CN102446239B - Gear transmission multidisciplinary reliability analysis method considering cognition and random uncertainty - Google Patents

Abstract

A gear transmission multidisciplinary reliability analysis method considering cognition and random uncertainty in an actual engineering design process belongs to the technical field of design optimization and reliability design of mechanical products and includes quantizing the cognition and the random uncertainty of gear modules, tooth number, tooth surface width, distance between bearings and gear diameter caused by the factors of processing, assembly, material characteristics, applied load, external environment and the like, establishing multidisciplinary reliability integrated evaluation index, integrating of the reliability analysis method and a multidisciplinary design optimization strategy, decoupling of a multidisciplinary reliability analysis process, and being based on a functional measurement method. The gear transmission multidisciplinary reliability analysis method considering the cognition and the random uncertainty provides a comprehensive quantitative method, establishes the multidisciplinary reliability integrated evaluation index which is more conformable to engineering practice, integrates a traditional reliability analysis method and the multidisciplinary design optimization strategy, and provides a new thought for reliability analysis of gear transmission products.

Description

Consider cognitive and the multidisciplinary analysis method for reliability of probabilistic gear drive at random

Technical field

The invention belongs to the reliability design technology field of complex product, particularly the multidisciplinary analysis method for reliability of a kind of multi-source condition of uncertainty lower gear transmission.

Background technology

Gear train assembly is such as common power transmission form in the Large Complex Equipment such as aerospace flight vehicle, aeromotor, boats and ships, giant mechanical and electrical equipment, weaponry and vehicle.Gear-driven performance especially reliability has directly determined the quality of Large Complex Equipment.Therefore, people have carried out correlative study from aspects such as product design, product manufacturing and product maintenances to gear-driven reliability.According to parties concerned's statistics, product design can reach 70%～80% to the contribution rate of product quality, and visible design has determined the inherent reliability of product, has given the intrinsic propesties of product " congenital quality ".Therefore, the traditional reliability design of gear has obtained extensive concern and further investigation, and fail-safe analysis is the part of core the most in the reliability design, has directly determined the success or failure of reliability design.Therefore, gear-driven Reliability Analysis Research is significant for the reliability management level that improves its reliability design level and Large-Scale Equipment.

At present, only consider uncertainty at random in the gear-driven reliability design process, it is ripe that its correlative study has been tending towards.Yet, in the Gear Transmission Design process of reality, not only had a large amount of uncertainties at random owing to make the influence of factors such as processing, product assembling, material behavior, imposed load and external environment, and have cognitive uncertainty to a certain degree.Cognitive uncertainty can't not make up probability density function owing to not possessing sufficient data and information, can't adopt traditional probability theory it is expressed and to quantize, if will cause the unreliable of design result to its hypothesis artificially, present this method has been subjected to unprecedented query.Therefore, how scientifically to select rational quantization method that the uncertainty in the Gear Transmission Design process is quantized, carries out the multidisciplinary fail-safe analysis that takes into account counting yield and design cost at different uncertainties and have important practical value and theory directive significance.

Summary of the invention

The present invention be to provide a kind of consideration cognitive with the multidisciplinary analysis method for reliability of probabilistic gear drive at random, the gear transmission structure sketch is shown in figure (1), this method takes into full account produce in the Gear Transmission Design process uncertain with cognition at random, single subject analysis method for reliability and parallel subspace design optimization strategy are carried out effective integration, provided gear-driven multidisciplinary analysis method for reliability, for the reliability that scientifically guarantees Gear Transmission Design provides guarantee.

The inventive method comprises integrated, the multidisciplinary fail-safe analysis modeling based on the function measure of multidisciplinary reliability comprehensive evaluation index, analysis method for reliability and parallel subspace optimisation strategy cognitive and probabilistic quantification, broad sense at random, the multidisciplinary fail-safe analysis process decoupling zero based on serializing thought, multidisciplinary probabilistic reliability analysis and multidisciplinary convex model extreme value analysis.Below introduce content of the present invention in detail:

One, cognitive and probabilistic quantification at random in the Gear Transmission Design process

At the stochastic variable (x with adequate data and information _r) adopt probability theory that it is quantized, choose probability density function p _i(x _r), provide average

And variances sigma _iIncomplete and the uncertain variable (x of the incomplete cognition of information for data _e) adopt convex model to be described and quantize, set average

Provide eigenmatrix W _jAnd parameter uncertainty degree ε _jTherefore, the quantification expression formula of stochastic variable is: x _r～[p _i(x _r), the quantization means of cognitive uncertain variable is: $x_{\mathrm{je}}~\left\{x_{\mathrm{je}}\right|{(x_{\mathrm{je}}-{\stackrel{\‾}{x}}_{\mathrm{je}})}^{T}W(x_{\mathrm{je}}-{\stackrel{\‾}{x}}_{\mathrm{je}})\≤{{\mathrm{\ϵ}}_j}^2\}.$

For ease of calculating, adopt the Rosenblatt conversion with skewed distribution and the stochastic variable x that is mutually related to uncertain variable at random _r=[x _R1x _R2X _Rm] be converted to separate standardized normal distribution form u=[u ₁u ₂U _i], conversion formula is μ _iAnd σ _iBe respectively average and the standard deviation of stochastic variable; For the uncertain variable of cognition, at first to eigenmatrix W _jDecompose,

Obtain eigenwert diagonal matrix Λ _jReach the matrix Φ that is formed by the orthogonalization proper vector _jIntroduce vector $v^T=\left[\begin{array}{cccc}{v}_1^T& v_2^T& \·\·\·& v_n^T\end{array}\right],$ Order $v=(1/{\mathrm{\θ}}_j)\sqrt{{\mathrm{\Λ}}_j}{\mathrm{\Φ}}_j^T(y_j-{\stackrel{\‾}{y}}_j),$ Then former many ellipsoid set Θ gather the many ellipsoids of the unit of being converted into, namely $\mathrm{\Θ}=\left\{v\right|v_j^T{v}_j\≤1(j=\mathrm{1,2},\·\·\·,n)\}.$

Two, make up the multidisciplinary reliability comprehensive evaluation index of broad sense

Use for reference geometry implication of probabilistic reliability index, it is expanded to comprise cognitive and probabilistic gear-driven multidisciplinary fail-safe analysis evaluation index at random simultaneously.Traditional probabilistic reliability evaluation index is The two tolerance of reliability after expansion index is respectively: ${\mathrm{\β}}_L=\underset{U}{\min }\sqrt{u{u}^T|\underset{V}{\min }\left(g(u,v)\right)}$ With ${\mathrm{\β}}_U=\underset{U}{\min }\sqrt{u{u}^T|\underset{V}{\max }\left(g(u,v)\right)},$ β wherein _UAnd β _LThe bound of representing reliability respectively, its difference DELTA β=β _U-β _LQuantitative response the influence degree of cognitive uncertainty to the gear drive reliability.

Three, based on the multidisciplinary fail-safe analysis modeling of function measure

For solving stability and the counting yield problem based on the iterative process of RELIABILITY INDEX method, the present invention adopts the multidisciplinary reliability analysis model of setting up gear train assembly based on the function measure.Minimum value based on limit state function is reliable principle greater than zero, and namely limit state function equals zero and is its inefficacy critical condition, is equivalent to RELIABILITY INDEX

Just satisfy the reliability index requirement, there is following relation in the numerical value of RELIABILITY INDEX and limit state function

Therefore, can be configured to based on the multidisciplinary reliability analysis model of the gear drive of function measure: $\mathrm{\α}=\underset{(U,V)}{\min }{g}_i(u,v,Y_{\mathrm{ij}})\≥0|\sqrt{u^{T}u}=\mathrm{\β},$ $v_j^T{v}_j\≤1(i=\mathrm{1,2},\·\·\·,11),$ I is the number of gear-driven limit state function, Y _IjBe interdisciplinary couple state variable.

Four, the multidisciplinary analysis method for reliability of the gear drive of serializing

For take into full account cognitive and at random probabilistic gear-driven multidisciplinary fail-safe analysis be typical three layers of nested loop optimization analytic process, as scheming shown in (2).Therefore, the multi-source uncertainty carried out comprehensive quantification in early stage, make up on the basis based on the multidisciplinary reliability analysis model of function measure of reliability evaluation index under the multi-source condition of uncertainty and structure, adopt the decoupling zero theory that its analysis process is carried out decoupling zero, to improve counting yield.The flow process of the gear-driven multidisciplinary analysis method for reliability of serializing of the present invention is shown in figure (3), and is specific as follows:

1, initialization setting

Provide the initial value of uncertain variable and cognitive uncertain variable at random,

2, probabilistic reliability analysis

(1) solidifies cognitive uncertainty, v=[v ⁰];

(2) integrated analysis method for reliability and parallel subspace optimisation strategy are carried out multidisciplinary analysis;

Gear drive comprises toothed wheel system and roller system, and state variable is y ₁And y ₂, with its simultaneous, as the formula (1).

$\left\{\begin{array}{c}{y}_1=y_1(u,v,y_{21})\\ y_2=y_2(u,v,y_{12})\end{array}\right.---\left(1\right)$

According to the type that will handle problems, can select different alternative manners to find the solution top system of equations, for example Gauss---Saden that process of iteration, Newton iteration method, Bao Weier polygometry etc.The value of the state variable that solves and this suboptimization loop iteration initial value substitution limit state function of design variable at random, can obtain all limite function g in the Gear Drive Optimization Design ₁～g ₁₁Value.The value of the state variable that is obtained by multidisciplinary systematic analysis can also directly be next step system sensitivity Analysis Service.

(3) integrated analysis method for reliability and parallel subspace optimisation strategy are carried out the system sensitivity analysis;

Global sensitivity equation (GSE) is adopted in the system sensitivity analysis in this method, and couple state variable y is about the individual design variable x at random of i _IrTotal derivative information be gradient information dy/dx _Ir, can try to achieve by following GSE equation.According to finding the solution the system of equations shown in the formula (1), carry out multidisciplinary analysis, obtain couple state function y ₁And y ₂Value, gear-driven global sensitivity equation is shown in the formula (2).

$\left[\begin{array}{cc}I& -\frac{\&PartialD;y_1}{\&PartialD;y_2}\\ -\frac{\&PartialD;y_2}{\&PartialD;y_1}& \mathrm{<figure-callout\; id="1"\; label="I\; dy"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">I</figure-callout>}& \\ \end{array},\right]\left[\begin{array}{c}\frac{{\mathrm{dy}}_{}}{}\end{array}\right]\left[\begin{array}{c}\frac{{}_1}{{\mathrm{dx}}_i}\\ \frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}\end{array}\right]=\left[\begin{array}{c}\frac{\&PartialD;y_1}{{\mathrm{dx}}_i}\\ \frac{\&PartialD;y_2}{{\mathrm{dx}}_i}\end{array}\right]---\left(2\right)$

Obtain dy/dx _IrAfter, the probability constraints condition in the gear drive fail-safe analysis, its limit state function g is about the individual design variable x at random of i _IrGradient information dg/dx _Ir, can be obtained by formula (3):

$\frac{\mathrm{dg}}{{\mathrm{dx}}_{\mathrm{ir}}}=\frac{\&PartialD;g}{\&PartialD;x_{\mathrm{ir}}}+\frac{\&PartialD;g}{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1}\frac{{\mathrm{<figure-callout\; id="1"\; label="dy"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">dy</figure-callout>}}_1}{{\mathrm{dx}}_{\mathrm{ir}}}+\frac{\&PartialD;g}{\&PartialD;y_2}\frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_{\mathrm{ir}}}---\left(3\right)$

Equally, limit state function g is about all gradient information dy/dx of design variable x at random _IrAll can be obtained by formula (3).Use below

Representative in luv space (x space) limit state function about the gradient information of design variable at random.If stochastic variable is obeyed standardized normal distribution, stochastic variable at the transformational relation of normed space and luv space as the formula (4):

$u=\frac{{\mathrm{\&mu;}}_x-x}{{\mathrm{\&sigma;}}_x}---\left(4\right)$

Therefore, the gradient information of limit state function is in normed space (u space):

${\&dtri;}_{u}g\left(u\right)=\frac{\mathrm{dg}}{\mathrm{dx}}\frac{\&PartialD;x}{\&PartialD;u}={\&dtri;}_{x}g\left(x\right)\&CenterDot;{\mathrm{\&sigma;}}_x---\left(5\right)$

(4) (Modified Advanced Mean Value MAMV) upgrades stochastic variable u based on improved Advanced mean value method;

(a) calculate u according to formula (6) _kWith

Between angle, if γ _k≤ ε jumps to step (6), otherwise jumps to step (b), and ε is a very little angle, for example 0.01 °.

${\mathrm{\&gamma;}}_k={\cos }^{-1}\frac{u_k\&CenterDot;{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|u_k\left|\right|\&CenterDot;\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|}---\left(6\right)$

(b) if g is (u _k)＞g (u _K-1), $u_{k+1}={\mathrm{\&beta;}}_t\frac{{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|},$ Otherwise

$u_{k+1}=\frac{{\mathrm{\&beta;}}_t}{\sin \left({\mathrm{\&gamma;}}_k\right)}(\sin ({\mathrm{\&gamma;}}_k-{\mathrm{\&delta;}}_k)\frac{u_k}{\left|\right|u_k\left|\right|}+\sin {\mathrm{\&delta;}}_k\frac{{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|})---\left(7\right)$

δ in the formula _kCan obtain by finding the solution an one dimension max problem, as the formula (8).

$\max g\left(u_{k+1}\right)=g\left\{\frac{{\mathrm{\&beta;}}_t}{\sin \left({\mathrm{\&gamma;}}_k\right)}(\sin ({\mathrm{\&gamma;}}_k-{\mathrm{\&delta;}}_k)\frac{u_k}{\left|\right|u_k\left|\right|}+\sin {\mathrm{\&delta;}}_k\frac{{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|})\right\}---\left(8\right)$

(5) with u _K+1Convert corresponding variable x in luv space (x space) to _K+1, change step (2);

(6) calculate g (u _k), obtain most probable failpoint (MPP), and order

Finish.

For the represented nothing constraint one dimension max problem of formula (8), can optimize algorithm by nothings such as Fibonacci method, overall Newton method, parabolic method constraint one dimension and try to achieve.

3, based on the extreme value analysis of convex model

(1) solidify uncertainty at random, $u_r=\left[u_r^0\right]=\left[u_r^{\mathrm{MPP}}\right];$

(2) adopt the method for step 2 (2) that the system of equations that only comprises cognitive uncertain v and state variable y is carried out multidisciplinary analysis, obtaining state variable is y ₁And y ₂Value;

(3) based on Lagrange multiplier constrained optimization problem is converted into unconfined optimization problem;

(4) for the majorized function of new structure, respectively to the uncertain v of cognition and λ differentiate, make

Obtain corresponding some v of limit state function extreme value _MinAnd v _Max

(5) with v _MinAnd v _MaxBring gear-driven limit state function into, ask its span

4, reliability decision

If

Illustrate that then this limit state function is discontented with sufficient reliability requirement, otherwise satisfy reliability requirement,

Quantitative response the influence degree of cognitive uncertainty to this limit state function.

5, finish.

The inventive method is used the reliability consideration of Large Complex Equipment middle gear kinematic trains such as aerospace flight vehicle, aeromotor, boats and ships, giant mechanical and electrical equipment, weaponry and vehicle.The inventive method has taken into full account gear drive in design process owing to be subjected to make processing, the product assembling, material behavior, imposed load, it is uncertain and cognitive uncertain at random in a large number that factor such as external environment and wearing and tearing causes, and proposed based on the uncertain comprehensive quantification method of the multi-source of probability theory and Convex Set Theory, set up the multidisciplinary reliability evaluation index of gear drive of broad sense, set up gear-driven multidisciplinary reliability analysis model based on the function measure, adopt serializing thought to carry out decoupling zero to having typical three layers of nested circulation, realized objective to the gear train assembly reliability, accurately estimate and succinct calculating, be more convenient for the engineering staff to the analysis of gear train assembly fiduciary level, very big actual application value is arranged.This technology can provide rational foundation for design, the maintenance of gear train assembly, and the revision that also can be corresponding standard simultaneously provides technical support.

Description of drawings

Fig. 1 gear transmission structure sketch

Fig. 2 comprises cognitive and the multidisciplinary fail-safe analysis flow process of probabilistic gear drive at random

The multidisciplinary fail-safe analysis flow process of the gear drive of Fig. 3 serializing

Embodiment

The present invention is further illustrated below in conjunction with the gear drive example.

The present invention considers cognitive and the multidisciplinary analysis method for reliability of probabilistic gear drive at random, at first clearly the ingredient of this gear drive fail-safe analysis comprises two subsystems of roller system and toothed wheel system, 11 limit state function, two couple state variablees and 7 design parameters.

One, at analyzing with gear drive shown in Figure 1, determine the degree of uncertainty of design parameter, fully take into account in actual design process owing to make the influence of factors such as processing, product assembling, material behavior, imposed load and external environment, the present invention considers face width of tooth (x _R1), module (x _R2), the pinion wheel number of teeth (x _R3), gearwheel bearing spacing (x _R4), Large Gear Shaft During diameter (x _R5), pinion shaft diameter (x _E1) and pinion bearing spacing (x _E2) uncertainty of 7 design parameters, set x according to design experiences and actual engineering design situation _R1～x _R5Obey standardized normal distribution, x _E1～x _E2Because data are insufficient, can't set up probability density function and be considered to cognitive uncertain.Known each uncertain information is as follows: μ (x _r)=[3.00.7207.85.0], σ (x _r)=[0.0250.0250.0250.0250.025], ${\stackrel{\&OverBar;}{x}}_e=\left[\begin{array}{cc}{x}_{e1}& x_{e2}\end{array}\right]=\left[\begin{array}{cc}7.8& 3.4\end{array}\right],$ $x_e~\left\{x_e\right|{(x_e-{\stackrel{\&OverBar;}{x}}_e)}^{T}W(x_e-{\stackrel{\&OverBar;}{x}}_{\mathrm{je}})\&le;{0.01}^2\},$ $W=\left[\begin{array}{cc}4& 0\\ 0& 1\end{array}\right].$

Two, set the RELIABILITY INDEX of limit state function

Three, make up gear-driven multidisciplinary reliability analysis model based on the multidisciplinary reliability evaluation index of carrying broad sense with based on the function measure, be respectively: $\mathrm{\&alpha;}=\underset{(U,V)}{\min }{g}_i(u,v,Y_{\mathrm{ij}})\&GreaterEqual;0|\sqrt{u^{T}u}=\mathrm{\&beta;},v_j^T{v}_j\&le;1(i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,11)$

${\mathrm{\&alpha;}}_1=\underset{(U,V)}{\min }{g}_1(-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1+\left(x_1{x}_2^2{x}_3\right)/27.0-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_2=\underset{(U,V)}{\min }{g}_2(-y_2+\left(x_1{x}_2^2{x}_3^2\right)/397.5-1.0\&GreaterEqual;0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,u_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_3=\underset{(U,V)}{\min }{g}_3(x_1/\left(5.0x_2\right)-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_4=\underset{(U,V)}{\min }{g}_4(12.0x_2/x_1-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_5=\underset{(U,V)}{\min }{g}_5(40.0/\left(x_2{x}_3\right)-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_6=\underset{(U,V)}{\min }{g}_6(\left(x_2{x}_3{x}_6^4\right)/{1.925x}_4^3-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_7=\underset{(U,V)}{\min }{g}_7(\left(x_2{x}_3{x}_7^4\right)/1.925x_5^3-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_8=\underset{(U,V)}{\min }{g}_8(110x_6^3/\sqrt{{\left(\frac{745x_4}{x_2{x}_3}\right)}^2+1.691\&times;{10}^7}-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_9=\underset{(U,V)}{\min }{g}_9(85x_7^3/\sqrt{{\left(\frac{745x_5}{x_2{x}_3}\right)}^2+1.575\&times;{10}^8}-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_{10}=\underset{(U,V)}{\min }{g}_{10}(-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1+x_4/(1.5x_6+1.9)-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1,$

${\mathrm{\&alpha;}}_{11}=\underset{(U,V)}{\min }{g}_{10}(x_5/(1.1x_7+1.9)-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1.$

Four, the multidisciplinary analysis method for reliability of the gear drive of serializing.

According to analyzing as can be known, gear-driven limit state function comprises two classes, and a class is only to comprise uncertain variable limit state function at random, and this class function only needs it is carried out the probabilistic reliability analysis; Another kind of is the limit state function that comprises at random uncertain and cognitive uncertain variable simultaneously, and this class function had both needed that it was carried out the probabilistic reliability analysis and also needed to carry out the convex model extreme value analysis, below it is launched elaboration respectively:

1, uncertain and cognitive probabilistic initialization at random: x=[3.0,0.75,23,7.8,7.8,3.4,5.25].

2, to only containing the limit state function α of uncertain variable at random ₁～α ₅, α ₇, α ₉And α ₁₁Carry out the probabilistic reliability analysis:

(1) with couple state function Simultaneous Equations, carries out multidisciplinary analysis.

$\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1=3.5y_1{x}_3+x_1+x_3+\frac{x_2{y}_2}{2}\\ y_2=0.6x_4{y}_2+x_6+x_5+\frac{0.6x_7}{y_1}-3\end{array}\right.\&DoubleRightArrow;\left\{\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1=3.5{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1*23+3.0+23+\frac{0.75y_2}{2}\\ y_2=0.6*7.8y_2+3.4+7.8+\frac{0.6*5.25}{y_1}-3\end{array}\right.$

Can try to achieve: y ₁=-0.3288, y ₂=0.3750.

(2) carry out the system sensitivity analysis based on the global sensitivity equation.

$\left[\begin{array}{cc}I& -\frac{\text{\&PartialD;}{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1}{\&PartialD;y_2}\\ -\frac{\&PartialD;y_2}{\&PartialD;y_1}& I\end{array}\right]\left[\begin{array}{c}\frac{{\mathrm{dy}}_1}{{\mathrm{dx}}_i}\\ \frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}\end{array}\right]=\left[\begin{array}{c}\frac{{\&PartialD;\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1}{{\mathrm{dx}}_i}\\ \frac{\&PartialD;y_2}{{\mathrm{dx}}_i}\end{array}\right]\&DoubleRightArrow;\left[\begin{array}{cc}I& -\frac{x_2}{2}\\ \frac{0.6x_7}{{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1^2}& I\end{array}\right]\left[\begin{array}{c}\frac{{\mathrm{dy}}_1}{{\mathrm{dx}}_i}\\ \frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}\end{array}\right]=\left[\begin{array}{c}\frac{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1}{{\mathrm{dx}}_i}\\ \frac{\&PartialD;y_2}{{\mathrm{dx}}_i}\end{array}\right]---\left(9\right)$

Obtain dy/dx _iAfter, the probability constraints condition in the gear drive fail-safe analysis, its limit state function g is about the individual design variable x at random of i _iGradient information dg/dx _i, just can obtain by through type (10):

$\frac{\mathrm{dg}}{{\mathrm{dx}}_i}=\frac{\&PartialD;g}{\&PartialD;x_i}+\frac{\&PartialD;g}{\&PartialD;{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1}\frac{{\mathrm{<figure-callout\; id="1"\; label="dy"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">dy</figure-callout>}}_1}{{\mathrm{dx}}_i}+\frac{\&PartialD;g}{\&PartialD;y_2}\frac{{\mathrm{dy}}_2}{{\mathrm{dx}}_i}---\left(10\right)$

(3) carry out the search of most probable failpoint based on improved Advanced mean value method (MAMV), carry out fail-safe analysis.

Limit state function α ₁～α ₅, α ₇, α ₉And α ₁₁Functional value and corresponding most probable failpoint as shown in table 1.As can be seen, gear-driven α ₃And α ₉The limit state function value is discontented with sufficient reliability requirement, α less than zero ₁, α ₂, α ₄, α ₅, α ₇And α ₁₁Limit state function all greater than zero, satisfies reliability requirement.

Table 1 α ₁～α ₅, α ₇, α ₉And α ₁₁The probabilistic reliability analysis result

3, to containing at random and the probabilistic gear drive limit state function of cognition α ₆, α ₈And α ₁₀Carry out fail-safe analysis.

Limit state function g ₆And g ₈Comprise uncertain variable and cognitive uncertain variable at random simultaneously, so need carry out probabilistic reliability analysis and convex model extreme value analysis; Limit state function g ₁₀Only contain cognitive uncertainty, so only need carry out the convex model extreme value analysis.The model of limit state function as the formula (11).

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_6=\underset{(U,V)}{\min }{g}_6(\left(x_2{x}_3{x}_6^4\right)/1.925x_4^3-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1\\ {\mathrm{\&alpha;}}_8=\underset{(U,V)}{\min }{g}_8(110x_6^3/\sqrt{{\left(\frac{745x_4}{x_2{x}_3}\right)}^2+1.691\&times;{10}^7}-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1\\ {\mathrm{\&alpha;}}_{10}=\underset{(U,V)}{\min }{g}_{10}(-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="HDA0000092337330000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_1+x_4/(1.5x_6+1.9)-1.0)\&GreaterEqual;0|\sqrt{u^{T}u}=3,v_j^T{v}_j\&le;1\end{array}\right.---\left(11\right)$

Here with g ₆For example is described in detail the fail-safe analysis process, limit state function g ₈And g ₁₀The same g of fail-safe analysis process ₆, concrete steps are as follows:

(1) fixing x ₄And x ₆Be x ₄=7.8, x ₆Be=3.4;

(2) probabilistic reliability analysis, multidisciplinary analysis and sensitivity analysis are with step 4, through stochastic variable x after the probabilistic reliability analysis ₂And x ₃Be respectively 0.6750,22.9978;

(3) fixing x ₂=0.6750 and x ₃=22.9978, according to eigenmatrix $W=\left[\begin{array}{cc}4& 0\\ 0& 1\end{array}\right]$ And parameter uncertainty degree ε=0.01 is with x ₄And x ₆Be converted to the variable in v space, therefore have:

And x ₆=v ₆+ 3.4, adopt method of Lagrange multipliers to α ₆Re-constructing function is:

${\mathrm{\&alpha;}}_6=\underset{(U,V)}{\min }{g}_6((0.6750*22.9978*{(v_6+3.4)}^4)/(1.925*{(\frac{v_4}{2}+7.8)}^3-1.0+\mathrm{\&lambda;}(v_4^2+v_6^2-1.0)).$

(4) adopt Karush-Kuhn-Tucker (KKT) condition respectively to λ, v ₄And v ₆Differentiate and to make it be zero; Namely have:

$\left\{\begin{array}{c}\frac{\&PartialD;{\mathrm{\&alpha;}}_6}{\&PartialD;v_4}=\frac{\&PartialD;g_6}{\&PartialD;v_4}=-193.5408*\frac{{(v_6+3.4)}^4}{{(v_4+15.6)}^4}+2\mathrm{\&lambda;}*v_4=0\\ \frac{\&PartialD;{\mathrm{\&alpha;}}_6}{\&PartialD;v_6}=\frac{\&PartialD;g_6}{\&PartialD;v_6}=-258.0544*\frac{{(v_6+3.4)}^3}{{(v_4+15.6)}^3}+2\mathrm{\&lambda;}*v_6=0\\ \frac{\&PartialD;{\mathrm{\&alpha;}}_6}{\&PartialD;\mathrm{\&lambda;}}=\frac{\&PartialD;g_6}{\&PartialD;\mathrm{\&lambda;}}=v_4^2+v_6^2-1.0=0\end{array}\right.$

(5) can ask α based on above-mentioned analysis result ₆Extreme value be [1.1816,1.3631].

As seen gear-driven limit state function α ₆Minimal value greater than zero, satisfy the reliability design requirement.

In like manner, can try to achieve gear-driven limit state function α ₈And α ₁₀Span be respectively: [0.0166,0.0781], [0.2063 ,-0.2226].As seen gear-driven limit state function α ₈Minimal value greater than zero, satisfy the reliability design requirement, and limit state function α ₈Minimal value less than zero, discontented can the property leaned on designing requirement.

4, at above result, take reasonable measure and improve gear-driven reliability.

Claims (1)

1. consider the cognition that produces in the Gear Transmission Design process and probabilistic multidisciplinary analysis method for reliability at random for one kind, it is characterized in that comprising integrated, the multidisciplinary fail-safe analysis modeling based on the function measure of multidisciplinary reliability comprehensive evaluation index, analysis method for reliability and parallel subspace optimisation strategy cognitive and probabilistic quantification, broad sense at random, the multidisciplinary fail-safe analysis process decoupling zero based on serializing thought, multidisciplinary probabilistic reliability analysis and multidisciplinary convex model extreme value analysis;

Described cognition and at random uncertain the quantification be cognitive uncertain and uncertain at random at what produce in the Gear Transmission Design process, adopt more suitable mathematical theory to quantize respectively, set up cognitive and probabilistic comprehensive quantification method at random, having probabilistic design parameter is module, the number of teeth, face width of tooth, distance between bearings and gear diameter, uncertain source comprises mismachining tolerance, rigging error, material behavior, the cognition that factor such as imposed load and external working environment causes and uncertain at random, quantizing process may further comprise the steps:

Step 1. beginning;

The uncertain data of step 2. design parameter whether complete is to enter step 3, otherwise enter step 4;

Step 3. is complete for data, the uncertainty of the sufficient design parameter of information adopts probability theory to quantize, and chooses probability distribution function, sets average μ and variances sigma;

The uncertainty of the design parameter that step 4. is incomplete for data, information is incomplete adopts convex model to quantize, and determines interval Δ I, eigenmatrix W and the uncertain degree ε of parameter;

Step 5. finishes;

The multidisciplinary reliability comprehensive evaluation index of described broad sense is the reliability evaluation index with robustness, has two tolerance indexs, the lower limit of RELIABILITY INDEX ${\mathrm{\&beta;}}_L=\underset{U}{\min }\sqrt{{\mathrm{uu}}^T|\underset{V}{\min }\left(g(u,v)\right)}$ The upper limit with RELIABILITY INDEX ${\mathrm{\&beta;}}_U=\underset{U}{\min }\sqrt{{\mathrm{uu}}^T|\underset{V}{\max }\left(g(u,v)\right)},$ U wherein, v represent the uncertain and cognitive uncertain design variable at random of normed space, the difference DELTA β=β of the reliability evaluation index of carrying respectively _U-β _LQuantitative reaction the influence degree of cognitive uncertainty to design result, the reliability evaluation of the more realistic engineering of this evaluation index;

The integrated of described analysis method for reliability and parallel subspace optimisation strategy is that the analysis method for reliability that will be applied to single subject expands to the multidisciplinary fail-safe analysis field with coupled relation, and integrating process may further comprise the steps:

Step 1. beginning;

Step 2. pair multidisciplinary fail-safe analysis process is analyzed, and comprises multidisciplinary analysis, global sensitivity analysis and most probable failpoint search (Most Probable Point, MPP point search);

The process of step 3. pair parallel subspace optimisation strategy is analyzed, and comprises multidisciplinary analysis, global sensitivity analysis, parallel subspace and system-level coordination optimization;

Step 4. is integrated with single subject analysis method for reliability and parallel subspace optimisation strategy, the multidisciplinary analysis of parallel subspace optimisation strategy and global sensitivity analysis provide limit state function for multidisciplinary fail-safe analysis value and gradient information;

Step 5. finishes;

Described multidisciplinary fail-safe analysis modeling based on the function measure is to adopt the fiduciary level Equivalent Thought to change former multidisciplinary reliability analysis model based on the RELIABILITY INDEX method into efficient, the stable multidisciplinary reliability analysis model based on the function measure, and transfer process may further comprise the steps:

Step 1. beginning;

Step 2. is set the reliability design index, β=3.0 wherein, and it is made as minimizes the optimization aim function, with constraint function g _i()=0 is defined as constraint condition, is namely satisfying g _iObtain the minimum value of fiduciary level under the situation of ()=0;

Step 3. adopts Equivalent Thought, with constraint function g _i()=0 is defined as the optimization aim function, the reliability design index of constraint condition for setting, and the minimum value of constraint function is namely obtained in β=3.0 wherein under the constraint of satisfying the reliability design index;

Step 4. judges whether to satisfy reliability requirement according to the optimal value of constraint function;

Step 5. finishes;

Described multidisciplinary fail-safe analysis process decoupling zero based on serializing thought is the multidisciplinary fail-safe analysis process at serious coupling, adopt serializing thought that it is carried out decoupling zero, and then form a multidisciplinary fail-safe analysis of single cycle recursion, detailed process may further comprise the steps:

Step 1. beginning;

Step 2. is fixing cognitive uncertain, carries out multidisciplinary analysis, carries out most possible failpoint (MPP point) search;

Step 3. is solidified uncertain at random, carries out multidisciplinary analysis, carries out the convex model extreme value analysis;

Step 4. arranges the condition of convergence, carries out the convergence checking;

Step 5. finishes;

Described multidisciplinary probabilistic reliability analysis, carry out single subject analysis method for reliability and parallel subspace optimisation strategy integrated, based on improved Advanced mean value method the multidisciplinary system with coupled relation is carried out fail-safe analysis, detailed process may further comprise the steps, wherein β _tBe the RELIABILITY INDEX of setting, k is the computation cycles number of times, For limit state function in the U space about the gradient information of design variable at random,

Step 1. beginning;

State variable y is obtained in the analysis of step 2. executive system _kWith limit state function g (x _k) value;

The sensitivity analysis of step 3. executive system utilizes global sensitivity equation (Global Sensitivity Equation, GSE) gradient of acquisition limit state function

Step 4. is carried out space conversion, uncertain variable x at random _kChange into the variable u in standard normal space _k, obtain limit state function in the gradient in standard normal space

Step 5. is upgraded u according to improved Advanced mean value method _k

Step 6. is calculated u according to formula (1) _kWith Between angle, if γ _k＜ε jumps to step 9, otherwise jumps to step 7, and ε is a very little angle,

${\mathrm{\&gamma;}}_k={\cos }^{-1}\frac{u_k\&CenterDot;{\&dtri;g}_u\left(u_k\right)}{\left|\right|u_k\left|\right|\&CenterDot;\left|\right|{\&dtri;g}_u\left(u_k\right)\left|\right|}---\left(1\right)$

If step 7. g is (u _k)＞g (u _K-1),

Otherwise

$u_{k+1}=\frac{{\mathrm{\&beta;}}_t}{\sin \left({\mathrm{\&gamma;}}_k\right)}(\sin ({\mathrm{\&gamma;}}_k-{\mathrm{\&delta;}}_k)\frac{u_k}{\left|\right|u_k\left|\right|}+\sin {\mathrm{\&delta;}}_k\frac{{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|})---\left(2\right)$

δ in the formula _kCan obtain by finding the solution an one dimension max problem, as the formula (3),

$\max g\left(u_{k+1}\right)=g\left\{\frac{{\mathrm{\&beta;}}_t}{\sin \left({\mathrm{\&gamma;}}_k\right)}(\sin ({\mathrm{\&gamma;}}_k-{\mathrm{\&delta;}}_k)\frac{u_k}{\left|\right|u_k\left|\right|}+{\sin \mathrm{\&delta;}}_k\frac{{\&dtri;}_{u}g\left(u_k\right)}{\left|\right|{\&dtri;}_{u}g\left(u_k\right)\left|\right|})\right\}---\left(3\right)$

Step 8. is u _K+1Convert corresponding variable x in luv space to _K+1, change step 1;

Step 9. is calculated g (u _k), finish;

Described multidisciplinary convex model extreme value analysis based on Karush-Kuhn-Tucker optimal condition equivalent substitution convex model extreme value analysis, improves efficient extensive, multiple coupled multidisciplinary fail-safe analysis, and detailed process may further comprise the steps, wherein v _kBe the cognitive uncertain design variable value of trying to achieve in the k time circulation of computation process,

Step 1. beginning;

State variable y is obtained in the analysis of step 2. executive system _kWith limit state function g (v _k) value;

Step 3. adopts method of Lagrange multipliers to make up the lagrangian optimization objective function;

Respectively to the uncertain variable of cognition and Lagrange multiplier differentiate, and to make it respectively be zero to step 4. based on partial differential;

Step 5. is obtained extreme point according to the Karush-Kuhn-Tucker optimal condition;

Step 6. is brought extreme point into limit state function g (v _k), obtain maximum value and minimal value;

Step 7. is determined cognitive uncertain influence degree to this limit state function according to maximum value and minimizing difference;

Step 8. is differentiated limit state function g (v _k) minimal value whether greater than zero, be then satisfy reliability requirement, otherwise discontented sufficient reliability requirement to adopt shift strategy to move to the safety zone uncertain at random, makes up new limit state function and carries out the multidisciplinary fail-safe analysis of a new round;

Step 9. finishes.

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