CN107563016A - A kind of wing box section Parameter Sensitivity Analysis method based on ellipsoidal model - Google Patents

A kind of wing box section Parameter Sensitivity Analysis method based on ellipsoidal model Download PDF

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CN107563016A
CN107563016A CN201710694426.5A CN201710694426A CN107563016A CN 107563016 A CN107563016 A CN 107563016A CN 201710694426 A CN201710694426 A CN 201710694426A CN 107563016 A CN107563016 A CN 107563016A
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CN107563016B (en
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张峰
骆凯亮
刘伟
翟伟昊
邓维维
敖良波
岳珠峰
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Northwestern Polytechnical University
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Abstract

The invention provides a kind of wing box section Parameter Sensitivity Analysis method based on ellipsoidal model, it is related to field of airplane structure, for the insufficient situation of aircraft wing wing box parameter of structure design information, it is proposed a kind of structural parameters Sensitivity Analysis Method based under ellipsoidal model, in the case of Structural functional equation is linear, it is deduced the analytic solutions of structural parameters sensitivity, invention defines the sensitivity index of the wing box section parameter under ellipsoidal model, derive the Analytical Solution method that wing box section power function is parametric sensitivity under linear case, with the box section structure sensitivity analysis of wing nine, demonstrate the correctness and validity of analytic method, fail-safe analysis and structure of the analysis result of the present invention to wing box section, which change design, to have great significance.

Description

A kind of wing box section Parameter Sensitivity Analysis method based on ellipsoidal model
Technical field
The present invention relates to field of airplane structure, especially a kind of Sensitivity Analysis Method.
Background technology
Wing box is the main load-carrying construction of aircraft wing, is mainly made up of front-axle beam, the back rest, internal rib, upper lower wall panels, the wing Box has important influence to whole wing as most important load part in outer wing structure.Before wing box front end is connected Edge and leading edge slat, rear end connection aileron, wing flap and spoiler, the hanging of lower end connection engine and undercarriage.Airplane operation mistake Load in journey in the case of all working can be all delivered in wing box, and this just needs the load path to wing box structure, geometry knot Structure makes preferable design.In the detailed design phase of wing, structure type and layout it has been determined that wing in aircraft flight During produce lift, be the basic guarantee that aircraft can fly.The structure design of wing box, to the shadow of the even whole aircraft of wing Sound has vital effect.Good structure design can not only ensure that wing produces normal aerodynamic lift, and wing The normal operation of interior system, and the performance advantage of material can be given full play to, mitigate construction weight, improve structural reliability. Parameter Sensitivity Analysis will be seen that relative importance of the variable to structural reliability, so as to be structural analysis and optimization design Reference is provided.At present, probability and reliability analysis method is using more fully, corresponding based on the ginseng under probabilistic model Number sensitivity analysis is also increasingly ripe.Parameter Sensitivity Analysis based on probabilistic model needs to assert that parameter is stochastic variable, and There are the information such as the regularity of distribution, average, variance of parameter.And in structured design process, lack sufficient information to determine structure The information such as the regularity of distribution of design parameter, average, variance, probability density function, limit the parameter-sensitive based on probabilistic model Spend the application of analysis method.
Non-probability model can efficiently solve that data volume is less, the inferior reliability of variable probability density function deletion condition is asked Inscribe and receive much concern.Since the 1990s, the concept of the Multidisciplinary systems based on Convex set model is suggested, and propose with What system can allow probabilistic at utmost measures reliability.Generally speaking, the Convex set model of non-probability can be divided into section Two kinds of model and ellipsoidal model.Compared to interval model, ellipsoidal model can describe the correlation between multiple uncertain variables, when When variable is uncorrelated, ellipsoidal model is degenerated to interval model, it follows that interval model can be considered the special case of ellipsoidal model.Mesh Before, for the existing more in-depth study of Analysis of structural reliability of ellipsoidal model, and the spirit of the structural parameters based on ellipsoidal model Sensitivity problems demand solves.
The content of the invention
In order to which overcome the deficiencies in the prior art, the present invention are insufficient for aircraft wing wing box parameter of structure design information Situation, a kind of structural parameters Sensitivity Analysis Method based under ellipsoidal model is proposed, be linear for Structural functional equation Situation, it is deduced the analytic solutions of structural parameters sensitivity.
The detailed step of the technical solution adopted for the present invention to solve the technical problems is as follows：
Step 1：If the power function of aircraft wing wing box structure is g (x), x is structural parameters, is a n dimension variable, remembers For x=(x1,x2,…,xi,…,xn), xiFor i-th of structural parameters, ellipsoidal model U (u, σ) is satisfied with, U (u, σ) is as follows：
Wherein, u=(u1,u2,…ui,…,un) be ellipsoidal model U (u, σ) location parameter, σ=(σ1,σ2,…, σi,…,σn) be ellipsoidal model U (u, σ) dimensional parameters, i=1,2 ..., n, uiFor ellipsoidal model U (u, σ) i-th of position Parameter, σiFor ellipsoidal model U (u, σ) i-th of dimensional parameters；
Step 2：When power function g (x) is equal to 0, the variable space is divided into security domain D by its curved surface g (x)=0 formeds (x:G (x) ＞ 0) and failure domain Df(x:G (x) ＜ 0), reliability of structure index η is expressed as formula (2)：
Wherein, λ is the scale factor of dimensional parameters；
Solution obtains reliability index η and optimization solution x*, wherein x*For design point corresponding to g (x)=0；
Step 3：Because structural parameters sensitivity definition is failure probability P under probabilistic modelfTo basic variable distributed constant θx Partial derivative, i.e.,Similarly, the parametric sensitivity based on structure under ellipsoidal model U (u, σ) is defined as can under ellipsoidal model Partial derivative by property index η to its parameter, you can join by property index η to the location parameter u in ellipsoidal model U (u, σ) and size Number σ partial derivative, is used respectivelyWithRepresent；
When the power function g (x) of wing boxes of wings structure is linear function, power function g (x) is as follows：
Wherein, x meets formula (1), a0It is constant term, biFor the coefficient in power function g (x)；
Define intermediate variable z=(z1,z2,…,zi,…,zn), zi=(xi-ui)/σi, then intermediate variable z is that n dimension units surpass Spheroid, i.e.,
By ziFormula (3) is updated to, then power function g (x) is rewritten as：
Because g (z) is linear function, obtaining reliability of structure index η under ellipsoidal model U (u, σ) is:
During then for wing box section power function g (x) forms as shown in formula (3), based on ellipsoidal model U's (u, σ) The analytic expression of structural parameters sensitivity is：
It can thus be concluded that ellipsoidal model U (u, σ) structural parameters sensitivity.
The beneficial effects of the present invention are refer to due to the sensitivity for defining the wing box section parameter under ellipsoidal model Mark, the Analytical Solution method that wing box section power function is parametric sensitivity under linear case is derived, with the box section of wing nine STRUCTURAL SENSITIVITY ANALYSIS INDESIGN, it was demonstrated that the correctness and validity of analytic method, analysis result of the invention is to wing box section Fail-safe analysis and structure change design and have great significance.
Brief description of the drawings
Fig. 1 is the box section structure schematic diagram of aircraft wing nine of the present invention, wherein x4And 2x4At the box section structure node of wing nine The load applied.
Embodiment
The present invention is further described with reference to the accompanying drawings and examples.
Step 1：If the power function of aircraft wing wing box structure is g (x), x is structural parameters, is a n dimension variable, remembers For x=(x1,x2,…,xi,…,xn), xiFor i-th of structural parameters, to be satisfied with ellipsoidal model U (u, σ), the following institutes of U (u, σ) Show：
Wherein, u=(u1,u2,…ui,…,un) be ellipsoidal model U (u, σ) location parameter, σ=(σ1,σ2,…, σi,…,σn) be ellipsoidal model U (u, σ) dimensional parameters, i=1,2 ..., n, uiFor ellipsoidal model U (u, σ) i-th of position Parameter, σiFor ellipsoidal model U (u, σ) i-th of dimensional parameters；
Step 2：When power function g (x) is equal to 0, the variable space is divided into security domain D by its curved surface g (x)=0 formeds (x:G (x) ＞ 0) and failure domain Df(x:G (x) ＜ 0), reliability of structure index η is expressed as formula (2)：
Wherein, λ is the scale factor of dimensional parameters；
Find out from formula (2), the solution of the structural reliability degree index η based on ellipsoidal model is actually the excellent of belt restraining Change problem, by simulated annealing optimization method or genetic algorithm, solution obtains reliability index η and optimization solution x*, wherein x*For Design point corresponding to g (x)=0；
Step 3：Because structural parameters sensitivity definition is failure probability P under probabilistic modelfTo basic variable distributed constant θx Partial derivative, i.e.,Similarly, the parametric sensitivity based on structure under ellipsoidal model U (u, σ) is defined as can under ellipsoidal model Partial derivative by property index η to its parameter, you can join by property index η to the location parameter u in ellipsoidal model U (u, σ) and size Number σ partial derivative, is used respectivelyWithRepresent；
When the power function g (x) of wing boxes of wings structure is linear function, power function g (x) is as follows：
Wherein, x meets formula (1), a0It is constant term, biFor the coefficient in power function g (x)；
Define intermediate variable z=(z1,z2,…,zi,…,zn), zi=(xi-ui)/σi, then intermediate variable z is that n dimension units surpass Spheroid, i.e.,
By ziFormula (3) is updated to, then power function g (x) is rewritten as：
Because g (z) is linear function, obtaining reliability of structure index η under ellipsoidal model U (u, σ) is:
During then for wing box section power function g (x) forms as shown in formula (3), based on ellipsoidal model U's (u, σ) The analytic expression of structural parameters sensitivity is：
It can thus be concluded that ellipsoidal model U (u, σ) structural parameters sensitivity.
Show the box section structure of wing nine being made up of 64 bars and 42 panel elements in Fig. 1, structure material is aluminium alloy, structure Intensity (the x of unit1,x2,x3) and the load (x that is applied4) it is variable.The dominant failure mode optiaml ciriterion of structural system Method obtains, and carries out variable (x1,x2,x3,x4) unit normalized, the power function g (x) finally given such as formulas (8) It is shown：
G (x)=4.0x1+4.0x2-3.9998x3-x4 (8)
When g (x) ＞ 0, structure safety；When g (x) ＜ 0, structural failure.The variable x of nine box section structures1,x2,x3,x4Meet Ellipsoidal model U (u, σ) is as shown in formula (9).
The location parameter of the box section structure parametric variable of ellipsoidal model U (u, σ) lower wing nine and the size parameter values such as institute of table 1 Show：
Table 1
The result that the box section structure parametric sensitivity of wing nine based on ellipsoidal model U (u, σ) calculates is as shown in table 2：
Table 2
The present embodiment analyzes the parametric sensitivity of the box section structure of wing nine, as shown in table 2, using finite difference calculus as testing Card solution, the difference of two methods acquired results are less than 1%, and this shows method proposed by the present invention for solving wing box section It is effective and feasible based on the parametric sensitivity problem under ellipsoidal model during linear function function situation.

Claims (1)

  1. A kind of 1. wing box section Parameter Sensitivity Analysis method based on ellipsoidal model, it is characterised in that including following steps Suddenly：
    Step 1：If the power function of aircraft wing wing box structure is g (x), x is structural parameters, is a n dimension variable, is designated as x =(x1,x2,…,xi,…,xn), xiFor i-th of structural parameters, ellipsoidal model U (u, σ) is satisfied with, U (u, σ) is as follows：
    <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>,</mo> <mi>&amp;sigma;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>{</mo> <mi>x</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>:</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msubsup> <mi>&amp;sigma;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mfrac> <mo>&amp;le;</mo> <mn>1</mn> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, u=(u1,u2,…ui,…,un) be ellipsoidal model U (u, σ) location parameter, σ=(σ1,σ2,…,σi,…,σn) For ellipsoidal model U (u, σ) dimensional parameters, i=1,2 ..., n, uiFor ellipsoidal model U (u, σ) i-th of location parameter, σiFor Ellipsoidal model U (u, σ) i-th of dimensional parameters；
    Step 2：When power function g (x) is equal to 0, the variable space is divided into security domain D by its curved surface g (x)=0 formeds(x:g (x) ＞ 0) and failure domain Df(x:G (x) ＜ 0), reliability of structure index η is expressed as formula (2)：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>F</mi> <mi>i</mi> <mi>n</mi> <mi>d</mi> <mi>&amp;eta;</mi> <mo>,</mo> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;eta;</mi> <mo>=</mo> <mi>&amp;lambda;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msqrt> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msubsup> <mi>&amp;sigma;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mfrac> </mrow> </msqrt> <mo>=</mo> <mi>&amp;lambda;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, λ is the scale factor of dimensional parameters；
    Solution obtains reliability index η and optimization solution x*, wherein x*For design point corresponding to g (x)=0；
    Step 3：Because structural parameters sensitivity definition is failure probability P under probabilistic modelfTo basic variable distributed constant θxIt is inclined Derivative, i.e.,Similarly, the parametric sensitivity based on structure under ellipsoidal model U (u, σ) is defined as reliability under ellipsoidal model Partial derivatives of the index η to its parameter, you can by property index η to the location parameter u's in ellipsoidal model U (u, σ) and dimensional parameters σ Partial derivative, use respectivelyWithRepresent；
    When the power function g (x) of wing boxes of wings structure is linear function, power function g (x) is as follows：
    <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, x meets formula (1), a0It is constant term, biFor the coefficient in power function g (x)；
    Define intermediate variable z=(z1,z2,…,zi,…,zn), zi=(xi-ui)/σi, then intermediate variable z is that n ties up unit hyper-sphere Body, i.e.,
    By ziFormula (3) is updated to, then power function g (x) is rewritten as：
    <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    Because g (z) is linear function, obtaining reliability of structure index η under ellipsoidal model U (u, σ) is:
    <mrow> <mi>&amp;eta;</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    During then for wing box section power function g (x) forms as shown in formula (3), the structure based on ellipsoidal model U (u, σ) The analytic expression of parametric sensitivity is：
    <mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>&amp;eta;</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <msub> <mi>b</mi> <mi>i</mi> </msub> <msup> <mrow> <mo>&amp;lsqb;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>&amp;eta;</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <msubsup> <mi>b</mi> <mi>i</mi> <mn>2</mn> </msubsup> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> <mfrac> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>+</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>&amp;sigma;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    It can thus be concluded that ellipsoidal model U (u, σ) structural parameters sensitivity.
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CN109766522A (en) * 2019-03-18 2019-05-17 西安科技大学 A kind of Multidisciplinary systems Sensitivity Analysis Method of Chain Wheel of Flight Bar Conveyor

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