Monitoring method of gear teeth failure of hydraulic gear pump based on improved VMD and DBN-DNN of electrical signal

YANG Sha,GU Lichen,SHI Yuan,GENG Baolong,LIU Jiamin,ZHAO Baojian,WU Haoyu

(School of Mechatronic Engineering,Xi’an University of Architecture and Technology,Xi’an 710055,China)

Abstract：Abundant system operation state information is included in the electrical signal of the hydraulic system motor.How to accurately extract and classify the operation information of electrical signal is the key to realize the condition monitoring of hydraulic system.The early fault characteristics of hydraulic gear pump hidden in the motor current signal are weak and difficult to extract by traditional time-frequency analysis.Based on the correlation coefficient and artificial bee colony algorithm (ABC),the parameter optimization of variational mode decomposition (VMD)is realized in this paper.At the same time,the principle of maximum signal correlation coefficient and kurtosis value is adopted to determine the effective intrinsic mode function (IMF).Moreover,the permutation entropy(PE)and root mean square(RMS)of the effective IMF components are input into the deep belief network (DBN-DNN)as high-dimensional feature vectors.The operation state of gear pump is monitored.The results show that the weak characteristics of current signal of gear pump fault are accurately and stably extracted by this method.The running state of gear pump is monitored and the accuracy of gear fault diagnosis is improved.

Key words：gear teeth fault；status monitoring；artificial bee colony algorithm (ABC)；variational mode decomposition (VMD)；deep belief network (DBN-DNN)

0 Introduction

In hydraulic system,gear pumps often work in overload,periodic fatigue stress and other bad working conditions.The gear is prone to root cracks,broken teeth and other faults[1].The essential monitoring and fault diagnosis of gear pump can realize the fundamental change from after-sale maintenance to early maintenance.It is of great significance to reduce the casualties and economic losses caused by industrial accidents[2].

In vibration diagnosis,vibration signals are collected by vibration sensors on the gear pump.It is one of the main methods of gear fault diagnosis.The quality of the vibration signal is very sensitive to the installation position and it is easily contaminated by industrial environmental vibration such as mechanical resonance of other equipment transmitted by the base.In order to overcome the disadvantages of vibration detection,foreign scholars have proposed motor current signature analysis (MCSA)to monitor the running state and reliability of gears[3].In addition to the normal operation information of the system,the health information of the system is contained in the monitored current signal.The fault of the motor itself can be monitored by the electrical signal of the motor.In 2007,Nagi[4]conducted signal decomposition and feature extraction of motor current signal through bispectral analysis.It is proved to be effective to use the current signal as the detection signal of the gear failure of the multi-stage transmission.The application range of motor current signal is extended.

In the sensorless diagnosis,most of the current scientific researchers adopt empirical mode decomposition (EMD)as diagnostic method.The gear fault information is included in the current signal.The information feature is weak and difficult to be decomposed and extracted by traditional empirical mode.Variational mode decomposition (VMD)is a new time-frequency analysis method proposed by Dragomiretskiy et al.in 2014[5],which is adaptive to signal processing and has fast computation,high SNR and effective suppression of modal aliasing.However,most authors used VMD to process some vibration signals.For example,Wu et al.[6-9]used VMD for gearbox fault diagnosis.There is little literature for processing other signals.In 2019,Men[10]studied the fault diagnosis method of planetary gearbox through VMD analysis of motor current.However,the decomposition layerKand penalty factorαwere not effectively screened.The IMFs are influenced by two parameters.If the choice is not good,the decomposition effect will be greater.The current signal after VMD decomposition is affected by the current frequency and its frequency doubling,and the signal components are still very complex and difficult to identify.Selecting appropriate pattern recognition algorithm can effectively play the advantages of VMD and complete the fault warning.Deep belief network (DBN-DNN)is a deep learning network proposed by Hinton et al[11].The advantages of supervised learning and unsupervised learning are combined by DBN-DNN.Li et al.[12]directly applied DBN to the bearing signal processing,the classification and identification of bearing faults is realized.Zheng et al.[13]proposed the parameter optimization of VMD by particle swarm optimization algorithm,and inputted each modal component into DBN for fault diagnosis of fan.Particle swarm optimization (PSO)converges quickly.However,due to poor diversity,it is easy to fall into local optimum.And IMFs do not contain all active components.

Therefore,artificial bee colony (ABC)is proposed to optimize the parameters of VMD.Firstly,the original electrical signal is processed by the improved VMD method to obtain some eigenmode functions.The effective IMFs are selected by calculating the correlation coefficient and kurtosis value of each component.Moreover,feature extractions of effective IMFs are carried out to form DBN-DNN high-dimensional input vectors.Finally,the gear state is recognized by DBN-DNN classifier.The method is verified by experiments,and can effectively identify the gear pump gear fault.

1 MCSA method

1.1 Electromechanical coupling mechanism between motor and gear pump

In theABCstator coordinate system,abcrotor coordinate system anddq0 coordinate system,the current and voltage signals of asynchronous motors are shown in Fig.1.The angular velocity ofdq0 coordinate system isω,which rotates synchronously with the air-gap field.The motor rotor rotates at an angular speed ofωr.The angle of thedaxis and theAaxis isθc.The angle of thedaxis andaaxis isβ.

Fig.1 ABC stator coordinate system,abc rotor coordinate system and dq0 coordinate system

The speed of an induction motor changes during dynamic operation.The motion equation of the motor is described by a nonlinear equation.For the convenience of using standard solution and computer implementation,motor motion equation is expressed in the form of state equation as

(1)

where

X=[idsiqsψdrψqr]T,

whereidsandiqsare the stator currents;ψdrandψqrare rotor fluxes;udsanduqsare the stator voltages;Rsis the single-phase stator resistance;Rris the single-phase rotor resistance;Lsis the single-phase stator inductance;Lris single-phase rotor inductance;Lmis the synchronous excitation inductance;ωsis the synchronous speed;ωmis the electric angular velocity of the rotor,ωs1=ωs-ωm.

In order to obtain a unique differential system composed of state variables of electromechanical signals,the Newmark time integration algorithm is used.

The coupling system is generated in the form of first-order differential system,which can be expressed as

(2)

Eq.(2)is transformed into a matrix form as

It can be seen from the above equations that the gear teeth failure of the gear pump will occur in the matrixB.The failure is an additional pulse forcing term.All components ofZare interfered byA.Mechanical,electrical and magnetic variables are connected byA.It can be inferred that the change caused by the defect depends on the strength of the coupling term in the state matrix and the amplitude of the related forcing term.

When the asynchronous motor drives the gear pump to run,the periodic mechanical vibration of the system is caused by the periodic meshing of the faulty gear.Periodic torque vibration is transmitted to the motor by the coupling.Motor air gap and abnormal torque fluctuations are generated.Finally,the vibration of the gear pump is converted into status information of the stator current signal.Therefore,the stator current contains abundant state information,and the analysis of stator current can realize the identification of gear fault frequency.

Blodt et al.[14]also deduced that the phase modulation of the stator current is caused by the periodic torque variation.An additional failure frequency is generated in the stator current.

2 VMD and its improvement

2.1 Introduction of VMD

VMD method is a new signal processing method with variable scale.This method can decompose multi-component amplitude modulation (AM)and frequency modulation (FM)signals into multiple single-component AM and FM signals at one time.The complex signalx(t)is decomposed by VMD into a series of basic mode functions,and each essential mode function fluctuates around its central frequency.The core of VMD is:assuming that most of the modes are located near a central frequency,the problem of solving the modal bandwidth is transformed into a constrained optimization problem,and finally the modes are obtained.By non-recursive decomposition,VMD can effectively avoid many problems such as boundary effect.

Assuming that the signalx(t)is composed of several components with different central frequencies and limited bandwidths,its decomposition problem can be transformed into a variational model for decomposition.Under the condition that the sum of each component is equal to the signalx(t),the sum of the bandwidth of each essential mode function is minimized.The constrained variational problem can be expressed as

(3)

whereukis the set of each modal function,uk={u1,u2,…,uk};ωkis the frequency set of each center,ωk={ω1,ω2,…,ωk};∂tis the partial derivative of time with respect to a function;δ(t)is the unit impulse function;j is an imaginary unit;* is convolution operator.

To solve the above equation,a quadratic penalty function term and a Lagrange multiplier term are introduced as VMD.

ζ({uk},{ωk},λ)=

(4)

whereαis the second penalty factor and can guarantee the reconstruction accuracy of the signal;λis the Lagrange multiplier and can guarantee the strictness of the constraints.

The multiplication operator is used to further solve Eq.(4)in alternating directions.The eigenmode function and its corresponding central frequency can be obtained by

(5)

(6)

Therefore,the VMD algorithm is described as

2.2 Parameter optimization of VMD

2.2.1 Determination of decomposition numberKbased on correlation coefficient

WhenKvalue is large,frequency aliasing will occur.WhenKvalue is small,signal information will be missing.Therefore,a method of selectingKvalue based on correlation coefficient is proposed.On the premise that IMFs have the maximum central frequency,the correlation coefficients between modal components are calculated.Frequency aliasing between modal components is judged by correlation coefficient.And thenKis determined.

(7)

The optimization steps ofKbased on correlation coefficient are shown in Fig.2.

Fig.2 Optimization flow chart of K value

2.2.2 Determination ofαbased on ABC algorithm

In VMD algorithm,the decomposition result will also be greatly affected byα.When the value ofαis smaller,IMF bandwidth is larger;conversely,whenαis larger,the bandwidth is smaller[15].

ABC[16]is a new global optimization algorithm for swarm intelligence.In the ABC algorithm,the foraging behavior of bees is simulated,and good global probabilistic searching ability is obtained.The penalty factor of VMD is optimized by ABC and the influence of human factors is avoided.

Suppose the number of bees isN,and each solution vector hasDdimensions.The whole process can be regarded as optimization inD-dimensional searching space.Nis the number of initial solutions andDis the number of optimizations.After the initialization,the following bees chose to follow the picking bees corresponding to theith nectar source,and it searches the neighborhood of the initial solution to find new food source.

xid=xmin,d+rand(0,1)(xmax,d-xmin,d).

(8)

With the decrease of the difference betweenxmax,dandxmin,d,the interference of nectar source will disappear,and the searching range of bees will gradually decrease and converge to the optimal solution.

ABC is adopted to optimize theαin the VMD algorithm.The specific steps are as:

① The initialization parameters of the ABC algorithm are set as shown in Table 1.

Table 1 Parameters initialization of ABC algorithm

② The fitness function is set during the optimization process.

Let [S1,S2,…,SV] be the information entropy ofKmodal components.The information difference coefficient between each modal component is defined asH.

S=(S1-S)2+(S2-S)+…+(SV-S)2,

(9)

(10)

The error coefficientebetween the initial signal and the reconstructed signal is defined as

e=|f-x1-x2-…-xV|.

(11)

The fitness function is

(12)

The difference in the modal components of the decomposition is reflected byH.The similarity between the reconstructed signal and the original signal is reflected bye.When the fitness valueρis larger,the VMD treatment results are better.When an equilibrium point is found betweenHande,the maximum value ofρis taken as the optimization target.

③ The fitness value of each solution is calculated,and the initial solution is updated by using Eq.(7).

④The maximum fitness valueρmaxand its solution are obtained by the number of cycles from iteration to termination

2.2.3 Simulated analysis

In order to test the effectiveness of the improved VMD algorithm,the method is compared with EMD method.Gear simulation signals are constructed for verification.There are

(13)

wherex1(t)andx2(t)are the harmonic signals;x3(t)is periodic attenuation signal;n(t)is Gaussian white noise signal.

The frequency of the pulse signal is 20 Hz.This simulation experiment is mainly to extract the pulse signal.The harmonic signal is suppressed,Gaussian white noise is filtered,and the amplitude of main frequency is increased.Fig.3 shows the time domain of the simulated signal.

Fig.3 Time domain diagram of simulated signal

Before the VMD decomposes the signal,the parameters of VMD are optimized,and the optimization results are shown in Table 2.

Table 2 Parameter initialization of improved VMD

The decomposition results of VMD are shown in Fig.4.

Fig.4 Decomposition of VMD

The decomposition results of EMD are shown in Fig.5.

As shown in Fig.6,in the envelope spectrum after EMD processing,the shock frequencies of 20 Hz and 40 Hz (2 times frequency)are extracted,but a large amount of white noise appear at the same time.

Fig.5 Decomposition of EMD

Fig.6 Envelope diagram after EMD

As shown in Fig.7,after the improved VMD method and envelope spectrum analysis of the simulation signal,the pulse frequencies of 20 Hz,40 Hz (2 times frequency)and 60Hz(3 times frequency)can be extracted.

Fig.7 Envelope diagram after improved VMD

Based on the above analysis,the extraction effect of improved VMD is better.The harmonic signal and noise are effectively suppressed and aliasing is not produced.

2.2.3 Principle of feature extraction

After VMD method decomposition,there are invalid components in IMFs.If all components are characterized,more work will be done.Therefore,the concepts of cross-correlation and kurtosis are introduced.Effective IMFs are selected according to correlation coefficient and kurtosis value.The efficiency of fault diagnosis has been improved[5].

According to Eq.(7),the mutual correlation between each IMF and the original signal is calculated.As the correlation coefficient increases,the modal components become closer to the original signal.More failure information is included by IMFs to determine valid components.

Kurtosis is a dimensionless parameter.It is independent of the speed,type and load of the tested gear,and only affected by the impact signal.The kurtosis value is very sensitive to pulse impact and suitable for local damage failure diagnose of gear.

(14)

The amplitude distribution of the current signal is close to the standard normal distribution.When there is gear fault,impulse impact will be produced in the current signal,and the current signal amplitude will deviate from the normal distribution.The deviation of the normal curve is generated.The kurtosis value also increases.Therefore,a high kurtosis value means more fault feature information.

From the above,valid IMFs in various states can be selected.However,different types of signals have different responses to signal changes of current and time complexity.Extracting a single feature will reduce the recognition accuracy.It can be seen that the amplitude change of the current signal in the time domain can be reflected by the root mean square (RMS).By comparing the data of adjacent time periods,the permutation entropy can be used to measure the complexity of one-dimensional time series.Therefore,RMS value and permutation entropy(PE)of the effective IMF are selected to construct the feature vector of the sample.The fault state of current signal can be effectively represented.

RMS is widely used as an indicator of signal fluctuation,and it is calculated by

(15)

PE has the advantages of simple calculation,strong anti-noise ability and stable calculated value[17].Its algorithm principle is described below.

The phase space reconstruction of the one-dimensional sequencexiis carried out.A two-dimensional matrix is obtained as

(16)

wheremis embedded dimension;τis delay time;Gis the number of reconstructed vectors.

Each row in the matrix is a reconstructed vector.Each reconstructed vector is rearranged in ascending order of magnitude.The sequence of element index values will be obtained.The number of occurrences of each permutation is counted.So the probability of each of these permutations will be given to form a sequence as [P1,P2,…,Pl,…,Pm!],wherePlis the probability of thelth permutation.The expression of PE is

(17)

The permutation entropy is normalized as

HPE=HPE(m)/ln(m!).

(18)

3 Fault identification based on improved VMD and DBN-DNN

DBN-DNN is composed of RBM network and BP neural network,as shown in Fig.8.RBN is a multi-layer unsupervised network,BP is a supervised network.Training algorithm of DBN-DNN is divided into two steps,pre-training and fine-tuning.This training mode enables DBN-DNN to avoid falling into a local minimum,and the training efficiency is greatly improved.

Fig.8 Structure of DBN-DNN

3.1 Pre-training stage

The energy function form of RBN is

(19)

whereθ={wij,ai,bj} is the model parameter to be determined for RBM；wijis the connection weight ofith neuron in the visible layer andjth neuron in the hidden layer;aiis the bias value ofith node in the visible layer;bjis the bias value ofjth node in the hidden layer.

The joint probability distribution of (v,h)can be obtained by

(20)

The input training data is fitted to trainθ.

3.2 Fine-tuning stage

The training of BP network includes forward propagation and backward propagation.In forward propagation,the input vectors are passed to the output layer by layer,and the output value is compared with the expected value.The difference between the actual output value and the expected output value is calculated.In backward propagation,the difference is propagated from the output to the input.The DBN-DNN parameters have been fine-tuned.The training error is further reduced.

Therefore,a fault signal identification method based on improved VMD and DBN-DNN is proposed.First,the original signal is processed with improved VMD and IMFs are obtained.Then,the PE and RMS of the effective IMFs are extracted.The PE and the RMS value are constituted as a feature vector.The feature vector is input into DBN-DNN classifier for state recognition.The flow chart is shown in Fig.9.

Fig.9 Identification flow chart of gear state

4 Experimental analysis

4.1 Introduction of experimental workbench and collection of current signals

In order to verify the validity of the method in this paper,a hydraulic testbench is used to verify the method.

The model and parameters of the asynchronous motor are shown in Table 3.

Table 3 Motor parameters

The three-phase asynchronous motor is connected byY-type and the motor is customized.The model of gear pump is Sichuan Changjiang CBK1004-AIFL.The rated working pressure of gear pump is 25 MPa;rated speed is 3 500 r/min;the theoretical displacement is 4.25×10-6m3.For three gear pumps of the same model,the gear teeth of the two gear pumps are specially treated.Thus the normal gear pump,the broken gear pump and the gear pump with a crack are obtained.The main structure of the test bed is shown in Fig.10.

1 Radiator;2 Electromagnetic relief valve;3 Proportional relief valve;4 Magnetic powder brake;5 Gearbox;6 Current transformer;7 Speed measuring sprocket;8 Magnetoelectric speed sensor;9 Hydraulic motor;10 Magnetic Exchange Valve;11 Combined pressure,flow and temperature sensor;12 Solenoid relief valve;13 Proportional relief valve controller;14 Filter;15 Asynchronous motor;16 Gear pumps;17 Servo controller/General frequency converter;18-1,18-2 Mechanical pressure gauge;18-3 Digital pressure transmitter;19 Check valve;20 Plunger pump;21 Permanent magnet motor;22 Hall voltage and current sensor;23 Servo controller;24 Temperature Sensor;25 D/A converter;26 Control computer;27 A/D converter;28 Gear motor

Fig.11 Gear failure test platform

The physical photo of test platfrom is shown in Fig.11.During the test,the load is simulated by the no-load proportional overflow valve.The voltage and current signals are collected synchronously by the signal collecting device.Through card A/D converter,signals are input into the computer.The data are processed and analyzed by the computer.The motor current test is carried out under load condition.The loading pressure of the system is set as 10 MPa during the loading test.The test wiring diagram is shown in Fig.12.

Fig.12 Wiring diagram of test site

Other test conditions are consistent.Motor current data are collected for normal pump,cracked pump and broken teeth pump respectively.The working frequency of the motor current is 50 Hz.The sampling point is 5 120.The sampling frequency is 1 000 Hz.A total of 900 sets of data are collected.Each type of signal is collected in 300 sets.

60% of the experimental data are randomly selected as training samples.The rest are used as test samples.Thus,the training samples of normal state,crack state and broken teeth are obtained in 200 groups.

The measured signals under three states are shown in Fig.13.It is difficult to distinguish gear state accurately from time-domain waveform.The signal processing method is needed to extract the key feature information.

(a)Normal state

4.2 Parameter optimization of VMD

A set of normal gear pump signals are randomly selected.WhenKis 3,there is the problem of missing information.SoKstarts at 3.The threshold is set to 0.1.As seen from Table 4,whenKis 6,the maximum correlation coefficient between signal components is greater than 0.1.WhenKis set at 3,4 and 5,the correlation coefficients are all less than 0.1.So the value ofKis 5.

Table 4 Maximum correlation coefficient of different K values

A set of normal gear pump signals are randomly selected.ABC algorithm is used to optimize penalty factors.It can be seen from Fig.14 that when the iteration reaches 5 times,the fitness value reaches the maximum of 3.47×10-4.The best penalty factor is 1 327.

Fig.14 Optimization result diagram of α

4.3 Feature extraction

After VMD algorithm parameter optimization,Kis 5,αis 1 327.The correlation coefficient and kurtosis values of IMF1～IMF5 are calculated as shown in Tables 5 and 6.

Table 5 Correlation coefficient and kurtosis of IMF1～IMF5 in crack state

Table 6 Correlation coefficient and kurtosis of IMF1～IMF5 in broken teeth state

According to the above tables,IMF2 and IMF3 are the effective components in the crack state.IMF2 and IMF5 are effective components in broken teeth state.In other words,IMF2,IMF3 and IMF5 are selected as the effective components in this paper.

Feature extraction of the effective IMFs is required.When PE of the effective IMFs is calculated,the time delayτis set as 1.Donoso et al.[18]pointed out that when the embedded dimensionMis between 5 and 7,the dynamic characteristics of the sequence can be accurately reflected by PE.So the embedding dimensionMis set as 5.

After PE and RMS are calculated,they are combined to form high-dimensional eigenvectors.A total of 600 sets of 10-dimensional eigenvectors are obtained.The average value of its eigenvectors is shown in Fig.15.

(a)Average PE of IMFs

The structure of DBN-DNN classifier is 10-20-3.The learning rate is set at 0.1.After the DBN-DNN classifier model is trained,300 sets of test data are input into the classifier for classification.The classification results are shown in Table 7.

Table 7 Gear fault identification based on improved VMD and DBN-DNN

In order to verify the effectiveness of the improved VMD in fault diagnosis,the same test signal is decomposed by EMD.The same feature extraction method and feature vector composition method are applied to the decomposition results.The fault diagnosis accuracy of EMD and DBN-DNN is shown in Table 8.

Table 8 Gear fault identification based on EMD and DBN-DNN

It can be seen from Table 8 that the accuracy of EMD in gear fault identification is significantly lower than that of improved VMD.This is because the current signal is easily affected by power frequency signals,power frequency harmonics and frequency conversion signals.VMD algorithm is embedded with adaptive Wiener filter bank and has good robustness.However,under complex low frequency conditions,the signal cannot be processed more effectively by the EMD algorithm.The improved VMD algorithm is further proved its superiority in gear fault diagnosis.

5 Conclusions

The electromechanical coupling mechanism of induction motor and gear pump is analyzed in this paper.The qualitative analysis of gear fault is verified and MCSA method is feasible.

A gear failure monitoring method of hydraulic pump is proposed based on the improved VMD method and DBN-DNN network.The correlation coefficient and ABC algorithm are used to optimize the VMD parameters.The PE and RMS of the effective IMFs of VMD are obtained to form the eigenvectors.They are then fed into the DBN-DNN classifier for training.The feasibility of this method in gear fault monitoring is verified by experimental signals.It has certain application value for monitoring operation state of hydraulic system.