CN104596767A - Method for diagnosing and predicating rolling bearing based on grey support vector machine - Google Patents

Abstract

The invention provides a method for diagnosing and predicating a rolling bearing based on a grey support vector machine. The method is characterized in that the rolling bearing is used as a key part of a mechanical device, and the advantages and disadvantages of the operation state influence the operation performances of the whole device. The method is the method for diagnosing and predicating the rolling bearing based on GM (1, 1)-SVM. The method comprises the steps of extracting a vibration signal time domain and frequency domain feature values of the rolling bearing under various fault and normal states; selecting important feature parameters to build a predicating model, namely, grey model; predicating the feature value; training a binary tree supporting vector machine according to various fault feature values and normal state feature values of the bearing; creating a rolling bearing decision making tree for determining the fault as well as classifying the fault type to diagnosis the fault of the bearing; then predicating the fault according to the predicating value and the trained supporting vector machine.

Description

A kind of based on the rolling bearing fault diagnosis of Grey support vector machine and the method for prediction

Technical field

The invention belongs to bearing failure diagnosis field, is the comprehensive fault diagnosis and fault prediction model GM of one (1, the 1)-SVM for rolling bearing exploitation.

Background technology

Rolling bearing is most popular mechanical component in electric power, petrochemical industry, metallurgy, machinery, Aero-Space and some war industry departments, is also one of parts of most easy damaged.It has that efficiency is high, frictional resistance is little, easy to assembly, the lubrication easily advantage such as realizations, and the application on rotating machinery very extensively, and plays key effect.Many faults of rotating machinery all have close associating with rolling bearing.According to relevant statistics, 70% of mechanical fault is vibration fault, and has 30% to be caused by rolling bearing in vibration fault.The consequence that rolling bearing fault causes gently then reduces and loses some function of system, heavy then cause serious or even catastrophic consequence.So the method for diagnosing faults of rolling bearing, be one of developing emphasis technology in mechanical fault diagnosis always, be devoted to the monitoring and predicting technology studying rolling bearing fault herein.

For solving the problem of bearing failure diagnosis and prediction, have already been proposed all kinds of algorithm model, but these methods effectively cannot realize predicting bearing fault.Therefore need to propose a kind of can not only realization the diagnosis of bearing fault and the model that will realize the effective early warning of fault.

Summary of the invention

The present invention is based on Grey support vector machine GM (1,1) bearing failure diagnosis of-SVM and the method for early warning, the fault diagnosis to rolling bearing can not only be realized, and effective early warning that can realize fault, contribute to improving the safe operation with the rotatory mechanical system of rolling bearing.

The technical solution used in the present invention is as follows,

Provided by the invention based on Grey support vector machine GM (1, the 1) bearing failure diagnosis of-SVM and the method for early warning, at least comprise following components:

The extraction of S1 characteristic variable and correlation analysis.Rolling bearing is typical rotating machinery, and the temporal signatures variable of its vibration signal has root-mean-square value, peak-to-peak value, average etc., and frequency domain character variable has fundamental frequency, 2 frequencys multiplication, 3 frequencys multiplication, 4 frequencys multiplication, 8 frequencys multiplication etc., and they comprise abundant failure message.All kinds of fault of comparative analysis rolling bearing and normal time vibration signal time domain and the characteristic variable of frequency domain, choose suitable characteristic variable.The characteristic variable chosen herein for fault distinguishing is:

X=(RMS, peak-to-peak value, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude).

RMS is the root-mean-square value of vibration signal, can the overall permanence of representation signal, choose RMS time series as with reference to sequence, all the other 6 sequences are as comparative sequences, obtain the grey relational grade of reference sequences and each comparative sequences, remove 2 characteristic variables that the degree of association is low.

If reference sequence Y={y (k) | k=1,2 ..., n},

Relatively ordered series of numbers X _i={ X _i(k) | k=1,2 ..., n}, i=1,2 ..., m

Nondimensionalization is carried out to variable: $x_i\left(k\right)=\frac{X_i\left(k\right)}{X_i\left(1\right)},k=\mathrm{1,2},\·\·\·,n;i=\mathrm{1,2},\·\·\·,m---\left(1\right)$

Reference sequence and the grey incidence coefficient comparing ordered series of numbers:

${\mathrm{\ξ}}_i\left(k\right)=\frac{\min \min |y\left(k\right)-x_i\left(k\right)|+\mathrm{\ρ}\max \max |y\left(k\right)-x_i\left(k\right)|}{|y\left(k\right)-x_i\left(k\right)|+\mathrm{\ρ}\max \max |y\left(k\right)-x_i\left(k\right)|}---\left(2\right)$

Compute associations degree: $r_i=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\ξ}}_i\left(k\right),k=\mathrm{1,2},\·\·\·,n---\left(3\right)$

Ranking associations degree, if r _i< r _j, so x _jk () compares x _ik () is tightr with reference sequence y (k).

S2 sets up forecast model, before using often kind of state respectively, 10 stack features values set up gray model and orthogonal polynomial makes least square fitting forecast model, rear 2 groups of correlative values as predicted value, calculate the error of two kinds of model predication values, choose the model that error is less---gray model.

(1) grey forecasting model

Gray system model GM (1,1) is by univariate time series { x _i(i=1,2,3 ...) carry out one-accumulate process, differential equation of first order is set up to disclose its in-house development rule to this formation sequence.

Defined feature gray system $X_i^{\left(o\right)}=[X^{\left(0\right)}\left(1\right),X^{\left(0\right)}\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^{\left(0\right)}\left(n\right)]$

Do: $X^{\left(1\right)}\left(t\right)=\underset{k=1}{\overset{t}{\mathrm{\&Sigma;}}}{X}^{\left(0\right)}\left(k\right),$ Have $\begin{array}{c}{X}^{\left(1\right)}=(X^1\left(1\right),X^{\left(1\right)}\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^{\left(1\right)}\left(n\right))\\ =(X^0\left(1\right),X^1\left(1\right)+X^0\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^1(n-1)+X^0\left(n\right))\end{array}$

To X ⁽¹⁾set up the following differential equation: $\frac{{\mathrm{dX}}^{\left(1\right)}}{\mathrm{dt}}+{\mathrm{aX}}^{\left(1\right)}=u---\left(4\right)$

Remember that the differential equation of this single order variable is GM (1,1)

In above formula, a and u can be obtained by least square fitting: $\left[\begin{array}{c}a\\ u\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_M---\left(5\right)$

In formula (5), Y _mfor column vector Y _m=[X ⁰(2), X ⁰(3) ..., X ⁰(n)] ^t,

B is matrix: $B=\left[\begin{array}{cc}-\frac{1}{2}[X^1\left(1\right)+X^1\left(2\right)]& 1\\ -\frac{1}{2}[X^1\left(2\right)+X^1\left(3\right)]& 1\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ -\frac{1}{2}[X^1(n-1)+X^1\left(n\right)]& 1\end{array}\right]$

Time respective function corresponding to the differential equation (5) $X^{\left(1\right)}(t+1)=[X^{\left(0\right)}\left(1\right)-\frac{u}{a}]e^{-\mathrm{at}}\text{+}\frac{u}{a}---\left(6\right)$

By formula (6), one-accumulate is generated to the predicted value of ordered series of numbers: X ⁽⁰⁾(t)=X ⁽¹⁾(t)-X ⁽¹⁾(t-1) (7)

(2) least square fitting prediction is done by orthogonal polynomial

Data fitting be according to measure data between mutual relationship, determine curve y=s (x; a ₀, a ₁..., a _n) type, and then reach minimum principle according to the quadratic sum of error on set point, namely solve unconstrained problem:

$\min F(a_0,a_1,\&CenterDot;\&CenterDot;\&CenterDot;,a_n)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{(s(x_i;a_0,a_1,\&CenterDot;\&CenterDot;\&CenterDot;,a_n)-y_i)}^2---\left(8\right)$

Determine optimized parameter thus obtain matched curve y=s ^*(x).

If φ ₀, φ ₁..., φ _nfor n+1 function, ω _ifor coefficient, meet:

<math> <mrow> <mrow> <mo>(</mo> <msub> <mi>&amp;phi;</mi> <mi>k</mi> </msub> <mtext>,</mtext> <msub> <mi>&amp;phi;</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>&amp;omega;</mi> <mi>i</mi> </msub> <msub> <mi>&amp;phi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;phi;</mi> <mtext>j</mtext> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='&lt;' close=''> <mtable> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>&amp;NotEqual;</mo> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo></mo> </mrow></math> I.e. φ ₀, φ ₁..., φ _nat X={x ₁, x ₂..., x _mupper orthogonal,

Wherein $A_k=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{\left[{\mathrm{\&phi;}}_k{x}_i\right]}^2,$ Then the solution of normal equations (8) is: $a_k^{*}=\frac{\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{y}_i{\mathrm{\&phi;}}_{\text{k}}\left(x_i\right)}{A_k}---\left(9\right)$

The typical neural network that the S3 support vector machine Corpus--based Method theories of learning build, it sets up a most optimal separating hyper plane, the distance between two class samples of these plane both sides is maximized, thus provides good generalization ability to classification problem.For sample (x _i, y _i), i=1,2 ..., s, wherein x _i∈ R ^m, y _i{+1 ,-1}, s is input variable dimension to ∈.Lineoid equation for classifying is: ω x+b=0

Sample is divided into two classes: ω x+b >=0, (y=+1)

ω·x+b≥0,(y＝-1) (10)

The optimal hyperlane of support vector machine is a lineoid making classifying edge maximum, namely makes maximum, so solve optimal hyperlane, namely $\min \mathrm{\&phi;}\left(\mathrm{\&omega;}\right)=\frac{1}{2}{\left|\right|\mathrm{\&omega;}\left|\right|}^2---\left(11\right)$ It should meet constraint condition: y _i(ω x _i+ b)-1>=0, i=1,2 ..., l

Under nonlinear condition, the maximization function of linearly inseparable support vector machine:

$\max W\left(\mathrm{\&alpha;}\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i-\frac{1}{2}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_j{y}_i{y}_{j}K<x_i\&CenterDot;x_j>---\left(12\right)$

Differentiate that objective function is:

Training Support Vector Machines, for the failure modes of rolling bearing, selects two class SVM to construct multi classifier herein.Because two class SVM 1 exist respective shortcoming to many algorithms and 1 to 1 algorithm, adopt the support vector machine multi-class classification method based on binary tree herein.Binary classifier constitution step based on binary tree is: i-th sorter by the i-th class and i-th+1, i+2 ..., N class separately, structure SVMi, until separate with N class by N-1 sorter by N-1 class.N-1 SVM is formed multi classifier, and structure SVM decision tree identifies N class fault.

Compared with prior art, this method can not only realize the fault diagnosis to rolling bearing, and can realize the effective early warning to fault, contributes to improving the safe operation with the rotatory mechanical system of rolling bearing.

Accompanying drawing explanation

Fig. 1 Grey support vector machine failure prediction process flow diagram.

Specific implementation method

Below in conjunction with accompanying drawing, enforcement of the present invention is specifically described.

As shown in Figure 1,1. the rolling bearing fault experiment of U.S. Xi Chu university, experiment porch comprises the motor (left side) of 2 horsepowers, a torque sensor (centre), a power meter (right side) and control electronics (not display), tested bearing supporting motor axle.Bearing designation is SKF bearing, and use spark technology to arrange Single Point of Faliure on bearing, fault diameter is respectively 0.007,0.014,0.028 inch.Use acceleration transducer to gather vibration signal in experiment, sensor is installed on the drive end of electric machine casing and fan end and motor support chassis respectively.Vibration signal is gathered by the DAT register of 16 passages, and digital signal samples frequency is 12000S/s.

The fault data adopted herein is for be respectively 0 horsepower and 1 horse-power-hour at motor load, the outer ring fault data of bearing, inner ring fault data, ball fault data, the diameter of often kind of fault is respectively 0.007,0.014,0.021 inch, data acquisition in often kind of situation 12 groups, totally 216 groups of data, choose 4 kinds of normal conditions totally 48 groups of data that motor load is 0 to 3 horsepowers, data count is 264 groups.

2. the extraction of characteristic variable and correlation analysis

Rolling bearing is typical rotating machinery, and the temporal signatures variable of its vibration signal has root-mean-square value, peak-to-peak value, average etc., and frequency domain character variable has fundamental frequency, 2 frequencys multiplication, 3 frequencys multiplication, 4 frequencys multiplication, 8 frequencys multiplication etc.The characteristic variable chosen herein for fault distinguishing is:

X=(RMS, peak-to-peak value, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude)

Record rolling bearing outer ring fault 0.007 inch, rotating speed be 1797,12 groups of time series datas of motor load 0 horsepower.Carrying out Fast Fourier Transform (FFT) to often organizing data, extracting its frequency domain character value.Extract characteristic variable and obtain 12x7 rank time series matrix, as table 1.

Table 1. rolling bearing outer ring fault 0.007 inch, rotating speed be 1797, motor load 0 horsepower of 12 groups of time series data

RMS represents the energy of whole signal, and choose RMS time series as reference sequence, all the other 6 sequences, as comparative sequences, obtain the grey relational grade of reference sequences and each comparative sequences, as table 2.

The grey relational grade of table 2. reference sequences and each comparative sequences

If reference sequence Y={y (k) | k=1,2 ..., n}, compares ordered series of numbers X _i={ X _i(k) | k=1,2 ..., n}, i=1,2 ..., m

Nondimensionalization is carried out to variable: $x_i\left(k\right)=\frac{X_i\left(k\right)}{X_i\left(1\right)},k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n;i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m---\left(14\right)$

Reference sequence and the grey incidence coefficient comparing ordered series of numbers:

${\mathrm{\&xi;}}_i\left(k\right)=\frac{\min \min |y\left(k\right)-x_i\left(k\right)|+\mathrm{\&rho;}\max \max |y\left(k\right)-x_i\left(k\right)|}{|y\left(k\right)-x_i\left(k\right)|+\mathrm{\&rho;}\max \max |y\left(k\right)-x_i\left(k\right)|}---\left(15\right)$

Compute associations degree: $r_i=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i\left(k\right),k=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n---\left(16\right)$

Ranking associations degree, if r _i< r _j, so x _jk () compares x _ik () is tightr with reference sequence y (k).

Calculate, with the RMS degree of association larger be peak-to-peak value, fundamental frequency amplitude, 8x amplitude, 2x amplitude, 3x amplitude and 4x amplitude and the RMS degree of association less, remove 3x and 4x variable, RMS, peak-to-peak value, fundamental frequency amplitude, 8x amplitude, 2x amplitude 5 variablees are used to set up gray model GM (1,1).

3. the foundation of forecast model

(1) grey forecasting model

Gray system model GM (1,1) is by univariate time series { x _i(i=1,2,3 ...) carry out one-accumulate process, differential equation of first order is set up to disclose its in-house development rule to this formation sequence.

Be provided with characteristic gray system $X_i^{\left(o\right)}=[X^{\left(0\right)}\left(1\right),X^{\left(0\right)}\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^{\left(0\right)}\left(n\right)]$

Do: $X^{\left(1\right)}\left(t\right)=\underset{k=1}{\overset{t}{\mathrm{\&Sigma;}}}{X}^{\left(0\right)}\left(k\right),$ Have $\begin{array}{c}{X}^{\left(1\right)}=(X^1\left(1\right),X^{\left(1\right)}\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^{\left(1\right)}\left(n\right))\\ =(X^0\left(1\right),X^1\left(1\right)+X^0\left(2\right),\&CenterDot;\&CenterDot;\&CenterDot;,X^1(n-1)+X^0\left(n\right))\end{array}$

To X ⁽¹⁾set up the following differential equation: $\frac{{\mathrm{dX}}^{\left(1\right)}}{\mathrm{dt}}+{\mathrm{aX}}^{\left(1\right)}=u---\left(17\right)$

Remember that the differential equation of this single order variable is GM (1,1)

In above formula, a and u can be obtained by least square fitting: $\left[\begin{array}{c}a\\ u\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_M---\left(18\right)$

In formula (5), Y _mfor column vector Y _m=[X ⁰(2), X ⁰(3) ..., X ⁰(n)] ^t,

B is matrix: $B=\left[\begin{array}{cc}-\frac{1}{2}[X^1\left(1\right)+X^1\left(2\right)]& 1\\ -\frac{1}{2}[X^1\left(2\right)+X^1\left(3\right)]& 1\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ -\frac{1}{2}[X^1(n-1)+X^1\left(n\right)]& 1\end{array}\right]$

Time respective function corresponding to the differential equation (5) $X^{\left(1\right)}(t+1)=[X^{\left(0\right)}\left(1\right)-\frac{u}{a}]e^{-\mathrm{at}}\text{+}\frac{u}{a}---\left(19\right)$

By formula (6), one-accumulate is generated to the predicted value of ordered series of numbers: X ⁽⁰⁾(t)=X ⁽¹⁾(t)-X ⁽¹⁾(t-1) (20)

Before using often kind of state, 10 stack features values set up gray model GM (1,1), rear 2 groups of correlative values as predicted value, the error of computational prediction value.The model that the 10 stack features values that bearing is 0.007 inch 1797 turns of outer ring faults are set up and precision of prediction are in table 3, and the average forecasting error of 10 groups of data is 4.94%.

(3) least square fitting prediction is done by orthogonal polynomial

Data fitting be according to measure data between mutual relationship, determine curve y=s (x; a ₀, a ₁..., a _n) type, and then reach minimum principle according to the quadratic sum of error on set point, namely solve unconstrained problem: $\min F(a_0,a_1,\&CenterDot;\&CenterDot;\&CenterDot;,a_n)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{(s(x_i;a_0,a_1,\&CenterDot;\&CenterDot;\&CenterDot;,a_n)-y_i)}^2---\left(21\right)$ Determine optimized parameter thus obtain matched curve y=s ^*(x).

If φ ₀, φ ₁..., φ _nfor n+1 function, ω _ifor coefficient, meet <math> <mrow> <mrow> <mo>(</mo> <msub> <mi>&amp;phi;</mi> <mi>k</mi> </msub> <mtext>,</mtext> <msub> <mi>&amp;phi;</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>&amp;omega;</mi> <mi>i</mi> </msub> <msub> <mi>&amp;phi;</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;phi;</mi> <mtext>j</mtext> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='&lt;' close=''> <mtable> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>&amp;NotEqual;</mo> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>,</mo> </mtd> <mtd> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mo></mo> </mrow></math> I.e. φ ₀, φ ₁..., φ _nat X={x ₁, x ₂..., x _mupper orthogonal, wherein

Then the solution of normal equations (8) is: $a_k^{*}=\frac{\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{y}_i{\mathrm{\&phi;}}_{\text{k}}\left(x_i\right)}{A_k}---\left(22\right)$

By orthogonal polynomial, least square fitting is done to 10 stack features values before often kind of state, rear 2 groups of data as predicted value correlative value, the error of computational prediction value.The 10 stack features values two that bearing is 0.007 inch 1797 turns of outer ring faults take advantage of matching predicted value and accuracy value in table 3, and the average forecasting error of 10 groups of data is 7.05%.By the comparison of two kinds of forecast models, can find out that the error mean 4.91% of gray model is less than orthogonal polynomial and makes least square fitting predicated error average 7.05%, therefore select gray model as forecast model.

The model that the 10 stack features values that table 3. bearing is 0.007 inch 1797 turns of outer ring faults are set up and precision of prediction

4. Training Support Vector Machines

The typical neural network that the support vector machine Corpus--based Method theories of learning build, its main thought sets up a most optimal separating hyper plane, the distance between two class samples of these plane both sides maximized, thus provides good generalization ability to classification problem.For sample (x _i, y _i), i=1,2 ..., s, wherein x _i∈ R ^m, y _i{+1 ,-1}, s is input variable dimension to ∈.Lineoid equation for classifying is: ω x+b=0

Sample is divided into two classes: ω x+b >=0, (y=+1)

ω·x+b≥0,(y＝-1) (23)

The optimal hyperlane of support vector machine is a lineoid making classifying edge maximum, namely makes 2/||w|| ²maximum.So solve optimal hyperlane, namely $\min \mathrm{\&phi;}\left(\mathrm{\&omega;}\right)=\frac{1}{2}{\left|\right|\mathrm{\&omega;}\left|\right|}^2---\left(24\right)$

It should meet constraint condition: y _i(ω x _i+ b)-1>=0, i=1,2 ..., l

Under nonlinear condition, the maximization function of linearly inseparable support vector machine:

$\max W\left(\mathrm{\&alpha;}\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i-\frac{1}{2}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_j{y}_i{y}_{j}K<x_i\&CenterDot;x_j>---\left(25\right)$

Differentiate that objective function is:

For the failure modes of rolling bearing, select two class SVM to construct multi classifier herein.With housing washer fault, inner ring fault, ball fault, normal condition structure multi classifier.60 groups of training samples are got to often kind of fault, 12 groups of test sample books, have 200 groups of training samples, 40 groups of test sample books; Normal condition 40 groups of training samples, 8 groups of test sample books.For outer ring fault, structure outer ring fault SVM1, all the other two SVM in like manner.Be { (x by every class fault 60 groups of training samples and 40 groups of normal sample composing training collection ₁, y ₁), (x ₂, y ₂) ..., (x ₂₂₀, y ₂₂₀), wherein, x _i∈ R ⁵, y _i=-1,1}, represent that as y=1 this sample has outer ring fault, there is no outer ring fault when y=-1 represents.

Adopt LIBSVM kit herein, the main task of training is the classifier parameters suitable according to samples selection, selects coef0 setting (c) in the type (s) of support vector machine, kernel function type (t), kernel function, gamma function setup (g) in kernel function.Vector machine is selected be C-SVC and s to be 0, kernel function is Gauss kernel function and t be 2, c be 1.2, g is 2.8.

For outer ring fault, can obtain support vector machine to the discrimination of training sample is 99%.Choose every 12 groups, class fault test sample totally 36 groups and normal condition test sample book 8 groups test, obtaining fault identification rate is 91%, in like manner, trains all the other two support vector machine, each vector machine to its discrimination of all test sample books as table 4.

Each vector machine of table 4. is to its discrimination of all test sample books

Finally, adopt the support vector machine multi-class classification method of the three class binary trees trained to carry out Fault Identification 12 groups of test sample books of single class fault, be recorded as table 5.As seen from Table 5, adopting the support vector machine classification method based on binary tree, can reach more than 90% to single class fault recognition rate, is therefore feasible, efficient to rolling bearing fault diagnosis.

The training result of table 5. single class fault test data

Bring 44 of gray model groups of predicted data (36 groups of fault datas and 8 groups of normal data) into train three class binary-tree support vector machines, record the result of often kind of bearing state training, as table 6.Can obtain from table 6, by extracting the eigenwert of the vibration data of each state collection of rolling bearing, set up gray model, predicted data, bring into and classify by three class binary-tree support vector machines of each malfunction eigenwert training, effectively can carry out the early warning of fault, greatly improve the safe operation state of plant equipment.

The training result of table 6. predicted data

Claims (1)

1. based on the rolling bearing fault diagnosis of Grey support vector machine and a method for prediction, it is characterized in that: the implementing procedure of the method is as follows,

The extraction of S1 characteristic variable and correlation analysis; Rolling bearing is typical rotating machinery, and the temporal signatures variable of its vibration signal has root-mean-square value, peak-to-peak value, average etc., and frequency domain character variable has fundamental frequency, 2 frequencys multiplication, 3 frequencys multiplication, 4 frequencys multiplication, 8 frequencys multiplication etc., and they comprise abundant failure message; All kinds of fault of comparative analysis rolling bearing and normal time vibration signal time domain and the characteristic variable of frequency domain, choose suitable characteristic variable; The characteristic variable chosen herein for fault distinguishing is:

X=(RMS, peak-to-peak value, 1x amplitude, 2x amplitude, 3x amplitude, 4x amplitude, 8x amplitude);

RMS is the root-mean-square value of vibration signal, can the overall permanence of representation signal, choose RMS time series as with reference to sequence, all the other 6 sequences are as comparative sequences, obtain the grey relational grade of reference sequences and each comparative sequences, remove 2 characteristic variables that the degree of association is low;

If reference sequence Y={y (k) | k=1,2 ..., n},

Relatively ordered series of numbers X _i={ X _i(k) | k=1,2 ..., n}, i=1,2 ..., m

Nondimensionalization is carried out to variable: $x_i\left(k\right)=\frac{X_i\left(k\right)}{X_i\left(1\right)},k=\mathrm{1,2},...,n;i=\mathrm{1,2},...,m---\left(1\right)$

Reference sequence and the grey incidence coefficient comparing ordered series of numbers:

${\mathrm{\&xi;}}_i\left(k\right)=\frac{\min \min |y\left(k\right)-x_i\left(k\right)|+\mathrm{\&rho;}\max \max |y\left(k\right)-x_i\left(k\right)|}{|y\left(k\right)-x_i\left(k\right)|+\mathrm{\&rho;}\max \max |y\left(k\right)-x_i\left(k\right)|}---\left(2\right)$

Compute associations degree: $r_i=\frac{1}{n}\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_i\left(k\right),k=\mathrm{1,2},...,n---\left(3\right)$

Ranking associations degree, if r _i< r _j, so x _jk () compares x _ik () is tightr with reference sequence y (k);

S2 sets up forecast model, before using often kind of state respectively, 10 stack features values set up gray model and orthogonal polynomial makes least square fitting forecast model, rear 2 groups of correlative values as predicted value, calculate the error of two kinds of model predication values, choose the model that error is less---gray model;

(1) grey forecasting model

Gray system model GM (1,1) is by univariate time series { x _i(i=1,2,3 ...) carry out one-accumulate process, differential equation of first order is set up to disclose its in-house development rule to this formation sequence;

Defined feature gray system $X_i^{\left(o\right)}=[X^{\left(0\right)}\left(1\right),X^{\left(0\right)}\left(2\right),...,X^{\left(0\right)}\left(n\right)]$

Do: $X^{\left(1\right)}\left(t\right)=\underset{k=1}{\overset{t}{\mathrm{\&Sigma;}}}{X}^{\left(0\right)}\left(k\right),$ Have $\begin{array}{c}{X}^{\left(1\right)}=(X^1\left(1\right),X^{\left(1\right)}\left(2\right),...,X^{\left(1\right)}\left(n\right))\\ =(X^0\left(1\right),X^1\left(1\right)+X^0\left(2\right),...,X^1(n-1)+X^0\left(n\right))\end{array}$

To X ⁽¹⁾set up the following differential equation: $\frac{{\mathrm{dX}}^{\left(1\right)}}{\mathrm{dt}}+{\mathrm{aX}}^{\left(1\right)}=u---\left(4\right)$

Remember that the differential equation of this single order variable is GM (1,1)

In above formula, a and u can be obtained by least square fitting: $\left[\begin{array}{c}a\\ u\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_M---\left(5\right)$

In formula (5), Y _mfor column vector Y _m=[X ⁰(2), X ⁰(3) ..., X ⁰(n)] ^t,

B is matrix: $B=\left[\begin{array}{cc}-\frac{1}{2}[X^1\left(1\right)+X^1\left(2\right)]& 1\\ -\frac{1}{2}[X^1\left(2\right)+X^1\left(3\right)]& 1\\ .& .\\ .& .\\ .& .\\ -\frac{1}{2}[X^1(n-1)+X^1\left(n\right)]& 1\end{array}\right]$

Time respective function corresponding to the differential equation (5) $X^{\left(1\right)}(t+1)=[X^{\left(0\right)}\left(1\right)-\frac{u}{a}]e^{-\mathrm{at}}+\frac{u}{a}---\left(6\right)$

By formula (6), one-accumulate is generated to the predicted value of ordered series of numbers: X ⁽⁰⁾(t)=X ⁽¹⁾(t)-X ⁽¹⁾(t-1) (7)

(2) least square fitting prediction is done by orthogonal polynomial

Data fitting be according to measure data between mutual relationship, determine curve y=s (x; a ₀, a ₁..., a _n) type, and then reach minimum principle according to the quadratic sum of error on set point, namely solve unconstrained problem:

$\min F(a_0,a_1,...,a_n)=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{(s(x_i;a_0,a_1,...,a_n)-y_i)}^2---\left(8\right)$

Determine optimized parameter thus obtain matched curve y=s ^*(x);

If φ ₀, φ ₁..., φ _nfor n+1 function, ω _ifor coefficient, meet:

i.e. φ ₀, φ ₁..., φ _nat X={x ₁, x ₂..., x _mupper orthogonal, wherein $A_k=\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{\left[{\mathrm{\&phi;}}_k{x}_i\right]}^2,$ Then the solution of normal equations (8) is: $a_k^{*}=\frac{\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&omega;}}_i{y}_i{\mathrm{\&phi;}}_k\left(x_i\right)}{A_k}---\left(9\right)$

The typical neural network that the S3 support vector machine Corpus--based Method theories of learning build, it sets up a most optimal separating hyper plane, the distance between two class samples of these plane both sides is maximized, thus provides good generalization ability to classification problem; For sample (x _i, y _i), i=1,2 ..., s, wherein x _i∈ R ^m, y _i{+1 ,-1}, s is input variable dimension to ∈; Lineoid equation for classifying is: ω x+b=0

Sample is divided into two classes: ω x+b >=0, (y=+1)

ω·x+b≥0,(y＝-1) (10)

The optimal hyperlane of support vector machine is a lineoid making classifying edge maximum, namely makes maximum, so solve optimal hyperlane, namely $\min \mathrm{\&phi;}\left(\mathrm{\&omega;}\right)=\frac{1}{2}{\left|\right|\mathrm{\&omega;}\left|\right|}^2---\left(11\right)$

It should meet constraint condition: y _i(ω x _i+ b)-1>=0, i=1,2 ..., l

Under nonlinear condition, the maximization function of linearly inseparable support vector machine:

$\max W\left(\mathrm{\&alpha;}\right)=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i-\frac{1}{2}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{\mathrm{\&alpha;}}_j{y}_i{y}_{j}K<x_i\&CenterDot;x_j>---\left(12\right)$

Differentiate that objective function is:

Training Support Vector Machines, for the failure modes of rolling bearing, selects two class SVM to construct multi classifier herein; Because two class SVM 1 exist respective shortcoming to many algorithms and 1 to 1 algorithm, adopt the support vector machine multi-class classification method based on binary tree herein; Binary classifier constitution step based on binary tree is: i-th sorter by the i-th class and i-th+1, i+2 ..., N class separately, structure SVMi, until separate by N-1 sorter by N-1 class and N class; N-1 SVM is formed multi classifier, and structure SVM decision tree identifies N class fault.