USING THE R-FUNCTIONS THEORY APPARATUS TO MATHEMATICALLY MODEL THE SURFACE OF THE SOYUZ-APPOLO SPACECRAFT MOCK-UP FOR 3D PRINTING

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14, Nauky Ave., Kharkiv, 61166, Ukraine Creation of mathematical models of objects to be 3D printed is of considerable interest, which is associated with the active introduction of 3D printing in various industries. The advantages of using modern 3D printers are: lower production costs and shorter periods of time for their appearance on the market, modeling objects of any shape and complexity, speed and high precision of manufacturing, their ability to use various materials. One of the methods for solving the problem of creating a mathematical and computer model of the object being designed is the application of the R-functions theory, with the help of which it is possible to describe geometric objects of complex shapes in a single analytical expression. The use of alphabetic parameters, when one specifies geometric information in analytical form, allows one to quickly change the size and shape of the object being designed, which helps to spend less time on building computational models. The proposed method can significantly reduce the complexity of work in CAD systems in those cases when one needs to view a large number of design options in search of an optimal solution. This gives a great effect on reducing labor intensity in the construction of computational models to determine aero-gas-dynamic and strength characteristics. Characterization is also often associated with the need to account for changes in aircraft shape. This leads to the fact that the determination of aerodynamic characteristics only due to the need to build a large number of computational models increases the duration of work by months. With parametric assignments, computational regions change almost instantly. In this paper, on the basis of the basic apparatus of the theory of R-functions as well as cylindrical, spherical, ellipsoidal, and conusoidal support functions, a multiparametric equation for the surface of a Soyuz-Apollo spacecraft model is constructed. A number of support functions were normalized according to a general formula, which made it possible to illustrate a new approach to constructing threedimensional equations for surfaces of a given thickness.

Introduction
One of the new technologies, that has recently been gaining increasing popularity, is 3D printing, which allows one to create three-dimensional models of any objects by using special 3D printing equipment. This is a high-tech equipment, requiring that one has certain knowledge and skills in order to use it. The heyday of the era of 3D printing began in the 21st century, although the principle of operation itself was formulated back in 1986. 3D printing in this century has already become one of the main achievements of mankind in the development of technology. Scientists have mastered the methods of creating three-dimensional models of any shape and content.
The 3D printing process consists of several stages. The first stage is preparatory. In the process of preparation, one creates a computer 3D model of the object to be printed. The creation of such a model is possible, for example, using a 3D modeling program. After the virtual model has been created, one needs to convert it to a set of instructions for the printer, i.e. generate a G-code, and then start the printing process. The potential and capabilities of 3D printers are enormous, most often they are used in mechanical engineering, architecture, film industry, and medicine. The main advantage of 3D printing is that it is a completely robotic process. Using a 3D modeling program, one can look at the prototype from all angles, see the real dimensions, proportions, and also correct errors and improve the product at the modeling stage. Experts from the Economist, one of the most influential magazines, have called 3D printing technology the third industrial revolution, which is now taking place quietly and insensibly right outside our windows. And while some Eastern European countries are investing billions in "traditional production", many have already realized the scale of changes ahead. For example, the United States plans to return production to its territory in the nearest future, but it will be mainly high-tech and versatile technologies based on 3D printing, for which it is planned to provide large-scale government subsidies for this process, similar to the current American breakthrough in the production of its own shale gas and oil. It turned out that it is economically feasible to 3D print entire rocket engines. 3D Systems Corporation (USA) announced a successful zero-gravity test of the BFB-3000 3D printer. The tests indicate the possibility of producing complex assemblies and parts in outer space. 3D printing will change the way we approach space exploration as well as how spacecraft operate. At present, all the items and parts necessary in space are manufactured on Earth, carefully adjusted and matched to spacecraft models for testing. Then they wait for the next spacecraft to fly, and only after docking in orbit, the astronauts receive the required items [1]. 3D printers make it easier to create and arrange living space in orbit, reducing the need to deliver a wide range of cargo from Earth. NASA engineers are even building a rocket from special plastic through using a 3D printer with selective laser sintering technology that can reach other planets in the solar system.
However, there is a problem of setting the print information, i.e. creating a mathematical and computer model of the object being designed. One of the methods for solving this problem is the application of the R-functions theory, which allows one to describe geometric objects of complex shapes in a single analytical expression. In [2], based on the R-functions theory, new approaches were developed to the analytical identification of the surfaces of unmanned aerial vehicles to be 3D printed. We used both well-known methods of standard primitives and various frame blending options. Multiparameter equations for the surfaces of unmanned aerial vehicles of various shapes and purposes have been constructed and visualized. The adequacy of the results obtained to the objects being designed is confirmed by visualization both in the operating conditions of the RFPreview program and by 3D printing. The use of alphabetic parameters, when one specifies geometric information in analytical form, allows one to quickly change the size and shape of the objects being designed, which helps to spend less time when building computational models. The proposed method can significantly reduce the complexity of work in CAD systems in those cases when one needs to view a large number of design options in search of an optimal solution. This can give a great effect on reducing labor intensity in the construction of computational models to determine aero-gas-dynamic and strength characteristics. Characterization is also often associated with the need to account for changes in aircraft shape. This leads to the fact that the determination of aerodynamic characteristics only due to the need to build a large number of computational models to take this factor into account increases the duration of work by months. With parametric assignments, computational regions change almost instantly.
In [3], on the basis of the theory of R-functions, equations of the surfaces of the models of the "Angara" launch vehicle, the "Buran" type spacecraft, and missiles for the delivery of objects to their destination points are constructed. Thus, at present, a certain experience has been accumulated in constructing equations for the surfaces of aerospace objects. However, each new spacecraft design requires, as a rule, the improvement of the basic apparatus of R-functions.
The purpose of this work is to create a mathematical and computer model of the surface of the Soyuz-Apollo type spacecraft mock-up.

Main part
In this work, we used the R-operations ; It should be noted that with a change in any of the presented alphabetic parameters, the shape of the corresponding fragments of the spacecraft will automatically change.

Conclusions
Creation of mathematical models of objects to be 3D printed is of considerable interest, which is associated with the active introduction of 3D printing in various industries. The advantages of using 3D printers are obvious: they make it possible to manufacture non-standard models, reduce time for creating new prototypes, simplify and significantly reduce cost of production even if modern ultra-strong materials are used. In this paper, based on the R-functions theory, methods are formulated for creating mathematical and computer models of 3D objects. Algorithms have been developed for the step-by-step construction of mathematical models of complex cross-sections, three-dimensional equations of surfaces, including those with the wall of a given thickness.
The reliability of the results obtained, their adequacy to the objects being designed is confirmed by visualization in the operating conditions of the RFPreview program. The analytical recording of the objects being designed makes it possible to use alphabetic geometric parameters, complex superposition of functions, which, in turn, allows one to quickly change their structural elements. The positivity property of the constructed functions at the internal points of the object is very convenient for 3D printing.