CN102661729B - Method for confirming sectional dimension of I-shaped hollow beam of high-speed fluttering model of airplane - Google Patents

Abstract

The invention belongs to the field of structural mechanics and relates to a method for determining the section size of an I-shaped hollow beam of an aircraft high-speed flutter model. It is characterized in that, the step of determining the section size of the thin-walled rectangular hollow beam with four lugs is as follows: calculate the vertical moment of inertia _Ix and the polar moment of inertia J of the thin-walled rectangular hollow beam with a predetermined value and the rectangular wall thickness t Equivalent width a ₁ and equivalent height b ₁ ; adjust the predetermined polar moment of inertia J; calculate equivalent width a ₂ and equivalent height b ₂ ; take b ₂ as the maximum range of b ₃ , iteratively obtain the cross section The characteristic governing equations are the section dimensions a ₃ , b ₃ and L ₃ . The invention improves the accuracy of model section design, reduces the uncertainty of model design, shortens the time for determining the section size, improves the design efficiency of the flutter model, and reduces the The overall height of the section satisfies the size restriction to a certain extent.

Description

A Method for Determining the Section Dimensions of I-shaped Hollow Beams of Aircraft High-speed Flutter Model

technical field

The invention belongs to the field of structural mechanics and relates to a method for determining the section size of an I-shaped hollow beam of an aircraft high-speed flutter model.

Background technique

The high-speed flutter model can be used to obtain the transonic flutter characteristics of the aircraft and its components, and the high-speed flutter model usually requires a small beam mass to provide a large airfoil stiffness, and in order to meet the three-dimensional stiffness design Rectangular thin-walled hollow beam section with lugs is an ideal design form.

In the past, it was usually necessary to obtain the required section size by trial and error. The trial and error method has the following disadvantages: First, the adjustment of the section size data requires the experience of the designer, and the size data given by experience often has a large error, and it may even be difficult to adjust the size that meets the design requirements, which affects the model. The accuracy of the section stiffness increases the uncertainty of the model design; second, to determine the section size through the trial and error method, multiple rounds of manual adjustment and judgment are required, which takes a long time and low efficiency, seriously affecting the model design cycle.

Refer to the Chinese patent "Method for Determining Sectional Dimensions of Thin-walled Rectangular Hollow Beams with Ears" (Application No. 201110232656.2), in order to ensure that the rectangular thin-walled hollow beams with ears have predetermined values of vertical moment of inertia I _x and lateral moment of inertia I When _y , polar moment of inertia J, and wall thickness and lug thickness are both t, the equivalent width a, equivalent height b, and total width L of the hollow beam section can be directly determined. This method does not require trial and error , with high accuracy and efficiency. However, the characteristic of the section corresponding to this method is that the lugs are located on the neutral plane of the section. This kind of section may cause troubles in model design and processing because the equivalent height b value is too large.

Contents of the invention

The purpose of the present invention is: to propose a method for determining the section size of an I-shaped hollow beam of an aircraft high-speed flutter model, to improve the accuracy of the model section stiffness, reduce the uncertainty of model design, shorten the time for determining the section size, and improve flutter The design efficiency of the model, and the lugs of the thin-walled rectangular hollow beam with four lugs proposed by the present invention are located at the four corners of the rectangle, and under the condition of the same section characteristic target value, the total height of the section is effectively reduced .

The technical solution of the present invention is: the method for determining the cross-sectional size of the I-shaped hollow beam of the high-speed flutter model of the aircraft, after ensuring that the I-shaped hollow beam of the high-speed flutter model of the aircraft has the vertical moment of inertia I _x and the lateral moment of inertia I _y and the polar moment of inertia J, and when the wall thickness of the rectangular thin-walled hollow beam is t and the thickness of the lug is t _r , determine its rectangular equivalent width a ₃ , equivalent height b ₃ and the total width L ₃ of the hollow beam section. All the length units are stipulated here as mm. It is characterized in that the steps of determining the section size of the I-shaped hollow beam of the aircraft high-speed flutter model are as follows:

1. Let t _r =n _r t;

2. Calculate the rectangular equivalent width a 1 and etc. of the thin-walled rectangular hollow beam when the vertical moment of inertia I _x and the polar moment of inertia J of the rectangular thin-walled hollow beam have a predetermined value, and the wall thickness of the rectangular thin-walled hollow beam is t ₁ and the thickness of the lug is _{t r} Effective height b ₁ :

2.1. Calculate the first intermediate variable p and the second intermediate variable q according to the following formula:

$\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span>$

2.2. Calculate the third intermediate variable s according to the following formula:

$\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ]</span>$

2.3. Calculate equivalent width a ₁ and equivalent height b ₁ :

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; -</span>\frac{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; <figure-callout\; id="3"\; label="s"\; filenames="HDA00001629009100012.png"\; state="\{\{state\}\}">s</figure-callout></span>}}}{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ {\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; <figure-callout\; id="3"\; label="s"\; filenames="HDA00001629009100012.png"\; state="\{\{state\}\}">s</figure-callout></span>}}\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ]</span>$

3. Adjust the predetermined polar moment of inertia J: Calculate the adjusted value J ₁ of the polar moment of inertia J according to the following formula:

J ₁ ＝J[1-t _r /(2b ₁ )]………………………[4]

4. Calculate the equivalent width a ₂ and equivalent height b ₂ : replace J in the formula [1] with J ₁ , and then calculate the equivalent width a ₂ and equivalent height b ₂ according to the method described in step 2;

5. Iteratively calculate the governing equation of the section properties and obtain the section size:

5.1. Let b _3k =b ₂ -0.1k, variable k=1,2,3,...,int(10b ₂ ), where int() is a rounding function;

5.2. Calculate a _3k according to b _3k :

${\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ]</span>$

5.3. Calculate L _3k according to a _3k and b _3k :

${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; ]</span>$

5.4. Calculate f _k according to a _3k , b _3k and L _3k :

5.5. Calculate err _k according to f _k :

${\mathrm{<span\; class="notranslate">\; err</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; |</span>}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 100</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; ]</span>$

5.6. Obtain the section size:

Find a _3k and b _3k corresponding to the minimum error value err _k and calculate L _3k , that is, calculate L ₃ for the final cross-sectional dimensions a ₃ and b ₃ .

The invention has the advantages of improving the accuracy of the section stiffness of the model, reducing the uncertainty of model design, shortening the time for determining the size of the section, and improving the design efficiency of the flutter model. Compared with the current trial and error method, the present invention only takes 10 minutes to determine the section size, while the trial and error method needs 25 hours, and the time spent by the present invention is only 150 minutes of the trial and error method. one. At the same time, the overall height of the section is reduced to meet the size constraints, which brings convenience to model design and processing.

Description of drawings

Figure 1 is a schematic cross-sectional view of a thin-walled rectangular hollow beam without lugs. In the figure, a ₁ is the equivalent width of the rectangle calculated in Step 1 of the present invention, and a ₁ = the width of the outer edge of the rectangle-thin-wall thickness t. b ₁ is the equivalent height of the rectangle calculated in step 1 of the present invention, b ₁ = height of the outer edge of the rectangle - thickness t of the thin wall. The origin 0 of the two-dimensional coordinate system in Figure 1 is the center of the rectangle, the x-axis is parallel to the width direction of the rectangle, the positive direction is to the right, and the positive direction of the y-axis is upward.

Fig. 2 is a cross-sectional schematic diagram of an I-shaped hollow beam of an aircraft high-speed flutter model. The two-dimensional coordinate system in Fig. 2 is the same as that in Fig. 1 .

Detailed ways

The present invention will be described in further detail below. Referring to Figures 1 and 2, the determination method of the section size of the I-shaped hollow beam of the aircraft high-speed flutter model is to ensure that the section of the I-shaped hollow beam of the aircraft high-speed flutter model has a predetermined value of vertical moment of inertia I _x and lateral moment of inertia I _y And the polar moment of inertia J, the wall thickness of the rectangular thin-walled hollow beam is t, and the thickness of the lug is t _r , determine its rectangular equivalent width a ₃ , equivalent height b ₃ and the total width L ₃ of the hollow beam section. All lengths are specified here in mm. It is characterized in that the steps of determining the section size of the I-shaped hollow beam of the aircraft high-speed flutter model are as follows:

1. Let t _r =n _r t;

2. Calculate the rectangular equivalent width a 1 and etc. of the thin-walled rectangular hollow beam when the vertical moment of inertia I _x and the polar moment of inertia J of the rectangular thin-walled hollow beam have a predetermined value, and the wall thickness of the rectangular thin-walled hollow beam is t ₁ and the thickness of the lug is _{t r} Effective height b ₁ :

2.1. Calculate the first intermediate variable p and the second intermediate variable q according to the following formula:

$\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span>$

2.2. Calculate the third intermediate variable s according to the following formula:

$\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ]</span>$

2.3. Calculate equivalent width a ₁ and equivalent height b ₁ :

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; -</span>\frac{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; <figure-callout\; id="3"\; label="s"\; filenames="HDA00001629009100012.png"\; state="\{\{state\}\}">s</figure-callout></span>}}}{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ {\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; <figure-callout\; id="3"\; label="s"\; filenames="HDA00001629009100012.png"\; state="\{\{state\}\}">s</figure-callout></span>}}\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ]</span>$

3. Adjust the predetermined polar moment of inertia J: Calculate the adjusted value J ₁ of the polar moment of inertia J according to the following formula:

J ₁ ＝J[1-t _r /(2b ₁ )]………………………[4]

4. Calculate the equivalent width a ₂ and equivalent height b ₂ : replace J in the formula [1] with J ₁ , and then calculate the equivalent width a ₂ and equivalent height b ₂ according to the method described in step 2;

5. Iteratively calculate the governing equation of section properties and obtain the section size:

5.1. Let b _3k =b ₂ -0.1k, variable k=1,2,3,...,int(10b ₂ ), where int() is a rounding function;

5.2. Calculate a _3k according to b _3k :

${\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ]</span>$

5.3. Calculate L _3k according to a _3k and b _3k :

${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; ]</span>$

5.4. Calculate f _k according to a _3k , b _3k and L _3k :

5.5. Calculate err _k according to f _k :

${\mathrm{<span\; class="notranslate">\; err</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; |</span>}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 100</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; ]</span>$

5.6. Obtain the section size:

Find a _3k and b _3k corresponding to the minimum value of err _k and calculate L _3k , that is, calculate _{L 3 for} the final section dimensions a ₃ and b 3 .

The working principle of the present invention is: through the derivation of the basic formula of material mechanics and parameter correction, (see "Material Mechanics" Shan Huizu Higher Education Press, 1999) obtains an I-shaped hollow beam section obtained from the section characteristics of the aircraft high-speed flutter model The size method is a semi-reverse design idea and method for determining the cross-section size. Therefore, compared with the previous method that requires manual trial and error to obtain the cross-section size, the efficiency and accuracy have been greatly improved. Moreover, compared with the previous thin-walled rectangular hollow beam with double lugs, the I-shaped hollow beam proposed by the present invention can meet certain size constraints.

Example

For the method described in the present invention, calculation verification is carried out.

Three sets of I-shaped hollow beam sections of the high-speed flutter main beam model of the aircraft are given, and the sections P1 to P3 are examples 1 to 3, respectively. Table 1 presents the predetermined values of the geometric properties of the three embodiments, ie the target values. For P1～P3, carry out cross-sectional dimension design. Table 1 also shows the geometric properties of the design section obtained by using the method of the present invention, that is, the design value and the error of the design value. Table 2 shows the dimensional design values of the three embodiments, wherein the thin wall and the thickness of the lugs are the values given in advance.

The design values of section geometric properties in Table 1 are calculated by FEMAP v9.31 based on the corresponding section dimensions in Table 2. Compared with the target value, the error of the characteristic value of the design section is not more than 2%. From an engineering point of view, this is a high-precision calculation result.

Table 1 Geometric properties of section mm ⁴

Table 2 Design value of section geometric dimension mm

Claims (1)

1. A method for determining the cross-sectional size of an aircraft high-speed flutter model I-shaped hollow beam, ensuring that the I-shaped thin-walled hollow beam has a vertical moment of inertia I _x of a predetermined value, a lateral moment of inertia I _y and a polar moment of inertia J and When the wall thickness of the I-shaped thin-walled hollow beam is t and the thickness of the lug is t _r , determine its rectangular equivalent width a ₃ , equivalent height b ₃ and the total width L ₃ of the hollow beam section; here all the length units are specified as mm, characterized in that the steps for determining the section size of a thin-walled rectangular hollow beam are as follows: 1.1. Let t _r =n _r t; 1.2. Calculate the vertical moment of inertia _Ix and polar moment of inertia J of the thin-walled rectangular hollow beam with predetermined values, and the rectangular equivalent width a ₁ and the _like when the wall thickness of the I-shaped thin-walled hollow beam is t and the thickness of the lug is tr Effective height b ₁ : 1.2.1. Calculate the first intermediate variable p and the second intermediate variable q according to the following formula: $\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 9</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span>$ 1.2.2. Calculate the third intermediate variable s according to the following formula: $\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; q</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ]</span>$ 1.2.3. Calculate equivalent width a ₁ and equivalent height b ₁ : $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}}{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}}<span\; class="notranslate">\; -</span>\frac{\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}{\mathrm{<span\; class="notranslate">\; 3</span>}}\\ {\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\sqrt[\mathrm{<span\; class="notranslate">\; 3</span>}]{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}\right.<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ]</span>$ 1.3. Adjust the predetermined polar moment of inertia J: Calculate the adjusted value J ₁ of the polar moment of inertia J according to the following formula: J ₁ ＝J[1-t _r /(2b ₁ )]………………………[4] 1.4. Calculate equivalent width a ₂ and equivalent height b ₂ : replace J in formula [1] with J ₁ , and then calculate equivalent width a ₂ and equivalent height b ₂ according to the method described in step 1.2; 1.5. Iteratively calculate the governing equation of the section properties and obtain the section size: 1.5.1. Let b _3k =b ₂ -0.1k, variable k=1,2,3,...,int(10b ₂ ), where int() is a rounding function; 1.5.2. Calculate a _3k according to b _3k : ${\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ]</span>$ 1.5.3. Calculate L _3k according to a _3k and b _3k : ${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}}{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 3</span>}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; ]</span>$ 1.5.4. Calculate f _k according to a _3k , b _3k and L _3k :

1.5.5. Calculate the error value err _k according to f _k : ${\mathrm{<span\; class="notranslate">\; err</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; |</span>}{{\mathrm{<span\; class="notranslate">\; 6</span>}\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 100</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; ]</span>$ 1.5.6. Obtain section size: Find a _3k and b _3k corresponding to the minimum error value err _k and calculate L _3k , that is, calculate L ₃ for the final cross-sectional dimensions a ₃ and b ₃ .

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