Computer variant dynamic forming of technical objects on the example of the aircraft wing

This article describes a mathematical apparatus of dynamic formation of technical objects on the basis of a study that has devised it with the aim to improve and develop computerized structural and parametric geometric models by appropriate integration with their available mathematical support. The practical value of the obtained results consists in creating a methodology for computer variant dynamic shaping, which helps flexibly combine the designing and manufacturing of technical objects, as is illustrated by the example of the wing of an aircraft. The proposed techniques provide an automated design of the wing surface and a computer simulation of such technological operations for manufacturing a centreplane longeron as cutting, pressure treatment, assembly, etc. The created structural and parametric geometric models contribute to the multicriteria optimization of technical objects throughout the lifecycle. The described approach can also be used for the computer variant dynamic formation of such structural units of the airframe as ribs, panels, sections, bends, and the like. Through further studying, the research materials can be distributed to diverse products of mechanical engineering and other industries


Introduction
The issues of improving technical facilities are always important. Progressive in this regard are computer information technologies. Complex automation of production shortens the time of creating new products, significantly increases the quality, and reduces the cost. An important task is to combine automated design and manufacturing of products on the basis of geometric modeling. Technological processes of mechanical engineering, particularly in the aviation industry, are characterized by a large number of assembly operations, cutting, pressure cutting, etc., with a large number of possible options. Taking into account the existing requirements for increasing the accuracy, flexibility, productivity, and shortening the time of production preparation, it is necessary to apply new approaches in the field of computer information technologies, which determines the relevance of the outlined scientific issues.

Literature review and problem statement
An important direction for the introduction of computer information technology is the use of a parametric approach to geometric modeling during automated design. Study [1], by the example of the variant three-dimensional design of the pin parts, outlines the general procedure to form a typical geometric model with the necessary dimensional parameters and the required interconnections between them.
The main focus is the integration of Pro/Engineer systems and Microsoft Excel spreadsheets, and the study justifies the possibility of increasing the productivity of automated construction at the expense of this approach. The universality and flexibility of parametric methods of geometric modeling is confirmed by article [2], where the corresponding computer constructs of such widespread engineering products as screw cylindrical wire springs are presented. Work [3] shows the effectiveness of information technology data for automated reproduction of assembly units of machine building on an example of a mechanism with two kinematic pairs. The study extends the methods of parametric geometric modeling to the assembly of the unit. However, the general disadvantage of the above-mentioned works is that they do not analyze the integration of the design of technical objects, for example, with calculations of durability, processes of manufacturing, operation, etc.
Study [4] highlights the kinematic optimization of mechanical systems on the basis of dynamic computer simulation. Article [5] shows the effectiveness of using parametric geometric means for combining automated solid-state shape forming with calculations of durability and development of design documentation. Thus, we see that the progressive modern trend is to expand the scope of the practical application of parametric geometric modeling during the creation of various technical objects.
A certain improvement of the analyzed approach to computer geometric modeling is parametric structural shape formation, the main provisions of which are given in article   COMPUTER VARIANT  DYNAMIC FORMING OF  TECHNICAL OBJECTS ON  THE EXAMPLE OF THE  AIRCRAFT WING V . V a n i n [6]. In the study, one parametric structural geometric model summarizes several parametric ones, which greatly increases the universality and performance of automated design. Work [7] describes some issues of the use of parametric structural shaping for the development of an aircraft, but the focus is on the stage of sketch design. In article [8], the emphasis is placed on efficient computer modeling of a large nomenclature of unified group of parts, but integration problems with the technology of their production, calculations of durability, etc. are not considered. In article [9], it is shown that further development of parametric structural geometric modeling is a computer variant dynamic formation. Thus, the analysis of the published data shows that the development of new approaches, methods, techniques and algorithms of the integrated computer variant dynamic formation of technical objects and mechanical engineering processes on the basis of structural and parametric geometric modeling can be regarded as promising in both theoretical and practical terms.

The aim and objectives of the study
The aim of the study is to develop, on the basis of structural and parametric geometric modeling, a method of computer variant dynamic formation of technical objects with illustration guidance on the example of an aircraft wing.
To achieve the aim, the following tasks should be solved: -to offer a mathematical apparatus of the dynamic formation of technical objects; -to perform an automated variant construction of the surface of the aircraft wing; -to carry out dynamic computer structural and parametric geometric modeling of the longeron of the central part (centripetal) of the aircraft wing.

The mathematical apparatus of dynamic shaping of technical objects
For the computer variant dynamic formation of technical objects, geometrical figures will be used according to their classification by the dimension ( ) where F 0 is points, F 1 is lines, F 2 is surfaces, and F 3 is bodies.
The geometric parameters will be presented as a tuple ( ) where P 1 , P 2 , and P 3 are position, size and shape parameters. According to expression (2), possible dynamic geometric modifications are described by the set where М 1 is the modification of the position (movement), where The object under consideration is a certain combinatorial configuration of the elements of set (1), and its modifications are obtained from the elements of tuple (3). It is necessary to take into account the further details of these components, for example in the form where where where М 1 1 =(parallel transfer)=m 1 , М 1 2 =(rotation)=m 2 , and М 1 3 =(symmetry)=m 3 ; where m 4 =(proportional scaling), and m 5 =(disproportionate scaling). On the basis of parametric structural shaping, an arbitrary simulated geometric object O is represented by an ordered set of elements: The possible types of o i are reproduced by tuples of variants and vectors of parameters where Np i j is the number of parameters of a j-th variant of an i-th element.
The structural correlation between the varieties of the n-th and the m-th components of the object O is determined by the adjacency matrices ( Fig. 1, а).  (10), and the edges are geometric models (11) that implement these elements with certain values of the parameters used in (12). Each type of О k of the object under study is a simple elementary chain with a beginning at the vertex О 1 and an end at the vertex О N ; the total number of N O is determined by the elements of matrices (13). The values of the parameters in (12) and their various combinations in the form of the necessary target analytic functions are placed in accordance with the length of the proper edges. Then the search for the optimal parametric structural variant of the technical object O is reduced to determining the extremal chain shown in Fig. 1, d of the graph. There are various algorithms for solving such problems -in particular, indexing of vertices, branches and boundaries, etc.
Based on the fact that geometric objects that are variable in time can reproduce certain technological processes of machine building, the developed method suggests that the parametric structural model components should combine the geometric shapes given by formulae (1)-(9), their parameters and dynamic modifications, with the above relations (10)-(14).
For a variant dynamic reproduction of technical objects and processes for their manufacturing, the specific composition of sets (1)-(14) is determined by the existing design conditions, which are illustrated later by specific examples of structural and parametric geometric modeling of the aircraft wing.

The results of variant dynamical geometric modeling of the aircraft wing
The following is a variant modeling of the wing surface, which largely determines such characteristics of the aircraft as aerodynamic, strength, weight, technological, etc., as well as computer dynamic structural and parametric shaping of the centreplane longeron.
The indicated information is important in the applied scientific plan because using modern computer information technologies provides an opportunity to increase the efficiency of conducting complex (multicriteria) optimization of the aircraft. This assertion is based on the fact that the tasks of aerodynamics, strength, layout, construction, technology of production and operation, etc. are not only closely related, but they also significantly affect one another. For example, tolerances in terms of the strength of the limit on the variation of the height of the longeron ends is determined by the shape and dimensions of the cross-sectional wings, selected in accordance with the required aerodynamic characteristics, and the latter, in turn, depend to a large extent on the stiffness of the structure of the bearing surface. Also, the complication of the shape of the wing leads to the improvement of its aerodynamics but worsens the manufacturing process. These and other features determine the iterative variant nature of the design of a modern aircraft. Among many other models (aerodynamics, strength, layout, weights, technological, operational, etc.), geometric models have a special place, which is related to the role of the model of the shape and size of the simulated technical object. As a result, the main requirements for computer geometric models ensure not only high accuracy of shaping but also flexible and productive construction of various design variants of products with a dynamic reflection of the included processes of their manufacture and operation, that is, for the entire life cycle.
The following specific examples confirm the validity of the above general methodology of the computer variant dynamic formation of technical objects on the basis of structural and parametric geometric modeling.

1. The variant construction of the surface of the aircraft wing
Let the surface as the output have an aerodynamic profile, represented in a rectangular coordinate system Oxy by a set of points The longest segment h i is called the thickness c of the profile, and the maximum distance from the points y av i to the chord is the concavity f of the profile f (Fig. 2). Under the conditions of using the initial symmetric profile, the required initial concavity f is obtained by multiplying the ordinates of the points of the upper and lower parts, respectively, by the coefficients where max u y and min l y are the maximum and minimum ordinates of the initial profile.
In the case of processing the lines, for example, of the F 1 2 shape in accordance with expression (5), the possible dynamic transformations are shown in Fig. 3, where the studied aerodynamic profile is presented as a composite line of arcs of the curves of the second order in the vector parametric form

2. The dynamic computer structural and parametric geometric modeling of the centerplane longeron of the aircraft wing
The parametric structural model of the longeron presented in [10] makes it possible to obtain the necessary design variants, but it does not allow reproducing in time technological operations of installing parts, drilling holes, riveting, etc. Let us consider some techniques for improving this model.
The composition of the projected longitudinal LN is represented by the set where LN 1 ={W} is the wall; LN 2 ={B 1 , B 2 } means the upper and lower belts; where According to the general approach described, the figures of (22) are the bodies, and the technological operations of (23) are geometrically reduced to the change in the parameters of the position and the shape of these objects. The parameters of the wall W as a rectangular parallelepiped are length, height, and thickness. For the T-shaped belts B 1 and B 2 , the parameters are the dimensions of the cross sections and the length. The operation of installing these parts consists in moving them to the necessary position of the proper base, which for the bodies can be carried out by the conjugation of face surfaces, edges, vertices, etc.
The operation T 2 is illustrated in Fig. 6, where the belts are attached to the wall of the longeron through drilling of the required holes and subsequent riveting.
For a dynamic simulation of the drilling of cylindrical holes, it is recommended to use a computer solid-state drill model and its geometric model presented in the Cartesian coordinate system Oxyz as a combination of a cone ( ) ,   1  cos 2  , 1  sin 2  ,  ,  2 2 and a straight circular cylinder  For the computer dynamic constructions of closing heads of rivets, we will use a geometric model of settling a direct circular cylinder, which is shown in Fig. 7.

Fig. 7. A model of settling a cylinder
The output cylinder has the base radius R and the height H. For the settled body, the relative compression is where ¢ H is the reduced height, and ¢ min R and ¢ max R are the increased radii of the bases in the contact planes and the median horizontal plane of symmetry.
The lower and upper parts of the lateral surface of the settled body, which are symmetrical in relation to the median horizontal plane, are formed in the rectangular coordinate system Oxyz by rotating around the axis z of the arcs of the second order curves: the extreme value of which is zero. Consequently, expressions (26)-(29) constitute the mathematical basis of the computer dynamic constructions of the closing heads of the rivets.
From formula (23), it is evident that the operations are the installation, drilling and riveting of stilts (Fig. 8). These actions are largely similar to those analyzed above for the belts. The finishing stage is the control T 5 of manufacturing the longeron, which may consist in checking the composition of the resulting node, the correctness of the values of the parameters of the shape, size and position of its elements, etc.

Discussion of the results of the proposed method of computer variant dynamic formation of technical objects on the example of the aircraft wing
The mathematical apparatus of dynamic formation of technical objects has been devised with the aim to improve and develop computerized structural and parametric geometric models by appropriate integration with their available mathematical support. This is the main scientific theoretical novelty of the results.
The automated variant construction of the aircraft wing surface and the dynamic solid-state modeling of the centerplane longeron are of practical importance. The results have confirmed the reliability of the proposed method of computer-based variant dynamic formation of technical objects on the basis of the parametric structural approach. This applies to the classification of applied geometric figures and their modifications, a combination of stages of sketch, technical and working designs, and the reproduction of certain technological processes of mechanical engineering.
Due to the controversial requirements of aerodynamics, strength, production and operation of the variation of the shape, size and position of the aerodynamic profiles of the projected wing, it is implemented by flexible means. The presented techniques of computer construction are aimed at ensuring the successful implementation of the appropriate multicriteria optimization. Thus, the presence of the possibility of parallel transfer of the final profile r k in the model (Fig. 4) along the z axis determines the required L width of the wing; along the x axis, the angle c is sagittal, and along the y-axis, the angle V is transverse. The profile rotation around the parallel z axis of the straight line provides a bearing for the aerofoil. It is also important to emphasize the variant nature of the created model of the wing longeron. Each element has its own geometric parameters of shape, size, and position. For example, the cross-section of the stilts can be not only angular (Fig. 8) but also T-shaped, and the stilts have certain dimensions and necessary quantity to be located properly.
Consequently, the fundamentally new features of the proposed dynamic parametric structural geometric models are the presence of their parameters related to the displacement and deformation of the constituent elements.
It is known that the development of complex technical objects, in particular in the aviation industry, has an iterative variant nature. Thus, during the technical task, the main characteristics of the product are determined, the possibility and expediency of its manufacturing are substantiated. The technical proposal contains the specified technical and economic characteristics, the draft design (the fundamental engineering solutions that give a general idea of the structure and operation of the product), and the technical design (the final technical solutions that give a complete picture of the product). In the aviation industry, at the stages of the sketch and technical projecting, various layouts of the technical object and its components are created, which is now often performed in a computerized form, and manufactured. In the course of the working design, the documentation for producing prototypes is first drawn up, and then, after proper correction on the basis of tests, mass production in launched.
In all of the above-mentioned stages of creating technical objects, geometric models are used, which are gradually complicated and refined in accordance with the stages of the lifecycle of mechanically engineered products.
The techniques and algorithms of parametric structural geometric modeling have been implemented at the Antonov Aviation Scientific and Technical Complex during the design of the AN-148 aircraft, the production of the trolley A promising direction of the modern development of parametric structural geometric modeling in the scientific and practical aspects consists in its further improvement by devising new methods, techniques and algorithms of dynamic shaping, which will significantly increase the accuracy and realism of the computer models. Some of the aforementioned issues have been elaborated in this paper by the proposed method of computer-based variant dynamic shape-forming of technical objects, which is illustrated by the example of the wing of an aircraft.