CN102682168B - Method for determining dimension of rectangular beam section of low-speed flutter main beam model of airplane - Google Patents

Abstract

The invention belongs to the field of structural mechanics and relates to a method for determining the dimension of a rectangular beam section of a low-speed flutter main beam model of an airplane. The method is characterized by adapting to the situation that the half width a of a rectangular section is smaller than the half height b of the rectangular section or the situation that J/Ix<1.69, wherein Ix refers to vertical inertia moment and J refers to polar inertia moment. The method for determining the dimension of the rectangular beam section with a lug piece comprises the following steps of: calculating the half width ar and the half height br of the rectangular section when the rectangular beam without the lug piece has the vertical inertia moment Ix of a preset value and the polar inertia moment J of a preset value; adjusting the polar inertia moment J with the preset value; calculating the half width a and the half height b; and calculating the half width 1 of the rectangular beam section with the lug piece when the rectangular beam with the lug piece has lateral inertia moment Iy of a preset value and the thickness of the lug piece is t. The method disclosed by the invention has the advantages of increasing the rigidity precision of the section of the model, reducing the uncertainty of model design, shortening the time for determining the dimension of the section, and increasing the design efficiency of the flutter model.

Description

A kind of aircraft low-speed flutter girder model rectangular beam sectional dimension is determined method

Technical field

The invention belongs to structural mechanics field, relate to a kind of aircraft low-speed flutter girder model rectangular beam sectional dimension and determine method.

Background technology

For the airplane flutter modelling of high aspect ratio wing, the version of single-beam model is widely used in the structures such as wing, fuselage, empennage.In order to meet the designing requirement of three-way rigidity, with the square-section of auricle, be a desirable design form.

The method that in the past conventionally need to gather by examination is obtained the sectional dimension meeting the demands.The method that examination is gathered has following shortcoming: the first, adjust the experience that sectional dimension data needs designer, the dimensional data providing by rule of thumb often error is very large, even there will be the situation that is difficult to adjust the size meeting design requirement, affect the precision of model section rigidity, increased the uncertainty of modelling.The second, by method of trial and error, will determine sectional dimension, need to carry out artificial adjustment and the judgement of many rounds, the time is long, and efficiency is low, has a strong impact on the modelling cycle.

Referring to Chinese patent < < band auricle rectangular beam sectional dimension, determine method > > (application number 201110232655.8), without examination, gather, only need provide and comprise auricle thickness t, vertically to moment of inertia I _x, side direction moment of inertia I _yin interior input data, can obtain sectional dimension data with polar moment of inertia J, comprise width 2a, height 2b and the cross-sectional width 2l of rectangle, but the applicable elements of this method be J/I _x>=1.69.Work as J/I _x< 1.69, and design error can increase.

Summary of the invention

The object of the invention is: propose a kind of aircraft low-speed flutter girder model rectangular beam sectional dimension and determine method, to improve the precision of model section rigidity, reduce the uncertainty of modelling, shorten the time of determining sectional dimension, improve the design efficiency of flutter model, and be applicable to cross section J/I _xrequirement during < 1.69.

Technical solution of the present invention is: aircraft low-speed flutter girder model rectangular beam cross section is the rectangular beam with auricle, guarantee with auricle rectangular beam have predetermined value vertically to moment of inertia I _x, side direction moment of inertia I _yin polar moment of inertia J and the auricle thickness situation that is t, determine width 2a, height 2b and the cross-sectional width 2l of its rectangle, it is characterized in that, determine as follows with the step of auricle rectangular beam sectional dimension:

1, calculate not with auricle rectangular beam have predetermined value vertically to moment of inertia I _xsquare-section half width a during with polar moment of inertia J _rwith cross section half height b _r:

1.1, set the initial value α of iteration variable α ₀=0.93 ~ 0.99:

1.2, according to following formula, calculate intermediate variable β and iteration variable α ₁:

$\left\{\begin{array}{c}24{\left(\frac{2}{\mathrm{\&pi;}}\right)}^5{\mathrm{\&alpha;}}_0{\mathrm{\&beta;}}^3-4{\mathrm{\&beta;}}^2+\frac{J}{I_x}=0\\ {\mathrm{\&alpha;}}_1=\tanh \frac{\mathrm{\&pi;}}{2\mathrm{\&beta;}}+\frac{1}{243}\tanh \frac{3\mathrm{\&pi;}}{2\mathrm{\&beta;}}\end{array}\right....\left[1\right]$

Attention: when solving β, relate to solving of simple cubic equation, suggestion adopts Secant Method, and β ∈ (0,1);

1.3, calculate iterative value α ₁with initial value α ₀scale error e ₁:

$e_1=\frac{{\mathrm{\&alpha;}}_1-{\mathrm{\&alpha;}}_0}{{\mathrm{\&alpha;}}_0}...\left[2\right]$

If | e ₁| < 0.000001, square-section half width a _rwith cross section half height b _rfor:

$\left\{\begin{array}{c}{a}_r=\sqrt[4]{\frac{{3I}_x{\mathrm{\&beta;}}^3}{4}}\\ b_r=\sqrt[4]{\frac{3I_x}{4\mathrm{\&beta;}}}\end{array}\right....\left[3\right]$

Step 1.1 finishes; Otherwise, carry out step 1.4;

1.4, make α ₀=α ₁, the method for repeating step 1.2～step 1.3, by iterative computation, until | e ₁| < 0.000001, and obtains square-section half width a now _rwith cross section half height b _r, step 1 finishes;

2, predetermined polar moment of inertia J is adjusted: according to following formula, calculate the value J after polar moment of inertia J adjusts ₁,

Note answering guaranteed conditions: t < a _r;

3, calculate half width a and half height b: change the J in formula [1] into J ₁, then according to the method described in step 1, calculate half width a and half height b;

4, calculate band auricle rectangular beam cross section half-breadth l:

4.1, according to following formula, calculate intermediate variable δ:

$\mathrm{\&delta;}=\frac{{3I}_y-{4a}^{3}b}{2t}...\left[5\right]$

4.2, calculate band auricle rectangular beam cross section half-breadth l:

$l=\sqrt[3]{\mathrm{\&delta;}+a^3}...\left[6\right]$

So far, obtain half width a, half height b and cross section half-breadth l with auricle rectangular beam, and then obtain width 2a, height 2b and the cross-sectional width 2l of corresponding rectangular beam.

Advantage of the present invention is: improved the precision of model section rigidity, reduced the uncertainty of modelling, shortened the time of definite sectional dimension, improved the design efficiency of flutter model, and met J/I _xthe requirement of < 1.69.One embodiment of the present of invention are compared with the current examination method of gathering, and the present invention determines that the time of sectional dimension is only 3 minutes, and the current examination method of gathering needs 4 hours, and the used time of the present invention is only for trying to gather 1/80th of method.

Accompanying drawing explanation

Fig. 1 is the schematic cross section of not being with the rectangular beam of auricle.A in figure _rthe half width of the rectangle that calculates of step 1 of the present invention, b _rit is half height of the rectangle that calculates of step 1 of the present invention.The center that the initial point 0 of the two-dimensional coordinate system in Fig. 1 is rectangle, x axle is parallel to the Width of rectangle, and positive dirction is towards the right side, and the positive dirction of y axle is upward.

Fig. 2 is the schematic cross section with the rectangular beam of auricle.Two-dimensional coordinate system in Fig. 2 is identical with Fig. 1.

Embodiment

Below the present invention is described in further details.Referring to Fig. 1,2, band auricle rectangular beam sectional dimension is determined method, guarantee with auricle rectangular beam have predetermined value vertically to moment of inertia I _x, side direction moment of inertia I _yin polar moment of inertia J and the auricle thickness situation that is t, determine width 2a, height 2b and the cross-sectional width 2l of its rectangle, it is characterized in that, determine as follows with the step of auricle rectangular beam sectional dimension:

1, calculate not with auricle rectangular beam have predetermined value vertically to moment of inertia I _xsquare-section half width a during with polar moment of inertia J _rwith cross section half height b _r:

1.1, set the initial value α of iteration variable α ₀=0.93 ~ 0.99:

1.2, according to following formula, calculate intermediate variable β and iteration variable α ₁:

$\left\{\begin{array}{c}24{\left(\frac{2}{\mathrm{\&pi;}}\right)}^5{\mathrm{\&alpha;}}_0{\mathrm{\&beta;}}^3-4{\mathrm{\&beta;}}^2+\frac{J}{I_x}=0\\ {\mathrm{\&alpha;}}_1=\tanh \frac{\mathrm{\&pi;}}{2\mathrm{\&beta;}}+\frac{1}{243}\tanh \frac{3\mathrm{\&pi;}}{2\mathrm{\&beta;}}\end{array}\right....\left[1\right]$

Attention: when solving β, relate to solving of simple cubic equation, suggestion adopts Secant Method, and β ∈ (0,1);

1.3, calculate iterative value α ₁with initial value α ₀scale error e ₁:

$e_1=\frac{{\mathrm{\&alpha;}}_1-{\mathrm{\&alpha;}}_0}{{\mathrm{\&alpha;}}_0}...\left[2\right]$

If | e ₁| < 0.000001, square-section half width a _rwith cross section half height b _rfor:

$\left\{\begin{array}{c}{a}_r=\sqrt[4]{\frac{{3I}_x{\mathrm{\&beta;}}^3}{4}}\\ b_r=\sqrt[4]{\frac{3I_x}{4\mathrm{\&beta;}}}\end{array}\right....\left[3\right]$

Step 1 finishes; Otherwise, carry out step 1.4;

1.4, make α ₀=α ₁, the method for repeating step 1.2～step 1.3, by iterative computation, until | e ₁| < 0.000001, and obtains square-section half width a now _rwith cross section half height b _r, step 1 finishes;

2, predetermined polar moment of inertia J is adjusted: according to following formula, calculate the value J after polar moment of inertia J adjusts ₁,

Note answering guaranteed conditions: t < a _r;

3, calculate half width a and half height b: change the J in formula [1] into J ₁, then according to the method described in step 1, calculate half width a and half height b;

4, calculate band auricle rectangular beam cross section half-breadth l:

4.1, according to following formula, calculate intermediate variable δ:

$\mathrm{\&delta;}=\frac{{3I}_y-{4a}^{3}b}{2t}...\left[5\right]$

4.2, calculate band auricle rectangular beam cross section half-breadth l:

$l=\sqrt[3]{\mathrm{\&delta;}+a^3}...\left[6\right]$

So far, obtain half width a, half height b and cross section half-breadth l with auricle rectangular beam, and then obtain width 2a, height 2b and the cross-sectional width 2l of corresponding rectangular beam.

Principle of work of the present invention is: by the mechanics of materials and Elasticity fundamental formular, derive and parameter correction, (seeing the logical Higher Education Publishing House 1997 of < < Elasticity > > Yang Gui and < < mechanics of materials > > Dan Hui ancestral Higher Education Publishing House 1999) obtained and a kind ofly from cross section property, directly obtained the method with auricle rectangular beam sectional dimension, for definite sectional dimension, this is a kind of reverse mentality of designing and method, therefore, than needed to gather by artificial examination the method for sectional dimension of obtaining in the past, efficiency and precision are greatly improved, and than the Reverse Design proposing before, this method is more suitable for J/I _x< 1.69 conditions.

Embodiment

To the method for the invention, calculate checking.

Provide three groups of aircraft low-speed flutter girder model band auricle rectangular beam cross sections, P1～P3 cross section is respectively embodiment 1～embodiment 3.Table 1 has provided the geometrical property predetermined value of three embodiment, i.e. desired value, and desired value J/I _x< 1.69.For P1, make t=2.0mm, for P2 and P3, make t=1.0mm, carry out sectional dimension design.Table 1 has also provided the geometrical property of using the design section that the inventive method obtains, the i.e. error of design load, and design load.Table 2 has provided the size design value of three embodiment.

The design load of the cross section geometric characteristic of table 1 is according to the sectional dimension of table 2 correspondence, by FEMAP v9.31, is calculated.The desired value of comparing, the error of design section characteristic value is all not more than 2%, and from engineering viewpoint, this is a kind of high-precision result of calculation.

Table 1 cross section geometric characteristic, unit: mm ⁴.

	I x	I y	J
				P1 desired value	227635.2	213104.0	18696.0
P1 design load	227761.0	213117.0	18535.4
				P1 error amount	0.06%	0.01%	-0.86%
P2 desired value	3528.3	52734.5	5835.9
				P2 design load	3531.3	52745.6	5892.3
P2 error amount	0.09%	0.02%	0.97%
				P3 desired value	2276.35	2131.04	236.96
P3 design load	2282.3	2132.4	232.7
				P3 error amount	0.26%	0.06%	-1.82%

Table 2 cross section geometry design load, unit: mm.

	t	2a	2b	2l
					P1	2.00	9.67	65.62	107.72
P2	1.00	14.08	14.43	84.12
					P3	1.00	3.30	20.24	29.19

Claims (1)

1. aircraft low-speed flutter girder model rectangular beam sectional dimension is determined a method, and aircraft low-speed flutter girder model rectangular beam cross section is the rectangular beam with auricle, guarantee with auricle rectangular beam have predetermined value vertically to moment of inertia I _x, side direction moment of inertia I _yin polar moment of inertia J and the auricle thickness situation that is t, and meet J/I _x< 1.69, determine its width 2a with auricle rectangular beam, with auricle rectangle depth of beam 2b with the cross-sectional width 2l of auricle, it is characterized in that, determine as follows with the step of auricle rectangular beam sectional dimension:

1.1, calculate not with auricle rectangular beam have predetermined value vertically to moment of inertia I _xduring with polar moment of inertia J not with the square-section half width a of auricle _rwith cross section half height b _r:

1.1.1, set the initial value α of iteration variable α ₀=0.93～0.99:

1.1.2, according to following formula, calculate intermediate variable β and iteration variable α ₁:

$\left\{\begin{array}{c}24{\left(\frac{2}{\mathrm{\&pi;}}\right)}^5{\mathrm{\&alpha;}}_0{\mathrm{\&beta;}}^3-4{\mathrm{\&beta;}}^2+\frac{J}{I_x}=0\\ {\mathrm{\&alpha;}}_1=\tanh \frac{\mathrm{\&pi;}}{2\mathrm{\&beta;}}+\frac{1}{243}\tanh \frac{3\mathrm{\&pi;}}{2\mathrm{\&beta;}}\end{array}\right....\left[1\right]$

Attention: when solving β, relate to solving of simple cubic equation, suggestion adopts Secant Method, and β ∈ (0,1);

1.1.3, calculate iterative value α ₁with initial value α ₀scale error e ₁:

$e_1=\frac{{\mathrm{\&alpha;}}_1-{\mathrm{\&alpha;}}_0}{{\mathrm{\&alpha;}}_0}...\left[2\right]$

If | e ₁| < 0.000001, square-section half width a _rwith cross section half height b _rfor:

$\left\{\begin{array}{c}{a}_r=\sqrt[4]{\frac{3I_x{\mathrm{\&beta;}}^3}{4}}\\ b_r=\sqrt[4]{\frac{3I_x}{4\mathrm{\&beta;}}}\end{array}\right....\left[3\right]$

Step 1.1 finishes; Otherwise, carry out step 1.1.4;

1.1.4, make α ₀=α ₁, the method for repeating step 1.1.2～step 1.1.3, by iterative computation, until | e ₁| < 0.000001, and obtains square-section half width a now _rwith cross section half height b _r, step 1.1 finishes;

1.2, predetermined polar moment of inertia J is adjusted: according to following formula, calculate the value J after polar moment of inertia J adjusts ₁,

Note answering guaranteed conditions: t < a _r;

1.3, calculate half width a and half height b: change the J in formula [1] into J ₁, then according to the method for described step 1.1, calculate half width a and half height b;

1.4, calculate band auricle rectangular beam cross section half-breadth l:

1.4.1, according to following formula, calculate intermediate variable δ:

$\mathrm{\&delta;}=\frac{3I_y-4a^{3}b}{2t}...\left[5\right]$

1.4.2, calculate band auricle rectangular beam cross section half-breadth l:

$l=\sqrt[3]{\mathrm{\&delta;}+a^3}...\left[6\right]$

So far, obtain half width a, half height b and cross section half-breadth l with auricle rectangular beam, and then obtain width 2a, rectangle depth of beam 2b and the cross-sectional width 2l of corresponding rectangular beam.