Gear fault diagnosis using transmission error and ensemble empirical mode decomposition

Mechanical Systems and Signal Processing 108 (2018) 262–275

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Gear fault diagnosis using transmission error and ensemble empirical mode decomposition Sungho Park a,1, Seokgoo Kim a,1, Joo-Ho Choi b,⇑ a Department of Aerospace & Mechanical Engineering, Korea Aerospace University, 100 Hanggongdae-gil, Hwajeon-dong, Deokyang-gu, Goyang-City, Gyeonggi-do 412-791, Republic of Korea b School of Aerospace & Mechanical Engineering, Korea Aerospace University, 100 Hanggongdae-gil, Hwajeon-dong, Deokyang-gu, Goyang-City, Gyeonggi-do 412-791, Republic of Korea

a r t i c l e

i n f o

Article history: Received 27 April 2017 Received in revised form 21 January 2018 Accepted 14 February 2018

Keywords: Gear fault Transmission error Gear spall Gear crack Diagnostics Ensemble empirical mode decomposition Fault classification

a b s t r a c t Classification of spall and crack faults of gear teeth is studied by applying the ensemble empirical mode decomposition (EEMD) to the transmission error (TE) measured by the encoders of the input and output shafts. Finite element models of the gears with the two faults are built, and TE’s are obtained by simulation of the faulty gears under loaded contact to identify the different characteristics. A simple test bed for a pair of spur gears is prepared to illustrate the approach, in which the TE’s are measured for the gears with seeded spall and crack, respectively. EEMD is applied to extract fault features under the noise from the measured TE. The differences of the spall and crack are clearly identified by the selected features of the intrinsic mode functions based on the class separability criterion. The knearest neighbor method is applied for the classification of the faults and normal gears using the features. The proposed method is advantageous over the existing practices in the sense that the TE signal measures the gear faults more directly with less noise, enabling successful diagnosis. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction For more than decades, great deal of research have been devoted to fault diagnostics of critical components in rotating machineries to achieve increased reliability of the system while reducing the operating cost. Among these, gearbox undergoing extreme loading conditions as used in helicopters and wind turbines has been the primary subject of intensive investigation. Typically, there are two types of gear faults leading to failure: one is the spall that chips off the surface of the teeth and the other is the crack formed at the root of tooth due to the repeated bending stress. Usually, crack is regarded as more critical since it grows suddenly to the tooth breakage, resulting in the whole system loss. This is why not only the faults detection but also their classification is important in the diagnostics. The diagnosis of gear fault has been mainly approached by the vibration sensors because of the ease of installation and cost efficiency. The related techniques are well established over the long periods of development. Numerous reviews are found in the literature with this topic; see, for example [1,2]. The vibration based technique, however, has its limitation that it is weak against the background noise and mechanical resonance, which usually exist in practice. In order to overcome ⇑ Corresponding author. 1

E-mail address: [email protected] (J.-H. Choi). Equally contributed as first authors.

https://doi.org/10.1016/j.ymssp.2018.02.028 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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these, studies using acoustic emission (AE) sensors have grown over the recent years since the AE mainly detects high frequency elastic waves, which is less affected by the noise and resonance (see [3–6]). This method however has drawbacks due to the cost for higher sampling rate and requirement for the sensors to be close to the source to avoid unwanted attenuation. Recently, transmission error (TE) has received another attention as an alternative way for the gear fault diagnosis. TE is defined as the angular difference between the ideal and actual position of the output gear, which results from the finite stiffness of the meshing gears and the manufacturing errors. TE is obtained by optical encoders attached to the input and output shafts that measure the rotational speed to very high accuracy. Considering the measurement ability with less disturbances, the TE based technique may offer more accurate diagnostics than those of the traditional ones. Li et al. [7] estimated crack size of tooth by employing the FEM to compute tooth stiffness, applied lumped parameters model to simulate the gear dynamic behavior, and obtained the TE over a cycle. Feki et al. [8] did the similar study for the TE’s with different loading levels by the simulation of dynamic model. While they have experimented to measure the TE, it was only to compare the raw signal with the simulation in an elementary manner, which is hard to speak of the diagnosis. Endo et al. [9,10] have made FEA simulations of the spall and cracked tooth to study the TE behavior and found that the effect of tooth cracks is load dependent, while the effect of spalls is geometry dependent. After examining the shapes of residual TEs and their derivatives by the FEA, however, they have used vibration signals in practice to classify the spall and crack, which is not the TE based technique. Park et al. [11] have proposed to use the peak to peak (P2P) TE as the feature based on the observations in the experiments that the P2P TE becomes pronounced under the fault conditions. As already mentioned, all of these studies on the TE have lacked the steps for the rigorous signal processing and feature extraction that is essential for the practical diagnosis. More recently, Fedala et al. [12,13] have carried out fault classification of gears using the classical time and frequency domain features obtained from the TE and acceleration signals. They have concluded that the TE shows superior performance over the acceleration signals. In this study, ensemble empirical mode decomposition (EEMD) technique [14] is proposed to extract features useful for the detection and classification of the gear faults: spall and crack. The EMD has recently emerged as a new time-frequency analysis technique suitable for nonlinear and non-stationary signals occurring in the faulted rotating machineries. Many publications on the use of EMD are available for gears (see [15]). Followed by the early applications of original EMD such as the paper by Loutridis [16], several papers have followed to improve the EMD. More lately, ensemble EMD, which is the most representative version of improved EMD was developed by Wu and Huang [14] by adding noise to the investigated signal. Since Ai and Li [17] has applied the EEMD to the diagnosis of gear crack and demonstrated the effectiveness by experiments, several studies have followed for more enhancement or application to the more complex cases [18–20]. However, all of the EMD studies have been based on the vibration signals, and none of the prior work has studied the TE signal as applied to the EEMD in the gear fault diagnosis. In this study, the EEMD, which was found feasible in the vibration based diagnosis, is applied to the TE signal for a single stage spur gears in motion. The objective is to develop an effective method that distinguishes the two faults: spall and crack from the normal ones in a systematic way. To achieve this goal, FEAs are first conducted to figure out the different behaviors of tooth deformation for normal and faulty gears in meshing. Virtual TE signals are then generated to simulate the gears in rotary motion by incorporating the low frequency waves. EEMD is applied to the residual TE (RTE) which is the difference between the faulty and normal signal, to obtain a set of intrinsic mode functions (IMF). The distinct behaviors of the spall and crack are easily identified from the resulting IMF simulations. A simple test bed is installed to illustrate the approach for a single stage gear system. TE’s are measured from the three states: normal, spalled and cracked gears, respectively. The profiles of residual signals of measured and simulated TE’s are in good agreement, thus suggesting a useful role of FEA model in the diagnostic process. This may be due to the use of encoders for the measurement, which is less affected by the intervening disturbance or noise. EEMD is applied to the real RTE to obtain the IMF’s, from which the features are selected and classification is performed using the k-nearest neighbor technique. The advantage of this approach is that the FEA enables the identification of the types of the fault in the EEMD process of the TE signal, which is not tractable by the other methods such as vibration or acoustic emission signals. This was already noticed by Endo et al. [9,10], although they did not continue to the diagnosis study with TE signals.

2. Overall procedure of the proposed method The overall procedure of the proposed method is displayed in Fig. 1, which consists of the steps: (1) TE data are acquired from the FEA as a virtual signal or from the test bed as a real signal; (2) the signals are preprocessed to obtain residual TE by removing unnecessary noise and frequency components; (3) EEMD is applied to decompose the signal into a set of IMFs which allows more chance to find fault information; (4) time based features are extracted from each IMF and best two features are selected that shows largest separability performance; (5) faults are classified using the chosen features data based on the classification algorithm. In Section 3, the procedure using simulated signal is explained, which explores the feasibility of TE signal for the fault diagnosis. In Section 4, the procedure is implemented using the real data from the test bed to obtain the classifier that can diagnose the faults. Conclusions are outlined in Section 4.

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Fig. 1. Overall procedure of the proposed method.

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3. Simulation study In this section, simulation procedure is presented for spur gears in motion that can extract features from the measured TE. Finite element model of the gears in meshing is constructed and TE profiles over a single tooth rotation are obtained for the three cases: normal, spalled and cracked tooth respectively. Virtual TE signal is then generated over the entire rotation by adding the artificial noise and manufacturing error to make it closer to the real signal. Then, signal processing is applied to obtain the RTE that removes the gear mesh and low frequency components. EEMD is applied to the RTE to obtain a set of IMF’s. 3.1. Finite element analysis Gear is a mechanical element that transmits rotational force via the meshing motion. Since the gears are not rigid, the gears undergo deformation, which results in the transmission error (TE), which is the angular difference between the ideal and actual position of the output gear:

TE ¼ hideal  h2 ¼  2

R1 h1  h2 R2

ð1Þ

where h1 and h2 are the angular displacements and R1 and R2 are the radii of input and output gears, respectively. It can be transformed to the following equation

TE ¼

T out

ð2Þ

K g R22

where T out is the torque applied at the output shaft and K g is the linear gear meshing stiffness (GMS) representing the rigidity of the gears in contact. The faulty gear may undergo more deformation, which leads to the higher TE, and this is the basic idea that constitutes the signal for the diagnosis. Finite element model is created by FEA software ANSYS as shown in Fig. 2 for the gears in contact, of which the material is stainless steel with elastic modulus 204 GPa and Poisson ratio 0.3, module is 4 mm, and the input and output pitch diameters are 140 and 280 mm respectively. Five teeth are generated for the both gears as shown in the figure, which are meshed with finer element size of 0.25 mm. Torque is applied to the input gear while the hub of the output gear is constrained. FEA is carried out over the range of five teeth distance to obtain the TE profile. Spall with the length and depth being 1.3 mm and 0.5 mm and the crack with the length 4 mm, as shown in Fig. 3, are applied to the tooth with the bottom width being 8 mm, and the analyses are performed as shown in Fig. 3(a) and (b). The resulting TE’s are given in Fig. 4(a) and (b), where the TE of normal gear is given by dotted line. In the figure, the profiles of the TE represents unique oscillating behavior as the gears alternate between single and double contact during a single pitch movement. In Fig. 4(a), the TE increases at the second half of the profile due to the contact with the spall, and the phase of the peak shifts a little to the right. In Fig. 4(b), the TE increases over the range of two profiles centered around the cracked tooth. The residual TE (RTE), which denotes the difference between faulty and normal tooth, are also given in Fig. 4(c) and (d). The difference is more evident in this figure, where the shape of spall is narrow and acute covering only a single angular pitch of 10.3° whereas that of crack is wider by more

pinion torque applied to the hub

Output gear restrained at the hub

Fig. 2. FEA model of gears in contact.

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Fig. 3. FEA results of faulty gears: (a) spall and (b) crack.

Fig. 4. Transmission error ((a) and (b)) and residual transmission error ((c) and (d)) by FEA for the spall and crack respectively.

than twice the pitch but shallow and exhibits a cusp at the right end. These two different characteristics are used to classify the faults from the measured signals in the diagnosis. The FEA is further carried out by varying the torque magnitude. Fig. 5(a) and (b) show that both RTEs for the spall and crack increase with rise of the torque from 50 to 450 N m. Comparing the two, the RTE of spall increases only by 1.4 times from 3.57  104 to 4.98  104 whereas that of crack is 6 times from 0.37  104 to 2.21  104 which is much greater under the same torque increase. Although not exactly the same, this is already observed by Endo et al. [9,10] in which the RTE of crack is affected by the torque at much greater degree than that of spall.

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Fig. 5. RTEs for the (a) spall and (b) crack for various torque conditions.

3.2. Signal processing of TE Theoretically, the TE profile of normal gear should repeat constantly during the meshing under rotation. In practice, however, it includes low frequency wave due to the tooth machining and assembly errors. Then the real TE signal consists of the normal TE added by the localized fault at a tooth given by Fig. 4(a) or (b) and the low frequency wave. This can be artificially made by simulation, and the results are given in Fig. 6(a) and (d) for the spall and crack, respectively, with the speed of the output gear at 60 rpm. The results show that the TE oscillates at the gear mesh frequency (GMF), and fluctuates with low frequency. The circles shown in the figures represent the location of the faulty tooth. Using these as the raw signal, the RTE can be obtained by subtracting the faulty from the normal signal according to its definition, to leave the faulty signal only. In practice, however, this is not feasible in the operation. Instead, signal processing is applied to obtain the RTE, which consists of notch filtering and high pass filtering to remove the GMF and its harmonics up to the 20th order and the low frequency component respectively. The order and cutoff frequency of each filter are 3 and 20 Hz and 2 and GMF ± 0.5 Hz, respectively. As a result, Fig. 6(c) and (f) are obtained for the RTE of the spall and crack, respectively. Comparing the close up views Fig. 6(c0 ) and (f0 ) with their counterparts Fig. 4(c) and (d), which are the original residual TE’s by the FEA, they look somewhat different each other. The reason can be attributed to the signal distortion after the high pass filtering because the profiles Fig. 6(b0 ) and (e0 ) after the notch filtering remain similar to the original shapes as those in Fig. 4(c) and (d). The final signals, however, still keep the unique behavior of spall and crack, which are characterized by the acute narrow shape with larger peak and wider plateau shape with smaller peak with a cusp at the right end. 3.3. Ensemble empirical mode decomposition (EEMD) Although the fault characteristics are easily identified by visual observation from the processed TE signals, it is necessary to extract proper feature that can represent the difference in quantitative way. To this end, empirical mode decomposition (EMD) is introduced in this study, which is to decompose a signal into a set of intrinsic mode functions (IMF) and one residue. While the EMD for the vibration signals has succeeded in various applications, particularly in the non-linear and nonstationary process [15], it was not applied to the TE signals in the gear fault diagnosis. The advantage of the EMD is that the original signal is decomposed into multiple IMF signals, out of which many more features can be extracted that allows more chances to find fault information. In the EMD, the original signals are decomposed as:

xðtÞ ¼

n X C i ðtÞ þ r n ðtÞ

ð3Þ

i¼1

where xðtÞ is the original signals, and C 1 ðtÞ, C 2 ðtÞ, . . ., C n ðtÞ are the 1st–nth IMFs respectively, while rn ðtÞ is the residue. One of the major drawbacks of the EMD, however, is the mode mixing that occurs when the signal includes intermittency such as the gear faults, and causes the decomposition of the IMF difficult. To overcome this, the ensemble EMD (EEMD) is proposed by Wu and Huang [14], which is to repeat the EMD several times with added white noise that populates the whole time-frequency space uniformly. Then the ensemble mean of the IMF’s becomes the true IMF that successfully extracts fault features. The process of the EEMD is given in Fig. 7. The EEMD is applied to the simulated RTE of spall and crack, and ten IMFs denoted as C 1 –C 10 are obtained as shown in Fig. 8(a) and (b), respectively. As was found in the original signals, the difference is also identified in the two sets of IMFs, in which the first four IMFs C 1 –C 4 tend to show more distinct characteristics at the fault location that differentiates the spall and crack. Comparing the two side by side, greater signals are observed at the IMFs

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deg

268

0.2 0.1

deg

0

0.2 0.1 0

deg

0.05

0

-0.05

deg

sec

0.2 0.1

deg

0

0.2 0.1 0

deg

0.05

0

-0.05

sec Fig. 6. Simulated TE of spalled gear (a–c) and cracked gear (d–f): original signal (a, d), after notch filtering (b, e), RTE after high-pass filtering (c, f) and their close up views in each circle (a0 –f0 ).

C 2 –C 4 for the spall due to the larger fault size. The IMF C 1 however has similar magnitude for the spall and crack. From the observations, it may be noted that the IMF’s can be used to distinguish the behaviors of the spalled and cracked RTE. 4. Experimental study 4.1. Test rig and experimental setup Test rig is prepared to measure the TE for the fault diagnosis as shown in Fig. 9. The system consists of a 2.9 kW motor to drive the gear, a pair of SCM440 steel spur gears, and powder brake to apply counter torque. Rotary encoders with resolution 8192 ppr are attached to the input and output shafts to measure the angular displacement. The gears are made of module 4, pitch diameters 280 and 140 mm and number of teeth 70 and 35, in which the larger one is the input gear. The TE are mea-

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Fig. 7. Algorithm of ensemble empirical mode decomposition (EEMD) [15].

sured with the input speed of 30 rpm and the output torque of 450 Nm. Two gears denoted as A and B are prepared for the output gear, in which the spall of width  depth being 1.3 mm  0.5 mm and the crack of length 4 mm are seeded using the wire cut electric discharge machining as shown in Fig. 10(a) and (b). The TE is measured over 40 rotary cycles in one operation. The measurement is made 10 times by repeating mounting and dismounting the gears to account for the associated uncertainty. Once the data are acquired, the fault is seeded, and measurements are repeated in the same manner with the faulty gears A and B respectively. Therefore, the number of data sets are 20 for the normal and 10 for each of the spalled and cracked conditions, respectively. 4.2. Signal processing and EEMD application Before applying the EEMD to the measured TE, time synchronous averaging (TSA) is implemented to the 40 sets of data to average out the unwanted noise. In practice, the TSA is carried out by averaging together a series of signal segments each corresponding to one period of a synchronizing signal [21]. Then the signal processing of notch and high pass filtering as stated in Section 3.2 is applied to obtain the RTE as shown in Fig. 11(c) and (f). Same characteristics are found, although the features are not as strong as those in the simulation given in Fig. 6(c0 ) and (f0 ): acute shape with larger peak and plateau shape with smaller peak and cusp at the right end. EEMD is applied to these signals to obtain the IMF’s C 1 –C 9 as did before, and they are in Fig. 12. The distinctive characteristics of the normal, spall and crack are found from the set of the IMF’s, which can be quantified by extracting suitable features for each of the IMF’s. 4.3. Fault classification In order to perform classification, one needs to employ a statistical measure that can represent the different characteristics of each IMF’s. For this purpose, five candidate features are examined out of dozens of popular features in the literature: they are the kurtosis, crest factor, shape factor, FM4, and energy operator (EOP) [1,22,23]. Other features like FM0, NA4 and NB4 are not applicable in our case since the TSA and notch and high pass filtering are already implemented prior to the feature extraction. The features are calculated from each of the 10 IMFs, which provides 50 features in total for the three classes: normal, spalled and cracked gears. The so-called J 3 criterion, which is a generalization of the Fisher’s discriminant ratio (FDR) criterion in the multiclass case [24], is introduced as follows to find the two best features that exhibit the largest separability of the three classes.

J 3 ¼ tracefS1 w Sb g

ð4Þ

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Fig. 8. IMFs of simulated TE: (a) spall and (b) crack.

where

Sx ¼

M X P i Si ; i¼1

Si ¼ E½ðx  li Þðx  li ÞT ;

and Sb ¼

M X i¼1

Pi ðli  l0 Þðli  l0 ÞT

ð5Þ

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Fig. 9. Test rig of single stage spur gears (Input gear with 70 teeth; output gear with 35 teeth).

Fig. 10. Gears with seeded faults: (a) gear A with spall of W  D 1.3 mm  0.5 mm and (b) gear B with crack of length 4 mm.

where the number of classes M is 3, and x and l are the feature vectors and their means consisting of two components, respectively. Exhaustive search is performed to find the vector out of all possible combinations of 2 features from the original 50. All the process is automated to obtain the optimal feature vector selection. The result is the crest factor of the IMF C 2 and shape factor of C 3 , which gives the maximum value of 80.63 for J 3 . The shape and crest factor are defined as follows [22,23]:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hP iffi N 2 1 ðx Þ i¼1 i N SF ¼ ; PN 1 i¼1 jxi j N

C¼

jxjpeak xrms

ð6Þ

Using the data of the two features, classification is carried out by applying the k-nearest neighbor (kNN), which is one of the simplest pattern recognition methods. The flow of the kNN algorithm is as follows: (1) set the k value, (2) calculate the distance between the test data point and the training data, (3) unlabeled test data is classified to the label which is most frequent among k training data point nearest to that test data point [24]. In our study, k is set at 3. The constructed decision hyperplane is visualized in Fig. 13(a). Note that the crest factor contributes to the distinction of normal and faulty conditions, whereas the shape factor to the crack and spall. Therefore, instead of classifying the three conditions at once, it may be better to employ two step procedures, one for the fault detection, and the other for the identification. The performance of the EEMD is compared with the result without EEMD, in which the 5 features are extracted from the original filtered signal directly, and 2 best features are chosen by J 3 criterion for the classification. The result is given in Fig. 13(b), in which the J 3 value is 29.13 and the two features are crest factor and EOP. Even without EEMD, the performance of fault detection is still favorable but

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Fig. 11. Measured TE of spalled gear (a–c) and cracked gear (d–f): original signal (a, d), after notch filtering (b, e), RTE after high-pass filtering (c, f) and their close up views in each circle (a0 –f0 ).

that of the fault identification is marginal since some points overlap the distinction boundary. The performance is also compared with those obtained by the same procedure for the vibration signal measured by the accelerometer as given in Fig. 9. The result is given in Fig. 13(c) where the J 3 value is 5.68 and the best two features are crest factor of C 2 and shape factor of C 1 . As shown in the figure, the method fails to classify the fault. The reason may be due to the remote location of the sensor from the faulty gear, which may hinder the accurate detection of the signal. 5. Concluding remark In this study, a new method for gear fault diagnosis is presented, which employs the transmission error as the sensor signal, ensemble EMD as the method for feature extraction, and kNN as the classification tool. The method has two important advantages over the existing practices: First is the use of the encoder sensor that measures the faulty signal of the gear in

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Fig. 12. IMFs of measured TE: (a) normal, (b) spall and (c) crack.

more direct manner, which is less disturbed by other intervening components, hence, leads to the higher SNR. Second is the use of FEA to identify the shape and size of the fault by evaluating the deformation of gears in mesh. The unique characteristics of each fault are well identified by the FEA and validated by the real signal, which may not be easy in the other vibration or acoustic emission based methods. The results of the study suggest the potential for the method to be more useful and effective in the gear fault diagnosis since the knowledge obtained from deformation physics of gears in mesh can be exploited not only for the classification but also for the severity estimation of the gear faults. This is contrasted with the purely data-driven methods, that requires many test data with seeded faults with different level of severity for the training and validation. The advantage however can be compromised by the cost for the two encoders necessary to obtain the TE. The Instantaneous Angular Speed (IAS) that can be obtained by a single encoder may be another option to overcome this [12]. The future study can be expanded to the identification and classification of more versatile types of spall and crack with different sizes, including the multiple faults occurrence. By taking advantage of the FEA to examine the TE behavior for these cases, the cost for the experiments will be greatly reduced because least necessary tests are implemented for validation. The approach can further be applied to the different scale in size, i.e., once the method is developed for the smaller scale system, it can be applied to the larger one.

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Fig. 13. Classification of normal, spall and crack conditions using kNN algorithm with the two best features: (a) result after applying EEMD for TE (b) result without EEMD for TE (c) result after applying EEMD for vibration signal: due to the failure to classify, only the data are plotted in this case.

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