CN108256254B - Unbalanced data multi-dimensional parameter estimation method based on sample expansion - Google Patents

Unbalanced data multi-dimensional parameter estimation method based on sample expansion Download PDF

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CN108256254B
CN108256254B CN201810089475.0A CN201810089475A CN108256254B CN 108256254 B CN108256254 B CN 108256254B CN 201810089475 A CN201810089475 A CN 201810089475A CN 108256254 B CN108256254 B CN 108256254B
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张超
王少萍
马仲海
孙旭
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Beihang University
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Abstract

The invention discloses a sample expansion-based unbalanced data multi-dimensional parameter estimation method, which is applied to the field of reliability of aviation airborne equipment. The method adopts a method of combining the multiplexing sample and the derivative sample, and can effectively expand the sample space; before the accelerated life test, the multiplexing sample has certain initial damage, and data statistics is carried out after the accumulated damage caused by the initial load spectrum is converted; the load spectrum of the derived sample is different from that of the normal sample, the data obtained under different load intensities should not adopt uniform data statistical weight, and different weights should be set according to the difference of the load intensities; by using the method of grouping resampling and normal weighting of the load intensity coefficient, different types of samples can be comprehensively subjected to parameter estimation, and the problem of small sample parameter estimation of unbalanced data is solved.

Description

Unbalanced data multi-dimensional parameter estimation method based on sample expansion
Technical Field
The invention belongs to the field of reliability of airborne equipment, and particularly relates to a sample expansion-based unbalanced data multi-dimensional parameter estimation method.
Background
Modern aircraft require ultra-high reliability and ultra-long life. The aviation hydraulic pump is used as a core component of an aircraft hydraulic system, and compared with a common hydraulic pump of the hydraulic system, the requirement is stricter. The hydraulic pump provides 21 ~ 35MPa high-pressure fluid for aircraft hydraulic system for drive flies control plane or receive and release undercarriage etc. its structure is complicated, powerful, life-span and reliability require more and more high. Once a hydraulic pump of an airplane breaks down, the hydraulic pump of the airplane is slightly out of order, and if the hydraulic pump of the airplane breaks down, the hydraulic system of the airplane is seriously damaged, and people die, so that huge economic loss and adverse social influence are brought, and therefore, the modern airplane has higher and higher requirements on the service life and the reliability of the hydraulic pump. The development of a highly reliable and long-life aircraft hydraulic pump is urgently needed. Generally, the life characteristics of the product are obtained by a method of life test under normal conditions. However, since the hydraulic pump of the airplane is a product with high reliability and long service life, the hydraulic pump has the advantages of precise design, complex weaving process, high production cost and strong specificity, and generally cannot be produced in large batch, and the situations of carrying out life tests of the whole life cycle and related destructive life tests are fewer. The conventional life test can cause the conventional life test period to be too long, so that the life test cost is huge, the development period is too long, the conventional life test cannot be matched with the rapid development requirement of the airplane, and the market competitiveness of the product is reduced. Therefore, the life evaluation of the hydraulic pump is generally carried out by a method of accelerated life test in engineering.
The basic idea of the accelerated life test is as follows: by adopting a more severe load spectrum, the hydraulic pump can be quickly failed under the condition that the failure mechanism is not changed, so that the test time is shortened; and then, analyzing and counting the accelerated life test result by adopting a proper accelerated life model, thereby estimating the service life of the hydraulic pump under the action of a normal load spectrum. The current universal accelerated life test method is mainly based on failure data under constant accelerated load and a classical statistical analysis model, and the method generally needs larger test sample size and long-term development experience accumulation. However, the civil aircraft hydraulic pump has various failure modes, complex failure mechanism, severe stress coupling and bears variable stress load spectrum, so that the traditional constant stress accelerated life test method is not suitable any more. Therefore, an accelerated life model suitable for the unique working condition of the aircraft hydraulic pump needs to be established on the basis of deeply analyzing the failure mechanism of the hydraulic pump, and a proper method is adopted to expand the sample space and carry out small sample statistics.
At present, the service life estimation of the domestic civil aircraft hydraulic pump is based on sufficient test samples and flight samples, the research of the domestic civil aircraft hydraulic pump is just started, the sample size is necessarily small due to the expensive cost and the short development period, how to fully utilize the existing information and how to maximally share the data of the same type of products is an effective way for effectively expanding the sample size, improving the parameter estimation precision and solving the small sample estimation difficulty in the service life test parameter statistics.
Disclosure of Invention
The invention aims to improve the parameter estimation precision under the limited test samples and improve the parameter estimation precision of an accelerated life test, and provides a sample expansion-based unbalanced data multi-dimensional parameter estimation method, which adopts samples of non-life tests such as a performance test, a quality assurance test and the like to continue the accelerated life test to form a multiplexing sample; and converting historical data of the military hydraulic pump, and applying the converted historical data to the accelerated life test statistics of the civil hydraulic pump to form a derivative sample. The initial damage characteristic of the multiplexing sample and the data imbalance problem of the derived sample caused by the difference of military and civil aircraft load spectrums are fully considered, and the small sample parameter estimation of the unbalanced data is carried out by adopting a method based on the grouping resampling and the normal weighting maximum likelihood estimation of the load intensity coefficient.
The invention provides a sample expansion-based unbalanced data multi-dimensional parameter estimation method, which specifically comprises the following steps:
step one, multiplexing samples and initial damage conversion of the samples.
And step two, analyzing the derivative sample and the load strength thereof.
And step three, resampling the unbalanced data packet and weighting maximum likelihood estimation.
The invention has the advantages and positive effects that:
(1) the problem of insufficient sample size caused by cost and development period limitation in an accelerated life test of a hydraulic pump is solved;
(2) the initial damage characteristic of the multiplexing sample and the problem of unbalanced sample amount of the derived sample due to the difference of military and civil aircraft load spectrums are considered;
drawings
FIG. 1 is a typical military and civil aircraft life test load spectrum;
FIG. 2 shows the weight w and the load intensity coefficient sjAnd the variance σ;
FIG. 3 is a flow chart of a packet sampling method;
FIG. 4 is a graph of reliability after sample expansion;
fig. 5 is a flow chart of a method of the present invention.
Detailed Description
The following describes the unbalanced data multidimensional parameter estimation method based on sample expansion in detail with reference to the accompanying drawings and embodiments.
As shown in fig. 5, the unbalanced data multidimensional parameter estimation method based on sample expansion is as follows:
step one, multiplexing samples and initial damage conversion of the samples.
The civil aircraft hydraulic pump can carry out performance test, quality assurance test and other non-life tests of product development processes such as in the development process, and after the test, some samples can also continue to use, as multiplexing sample, continue the test under the experimental load spectrum of accelerated life. For example, since a certain hydraulic pump is different from a normal accelerated life test sample in terms of first, these multiplexed samples have a certain initial damage before the accelerated life test, the accumulated damage caused by the initial load spectrum should be converted, and then data statistics should be performed.
Since the main failure mode of the hydraulic pump is wear failure of internal components, the initial damage should be reduced according to the cumulative damage law of frictional wear. The volume loss of the rotor-valve plate contact surface caused by the abrasive grains in unit time meets the following relational expression:
v＝aPbωcexp{dP+eω} (1)
wherein a, b, c, d and e are undetermined parameters, v is the volume loss of a rotor-valve plate contact surface caused by abrasive particles in unit time, omega is the rotating speed of the rotor, and P is the working pressure of the hydraulic pump.
Suppose that the sample underwent S before accelerated life testing1(P1,ω1)→S2(P2,ω2)→...→Sk(Pk,ωk) The corresponding test time is 0 → t1→t2→...→tk. Wherein S iskIs the kth load, PkTo under a load SkHydraulic pump pressure, omegakTo under a load SkLower rotor speed, then S1(P1,ω1) Under the action, the accumulated damage caused by abrasion is as follows:
v1t1＝aP1 bω1 cexp{dP1+eω1}t1(2)
if it is to be S1(P1,ω1) Cumulative damage of S2(P2,ω2) Then its equivalent reduced time τ1Can be calculated according to the following formula:
v1t1＝v2τ1(3)
τ1is S1(P1,ω1) Cumulative damage of S2(P2,ω2) Equivalent reduced time under load.
Then
aP1 bω1 cexp{dP1+eω1}t1＝aP2 bω2 cexp{dP2+eω2}τ1(4)
Is finished to obtain
Figure BDA0001563278270000041
Similarly, the general formula of S1(P1,ω1)、S2(P2,ω2) Cumulative damage of S3(P3,ω3) Equivalent reduced time τ of2Can be calculated according to the following formula:
v2(t2-t1+τ1)＝v3τ2(6)
then
aP2 bω2 cexp{dP2+eω2}(t2-t1+τ1)＝aP3 bω3 cexp{dP3+eω3}τ2(7)
Is finished to obtain
Figure BDA0001563278270000042
By analogy, the initial load spectrum S1(P1,ω1)→S2(P2,ω2)→...→Sk(Pk,ωk) The cumulative damage caused is:
vk(tk-tk-1+τk-1)＝aPk bωk cexp{dPk+eωk}(tk-tk-1+τk-1) (9)
wherein the content of the first and second substances,
Figure BDA0001563278270000043
if the accelerated life test is continued, assume that the elapsed time of the jth sample is
Figure BDA0001563278270000044
The j-th sample underwent an accelerated test loading spectrum of Sj,1→Sj,2→...→
Figure BDA0001563278270000045
According to the theory of cumulative damage, the initial damage is converted to Sj,1(Pj,1,ωj,1) Equivalent reduced time τ ofkCan be calculated according to the following formula:
vk(tk-tk-1+τk-1)＝vj,1τk(11)
then
aPk bωk cexp{dPk+eωk}(tk-tk-1+τk-1)＝aPj,1 bωj,1 cexp{dPj,1+eωj,1}τk(12)
Is finished to obtain
Figure BDA0001563278270000051
The failure probability of the damage at this time can be expressed as:
Figure BDA0001563278270000052
Figure BDA0001563278270000053
Figure BDA0001563278270000054
wherein: fre(t) is the probability of failure of the impairment of the multiplexed samples,
Figure BDA0001563278270000055
ith sample of jth samplejThe duration of each of the plurality of time periods,
Figure BDA0001563278270000056
for historical loads
Figure BDA0001563278270000057
To the current load
Figure BDA0001563278270000058
The equivalent reduced time of (a) is,
Figure BDA0001563278270000059
for the jth sample in
Figure BDA00015632782700000510
Characteristic lifetime of time, m is the shape parameter.
Sample of faults ZjCumulative failure density fre(Zj) Comprises the following steps:
Figure BDA00015632782700000511
truncated sample YjAccumulated reliability of Rre(Yj) Comprises the following steps:
Figure BDA00015632782700000512
maximum likelihood function L of multiplexed samplesreComprises the following steps:
Figure BDA00015632782700000513
wherein n is1Number of fault samples, n2The number of truncated samples.
Step two, derivative sample and load strength analysis thereof
The derivative sample is converted into a test sample of the civil aircraft hydraulic pump by adopting a service life test sample of the conventional historical military aircraft hydraulic pump or a load spectrum of the actual flight in an outfield according to the corresponding relation between different load spectrums.
The load-time course is set as follows:
Figure BDA0001563278270000061
to introduce the load spectrum into a load spectrum process, load conversion is firstly needed, and the load conversion is converted into the related accumulated damage amount under the load spectrum of an accelerated life test:
Figure BDA0001563278270000062
Figure BDA0001563278270000063
Figure BDA0001563278270000064
wherein Fde(t) is the probability of failure of the damage of the derivative sample,
Figure BDA0001563278270000065
for historical loads
Figure BDA0001563278270000066
To the current load
Figure BDA0001563278270000067
The equivalent reduced time of (a) is,
failure sample Z of derivative samplejThe cumulative failure density of (a) is:
Figure BDA0001563278270000068
truncated sample Y of the derived samplejThe cumulative reliability of (d) is:
Figure BDA0001563278270000069
the maximum likelihood function of the derived samples is:
Figure BDA00015632782700000610
wherein n is1Number of fault samples, n2The number of truncated samples.
The main concerns of military hydraulic pumps are high pressure, high speed and high power of the hydraulic pumps, while the design concept of civil hydraulic pumps is completely different. The civil aircraft hydraulic pump is designed and developed according to airworthiness regulations strictly, and the research emphasis is on safety, economy and comfort. As shown in FIG. 1, the test load spectra of hydraulic pumps of military and civil aircraft are very different in load strength.
Therefore, when the military machine hydraulic pump derivative sample is used for data statistics, the strength of the military machine load spectrum is considered to be greater than that of the civil machine load spectrum, the data obtained under different load strengths should not adopt uniform data statistics weight, and different weights should be set according to the difference of the load strengths. It is generally believed that the closer the test sample load spectrum is to the nominal load spectrum, the higher its confidence, and the greater the contribution to the parameter estimate should be.
Firstly, defining a load intensity coefficient to describe the similarity degree of a load spectrum and a rated load spectrum of a test sample, and setting the load intensity coefficient of a jth sample as an average value of the ratio of the pressure and the rotating speed of the load spectrum to the rated pressure and the rotating speed of the jth sample:
Figure BDA0001563278270000071
obviously, in the nominal load spectrum S0(P0,ω0) The load intensity coefficient of the sample is s0＝1。
Suppose the weight w(s) of the jth sample in the maximum likelihood estimatej) Coefficient of strength under load sjObeying a normal distribution with mean 1 and variance σ, i.e.:
Figure BDA0001563278270000072
as shown in FIG. 2, the normal distribution variance σ reflects the weight w(s)j) Coefficient of load strength sjIs sensitive to changes in the signal. The smaller sigma is, the steeper the curve of the weight function is, and the larger the sample weight difference of different load strengths is; a larger value of σ means that the curve of the weight function is more gradual and the sample weight differences for different load strengths are not large.
And step three, resampling the unbalanced data packet and weighting maximum likelihood estimation.
For a typical accelerated life test sample, assume that the jth sample experiences Sj,1→Sj,2→...→
Figure BDA0001563278270000076
With a corresponding cumulative failure time of
Figure BDA0001563278270000073
Failure sample Z of accelerated life test samplejThe cumulative failure density of (a) is:
Figure BDA0001563278270000074
accelerated life test specimenTruncated sample YjThe cumulative reliability of (d) is:
Figure BDA0001563278270000075
wherein the content of the first and second substances,
Figure BDA0001563278270000081
for historical loads
Figure BDA0001563278270000082
To the current load
Figure BDA0001563278270000083
The equivalent reduced time of (a) is,
Figure BDA0001563278270000084
because the number of the accelerated life test samples is usually much smaller than that of the multiplexed samples and the derivative samples, in order to eliminate estimation errors caused by unbalanced sample number, a grouping resampling method is adopted to solve the unbalanced data statistics problem.
And respectively extracting the samples with the same quantity from each group of samples to form a new sampling population, and then performing parameter estimation on the sampling population. Assuming that m groups of samples are sampled, the number of samples in the jth group of samples is njThe observed value of the sample is
Figure BDA0001563278270000085
The predetermined number of samples is N, and the sampling method is shown in fig. 3, and specifically includes:
1. generating N random numbers between 0 and 1 by using a computer, and recording the random numbers as a1,a2,…,aN；
2. Will be interval [0,1]Is divided into njSegments, each segment being equal in inter-segment length:
Figure BDA0001563278270000086
Figure BDA0001563278270000087
3. sampling according to the value of random number, if ak∈Cl，k＝1,2,…,N，l＝1,2,…,njThen the k sample is Zj,k′＝Zj,lThe obtained sampling sample sequence is Zj,1′,Zj,2′,…,Zj,N′；
4. The overall sequence of samples is Z1,1′,Z1,2′,…,Z1,N′…Zj,1′,Zj,2′,…,Zj,N′…Zm,1′,Zm,2′,…,Zm,N′。
Considering all the accelerated life test samples, the multiplexed samples and the derived samples together, the jointly weighted log-maximum likelihood function can be expressed as:
Figure BDA0001563278270000088
wherein n is1To accelerate the life test failure sample size, n2To accelerate the life test by truncating the sample size, n1For the number of multiplexed sample failure samples, n2' for the amount of truncated samples of the multiplexed sample, n1"is the derived sample failure sample size, n2"is the derived sample truncated sample size. The parameter to be estimated is
Figure BDA0001563278270000089
Rated load S0(P0,ω0) The following reliability function is:
Figure BDA0001563278270000091
the failure probability density function at rated load is:
Figure BDA0001563278270000092
mean Time To Failure (MTTF) of
Figure BDA0001563278270000093
For convenience of expression and calculation, unified marks are adopted to represent accelerated life test samples, multiplexing samples and derivative samples, and the jth fault sample Z is usedjIs recorded as fj(Zj) The jth truncated sample YjThe cumulative reliability of (D) is denoted as Rj(Yj) Then the formula (27) can be described as
Figure BDA0001563278270000094
Wherein N is1Is the total fault sample size, N1＝n1+n1′+n1″，N2Is the total truncated sample size, N2＝n2+n2′+n2″。
The invention discloses a sample expansion-based unbalanced data multi-dimensional parameter estimation method, which is applied to the field of reliability of airborne equipment. The method adopts a method of combining the multiplexing sample and the derivative sample, and can effectively expand the sample space; before the accelerated life test, the multiplexing sample has certain initial damage, and data statistics is carried out after the accumulated damage caused by the initial load spectrum is converted; the load spectrum of the derived sample is different from that of the normal sample, the data obtained under different load intensities should not adopt uniform data statistical weight, and different weights should be set according to the difference of the load intensities; by using the method of grouping resampling and normal weighting of the load intensity coefficient, different types of samples can be comprehensively subjected to parameter estimation, and the problem of small sample parameter estimation of unbalanced data is solved.
Examples
The effectiveness of the above method is demonstrated by the accelerated life test example below.
(1) Accelerated life test sample
The 1 st group of test samples are accelerated life test samples, and 2 pumps are used for accelerated life test:
pump 1 was a new pump that failed at 187.1 hours accelerated according to the accelerated life test load spectrum shown in table 1.
TABLE 1 accelerated Life test load Spectrum for Pump 1
Figure BDA0001563278270000101
Pump 2 was a new pump that failed at 188.6 hours accelerated according to the accelerated life test load spectrum shown in table 2.
TABLE 2 accelerated life test load spectra for pump 2
Figure BDA0001563278270000102
(2) Multiplexing samples
In order to improve the accuracy of parameter estimation, 3 non-life test samples are introduced as multiplexing samples:
pump 3 was run for 425 hours according to the long test load spectrum shown in table 3 and then accelerated 357.3 hours to fail according to the accelerated life test load spectrum shown in table 4.
TABLE 3 Long trial test load Spectrum
Figure BDA0001563278270000111
TABLE 4 accelerated life test load spectra for pump 3
Figure BDA0001563278270000112
Pump 4 was run for 425 hours according to the long test load spectrum shown in Table 3 and then accelerated for 64 hours to fail according to the accelerated life test load spectrum shown in Table 5
TABLE 5 accelerated life test load spectra for pump 4
Figure BDA0001563278270000121
Pump 5 failed after 1275 hours of operation according to the long test load spectrum shown in table 3.
(3) Derivative sample
An 8-force machine sample was introduced as a derivative sample, and its accelerated life test load spectrum is shown in table 6. The test result is that 7 samples have faults, 1 sample has a tail, and the specific failure time and tail cutting time of each sample are shown in table 7.
TABLE 6 derived sample accelerated life test load spectra
Figure BDA0001563278270000122
Figure BDA0001563278270000131
TABLE 7 derivative sample failure or tailgating time
Figure BDA0001563278270000132
The packet sampling method is adopted, 3 groups of data are shared, and the number of samples in each group is 15. The sampling interval corresponding to the 2 accelerated life experiment samples is C1＝[0,0.5)，C2＝[0.5,1](ii) a The sampling interval corresponding to the 3 multiplexing samples is C1＝[0,0.333)，C2＝[0.333,0.666)，C3＝[0.666,1](ii) a The sampling interval corresponding to 8 derived samples is C1＝[0,0.125，)C2＝[0.125,0.25)，…，C8＝[0.875,1]. The packet sampling results are shown in table 8.
TABLE 8 packet sampling results
Figure BDA0001563278270000133
Figure BDA0001563278270000141
The data was subjected to weighted maximum likelihood estimation using a genetic algorithm according to equation (31), and the parameter estimation results are shown in table 9.
TABLE 9 weighted maximum likelihood estimation results
Figure BDA0001563278270000142
Figure BDA0001563278270000151
According to the parameter estimation results in Table 9, the reliability curve under the rated working condition (pressure 21MPa, rotation speed 4000r/min) is calculated and shown in FIG. 4
The load intensity coefficients of the respective experimental samples and the weights in the maximum likelihood estimation are shown in table 10, and the closer the load intensity coefficient is to 1, the greater the weight occupied in the maximum likelihood estimation is, that is, this method can increase the weight of the sample closer to the rated load. It can be seen that the weight of the multiplexed samples is the largest, while the weight of the derived samples is the smallest.
TABLE 10 load Strength coefficients for the experimental samples and weights in maximum likelihood estimation
Figure BDA0001563278270000152

Claims (1)

1. A multidimensional parameter estimation method of unbalanced data based on sample expansion is characterized by comprising the following steps:
multiplexing samples and initial damage conversion thereof;
converting the initial damage according to the accumulated damage rule of the friction and the wear, and satisfying the following relational expression for the volume loss of the rotor-valve plate contact surface caused by the abrasive particles in unit time:
v＝aPbωcexp{dP+eω} (1)
the method comprises the following steps that a, b, c, d and e are undetermined parameters, v is the volume loss of a rotor-valve plate contact surface caused by abrasive particles in unit time, omega is the rotating speed of a rotor, and P is the working pressure of a hydraulic pump;
suppose that the sample underwent S before accelerated life testing1(P1,ω1)→S2(P2,ω2)→...→Sk(Pk,ωk) The corresponding test time is 0 → t1→t2→...→tk(ii) a Wherein S iskIs the kth load, PkTo under a load SkHydraulic pump pressure, omegakTo under a load SkLower rotor speed, then S1(P1,ω1) Under the action, the accumulated damage caused by abrasion is as follows:
v1t1＝aP1 bω1 cexp{dP1+eω1}t1(2)
if it is to be S1(P1,ω1) Cumulative damage of S2(P2,ω2) Then its equivalent reduced time τ1Comprises the following steps:
v1t1＝v2τ1(3)
τ1is S1(P1,ω1) Cumulative damage of S2(P2,ω2) Equivalent reduced time under load;
then
aP1 bω1 cexp{dP1+eω1}t1＝aP2 bω2 cexp{dP2+eω2}τ1(4)
Is finished to obtain
Figure FDA0002450418190000011
Will S1(P1,ω1)、S2(P2,ω2) Cumulative damage of S3(P3,ω3) Equivalent reduced time τ of2Comprises the following steps:
v2(t2-t1+τ1)＝v3τ2(6)
then
aP2 bω2 cexp{dP2+eω2}(t2-t1+τ1)＝aP3 bω3 cexp{dP3+eω3}τ2(7)
Is finished to obtain
Figure FDA0002450418190000021
Then the initial load spectrum S1(P1,ω1)→S2(P2,ω2)→...→Sk(Pk,ωk) The cumulative damage caused is:
vk(tk-tk-1+τk-1)＝aPk bωk cexp{dPk+eωk}(tk-tk-1+τk-1) (9)
wherein the content of the first and second substances,
Figure FDA0002450418190000022
if the accelerated life test is continued, assume that the elapsed time of the jth sample is
Figure FDA0002450418190000023
The j-th sample undergoes an accelerated test loading spectrum of
Figure FDA0002450418190000028
According to the theory of cumulative damage, the initial damage is converted to Sj,1(Pj,1,ωj,1) Equivalent reduced time τ ofkComprises the following steps:
vk(tk-tk-1+τk-1)＝vj,1τk(11)
then
aPk bωk cexp{dPk+eωk}(tk-tk-1+τk-1)＝aPj,1 bωj,1 cexp{dPj,1+eωj,1}τk(12)
Is finished to obtain
Figure FDA0002450418190000024
The failure probability of the damage is:
Figure FDA0002450418190000025
wherein: fre(t) is the probability of failure of the impairment of the multiplexed samples,
Figure FDA0002450418190000026
ith sample of jth samplejThe duration of each of the plurality of time periods,
Figure FDA0002450418190000027
for historical loads
Figure FDA0002450418190000031
To the current load
Figure FDA0002450418190000032
The equivalent reduced time of (a) is,
Figure FDA0002450418190000033
for the jth sample in
Figure FDA0002450418190000034
Is a characteristic lifetime, m is a shape parameter;
sample of faults ZjCumulative failure density fre(Zj) Comprises the following steps:
Figure FDA0002450418190000035
truncated sample YjAccumulated reliability of Rre(Yj) Comprises the following steps:
Figure FDA0002450418190000036
maximum likelihood function L of multiplexed samplesreComprises the following steps:
Figure FDA0002450418190000037
wherein n is1To accelerate the number of failure samples in the life test, n2The number of truncated samples is used for the accelerated life test;
step two, derivative sample and load strength analysis thereof
The load-time history is set as follows:
Figure FDA0002450418190000038
introducing the load spectrum into a load spectrum process, firstly carrying out load conversion to the accumulated damage amount under the load spectrum of the accelerated life test:
Figure FDA0002450418190000039
wherein: fde(t) is the probability of failure of the damage of the derivative sample,
Figure FDA00024504181900000310
for historical loads
Figure FDA00024504181900000311
To the current load
Figure FDA00024504181900000312
Equivalent reduced time of (d);
failure sample Z of derivative samplejAll right ofThe cumulative failure density is:
Figure FDA0002450418190000041
truncated sample Y of the derived samplejThe cumulative reliability of (d) is:
Figure FDA0002450418190000042
the maximum likelihood function of the derived samples is:
Figure FDA0002450418190000043
wherein n is1To accelerate the number of failure samples in the life test, n2The number of truncated samples is used for the accelerated life test;
and setting the load intensity coefficient of the jth sample as the average value of the ratios of the load spectrum pressure and the rotating speed to the rated pressure and the rotating speed:
Figure FDA0002450418190000044
at rated load spectrum S0(P0,ω0) The load intensity coefficient of the sample is s0＝1；
Suppose the weight w(s) of the jth sample in the maximum likelihood estimatej) Coefficient of strength under load sjObeying a normal distribution with mean 1 and variance σ, i.e.:
Figure FDA0002450418190000045
step three, resampling unbalanced data groups and weighting maximum likelihood estimation;
suppose that the jth sample is subjected to
Figure FDA0002450418190000048
The load history of (a) is obtained,with a corresponding cumulative failure time of
Figure FDA0002450418190000046
Failure sample Z of accelerated life test samplejThe cumulative failure density of (a) is:
Figure FDA0002450418190000047
truncated sample Y of accelerated life test samplejThe cumulative reliability of (d) is:
Figure FDA0002450418190000051
wherein the content of the first and second substances,
Figure FDA0002450418190000052
for historical loads
Figure FDA0002450418190000053
To the current load
Figure FDA0002450418190000054
The equivalent reduced time of (a) is,
Figure FDA0002450418190000055
respectively sampling the samples in each group to form a new sampling population, and performing parameter estimation on the sampling population, wherein J groups of samples are assumed to be sampled, and the number of the jth sample is αjThe observed value of the sample is
Figure FDA0002450418190000056
The predetermined sampling number is N, and the sampling method specifically comprises the following steps:
1. generating N random numbers between 0 and 1 by using a computer, and recording the random numbers as a1,a2,…,aN；
2. Will be interval [0,1]Division into αjSegments, each segment being equal in inter-segment length:
Figure FDA0002450418190000057
Figure FDA0002450418190000058
3. sampling according to the value of random number, if ak'∈Cl，k'＝1,2,…,N，l＝1,2,…,αjThen the k' th sample is Zj,k'′＝Zj,lThe obtained sampling sample sequence is Zj,1′,Zj,2′,…,Zj,N′；
4. The overall sequence of samples is Z1,1′,Z1,2′,…,Z1,N′…Zj,1′,Zj,2′,…,Zj,N′…ZJ,1′,ZJ,2′,…,ZJ,N′；
Comprehensively considering all accelerated life test samples, multiplexed samples and derivative samples, the combined weighted log-maximum likelihood function is as follows:
Figure FDA0002450418190000059
wherein n is1To accelerate the number of failure samples in the life test, n2Number of truncated samples, n, for accelerated life test1For the number of multiplexed sample failure samples, n2' for the amount of truncated samples of the multiplexed sample, n1"is the derived sample failure sample size, n2"is derived sample truncated sample size; the parameter to be estimated is
Figure FDA00024504181900000510
Rated load S0(P0,ω0) The following reliability function is:
Figure FDA0002450418190000061
the failure probability density function at rated load is:
Figure FDA0002450418190000062
mean Time To Failure (MTTF) of
Figure FDA0002450418190000063
Let j fault sample ZjIs recorded as fj(Zj) The jth truncated sample YjThe cumulative reliability of (D) is denoted as Rj(Yj) When formula (27) is as
Figure FDA0002450418190000064
Wherein N is1Is the total fault sample size, N1＝n1+n1′+n1″，N2Is the total truncated sample size, N2＝n2+n2′+n2″。
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