Study on planetary gear fault diagnosis bad on entropy feature fusion of enmble empirical mode
decomposition
Gang Cheng,Xihui Chen ⇑,Hongyu Li,Peng Li,Houguang Liu
School of Mechatronic Engineering,China University of Mining and Technology,221116Xuzhou,China
a r t i c l e i n f o Article history:
Received 18August 2015
Received in revid form 12May 2016Accepted 13May 2016
Available online 14May 2016Keywords:
Planetary gear Fault diagnosis EEMD
Entropy feature fusion KPCA
LVQ neural network
a b s t r a c t
Becau planetary gear is characterized by its small size,light weight and large transmission ratio,it is widely ud in large-scale,low-speed and heavy-duty mechanical systems.Therefore,the fault diagnosis of planetary gear is a key to ensure the safe and reliable operation of such mechanical equipment.A fault diagnosis method of planetary gear bad on the entropy feature fusion of enmble empirical mode decomposition (EEMD)is propod.The intrinsic mode functions (IMFs)with small modal aliasing are obtained by EEMD,and the original feature t is compod of v
arious entropy features of each IMF.To address the innsitive features in the original feature t and the excessive feature dimension,kernel principal component analysis (KPCA)is ud to process the original feature t.Kernel principal compo-nent extraction and feature dimension reduction are performed.The fault diagnosis of planetary gear is eventually realized by applying the extracted kernel principal components and learning vector quantiza-tion (LVQ)neural network.The experiments under different operation conditions are carried out,and the experimental results indicate that the propod method is capable of extracting the nsitive features and recognizing the fault status.The overall recognition rate reaches to 96%when the motor output fre-quency is 45Hz and the load is 13.5N m,and the fault recognition rates of the normal gear,the gear with one missing tooth and the broken gear can reach to 100%.The recognition rates of different fault gears under other operation conditions also can achieve better results.Thus,the propod method is effective for the diagnosis of planetary gear faults.
Ó2016Elvier Ltd.All rights rerved.
1.Introduction
Planetary gear transmission has many advantages,including small size,light weight and large transm
ission ratio,so it is widely ud in the drivetrain of mechanical equipment.However,the planetary gear often subject to heavy duty operation,inten impacts and large amounts of pollution,so it is prone to appear various faults.The faults can result in the catastrophic failure of mechanical equipment,therefore,the fault diagnosis of plane-tary gear is important.The planetary gear is compod of a sun gear,a ring gear and multiple planet gears.Usually,the sun gear is a fixed axis gear,and the ring gear is stationary.The planet gears meshing with the sun gear and ring gear simultaneously not only rotate around the planet gear shaft but also revolute around sun gear shaft.Due to complex movement of planetary gear,working conditions,manufacturing errors and installation errors,the vari-ous complex frequency components are produced.The vibration signal of planetary gear has the frequency modulation (FM)charac-teristic when various frequency components are coupled.Mean-while,in general,the acceleration nsors are installed on the fixed position of planetary gearbox to collect vibration signals.The relative distances among the acceleration nsors and the meshing positions of sun gear with planet gears and planet gears with ring gear are time-varying,and the changing of vibration transmission path would produce the amplitude modulation (AM)effect on the vibration signals of planetary gear.So the vibra-tion signals of planetary gear exhibit the AM and FM phenomena,which show the nonlinear and non-stationary characteristics increasing the difficulty of planetary gear fault diagnosis [1].The fault feature extractio
n method adopted to process the vibration signal of planetary gear could be studied,and the fault diagnosis of planetary gears should be achieved through the application of the advanced classification methods [2].
The traditional fault diagnosis methods are performed such that the monitored time-domain signal is transformed into the fre-quency domain by comparing feature frequencies and other fre-quency indicators.However,tho methods are only suitable to distinguish the fault occurring part of planetary gear (sun gear fault,planet gear fault or ring gear fault),that is becau the fea-
dx.doi/10.asurement.2016.05.0590263-2241/Ó2016Elvier Ltd.All rights rerved.
⇑Corresponding author.
E-mail address:chenxh@cumt.edu (X.Chen).
ture frequency components generated by sun gear fault,planet gear fault and ring gear fault are significantly different.But the fre-quency components generated by different gear fault status occurring on the same part of planetary gear are only have a little difference in terms of the amplitud
es and the sidebands of feature frequencies,and the difference would be usually very weak[3,4]. So the fault status of planetary gear cannot be diagnod bad on the changes in the feature frequencies and other frequency indi-cators.Time–frequency analysis methods not only reflect the fre-quency components but also show their time-varying features, and the time–frequency analysis methods include the short-time Fourier transform,wavelet transform and empirical mode decom-position(EMD).The short-time Fourier transform is a type of linear analysis method,and the wavelet transform requires that the wavelet basis and decomposition layer should be determined in advance.Thus,it is not an adaptive decomposition method,and it suffers from the shortcoming of frequency leakage.EMD is a time–frequency analysis method propod by Huang in1998, wherein the vibration signal is decompod into many IMFs with strict definitions.The definitions of the IMF are as follows:(1) For all of the data,the number of extreme points and the number of zero crossing points must be equal or differ by no more than one.
(2)At any point,the average of the upper envelope value formed by the local maximum point and the low envelope value formed by the local minimum point is zero[5,6].However,EMD suffers from modal aliasing.To solve this problem,EEMD method was pro-pod by Wu and Huang[7].The extreme point distribution of the original signal is varied by adding Gaussian white noi,and EMD is ud to deco
mpo the corrected vibration signal.Adding multi-ple Gaussian white noi,multiple IMF ts can be obtained. According to the frequency homogeneity characteristic of Gaussian white noi,pure and effective IMFs can be calculated by comput-ing the average of multiple IMF ts.In addition,EEMD has no basis function,so it is an adaptive decomposition process.The effective IMFs obtained by EEMD,therefore,how to extract the feature information included in IMFs is the important topics of further rearch.
Becau the decompod IMFs have nonlinear and non-stationary characteristics,the traditional statistical properties and frequency-domain features cannot meet the requirements. Some non-linear parameter estimation methods have proved to be excellent methods to obtain the feature information contained in IMFs,and the entropy theory ud to estimate the complexity and stationarity of signal is introduced in the signal processing field.Zhang et al.[8]prented a hybrid model that combines EEMD and integrated permutation entropy and ud this model for the fault detection and classification of motor bearings.Zeng et al.[9]propod the wavelet correlation feature scale entropy that combined entropy theory and a wavelet correlationfilter achieving the fault feature information extraction of rolling bear-ings under various degraded conditions.However,when the entropy features are ud to indicate the complexity and stationar-ity of signal,the signal can be expresd from multi
ple angles,such as signal energy and signal similarity.Regarding the feature extrac-tion procedure of fault diagnosis,the esnce of feature extraction is that the most distinguishing features should be extracted as much as possible.When different faults occur,a more comprehen-sive expression for the vibration signal can be obtained by extract-ing the entropy features from multiple angles.However,for multiple decompod IMFs,the dimension of entropy features is excessive.In addition,not all entropy features are beneficial to dis-tinguish planetary gear faults;for example.For multiple planetary gear faults,the differences between some entropy features of some IMFs are small,and the innsitive features would affect the accu-racy of the fault diagnosis.The fusion of the original feature t composing of entropy features,data dimension reduction and innsitive feature elimination must be applied.In terms of data dimension reduction and nsitive feature extraction,the more common ud method is principal component analysis(PCA). However,PCA is bad on a linear space transformation,so it can-not obtain the required results for addressing nonlinear problems. KPCA is a nonlinear feature extraction method bad on a kernel function that is ud to map the original data to a high-dimensional feature space.Then,the linear principal components of the mapped data are extracted in the high-dimensional feature space,and the dimension reduction and nonlinear fault feature extraction are obtained[10].
The recognition precision is important after the effective extrac-tion of fault feature information,and there are many rearch works studied for fault recognition,for example support vector machine(SVM),extreme learning machine(ELM),nearest neigh-borhood model and neural network,etc.[11–13].SVM is a fault recognition model that is suitable for processing the problem with small samples.It takes the inner product kernel function to replace the nonlinear mapping of the high-dimensional space,and the nonlinear problem can be solved.But the recognition effect of SVM is greatly affected by the lection of kernel function and its parameters.The optimal parameters are difficult tofind,and the inappropriate parameters will cau the model perform badly in generalization ability.To solve this problem,some parameter opti-mization methods such as artificial colony bee algorithm and genetic algorithm are ud to improve the performance of SVM
[14].But that will increa the complexity of the model,and cau
a reduction in computational speed.ELM is a new leaning algo-rithm for feed forward neural network with single-hidden layer. The weights between the input layer and the hidden layer are ran-domly generated,and there is no need to adjust in the training pro-cess.It has the advantages of easy parameter lection,fast learning speed and good generalization performance[15].How-ever,ELM still has some disadvantages.The parameters of hidden layer are randomly generated,that may caus
e the overfitting phe-nomenon of hidden layer in the training process,and the accuracy of training process is not consistent with that of testing process. Meanwhile,the classification effect of ELM also depends on the activation function of hidden layer,and it has a weak robustness for the blended data.On the other hand,the output weights of ELM are directly calculated by the least square method,which can-not be adjusted according to the specific circumstances of the data t[16,17].LVQ neural network is an input-forward neural net-work with supervid learning[18],and it is widely ud in the field of pattern recognition and optimization.The competition layer of LVQ neural network can automatically complete the learn-ing process bad on the training data,and the complex classifica-tion can be achieved by the interaction of the competition layer neurons and the distance between the competitive layer and the input layer.In addition,LVQ neural network does not require the normalization and orthogonalization for the input vectors.So LVQ neural network has the advantages such as simple structure, fast learning speed,strong lf-learning ability and easy to realize in the practical application,the disadvantages of other neural net-works can be overcome[19].Through the above analysis,ELM and LVQ neural network are superior to SVM in the aspects of param-eter lection and learning speed.At prent,SVM is often ud in the combination with some parameter optimization algorithms. Even though LVQ neural network needs to t more parameters than ELM,but LVQ neural network has a better lf-learning ability and does not require t
he preprocess for input vectors.It is easy to realize in the practical application.So LVQ neural network can be ud to recognize the planetary gear status bad on the fault fea-ture information.
The remainder of this paper is as follows:In Section2,the mathematical fault diagnosis model of planetary gear bad on
G.Cheng et al./Measurement91(2016)140–154141
the entropy feature fusion of EEMD is established.In Section3,the experiments are performed on a drivetrain dynamics simulator (DDS).In Section4,the vibration signal is decompod into multi-ple IMFs,wherein the original feature t is compod of the entropy features extracted from multi angles of each IMF.For the excessive dimensions of original feature t,KPCA is ud to pro-cess the original feature t,and LVQ neural network is applied to recognize planetary gear status.In thefinal ction,some con-clusions would be drawn.
2.Model analysis
2.1.Enmble empirical mode decomposition
EEMD was propod to resolve the problem of modal aliasing [20],and the decomposition process of
EEMD is as follows[21]:
Step1:Two parameters of EEMD method should be initialized: the number of added white noi signals M,and the amplitude of added white noi.
Step2:The m-th signal-added white noi can be obtained as follows:
x mðtÞ¼xðtÞþn mðtÞð1Þ
where x mðtÞis m-th signal-added white noi,xðtÞis original vibra-tion signal,and n mðtÞis m-th added white noi.
Step3:For signal-added white noi,EMD method is ud.The decomposition process of EMD is as follows:
(1)The upper and lower envelope values of the signal x mðtÞ
are calculated and expresd as wðtÞand zðtÞ.The mean
of wðtÞand zðtÞis lðtÞ,and t1ðtÞcan be described as
follows:
t1ðtÞ¼x mðtÞÀlðtÞð2Þ
(2)t1ðtÞis defined as a new signal,and its corresponding upper
and lower envelope values are w1ðtÞand z1ðtÞ.The following process are repeated.
l1ðtÞ¼f w1ðtÞþz1ðtÞg=2ð3Þt2ðtÞ¼t1ðtÞÀl1ðtÞð4Þ......金利来男包
l kÀ1ðtÞ¼f w kÀ1ðtÞÀz kÀ1ðtÞg=2ð5Þt kðtÞ¼t kÀ1ðtÞÀl kÀ1ðtÞð6Þ
If t kðtÞcan satisfy two definitions of an IMF,t kðtÞshould be the first IMF c1ðtÞ.The difference between the original vibration signal xðtÞand thefirst IMF c1ðtÞis defined as a new signal,and that is ud to repeat EMD process until the resulting signal is below a pre-t value.The original signal can be expresd as follows:
xðtÞ¼
X I
i¼1
c iðtÞþr IðtÞð7Þ
where c iðtÞis i-th IMF,I is the total number of IMFs,and r IðtÞis a residual signal.
Step4:If m<M,then m¼mþ1.Then,repeat Step2and Step3 until m¼M.
Step5:Add a Gaussian white noi and obtain an IMF t.Then, calculate the average of M groups of IMF ts obtained by add-ing M times Gaussian white noi.
IMF i¼1
M
X M
m¼1
c i;mðtÞi¼1;2;...;I m¼1;2;...;Mð8Þ
Step6:Eventually,except for IMF i,a residual rðtÞcan be
obtained.For convenient reprentation,rðtÞis expresd as雪团
IMF I.
2.2.Entropy feature extraction
The entropy feature can be ud to quantify the malfunction
and reflect the complexity and uncertainty of vibration signals
[22].The original feature t of vibration signal is generated by cal-
culating the entropy features of each IMF from multiple angles
including the singular spectrum entropy,time-domain energy
entropy,power spectrum entropy and sample entropy.The com-
mon entropy definition is as follow.Assuming X is a generated
space of measurable t H,this space has l measure,and
lðXÞ¼1.X is divided into afinite compatible t
A¼ðA jÞj¼1;2;...;J,namely,X¼
S n
j¼1
A j and A i
T
A j¼0.The
mathematical expression of entropy is defined as follows:
HðAÞ¼À
X J
j¼1
lðA
j
Þln lðA jÞð9Þ
2.2.1.Singular spectrum entropy
糕点品牌
The singular spectrum entropy is calculated bad on the singu-
lar spectrum analysis,and it reflects the uncertainty degree of the
vibration signal energy divided by singular spectrum[23].The
basic idea of singular spectrum analysis is that the time-domain
signal is constructed into a pha space.In this pha space,singu-
lar valve decomposition(SVD)is ud,and the singular spectrum
composing of all singular values is obtained.If the distribution of
signal energy is more concentrated,the distribution of singular
spectrum would be simpler.On the contrary,if the distribution
of signal energy is more disperd,the distribution of singular
spectrum would be more complex.So the status of vibration signal
can be described by singular spectrum.
For IMF i i¼1;2;...;I decompod by EEMD,where I is the total
number of IMFs.Assume that IMF i¼f x1;x2;...;x N g,where N is the
number of data points of IMF i.If the data analysis length K and the
delay constant s are determined,the pha space could be con-
structed as follows:
A¼
x1x2ÁÁÁx K
x sþ1x sþ2ÁÁÁx sþK
.........
x NÀKþ1x NÀKþ2ÁÁÁx K
2
66
66
4
3
77
77
5
ð10Þ
The SVD of the pha space is conducted,and the relations
among the singular values can be ordered as k1P k2PÁÁÁP k K.
The singular spectrum is compod of all singular values,and each
singular value is defined as an effective division for singular spec-
trum.The sum of K singular values is k sum¼
P K
i¼1
k i,and the singu-
lar spectrum entropy can be defined as follows:
H SSE¼À
X K
i¼1
ðk i=k sumÞlnðk i=k sumÞð11Þ
2.2.2.Time-domain energy entropy
When planetary gear faults occur,a complex surge signal is pro-
duced.The time-domain energy entropy reflects the complexity
and uncertainty bad on time-domain signal energy.
Each data point energy and the total energy of IMF i are
expresd as E n¼x2n and E sum¼
P N
n¼1
x2
n
,respectively.By assuming
each data point energy to be a fraction of the total energy,the
time-domain energy entropy can be defined as follows:
142G.Cheng et al./Measurement91(2016)140–154
H TEE ¼ÀX
N n ¼1
ðE n =E sum Þln ðE n =E sum Þ
ð12Þ
2.2.
3.Power spectrum entropy
The power spectrum entropy reflects the complexity and stabil-ity from frequency-domain signal.The spectrum structure of the signal to be analyzed is displayed,and the complexity of the energy distribution in frequency domain can be obrved [24].
F i ðx Þis obtained by using the discrete Fourier transform for each IMF i ,so the power spectrum is expresd as follows:
S ðx Þ¼
1
N
j F i ðx Þj 2x ¼1;2;...;N
ð13Þ
where f S ð1Þ;S ð2Þ;...;S ðN Þg is defined as a fraction of the power energy in frequency domain,and the total power energy is
S sum ¼P
N x ¼1S ðx Þ.The proportional distribution of different fre-quencies in power spectrum is defined as an information probabil-ity distribution,and the power spectrum entropy can be defined as follows:
H PSE
¼ÀX
N x ¼1
ðS ðx ÞS sum Þln ðS ðx Þ=S sum Þ
ð14Þ
2.2.4.Sample entropy
The sample entropy reflects the complexity and stability from quence approximation.The calculatio
n of sample entropy needs to produce a pha space matrix such as Eq.(10),and the obtained matrix can be written as follows [25]:
A ¼X X N ÀK þ1
26666
4
377775¼x 1x 2ÁÁÁ
x K
x
s þ1x s þ2ÁÁÁx s þK ..
.......x N ÀK þ1
x N ÀK þ2
ÁÁÁ
x N
2
666643
77
775ð15Þ
The distances between rows are defined as follows:
d ðX i ;X j Þ¼max ðj X i ðl ÞÀX j ðl ÞjÞl ¼1;2;...;K ð16Þ
The distances between row vector X i and row vector X j are cal-culated.A similar tolerance is t as r ,in general,r usually takes
0.1std –0.25std ,where std is the standard deviation of original data [26].The ratio of the number d ðX i ;X i Þ<r and the number of row vectors can be calculated.
B s i ¼
1
N ÀK
num f d ðX i ;X j Þ<r g i ¼1;2;...;N ÀK þ1;j ¼1;2;...;N ÀK þ1;i –j
ð17Þ
The average for all i is taken,and the result is as follows:
B s
¼1
N ÀK À1X N ÀK þ1
i ¼1B s
i
ð18Þ
石太铁路
Letting
s ¼s þ1,the above process is repeated:
B
s þ1
¼
1N ÀK
社会调查选题X N ÀK i ¼1
B s þ1i ð19Þ
The sample entropy can be defined as follows:
H SE ¼Àln ½B s þ1=B s
ð20Þ
The entropy features extracted from different information
processing angles are realized.However,the original feature t composing of the entropy features of each IMF from four angles has a large dimension,and there are some innsitive features that have a smaller effect on distinguishing planetary gear faults.To
conduct a comprehensive analysis and effective diagnosis,the fusion analysis and dimension reduction of original features t must be performed.
2.3.Kernel principal component analysis
There are additional innsitive features that have a smaller effect on the ability to categorize the planetary gear status in orig-inal feature t.The features strongly degrade the accuracy of fault diagnosis and result in the features with high dimension.Therefore,KPCA method is ud to fu the original feature t,and the kernel principal components can be extracted.
The original feature t is converted into the feature space by a non-linear transformation of kernel function,and PCA method is ud in feature space [27].Assume a total of N samples,and the sample t is X ¼f x 1;x 2;...;x N g ,where x k 2R m .Each sample con-tains m features.Sample x k converte
d to the high-dimensional fea-ture space can be expresd as a nonlinear mapping /ðx k Þ,and P N
k ¼1/ðx k Þ¼0.Then,the covariance matrix of /ðx k Þcan be expresd as follows:
C /¼
1N X
N k ¼1
/ðx k Þ/ðx k ÞT
ð21Þ
The eigenvalue decomposition of C /can be written as follows:
k v ¼C /v
ð22Þ
where k and v are the eigenvalue and eigenvector of C /,
respectively.
All of the data can be mapped to the eigenvectors corresponding to non-zero eigenvalues,and the eigenvector can be expresd as follows:
v ¼
X N i ¼1
a i /ðx i Þ
ð23Þ
where a i is the coefficient corresponding to /ðx i Þ./ðx j Þis multiplied by Eq.(22),
producing
k ð/ðx j ÞÁv Þ¼ð/ðx j ÞÁC /
v Þ.
Combining the above equations gives
k X N i ¼1a i ð/ðx j ÞÁ/ðx i ÞÞ¼1N X N i ¼1
a i ð/ðx j ÞÁX N k ¼1
/ðx k ÞÞð/ðx k ÞÁ/ðx i ÞÞ
ð24Þ
After defining the kernel function k ðx ;y Þ¼h /ðx Þ;/ðy Þi ,the ker-nel transformation matrix k ðx i ;y i Þ¼h /ðx i Þ;/ðy i Þi i ;j ¼1;2;...;N can be obtained.The inner product of two vectors in feature space is expresd as the kernel function of two vectors in input space,and the non-linear mapping that converts the input space into the feature space is replaced by the kernel function.Then,the eigenvectors and eigenvalues of the matrix k ðx i ;y i Þcould be calcu-lated [28].
To meet the requirements of the initial conditions P N
k ¼1/ðx k Þ¼0,the kernel matrix k ðx i ;y i Þshould be normalized.
k ðx i ;y i Þ¼k ðx i ;y i ÞÀL N k ðx i ;y i ÞÀk ðx i ;y i ÞL N þL N k ðx i ;y i
ÞL N ð25Þ
where L N is an N ÂN unit matrix and the coefficient is 1/N .Then,the variance normalization can be written as follows:
k s ðx i ;y i
Þ¼ k s ðx i ;y i
Þtr ð k s ðx i ;y i
ÞÞ=ðN À1Þð26Þ
k ðx i ;y i Þis replaced by k s ðx i ;y i
Þ,so the eigenvalues and eigenvectors can be calculated as follows:
光身女人N k k
s v ¼ k s k s v )N k v ¼ k s v ð27Þ
G.Cheng et al./Measurement 91(2016)140–154
143
The relation of eigenvalues and eigenvectors can be shown as k1P k2PÁÁÁP k n and v1;v2;...;v n.The directions of the kernel principal components are compod of eigenvectors v1;v2;...;v p corresponding to thefirst p larger eigenvalues,which are normal-ized.For an arbitrary test vector x,its kernel principal components t¼ðt1;t2;...;t pÞare the projection of/ðxÞmapping to the eigen-vectors v1;v2;...;v p.
t lðxÞ¼h v l;/ðxÞi¼X N
i¼1
a l
i
h/ðx iÞ;/ðxÞi¼
X N
i¼1
a l
i
kðx i;xÞl¼1;2;...;p
ð28Þ
So the inner product and non-linear mapping can be solved by introducing the kernel function.
2.4.LVQ neural network
LVQ neural network developed from a lf-organizing mapping algorithm is a supervid learning algorithm and is compod of an input layer,a competitive layer and an output layer[29–31].The structure of LVQ neural network is shown in Fig.1.
Where X is the R-dimensional input,S1is the neuron number of competitive layer,k ndist k is the distance between competitive layer and input layer,I x1;1is the weight vector between input layer and competitive layer,n1is the input of competitive layer, a1is the output of competitive layer,L x2;1is the weight vector between competitive layer and linear output layer,n2is the input of linear output layer,and a2is the output of linear output layer.
The learning step of LVQ neural network is as follows:
Step1:The weight vector I x i;j of input layer and competitive layer is initialized to a random value in[0,1].
Step2:The fault feature X¼f x1;x2;...;x R g T is defined as the input of LVQ neural network,and the distance between com-petitive layer and inputted fault features can be calculated.
d i¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
斗智斗勇X R
j¼1
ðx jÀI x ijÞ2
r
i¼1;2;...;S1;j¼1;2;...;Rð29Þ
where I x i;j is the weight vector between neuron j of input layer and neuron i of competitive layer.
Step3:Two neurons a and b of competitive layer that provide the nearest distance to the inputted features are lected,where the distance d a between neuron a and the inputted features is nearer than the distance d b between neuron b and the inputted features.
Step4:The judgment conditions of the neurons are t as fol-lows:(1)Neuron a and neuron b of competitive layer do not
belong to the same category.(2)A window is t to reflect the difference between d a and d b,and it is expresd as min f d a=d b;d b=d a g>q¼ð1ÀwÞ=1þw,q is a window for the midplane of the weight vectors corresponding to neuron a
and neuron b,and w is the window width.The optimal size of the window width w depends on the number of training sam-ples.If there are more training samples,a small window width would have a good performance.In general,the window width w is lected between0.15and0.25[32].In this paper,the win-dow width w is t as0.2,so q is2/3.
Step5:If the lected neurons a and b satisfy the conditions
established in Step4,the weight vector of two winning neurons should be updated.The update rules of the weight vector are as follows:
(1)If the category C a corresponding to a is consistent with
the input category C R,the weight vector correction equa-
tions of neuron a and neuron b should be as follows: L x new j a¼L x old j aþg xÀL x old j a
L x new j b¼L x old j bÀg xÀL x old j b
8
><
>:ð30Þ
where g is the learning rate of LVQ neural network.
(2)If the category C b corresponding to neuron b is consistent
with the input category C R,the weight vector correction equations of neuron a and neuron b should be as follows: L x new j a¼L x old j aÀg xÀL x old j a
L x new j b¼L x old j bþg xÀL x old j b
8
><
>:ð31Þ
Step6:If the lected neurons a and b do not satisfy the condi-tions established in Step4,the weight vector of only one win-ning neuron should be updated.The update rules of the weight vector are as follows:
(1)If the category C a corresponding to neuron a is consis-
tent with the input category C R,the weight vector cor-
rection equation of neuron a should be as follows:
L x new j a¼L x old j aþg xÀL x old j a
ð32Þ(2)If the category C a corresponding to neuron a is not consis-
tent with the input category C R,the weight vector correction equation of neuron a should be as follo
ws:
L x new j a¼L x old j aÀg xÀL x old j a
ð33ÞStep7:The training process of LVQ neural network is completed.
144G.Cheng et al./Measurement91(2016)140–154
3.Test equipment and data acquisition
The fault experiment for planetary gear is shown in Fig.2.The drivetrain dynamics simulator is a mec
hanical fault comprehensive simulation bench produced by Spectra Quest Company in the Uni-ted States.The drivetrain is compod of a programmable control motor,a two-stage planetary gearbox,a two-parallel-shaft gearbox supported by some rolling bearings and a programmable magnetic brake component.The layout of the acceleration nsors on plane-tary gearbox is shown in Fig.3.The sun gear faults of cond-stage planetary gear are prone to appear in planetary gear transmission,so they are ud as the examples in this study.The following com-mon faulty sun gears are prepared and shown in Fig.4:a normal gear,a gear with one missing tooth,a broken gear,a gear with wear and a gear with a tooth root crack.To verify the robustness and performance of the propod method,the experiments are per-formed under different loads and different motor output frequen-cies.The output frequency of the motor controlled by a programmable control component is t as 40Hz and 45Hz,respectively.And the load controlled by a programmable magnetic brake component is t as 6.5N m and 13.5N m,respectively.In the experiments,the sampling frequency is t as 12,800Hz.The vibration signals of five types of gears under different loads and different motor output frequencies have been collected.The fault diagnosis method for planetary gear is propod and tested by ana-lyzing the collected vibration signals.The analysis flowchart of the propod method bad on the entropy feature fusion of EEMD is shown in Fig.5.
4.Experimental analysis
郭志安The basic parameters of two-stage planetary gear of the mechanical fault comprehensive simulation bench are listed in Table 1.Due to the space limitations,in the following,the collected vibration signals when the motor output frequency is 45Hz and the load is 13.5N m are lected as the examples to complete the process of original entropy feature extraction and feature dimen-sion reduction.The feature frequencies of partial gear faults are For the gear with one missing tooth,the sun gear has lost a tooth,so the meshing frequency between sun gear and planet gear changes to affect the periodicity of vibration signal.Becau each tooth of the gear with wear has suffered by varying degrees of wear,the high-frequency components are introduced in the mesh-ing process.It can be en from the time-domain signal that the high-frequency components even cover up the characteristics of the periodic impul.Similarly,comparing with the normal gear,the time-domain signals of the broken gear and the gear with a tooth root crack exhibit the corresponding changes.But the changes are vague,and the gear status cannot be accurately distin-guished by analyzing the time-domain signals.
Next,the propod fault diagnosis method bad on the entropy feature fusion of EEMD is ud to process vibration signals.EEMD has two important parameters including the amplitude of added white noi A and the number of added white noi M .When A is too small,the added white noi m
ay not be enough to cau the change of the local extreme points of the signal.When A is too large,the added white noi will increa the decomposition error.And in theory,the larger the M value is,the smaller decomposition error is.But too large M will increa the computa-burden.Wu and Huang proposing EEMD suggested that amplitude of added white noi A is 0.2times the standard devia-of original signal,and the amplitude of added white noi the number of added white noi M meet the following tionships:A =ffiffiffiffiffi
M p ¼e ,where e reprents the standard deviation Fig.2.Fault experiment for the planetary gear.
Triaxial nsor
Uniaxial nsor
Fig.3.Layout of acceleration nsors on planetary gearbox.
G.Cheng et al./Measurement 91(2016)140–154145
