CN110059447A - A kind of heavy tractor front rack analysis method for reliability - Google Patents

A kind of heavy tractor front rack analysis method for reliability Download PDF

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CN110059447A
CN110059447A CN201910421412.5A CN201910421412A CN110059447A CN 110059447 A CN110059447 A CN 110059447A CN 201910421412 A CN201910421412 A CN 201910421412A CN 110059447 A CN110059447 A CN 110059447A
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姜潮
刘凯
茅冬琳
龙湘云
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Abstract

The invention discloses a kind of heavy tractor front rack analysis method for reliability.This method uses finite element software ABAQUS to calculate the maximum stress value born under bracket applying working condition first.Then front rack loaded, the material properties Young's modulus of institute and Poisson's ratio are considered as stochastic variable, establish the system function function of characterization heavy tractor front rack use reliability, utilizes the preceding Fourth-order moment of dimensionality reduction integration method computing function function.The probability density function that system function function is finally solved using principle of maximum entropy, carries out the reliability of Integration Solving front rack.The present invention makes full use of the Efficient Solution efficiency of dimensionality reduction integration method, it is only necessary to call a small amount of finite element model that the accurate reliability of heavy tractor front rack can be obtained, this has Important Project meaning with economy to safety of guarantee front rack during military service.

Description

A kind of heavy tractor front rack analysis method for reliability
Technical field
The present invention relates to heavy tractor intelligence manufacture more particularly to a kind of heavy tractor front rack fail-safe analysis sides Method.
Background technique
Preceding tractor is connected with counterweight and counter-balanced carriage forward as the key components and parts in heavy tractor, connects backward Engine.Bear a variety of load such as bending, torsion simultaneously during heavy tractor is on active service, extreme operating condition is extremely complex, institute It is most important with the check to heavy tractor front rack intensity.In Practical Project, due to processing and manufacturing error and environment with Machine etc., exist in heavy tractor front rack structure it is relevant to load, material properties, geometrical property etc. it is a large amount of it is uncertain because Element, these uncertain factors will directly affect the reliability of front rack, seriously threaten the military service safety of heavy tractor and use The personal safety of personnel.Therefore, the heavy tractor front rack reliability consideration work tool of these uncertain factors is fully considered There is very important engineering significance.Currently, the analysis method for bracket mainly includes Monte-Carlo Simulation Method (MCS) and one Rank analysis method for reliability (FORM) etc..MCS method is a kind of method being widely used, but since it is needed based on a large amount of samples Original accurate estimation structural reliability, causes its computational efficiency lowly and with high costs.Although FORM method can reduce simulation Sample size, but it needs successive ignition and gradient to calculate.Heavy tractor Control system resolution is higher, work operating condition more pole End, nonlinear degree are higher.It is therefore proposed that a kind of analysis method for reliability for being adapted to heavy tractor front rack Efficient robust It is very necessary.
Summary of the invention
For above-mentioned low efficiency, time-consuming the problems such as, the invention proposes a kind of heavy tractor front rack reliabilities point Analysis method.So as to the reliability of Efficient Solution heavy tractor front rack, increase economic efficiency.
According to an aspect of the present invention, a kind of heavy tractor front rack reliability estimation method, the method are provided Include the following steps:
Step 1: establishing the power function Z of rack system according to the definition of heavy tractor front rack use reliability (X), expression formula are as follows:
Z (X)=σ-σm(X) (1)
In formula, σmIt (X) is the maximum stress born in rack system applying working condition, σ is the yield strength of bracket, and X is table Show that the Q in bracket model ties up independent Uncertain Stochastic vector, tractor bracket uncertainty variable is the Young's modulus of material E, Poisson's ratio V and applying working condition load p, i.e. X={ P, E, V }.
Step 2: according to the definition of moment of the orign, the preceding l rank moment of the orign expression formula of rack system power function are as follows:
E is mathematic expectaion operator, f in formulax(X) PDF (probability density function) of X is indicated.
Step 3: converting stochastic model to the subsystem of single random parameter by introducing one-dimensional dimensionality reduction integration method Higher Dimensional Integration is converted to low-dimensional integral to solve by system, and quadravalence moment of the orign has following expression before rack system power function:
μ in formulajIndicate stochastic variable XjMean value, Q indicate stochastic variable number.Above formula is unfolded by binomial theorem It can be expressed as follows again:
Definition:
It is solved by following recurrence formula
Wherein, one-dimensional Integral ProblemIt is solved using Gauss integration:
In formula,For XjPDF.
Step 4: being the Gauss product for enabling dimensionality reduction integration method suitable for any continuously distributed, while obtaining succinct Divide expression formula, stochastic variable need to be converted to N (0,1/2) distribution:
Φ in formula-1[] indicates the inverse function of standardized normal distribution cumulative distribution function,Indicate XjIterated integral Cloth function.αjPDF expression formula it is as follows, wherein e be natural constant:
Step 5: after uncertain variable conversion, one-dimensional integralIt is expressed as follows:
Had using Gauss-Hermite quadrature formula:
W in formulaqAnd hqGauss integration weight and Gauss integration point are respectively indicated, r indicates Gauss integration points, l=4.
Step 6: one-dimensional integral formula is transformed into former space from normed space:
X in formulajqIt can be represented by the formula:
In formulaRepresent XjCumulative distribution function inverse function.
Step 7: then each stochastic variable is both needed in each Gauss integration point using 3 Gauss integration point calculating formulas (12) It calls expression of first degree (13), in addition when stochastic variable is mean value, it is only necessary to call 3Q+1 formula (12) that can solve.
Step 8: value of the stochastic variable solved according to the 7th step in different Gauss integration points in former space, utilizes ABAQUS carries out finite element analysis to bracket, acquires its maximum stress value σmi, i=1,2 ..., 3Q+1 calculate 3Q+1 support The maximum stress that frame is born.
Step 9: the σ calculated by the 8th stepmiZ is calculated with rack system power functioni, i=1,2 ..., 3Q+1, i.e., One-dimensional integral in three stepsUsing in second stepAnd mZlCalculation formula obtain rack system power function preceding 4 rank it is former Point square.
Step 10: solving central moment k by moment of the orign.
Step 11: being the canonical statistics Y that mean value is 0, variance is 1, transformation for mula by power function value transform Are as follows:
μ in formulaZAnd σZThe mean value and standard deviation of rack system power function are respectively indicated, formula is as follows:
Step 12: calculating the preceding quadravalence moment of the orign of canonical statistics Y, calculation formula are as follows:
In formula, i indicates specific order, value i=0,1,2,3,4.
Step 13: the comentropy for defining stochastic variable Y is H [fY(y)], comentropy expression are as follows:
In formula, fY(y) indicate that the probability density function of stochastic variable Y, c > 0 are constant；Before canonical statistics Y Quadravalence moment of the orign constructs Lagrangian, Lagrangian expression formula as constraint condition, according to principle of maximum entropy are as follows:
Wherein, λiFor the corresponding Lagrange coefficient of the i-th rank, λi=(λ0,λ1,λ2,λ3,λ4)；
Using LagrangianL to probability density function fY(y) partial derivative is sought, so that partial derivative is 0 at extreme point, Derivation process expression are as follows:
Step 14: being calculated using the following equation the probability density letter for obtaining stochastic variable Y according to previous step derivation process Number fY(y), expression formula specifically:
In formula, a0,a1,...,aiFor undetermined parameter,
Step 15: according to the probability density function f of stochastic variable YY(y), the reliability of bracket is calculatedExpression formula Are as follows:
Wherein, σZIndicate the standard deviation of rack system power function, μZIndicate the mean value of the power function of rack system
Advantage and beneficial effect of the invention is the following:
1. the method for the present invention makes full use of the characteristics of lowering dimension decomposition method and Gauss integration, the statistical moment of subsystem is derived It is assembled into the statistical moment of original system, avoids and directly calculates Higher Dimensional Integration problem, has been had both in high precision and efficient dual excellent Point.
2. the method for the present invention is guaranteeing compared to now widely used Monte Carlo Analogue Method in fail-safe analysis As a result under the premise of accuracy, the demand to sample size can be greatly lowered, it is only necessary to which few sample size just obtains Higher computational accuracy.
3. the method for the present invention, compared to traditional single order fail-safe analysis method in fail-safe analysis, the information utilized is more It is more, the first order and second order moments of stochastic variable are not only utilized, three ranks and the fourth central of stochastic variable is also further utilized The statistical information of square, although sample size needed for the two is less, comparatively the method for the present invention can obtain more Accurate calculated result.
Detailed description of the invention
The flow diagram of Fig. 1 the method for the present invention
Fig. 2 bracket finite element model
Fig. 3 bracket failure probability is with safety coefficient variation tendency
Specific embodiment
The present invention is based on lowering dimension decomposition method, Gauss integration is theoretical and principle of maximum entropy, before proposing a kind of heavy tractor Bracket analysis method for reliability.The present invention is with traditional reliability calculation method difference, and Higher Dimensional Integration is transformed into low-dimensional Space integral effectively reduces the difficulty in computation of higher-dimension Gauss integration.And it can effectively calculate the system containing probabilistic system Moment problem is counted, high-precision and efficient two-fold advantage are had both.
The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:
Step 1: establishing the power function Z of rack system according to the definition of heavy tractor front rack use reliability (X), expression formula are as follows:
Z (X)=σ-σm(X)
In formula, σmIt (X) is the maximum stress born in rack system applying working condition, σ=310Mpa is that the surrender of bracket is strong Degree.
Step 2: obtaining the distribution such as table 1 of stochastic variable by the statistical information of the medium and heavy tractor front rack of Practical Project It is shown:
1 heavy tractor front rack uncertain parameters distribution table of table
The then preceding 4 rank moment of the orign expression formula of rack system power function Z (X) are as follows:
E is mathematic expectaion operator, f in formulax(X) PDF of X is indicated.
Step 3: quadravalence moment of the orign has following expression before power function by introducing one-dimensional dimensionality reduction integration method:
μ in formulajIndicate stochastic variable XjMean value, Z (μ1,...,μj-1,Xj,μj+1,...,μQ) indicate stochastic variable Xj's It rings
It answers, has following expression by binomial theorem expansion:
Definition
It is solved by following recurrence formula
Wherein, one-dimensional Integral ProblemIt is solved using Gauss integration:
In formula,For XjPDF.
Step 4: being the Gauss product for enabling dimensionality reduction integration method suitable for any continuously distributed, while obtaining succinct Divide expression formula, stochastic variable need to be converted to N (0,1/2) distribution:
Φ in formula-1[] indicates the inverse function of standardized normal distribution cumulative distribution function,Indicate XjIterated integral Cloth function.αjPDF expression formula it is as follows:
Step 5: after uncertain variable conversion, one-dimensional integralIt is expressed as follows:
Had using Gauss-Hermite quadrature formula:
W in formulaqAnd hqGauss integration weight and Gauss integration point are respectively indicated, r indicates Gauss integration points, l=4.
Step 6: one-dimensional integral formula is transformed into former space from normed space:
X in formulajqIt can be represented by the formula:
In formulaRepresent XjCumulative distribution function inverse function.
Step 7: calculating by MATLAB software using three Gauss integration points, stochastic variable is obtained in each Gauss product Value of the branch in former space:
Step 8: value of 3 stochastic variables acquired according to the 7th step in 3 different Gauss integration point originals spaces, and with When machine variable is mean value, it is as follows to amount to 3*3+1=10 group data:
Finite element analysis is carried out to bracket using ABAQUS, obtains the maximum stress value σ of bracketmi, as a result following (unit MPa):
The maximum stress of rack system power function is (units MPa):
Power function value Z when stochastic variable takes mean value1=102.9Mpa.
Step 9: the one-dimensional Integration Solving multidimensional integral calculated by the 8th step, using in third stepAnd mZlCalculating it is public Formula obtains the preceding 4 rank moment of the orign m of rack system power functionZ1, mZ2, mZ3, mZ4:
μ in formulajIndicate stochastic variable XjMean value, Z (μ1,...,μj-1,Xj,μj+1,...,μQ) indicate stochastic variable Xj's Response, Q=3, l=1,2,3,4；
mZ1=103.45
mZ2=1.1116e+04
mZ3=1.2362e+06
mZ4=1.4189e+08
Step 10: according to 4 rank moment of the orign m before the rack system power function of the 9th step calculating acquisitionZ1, mZ2, mZ3, mZ4, The preceding 4 rank central moment k of computing function functionZ1, kZ2, kZ3, kZ4.The relational expression of central moment and moment of the orign is as follows:
It calculates: kZ1=103.45, kZ2=413.8269, kZ3=662.4742,
Step 11: being the canonical statistics Y that mean value is 0, variance is 1, transformation for mula by power function value transform Are as follows:
μ in formulaZAnd σZThe mean value and standard deviation of rack system power function are respectively indicated,
μZ=kZ1=103.45
σZ=(kZ2)1/2=20.3427
Step 12: calculating the preceding quadravalence moment of the orign of canonical statistics Y, calculation formula are as follows:
In formula, i indicates specific order, value i=0,1,2,3,4, it calculates:
mY0=1；mY1=0；mY2=1；mY3=0.0787；mY4=3.0021.
Step 13: the comentropy for defining stochastic variable Y is H [fY(y)], comentropy expression are as follows:
In formula, fY(y) indicate that the probability density function of stochastic variable Y, c > 0 are constant.
Using the preceding quadravalence moment of the orign of canonical statistics Y as constraint condition, Lagrange is constructed according to principle of maximum entropy Function, Lagrangian expression formula are as follows:
Wherein, λiFor the corresponding Lagrange coefficient of the i-th rank, λi=(λ0,λ1,λ2,λ3,λ4)；
Using LagrangianL to probability density function fY(y) partial derivative is sought, so that partial derivative is 0 at extreme point, Derivation process expression are as follows:
Step 14: being calculated using the following equation the probability density letter for obtaining stochastic variable Y according to previous step derivation process Number fY(y), expression formula specifically:
In formula, a0,a1,...,aiFor undetermined parameter,
Step 15: according to the probability density function f of stochastic variable YY(y), the reliability of bracket is calculatedExpression formula Are as follows:
Wherein, σZIndicate the standard deviation of rack system power function, μZIndicate the mean value of the power function of rack system.Meter Calculation obtains bracket reliability
Step 16: the reliability and failure probability for calculating bracket in the case where taking different safety coefficients that repeat the above steps, knot Fruit is as shown in Fig. 3.
The present invention overcomes computational efficiencies in the prior art lowly with technical problem with high costs, realizes Efficient Solution The reliability of heavy tractor front rack, increases economic efficiency.

Claims (5)

1. a kind of heavy tractor front rack analysis method for reliability, which is characterized in that described method includes following steps: first Step: according to the definition of heavy tractor front rack use reliability, the power function Z (X) of rack system, expression formula are established are as follows:
Z (X)=σ-σm(X) (1)
In formula, σmIt (X) is the maximum stress born in rack system applying working condition, σ is the yield strength of bracket, and X is to indicate bracket Q in model ties up independent Uncertain Stochastic vector, and tractor bracket uncertainty variable is the Young's modulus E of material, Poisson Than V and applying working condition load p, i.e. X={ P, E, V }；
Step 2: according to the definition of moment of the orign, the preceding l rank moment of the orign expression formula of rack system power function are as follows:
E is mathematic expectaion operator, f in formulax(X) PDF (probability density function) of X is indicated；
Step 3: stochastic model is converted to the subsystem of single random parameter by introducing one-dimensional dimensionality reduction integration method, it will Higher Dimensional Integration is converted to low-dimensional integral to solve, and quadravalence moment of the orign has following expression before rack system power function:
μ in formulajIndicate stochastic variable XjMean value, Q indicate stochastic variable number；
Step 4: being to enable dimensionality reduction integration method suitable for any continuously distributed, while obtaining succinct Gauss integration table Up to formula, stochastic variable need to be converted to N (0,1/2) distribution:
Φ in formula-1[] indicates the inverse function of standardized normal distribution cumulative distribution function,Indicate XjCumulative distribution letter Number.αjPDF expression formula it is as follows, wherein e be natural constant:
Step 5: after uncertain variable conversion, one-dimensional integralIt is expressed as follows:
Had using Gauss-Hermite quadrature formula:
W in formulaqAnd hqGauss integration weight and Gauss integration point are respectively indicated, r indicates Gauss integration points, l=4.
Step 6: one-dimensional integral formula is transformed into former space from normed space:
X in formulajqIt can be represented by the formula:
In formulaRepresent XjCumulative distribution function inverse function；
Step 7: then each stochastic variable is both needed to call in each Gauss integration point using 3 Gauss integration point calculating formulas (12) Expression of first degree (13), in addition when stochastic variable is mean value, it is only necessary to call 3Q+1 formula (12) that can solve.
Step 8: utilizing ABAQUS pairs in the value in former space in different Gauss integration points according to the stochastic variable of the 7th step solution Bracket carries out finite element analysis, acquires its maximum stress value σmi, i=1,2 ..., 3Q+1 calculate what 3Q+1 bracket was born Maximum stress.
Step 9: the σ calculated by the 8th stepmiZ is calculated with rack system power functioni, i=1,2 ..., 3Q+1, i.e. third step In one-dimensional integralUsing in second stepAnd mZlCalculation formula obtain the preceding 4 rank origin of rack system power function Square.
Step 10: solving central moment by moment of the orign.
Step 11: being the canonical statistics Y that mean value is 0, variance is 1, transformation for mula by power function value transform are as follows:
μ in formulaZAnd σZThe mean value and standard deviation of rack system power function are respectively indicated, formula is as follows:
Step 12: calculating the preceding quadravalence moment of the orign of canonical statistics Y, calculation formula are as follows:
In formula, i indicates specific order, value i=0,1,2,3,4；
Step 13: the comentropy for defining stochastic variable Y is H [fY(y)], comentropy expression are as follows:
In formula, fY(y) indicate that the probability density function of stochastic variable Y, c > 0 are constant；The preceding quadravalence of canonical statistics Y is former Point square constructs Lagrangian, Lagrangian expression formula as constraint condition, according to principle of maximum entropy are as follows:
Wherein, λiFor the corresponding Lagrange coefficient of the i-th rank, λi=(λ0,λ1,λ2,λ3,λ4)；
Using LagrangianL to probability density function fY(y) partial derivative is sought, so that partial derivative is 0 at extreme point, derivation Process expression are as follows:
Step 14: being calculated using the following equation the probability density function f for obtaining stochastic variable Y according to previous step derivation processY (y), expression formula specifically:
In formula, a0,a1,...,aiFor undetermined parameter,
Step 15: according to the probability density function f of stochastic variable YY(y), the reliability of bracket is calculatedExpression formula are as follows:
Wherein, σZIndicate the standard deviation of rack system power function, μZIndicate the mean value of the power function of rack system.
2. the method as described in claim 1, which is characterized in that the third step further include: pass through binomial theorem expansion the Formula in three steps can be expressed as follows again:
Definition:
It is solved by following recurrence formula
Wherein, one-dimensional Integral ProblemIt is solved using Gauss integration:
In formula,For XjPDF.
3. the method as described in claim 1, which is characterized in that the 7th step further include: by MATLAB software, using three A Gauss integration point calculates, and obtains value of the stochastic variable in each Gauss integration point in former space.
4. the method as described in claim 1, which is characterized in that further include existing step 16: repeating the above steps and calculating bracket Take the reliability and failure probability under different safety coefficients.
5. the method as described in claim 1, which is characterized in that in the second step, before the medium and heavy tractor of Practical Project The statistical information of bracket obtains the distribution of stochastic variable.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113420447A (en) * 2021-06-25 2021-09-21 中国人民解放军空军工程大学 Efficient solution algorithm for importance measure of system safety life

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Publication number Priority date Publication date Assignee Title
US20030061152A1 (en) * 2001-09-26 2003-03-27 De Rabi S. System and method for determining Value-at-Risk using FORM/SORM
CN106844976A (en) * 2017-01-23 2017-06-13 沈阳航空航天大学 It is a kind of based on the STRUCTURES WITH RANDOM PARAMETERS reliability estimation method with point-type algorithm
CN107443370A (en) * 2017-07-07 2017-12-08 浙江大学 A kind of industrial robot reliability calculation method based on quadravalence moments estimation

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20030061152A1 (en) * 2001-09-26 2003-03-27 De Rabi S. System and method for determining Value-at-Risk using FORM/SORM
CN106844976A (en) * 2017-01-23 2017-06-13 沈阳航空航天大学 It is a kind of based on the STRUCTURES WITH RANDOM PARAMETERS reliability estimation method with point-type algorithm
CN107443370A (en) * 2017-07-07 2017-12-08 浙江大学 A kind of industrial robot reliability calculation method based on quadravalence moments estimation

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113420447A (en) * 2021-06-25 2021-09-21 中国人民解放军空军工程大学 Efficient solution algorithm for importance measure of system safety life

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