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Chinese Journal of Aeronautics 23(2010) 573-577

Chinese

Journal of Aeronautics

www.elsevier.com/locate/cj a

Winding Pattern Design and Simulation of S-elbow

Wang Xianfeng, Xiao Jun*, Wen Liwei

College of Material Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 15 October 2009; accepted 23 February 2010

Abstract

Aimed at the S-elbow composed of two elbows with different radii, this article proposes a winding pattern design method combined with patch winding method and traditional winding method. It proposes an optimal combination of calculating the tangential point amount and skip point amount to make the pattern distribution even and keep the minimal adjusting angle. The S-elbow overall winding pattern plan and simulation module are designed to verify the combined winding pattern design method and the calculation algorithm of the tangential point amount and skip point amount. From the pattern distribution and the simulation effect analysis, it shows that this combined winding pattern design method is a good solution to the S-elbow combined winding pattern design. Aimed at the S-elbow winding pattern based on the patch winding method, it carries out the precision error analysis and points out the correspondence between the error and mesh size. Generally speaking, the bigger the mesh size is, the quicker the program calculation speed is; the smaller the mesh size is, the smaller the winding pattern error is.

Keywords: S-elbow; winding; mesh; composite; pattern

1. Introduction

The use of pipelines accounts for a large proportion

in the composite material applications. However, as

part of the matching connecting pipes, elbow is indis-

pensable. Because of severe corrosion, composite el-

bows and joints are urgently needed in the areas of

aerospace, petrochemical industry, building industry,

ocean development and food processing. The S-elbow winding pattern design is always a difficult issue. Many researchers have done a great deal of researches.

They proposed many valuable methods and theories

about rotation model and elbow based on equa-tions[1-13]. H. S. Li from Zhejiang University proposes

"the elbow winding pattern method based on the traditional winding method"[14]. Z. Y. Han from Harbin In-

stitute of Technology realizes "the elbow winding with

one pattern" successfully[15]. However, the overall pattern coverage is not ideal. The pattern layout is uneven; there is accumulation on the inner part and gap on the

outer part; the transition part is too long or the winding

angle is the same with only one direction. Combining with the patch winding method and the traditional wind-

ing method, this article realizes the S-elbow winding pattern design and overall even coverage.

2. Pattern Design

As for the single reciprocating winding pattern, the traditional winding method based on the equation can resolve the problem very well. Its pattern design is accurate and it is suitable for the elbow's end transitional pattern design. However, as for the overall full coverage of the elbow, the pattern is uneven and messy. Therefore, the traditional winding method is not suitable for the full coverage of the S-elbow's body. The winding pattern design method based on the patch winding has good applicability. It can be used in winding pattern design of various elbows, straight pipes or the combination of the two. The combination elbow is shown in Fig.1. It is ideal to design the S-elbow's body winding pattern with the patch winding method. The following is the discussion of the S-elbow's body winding pattern design based on the patch winding method and the end winding pattern design based on the traditional winding method.

♦Corresponding author. Tel.: +86-25-84892980-801. E-mail address: j. xiao@nuaa.edu.cn

Foundation items: National Natural Science Foundation of China (50905088); Science and Technoloy Major Project (2009ZX04004-102)

1000-9361/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(09)60256-9

Fig. 1 Combined elbow.

2.1. Pattern design of elbow body

The patch winding pattern design method is based on the mesh points on the elbow surface. Therefore, the S-elbow surface needs to be meshed. The elbow is divided into straight part and elbow part. Because the center point and radii of every elbow are different, the elbow needs to be meshed firstly. The S-elbow meshing includes segmentation and cross-section division. The S-elbow with 90° is taken as an example to explain the patch winding method. Fig.2 is the segmentation of the S-elbow. The S-elbow with 90° is divided into n segments. Fig.3 is the cross-section division of the S-elbow. The cross-section circle is divided into M sections. The mesh nodes are the pattern source codes, that is, the winding pattern points are generated from the mesh nodes. The denser the meshing nodes are, the more accurately model surface features are reflected; the denser the meshing nodes are, the smaller the deviation between the actual pattern and the meshed pattern is. If the calculation efficiency is taken into account, the meshing nodes density cannot be unlimited intensive. Generally speaking, if the S-elbow's radius R is small, its corresponding segment amount is small. However, compared with the S-elbow with bigger radius, the segment length of S-elbow with smaller radius is small. If the cross-section circle's

Fig.2 S-elbow segmentation.

Fig.3 S-elbow's cross-section division.

diameter d is small, its corresponding section amount M is small. However, compared with cross-section circle with bigger diameter, the distance between the nodes is small.

The winding angle a in Fig.4 is determined by the meshing node amount in the horizontal direction and the vertical direction. The winding angle is adjusted by adjusting the doffing points to keep the pattern stable and realize the full coverage. Test method is used in the practice[16]. Assume the pattern goes to point O and select point B as the next doffing point. If the winding angle needs to be increased, the pattern can go horizontally or vertically to point C2 or C1. If the winding angle needs to be reduced, the pattern can go from point B horizontally or vertically to point A2 or A1 until the requirements are met. As long as the doffing points are well controlled, the stable pattern without slip-line can be got. Therefore the product quality is ensured. From the above analysis, the patch winding method has the following characteristics:

1) Good applicability. It can be applied to both gyration object and non-gyration object.

2) It is easy to control the pattern.

3) It is easy to adjust the winding angle to improve design.

A- B C,

-—„ a Pi

Fig.4 Wiring theory.

2.2. Pattern design of head transition

The elbow is actually part of annulus. The non-geodesic pattern on the elbow surface can be got through traditional winding design theory. The end transitional pattern can be got through non-geodesic pattern. The following is the calculation process of non-geodesic pattern on the annulus surface. The an-nulus is got by rotating the circle whose center point is O' around point O for one round. Fig. 5 is the cross-section picture of the annulus. The annulus equation expression is

r (<j>,0) = {(a cos <j> + b)cos 0,

(a cos <j> + b)sin d, a sin <j>+ c} (1)

where a is the radius, b the circle center pattern radius, c annulus center's position on the Z axis, <f> the rotation angle around center point O', and 0the revolution angle around center point O.

From Eqs.(2)-(3) and Eq.(6), We can get dd a tan a

d^ a cos </> + b

da Âa

— = 2cos a -\--

d^ a cos $ + b

tan a cos é —

sin a-

a cos ^ + b

tan a sin ^

Fig.5 Annulus cross-section.

It is difficult to derive the exact solution from Eq.(7) through calculus. An approximate solution can be got by using iteration technique. In Eq.(7), the value range of a can be got according to the design requirements. When the winding angle a changes, the variance range of the revolution angle 0 and rotation angle ^ can be well controlled through slip line factor 2. These parameters' characteristics are used to control the transitional pattern accuracy.

The first fundamental variable and the second fundamental variable are[17]

E = r^ r = a2

F = V re= 0

G = rg • rg = (acos </> + b)2 L = n = a

M = r^g- n = 0 N = rgg • n = (acos </> + b)cos ^

where E, F and G are the first fundamental quantities; L, M and N the second fundamental quantities; r^, re, r^g, rgg and r^ the r's first and second partial derivatives about <f> and 6; n is the normal vector of the ring.

Substitute the above six equations into the Liouville equation, the following equations can be got:

d^ _ cos a ds a dd _ sin a

a cos $ + b

(2) (3)

da ö(lnE) ö(lnG) .

---¡=— cos a--¡=— sin a =

ds 2ylGd0 2-JEÖ0

Ld2fi + 2Md0d0 + Nd20 Ed2$ + 2 Fd0d$ + Gd20

da sin é sin a — +---

ds a cos tj> + b

cos2 a sin2 a cos <b -+--

a a cos </> + b

A= i =

da sin d> sin a

— +---

ds a cos </> + b

k„ cos2 a sin2 a cos ^

a cos </> + b

cos2 a + sin2 acos<

. sin ©sin a )--;—r (6)

a cos </> + b a cos </> + b

where s is the natural parameter, kg the geodesic curvature, kn the normal curvature, and 2 the slip line factor.

3. Pattern Simulation and Analysis

The S-elbow modeling is made according to the an-nulus equation. Two parameters are needed: circle radius and gyration radius.

In order to determine the elbow's corresponding rotation angle, the terminal angle is set as zero. The other starting angle is input from the input window. In order to determine the pattern amount for the full coverage, the fiber width is needed to be input; in order to ensure the winding pattern stability, the friction factor is induced. In this way, necessary constrains for the pattern design are provided. From the point of view of component strength, a reasonable winding angle is important for the weight reducing and component load optimizing. As a result, the above parameters are set in the input window, as shown in Fig.6. According to the known condition, input the given parameters in order.

Stan Angle

r{0,9) = {(acos^ : È)cosfl,(acos^ + 6)sinö,iisin(f + c)

Friction [i

Ca n er J

HoTalive jj

Rullui : End

Modulus Winding ; Angle U

Fig.6 Input window.

After the design parameters are input, the program modules will carry on the single-iterative pattern planning operations according to the above pattern design method. Then, the overall winding coverage pattern is designed according to the single cycle pattern and the design parameters. The key technology is to select circulation tangential point amount and skip point amount. The following program segment is used to determine the amounts of tangential point and skip point, which can ensure the minimal error and ideal overall full coverage: for (i =1; i<=20; i++)

{for ( j=1; j<=i; j++) {for (int j1=1, k=1; j1<=j; j1++, k=k+1) if(i/k-i/j1<1e-4 && j/k-j/j1<1e-4 && (i!=1 && j!=1 && j1!=1))

biaoji1=true; if (biaoji1==false) {sita0=sita3-2*PI*j/i; if (fabs(sitatiao)>fabs(sita0) && sita0<0) {sitatiao=-sita0; n1=i;

m1=j; }}}

biaoji1=false;}

In order to ensure the full coverage of the elbow surface and optimize the end pattern, the skip point amount and tangential point amount should be prime numbers. At the same time, the tangential point amount cannot be divided by the skip point amount. In the above program segment, i is the tangential point amount and j is the skip point amount. After the pattern calculation, the overall pattern simulation is conducted. The overall pattern simulation is shown in Fig.7. The model in Fig.7 is made by docking two elbows. The gyration radii of the two elbows are different. It shows that this pattern design method can be applied not only to the single elbow winding pattern design, but also to the overall pattern design of the elbow combination.

Fig.7 Pattern simulation.

4. Experiment and Discussion

According to the design pattern, S-elbow winding experiment can be carried out after pattern post-processing. The winding picture is shown in Fig.8.

Fig. 8 S-elbow winding pattern.

From Fig.8, it can be found that the winding pattern is stable, and there are no slip-line and bridge condition. The experiment shows that the combined winding pattern design method can be applied to the successful design of S-elbow winding pattern.

The model winding codes are generated by using the doffing points through the patch winding method which depends largely on the meshing nodes on the model surface. The meshing nodes' density and level of

uniformity influence the deviation between the actual winding pattern and the theoretical winding pattern directly[18-19]. When using the patch winding theory in the post disposal, the denser the nodes are, the closer the theoretical yarn vector and the actual yarn vector are. The evener the meshing nodes are on the surface, the more precise the tangent is at the doffing points. When the model radius a is 20, 70, 450 mm respectively, the winding error which enlarges with the increase of the node distance is shown in Fig.9. Therefore, when the model radius is small, smaller mesh node is needed to design the winding pattern as far as possible; when the model radius is greater than 450 mm, the mesh node should not exceed 2 mm.

0.06 r ♦ £7=20 mm

a (i=70 mm

Is 1 0.04 ii=450 mm

:: 0.02 -

(1 0.5 1.0 1.5 2.0

Points distribution density/ram

Fig.9 Trend line between gyration error and mesh size. 5. Conclusions

As for the non-gyration feature of the S-elbow, this article firstly proposes a winding method combined with the patch winding method and the traditional winding method. Then, it also programs the optimal calculation module to get the tangential point amount and skip point amount. According to the S-elbow design method, it programs the S-elbow pattern design and simulation module. The pattern design and simulation are realized successfully, which provides technical support for the combined elbow winding pattern design. Finally, this article makes a precision error analysis and points out the correspondence between mesh size and the error, which provides a basis for the meshing.

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Biographies:

Wang Xianfeng Born in 1980, he is a lecturer in Nanjing University of Aeronautics and Astronautics. His main research interests are composite winding technology and fiber placement technology. E-mail: wangxf@nuaa.edu.cn

Xiao Jun Born in 1959, he is a professor in Nanjing University of Aeronautics and Astronautics. His main research interests are composite winding technology and fiber placement technology. E-mail: j.xiao@nuaa.edu.cn