CN106224224B - A kind of Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy - Google Patents

A kind of Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy Download PDF

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CN106224224B
CN106224224B CN201610550146.2A CN201610550146A CN106224224B CN 106224224 B CN106224224 B CN 106224224B CN 201610550146 A CN201610550146 A CN 201610550146A CN 106224224 B CN106224224 B CN 106224224B
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刘红梅
李连峰
吕琛
马剑
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Beihang University
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04B—POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS
    • F04B51/00—Testing machines, pumps, or pumping installations

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Abstract

The present invention relates to a kind of Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy, hydraulic pump is in the oil circuit of closing, the fluid structure interaction between the compressibility and hydraulic pump and servo-drive system of hydraulic oil, so that the fault signature unobvious of hydraulic pump, fault message extraction is more difficult.For the problem, the present invention proposes a kind of Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy：Empirical mode decomposition is carried out to hydraulic pump vibration signal first, obtains its natural mode of vibration component；Then, Hilbert transform is carried out to each natural mode of vibration component, obtains hilbert spectrum；Finally, the moment of mass entropy of fault-signal Hilbert time-frequency distributions is calculated.Experiment proves that the Hydraulic pump fault feature acquired in the method for proposition has excellent sort feature, well Fault Diagnosis of Hydraulic Pump can be supported to work.

Description

A kind of Hydraulic pump fault feature extraction based on Hilbert-Huang transform and moment of mass entropy Method
Technical field
The present invention relates to a kind of Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy, category In Hydraulic system fault diagnosis technologies field.
Background technology
Hydraulic system is widely used in the systems such as Aero-Space, naval vessel, vehicle, and current positive high pressure, high power are close Degree and extensive, integrated direction development, the safety and reliability of hydraulic system are increasingly subject to the attention of people.Hydraulic pump quilt " heart " of hydraulic system is described as, is the dynamical element of whole hydraulic system, the quality of its service behaviour will directly affect hydraulic pressure The overall work state of system.As high-speed rotating machine, hydraulic pump run time in hydraulic system is most long, and workload is most Greatly, so the abrasion degradation speed of hydraulic pump is very fast.Statistics shows, in the failure of all engineering machinery, the failure of hydraulic pump Proportion accounts for 30%~40%, therefore the fault diagnosis of hydraulic pump is the pith of Failure Diagnosis of Hydraulic System.Hydraulic pump one Denier breaks down, and easily causes the failure of whole hydraulic system, causes irremediable loss.As applied hydraulic system on aircraft Posture manipulation is carried out to aircraft, when fuel feeding failure occurs in hydraulic system, aircraft will be caused out of hand, gently then emergency landing, Major accident that is heavy then causing fatal crass.
Feature extraction is the core of fault diagnosis, and good fault signature is particularly significant to the precision for improving fault diagnosis.So And because hydraulic pump is in the oil circuit of closing, the fluid structurecoupling between the compressibility and hydraulic pump and servo-drive system of hydraulic oil Effect so that Hydraulic pump fault feature unobvious, fault message extraction is more difficult, and causing it to diagnose, ambiguity is strong, difficulty is big.
The content of the invention
The technology of the present invention solves problem：Overcome the deficiencies in the prior art, propose that one kind is based on Hilbert-Huang transform With the Hydraulic pump fault feature extracting method of moment of mass entropy, it is special that it can extract the failure of sensitivity in the vibration signal of hydraulic pump Sign, support is provided for fault diagnosis.
The technology of the present invention solution：A kind of Hydraulic pump fault feature based on Hilbert-Huang transform and moment of mass entropy carries Method is taken, its step is as follows：
(1) the moment of mass entropy suitable for troubleshooting issue is proposed, is filled when quantifying fault-signal time-frequency distributions complexity Divide the positional information for considering time frequency block, three moment of mass entropy (s of fault-signal two dimension time-frequency distributionst(q),sf(q),so(q)) It is defined as follows：
Time-frequency plane is divided into the time frequency block of N number of area equation, the energy in every piece is Ei, during the time-frequency block energy pair Countershaft t, frequency axis f and origin O moment of mass is respectively：
In formula, dti, dfiAnd doiRepresent i-th of time frequency block to the distance of time shaft, frequency axis and origin respectively.
Entirely time-frequency plane is respectively to the moment of mass of time shaft, frequency axis and origin：
The moment of mass of each time-frequency block energy is normalized, obtained：
Then have：
Fault-signal time-frequency distributions are to time shaft moment of mass entropy st(q), to frequency axis moment of mass entropy sf(q) and to origin O's Moment of mass entropy so(q) it is defined respectively as：
In formula, qti, qfiAnd qoiWhen respectively i-th of time-frequency block energy accounts for whole to the moment of mass of each reference axis or origin Frequency distribution energy is relative to respective coordinates axle or the ratio of the moment of mass of origin；
To the moment of mass entropy s of time shaftt(q) time-frequency distributions are characterized to frequency f complexity, i.e. fault-signal energy is not The distribution situation measurement of same frequency section；To the moment of mass entropy s of frequency axisf(q) complexity of the time-frequency distributions to the time, i.e. event are characterized Hinder the time-varying characteristics measurement of signal energy distribution；To origin O moment of mass entropy so(q) general complexity of time-frequency distributions is characterized；
(2) Hilbert-Huang transform and moment of mass entropy are combined, proposes a kind of failure for being applied to processing non-stationary signal Feature extracting method, empirical mode decomposition are used to adaptively for vibration signal to be decomposed into a series of natural mode of vibration component, wished Your Bert is converted for calculating instantaneous amplitude and instantaneous frequency so as to obtain hilbert spectrum, when last use quality square entropy quantifies The complexity of frequency division cloth, as Hydraulic pump fault feature.
Above-mentioned specific implementation step is as follows：
The first step, data prediction：Hydraulic pump vibration signal is gathered, and outlier rejecting and noise reduction are carried out to vibration signal Processing；
Second step, empirical mode decomposition EMD：Vibration signal is adaptively decomposed into a series of intrinsic mode function IMF Component and trend term；
3rd step, Hilbert transform：Hilbert transform is implemented to each intrinsic mode function IMF components, obtains wink When amplitude and instantaneous frequency, so as to obtain hilbert spectrum；
4th step, calculate the moment of mass entropy of fault-signal time-frequency distributions：The Hilbert obtained according to Hilbert transform Spectrum, three moment of mass entropys of fault-signal time-frequency distributions are calculated, i.e., time-frequency distributions are to time shaft moment of mass entropy st(q), to frequency Axle moment of mass entropy stAnd the moment of mass entropy s to origin O (q)o(q)；
5th step, draw the dendrogram of different faults state sample characteristic value, analysis method validity.
The second step, the process that EMD is decomposed are as follows：By the composition of signal highest frequency, decompose what is obtained one by one IMF frequency range reduces successively, and decomposition method builds the local maximum and local pole of signal sequence by cubic spline interpolation The envelope of small value, removed after calculating the average of upper and lower envelope from primary signal；Remaining residual error is repeated using same procedure Untill average envelope of the progress at each point goes to zero, first IMF is resulting in, first is subtracted from primary signal IMF, other IMF are decomposited one by one using identical screening technique, stop when residual signals amplitude very little or when becoming dullness point Solution.
In 4th step, three moment of mass entropy (s of fault-signal time-frequency distributions are calculatedt(q),sf(q),so(q)) process For：Time-frequency plane is divided into the time frequency block of N number of area equation, the energy in every piece is Ei, the time frequency block self-energy is to the time Axle t, frequency axis f and the moment of mass to origin O are respectively：
In formula, dti, dfiAnd doiRepresent i-th of time frequency block to the distance of time shaft, frequency axis and origin respectively.
Entirely time-frequency plane is respectively to the moment of mass of two reference axis and origin：
The moment of mass of each time-frequency block energy is normalized, obtained：
Then have：
Fault-signal time-frequency distributions are to time shaft moment of mass entropy st(q), to frequency axis moment of mass entropy sf(q) and to origin O Moment of mass entropy so(q) it is defined respectively as：
In formula, qti, qfiAnd qoiEach moment of mass of respectively i-th time-frequency block energy accounts for the matter of whole time-frequency distributions energy Measure the ratio of square；
To the moment of mass entropy s of time shaftt(q) time-frequency distributions are characterized to frequency f complexity, i.e. fault-signal energy is not The distribution situation of same frequency section；To the moment of mass entropy s of frequency axisf(q) complexity of the time-frequency distributions to the time is characterized, i.e. failure is believed The time-varying characteristics of number Energy distribution；To origin O moment of mass entropy so(q) general complexity of time-frequency distributions is characterized.
The present invention compared with prior art the advantages of be：Hydraulic pump is in the oil circuit of closing, the compressibility of hydraulic oil And the fluid structure interaction between hydraulic pump and servo-drive system so that Hydraulic pump fault feature unobvious, fault message extraction compared with For difficulty.For the problem, the present invention proposes a kind of special based on the Hydraulic pump fault of Hilbert-Huang transform and moment of mass entropy Levy extracting method：Empirical mode decomposition is carried out to hydraulic pump vibration signal first, obtains natural mode of vibration component；Then, to each solid There are modal components to carry out Hilbert transform, obtain hilbert spectrum；Finally, the massic entropy of fault-signal time-frequency distributions is calculated. Experiment proves that the fault signature acquired in proposition method of the present invention has excellent sort feature, hydraulic pressure can be supported well Failure of pump diagnostic work.
Brief description of the drawings
Fig. 1 is the Hydraulic pump fault feature extraction flow chart based on Hilbert-Huang transform and moment of mass entropy of the present invention；
Fig. 2 is EMD algorithm flow charts；
Fig. 3 is time-frequency Entropy principle figure；
Fig. 4 is moment of mass Entropy principle figure；
Fig. 5 is plunger hydraulic Test-bed for pump；
Fig. 6 is the vibration signal of each malfunction, i.e., under normal condition, valve plate wear-out failure and piston shoes wear-out failure Vibration signal figure；(a) normal (b) valve plate failure (c) piston shoes failure；
Fig. 7 is the hilbert spectrum of each malfunction：(a) normal (b) valve plate failure (c) piston shoes failure；
Fig. 8 is fault signature dendrogram.
Embodiment
As shown in figure 1, the Hydraulic pump fault feature extracting method of the invention based on Hilbert-Huang transform and moment of mass entropy Mainly comprise the steps of：
The first step, data prediction.Hydraulic pump vibration signal is gathered, and outlier rejecting and noise reduction are carried out to vibration signal Processing；
Second step, EMD are decomposed.Vibration signal is adaptively decomposed into a series of intrinsic mode function IMF components and become Gesture item；
3rd step, Hilbert transform.Hilbert transform is implemented to each intrinsic mode function IMF components, obtains it Instantaneous amplitude and instantaneous frequency, so as to obtain hilbert spectrum；
4th step, calculate the moment of mass entropy of fault-signal time-frequency distributions：The Hilbert obtained according to Hilbert transform Spectrum, three moment of mass entropys of fault-signal time-frequency distributions are calculated, i.e., time-frequency distributions are to time shaft moment of mass entropy st(q), to frequency Axle moment of mass entropy stAnd the moment of mass entropy s to origin O (q)o(q)；
5th step, draw the dendrogram of different faults state sample characteristic value, analysis method validity.
1. the Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy
1.1 Hilbert-Huang transform
Hilbert-Huang transform (HHT) is mainly made up of 2 parts：Empirical mode decomposition (Empirical Mode Decomposition, EMD) and Hilbert transform.
(1) empirical mode decomposition
Time Series are referred to as intrinsic mode function (Intrinsic mode by empirical mode decomposition into a series of Function, IMF) simple component signal, a simple component signal represents one and is similar to most universal, most basic harmonic function Oscillating function.Each IMF has different frequency components, comprising, by up to minimum frequency component, there is instruction in signal The potential of different faults information.They need to meet following two conditions：
First, extreme point is identical with the number of zero crossing in signal or at most differs one；Second, for appointing on signal A bit, the average of the envelope defined respectively by local maximum and local minimum is zero to meaning.First for oscillation data Individual condition is very necessary, to meet that the stringent condition for calculating instantaneous frequency limits, i.e., provides shaking for signal at specific time point Swing frequency.The upper and lower envelope that second condition requires an IMF enables signal with decomposition relative to time shaft Local Symmetric IMF out is modulated.
EMD is assumed based on following three points：
(a) at least one minimum of signal and a maximum (Non-monotonic function)；
(b) time difference between continuous threshold point defines characteristic time scale；
If (c) only flex point will be differentiated without extreme point, data, then using EMD methods and will obtain point Amount is integrated to obtain final result.
The decomposable process of EMD methods is by the composition of signal highest frequency, obtained IMFs frequency model is decomposed one by one Enclose reduces successively.Decomposition method builds the local maximum of signal sequence and the envelope of local minimum by cubic spline interpolation, Removed after calculating the average of upper and lower envelope from primary signal.Remaining residual error is repeated until each using same procedure Untill average envelope at point reasonably goes to zero, first IMF resulting in.First IMF is subtracted from primary signal, Other IMF are decomposited one by one using identical screening technique, stop when residual signals amplitude very little or when becoming dullness decomposing.According to This, it is as follows can to sum up corresponding arthmetic statement：
(a) initialize：r0(t)=x (t), i=1；
(b) i-th of intrinsic mode function IMF is soughti=ci(t)：
A) initialize：h0(t)=ri-1(t), j=1；
B) h is found outj-1(t) whole Local Extremums；
C) using cubic spline interpolation difference interpolation fitting hj-1(t) very big and minimum point, tries to achieve lower envelope e+ And e (t)-(t), and its average is calculated
D) average of envelope is therefrom subtracted, tries to achieve hj(t)=hj-1(t)-mj-1(t)；
E) judge whether the condition of convergence meets, if satisfied, there is ci(t)=hi(t)；If not satisfied, making j=j+1, step is returned Suddenly (b)；
(c)ri(t)=ri-1(t)-ci(t)；
If (d) ri(t) extreme point number more than one, order i=i+1, return to step (b), otherwise decompose and complete.
The idiographic flow of EMD algorithms is as shown in Figure 2.
By EMD methods, original signal is broken down into：
Wherein ci(t) it is an IMF component, rn(t) be residual components, generally signal average tendency, be constant sequence Or monotonic sequence.
(2) Hilbert transform
Decomposed by EMD after obtaining IMF, it is possible to Hilbert transform is done to each component, obtains its instantaneous frequency And instantaneous amplitude.If IMF components are c (t), then its complex analytic signal is：
WhereinFor：
A (t) is magnitude function：
φ (t) is phase function：
Instantaneous frequency is：
Wherein magnitude function represents the instantaneous amplitude energy of each sampled point of signal；Phase function represents that signal each samples The instantaneous phase of point, instantaneous frequency is just obtained to its derivation.Hilbert is to each IMF components to convert and ignore decomposition remainder, Data can be expressed as：
According to formula (7) can using amplitude and instantaneous phase as the function representation of time in three-dimensional planar, amplitude this Kind time-frequency distributions are referred to as Hilbert amplitude spectrum, referred to as hilbert spectrum.Traditionally square energy is represented with amplitude Density, if replacing the amplitude in Hilbert amplitude spectrum with amplitude square, Hilbert energy spectrum will be obtained.
1.2 moment of mass entropys
(1) comentropy
The mathematical definition of comentropy is：If p (p1,p2,...,pn) be a chance event probability distribution, k for it is arbitrary often Number, is typically taken as 1, comentropy possessed by the distribution is defined as：
The size of comentropy can be used for portraying the average degree of uncertainty of probability system.If certain in a certain probability system Probability caused by one event is 1, and probability caused by other events is 0, after being calculated from formula (12), the comentropy s of the system =0, thus be a determination system, uncertainty 0.If in a certain system, its probability distribution is uniform, then it represents that should Probability caused by each event is equal in system, and the comentropy of the system has the uncertainty maximum of maximum, the i.e. system. Theoretical according to this, most uncertain probability distribution has maximum entropy, and information entropy reflects the inequality of its probability distribution Even degree.
(2) time-frequency entropy
The time-frequency distributions of signal describe Energy distribution situation of the signal within the sampling time at each frequency, different operating shape The time-frequency distributions of hydraulic pump are different under state, and for quantitative this difference degree of description, information entropy theory is incorporated into fault-signal In time-frequency distributions.Difference of the different faults signal in time-frequency distributions shows as time-frequency fragment energy point different on time-frequency plane The difference of cloth, time-frequency entropy can quantify this species diversity, and then reflect the running status of machine.As shown in figure 3, by time-frequency plane etc. It is divided into the time frequency block of N number of area equation, the energy in every piece is Ei(i=1 ..., N), the energy of whole time-frequency plane is A, right Every piece of progress energy normalized, obtains qi=Ei/ A (i=1 ..., N), then hasMeet the normalizing for calculating comentropy Change condition, copies the calculation formula of comentropy, and the calculation formula of the time-frequency entropy of signal is defined as：
(3) moment of mass entropy
With the definition from comentropy, time-frequency entropy carried out under the hypothesis of stochastic variable, namely without order between variable Difference.However, after comentropy is introduced into fault diagnosis field, the energy size of each energy block is not only distinguished, should also be closed The position where the energy block is noted, comprehensive coordinate and magnitude information weigh the distribution of fault-signal exactly.Conversely, It is discounting for the position of each time frequency block, the energy value of each time frequency block of time-frequency plane is constant, upset original order, The time-frequency entropy being then calculated is constant, and order difference exactly usually reflects different fault messages, and this explanation is concerned only with value Comentropy form of Definition can not portray fault signature exactly.
For the comprehensive magnitude information and positional information for portraying fault-signal distribution, the present invention examines during entropy is defined Consider the position where current time frequency block, propose a kind of moment of mass entropy for being suitable for troubleshooting issue.As shown in figure 4, by time-frequency Plane is divided into the time frequency block of N number of area equation, and the energy in every piece is Ei(i=1 ..., N), during the time frequency block self-energy pair Countershaft t, frequency axis f and the moment of mass to origin O are respectively：
In formula, dti, dfiAnd doiRepresent i-th of time frequency block to the distance of time shaft, frequency axis and origin respectively.
Entirely time-frequency plane is respectively to two reference axis and to the moment of mass of origin：
The moment of mass of each time-frequency block energy is normalized, obtained：
Then have：
Fault-signal time-frequency distributions are defined respectively as to time shaft, frequency axis and moment of mass entropy to origin O：
In formula, qti, qfiAnd qoiRespectively i-th of time frequency block energy quality square accounts for whole time-frequency distributions energy quality square Ratio.
To the moment of mass entropy s of time shaftt(q) time-frequency distributions are characterized to frequency f complexity, i.e. fault-signal energy is not The distribution situation of same frequency section；To the moment of mass entropy s of frequency axisf(q) complexity of the time-frequency distributions to the time is characterized, i.e. failure is believed The time-varying characteristics of number Energy distribution；To origin O moment of mass entropy so(q) general complexity of time-frequency distributions is characterized.Moment of mass entropy (st(q),sf(q),so(q) complexity of fault-signal time-frequency distributions, and the relatively low suitable visualization of dimension) can comprehensively be measured Analysis, thus the present invention as Fault Diagnosis of Hydraulic Pump when fault feature vector.
2. case is verified
The present invention uses vibration data during hydraulic plunger pump operation to verify the validity and feasibility of proposition method, tests Data acquisition is from plunger pump trouble injection testing platform.Testing stand leads to as shown in figure 5, in motor speed stabilization after 528r/min The vibrating sensor for being installed at plunger pump end is crossed, the vibration signal of testing stand is obtained with 1000Hz sample frequency.Divide successively Not Cai Ji vibration data of the hydraulic pump system under normal condition, valve plate wear-out failure and piston shoes wear-out failure analyzed.
Vibration signal under normal condition, valve plate wear-out failure and piston shoes wear-out failure is as shown in fig. 6, (a) is normal (b) Valve plate failure (c) piston shoes failure.
Hilbert-Huang transform is carried out to the vibration signal under each state, obtains the yellow spectrum of Hilbert as shown in fig. 7, (a) Normally (b) valve plate failure (c) piston shoes failure.
The moment of mass entropy of each state Hilbert spectrum is calculated, makes the fault signature dendrogram of each malfunction as schemed Shown in 8.
As it can be observed in the picture that the fault sample of health status of the same race flocks together, square between the fault sample of different faults state From larger, moment of mass entropy is had excellent sort feature as fault signature by this explanation, can be follow-up fault diagnosis work Good fault signature is provided to support.
Above case study on implementation is provided just for the sake of the description purpose of the present invention, and is not intended to limit the scope of the present invention. The scope of the present invention is defined by the following claims.The various equivalent substitutions that do not depart from spirit and principles of the present invention and make and Modification, all should cover within the scope of the present invention.

Claims (1)

  1. A kind of 1. Hydraulic pump fault feature extracting method based on Hilbert-Huang transform and moment of mass entropy, it is characterised in that：
    (1) the moment of mass entropy suitable for troubleshooting issue is proposed, is fully examined when quantifying fault-signal time-frequency distributions complexity Consider the positional information of time frequency block, three moment of mass entropy (s of fault-signal two dimension time-frequency distributionst(q),sf(q),so(q)) specific It is defined as follows：
    Time-frequency plane is divided into the time frequency block of N number of area equation, the energy in every piece is Ei, the time-frequency block energy is to time shaft T, frequency axis f and origin O moment of mass are respectively：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mi>i</mi> </msub> <mo>.</mo> <msub> <mi>d</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mi>i</mi> </msub> <mo>.</mo> <msub> <mi>d</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mi>i</mi> </msub> <mo>.</mo> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> </mrow>
    In formula, dti, dfiAnd doiRepresent i-th of time frequency block to the distance of time shaft, frequency axis and origin respectively；
    Entirely time-frequency plane is respectively to the moment of mass of time shaft, frequency axis and origin：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>t</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>f</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>o</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>M</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> </mrow>
    The moment of mass of each time-frequency block energy is normalized, obtained：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>M</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>M</mi> <mi>f</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>M</mi> <mi>o</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> </mrow>
    Then have：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mstyle> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> </mstyle> <msub> <mi>q</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mstyle> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> </mstyle> <msub> <mi>q</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mstyle> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> </mstyle> <msub> <mi>q</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> </mrow>
    Fault-signal time-frequency distributions are to time shaft moment of mass entropy st(q), to frequency axis moment of mass entropy sfAnd the quality to origin O (q) Square entropy so(q) it is defined respectively as：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>q</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>lnq</mi> <mrow> <mi>t</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>q</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>lnq</mi> <mrow> <mi>f</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mi>q</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mi>q</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>lnq</mi> <mrow> <mi>o</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>N</mi> </mrow>
    In formula, qti, qfiAnd qoiFrequency division when respectively i-th of time-frequency block energy accounts for whole to the moment of mass of each reference axis or origin Cloth energy is relative to respective coordinates axle or the ratio of the moment of mass of origin；
    To the moment of mass entropy s of time shaftt(q) time-frequency distributions are characterized to frequency f complexity, i.e., fault-signal energy is in different frequencies The distribution situation measurement of rate section；To the moment of mass entropy s of frequency axisf(q) complexity of the time-frequency distributions to the time is characterized, i.e. failure is believed The time-varying characteristics measurement of number Energy distribution；To origin O moment of mass entropy so(q) general complexity of time-frequency distributions is characterized；
    (2) Hilbert-Huang transform and moment of mass entropy are combined, proposes a kind of fault signature for being applied to processing non-stationary signal Extracting method, empirical mode decomposition are used to adaptively for vibration signal to be decomposed into a series of natural mode of vibration component, Martin Hilb Spy is converted for calculating instantaneous amplitude and instantaneous frequency so as to obtain hilbert spectrum, frequency division when last use quality square entropy quantifies The complexity of cloth, as Hydraulic pump fault feature.
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