CN105224766A - A kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics - Google Patents
A kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics Download PDFInfo
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Abstract
A kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics of the present invention, belong to Mechanical Reliability field, the present invention establishes gear life distribution shifts model by Minimal sequence statistics concept, this model using gear life test figure as known conditions, by transforming the gear life distribution can predicting optional tooth number (other parameter constant of gear), and use simple; , a large amount of test figures is verified meanwhile, model to process small sample amount or the larger test figure of dispersiveness effective equally, be applicable to very much the feature of gear life test figure; The life information of gear combines with forecast model by the present invention, significantly can reduce the workload of Gear Experimentation under guarantee probability life prediction very reliably prerequisite, especially obvious to long-life high class gear test meaning.
Description
Technical field
The invention belongs to Mechanical Reliability field, be specifically related to a kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics.
Background technology
Gear uses one of mechanical core part the most general, along with the development of mechanical industry, serviceable life of gear and reliability are had higher requirement, the gear that particularly aircraft industry is used, any inefficacy under duty, all may cause serious consequence; Therefore, evaluate accurately and efficiently, predict gear life, especially expectancy life is most important;
In general, the life-span distributed intelligence of gear can be obtained by the durability test of gear; But the durability test of gear needs a large amount of samples, time and loaded down with trivial details analytic process, and test the data obtained and also there is a large amount of not exclusively clear and definite information; Along with the aggravation of market competition, the time that research and development gear product allows is more and more short, and the simple test acquisition gear life data that rely on are difficult to meet requirement of engineering; Therefore, by suitable model prediction gear life, for reduce Gear Experimentation workload, to shorten the lead time meaning of gear obvious;
The existing life-span prediction method of a large amount of scholar to gear is studied both at home and abroad; Model greatly mainly with based on fracturing mechanics, according to Optimization of Material Property part performance; But, the difference between material and part performance make models applying complicated, predict the outcome unreliable.
Summary of the invention
For the deficiencies in the prior art, the present invention proposes a kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics, to reach the workload alleviating Gear Experimentation, ensures the object of life prediction reliable results simultaneously.
Based on a gear Probabilistic Life Prediction method for Minimal sequence statistics, comprise the following steps:
Step 1, torture test is carried out to multipair gear, obtain the testing data of life-span of gear;
Step 2, matching is carried out to the testing data of life-span of gear, obtain the life-span estimation of distribution parameters of gear;
Step 3, based on Minimal sequence statistics Modling model, by gear life-span estimation of distribution parameters substitute into model, obtain the single gear teeth life-span distribution;
Step 4, be the gear life distribution of the target number of teeth by the life-span distribution shifts of the single gear teeth, complete the Probabilistic Life Prediction of gear.
The testing data of life-span to gear described in step 2 carries out matching, obtains the life-span estimation of distribution parameters of gear, is specially: adopt two parameter Weibull distribution function to carry out matching, and the form parameter obtaining Weibull Function is estimated and scale parameter estimation.
Described in step 3 based on Minimal sequence statistics Modling model, specific as follows:
The gear life cumulative distribution function G of the specific number of teeth
(1)t the expression formula of () is:
G
(1)(t)＝1-[1-F(t)]
n(1)
The life-span cumulative distribution function expression formula obtaining single tooth according to formula (1) is:
F(t)＝1-[1-G
(1)(t)]
1/n(2)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; N represents the number of teeth of particular gear.
Described in step 4 by the life-span distribution shifts of the single gear teeth be the target number of teeth gear life distribution, specific as follows:
The gear life cumulative distribution function G ' of the target number of teeth
(1)t () expression formula is as follows:
G′
(1)(t)＝1-[1-F(t)]
n′(3)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; The number of teeth of n ' expression target gear;
By F (t)=1-[1-G
(1)(t)]
1/nin substitution formula (3), the distribution of the gear life of the specific number of teeth is converted to the gear life distribution of the target number of teeth:
G′
(1)(t)＝1-[1-G
(1)(t)]
n′/n(4)
In formula, G
(1)t () represents the gear life cumulative distribution function of the specific number of teeth, n represents the number of teeth of particular gear.
Advantage of the present invention:
The present invention proposes a kind of gear Probabilistic Life Prediction method based on Minimal sequence statistics, the present invention establishes gear life distribution shifts model by Minimal sequence statistics concept, this model using gear life test figure as known conditions, by transforming the gear life distribution can predicting optional tooth number (other parameter constant of gear), and use simple; , a large amount of test figures is verified meanwhile, model to process small sample amount or the larger test figure of dispersiveness effective equally, be applicable to very much the feature of gear life test figure; The life information of gear combines with forecast model by the present invention, significantly can reduce the workload of Gear Experimentation under guarantee probability life prediction very reliably prerequisite, especially obvious to long-life high class gear test meaning.
Accompanying drawing explanation
Fig. 1 is the gear Probabilistic Life Prediction method flow diagram based on Minimal sequence statistics of an embodiment of the present invention;
Fig. 2 is the monodentate life-span probability density curve figure of an embodiment of the present invention, figure (a) is the transformation model result based on 10 gear life test figures and the comparison diagram based on the result of 250 Random censoring datas, figure (b) is the transformation model result based on 15 gear life test figures and the comparison diagram based on the result of 375 Random censoring datas, figure (c) is the transformation model result based on 20 gear life test figures and the comparison diagram based on the result of 500 Random censoring datas, figure (d) is the transformation model result based on 25 gear life test figures and the comparison diagram based on the result of 625 Random censoring datas,
Fig. 3 be an embodiment of the present invention be 4.1279 × 10 based on standard deviation
5the monodentate life-span distribution shifts model result of 25 gear life data and Random censoring data the result comparison diagram;
Fig. 4 be an embodiment of the present invention be 1.1097 × 10 based on standard deviation
6the monodentate life-span distribution shifts model result of 25 gear life data and Random censoring data the result comparison diagram;
Fig. 5 be an embodiment of the present invention be 3.7968 × 10 based on standard deviation
6the monodentate life-span distribution shifts model result of 25 gear life data and Random censoring data the result comparison diagram.
Embodiment
Below in conjunction with accompanying drawing, an embodiment of the present invention is described further.
In the embodiment of the present invention, based on the gear Probabilistic Life Prediction method of Minimal sequence statistics, method flow diagram as shown in Figure 1: comprise the following steps:
Step 1, torture test is carried out to multipair gear, obtain the testing data of life-span of gear;
Broken teeth is that gear breaks down the most serious main damage type, and gear teeth fracture can directly cause dynamic transfer system to lose efficacy, and the broken tooth failure of Aeronautical Gears also may cause the tragedy of fatal crass;
In the embodiment of the present invention, the object of torture test is the tooth root flexible life data of acquisition gear, verifies relevant model; Standard straight spur geer is tested, by GB/T14230 requirement design, modulus m=6mm, number of teeth n=25, facewidth b=16mm, gear material is 20CrMnTi; Testing equipment adopts poower flow closed-end gear testing machine, adopts mechanical close lever-loading mode, and each axle head adopts duplex bearing to support; The centre distance of test gear is 150mm, ratio of gear 1
:1, test adopts widths over teeth contact, and gear rotational speed is constant is 1460r/min;
In the embodiment of the present invention, tooth bending data of fatigue life is as shown in table 1:
Table 1
Step 2, matching is carried out to the testing data of life-span of gear, obtain the life-span estimation of distribution parameters of gear;
In the embodiment of the present invention, use the data in multiple distribution function his-and-hers watches 1 to carry out matching, find that two parameter Weibull distribution function obtains better to data fitting; For choosing of estimation of distribution parameters method, consider that Maximum-likelihood estimation needs to solve very complicated transcendental equation, and estimation has under small sample situation inclined; Linear minimum-variance estimation and least square are estimated all to need to use special mathematic(al) table (linear unbiased estimate coefficient table or Median rank table), apply loaded down with trivial details; Test figure in table 1 is all complete lifetime data, simple in order to calculate, and adopts moments estimation method to carry out parameter estimation to two parameter Weibull distribution;
In the embodiment of the present invention, the FATIGUE LIFE DISTRIBUTION of gear adopts two parameter Weibull distribution to represent, its cumulative distribution function is:
G
(1)(t)＝1-exp[-(t/θ)
β]，t＞0(5)
In formula: β represents form parameter, θ represents scale parameter, and t represents the life-span;
In the embodiment of the present invention, the gear life test figure in table 1 substituted in two parameter Weibull distribution function, the form parameter obtaining Weibull Function is estimated
scale parameter is estimated
Step 3, based on Minimal sequence statistics Modling model, by gear life-span estimation of distribution parameters substitute into model, obtain the single gear teeth life-span distribution;
The concept of Minimal sequence statistics is, if X
1, X
2..., X
nfor taking from the sample of overall X, by its sequential arrangement X by size
(1)≤ X
(2)≤ ...≤X
(n), claim X
(1)=min (X
1, X
2..., X
n) be Minimal sequence statistics; If the probability density function of X is f (x), cumulative distribution function is F (x), then the probability density function g of Minimal sequence statistics
(1)(x) be:
g
(1)(x)＝n[1-F(x)]
n-1f(x)(6)
For Probabilistic Life Prediction, can regard a gear as a cascade system, each tooth regards the part in system as; If tooth lost efficacy arbitrarily, make gear cannot complete the function transmitting power or motion, then this cascade system of gear lost efficacy; Consider gear cascade system, from failure mode, gear failure only can be lost efficacy by a tooth and cause, and also can be lost efficacy by more than one tooth causes simultaneously; Again because the inefficacy of gear is that the most weak tooth lost efficacy at first naturally, in gear, the inefficacy of the most weak tooth just means the inefficacy of whole gear; According to the definition of order statistic, the life-span distribution of gear is equal to the life-span distribution of the most weak tooth on probability meaning, the i.e. distribution of gear each tooth life-span Minimal sequence statistics; In general, for a collection of gear product, its material, process equipment, manufacture and Technology for Heating Processing etc. are all identical, therefore, can be regarded as I.i.d. random variables the life-span of each tooth under specified load course of a gear;
By above analysis, by formula (6) both sides integration, obtain the cumulative distribution function G of the life-span Minimal sequence statistics of n tooth
(1)(t), i.e. the gear life cumulative distribution function G of the specific number of teeth (namely testing the number of teeth of gear in step 1)
(1)t the expression formula of () is:
G
(1)(t)＝1-[1-F(t)]
n(1)
The life-span cumulative distribution function expression formula obtaining single tooth according to formula (1) is:
F(t)＝1-[1-G
(1)(t)]
1/n(2)
Formula (6) is substituted in formula (2), obtains:
F(t)＝1-exp[-(t/(θn
1/β))
β]，t＞0(7)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; N represents the number of teeth of gear;
In the embodiment of the present invention, the life-span distribution of to be the gear life distribution shifts of n by the number of teeth be monodentate; From the form of distribution function, the life-span distribution of monodentate is similarly two parameter Weibull distribution; Wherein, form parameter is still β, and scale parameter becomes θ n
1/ β, be the life-span distribution of monodentate by the life-span distribution shifts of gear, its distribution parameter becomes
Step 4, be the gear life distribution of the target number of teeth by the life-span distribution shifts of the single gear teeth, complete the Probabilistic Life Prediction of gear;
In the embodiment of the present invention, for the gear that the number of teeth is n ' (n ' ≠ n), other design parameter and the number of teeth are the gear of n identical (namely available F (t) is simultaneously as the single gear tooth life-span cumulative distribution function of two kinds of numbers of teeth), the gear life cumulative distribution function G ' of the target number of teeth
(1)t () expression formula is as follows:
G′
(1)(t)＝1-[1-F(t)]
n′(3)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; The number of teeth of n ' expression target gear;
By F (t)=1-[1-G
(1)(t)]
1/nin substitution formula (3), the distribution of the gear life of the specific number of teeth is converted to the gear life distribution of the target number of teeth:
G′
(1)(t)＝1-[1-G
(1)(t)]
n′/n(4)
In the embodiment of the present invention, formula (4) is the gear life distribution shifts model utilizing Minimal sequence statistics concept to set up, gear life distribution (two kinds of other design parameters of gear are identical) of the number of teeth is the gear life distribution shifts of n by it to be the number of teeth be n '; Therefore, as long as the life-span distribution of certain gear known just can obtain the gear life distribution that the number of teeth is different and other design parameter is identical; Model shows, along with the increase of number of teeth n ', and G '
(1)t () also increases, illustrate in cascade system, and the unit number of construction system is more, the rule that the failure probability of system is higher;
Formula (7) is substituting in formula (3), obtains following expression:
G′
(1)(t)＝1-exp[-(t/(θ(n/n′)
1/β))
β]，t＞0(8)
In the embodiment of the present invention, the gear life distribution of be the gear life distribution shifts of n by the number of teeth to be the number of teeth be n '; Life-span distribution after conversion is still two parameter Weibull distribution, and form parameter is still β, and scale parameter becomes θ (n/n ')
1/ β.
In the embodiment of the present invention, adopt Random censoring data estimation of distribution parameters methods combining monodentate testing data of life-span to transform the first time of model and verify, be i.e. verification model F (t)=1-[1-G
(1)(t)]
1/n:
For the gear mesh engaged transmission in test, because in operation process, driving gear is different with the motion torque on follower gear, and finding in process of the test, is all driving gear generation broken tooth failure, therefore, using driving gear as life search object.
If driving gear is made up of n tooth, can regard the life-span of each tooth as I.i.d. random variables, any one tooth lost efficacy and then tested stopping.Write down Life Cycle number of times, this cycle index is the fail data of 1 tooth and the censored data (life-span of n-1 tooth is greater than the cycle index of record) of n-1 tooth.If test obtains k " Life Cycle number of times " altogether, just obtain the fail data of k tooth and the censored data of k (n-1) individual tooth.
In the embodiment of the present invention, gear life test figure in table 1 is transformed into monodentate life-span Random censoring data, use Random censoring data disposal route to calculate monodentate life-span estimation of distribution parameters value, and compare with the monodentate life-span distribution results that the present invention is obtained by life-span distribution shifts model;
The parameter estimation contrast of different sample size test figure is as follows:
First front 10 gear life test figures in table 1 are considered, through changing 250 monodentate life-span Random censoring datas that can obtain being made up of 10 fail datas and 240 censored datas; Transformation model result based on 10 fail datas and the result based on 250 Random censoring datas are carried out distribution parameter contrast; In like manner consider 15,20 and 25 gear life test figures, monodentate life-span distribution parameter comparing result is as shown in table 2:
Table 2
As can be seen from Table 2, based on the gear life test figure of different sample size, the monodentate life-span estimation of distribution parameters that two kinds of methods obtain has good degree of closeness (as schemed in Fig. 2 shown in (a), figure (b), figure (c) and figure (d)), shows the rationality of transformation model of the present invention; About the difference between the monodentate life-span distribution parameter drawn by two kinds of distinct methods, can see sample size insensitive, reason is in the gear sample range selected by table 1, and the censored data amount of monodentate is all very large; In addition, because censored data amount is more much bigger than fail data, can not be very high according to the life-span distribution parameter precision that Censored Test data estimation goes out.
In the embodiment of the present invention, the impact of difference on life-span distribution shifts the model calculation of gear life test figure dispersiveness is as follows:
Use computing machine to generate two groups of random numbers under the condition of specific data average and standard deviation, sample size is all 25; Using the average of the average of table 1 middle gear testing data of life-span as two groups of random numbers, the standard deviation of data in table is reduced 3 times and amplification 3 times of standard deviations as two groups of random numbers; These two groups of random numbers are regarded as the lifetime data of gear, the above-mentioned two kinds of methods of same use calculate the monodentate life-span distribution parameter comparing result (a middle column data is the result of gear life test figure, takes from table 2) as shown in table 3 based on 25 gear life data:
Table 3
Can be found by table 3 and Fig. 3 ~ Fig. 5, when the sample size of gear life data is constant, obviously do not changed the relative result of two kinds of methods by the dispersiveness changing data, visible conversion model requires not harsh to the dispersiveness of test figure;
In sum, the concept that the present invention is based on Minimal sequence statistics establishes gear fatigue life distribution shifts model, and the gear life distribution shifts of the specific number of teeth can be the gear life distribution of optional tooth number (other design parameter is identical) by model; Under constant stress level, carry out gear torture test, obtain the tooth root flexible life data of 20CrMnTi carburized gears; Demonstrate model by parameter comparison and there is good life prediction ability, demonstrate model to the validity processing small sample amount or different dispersed test figure simultaneously.Therefore, life-span distribution shifts model is well suited for that gear torture test sample size is little, the dispersed large feature of lifetime data; The life information of gear is combined with transformation model, under the prerequisite ensureing life-span forecast of distribution precision, significantly can reduce the workload of reliability of gears test, especially obvious for meaning long-life high class gear test.
Claims (4)
1., based on a gear Probabilistic Life Prediction method for Minimal sequence statistics, it is characterized in that, comprise the following steps:
Step 1, torture test is carried out to multipair gear, obtain the testing data of life-span of gear;
Step 2, matching is carried out to the testing data of life-span of gear, obtain the life-span estimation of distribution parameters of gear;
Step 3, based on Minimal sequence statistics Modling model, by gear life-span estimation of distribution parameters substitute into model, obtain the single gear teeth life-span distribution;
Step 4, be the gear life distribution of the target number of teeth by the life-span distribution shifts of the single gear teeth, complete the Probabilistic Life Prediction of gear.
2. the gear Probabilistic Life Prediction method based on Minimal sequence statistics according to claim 1, it is characterized in that, the testing data of life-span to gear described in step 2 carries out matching, obtain the life-span estimation of distribution parameters of gear, be specially: adopt two parameter Weibull distribution function to carry out matching, the form parameter obtaining Weibull Function is estimated and scale parameter estimation.
3. the gear Probabilistic Life Prediction method based on Minimal sequence statistics according to claim 1, is characterized in that, described in step 3 based on Minimal sequence statistics Modling model, specific as follows:
The gear life cumulative distribution function G of the specific number of teeth
(1)t the expression formula of () is:
G
(1)(t)＝1-[1-F(t)]
n(1)
The life-span cumulative distribution function expression formula obtaining single tooth according to formula (1) is:
F(t)＝1-[1-G
(1)(t)]
1/n(2)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; N represents the number of teeth of particular gear.
4. the gear Probabilistic Life Prediction method based on Minimal sequence statistics according to claim 1, is characterized in that, described in step 4 by the life-span distribution shifts of the single gear teeth be the target number of teeth gear life distribution, specific as follows:
The gear life cumulative distribution function G' of the target number of teeth
(1)t () expression formula is as follows:
G'
(1)(t)＝1-[1-F(t)]
n'(3)
In formula, F (t) represents the life-span cumulative distribution function of single tooth; The number of teeth of n ' expression target gear;
By F (t)=1-[1-G
(1)(t)]
1/nin substitution formula (3), the distribution of the gear life of the specific number of teeth is converted to the gear life distribution of the target number of teeth:
G'
(1)(t)＝1-[1-G
(1)(t)]
n'/n(4)
In formula, G
(1)t () represents the gear life cumulative distribution function of the specific number of teeth, n represents the number of teeth of particular gear.
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Cited By (2)
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| CN111791237A (en) * | 2020-07-22 | 2020-10-20 | 东北大学 | Accumulated deformation reliability evaluation method for tool changing manipulator finger claw taking mechanism |
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| CN108564282B (en) * | 2018-04-13 | 2021-08-27 | 东北大学 | Truncation life data accepting and rejecting method for reliability evaluation |
| CN111791237A (en) * | 2020-07-22 | 2020-10-20 | 东北大学 | Accumulated deformation reliability evaluation method for tool changing manipulator finger claw taking mechanism |
| CN111791237B (en) * | 2020-07-22 | 2023-01-24 | 东北大学 | Accumulated deformation reliability evaluation method for tool changing mechanical finger grabbing mechanism |
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