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Abstract A theory of composite material patch winding is proposed to determine the winding trajectory with a meshed data model. Two dif-ferent conditions are considered in this study. One is Bridge condition on the concave surface and the other is Slip line condition in the process of patch winding. This paper presents the judgment principles and corresponding solutions by applying differential geometry theory and space geometry theory. To verify the feasibility of the patch winding method, the winding control code is programmed. Fur-thermore, the winding experiments on an airplane inlet and a vane are performed. From the experiments, it shows that the patch winding theory has the advantages of flexibility, easy design and application. Keywords: filament winding; fiber technology; composite material; airplane inlet; vane 51997
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1 Introduction* In recent years, composite materials have been found wide application in the fields of aerospace industry, weapon manufacturing industry and che- mical industry thanks to its strong design ability, high carrying capacity, high reliability, light weight and low cost. In the previous researches, most of researchers’ work was focused on the model equa-tions[1-7]. However, with the development of tech-nology, the abnormal shape mould winding is draw-ing more attention because some special parts, such as airplane inlets and vanes, can hardly be produced with traditional winding methods. This paper aims at tackling the problem of abnormal shape mould winding with the patch winding method. Different from the traditional winding methods, it is not founded on the base of model equations, but establishes a data model after meshing the mould and determines the winding trajectory by the points by the points on the mould surface. 2 Patch Winding The node coordinates can be obtained from the meshed mould surface. In winding, it is important to achieve the doffing points, the points where the fi-ber falls on the mould surface. The doffing points can be found following the basic winding theory to ensure that the Slip line condition and the Bridge condition will not happen during winding. Obtained according to the doffing points through coordinate change, the spinning points are used to enable NC code to be output to control the winding trajectory. This winding design principle is called patch wind-ing theory. To obtain the original doffing points before winding, exist two ways: one is to choose any point at one end of the mould; the other is to choose the point on the mould surface where the Slip line con-dition and the Bridge condition tend to occur. Gen-erally speaking, the first method is applied to the normal non-gyration mould, while the second is for those moulds whose surfaces are extremely irregu-lar. The following work is to find the other doffing points after the original doffing points have been determined. As shown in Fig.1, point A is taken as the original doffing point in order to explain the patch winding theory. If the winding fiber arrives at B, the next doffing point C can be obtained from the first quadrant in X1BY1 coordinate system to ensure that no Slip line condition and Bridge condition would happen at B. And then, the new doffing point C is taken as the origin of the new coordinate sys-tem. The straight line CX2, which is parallel with mould axis through C, is taken as X- axis.
The Y- axis is the straight line CY2, which, in the mould, is in a tangential plane at C and in a vertical plane about the line CX2. Fig.1 shows the directions of the two lines. By repeating the process of finding the point C, the other doffing points (for example, point D etc.) can be found. The winding trajectory design is finished after all the doffing points have been found. Fig.1 Patch winding theory. 2.1 The judgment of Slip line Slip line is a phenomenon that the fiber cannot remain stable at the doffing point on the mould sur-face or on the wound fiber. A relative motion exists between the fiber and the mould surface or the wound fiber. The Slip line condition depends not only on the degree of the winding angle, but also on the superficial friction coefficient. The following illustrates the detailed analysis. As shown in Fig.2, P1P2 = ∆s is a tiny section of fiber on the curved surface S (u, θ ), with P being its midpoint and T1, T2 the tension vectors at the points P1 and P2 respectively. α, β are the tangen-tial vector and the main normal one across the point P on the fiber trajectories, respectively. n is the normal vector across P on the curved surface (pointing to the outside of the mould). v = n×α is the unit vector through point P on the tangential plane. e1 is the unit tangential vector of the parame-ter curve u across point P, ϕ the angle between e1 and α, the winding angle at point P, Ff the friction between the fiber and the mould surface or the wound fiber, Fn the reacting normal force exerted on the fiber by the mould. Fig.2 Force analysis of mini fiber. To keep the fiber stable on the mould surface, the following requirements must be met fn12 0 + ++ = FFTT (1) The fiber weight is so small that no attention should be attached to its gravity. After deduction[1], Eq.(1) is developed into 2gn1dd iiTk s Tk s== ⋅+ ⋅ ∑Tn ν (2) Define the force in the direction v as sgd Tk s = ⋅ F ν (3) Define the force in the direction n as pnd Tk s = ⋅ F n (4) It can be found that Fs is the force that makes the fiber move on the mould surface, while Fp the force making it adhere to or depart from the mould. In order to keep the fiber stable on the mould surface, the requirements of the following Eq.(2) must be met. sf pmax maxµ =⋅ ≤ FF F (5) where µmax is the maximal friction coefficient be-tween the fiber and the mould or the wound fiber. By deleting |Tds| from Eq.(3) and Eq.(4), these two equations can be developed into gspnn 0kkk⎫⎪ =⋅⎬⎪≠ ⎭FF (6) Thus, in order to make the fiber remain stable without the Slip line condition, the following re-quirements should be met: gmaxn nn0 or 0kk kkµ⎧⎪ ≤= ⎨⎪≠ ⎩ (7) Taking the specific condition in patch winding into account, the normal curvature kn and the geo-desic one kg on the surface can be acquired with data model. The two curvatures through B on the mould surface can be acquired from Fig.3. The points P1 and P2 are the two points closest to the point B. If the mesh points on the surface are dense enough, the distance between P1 and P2 may be small enough as to be neglected compared with the mould radius, which enables P1 and P2 to be con-sidered on the same plane. As a result, the normal vector n can be determined by P1, P2 and B in Fig.3. Then, in Fig.1, the tangential vector α of fiber tra-jectory at the point B can be determined by B and C. At last, the vice normal vector at B can be found through A, B and C. |kg/kn| can be obtained from Eq.(10), which is derived from Eq.(8)[8] and Eq.(9). =× β γα (8) arccos θ⋅=⋅nnββ (9) gntankkθ = (10) where θ is the angle between the normal vector n of the tangential plane and the main normal vector β of the osculation plane.