CN103246803B - A kind of significance test method of rolling bearing performance variation process - Google Patents

Abstract

The invention discloses a kind of significance test method of rolling bearing performance variation process, by during one's term of military service its performance being carried out to periodic sampling at rolling bearing, obtain r time series, r time series is divided into D subsequence, d sub-sequence data is X_rd; With bootstrap, each subsequence is processed, obtain each rank moment of the orign of each subsequence; Build the probability density function of each subsequence with principle of maximum entropy; Choose reference time array section; Set up posterior probability density function; Set up registration; Given level of significance α; If registration is less than 1-α, assert r seasonal effect in time series d sub-sequence X_rdSignificant variation; Otherwise assert its not significant variation. The present invention, lacking under probability distribution and trend prior information condition, can check out the variation information of rolling bearing performance during one's term of military service in time, finds bearing failure hidden danger, to take measures as early as possible, avoids serious accident to occur.

Description

A kind of significance test method of rolling bearing performance variation process

Technical field

The invention belongs to rolling bearing performance inspection field, relate to a kind of conspicuousness inspection of rolling bearing performance variation processProved recipe method.

Background technology

Rolling bearing is widely used crucial Mechanical Fundamentals parts, in national basis construction, national economy operation and stateIn anti-safety guarantee, there is the effect of important joint. Along with the fast development in the fields such as Aero-Space, bullet train and new forms of energy, rightMany rolling bearings, as space flight and aviation bearing, naval vessels bearing, aircraft carrier bearing, nuclear reactor bearing, high-speed railway bearing withAnd bearing of wind power generator etc., in engineering circles and academia, its military service performance receives publicity day by day, to guarantee working host safetyReliability service. Meet the demands during one's term of military service in the normal operation of rolling bearing and performance, active demand can be checked its performance in timeVariation information, find bearing failure hidden danger, to take measures as early as possible, avoid serious accident to occur.

For a long time, rolling bearing performance research mainly depends on the known prior information such as probability distribution and trend, asSuppose that performance meets normal distribution, Weibull distribution or Poisson distribution etc., suppose that performance trend is given potential function and littleRipple basic function, given kernel function, suppose piece-wise linearization etc. But rolling bearing has a lot of performance indications requirements, purposes is notWith, the main performance difference of examination. At rolling bearing during one's term of military service, some probability of failure, performance distributes and performance degradation information quiltThink knownly, it is unknown or unascertained also having a lot of probability of failure, performances to distribute with performance degradation information. For example, frictionTrend rule and the failure probability distributions such as wearing and tearing, part breaking, bonding and burn, remain unknown or unascertained so far. ?Making is same performance, in the time of new bearing development and the improvement of existing bearing, and deterioration law and the probability of failure, performance of new bearing performanceDistributing all may be from original different.

Especially, rolling bearing performance is degenerated and is belonged to the nonstationary random process of Nonlinear Dynamical Characteristics, experience conventionallyIn the stages such as initial stage degeneration, progressive degeneration, quick degeneration, the information such as performance trend, performance failure track and probability distribution thereuponBecome. Like this, the rolling bearing performance analysis theories that depends on the prior informations such as known probability distribution and trend meets with serious challenge,Be difficult to address this problem.

Summary of the invention

The object of this invention is to provide a kind of significance test method of rolling bearing performance variation process, to solve probabilityThe check problem that the prior informations such as distribution and trend are unknown and whether unascertained rolling bearing performance worsens.

In order to realize above object, the technical solution adopted in the present invention is: a kind of rolling bearing performance variation processSignificance test method, the method specifically comprises the following steps:

(1) selection can reflect the parameter of rolling bearing service behaviour, carries out the sampling of R time quantum, obtains this propertyR time series of energy data, more each time series is divided into D subsequence, establish d son of r seasonal effect in time seriesSequence is X_rd；

(2) process each subsequence of each seasonal effect in time series with bootstrap, obtain X_rdEach rank moment of the orign;

(3) build the each subsequence X of each time series with principle of maximum entropy_rdProbability density function f_rd(x), x is the axis of rollingHold the stochastic variable of performance data;

(4) choose reference time array section: make r=1, obtain the 1st seasonal effect in time series d sub-sequence X_1dProbability closeDegree function f_1d(x), make f_1d(x) being both prior distribution, is again current sample distribution, by the Bayesian statistics X that learns_1dPosteriorityProbability density functionCalculate its corresponding variance D_1d, choosing wherein minimum one is D_1min, establish corresponding to its varianceLittle subsequence and posterior probability density function thereof are respectively X_1minWithX_1minBe reference time array section, f_1min(x) be reference distribution;

(5) set up posterior probability density function: investigation r (r=2,3 ..., R) and individual time series, make X_rdFor current sample,f_rd(x) be current sample distribution, by Bayesian statistics r seasonal effect in time series d the sub-sequence X that learns_rdPosterior probabilityDensity function

(6) obtain according to step (4) and (5)WithCommon factor edge curve S (x):Area under common factor edge curve S (x) is registration α_1,rd：Wherein R_SFor x about common factor edge songThe integrating range of line S (x);

(7) given level of significance α ∈ [0,1], getting α=0.1 is with reference to significance; If registration α_1,rdBe less than 1-α, assert r seasonal effect in time series d sub-sequence X_rdSignificant variation; Otherwise, assert d son of r seasonal effect in time seriesSequence X_rdNot significant variation.

In step (2), obtain X_rdThe process of each rank moment of the orign as follows:

C) according to bootstrap to X_rdCarrying out equiprobability can sampling with replacement, forms generated data sequence Y_rd：Y_rd＝(y_rd(1),y_rd(2),…,y_rd(b),…,x_rd(B)), whereinIn formula, b represents that the b time equiprobability canPut back to Bootstrap sampling, B is Bootstrap sampling number of times, θ_b(i) i data that obtain while being the b time sampling, y_rd(b) be to take out for the b timeThe average of the I obtaining a when sample data from the sample survey;

D) calculate X_rdM rank moment of the orign M_rdmFor:m＝1,2,…,M_rd，M_rdFor X_rd?The exponent number of High Order Moment.

X in step (3)_rdProbability density function f_rd(x) be:

$f_{\mathrm{rd}}\left(x\right)=\exp \left(\underset{k=0}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)$

In formula, x is the stochastic variable of rolling bearing performance data, c_rdkAbout X_rdK+1 Lagrange multiplier;

The 1st Lagrange multiplier c_rd0For:

$c_{\mathrm{rd}0}=-\ln \left(\underset{R_{\mathrm{rd}}}{\∫}\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}\right)$

In formula, R_rdFor about X_rdThe integrating range of x;

Other Lagrange multipliers are obtained by following formula:

$M_{\mathrm{rdm}}=\frac{\underset{R_{\mathrm{rd}}}{\∫}{x}^m\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}}{\underset{R_{\mathrm{rd}}}{\∫}\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}};m=\mathrm{1,2},\·\·\·,M_{\mathrm{rd}}.$

The 1st seasonal effect in time series d sub-sequence X in step (4)_1dProbability density function f_1d(x) be:

$f_{1d}\left(x\right)=\exp (c_{1d0}+\underset{m=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{1\mathrm{dm}}{x}^m)$

For the 1st time series, make X_1dBoth being priori sample, is again current sample, f_1d(x) be both prior distribution, againFor current sample distribution, by Bayesian statistics, obtain X_1dPosterior probability density functionFor:

${\mathrm{\φ}}_{1d}\left(x\right)=\frac{f_{1d}\left(x\right)f_{1d}\left(x\right)}{\underset{R_{1d}}{\∫}{f}_{1d}\left(x\right)f_{1d}\left(x\right)\mathrm{dx}}$

Obtain X according to as above formula_1dExpectation E_1dFor:

X_1dVariance D_1dFor:

R seasonal effect in time series d sub-sequence X in step (5)_rdPosterior probability density function

In formula, R₀For the integrating range of x, concrete value region is R_1minWith R_rd(r=2,3 ..., R) common factor; R_1minFor closingIn f_1min(x) integrating range of x.

The significance test method of rolling bearing performance variation process of the present invention at rolling bearing during one's term of military service, to its propertyCan carry out periodic sampling, obtain r time series; R time series is divided into D subsequence, d subsequenceData are X_rd; With bootstrap, each subsequence is processed, obtain each rank moment of the orign of each subsequence; Use principle of maximum entropyBuild the probability density function of each subsequence; Choose reference time array section; Set up posterior probability density function; Set up and overlapDegree; Given level of significance α; If registration is less than 1-α, assert r seasonal effect in time series d sub-sequence X_rdSignificantly becomeDifferent; Otherwise, assert r seasonal effect in time series d sub-sequence X_rdNot significant variation. The present invention is lacking probability distribution and trendUnder prior information condition, the present invention can check out the variation information of rolling bearing performance during one's term of military service in time, finds that bearing losesEffect hidden danger, to take measures as early as possible, avoids serious accident to occur.

Brief description of the drawings

Fig. 1 is front 5 the time series vibration datas of embodiment of the present invention rolling bearing;

Fig. 2 is 5 time series vibration datas after embodiment of the present invention rolling bearing;

Fig. 3 is the registration collection of illustrative plates of embodiment of the present invention rolling bearing each subsequence.

Detailed description of the invention

Below in conjunction with accompanying drawing and specific embodiment, the present invention is described further.

The step of the significance test method of rolling bearing performance variation process provided by the invention is as follows:

1 obtains R time series of rolling bearing performance

At rolling bearing during one's term of military service, certain performance is carried out to the sampling of R time quantum, the R that obtains this performance data is individualTime series.

Each time series is divided into D subsequence by 2

Form r time series by the performance data that r time quantum gathers, r=1,2 ..., R.

R time series is X_r：

X_r＝(x_r(1),x_r(2),…,x_r(h),…,x_r(H))(1)

In formula, x_r(h) represent X_rIn h data, H is X_rIn data amount check.

By X_rBe divided into D subsequence. If d subsequence of r seasonal effect in time series is X_rd：

X_rd＝(x_rd(1),x_rd(2),…,x_rd(i),…,x_rd(I));d＝1,2,…,D(2)

In formula, x_rd(i) represent X_rdIn i data, I is X_rdIn data amount check, and have

$I=\frac{H}{D}---\left(3\right)$

3 use bootstraps are processed subsequence, obtain each rank moment of the orign

With bootstrap to X_rdCarrying out equiprobability can sampling with replacement, and concrete steps are

(1) make constant B=500000, and establish variable b and get initial value 1, wherein, B is Bootstrap sampling number of times, and b is that b is inferior generalRate can be put back to Bootstrap sampling.

(2) from X_rdMiddle equiprobability can be put back to and extract 1 data.

(3) step (2) is repeated I time, obtain I data from the sample survey.

(4) the average y of I data from the sample survey of calculating_rd(b), and by y_rd(b) as generated data sequence Y_rdOne of data.

(5) make b add 1.

(6) if b > B, proceed to step (7); Otherwise, proceed to step (2).

(7) establish the dimension B=500000 of generated data sequence, just obtained a large amount of generated datas.

According to bootstrap method, to X_rdCarrying out equiprobability can sampling with replacement, obtains a large amount of generated datas, forms generated dataSequence Y_rd：

Y_rd＝(y_rd(1),y_rd(2),…,y_rd(b),…,x_rd(B))(4)

In formula

$y_{\mathrm{rd}}\left(b\right)=\frac{1}{I}\underset{i=1}{\overset{I}{\mathrm{\Σ}}}{\mathrm{\θ}}_b\left(i\right);b=\mathrm{1,2},\·\·\·,B---\left(5\right)$

In formula, b represents that the b time equiprobability can put back to Bootstrap sampling, and B is Bootstrap sampling number of times, θ_b(i) be the b time samplingTime obtain i data, y_rd(b) average of I the data from the sample survey obtaining while being the b time sampling.

X_rdM rank moment of the orign M_rdmFor

$M_{\mathrm{rdm}}=\frac{1}{B}\underset{b=1}{\overset{B}{\mathrm{\Σ}}}{\left(y_{\mathrm{rd}}\left(b\right)\right)}^m;m=\mathrm{1,2},\·\·\·,M_{\mathrm{rd}}---\left(5\right)$

In formula, M_rdFor X_rdThe exponent number of High Order Moment.

4 use principle of maximum entropy build the probability density function of subsequence

Suppose that x is the stochastic variable of describing rolling bearing performance data. According to principle of maximum entropy, obtain about X_rdProbabilityDensity function f_rd(x)：

$f_{\mathrm{rd}}\left(x\right)=\exp \left(\underset{k=0}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)---\left(6\right)$

In formula, c_rdkAbout X_rdK+1 Lagrange multiplier.

In formula (6), the 1st Lagrange multiplier c_rd0For

$c_{\mathrm{rd}0}=-\ln \left(\underset{R_{\mathrm{rd}}}{\∫}\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}\right)---\left(7\right)$

In formula, R_rdFor about X_rdThe integrating range of x.

Other Lagrange multipliers are obtained by formula (8):

$M_{\mathrm{rdm}}=\frac{\underset{R_{\mathrm{rd}}}{\∫}{x}^m\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}}{\underset{R_{\mathrm{rd}}}{\∫}\exp \left(\underset{k=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{\mathrm{rdk}}{x}^k\right)\mathrm{dx}};m=\mathrm{1,2},\·\·\·,M_{\mathrm{rd}}---\left(8\right)$

5 choose reference time array section

In formula (6), make r=1, obtain the 1st seasonal effect in time series d sub-sequence X_1dProbability density function f_1d(x)：

$f_{1d}\left(x\right)=\exp (c_{1d0}+\underset{m=1}{\overset{M_{\mathrm{rd}}}{\mathrm{\Σ}}}{c}_{1\mathrm{dm}}{x}^m)---\left(9\right)$

For the 1st time series, make X_1dBoth being priori sample, is again current sample, f_1d(x) be both prior distribution, againFor current sample distribution, by Bayesian statistics, obtain X_1dPosterior probability density function

${\mathrm{\φ}}_{1d}\left(x\right)=\frac{f_{1d}\left(x\right)f_{1d}\left(x\right)}{\underset{R_{1d}}{\∫}{f}_{1d}\left(x\right)f_{1d}\left(x\right)\mathrm{dx}}---\left(10\right)$

According to statistics, X_1dExpectation E_1dFor

Variance D_1dFor

The 1st total D the subsequence of time series, can obtain D variance. Observe this D variance, choose wherein minimumOne be D_1min：

D_1min＝min(D_1,1,D_1,2,…,D_1d,…,D_1D)(13)

In the 1st time series, establish corresponding to subsequence and the posterior probability density function thereof of variance minimum and be respectivelyX_1minWithDefinition X_1minFor reference time array section, definition f_1min(x) be reference distribution.

The 1st time series is the initial time sequence between probation, the variance minimum of reference time array section. CauseThis, can regard rolling bearing performance as the subsequence of behaving oneself best of investigating initial stage with reference to time series section.

6 set up posterior probability density function

Investigate r (r=2,3 ..., R) and individual time series. Make X_rdFor current sample, f_rd(x) be current sample distribution, byBayesian statistics, obtains r seasonal effect in time series d sub-sequence X_rdPosterior probability density function

In formula, R₀For the integrating range of x, concrete value region is R_1minWith R_rd(r=2,3 ..., R) common factor; R_1minFor closingIn f_1min(x) integrating range of x.

7 set up registration

According to the common factor concept of fuzzy set, obtainWithCommon factor edge curve S (x):

Area under definition common factor edge curve S (x) is registration α_1,rd：

${\mathrm{\α}}_{1,\mathrm{rd}}=\underset{R_S}{\∫}S\left(x\right)\mathrm{dx}---\left(16\right)$

In formula, R_SFor the integrating range about common factor edge curve S (x) of x.

8 set up null hypothesis H₀With alternative hypothesis H₁, implement significance test

Definition null hypothesis H₀With alternative hypothesis H₁。H₀Represent r seasonal effect in time series d sub-sequence X_rdNot significant variation;H₁Represent r seasonal effect in time series d sub-sequence X_rdSignificant variation.

Given level of significance α ∈ [0,1], according to Little Probability Event Princiole, getting α=0.1 is with reference to significance. IfRegistration α_1,rdBe less than 1-α, assert r seasonal effect in time series d sub-sequence X_rdSignificant variation, should refuse H₀; Otherwise, recognizeA fixed r seasonal effect in time series d sequence X_rdNot significant variation, should accept H₀。

Significant variation represents that rolling bearing performance performance is abnormal, and significant variation does not represent that rolling bearing performance acts normally.

Embodiment:

Cause that taking wear Simulation bearing vibration makes a variation as case study on implementation. The experiment of research bearing vibration accelerationData (unit: m/s²), with the wear process of check bearing, the simulated experiment time is that on November 8th, 2010 was to 2010 12 years23 days. Rolling bearing continuous operation 46 days (axial load 49N, rotating speed 1000r/min), every day is to bearing timing acquiring data65000. Since November 8, within every 5 days, get 1 secondary data, obtain altogether the data of 10 days. Choose front 4000 numbers of every dayAccording to as research object, therefore whole experimentation has 40000 original vibration datas, (totally 10 time quantums, each time is singleCorresponding 1 time series of unit, each time series includes 4000 data, has 10 time serieses), as Fig. 1 and Fig. 2 instituteShow.

Observe Fig. 1 and Fig. 2, choose every 400 vibration datas as a subsequence, choose the 1st time series (November8 days) 4000 data as prior information (comprise 10 subsequences, each subsequence has 400 data).

Utilize bootstrap to 10 subsequences of the 1st seasonal effect in time series, self-service extraction 500000 times respectively, then utilizesLarge entropy principle, to each subsequence structure maximum entropy probability density function, makes each subsequence self not only for priori sample but also for working asFront sample, in conjunction with Bayesian statistics, constructs the posterior probability density function of priori sample, then calculation expectation value and sidePoor, to determine to choose which subsequence as with reference to time series section, result is as shown in table 1.

The selection of table 1 reference time array section

Priori sample (November 8)	Current sample (November 8)	Desired value	Variance/10-5
				The 1st subsequence	The 1st subsequence	-0.003	5.8993
The 2nd subsequence	The 2nd subsequence	-0.0045	16.294
				The 3rd subsequence	The 3rd subsequence	-0.0056	3.7922 6 -->
The 4th subsequence	The 4th subsequence	-0.0054	1.4836
				The 5th subsequence	The 5th subsequence	-0.0048	2.4824
The 6th subsequence	The 6th subsequence	-0.0012	8.7588
				The 7th subsequence	The 7th subsequence	-0.0070	9.4744
The 8th subsequence	The 8th subsequence	-0.0024	43.029
				The 9th subsequence	The 9th subsequence	0.0052393	31.533
The 10th subsequence	The 10th subsequence	-0.0035	22.149

By table 1, choose the 4th subsequence in the middle of November 8 data as with reference to time series section, because its varianceLittle.

Utilize bootstrap to rear 9 each subsequences of seasonal effect in time series, carry out respectively Bootstrap sampling 500000 times, then profitTo each subsequence structure probability density function, obtain the current probability density of 90 current samples of subsequence with principle of maximum entropyFunction. Utilize Bayesian statistics, selected current sample is 90 subsequences in rear 9 time serieses, structure posterior probabilityDensity function, calculates registration, as shown in Figure 3.

As shown in Figure 3, first and last, since the 2nd time series (subsequence is 1-10), registration presents decline and becomesGesture drops to minimum point before the 3rd time series (subsequence is 11-20) finishes; From the 4th time series, (subsequence is21-30) start, until the 8th time series (subsequence is 61-70), registration remains near 0.9 substantially, and fluctuation is notKeep stable state greatly; To the 9th time series (subsequence is 71-80), there is once in a while great fluctuation process; To the 10th time series (sub-orderClassify 81-90 as), registration returns near 0.9 again, and fluctuation is very little. Can check out thus rolling bearing wear process withVibration performance mutation process is as follows:

8～November 18 November: initial wear is from little increase gradually (vibration performance variation is remarkable gradually);

23～December 18 November: the transitional period (vibration performance variation is not remarkable) from initial wear to normal wear;

December 23: normal wear starts (vibration performance variation is not remarkable).

Claims (5)

1. a significance test method for rolling bearing performance variation process, is characterized in that: the method specifically comprises followingStep:

(1) selection can reflect the parameter of rolling bearing service behaviour, carries out the sampling of R time quantum, obtains this performance numberAccording to R time series, more each time series is divided into D subsequence, establish d subsequence of r seasonal effect in time seriesFor X_rd；

(2) process each subsequence of each seasonal effect in time series with bootstrap, obtain X_rdEach rank moment of the orign;

(3) build the each subsequence X of each time series with principle of maximum entropy_rdProbability density function f_rd(x), x is rolling bearingThe stochastic variable of energy data;

(4) choose reference time array section: make r=1, obtain the 1st seasonal effect in time series d sub-sequence X_1dProbability density letterNumber f_1d(x), make f_1d(x) being both prior distribution, is again current sample distribution, by the Bayesian statistics X that learns_1dPosterior probabilityDensity functionCalculate its corresponding variance D_1d, choosing wherein minimum one is D_1min, establish corresponding to its variance minimumSubsequence and posterior probability density function thereof be respectively X_1minWithX_1minBe reference time array section, f_1min(x)For reference distribution;

(5) set up posterior probability density function: investigate r time series, make X_rdFor current sample, f_rd(x) be current sampleDistribute, by the Bayesian statistics individual sub-sequence X of r seasonal effect in time series d that learns_rdPosterior probability density function

(6) obtain according to step (4) and (5)WithCommon factor edge curve S (x):Hand overArea under collection edge curve S (x) is registration α_1,rd：Wherein R_SFor x about common factor edge curve S(x) integrating range;

(7) given level of significance α ∈ [0,1], getting α=0.1 is with reference to significance; If registration α_1,rdBe less than 1-α,Assert r seasonal effect in time series d sub-sequence X_rdSignificant variation; Otherwise, assert d subsequence of r seasonal effect in time seriesX_rdNot significant variation.

2. the significance test method of rolling bearing performance variation process according to claim 1, is characterized in that: step(2) in, obtain X_rdThe process of each rank moment of the orign as follows:

A) according to bootstrap to X_rdCarrying out equiprobability can sampling with replacement, forms generated data sequence Y_rd：Y_rd＝(y_rd(1),y_rd(2),…,y_rd(b),…,x_rd(B)), whereinIn formula, b represents that the b time equiprobability can putReturn Bootstrap sampling, B is Bootstrap sampling number of times, θ_b(i) i data that obtain while being the b time sampling, y_rd(b) be the b time samplingTime the average of I data from the sample survey that obtains; x_rd(B) be X_rdIn B data;

B) calculate X_rdM rank moment of the orign M_rdmFor:M_rdFor X_rdHigh Order MomentExponent number.

3. the significance test method of rolling bearing performance variation process according to claim 2, is characterized in that: step(3) X in_rdProbability density function f_rd(x) be:

$f_{rd}\left(x\right)=ex\left(\underset{k=0}{\overset{M}{\Σ}}{c}_{rdk}{x}^k\right)$

In formula, x is the stochastic variable of rolling bearing performance data, c_rdkAbout X_rdK+1 Lagrange multiplier;

The 1st Lagrange multiplier c_rd0For:

$c_{rd0}=-ln\left(\underset{R_{rd}}{\∫}\exp \right(\underset{k=1}{\overset{M_{rd}}{\Σ}}{c}_{rdk}{x}^k\left)dx\right)$

In formula, R_rdFor about X_rdThe integrating range of x;

Other Lagrange multipliers are obtained by following formula:

$M_{rdm}=\frac{\underset{R_{rd}}{\∫}{x}^m\exp \left(\underset{k=1}{\overset{M_{rd}}{\mathrm{\Σ}}}{c}_{rdk}{x}^k\right)dx}{\underset{R_{rd}}{\∫}\exp \left(\underset{k=1}{\overset{M_{rd}}{\mathrm{\Σ}}}{c}_{rdk}{x}^k\right)dx};m=1,2,...,M_{rd}.$

4. the significance test method of rolling bearing performance variation process according to claim 2, is characterized in that: step(4) the 1st seasonal effect in time series d sub-sequence X in_1dProbability density function f_1d(x) be:

$f_{1d}\left(x\right)=\exp (c_{1d0}+\underset{m=1}{\overset{M_{rd}}{\Σ}}{c}_{1dm}{x}^m)$

Wherein, c_1d0For about X_1dThe 1st Lagrange multiplier, c_1dmFor about X_1dM+1 Lagrange multiplier;

For the 1st time series, make X_1dBoth being priori sample, is again current sample, f_1d(x) be both prior distribution, again for working asFront sample distribution, by Bayesian statistics, obtains X_1dPosterior probability density functionFor:

${\mathrm{\φ}}_{1d}\left(x\right)=\frac{f_{1d}\left(x\right)f_{1d}\left(x\right)}{\underset{R_{1d}}{\∫}{f}_{1d}\left(x\right)f_{1d}\left(x\right)dx}$

Wherein R_1dFor about X_1dThe integrating range of x;

Obtain X according to as above formula_1dExpectation E_1dFor:

X_1dVariance D_1dFor:

5. the significance test method of rolling bearing performance variation process according to claim 3, is characterized in that: step(5) r seasonal effect in time series d sub-sequence X in_rdPosterior probability density function

In formula, R₀For the integrating range of x, concrete value region is R_1minWith R_rdCommon factor; R_1minFor about f_1min(x) x's is long-pendingBy stages.