Geometric modeling of surfaces dependent cross sections in the tasks of spinning and laying

The article is devoted to the development of a geometric model of surfaces of dependent sections to solve the problems of winding by continuous fibers in the direction of the force and its related process of automated winding of composite materials. A uniform method for specifying the surfaces of dependent sections with a curvilinear generator and a method for solid modeling of the shell obtained by winding or calculation methods are described.


Introduction
Methods of automated winding and calculations are one of the main methods for obtaining structures from composite materials. In the process of winding, carried out on machines with numerical control, the surface of the mandrel is laid with tension continuous tape composed of unidirectional fibers, threads, strands or bundles impregnated with a binder. After obtaining the required thickness and structure of the shell, polymerization is performed, the final curing of the binder.

Problem definition
In the process of winding, various defects may occur: the tape does not fit to the surface or the tape slides from a given curve along which it is laid. You can track these defects in the virtual model of the process for which you want to create a geometric model. The construction of a generalized geometric model of tape laying on a curved surface was described in [1]. In this paper, it is assumed that the simulation of winding ("dry" and "wet") is carried out in a single way -by means of a smooth mapping of a rectangle into a three-dimensional Euclidean space. Also describes methods of analyzing a circuit, the belt laying on the subject of equilibrium filaments Lena and their contact with the surface. All builds use the surface of class C2. It should be noted that in the process of laying the tape on the surface only its first layers are laid on the surface. The remaining layers are placed on the surface formed by the previous layers. Thus, the formed surface is continuously changing. These changes naturally affect both the analysis of the packing patterns of the * Corresponding author: belyaeva-sv@mail.ru  10 10 SPbWOSCE 57 57 ribbon and the law of motion naturallanguage mechanism of the machine, which was described in the article [2]. To take this fact into account, it is necessary to modify the surface shape in accordance with the variable thickness of the tape [3]. Note that the presented in the paper [4] the method of surface modification is applied to the class of twice continuously differentiable surfaces, the dependent variable of the closed cross sections with curvilinear generatrix, which is in the process of change is incident to the plane parallel to the coordinate plane. We denote this class of surfaces [3] . In this paper [4] we propose a uniform method for describing class surfaces . The surface of technological mandrels often consist of several parts -constructive (surface of the product) and technological (serving for the reversal of the tape). Such surfaces can be specified by different parametric representations. Therefore, to work with such a surface in a virtual model, there is a problem of smooth connection of different parts, coordination of parametrization on the structural and technological parts. With a uniform specification of class surfaces, all these problems are automatically eliminated, since the entire surface of the technological mandrel (all its parts) will be described by one twice continuously differentiable, explicitly given vector function. This greatly simplifies the subsequent computer task of the surfaces of the class in question. When calculating the parameters of the shell obtained by winding or laying out, there are important geometric problems of constructing intermediate surfaces of deformable solids [6] of a multilayer structure. The article offers a method of modeling a solid body obtained by winding or calculation methods. ; on each segment

On one method of approximation of functions
a polynomial of degree m. It is known [7] that the dimension of the space S m, () is m+1+(n-1). Next, we consider the space S 3,1 (), since the surfaces used must be twice continuously differentiable. In the space S m,1 () there is a basis consisting of finite functions N m+1,i (x), called B-splines [8]. Extend grid , adding Then the functions N m,i (x) can be defined by the following relation [4]: can be represented as a linear combination of B-splines Let the values of the function f in the grid nodes be known ( ) Then in the space S 3,1 () we can find a single function ( ) x s 1 , 3 satisfying the conditions [4]: We introduce ( ) , then the estimates are valid [4]: As shown in [5], if the grid  is uniform, can be written explicitly: It should be noted that the vector is the solution of the equation    10 10 SPbWOSCE 57 57 ... : , it is a fair assessment: . Enter the symbol We introduce the following notations , , ,..., , Then, obviously, there is equality Since the matrix of the system has a strict diagonal predominance [6], the estimates are valid ).
From equality Since cubic B-splines are non-negative and form a partition   b a; of one on the segment, Then for it to be a fair assessment (2).
. Then, obviously, the following equality holds . Then, using expressions for derivatives of the Bspline [4], Similarly, the inequality is obtained . Therefore, , it is a fair assessment: Evidence. Imagine both the spline in the form The Lemma is proved.
Consider another type of conditions imposed on the spline s 3,1 (x), which arise in the interpolation of periodic functions ( As it is known [12], there is a single function ( ) ) . ,..., 2 , 1 it satisfy the condition: ). We choose a uniform mesh [14] ( ) For m=2 inequality holds: Evidence. For each fixed value [15]  interpolation spline that satisfies the interpolation conditions a and the boundary conditions By virtue of Lemma 1, the estimate is fair Note that for a fixed value it is a cubic [16] spline that satisfies the interpolation conditions and boundary condition Since the ( ) Therefore, From here, on the basis of Lemma 2, we can conclude that So, we come to the following assessment , it is a cubic spline that [17] satisfies the interpolation Due to inequality (4) we have Based on Lemma 2, we conclude that

By virtue of Lemma 1, there is inequality
We consider separately the case. By virtue of Lemma 1, inequality holds true Consider a surface k n,  with a parametric representation      [19], and the algorithmic part of the determinant is given by the vector function (5).
The results of section 1 can be applied to the construction of a vector function that determines the winding body for some of its intermediate layers. Moreover, such a vector function can be written out explicitly.
,..., 1 , 0 , ; ; , , ,      Figure 1 shows the simulation of the winding surface of a rectangular profile and the surface of rotation. Figure 2 shows the results of solid modeling [20].

Conclusion
The article presents the method of approximation of functions of two arguments, the approximation error is found. The main advantage of the developed method is that the approximating function is written explicitly. The proposed method is applied to the construction of the determinant of the surface of dependent sections with variable generatrix. Also, the method of layer-by-layer modeling of a solid body is presented, the distinctive feature of which is the ability to specify only the points of sections of layers using an explicitly given vector function. The application of the developed methods for geometric modeling of bodies obtained by winding and laying out of composite materials is shown.