
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Email: [email protected]
Jing Lin1, PhD, Professor
School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China Email: [email protected]
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Corresponding author
*Highlights (for review)
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Highlights 1. This paper originally designs SOVMD for gearbox fault diagnosis based on built-in encoder information. It provides an alternative scheme for the encoder signal analysis.
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2. With least input parameter, the proposed SOVMD is more suitable for the encoder analysis than the traditional decomposition methods.
3. The sparsity operation is introduced to significantly improve the ability of denoising of VMD,
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which broadens the application range of the proposed method.
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*Blinded Manuscript - without Author Details Click here to view linked References
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Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information
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always used for the speed and motion control. Meanwhile, it has remarkable superiority in the fault diagnosis of gearbox compared with the popular vibration signal. Traditional decomposition method, such as EMD, gradually loses competitiveness with the increase of the complexity of the encoder signal. To solve the problem, with aid of the unique characteristic of encoder signal and the decomposition performance of variational mode decomposition (VMD), a new sparsity-oriented VMD (SOVMD), is originally designed and initially introduced for encoder signal analysis in this paper. Firstly, SOVMD is free from the selection of mode number and initial center frequency (ICF), which troubles seriously the application of VMD. Since a prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD, ICF = 0 is more appropriate and easier for the extraction of fault information concentrated in the low frequency region. Benefiting from the characteristics of distribution, the optimization of the mode number is unnecessary since the fault mode will generate in the first mode. Secondly, with the proposed selection criterion of the balance parameter, SOVMD can decompose the mode with most fault information more effectively and accurately. Furthermore, a sparsity operation which is originally designed for the encoder signal analysis can further suppress noise and enhance the fault impulses. Through the simulation and experimental cases from the planet gearbox bench, the feasibility and effectiveness of SOVMD can be verified. Therefore, it is reasonable to conclude that the proposed SOVMD is an alternative scheme for gearbox fault diagnosis based on built-in encoder information. Keywords: Encoder signal, variational mode decomposition (VMD), Balance parameter, Gearbox fault diagnosis, Sparsity
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Abstract: Encoder signal as the built-in information is
Planetary gearboxes have been widely applied in many industries, including aerospace, mining, railway, automobile, manufacturing and wind energy, for their remarkable superiorities such as large loadbearing capacity and compact structure [1]. Due to the long-term service especially under tough operating conditions, planetary gearboxes are susceptible to different kinds of faults [2], for example wear, spalling, crack etc.. It is reported that the gearbox faults account for about 80% of all the failures in the transmission machinery, and the gear faults account for 60% in the gearboxes [3, 4]. Therefore, the fault diagnosis of planetary gearboxes is of great significance for the safety of machinery equipment [5]. Vibration analysis is always considered to be the most powerful tool for the gearbox fault diagnosis [6]. Numerous signal processing techniques, such as deconvolution methods [7, 8], spectral kurtosis [9, 10], wavelet transform [11], decomposition methods [12, 13], intelligent methods [14] and the hybrid methods [15, 16] have been introduced in the field. However, with the advancement of machinery equipment towards large-scale and high-complexity, the vibration-based methods are facing increasing challenges [17]. In recent years, the built-in information of the machinery equipment, for example the encoder signal, is thoroughly studied and its application potential is being gradually explored. Although the raw encoder signal is the accumulated position series which manifested as a gradient straight line, by the difference operation, it could be converted into more meaningful kinetic variables, such as instantaneous angular speed (IAS), instantaneous angular acceleration (IAA) and jerk. Since IAS, IAA and jerk signals carry abundant information related to the fault, they have been considered and proven to be an alternative mean for the fault
1.
Introduction
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other interference. Lin et al. [22] successfully diagnosed the fault of diesel engines by IAS. Zhao et al. [22] found that IAA signal can directly reflect the torsional vibration introduced by faults, then designed a low-pass filter to denoise and further enhance the gearbox fault feature. Although the encoder signal has a higher SNR than the vibration signal under the same scenario [23], noise and interference from the measurement and the errors of difference operation inevitably undermine the usefulness of encoder signal. These researches above just provided the simple and coarse denoising schemes which usually lose effectiveness in identifying faults of different categories both in the presence of environmental noise. To address the bottle neck issue from the complicated interference, decomposition methods which can separate the different components into the different modes with appropriate criterion are introduced. To achieve the feed-axis gearbox condition monitoring, Zhou et al. [18] proposed to use ensemble empirical mode decomposition (EEMD) to decompose IAS signal and combine with power spectrum to extract the fault feature under the non-linear and nonstationary process. Li et al. [24] introduced a hybrid method by combining empirical mode decomposition (EMD), kernel independent component analysis, Wigner bi-spectrum with support vector machine for multi-class fault recognitions of the marine diesel engine with IAS signal. Similarly, to separate the information induced by the gearbox fault, Li et al. [25] proposed auto-correlation local cepstrum and applied it to enhance the performance of EMD in the IAS signal. These works have confirmed the feasibility of decomposition methods in processing encoder signal. Actually, some new decomposition methods, such as local mean decomposition, ensemble local mean decomposition and their variants, have achieved much in terms of machinery fault diagnosis. Especially, a novel adaptive decomposition method, called variational mode decomposition (VMD), is proposed for the
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diagnosis of machinery [17, 18]. Compared with vibration analysis, the research based on built-in encoder information has some remarkable advantages: 1) Different from structural vibration measured by the accelerometer, the encoder signal mainly carries the information regarding the torsional vibration of the system which is more directly related to machine dynamics and fault extraction [19]. Since the fault of the machinery system, especially the gearboxes, has a direct expression of stiffness loss, encoder signal is more sensitive to the change than vibration signals, which indicates the encoder signal has a higher fault-sensibility. 2) Vibration signal is collected by the sensors which must be installed on the casing of equipment. Furthermore, the effect from the complex transmission path between the fault and the sensors becomes one of the most challenging problems in the vibration analysis [20]. By contrast, the signals measured by encoders which are built in the equipment experience shorter transmission path. Therefore, it is easier to extract the fault information based on encoder information with less effect from transmission path. 3) As the vital part, encoders have been widely configured in most machinery equipment for the motion and speed control. Therefore, the measurement of encoder signal is cheaper without the external sensors [19]. In addition, due to the limitations from the install space and wiring requirement of external sensors, the vibration measurement is obviously forbidden under these conditions, such as the cutting of CNC machine tools and the operational process of robots. Without these limitations above, the encoder information measurement has a broader applicability scope. Benefiting from these merits, the research based on built-in encoder information for the fault diagnosis of machines has aroused increasing attention. Gu et al. [19, 21] found IAS has less noise contamination and is more directly related to machine dynamics than the vibration analysis, then applied IAS for the rotor bar fault diagnosis. Different from the vibration signal, IAS is not affected by the engine operation noise and
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been verified that the selection of ICFs affects the decomposition performance of VMD. Since different components may occupy different regions in the frequency domain and even overlap each other, the frequency band with most fault information is mostly concealed. Ref.[36] found a prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD. Fortunately, different from the fault in vibration signal, the fault information concentrates in the low frequency domain of the encoder signal. Benefiting from the feature, the prior ICF is set as 0 which avoids the problems from the estimation error. Therefore, the fault mode of the encoder signal will generate in the first mode when the balance parameter is appropriate. That is, in this paper K = 1 is suitable for the decomposition of encoder signal and the optimal selection of the mode number K will become unnecessary. Secondly, the change rule between the balance parameter and the decomposition performance is explored. By virtue of this, an optimization criterion of balance parameter is proposed to efficiently choose the optimal even under different scenarios. Thirdly, in general the fault information as a mild fluctuation reflects in the encoder signal, such as IAA signal. By contrast, the fluctuation from the gear meshing may have the better visual inspection ability than the fault even after VMD. To resolve the problem, a sparsity operation is applied in the update process of the selection of balance parameter which makes the fault information easy to detect. The remainder of this paper is organized as follows. Section 2 gives a brief review of VMD. In Section 3, the procedure of the proposed sparsity-oriented VMD is introduced in detail. To highlight the advantage of the proposed method, the validation study using the simulation and experiment cases is carried out in Sections 4 and 5, respectively. Finally, the conclusions are drawn in Section 6.
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vibration analysis in the field of machinery fault diagnosis [26]. Since then, the research and improvements about VMD become the hot topic. Firstly, Wang et al. [27, 28] conducted a thorough investigation of VMD from its filter bank property and applications. Through the simulated and experimental analysis, the superiority of VMD over the traditional methods, such as empirical wavelet transform, EMD and EEMD, was validated. More importantly, they found the success of VMD lies in the appropriate choice of the input parameters, including mode number K, balance parameter and the initial center frequencies (ICFs). Subsequently, in the problem of the selection for the parameters K and , various criteria, for example correlation coefficient [29], multi-Teager energy operator [30], scaling exponent [31] and etc., are applied for the decomposition of different signal styles. In addition, to obtain the optimal parameters K and simultaneously, lots of intelligence optimization algorithms, including genetic algorithm [32], grasshopper optimization algorithm [33, 34] and artificial fish swarm algorithm [35] are introduced. Recently, Jiang et al. [36, 37] highlighted the effect of ICFs for the performance of VMD. They found the converging U-shape phenomenon and proposed initial center frequency-guided VMD (ICFVMD). Numerous efforts on VMD have been devoted to the development of the vibration analysis. Meanwhile, they also provide profound foundation and guideline for the extended application of VMD. However, from the signal style to the fault nature, the vibration signal is completely different from the encoder signal. Therefore, it is impractical to directly use VMD to the analysis of the encoder signal just as in the vibration. Motivated by this, considering the unique characteristic of encoder signal and the decomposition performance of VMD, a new VMD method, called sparsity-oriented VMD (SOVMD), is originally designed and introduced for the gearbox fault diagnosis by using built-in encoder information in this paper. Firstly, it has
2. Brief review of VMD
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j K min t [ (t ) uk (t )]e jk t uk ,k k 1 t
represents the L2 norm. Subsequently, by using the alternate direction method of multipliers (ADMM), the equation (2) can be resolved. Then, the decomposed modes uk in Eq.(3) and their center frequencies k in Eq.(4) can be derived by their iterative update until iteration termination condition in Eq.(5) is satisfied.
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VMD is a new non-recursively decomposition method proposed recently as an alternative scheme of EMD. The idea behind the decomposition pattern is to generalize of the classic Wiener filter into multiple, adaptive bands. Through the constraint condition by minimizing the sum of the estimated bandwidth of each mode, the input real value signal can be decomposed into the limited-bandwidth modes with specified numbers [38]. Actually, the composition process of VMD can be considered as the solution process of the variational problem. The constrained variational problem can be described as follows [26]:
ˆ n ( ) fˆ ( ) i k uˆin 1 ( ) i k uˆin ( ) 2 uˆkn1 ( ) 1 2 ( kn ) 2
2 2
n 1 k
(1)
K
subject to uk (t ) f (t )
uˆ uˆ 0
0
n 1 k
uˆkn
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uˆ
k 1
k
where f(t) represents the raw data and t is the time
script. uk is the decomposed mode with the
n 1 k 2
2
( ) d 2
( ) d
/ uˆkn
2
transform of uk (t ) , f (t ) and (t ) , respectively.
is tolerance parameter of the convergence criterion and is set as 10-7.
denotes the convolution operation. k is the
3. Proposal of sparsity-oriented VMD
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3.1. Introduction of encoder signal
Generally, the solution of constrained variational problem to apply quadratic penalty term and Lagrangian multiplier and make the problem unconstrained: K j L(uk , k , ) t [ (t ) uk (t )]e jk t t k 1 K
2
k 1
2
2
2
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f (t ) uk (t )
(5)
where uˆk ( ) , fˆ ( ) and ˆ ( ) are the Fourier
with time t. is the Dirac distribution and *
corresponding center frequency of uk .
(4)
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number K. t () represents the partial derivative
n 1 k
(3)
(2)
K
(t ), f (t ) uk (t ) k 1
where is the balance parameter which can control the bandwidth of the decomposed mode. denotes Lagrangian multiplier coefficient.
.
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As mentioned, the rotary encoders have been widely configurated in the machinery equipment, such as CNC machine tools, industrial robots, etc., for speed and motion control. Generally, encoders are installed on the input and output shaft to measure the angular position or motion of a shaft. The measure principle of encoders has been described in detail in Ref. [39]. Therefore, the signal obtained directly from the encoder sensor is the position series as shown in Fig. 1. Although it is approximatively a gradient straight line, the composition of the encoder signal is considerably complicated. The simulated encoder signal mode can be represented as follows [17]:
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i
(6)
Cm sin 2 mf mt m n(t ) m
g (t ) A exp(t 2 / 2 2 )
(7)
where s(ti ) denotes the estimated IAA at ti . t
is the time interval between samples. To ensure the accuracy of estimation, it is typically set as the reciprocal of sampling rate. It can be found that the noise, especially for the nonstationary noise, is easy to be magnified with the second difference operation. To clearly show the effect, a simulated signal is generated and used to test. The mode parameters are listed in Table 1. Fig. 2 depicts transient impulse induced by fault, the periodic interference and noise with SNR = -8 dB according to the provided parameters. To obviously display these components, the local zoomed figure of transient impulse by fault and the part waveform of periodic interference are presented. Fig. 3 illustrates the IAA and the local zoomed figure of the transient impulse by fault. One can observe that the component caused by the fault is mostly concealed by the noise. It is rather difficult to extract the weak fault information from the noisy signal. Yet, some important features of IAA signal which creates much possibility for VMD in the application of using encoder signal can be summarized as follows: 1) The components from the fault are obviously different from other components, which meets the requirements of decomposition methods to divide the different components into the different modes. 2) It has been verified that the fault information is located the low frequency domain of the frequency spectrum. It provides guideline for the selection of ICF and the mode number using VMD
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As in Eq.(6), the simulated signal can be divided into four parts. The first part caused by the accumulated angular position of a rotating machine with speed v0 is the main component. It commonly dominates the landscape of the temporal waveform. By contrast, other components can be considered as the slight fluctuations according to their amplitudes. The second part is the periodic impacts which are used to simulate the position change induced by the gear fault. T is the period of fault transient impulses and iT denotes the time of the ith impulse occurrence. A and are used to control the amplitude and width of g(t), respectively. The effect from other rotating parts, such as gear meshing, the rotating of shaft and bearing, the load and speed fluctuation also generate change of the position of encoder signal. Therefore, the third part is the combination of the sine signals which are used to model the periodic position oscillation caused by these interferences. Cm, f m and m are the amplitude, frequency and phase of the mth modulation components, respectively. The last part n(t) is the background noise and the measurement noise. Compared with the position sequence, IAS, IAA and jitter signal are more meaningful and interpretable for the fault diagnosis using the signal processing methods [39]. Among these variables, IAA not only can directly reflect the change of the torsional vibration generated by mechanical faults, but has less affected by the interference from the speed fluctuation [17]. Hence, this paper chooses IAA to verify the feasibility of the proposed method. It is easy to convert the angular position to IAA in MATLAB by the second difference operation. Through using the N+1 adjacent data
points to fit an Nth-order polynomial, IAA is obtained by taking the second derivative of polynomial. The process with N=3 can be denoted as follows: s (t +t ) 2s (ti ) s (ti -t ) s(ti )= i (8) t 2
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s(t ) v0t g (t iT )
Table 1. The parameters of simulated encoder signal.
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Value
Parameters
Value
Parameters
Value
v0 (r/min) T (s) f1 (Hz)
300 0.1 300
C2 (degree)
0.01 0.000025
1 (rad)
7/18
-1/3
0.08 0.000025 300 -8
(s)
2 (rad)
A (degree) C1 (degree) f2 (Hz) SNR (dB)
b) ICF is the spaced distribution with specific constraint:
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k0 sort{exp[ln( f ) ln(1/ 2)
ln( f )rand (1, K )]}, k 1, 2,..., K
Fig. 1. Simulated encoder signal. (a)
(10)
where f is the reciprocal of the length of the input signal. sort and rand denote the rank function and rand function, respectively. c) ICF is initialized as zero, i.e.
(b)
k0 0, k 1, 2,..., K
(11)
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The effect of ICF is discussed rarely. Recently, Jiang et al. [36] thoroughly investigated the relationship between the selection of ICF and the decomposition performance. They found the converging U-shape phenomenon from the analysis of vibration signal with machinery fault. Based upon this, some important conclusions are given and verified by the simulation and experiments: 1) A prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD. 2) The optimization of the mode number K is not necessary since the fault mode will generate in the first mode when the balance parameter is appropriate.
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Fig. 2. (a) The transient impulse by fault and its local zoomed figure, (b) the periodic interference with 0-0.5s, (c) noise.
Fig. 3. IAA and the local zoomed figure of the transient impulse by fault. 3.2. Theories and algorithms of sparsity-oriented VMD
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The significance of the appropriate selection of the input parameters, including ICF, the mode number and balancing parameter, has been widely highlighted in the previous articles about VMD. In Ref. [26], three schemes of ICF are provided and can be illustrated as follows: a) ICF is uniformly spaced distribution: k 1 k0 , k 1, 2,..., K (9) 2K
Therefore, the scheme c), i.e. k0 0 , can be applied in the encoder signal analysis. The encoder signal has a relatively simpler spectrum distribution than the vibration signal with the same fault, since the desired information is concentrated in the low frequency domain which can be considered close to 0. And, K=1 is fixed when VMD is applied to decompose the encoder signal. However, the satisfactory decomposition results need more accurate selection of balance parameter which controls the width of the decomposed mode. In the measured encoder signal with gear fault, the
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K=1, ICF k0 0 , 500 and 0 =5000;
paper, 0 =5000 and =500 . Secondly, CK which can reflect the periodicity and impulsiveness of the signal at the same time has been widely applied to evaluate the fault information quantificationally [8, 20]. In the fault diagnosis of gearbox, it is considerably easy to estimate the period of the specified gear fault. Similarly, CK is used as the evaluation index for the selection of the desired fault mode. Although the mode with most fault information can be extracted with least input parameters, in the encoder signal the weak fluctuation induced by gear fault is still inconspicuous because of the overlap of the periodic oscillation caused by the gear meshing. It is rather necessary to improve the SNR of the decomposed mode to make the diagnosis decision easy to give. A sparsity operation with l0 norm is applied to suppress the oscillation interference and further highlight the impulses generated from the gear fault. With the update process of balance parameter, the sparsity operation is added into the loop to enhance the fault impulse. Step 1: Initialization of loop number n=1.
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Step 2: Set the loop number i=1. Run VMD with the input parameters to obtain the decomposed mode ui-1. Calculate the correlated kurtosis (CK) value of ui-1 and define it as CKi-1; Step 3: Start loop. Change the value of balance parameter to obtain the new decomposed mode.
4.2. Without any prior knowledge about , in this
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periodic oscillation caused by the gear meshing are also the low frequency component. To divide the fault component or the mode with most fault information into the first mode, the optimization of balance parameter becomes rather critical. Fortunately, the relationship between the balance parameter and the corresponding decomposition performance is monotonous until the optimal point which will be verified in the following simulation in Section 4.2. Based upon this, a selection criterion of the balance parameter is proposed. The procedure can be elaborated as follows: Step 1: Initialization of the input parameters. Set
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Obtain the mode ui when i i 1 and further calculate its CK, i.e. CKi . And obtain the mode
ui
when
i i 1 i
further
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calculate its CK, i.e. CK ;
and
Step 4: If CKi 1 max[CKi 1 , CKi , CKi ] , break
Calculate CK of the candidate uk , i.e. CKn1 ; Step 2: w 2n if CKn1 CKn ; else end the process;
the loop. If CKi max[CKi 1 , CK i , CK i ] , update
Step 3: uk uk 1 exp uk / 2 w rms(uk )
i i . i=i+1 and go back to Step 3. Else, update
and n=n+1; Step 4: End until n reaches the update number of balance parameter. Fig. 4 illustrates the relationship of the sparsity
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i i . i=i+1 and go back to Step 3;
Step 5: End loop. Obtain the candidate and run VMD. Note that some about the selection process of balance parameter need to be described. Firstly, the initialization of may affect the efficiency of SOVMD, but the result will be free from it, which will be verified by the simulation case in Section
2
2
factor L( x) 1 exp( x 2 / 2 2 ) and the amplitude of signal x. When the signal is applied to multiply the sparsity factor L(x), some significant results can be found [40-42]. Firstly, it can be observed that the points with less amplitude will suppressed with a higher degree. Compared with most noise and
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Section 3.1 has introduced the simulated model and the features of the encoder signal. Firstly, the IAA signal in Fig. 3 is processed by TSA to elimination of asynchronous components. The result is displayed in Fig. 5. Generally, with the strong filtering ability, most noise can be suppressed in the encoder signal. Yet, from the result, the periodic impulses are barely visible which indicates the existence of the heavy noise. ICFVMD with the initial balance parameter 0 =5000 is applied to decompose the signal and the result is illustrated in Fig. 6. Aside from the change of the amplitude, any information about periodic impulses is hardly found. Using the optimization criterion of balance parameter, the decomposition mode with =2000 is shown in Fig. 7. The weak impulses with same interval (20 teeth) are found even though the oscillation component dominates the landscape of the waveform. This may be attributed to the fact that the fault components and the oscillation component share the same domain in the frequency spectrum. However, it is difficult for the Wiener filter designed by VMD to remove the oscillation component. Therefore, the importance of the sparsity operation originally proposed in this paper can be highlighted. With the sparsity operation, the result of the proposed SOVMD is depicted in Fig. 8. From the figure, 4 prominent impulses with 20 teeth interval are clearly observed which indicates the fault generated in the planet gear since the number of planet gear tooth is set as 20 teeth in the simulation. Fig. 9 presents the decomposition result of EMD, from which one can see that EMD fails to identify the periodic impulses from the gear fault.
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interference, the fault impulses have bigger amplitude in the encoder signal. Therefore, the impulses caused by fault are easy to be reserved. Secondly, the sparsity parameter can control the shape of the function of L(x), that is a bigger can obtain a sparer signal. Yet, the fixed sparsity parameter is obviously unsuitable for different conditions with different signals. Therefore, in this paper, we choose the parameters related to the signal feature, such as RMS, Mean and etc., to construct the sparsity factor. After a trial-and-error in the encoder signal, RMS value is considered to be the appropriate sparsity parameter to ensure the feasibility and effectiveness. In addition, an increasing weigh value w is applied to further improve the sparsity by increasing the value of . In all steps, CK values of the candidate signals is used as the assessment criteria, since the increase of CK values means the success of the sparsity operation.
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Fig. 4. The relationship of sparsity factor and its sparsity parameter. Based on the introduction, SOVMD is proposed for the gearbox fault diagnosis using built-in encoder information. To highlight the superiority of the proposed method, a simulation and three experimental data are used to test with ICFVMD and EMD used in the comparative study. Additionally, time synchronous averaging (TSA) [43], which is the effective tool for the enhancement of synchronous components and the elimination of asynchronous components by taking the ensemble average of the segments with the specified period, is used as pretreatment method. And the number of periods being averaged is chosen to be 4 for the measurement of periodicity in this paper.
Fig. 5. IAA after TSA in the simulation.
Journal Pre-proof efficiency change of VMD with the increase of the initial balance parameter (from 1000 to 30000 and =500 ) is obtained in Fig. 10. One can see that iteration times is minimum, i.e. 12, when =2000 . Actually, using the proposed selection criterion, the optimal balance parameter is 2000. Therefore, it is reasonable that iteration times of VMD can reflect the efficiency of VMD. Meanwhile, it is easy to find that the curve of the balance parameter and the corresponding decomposition performance is monotonous. So, the optimal candidate can be found regardless the initial balance parameter. Fig. 11 uses three different initial balance parameters,
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Fig. 6. The result of ICFVMD with the initial balance parameter in the simulation.
Fig. 7. The result of ICFVMD with the optimal balance parameter in the simulation. 20 teeth
including 0 =10000 , 0 =5000 and 0 =1000 to
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IMF#6
IMF#3
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Fig. 8. The result of SOVMD in the simulation.
test the performance. The X-axis is the update times of balance parameter of SOVMD and Y- axis denotes the CK value of the decomposed mode with the corresponding balance parameter. Note that all the update process is terminated when =2000 with different update times. Therefore, the feasibility of the proposed selection criterion of balance parameter is validated. 0 =1000
0 =10000 0 =5000
Fig. 9. The result of EMD in the simulation. 4.2. Performance analysis
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To further clarify the essence of the proposed method in the analysis of encoder signal, performance analysis about the simulation using SOVMD is provided. As mentioned, the initialization of may affect the efficiency of SOVMD, but the result will be free from it. To verify the conclusion, the efficiency of VMD is tested by changing the initial balance parameter
0 . It has been reported [36] that the iteration times of VMD is found to be a reliable evaluation index for the efficiency of VMD. With the input parameters, that is K=1 and ICF k0 0 , the
Fig. 10. The change of iteration times of VMD with the increase of . = 2000
(a) 0 =10000
(b)
Update Times
= 2000
Update Times
= 2000
0 =5000
(c) 0 =1000
Update Times
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crack; (b) surface spalling and (c) half-tooth broken.
5.
5.1. Case 1: root crack Generally, root crack of the gear is the early fault which is rather challenging for the vibration analysis, since the impulses induced by root crack are weak and easily buried by heavy noise. Similarly, the change by root crack in the encoder signal is also mild. To highlight the superiority of SOVMD, the challenging experimental case firstly is applied. Fig. 14 illustrates the measured encoder signal. one cannot find any fault information from the figure. Through the second difference operation and TSA, the result is displayed in Fig. 15. Numerous impulses with small interval caused by the gear meshing can be observed. Yet, the periodic impulses with 31 teeth are hardly visible which can indicate a lack of planet gear fault. ICFVMD with the optimal balance parameter is applied to decompose the signal. Fig. 16 presents the result, from which 4 humps with fix interval are clearly identified. However, it is difficult to determine the position of the fault impulses. By contrast, 4 individual impulses dominate the landscape of the waveform from the result of SOVMD in Fig. 17. It clearly shows the positions of fault impulses and the interval with 31 teeth is easy to be measured. Therefore, it is considerably explicit to give the diagnosis about the fault of planet gear. Fig. 18 displays the result of EMD which does not achieve the expected goal.
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This section uses the experimental data from the planetary gearbox to further verify the advantage of SOVMD. Fig. 12 displays the experimental bench and schematic view of the planetary gearbox. The driving motor provides the torque for the gearbox and the magnetic break is used for loading or braking. The input and output encoders are installed on the shaft of the gearbox to collect the encoder signal. In Fig. 12, the configuration of the gearbox, such as the tooth number of different gears are presented. Three faults, including root crack, surface spalling and half-tooth broken are artificially rooted on the planet gear to model the real gear fault. Through replacing these fault gear as shown in Fig. 13 of the normal planet gear, different encoder signals with planet gear fault are collected. The following subsections will introduce the decomposition performance of SOVMD in processing these signals. In addition, in the following experimental cases, the input parameters of SOVMD are set as K=1;
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Experiment study
0 =5000 ; =500 and ICF k0 0 . Output Planetary Input Encoder Gearbox Encoder
Driving Motor
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Planet gear Zp = 31
Sun gear Zs = 20
Ring gear Zr = 82
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(b)
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Fig. 12. Experimental bench:(a) the physical map; (b) schematic view of the planetary gearbox.
(a)
(b)
Fig. 14. Measured encoder signal with root crack of the gear.
(c)
Fig. 13. Physical maps of different faults: (a) root
Journal Pre-proof Fig. 15. IAA after TSA in the case 1.
demonstrated by remarkable visual inspection ability for fault impulses with 31 teeth interval. Yet, EMD still fails to provide any fault related signatures from Fig. 22.
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Fig. 16. The result of ICFVMD with the optimal balance parameter in the case 1. 31 teeth
Fig. 19. IAA after TSA in the case 2.
Fig. 17. The result of SOVMD in the case 1. IMF#5
IMF#6
IMF#3
IMF#7
IMF#4
IMF#8
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IMF#2
Fig. 20. The result of ICFVMD with the optimal balance parameter in the case 2.
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IMF#1
31 teeth
Fig. 21. The result of SOVMD in the case 2.
Fig. 18. The result of EMD in the case 1.
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5.2. Case 2: surface spalling
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In this subsection, the encoder signal with surface spalling of planet gear is used to test the proposed method. Fig. 19 gives the IAA signal after TSA. One can clearly find the periodic impulses caused by the gear meshing from the figure. Meanwhile, they obfuscate the fault information in turn. To extract the impulses generated by planet gear fault, ICFVMD with the optimal balance parameter is applied to decompose the signal and the result is presented in Fig. 20. Similarly, 4 humps appear in the decomposed mode. The oscillation effect from the gear meshing also occupies the waveform. To suppress the oscillation component and enhance the fault information, SOVMD is applied. From the result in Fig. 21, the advantage of SOVMD is clearly
IMF#1
IMF#5
IMF#2
IMF#6
IMF#3
IMF#7
IMF#4
IMF#8
Fig. 22. The result of EMD in the case 2. 5.3. Case 3: half-tooth broken Compared with the previous cases, the half-tooth broken fault is easier to be found. From the IAA signal after TSA in Fig. 23, the periodic impulses caused by gear meshing are rather clear. Furthermore, by virtue of the advantage of encoder signal, that is it is more sensitive to the fault than
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IMF#1
IMF#2
Fig. 24. The result of ICFVMD with the optimal balance parameter in the case 3.
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Fig. 25. The result of SOVMD in the case 3.
IMF#6
IMF#4
IMF#8
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IMF#7
Fig. 26. The result of EMD in the case 3. 6. Conclusion
From the measurement cost and convenience perspective, the research on the encoder signal has more advantages in the fault diagnosis of gearbox. Even though the traditional decomposition methods, such as EMD, have been widely applied to separate the fault information from encoder signal. Yet, diverse noises undermine the usefulness of EMD. To solve the problem, this paper proposed a novel sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information. Adequately considering the unique characteristic of encoder signal and the decomposition performance of VMD, the proposed SOVMD shows remarkable performance superiority compared with the traditional decomposition methods. Through the comparison of simulation and experimental cases, including root crack fault, surface spalling fault and half-tooth broken fault on planet gear, the feasibility and effectiveness of SOVMD can be verified. Based upon the analysis, the known contributions of this paper are summarized as follows: (1) This paper originally designs SOVMD for gearbox fault diagnosis based on built-in encoder information. It provides an alternative scheme for the encoder signal analysis. (2) With least input parameter, the proposed
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Fig. 23. IAA after TSA in the case 3.
IMF#5
IMF#3
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vibration signal, the change of planet gear fault in the encoder signal could be observed faintly. The removement of the interference from the gear meshing and other noise is of great significance for the correct diagnosis decision. Therefore, to avoid the wrong conclusion and the omission of diagnosis, more efforts should be devoted to denoising of encoder signal. Fig. 24 displays the result of ICFVMD with the optimal balance parameter in processing of the encoder signal with half-tooth broken fault. The fault impulses, including their positions and interval can be identified. By contrast, the result of SOVMD in Fig. 25 has a higher SNR. The remarkable distinction highlights the importance of the sparsity operation proposed in this paper. The result of EMD is shown in Fig. 26. It is difficult to find the fault information even though two obvious impulses generate in mode 2 with 62 interval which means EMD just extracts partial information.
Journal Pre-proof SOVMD is more suitable for the encoder analysis than the traditional decomposition methods. (3) The sparsity operation is introduced to significantly improve the ability of denoising of VMD, which broadens the application range of the proposed method.
model based gear tooth fault detection technique by the use of minimum entropy deconvolution filter, Mechanical Systems and Signal Processing, 21 (2007) 906-919. [8] G.L. McDonald, Q. Zhao, M.J. Zuo, Maximum correlated Kurtosis deconvolution and application on gear tooth chip fault detection, Mechanical Systems and Signal Processing, 33 (2012) 237-255. [9] T. Barszcz, R.B. Randall, Application of spectral kurtosis
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for detection of a tooth crack in the planetary gear of a wind
Acknowledgments
turbine, Mechanical Systems and Signal Processing, 23 (2009)
This work is supported by the National Natural Science Foundation of China (91860205, 51875434), and the Defense Industrial Technology Development Program (JCKY2018601C013), which are highly appreciated by the authors.
1352-1365.
[10] F. Combet, L. Gelman, Optimal filtering of gear signals for early damage detection based on the spectral kurtosis, Mechanical Systems and Signal Processing, 23 (2009) 652-668.
[11] B. Chen, Z. Zhang, C. Sun, B. Li, Y. Zi, Z. He, Fault feature extraction of gearbox by using overcomplete rational
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Journal Pre-proof *Conflict of Interest
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII: DOI: Reference:
S0019-0578(19)30459-8 https://doi.org/10.1016/j.isatra.2019.10.005 ISATRA 3378
To appear in:
ISA Transactions
Received date : 30 April 2019 Revised date : 8 October 2019 Accepted date : 8 October 2019 Please cite this article as: Y. Miao, M. Zhao, Y. Yi et al., Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information. ISA Transactions (2019), doi: https://doi.org/10.1016/j.isatra.2019.10.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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*Title page showing Author Details
Journal Pre-proof Yonghao Miao, PhD 1. School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China 2. Science & Technology on Reliability and Environmental Engineering Laboratory, Beihang
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University, Xueyuan Road No. 37, Haidian District, Beijing, China Email: [email protected]
Ming Zhao, PhD, Associate Professor
1. Shaanxi Key Laboratory of Quality Assurance and Diagnosis of Mechanical Products, School of
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Mechanical Engineering, Xi’an Jiaotong University, Xi'an 710049, China
2. School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37,
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Haidian District, Beijing 100191, China Email: [email protected]
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Yinggang Yi, PhD candidate
School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China
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Email: [email protected]
Jing Lin1, PhD, Professor
School of Reliability and Systems Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China Email: [email protected]
1
Corresponding author
*Highlights (for review)
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Highlights 1. This paper originally designs SOVMD for gearbox fault diagnosis based on built-in encoder information. It provides an alternative scheme for the encoder signal analysis.
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2. With least input parameter, the proposed SOVMD is more suitable for the encoder analysis than the traditional decomposition methods.
3. The sparsity operation is introduced to significantly improve the ability of denoising of VMD,
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which broadens the application range of the proposed method.
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*Blinded Manuscript - without Author Details Click here to view linked References
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Application of sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information
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always used for the speed and motion control. Meanwhile, it has remarkable superiority in the fault diagnosis of gearbox compared with the popular vibration signal. Traditional decomposition method, such as EMD, gradually loses competitiveness with the increase of the complexity of the encoder signal. To solve the problem, with aid of the unique characteristic of encoder signal and the decomposition performance of variational mode decomposition (VMD), a new sparsity-oriented VMD (SOVMD), is originally designed and initially introduced for encoder signal analysis in this paper. Firstly, SOVMD is free from the selection of mode number and initial center frequency (ICF), which troubles seriously the application of VMD. Since a prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD, ICF = 0 is more appropriate and easier for the extraction of fault information concentrated in the low frequency region. Benefiting from the characteristics of distribution, the optimization of the mode number is unnecessary since the fault mode will generate in the first mode. Secondly, with the proposed selection criterion of the balance parameter, SOVMD can decompose the mode with most fault information more effectively and accurately. Furthermore, a sparsity operation which is originally designed for the encoder signal analysis can further suppress noise and enhance the fault impulses. Through the simulation and experimental cases from the planet gearbox bench, the feasibility and effectiveness of SOVMD can be verified. Therefore, it is reasonable to conclude that the proposed SOVMD is an alternative scheme for gearbox fault diagnosis based on built-in encoder information. Keywords: Encoder signal, variational mode decomposition (VMD), Balance parameter, Gearbox fault diagnosis, Sparsity
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Abstract: Encoder signal as the built-in information is
Planetary gearboxes have been widely applied in many industries, including aerospace, mining, railway, automobile, manufacturing and wind energy, for their remarkable superiorities such as large loadbearing capacity and compact structure [1]. Due to the long-term service especially under tough operating conditions, planetary gearboxes are susceptible to different kinds of faults [2], for example wear, spalling, crack etc.. It is reported that the gearbox faults account for about 80% of all the failures in the transmission machinery, and the gear faults account for 60% in the gearboxes [3, 4]. Therefore, the fault diagnosis of planetary gearboxes is of great significance for the safety of machinery equipment [5]. Vibration analysis is always considered to be the most powerful tool for the gearbox fault diagnosis [6]. Numerous signal processing techniques, such as deconvolution methods [7, 8], spectral kurtosis [9, 10], wavelet transform [11], decomposition methods [12, 13], intelligent methods [14] and the hybrid methods [15, 16] have been introduced in the field. However, with the advancement of machinery equipment towards large-scale and high-complexity, the vibration-based methods are facing increasing challenges [17]. In recent years, the built-in information of the machinery equipment, for example the encoder signal, is thoroughly studied and its application potential is being gradually explored. Although the raw encoder signal is the accumulated position series which manifested as a gradient straight line, by the difference operation, it could be converted into more meaningful kinetic variables, such as instantaneous angular speed (IAS), instantaneous angular acceleration (IAA) and jerk. Since IAS, IAA and jerk signals carry abundant information related to the fault, they have been considered and proven to be an alternative mean for the fault
1.
Introduction
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other interference. Lin et al. [22] successfully diagnosed the fault of diesel engines by IAS. Zhao et al. [22] found that IAA signal can directly reflect the torsional vibration introduced by faults, then designed a low-pass filter to denoise and further enhance the gearbox fault feature. Although the encoder signal has a higher SNR than the vibration signal under the same scenario [23], noise and interference from the measurement and the errors of difference operation inevitably undermine the usefulness of encoder signal. These researches above just provided the simple and coarse denoising schemes which usually lose effectiveness in identifying faults of different categories both in the presence of environmental noise. To address the bottle neck issue from the complicated interference, decomposition methods which can separate the different components into the different modes with appropriate criterion are introduced. To achieve the feed-axis gearbox condition monitoring, Zhou et al. [18] proposed to use ensemble empirical mode decomposition (EEMD) to decompose IAS signal and combine with power spectrum to extract the fault feature under the non-linear and nonstationary process. Li et al. [24] introduced a hybrid method by combining empirical mode decomposition (EMD), kernel independent component analysis, Wigner bi-spectrum with support vector machine for multi-class fault recognitions of the marine diesel engine with IAS signal. Similarly, to separate the information induced by the gearbox fault, Li et al. [25] proposed auto-correlation local cepstrum and applied it to enhance the performance of EMD in the IAS signal. These works have confirmed the feasibility of decomposition methods in processing encoder signal. Actually, some new decomposition methods, such as local mean decomposition, ensemble local mean decomposition and their variants, have achieved much in terms of machinery fault diagnosis. Especially, a novel adaptive decomposition method, called variational mode decomposition (VMD), is proposed for the
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diagnosis of machinery [17, 18]. Compared with vibration analysis, the research based on built-in encoder information has some remarkable advantages: 1) Different from structural vibration measured by the accelerometer, the encoder signal mainly carries the information regarding the torsional vibration of the system which is more directly related to machine dynamics and fault extraction [19]. Since the fault of the machinery system, especially the gearboxes, has a direct expression of stiffness loss, encoder signal is more sensitive to the change than vibration signals, which indicates the encoder signal has a higher fault-sensibility. 2) Vibration signal is collected by the sensors which must be installed on the casing of equipment. Furthermore, the effect from the complex transmission path between the fault and the sensors becomes one of the most challenging problems in the vibration analysis [20]. By contrast, the signals measured by encoders which are built in the equipment experience shorter transmission path. Therefore, it is easier to extract the fault information based on encoder information with less effect from transmission path. 3) As the vital part, encoders have been widely configured in most machinery equipment for the motion and speed control. Therefore, the measurement of encoder signal is cheaper without the external sensors [19]. In addition, due to the limitations from the install space and wiring requirement of external sensors, the vibration measurement is obviously forbidden under these conditions, such as the cutting of CNC machine tools and the operational process of robots. Without these limitations above, the encoder information measurement has a broader applicability scope. Benefiting from these merits, the research based on built-in encoder information for the fault diagnosis of machines has aroused increasing attention. Gu et al. [19, 21] found IAS has less noise contamination and is more directly related to machine dynamics than the vibration analysis, then applied IAS for the rotor bar fault diagnosis. Different from the vibration signal, IAS is not affected by the engine operation noise and
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been verified that the selection of ICFs affects the decomposition performance of VMD. Since different components may occupy different regions in the frequency domain and even overlap each other, the frequency band with most fault information is mostly concealed. Ref.[36] found a prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD. Fortunately, different from the fault in vibration signal, the fault information concentrates in the low frequency domain of the encoder signal. Benefiting from the feature, the prior ICF is set as 0 which avoids the problems from the estimation error. Therefore, the fault mode of the encoder signal will generate in the first mode when the balance parameter is appropriate. That is, in this paper K = 1 is suitable for the decomposition of encoder signal and the optimal selection of the mode number K will become unnecessary. Secondly, the change rule between the balance parameter and the decomposition performance is explored. By virtue of this, an optimization criterion of balance parameter is proposed to efficiently choose the optimal even under different scenarios. Thirdly, in general the fault information as a mild fluctuation reflects in the encoder signal, such as IAA signal. By contrast, the fluctuation from the gear meshing may have the better visual inspection ability than the fault even after VMD. To resolve the problem, a sparsity operation is applied in the update process of the selection of balance parameter which makes the fault information easy to detect. The remainder of this paper is organized as follows. Section 2 gives a brief review of VMD. In Section 3, the procedure of the proposed sparsity-oriented VMD is introduced in detail. To highlight the advantage of the proposed method, the validation study using the simulation and experiment cases is carried out in Sections 4 and 5, respectively. Finally, the conclusions are drawn in Section 6.
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vibration analysis in the field of machinery fault diagnosis [26]. Since then, the research and improvements about VMD become the hot topic. Firstly, Wang et al. [27, 28] conducted a thorough investigation of VMD from its filter bank property and applications. Through the simulated and experimental analysis, the superiority of VMD over the traditional methods, such as empirical wavelet transform, EMD and EEMD, was validated. More importantly, they found the success of VMD lies in the appropriate choice of the input parameters, including mode number K, balance parameter and the initial center frequencies (ICFs). Subsequently, in the problem of the selection for the parameters K and , various criteria, for example correlation coefficient [29], multi-Teager energy operator [30], scaling exponent [31] and etc., are applied for the decomposition of different signal styles. In addition, to obtain the optimal parameters K and simultaneously, lots of intelligence optimization algorithms, including genetic algorithm [32], grasshopper optimization algorithm [33, 34] and artificial fish swarm algorithm [35] are introduced. Recently, Jiang et al. [36, 37] highlighted the effect of ICFs for the performance of VMD. They found the converging U-shape phenomenon and proposed initial center frequency-guided VMD (ICFVMD). Numerous efforts on VMD have been devoted to the development of the vibration analysis. Meanwhile, they also provide profound foundation and guideline for the extended application of VMD. However, from the signal style to the fault nature, the vibration signal is completely different from the encoder signal. Therefore, it is impractical to directly use VMD to the analysis of the encoder signal just as in the vibration. Motivated by this, considering the unique characteristic of encoder signal and the decomposition performance of VMD, a new VMD method, called sparsity-oriented VMD (SOVMD), is originally designed and introduced for the gearbox fault diagnosis by using built-in encoder information in this paper. Firstly, it has
2. Brief review of VMD
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j K min t [ (t ) uk (t )]e jk t uk ,k k 1 t
represents the L2 norm. Subsequently, by using the alternate direction method of multipliers (ADMM), the equation (2) can be resolved. Then, the decomposed modes uk in Eq.(3) and their center frequencies k in Eq.(4) can be derived by their iterative update until iteration termination condition in Eq.(5) is satisfied.
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VMD is a new non-recursively decomposition method proposed recently as an alternative scheme of EMD. The idea behind the decomposition pattern is to generalize of the classic Wiener filter into multiple, adaptive bands. Through the constraint condition by minimizing the sum of the estimated bandwidth of each mode, the input real value signal can be decomposed into the limited-bandwidth modes with specified numbers [38]. Actually, the composition process of VMD can be considered as the solution process of the variational problem. The constrained variational problem can be described as follows [26]:
ˆ n ( ) fˆ ( ) i k uˆin 1 ( ) i k uˆin ( ) 2 uˆkn1 ( ) 1 2 ( kn ) 2
2 2
n 1 k
(1)
K
subject to uk (t ) f (t )
uˆ uˆ 0
0
n 1 k
uˆkn
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uˆ
k 1
k
where f(t) represents the raw data and t is the time
script. uk is the decomposed mode with the
n 1 k 2
2
( ) d 2
( ) d
/ uˆkn
2
transform of uk (t ) , f (t ) and (t ) , respectively.
is tolerance parameter of the convergence criterion and is set as 10-7.
denotes the convolution operation. k is the
3. Proposal of sparsity-oriented VMD
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3.1. Introduction of encoder signal
Generally, the solution of constrained variational problem to apply quadratic penalty term and Lagrangian multiplier and make the problem unconstrained: K j L(uk , k , ) t [ (t ) uk (t )]e jk t t k 1 K
2
k 1
2
2
2
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f (t ) uk (t )
(5)
where uˆk ( ) , fˆ ( ) and ˆ ( ) are the Fourier
with time t. is the Dirac distribution and *
corresponding center frequency of uk .
(4)
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number K. t () represents the partial derivative
n 1 k
(3)
(2)
K
(t ), f (t ) uk (t ) k 1
where is the balance parameter which can control the bandwidth of the decomposed mode. denotes Lagrangian multiplier coefficient.
.
2
As mentioned, the rotary encoders have been widely configurated in the machinery equipment, such as CNC machine tools, industrial robots, etc., for speed and motion control. Generally, encoders are installed on the input and output shaft to measure the angular position or motion of a shaft. The measure principle of encoders has been described in detail in Ref. [39]. Therefore, the signal obtained directly from the encoder sensor is the position series as shown in Fig. 1. Although it is approximatively a gradient straight line, the composition of the encoder signal is considerably complicated. The simulated encoder signal mode can be represented as follows [17]:
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i
(6)
Cm sin 2 mf mt m n(t ) m
g (t ) A exp(t 2 / 2 2 )
(7)
where s(ti ) denotes the estimated IAA at ti . t
is the time interval between samples. To ensure the accuracy of estimation, it is typically set as the reciprocal of sampling rate. It can be found that the noise, especially for the nonstationary noise, is easy to be magnified with the second difference operation. To clearly show the effect, a simulated signal is generated and used to test. The mode parameters are listed in Table 1. Fig. 2 depicts transient impulse induced by fault, the periodic interference and noise with SNR = -8 dB according to the provided parameters. To obviously display these components, the local zoomed figure of transient impulse by fault and the part waveform of periodic interference are presented. Fig. 3 illustrates the IAA and the local zoomed figure of the transient impulse by fault. One can observe that the component caused by the fault is mostly concealed by the noise. It is rather difficult to extract the weak fault information from the noisy signal. Yet, some important features of IAA signal which creates much possibility for VMD in the application of using encoder signal can be summarized as follows: 1) The components from the fault are obviously different from other components, which meets the requirements of decomposition methods to divide the different components into the different modes. 2) It has been verified that the fault information is located the low frequency domain of the frequency spectrum. It provides guideline for the selection of ICF and the mode number using VMD
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As in Eq.(6), the simulated signal can be divided into four parts. The first part caused by the accumulated angular position of a rotating machine with speed v0 is the main component. It commonly dominates the landscape of the temporal waveform. By contrast, other components can be considered as the slight fluctuations according to their amplitudes. The second part is the periodic impacts which are used to simulate the position change induced by the gear fault. T is the period of fault transient impulses and iT denotes the time of the ith impulse occurrence. A and are used to control the amplitude and width of g(t), respectively. The effect from other rotating parts, such as gear meshing, the rotating of shaft and bearing, the load and speed fluctuation also generate change of the position of encoder signal. Therefore, the third part is the combination of the sine signals which are used to model the periodic position oscillation caused by these interferences. Cm, f m and m are the amplitude, frequency and phase of the mth modulation components, respectively. The last part n(t) is the background noise and the measurement noise. Compared with the position sequence, IAS, IAA and jitter signal are more meaningful and interpretable for the fault diagnosis using the signal processing methods [39]. Among these variables, IAA not only can directly reflect the change of the torsional vibration generated by mechanical faults, but has less affected by the interference from the speed fluctuation [17]. Hence, this paper chooses IAA to verify the feasibility of the proposed method. It is easy to convert the angular position to IAA in MATLAB by the second difference operation. Through using the N+1 adjacent data
points to fit an Nth-order polynomial, IAA is obtained by taking the second derivative of polynomial. The process with N=3 can be denoted as follows: s (t +t ) 2s (ti ) s (ti -t ) s(ti )= i (8) t 2
pro of
s(t ) v0t g (t iT )
Table 1. The parameters of simulated encoder signal.
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Value
Parameters
Value
Parameters
Value
v0 (r/min) T (s) f1 (Hz)
300 0.1 300
C2 (degree)
0.01 0.000025
1 (rad)
7/18
-1/3
0.08 0.000025 300 -8
(s)
2 (rad)
A (degree) C1 (degree) f2 (Hz) SNR (dB)
b) ICF is the spaced distribution with specific constraint:
pro of
k0 sort{exp[ln( f ) ln(1/ 2)
ln( f )rand (1, K )]}, k 1, 2,..., K
Fig. 1. Simulated encoder signal. (a)
(10)
where f is the reciprocal of the length of the input signal. sort and rand denote the rank function and rand function, respectively. c) ICF is initialized as zero, i.e.
(b)
k0 0, k 1, 2,..., K
(11)
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The effect of ICF is discussed rarely. Recently, Jiang et al. [36] thoroughly investigated the relationship between the selection of ICF and the decomposition performance. They found the converging U-shape phenomenon from the analysis of vibration signal with machinery fault. Based upon this, some important conclusions are given and verified by the simulation and experiments: 1) A prior ICF which coarsely indicates the location of the fault band can enhance the decomposing efficiency of VMD. 2) The optimization of the mode number K is not necessary since the fault mode will generate in the first mode when the balance parameter is appropriate.
(c)
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Fig. 2. (a) The transient impulse by fault and its local zoomed figure, (b) the periodic interference with 0-0.5s, (c) noise.
Fig. 3. IAA and the local zoomed figure of the transient impulse by fault. 3.2. Theories and algorithms of sparsity-oriented VMD
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The significance of the appropriate selection of the input parameters, including ICF, the mode number and balancing parameter, has been widely highlighted in the previous articles about VMD. In Ref. [26], three schemes of ICF are provided and can be illustrated as follows: a) ICF is uniformly spaced distribution: k 1 k0 , k 1, 2,..., K (9) 2K
Therefore, the scheme c), i.e. k0 0 , can be applied in the encoder signal analysis. The encoder signal has a relatively simpler spectrum distribution than the vibration signal with the same fault, since the desired information is concentrated in the low frequency domain which can be considered close to 0. And, K=1 is fixed when VMD is applied to decompose the encoder signal. However, the satisfactory decomposition results need more accurate selection of balance parameter which controls the width of the decomposed mode. In the measured encoder signal with gear fault, the
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K=1, ICF k0 0 , 500 and 0 =5000;
paper, 0 =5000 and =500 . Secondly, CK which can reflect the periodicity and impulsiveness of the signal at the same time has been widely applied to evaluate the fault information quantificationally [8, 20]. In the fault diagnosis of gearbox, it is considerably easy to estimate the period of the specified gear fault. Similarly, CK is used as the evaluation index for the selection of the desired fault mode. Although the mode with most fault information can be extracted with least input parameters, in the encoder signal the weak fluctuation induced by gear fault is still inconspicuous because of the overlap of the periodic oscillation caused by the gear meshing. It is rather necessary to improve the SNR of the decomposed mode to make the diagnosis decision easy to give. A sparsity operation with l0 norm is applied to suppress the oscillation interference and further highlight the impulses generated from the gear fault. With the update process of balance parameter, the sparsity operation is added into the loop to enhance the fault impulse. Step 1: Initialization of loop number n=1.
re-
Step 2: Set the loop number i=1. Run VMD with the input parameters to obtain the decomposed mode ui-1. Calculate the correlated kurtosis (CK) value of ui-1 and define it as CKi-1; Step 3: Start loop. Change the value of balance parameter to obtain the new decomposed mode.
4.2. Without any prior knowledge about , in this
pro of
periodic oscillation caused by the gear meshing are also the low frequency component. To divide the fault component or the mode with most fault information into the first mode, the optimization of balance parameter becomes rather critical. Fortunately, the relationship between the balance parameter and the corresponding decomposition performance is monotonous until the optimal point which will be verified in the following simulation in Section 4.2. Based upon this, a selection criterion of the balance parameter is proposed. The procedure can be elaborated as follows: Step 1: Initialization of the input parameters. Set
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Obtain the mode ui when i i 1 and further calculate its CK, i.e. CKi . And obtain the mode
ui
when
i i 1 i
further
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calculate its CK, i.e. CK ;
and
Step 4: If CKi 1 max[CKi 1 , CKi , CKi ] , break
Calculate CK of the candidate uk , i.e. CKn1 ; Step 2: w 2n if CKn1 CKn ; else end the process;
the loop. If CKi max[CKi 1 , CK i , CK i ] , update
Step 3: uk uk 1 exp uk / 2 w rms(uk )
i i . i=i+1 and go back to Step 3. Else, update
and n=n+1; Step 4: End until n reaches the update number of balance parameter. Fig. 4 illustrates the relationship of the sparsity
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i i . i=i+1 and go back to Step 3;
Step 5: End loop. Obtain the candidate and run VMD. Note that some about the selection process of balance parameter need to be described. Firstly, the initialization of may affect the efficiency of SOVMD, but the result will be free from it, which will be verified by the simulation case in Section
2
2
factor L( x) 1 exp( x 2 / 2 2 ) and the amplitude of signal x. When the signal is applied to multiply the sparsity factor L(x), some significant results can be found [40-42]. Firstly, it can be observed that the points with less amplitude will suppressed with a higher degree. Compared with most noise and
Journal Pre-proof 4. Simulation analysis 4.1. Signal decomposition
pro of
Section 3.1 has introduced the simulated model and the features of the encoder signal. Firstly, the IAA signal in Fig. 3 is processed by TSA to elimination of asynchronous components. The result is displayed in Fig. 5. Generally, with the strong filtering ability, most noise can be suppressed in the encoder signal. Yet, from the result, the periodic impulses are barely visible which indicates the existence of the heavy noise. ICFVMD with the initial balance parameter 0 =5000 is applied to decompose the signal and the result is illustrated in Fig. 6. Aside from the change of the amplitude, any information about periodic impulses is hardly found. Using the optimization criterion of balance parameter, the decomposition mode with =2000 is shown in Fig. 7. The weak impulses with same interval (20 teeth) are found even though the oscillation component dominates the landscape of the waveform. This may be attributed to the fact that the fault components and the oscillation component share the same domain in the frequency spectrum. However, it is difficult for the Wiener filter designed by VMD to remove the oscillation component. Therefore, the importance of the sparsity operation originally proposed in this paper can be highlighted. With the sparsity operation, the result of the proposed SOVMD is depicted in Fig. 8. From the figure, 4 prominent impulses with 20 teeth interval are clearly observed which indicates the fault generated in the planet gear since the number of planet gear tooth is set as 20 teeth in the simulation. Fig. 9 presents the decomposition result of EMD, from which one can see that EMD fails to identify the periodic impulses from the gear fault.
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interference, the fault impulses have bigger amplitude in the encoder signal. Therefore, the impulses caused by fault are easy to be reserved. Secondly, the sparsity parameter can control the shape of the function of L(x), that is a bigger can obtain a sparer signal. Yet, the fixed sparsity parameter is obviously unsuitable for different conditions with different signals. Therefore, in this paper, we choose the parameters related to the signal feature, such as RMS, Mean and etc., to construct the sparsity factor. After a trial-and-error in the encoder signal, RMS value is considered to be the appropriate sparsity parameter to ensure the feasibility and effectiveness. In addition, an increasing weigh value w is applied to further improve the sparsity by increasing the value of . In all steps, CK values of the candidate signals is used as the assessment criteria, since the increase of CK values means the success of the sparsity operation.
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Fig. 4. The relationship of sparsity factor and its sparsity parameter. Based on the introduction, SOVMD is proposed for the gearbox fault diagnosis using built-in encoder information. To highlight the superiority of the proposed method, a simulation and three experimental data are used to test with ICFVMD and EMD used in the comparative study. Additionally, time synchronous averaging (TSA) [43], which is the effective tool for the enhancement of synchronous components and the elimination of asynchronous components by taking the ensemble average of the segments with the specified period, is used as pretreatment method. And the number of periods being averaged is chosen to be 4 for the measurement of periodicity in this paper.
Fig. 5. IAA after TSA in the simulation.
Journal Pre-proof efficiency change of VMD with the increase of the initial balance parameter (from 1000 to 30000 and =500 ) is obtained in Fig. 10. One can see that iteration times is minimum, i.e. 12, when =2000 . Actually, using the proposed selection criterion, the optimal balance parameter is 2000. Therefore, it is reasonable that iteration times of VMD can reflect the efficiency of VMD. Meanwhile, it is easy to find that the curve of the balance parameter and the corresponding decomposition performance is monotonous. So, the optimal candidate can be found regardless the initial balance parameter. Fig. 11 uses three different initial balance parameters,
pro of
Fig. 6. The result of ICFVMD with the initial balance parameter in the simulation.
Fig. 7. The result of ICFVMD with the optimal balance parameter in the simulation. 20 teeth
including 0 =10000 , 0 =5000 and 0 =1000 to
IMF#5
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IMF#6
IMF#3
IMF#7
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IMF#8
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IMF#1
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Fig. 8. The result of SOVMD in the simulation.
test the performance. The X-axis is the update times of balance parameter of SOVMD and Y- axis denotes the CK value of the decomposed mode with the corresponding balance parameter. Note that all the update process is terminated when =2000 with different update times. Therefore, the feasibility of the proposed selection criterion of balance parameter is validated. 0 =1000
0 =10000 0 =5000
Fig. 9. The result of EMD in the simulation. 4.2. Performance analysis
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To further clarify the essence of the proposed method in the analysis of encoder signal, performance analysis about the simulation using SOVMD is provided. As mentioned, the initialization of may affect the efficiency of SOVMD, but the result will be free from it. To verify the conclusion, the efficiency of VMD is tested by changing the initial balance parameter
0 . It has been reported [36] that the iteration times of VMD is found to be a reliable evaluation index for the efficiency of VMD. With the input parameters, that is K=1 and ICF k0 0 , the
Fig. 10. The change of iteration times of VMD with the increase of . = 2000
(a) 0 =10000
(b)
Update Times
= 2000
Update Times
= 2000
0 =5000
(c) 0 =1000
Update Times
Journal Pre-proof Fig. 11. The change of update times of SOVMD with the different initial balance parameters.
crack; (b) surface spalling and (c) half-tooth broken.
5.
5.1. Case 1: root crack Generally, root crack of the gear is the early fault which is rather challenging for the vibration analysis, since the impulses induced by root crack are weak and easily buried by heavy noise. Similarly, the change by root crack in the encoder signal is also mild. To highlight the superiority of SOVMD, the challenging experimental case firstly is applied. Fig. 14 illustrates the measured encoder signal. one cannot find any fault information from the figure. Through the second difference operation and TSA, the result is displayed in Fig. 15. Numerous impulses with small interval caused by the gear meshing can be observed. Yet, the periodic impulses with 31 teeth are hardly visible which can indicate a lack of planet gear fault. ICFVMD with the optimal balance parameter is applied to decompose the signal. Fig. 16 presents the result, from which 4 humps with fix interval are clearly identified. However, it is difficult to determine the position of the fault impulses. By contrast, 4 individual impulses dominate the landscape of the waveform from the result of SOVMD in Fig. 17. It clearly shows the positions of fault impulses and the interval with 31 teeth is easy to be measured. Therefore, it is considerably explicit to give the diagnosis about the fault of planet gear. Fig. 18 displays the result of EMD which does not achieve the expected goal.
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This section uses the experimental data from the planetary gearbox to further verify the advantage of SOVMD. Fig. 12 displays the experimental bench and schematic view of the planetary gearbox. The driving motor provides the torque for the gearbox and the magnetic break is used for loading or braking. The input and output encoders are installed on the shaft of the gearbox to collect the encoder signal. In Fig. 12, the configuration of the gearbox, such as the tooth number of different gears are presented. Three faults, including root crack, surface spalling and half-tooth broken are artificially rooted on the planet gear to model the real gear fault. Through replacing these fault gear as shown in Fig. 13 of the normal planet gear, different encoder signals with planet gear fault are collected. The following subsections will introduce the decomposition performance of SOVMD in processing these signals. In addition, in the following experimental cases, the input parameters of SOVMD are set as K=1;
pro of
Experiment study
0 =5000 ; =500 and ICF k0 0 . Output Planetary Input Encoder Gearbox Encoder
Driving Motor
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Magnetic Break
Planet gear Zp = 31
Sun gear Zs = 20
Ring gear Zr = 82
(a)
(b)
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Fig. 12. Experimental bench:(a) the physical map; (b) schematic view of the planetary gearbox.
(a)
(b)
Fig. 14. Measured encoder signal with root crack of the gear.
(c)
Fig. 13. Physical maps of different faults: (a) root
Journal Pre-proof Fig. 15. IAA after TSA in the case 1.
demonstrated by remarkable visual inspection ability for fault impulses with 31 teeth interval. Yet, EMD still fails to provide any fault related signatures from Fig. 22.
pro of
Fig. 16. The result of ICFVMD with the optimal balance parameter in the case 1. 31 teeth
Fig. 19. IAA after TSA in the case 2.
Fig. 17. The result of SOVMD in the case 1. IMF#5
IMF#6
IMF#3
IMF#7
IMF#4
IMF#8
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IMF#2
Fig. 20. The result of ICFVMD with the optimal balance parameter in the case 2.
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IMF#1
31 teeth
Fig. 21. The result of SOVMD in the case 2.
Fig. 18. The result of EMD in the case 1.
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5.2. Case 2: surface spalling
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In this subsection, the encoder signal with surface spalling of planet gear is used to test the proposed method. Fig. 19 gives the IAA signal after TSA. One can clearly find the periodic impulses caused by the gear meshing from the figure. Meanwhile, they obfuscate the fault information in turn. To extract the impulses generated by planet gear fault, ICFVMD with the optimal balance parameter is applied to decompose the signal and the result is presented in Fig. 20. Similarly, 4 humps appear in the decomposed mode. The oscillation effect from the gear meshing also occupies the waveform. To suppress the oscillation component and enhance the fault information, SOVMD is applied. From the result in Fig. 21, the advantage of SOVMD is clearly
IMF#1
IMF#5
IMF#2
IMF#6
IMF#3
IMF#7
IMF#4
IMF#8
Fig. 22. The result of EMD in the case 2. 5.3. Case 3: half-tooth broken Compared with the previous cases, the half-tooth broken fault is easier to be found. From the IAA signal after TSA in Fig. 23, the periodic impulses caused by gear meshing are rather clear. Furthermore, by virtue of the advantage of encoder signal, that is it is more sensitive to the fault than
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IMF#1
IMF#2
Fig. 24. The result of ICFVMD with the optimal balance parameter in the case 3.
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31 teeth
Fig. 25. The result of SOVMD in the case 3.
IMF#6
IMF#4
IMF#8
pro of
IMF#7
Fig. 26. The result of EMD in the case 3. 6. Conclusion
From the measurement cost and convenience perspective, the research on the encoder signal has more advantages in the fault diagnosis of gearbox. Even though the traditional decomposition methods, such as EMD, have been widely applied to separate the fault information from encoder signal. Yet, diverse noises undermine the usefulness of EMD. To solve the problem, this paper proposed a novel sparsity-oriented VMD for gearbox fault diagnosis based on built-in encoder information. Adequately considering the unique characteristic of encoder signal and the decomposition performance of VMD, the proposed SOVMD shows remarkable performance superiority compared with the traditional decomposition methods. Through the comparison of simulation and experimental cases, including root crack fault, surface spalling fault and half-tooth broken fault on planet gear, the feasibility and effectiveness of SOVMD can be verified. Based upon the analysis, the known contributions of this paper are summarized as follows: (1) This paper originally designs SOVMD for gearbox fault diagnosis based on built-in encoder information. It provides an alternative scheme for the encoder signal analysis. (2) With least input parameter, the proposed
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Fig. 23. IAA after TSA in the case 3.
IMF#5
IMF#3
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vibration signal, the change of planet gear fault in the encoder signal could be observed faintly. The removement of the interference from the gear meshing and other noise is of great significance for the correct diagnosis decision. Therefore, to avoid the wrong conclusion and the omission of diagnosis, more efforts should be devoted to denoising of encoder signal. Fig. 24 displays the result of ICFVMD with the optimal balance parameter in processing of the encoder signal with half-tooth broken fault. The fault impulses, including their positions and interval can be identified. By contrast, the result of SOVMD in Fig. 25 has a higher SNR. The remarkable distinction highlights the importance of the sparsity operation proposed in this paper. The result of EMD is shown in Fig. 26. It is difficult to find the fault information even though two obvious impulses generate in mode 2 with 62 interval which means EMD just extracts partial information.
Journal Pre-proof SOVMD is more suitable for the encoder analysis than the traditional decomposition methods. (3) The sparsity operation is introduced to significantly improve the ability of denoising of VMD, which broadens the application range of the proposed method.
model based gear tooth fault detection technique by the use of minimum entropy deconvolution filter, Mechanical Systems and Signal Processing, 21 (2007) 906-919. [8] G.L. McDonald, Q. Zhao, M.J. Zuo, Maximum correlated Kurtosis deconvolution and application on gear tooth chip fault detection, Mechanical Systems and Signal Processing, 33 (2012) 237-255. [9] T. Barszcz, R.B. Randall, Application of spectral kurtosis
pro of
for detection of a tooth crack in the planetary gear of a wind
Acknowledgments
turbine, Mechanical Systems and Signal Processing, 23 (2009)
This work is supported by the National Natural Science Foundation of China (91860205, 51875434), and the Defense Industrial Technology Development Program (JCKY2018601C013), which are highly appreciated by the authors.
1352-1365.
[10] F. Combet, L. Gelman, Optimal filtering of gear signals for early damage detection based on the spectral kurtosis, Mechanical Systems and Signal Processing, 23 (2009) 652-668.
[11] B. Chen, Z. Zhang, C. Sun, B. Li, Y. Zi, Z. He, Fault feature extraction of gearbox by using overcomplete rational
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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: