Using the R-Functions Theory Apparatus to Mathematically Model the Surface of the Soyuz-Appolo Spacecraft Mock-up for 3D Printing

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Podgorny Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN 0131-2928 (Print), 2411-0779 (Online)
Issue Vol. 23, no. 3, 2020 (September)
Pages 55-60
Cited by J. of Mech. Eng., 2020, vol. 23, no. 3, pp. 55-60



Tetiana I. Sheiko, A. Pidhornyi Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), e-mail:, ORCID: 0000-0003-3295-5998

Kyrylo V. Maksymenko-Sheiko, A. Pidhornyi Institute of Mechanical Engineering Problems of NASU (2/10, Pozharskyi St., Kharkiv, 61046, Ukraine), V. N. Karazin Kharkiv National University (4, Svobody Sq., Kharkiv, 61022, Ukraine), e-mail:, ORCID: 0000-0002-7064-2442

Anna I. Morozova, Kharkiv National University of Radio Electronics (14, Nauky Ave., Kharkiv, 61166, Ukraine), ORCID: 0000-0002-7082-4115



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 three-dimensional equations for surfaces of a given thickness.


Keywords: R-functions, alphabetic parameters, standard primitives, Soyuz-Apollo spacecraft model.


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  1. (2020). 3D-printer budet “pechatat'” detali pryamo v kosmose [The 3D printer will “print” parts right in space.]. Nano News Net: Official site, 2020. URL:
  2. Sheyko, T., Maksymenko-Sheyko, K., Sirenko, V., Morozova, A., & Petrova, R. (2019). Analytical identification of the unmanned aerial vehicles’ surfaces for the implementation at a 3D printer. Eastern-European Journal of Enterprise Technologies, vol. 1, no. 2 (97), pp. 48–56.
  3. Sheyko, T. I., Maksimenko-Sheyko, K. V., Tolok, A. V., & Morozova, A. I. (2019). Matematicheskoye i kompyuternoye modelirovaniye aerokosmicheskikh obyektov dlya realizatsii tekhnologii 3D-pechati [Mathematical and computer modeling of aerospace objects for the implementation of 3D printing technology]. Informatsionnyye tekhnologii v proyektirovanii i proizvodstveInformation technology of CAD/CAM/CAE, iss. 2 (174), pp. 16–20 (in Russian).
  4. Rvachev, V. L. (1982). Teoriya R-funktsiy i nekotoryye yeye prilozheniya [R-functions theory and some of its applications]. Kiyev: Naukova dumka, 552 p. (in Russian).
  5. Rvachev, V. L. & Sheiko, T. I. (1995). R-functions in boundary value problems in mechanics. Applied Mechanics Reviews, vol. 48, no. 4, pp. 151–188.
  6. Maksimenko-Sheyko, K. V. (2009). R-funktsii v matematicheskom modelirovanii geometricheskikh obyektov i fizicheskikh poley [R-functions in mathematical modeling of geometric objects and physical fields]. Kharkov: IPMash NAN Ukrainy, 306 p. (in Russian).


Received 11 May 2020

Published 30 September 2020