CN1868807A - Layer spreading design calculating method of composite material according to rigidity requirement - Google Patents

Abstract

A design calculation method based on the rigidity requirement for laminated composite material, such as the vertical tail fin of airplane features that when the relative angle between layers of a laminated plate with equal rigidity in three directions is changed, its thickness or weight characteristic is not changed, and the relation between the rigidity in three directions and the relative weight is used in the aerodynamic elasticity calculation to replace complex calculation model.

Description

The layer spreading design calculating method of composite material according to rigidity requirement

Technical field:

The present invention relates to the design of aircraft structure, especially the method for designing of compound material laying layer.

Background technology:

In the airfoil structure design, except that satisfying requirement of strength, also must satisfy and calculate the rigidity index that proposes by aeroelasticity.Particularly concerning superfighter, often the rigidity requirement strength-to-density ratio requires more difficult satisfied.Because the characteristics that composite material is better than metallic material are that its axial stiffness and shear rigidity is respectively programmable, thus we can to utilize these characteristics to design lighter than metal construction, more rational aerofoil comes.In order to carry out shop layer design, thereby this important function for of research has been proposed by rigidity requirement.

When aeroelasticity is calculated the airfoil structure rigidity requirement is often provided with following two kinds of forms.

Regarding aerofoil as the engineering beam when (1) aeroelasticity is calculated simplifies

After aerofoil is simplified to the engineering beam,, propose along the distribution curve of each tangent plane bending stiffness EI1 of axis and torsional stiffness GJ requirement according to aerofoil aeroelasticity result of calculation.Like this in structure design satisfying under the prerequisite of requirement of strength, also should satisfy rigidity requirement.

According to present aerofoil design-calculated reality, we are before the covering parameter is selected, and the principal parameter of topology layout and inner structure is all definite.Covering is except that wing root portion, along tangential constant substantially, just along opening up to just changing.So we can be the require EI of aeroelasticity to rigidity ₁, GJ changes into the requirement to the axial stiffness E δ of covering and shear rigidity G δ.

If establishing the bending stiffness of beam and stiffening rib is EIa, the bending stiffness of section just can be write as following relational expression so:

EI ₁＝EI _a+∫Eδ ₁·h ²ds (1)

H represents aerofoil profile half height of tangent plane ds place covering line of centers in the equation.E δ represents the axial stiffness of covering.

If each tangent plane covering thickness is unmodified, and only along opening up to changing, equation (1) can be write as:

Eδ ₁＝(EI ₁-EI _a)/∫h ²ds (2)

We just can change into aeroelasticity to the requirement of covering axial stiffness E δ 1 by equation (2) the requirement of section bending stiffness EI1 like this.

Equally, if thinking, we during web thickness δ W, just can set up following equation when the torsional stiffness of section is GJ by section closed chamber area F

$\mathrm{GJ}=\frac{4{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{\&Integral;\frac{\mathrm{ds}}{G{\mathrm{\&delta;}}_s}}---\left(3\right)$

Wherein $\&Integral;\frac{\mathrm{ds}}{G{\mathrm{\&delta;}}_s}=\mathrm{\&Sigma;}\frac{h_w}{G{\mathrm{\&delta;}}_w}+2\underset{s}{\&Integral;}\frac{\mathrm{ds}}{\mathrm{G\&delta;}}---\left(4\right)$

From equation (3), solve in (4)

$\mathrm{G\&delta;}=\frac{2\underset{s}{\&Integral;}\mathrm{ds}}{\frac{4{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">F</figure-callout>}}^2}{\mathrm{GJ}}-\mathrm{\&Sigma;}\frac{h_w}{G{\mathrm{\&delta;}}_w}}---\left(5\right)$

We have changed into aeroelasticity to the requirement of the shear rigidity G δ of covering to the requirement of the torsional stiffness of section again like this.

Press in the shop layer design of rigidity requirement at the composite material aerofoil, as long as the axial stiffness EI that formula (2) and (5) provide is satisfied in the shop layer design of covering ₁After the requirement of shear rigidity GJ, promptly satisfy the desired aerofoil bending of aeroelasticity stiffness curve.

With finite element method (FEM) aerofoil being carried out aeroelasticity calculates

After this form aeroelasticity is calculated, will provide the requirement of axial stiffness E and shear rigidity to each piece covering of aerofoil, we just can directly carry out having designed by the shop layer of rigidity to covering according to this requirement like this.

Summary of the invention:

The present invention according to rigidity requirement carries out the equivalent stiffness conversion of compound material laying layer and meter and shop layer.Drawn by the shop layer method of designing of three-way rigidity requirement and concrete formula and calculated.When the three-way rigidity of known layer plywood, can obtain the shop numbers of plies under any three kinds of shop layer angles, thereby obtain the new layering type that engineering is used that is applicable to.

1. pressing the shop layer design equation of rigidity requirement derives

1.1 to know condition

Provided covering rigidity requirement E δ and G δ by aeroelasticity.

The thickness that provides every layer of the key property of unidirectional composite material and composite panel by composite material key property experiment promptly

Ex-----composite material longitudinal stretching modulus

Ey-----composite material cross directional stretch modulus

Es-----composite material longitudinal shear modulus

The vertical Poisson's ratio of Vx-----

The every layer thickness of TB-----composite panel

1.2 the derivation of stiffness equations in the RATE OF COMPOSITE LAMINATES face

Along the fiber master to rigidity component calculating formula:

$Q_{\mathrm{xx}}=\frac{E_x}{1-v_x{v}_y}$

$Q_{\mathrm{yy}}=\frac{E_y}{1-v_x{v}_y}$

$Q_{\mathrm{xy}}=\frac{v_y{E}_x}{1-v_x{v}_y}---\left(6\right)$

$Q_{\mathrm{yx}}=\frac{v_x{E}_y}{1-v_x{v}_y}$

Q _ss＝E _s

Wherein

$v_y=\frac{E_y}{E_x}{v}_x---\left(7\right)$

The numerical invariants calculating formula:

U ₁＝(3Q _xx+3Q _yy+2Q _xy+4Q _ss)/8

U ₂＝(Q _xx-Q _yy)/2

U ₃＝(Q _xx+Q _yy-2Q _xy-4Q _ss)/8

U ₄＝(Q _xx+Q _yy+6Q _xy-4Q _ss)/8

U ₅＝(Q _xx+Q _yy-2Q _xy+4Q _ss)/8 (8)

The rigidity component of shop layer rotational angle theta is:

Q ₁₁＝U ₁+U ₂cosθ+U ₃cos4θ

Q ₂₂＝U ₁-U ₂cosθ+U ₃cos4θ

Q ₁₂＝U ₄-U ₃cos4θ

Q ₆₆＝U ₅-U ₃cos4θ (9)

${\mathrm{<figure-callout\; id="16"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{16}=\frac{1}{2}{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">U</figure-callout>}}_2\sin 2\mathrm{\&theta;}+{\mathrm{<figure-callout\; id="3"\; label="U"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">U</figure-callout>}}_3\sin 4\mathrm{\&theta;}$

$Q_{26}=\frac{1}{2}{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">U</figure-callout>}}_2\sin 2\mathrm{\&theta;}-U_3\sin 4\mathrm{\&theta;}$

Stress-strain relation behind the single-skin panel corner is:

σ ₁＝Q ₁₁ε ₁+Q ₁₂ε ₂+Q ₁₆ε ₆

σ ₂＝Q ₂₁ε ₁+Q ₂₂ε ₂+Q ₂₆ε ₆ (10)

σ ₆＝Q ₆₁ε ₁+Q ₆₂ε ₂+Q ₆₆ε ₆

If suppose that the strain of three each layers of direction was consistent when each layer of composite panel was out of shape under external force, then strain is expressed as ε respectively ₁ ⁰, ε ₂ ⁰, ε ₆ ⁰, the steady component of stress σ of composite panel ₁ ⁰, σ ₂ ⁰, σ ₆ ⁰, represent that respectively composite panel General Logistics Department degree is δ.

Actual composite panel covering is when airfoil structure is stressed, and each ply strain is identical.Now the single-skin panel of various angles shop layer is superimposed together and sets up following relational expression

${\mathrm{\&sigma;}}_1^0=\frac{1}{\mathrm{\&delta;}}[A_{11}{\mathrm{\&epsiv;}}_1^0+A_{12}{\mathrm{\&epsiv;}}_2^0+A_{16}{\mathrm{\&epsiv;}}_6^0]$

${\mathrm{\&sigma;}}_2^0=\frac{1}{\mathrm{\&delta;}}[A_{21}{\mathrm{\&epsiv;}}_1^0+A_{22}{\mathrm{\&epsiv;}}_{22}^0+A_{26}{\mathrm{\&epsiv;}}_6^0]---\left(11\right)$

${\mathrm{\&sigma;}}_6^0=\frac{1}{\mathrm{\&delta;}}[A_{61}{\mathrm{\&epsiv;}}_1^0+A_{62}{\mathrm{\&epsiv;}}_2^0+A_{66}{\mathrm{\&epsiv;}}_6^0]$

In the formula:

$A_{11}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{11j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

$A_{12}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{12j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

$A_{22}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{22j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

$A_{16}=A_{61}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{16j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}---\left(12\right)$

$A_{26}=A_{62}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{26j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

$A_{66}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{66j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

Relational expression between the strain and stress can be write as:

${\mathrm{\&epsiv;}}_1^0=a_{11}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_1^0+a_{12}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_2^0+a_{16}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_6^0$

${\mathrm{\&epsiv;}}_2^0=a_{21}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_1^0+a_{22}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_2^0+a_{26}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_6^0---\left(13\right)$

${\mathrm{\&epsiv;}}_6^0=a_{61}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_1^0+a_{62}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_2^0+a_{66}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_6^0$

The flexibility component of aij display plate in the formula.According to modulus definition in the face of plate following relational expression is arranged like this

$E_1^0=\frac{1}{a_{11}\mathrm{\&delta;}}$ Longitudinal modulus

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^0=\frac{1}{a_{22}\mathrm{\&delta;}}$ Transverse modulus

${\mathrm{<figure-callout\; id="6"\; label="E"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">E</figure-callout>}}_6^0=\frac{1}{a_{66}\mathrm{\&delta;}}$ Modulus of shearing (14)

$v_{21}^0=-\frac{a_{12}}{a_{11}}$ Poisson's ratio in the face

$v_{61}^0=\frac{a_{61}}{a_{11}}$ The inplane shear coefficient of coupling

${\mathrm{<figure-callout\; id="16"\; label="v"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">v</figure-callout>}}_{16}^0=\frac{a_{16}}{a_{66}}$ Normal direction coefficient of coupling in the face

During balancing disk, a16=0, a26=0, so

${\mathrm{\&sigma;}}_1^0\mathrm{\&delta;}=A_{11}{\mathrm{\&epsiv;}}_1^0+A_{12}{\mathrm{\&epsiv;}}_2^0$

${\mathrm{\&sigma;}}_2^0\mathrm{\&delta;}=A_{21}{\mathrm{\&epsiv;}}_1^0+A_{22}{\mathrm{\&epsiv;}}_{22}^0---\left(15\right)$

${\mathrm{\&sigma;}}_6^0\mathrm{\&delta;}=A_{66}{\mathrm{\&epsiv;}}_6^0$

With the ε in the equation ₁ ⁰Solve, be respectively:

${\mathrm{\&epsiv;}}_1^0=\frac{A_{22}}{A_{11}{A}_{22}-{\mathrm{<figure-callout\; id="12"\; label="A"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">A</figure-callout>}}_{12}^2}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_1^0-\frac{A_{12}}{A_{11}{A}_{22}-A_{21}^2}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_2^0$

${\mathrm{\&epsiv;}}_2^0=\frac{A_{21}}{A_{21}^2-A_{11}{A}_{22}}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_1^0+\frac{A_{11}}{A_{11}{A}_{12}-A_{21}^2}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_2^0---\left(16\right)$

${\mathrm{\&epsiv;}}_6^0=\frac{1}{A_{66}}\mathrm{\&delta;}\&CenterDot;{\mathrm{\&sigma;}}_6^0$

Thereby solve flexibility component relational expression and be:

$a_{11}=\frac{A_{22}}{A_{11}{A}_{22}-{\mathrm{<figure-callout\; id="12"\; label="A"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">A</figure-callout>}}_{12}^2}$

$a_{22}=\frac{A_{11}}{A_{11}{A}_{22}-A_{21}^2}---\left(17\right)$

$a_{12}=a_{21}=\frac{A_{12}}{{\mathrm{<figure-callout\; id="12"\; label="A"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">A</figure-callout>}}_{12}^2-A_{11}{A}_{22}}$

$a_{66}=\frac{1}{A_{66}}$

Get the interior stiffness equations of face of ejecting plate:

$E_1^0\mathrm{\&delta;}=\frac{1}{a_{11}}=A_{11}-\frac{{\mathrm{<figure-callout\; id="12"\; label="A"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">A</figure-callout>}}_{12}^2}{A_{22}}$

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^0\mathrm{\&delta;}=\frac{1}{a_{22}}=A_{22}-\frac{A_{21}^2}{A_{11}}---\left(18\right)$

${\mathrm{<figure-callout\; id="6"\; label="E"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">E</figure-callout>}}_6^0\mathrm{\&delta;}=\frac{1}{a_{66}}=A_{66}$

The above-mentioned equation of Aij relational expression substitution is drawn:

$E_1^0\mathrm{\&delta;}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{11j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}-\frac{{(\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{12j}\&CenterDot;N_j\&CenterDot;\mathrm{TB})}^2}{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{22j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}}$

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^0\mathrm{\&delta;}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{22j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}-\frac{{(\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{12j}\&CenterDot;N_j\&CenterDot;\mathrm{TB})}^2}{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{11j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}}---\left(19\right)$

$\mathrm{G\&delta;}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{Q}_{66j}\&CenterDot;N_j\&CenterDot;\mathrm{TB}$

2. satisfy by two to the layer design of the RATE OF COMPOSITE LAMINATES shop of rigidity requirement

2.1 the derivation of equation

We obtain from the derivation of first, when aerofoil by the engineering beam carried out providing when aeroelasticity is calculated aerofoil along exhibition to bending stiffness EI1 and torsional stiffness GJ.Conversion through formula (2) and (6) just can provide on each tangent plane covering two to rigidity requirement, and promptly axial stiffness requires E δ and shear rigidity requirement G δ.Rigidity should satisfy this simultaneously two to rigidity requirement in the face of the RATE OF COMPOSITE LAMINATES of being set up by equation (19), thereby the set of equations below having set up.

$E{\mathrm{\&delta;}}_1=E_1^0\mathrm{\&delta;}$

$\mathrm{G\&delta;}={\mathrm{<figure-callout\; id="6"\; label="E"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">E</figure-callout>}}_6^0\mathrm{\&delta;}$

Satisfy two under the rigidity situation a minimum shop layer angle be two groups owing to be balancing disk, thereby every group removed 0 °, outside 90 °, all be ± α or ± β.And the stiffness coefficient of the positive negative angle of balanced composite panel is identical, thereby we just use α, β to represent these two groups positive negative angle shop layer angle groups.Corresponding every kind of angle shop number of plies uses N1 and N2 to represent, the rigidity component Qik α of corresponding every kind of angle, and Qik β represents respectively.Set of equations (20) can be write as:

$\mathrm{E\&delta;}=({\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&beta;}}{N}_2)\&CenterDot;\mathrm{TB}-\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{N}_2)}^2\mathrm{TB}}{Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2}$

$\mathrm{G\&delta;}=(Q_{66}^{\mathrm{\&alpha;}}{N}_1+Q_{66}^{\mathrm{\&beta;}}{N}_2)\mathrm{TB}$

2.2 result of calculation discussion

Our corresponding as can be seen a certain β angle time shop layer N1 or N2 are zero in the calculating; Be that corresponding this angle satisfied two only needs a kind of shop layer to get final product to rigidity requirement.Here we cry its " list is spread a layer angle γ ".Corresponding each shear rigidity requires than having only a γ angle with axial stiffness.But along with the variation γ of G δ/E δ also changes thereupon.

By to calculated result analysis, we can obtain following some conclusion:

Result of calculation shows, is satisfying two under the prerequisite of rigidity requirement, is not that any two kinds of angles shop laminate is all separated.But γ just separates when a kind of shop layer angle α spreads layer angle less than list.Otherwise when α＞γ, then must just separate less than γ by β.When we known two behind rigidity requirement, can from figure, find the γ angle, just can select α then, the scope at β angle.

The total result of calculation of minimum shop layer about various combination of angles shows, obtains the minimum shop number of plies when γ is got at layer angle, pawnshop.Be and satisfy two to the minimum weight shop of rigidity requirement layer.But we see that also it is all very little always to spread number of plies variable quantity when the α angle changes in 0 ° to 45 ° scope.In calculating example, from 0 ° to 45 ° range shop layer sum only from minimum γ angle 19.5 layers change to 45 ° 20.3 layers.Variable quantity is 4%.Thereby in this scope, be optimized to calculate and have little significance.After α or β angle are greater than 60 °, a shop layer sum will sharply increase, thereby should not select the layer angle, shop greater than 60 °.

Because this program is to satisfy the shop layer design that longitudinal rigidity and shear rigidity require, from Fig. 4, lateral stiffness E δ 2 finds out with layer angle, a shop β situation of change in 6, though always in 0 ° to 45 ° scope, spread the number of plies change little, and lateral stiffness to change be very greatly.Locate ratio of rigidity K=24.9% in length and breadth when α=γ angle in the example of being calculated; And when α=45 ° K=45.7%.Shop layer sum only increases by 4%.Thereby it is good by 0 ° and the 45 ° shop layer of forming to consider that from Combination property shop, γ angle layer instead is not so good as.We see when selecting α=0 ° for use, K ≈ 1 during β=60 °.Therefore be isotropic plate.Elected β angle will increase sharply greater than lateral stiffness after 60 °, and a layer gross thickness in shop also increases sharply thereupon.

From Fig. 7 we as can be seen the γ angle be with given two change to ratio of rigidity G δ/E δ 1, and along with ratio of rigidity increases and increases.The γ angle can only change between 0 ° and 45 ° during calculating, otherwise does not just separate.Because unidirectional composite material G δ/E δ 1=0.017 in this example, 45 ° of shop layer G δ/E δ 1=1.155, thereby two also must just separating of requiring at following formula to ratio of rigidity.Have the scope of separating to be for this routine situation:

$0.017\&le;\frac{\mathrm{G\&delta;}}{E{\mathrm{\&delta;}}_1}\&le;1.155$

Here to illustrate a bit that different materials G δ/E δ 1 scope also is different.

3 shop layer design by the three-way rigidity requirement

3.1 the derivation of equation

Satisfy equation by the three-way rigidity requirement.We establish layer angle, three kinds of shops is α, β, and the θ and the corresponding shop number of plies are N1, N2, N3.

$E_1\mathrm{\&delta;}=({\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&theta;}}{N}_3)\&CenterDot;\mathrm{TB}-\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&theta;}}{N}_3)}^2\mathrm{TB}}{Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{22}^{\mathrm{\&theta;}}{\mathrm{<figure-callout\; id="3"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_3}$

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2\mathrm{\&delta;}=(Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{22}^{\mathrm{\&theta;}}{N}_3)\&CenterDot;\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&theta;}})}^2\mathrm{<figure-callout\; id="11"\; label="TB\; Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">TB</figure-callout>}}{Q_{}^{}}$

$\mathrm{G\&delta;}=(Q_{66}^{\mathrm{\&alpha;}}{N}_1+Q_{66}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{66}^{\mathrm{\&theta;}})\mathrm{TB}$

For separating above-mentioned nonlinear simultaneous equations group, thereby solve the shop number of plies N1 under the various combination of angles, N2, N3.

3.2 result of calculation and verification experimental verification

The three-way rigidity value is: E δ 1=72670kg/cm2; E δ 2=79330kg/cm2;

Gδ＝38700kg/cm2；

The unidirectional composite material key property:

EX＝1180000kg/cm2；EY＝98000kg/cm2；

ES＝98000kg/cm2；ve＝0.28；

TB＝0.0134cm。

We utilize program to α=0 °, and β changes to 90 ° by 0 ° and increases progressively with 5 °, and θ changes to 90 ° by 0 ° and increases progressively with 1 °, has calculated the shop layer situation of 1600 multiple angles combinations altogether.Program has provided N1, N2, and the numerical value of N3 and N1+N2+N3 has also provided the N1 after rounding, N2, the numerical value of N3 and N1+N2+N3.And the three-way rigidity situation of change after rounding.

Result of calculation is analyzed:

Satisfying the three-way rigidity requirement is not that any three kinds of combination of angles all have normal solution.As can be seen, these two kinds of combination of angles are not all separated between θ is from 35 ° to 50 ° among Figure 11.Promptly when α=0 °, during β=90 ° during θ＞35 ° N3 separate and be negative shop layer. when α=θ °, during β=45 ° during θ＜45 ° N3 separate and also be negative shop layer.Thereby must select shop layer angle group could satisfy the three-way rigidity requirement.

Satisfy three-way rigidity requirement general run of thins and need three kinds of angle shop layers.Need positive and negative shop layer as every kind of angle is another, be actually six kinds of angle shop layers.But N3=0 during second group of θ=50 ° when first group of θ=35 ° promptly just can satisfy the three-way rigidity requirement with two kinds of angles shop layers as can be seen from Figure.We are called " layer angle, two shop " γ these θ.This angle is again to distinguish the layer angle of selecting, shop whether the critical angle of separating is arranged.

Result of calculation shows, satisfies given three-way rigidity and requires situation, no matter how three kinds of angles make up, be that N1+N2+N3 is a constant always spread the number of plies.So just there is not shop layer angle computation optimization problem.Below we prove with mathematical derivation:

$E_1^0\mathrm{\&delta;}=\frac{1}{a_{11}}=A_{11}-\frac{{\mathrm{<figure-callout\; id="12"\; label="A"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">A</figure-callout>}}_{12}^2}{A_{22}}---\left(a\right)$

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^0\mathrm{\&delta;}=\frac{1}{a_{22}}=A_{22}-\frac{A_{21}^2}{A_{11}}---\left(b\right)$

${\mathrm{<figure-callout\; id="6"\; label="E"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">E</figure-callout>}}_6^0\mathrm{\&delta;}=\frac{1}{a_{66}}=A_{66}---\left(c\right)$

Obtaining A66 from equation (c) is constant A 66=G δ.(d)

Equation (a) * A22-(b) * A11 is obtained

Eδ ₁·A ₂₂-Eδ ₂·A ₁₁＝0 (e)

Thereby $\frac{A_{11}}{A_{22}}=\frac{{\mathrm{E\&delta;}}_1}{{\mathrm{E\&delta;}}_2}=C---\left(f\right)$

Provide following relation by the Compound Material Engineering formula:

A ₆₆-A ₁₂＝(U ₅-U ₄)·δ (g)

A ₁₁+A ₂₂+2A ₁₂＝2(U ₁+U ₄)·δ (h)

Equation (d) substitution (g) is got:

A ₁₂＝Gδ-(U ₅-U ₄)·δ (i)

Equation (i) substitution (h) is got

A ₁₁+A ₂₂＝2(U ₁+U ₅)·δ-Gδ (j)

Equation (f) substitution (j) is got

$A_{11}=\frac{2C}{1+C}[(U_1+U_5)\&CenterDot;\mathrm{\&delta;}-\mathrm{G\&delta;}]---\left(k\right)$

$A_{22}=\frac{2}{1+C}[(U_1+U_5)\&CenterDot;\mathrm{\&delta;}-\mathrm{G\&delta;}]---\left(l\right)$

With equation (i), (l), (k) obtain in the substitution equation (a):

$E{\mathrm{\&delta;}}_1=\frac{2C}{1+C}[(U_1+U_5)\&CenterDot;\mathrm{\&delta;}-\mathrm{G\&delta;}]-\frac{(1+C)[(U_5-U_4)\&CenterDot;\mathrm{\&delta;}-\mathrm{G\&delta;}{]}^2}{2\left[\right(U_1+U_5]\&CenterDot;\mathrm{\&delta;}-\mathrm{G\&delta;}]}$

Because δ=(N ₁+ N ₂+ N ₃) TB

Thereby from last equation, can find out, a δ and a shop layer angle α, β, θ is irrelevant, and only with E δ 1, E δ 2, G δ and engineering constant U1, U4, U5 is relevant.

Experimental result and the result of calculation trend at layer angle, three kinds of shops more as can be seen conform to.Total shop number of plies error is 11.4%, and owing to factor on technology, the performance, some error is normal on every kind of angle shop number of plies.

Description of drawings:

The three-way rigidity of Fig. 1 composite material skin is represented

The three-dimensional stress strain of Fig. 2 composite material unit

Fig. 3 satisfies two under the rigidity situation, and the shop number of plies is with β angle change curve

Fig. 4 satisfy two under the rigidity situation k=E20 δ/E10 δ with a shop layer angle β change curve (only providing α=00,450 two kind of situation)

Shop layer under several angle [alpha] situations of Fig. 5 is with β angle change curve

The k of ratio of rigidity in length and breadth under several angle [alpha] situations of Fig. 6 is with β angle change curve

Fig. 7 satisfy two under the rigidity situation γ angle with G δ/E1 δ change curve

The section of the main stress box of Fig. 8 aerofoil number

Fig. 9 aeroelasticity is to the bending of aerofoil and the requirement of torsional stiffness

Figure 10 compares the requirement of aerofoil according to two rigidity and aeroelasticities to layer back, rigidity shop composite material aerofoil

Figure 11 three-way rigidity shop layer result and test results

The specific embodiment

Step 1: regard aerofoil as the engineering beam when aeroelasticity is calculated and simplify

After aerofoil is simplified to the engineering beam,, propose along the distribution curve of each tangent plane bending stiffness EI1 of axis and torsional stiffness GJ requirement according to aerofoil aeroelasticity result of calculation.Like this in structure design satisfying under the prerequisite of requirement of strength, also should satisfy rigidity requirement.

Step 2: satisfy by two and design to the RATE OF COMPOSITE LAMINATES shop of rigidity requirement layer:

We obtain from the derivation of first, when aerofoil by the engineering beam carried out providing when aeroelasticity is calculated aerofoil along exhibition to bending stiffness EI1 and torsional stiffness GJ.Conversion through formula (2) and (6) just can provide on each tangent plane covering two to rigidity requirement, and promptly axial stiffness requires E δ and shear rigidity requirement G δ.Rigidity should satisfy this simultaneously two to rigidity requirement in the face of the RATE OF COMPOSITE LAMINATES of being set up by equation (19), thereby the set of equations below having set up.

$E{\mathrm{\&delta;}}_1=E_1^0\mathrm{\&delta;}$

$\mathrm{G\&delta;}={\mathrm{<figure-callout\; id="6"\; label="E"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">E</figure-callout>}}_6^0\mathrm{\&delta;}$

Satisfy two under the rigidity situation a minimum shop layer angle be two groups owing to be balancing disk, thereby every group removed 0 °, outside 90 °, all be ± α or ± β.And the stiffness coefficient of the positive negative angle of balanced composite panel is identical, thereby we just use α, β to represent these two groups positive negative angle shop layer angle groups.Corresponding every kind of angle shop number of plies uses N1 and N2 to represent, the rigidity component Qik α of corresponding every kind of angle, and Qik β represents respectively.Set of equations (20) can be write as:

$\mathrm{E\&delta;}=({\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&beta;}}{N}_2)\&CenterDot;\mathrm{TB}-\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{N}_2)}^2\mathrm{TB}}{Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2}$

$\mathrm{G\&delta;}=(Q_{66}^{\mathrm{\&alpha;}}{N}_1+Q_{66}^{\mathrm{\&beta;}}{N}_2)\mathrm{TB}$

Establishment corresponding calculated program, in aerofoil each section bending and torsional stiffness that input needs, layer angle spread in given design, can calculate the shop number of plies of each angle of each section.According to the shop layer thickness, carry out former standard.

Step 3: by the shop layer design of three-way rigidity requirement

A kind of shop laminate is calculated, provide its three-way rigidity value.

The exploitation of following software system is carried out in the requirement at the layer angle, another kind of situation shop that converts to as required.

Satisfy the equation that the three-way rigidity of given plate requires.We establish layer angle, three kinds of shops is α, β, and the θ and the corresponding shop number of plies are N1, N2, N3.Draw down and establish an equation:

$E_1\mathrm{\&delta;}=({\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="11"\; label="Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{11}^{\mathrm{\&theta;}}{N}_3)\&CenterDot;\mathrm{TB}-\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&theta;}}{N}_3)}^2\mathrm{TB}}{Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{22}^{\mathrm{\&theta;}}{\mathrm{<figure-callout\; id="3"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_3}$

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">E</figure-callout>}}_2\mathrm{\&delta;}=(Q_{22}^{\mathrm{\&alpha;}}{N}_1+Q_{22}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{22}^{\mathrm{\&theta;}}{N}_3)\&CenterDot;\mathrm{TB}-\frac{{({\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&alpha;}}{N}_1+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+{\mathrm{<figure-callout\; id="12"\; label="Q"\; filenames="A20051007113500171.png,A20051007113500181.png"\; state="\{\{state\}\}">Q</figure-callout>}}_{12}^{\mathrm{\&theta;}})}^2\mathrm{<figure-callout\; id="11"\; label="TB\; Q"\; filenames="A20051007113500181.png"\; state="\{\{state\}\}">TB</figure-callout>}}{Q_{}^{}}$

$\mathrm{G\&delta;}=(Q_{66}^{\mathrm{\&alpha;}}{N}_1+Q_{66}^{\mathrm{\&beta;}}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A20051007113500171.png"\; state="\{\{state\}\}">N</figure-callout>}}_2+Q_{66}^{\mathrm{\&theta;}})\mathrm{TB}$

Worked out " the compound material laying layer by the three-way rigidity requirement is designed program " for separating above-mentioned nonlinear simultaneous equations group.

At the three-way rigidity, the layer angle, shop of requirement of this plate of input, utilize this software can solve shop number of plies N1 under the various combination of angles, N2, N3.

Step 4: former accurate result of calculation

Because composite material is to spread layer by certain requirement, particularly guarantees its symmetry, need be with the former standard of its result of calculation, for example, 45 ° of shop layers that provide are 3 layers, need former standard to become 4 layers, to guarantee its symmetry.

Claims (6)

1. carry out the method for composite material aerofoil shop layer by rigidity requirement, the gross thickness of the laminate that the composite panel three-way rigidity at layer angle, several different shop is identical equates that promptly weight characteristic is identical.

2. according to the method for claim 1, can realize aerofoil is carried out according to rigidity requirement shop layer.Can solve the shop layer thickness and the shop layer gross thickness of given several shops layer angle.Comprising according to two to rigidity requirement spread the layer and according to the three-way rigidity requirement spread the layer method.

3. according to the method that with all strength requires 1 and 2, the complicated calculations model that can be in aeroelasticity is calculated replaces the multiple shop of multiple angles number of plies amount with the three-way rigidity and the cooresponding weight relationships of covering.

4. according to claim 1,2,3 method, set up the equivalent stiffness conversion of composite panel.Be about to a kind of multi-angle shop laminate, according to the principle that three-way rigidity equates, weight is identical, the three-way rigidity that can be converted to another group angle equates and equiponderant plate.This can adjust the result after the optimization after aerofoil is optimized, make its layer angle, shop convert given layer angle, shop to.Can solve the possibility that realizes on the technology like this.Its concrete steps are: at first obtain the three-way rigidity of plate according to the shop layer at each the layer angle, shop after optimizing, utilize the method for this claim to obtain the shop number of plies at the layer angle, given shop of equivalent three-way rigidity again.According to claim 1,2,3 method, the weight of newly spreading laminate is identical with former shop laminate.

5. the shop layer of an aircraft vertical fin, it designs by using each method of claim 1-4.

6. according to the aircraft vertical fin of claim 5 shop layer, its aerofoil bending stiffness and turn round rigidity as shown in figure 10.Well satisfy mutually with the designing requirement of Fig. 9.This vertical fin test shows, meets design requirement fully.

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