2020, issue 1, p. 23-31

Received 09.02.2020; Revised 18.02.2020; Accepted 10.03.2020

Published 31.03.2020; First Online 26.04.2020


Previous  |  Full text (in Ukrainian)  |  Next



Petro Stetsyuk 1 * ORCID ID favicon Big,   Oleksandr Tkachenko 2,   Olga Gritsay 2

1 V.M. Glushkov Institute of Cybernetics, Kyiv, Ukraine.

2 SE "Ivchenko-Progress", Zaporozhye, Ukraine.

* Correspondence: This email address is being protected from spambots. You need JavaScript enabled to view it.


The aim of the article is to develop a method, an algorithm, and appropriate software for constructing the external contour of the Frankl nozzle in the supersonic part using S-shape curves. The method is based on the problem of constructing a curve with the natural parameterization. The curve passes through two given points with the given inclination angles of the tangents and provides the given inclination angle of the tangent at the point with the given abscissa [4]. To control the inflection point of the S-shaped curve, the inclination angle of the tangent at a point with the known abscissa is used.

In the case, when the curvature is given by a quadratic function, the system of five nonlinear equations is formulated, among which three equations are integral. The system has five unknown variables three coefficients of the quadratic function, the total length of the curve and the length of the curve to the point with a known abscissa.

The lemma on the relation between solutions of the original and the scalable systems, in which the coordinates of the points are multiplied by the same value, is proved. Due to this lemma, it becomes possible, using the obtained solution of the well-scalable system, to find easily the corresponding solution of a bad-scalable (singular) system.

To find a solution to the system, we suggest to use the modification of the r-algorithm [5] solving special problem on minimization of the nonsmooth function (the sum of the modules of the residuals of the system), under controlling of the constraints on unknown lengths, in order to guarantee their feasible values.

The algorithm is implemented using the multistart method and the ralgb5a octave function [6]. It finds the best local minimum of nonsmooth function by starting the modification of the r-algorithm from a given number of starting points. The algorithm uses an analytical computation of generalized gradients of the objective function and the trapezoid rule to calculate the integrals.

The computational experiment was carried out to design the fragment of supersonic part in the external contour of a Frankl-type nozzle. The efficiency of the algorithm, developed for constructing S-shape curves, is shown.


Keywords: nozzle contour, natural parameterization, curvature, nonsmooth optimization, r-algorithm


Cite as: Stetsyuk P., Tkachenko O., Gritsay O. On Construction of the External Frankl Nozzle Contour Using Quadratic Curvature. Cybernetics and Computer Technologies. 2020. 1. 23–31. (in Ukrainian) https://doi.org/10.34229/2707-451X.20.1.3



           1.     Alemasov V.E. Theory of rocket engines: Textbook for students of higher technical educational institutions/ Alemasov V.E., A.F. Dregalin, A.P. Tishin; Ed. V.P. Glushko.  M. Mashynostroenie, 1989. 464 p. (in Russian)

           2.     Rashevskii P.K. Differential Geometry Course. 4 edition. M. Gostekhizdat, 1956. 420 p. (in Russian)

           3.     Borysenko V., Agarkov O.,Pal’ko K, Pal’ko M. Modeling of plane curves in natural parameterization. Geometrychne modeliuvania ta informaciini tekhnologii. 2016. 1. P. 2127. (in Ukrainian) http://nbuv.gov.ua/UJRN/gmtit_2016_1_6

           4.     Borysenko V. D., Ustenko S.A., Ustenko I.V. Geometric modeling of s-shaped skeletal lines of axial compressor blades profiles. Vestnik dvigatelestroeniia. 2018. 1. P. 4552. (in Russian) https://doi.org/10.15588/1727-0219-2018-1-7

           5.     Shor N.Z., Stetsyuk P.I. Modified r-algorithm to find the global minimum of polynomial functions. Cybernetics and Systems Analysis. 1997. 33(4) P. 482 497. https://doi.org/10.1007/BF02733104

           6.     Stetsyuk P.I. Computer program "Octave program ralgb5a: r(α)-algorithm with adaptive step". Svidotstvo pro rejestratsiju avtorskogo prava na tvir 8501. Ukraine. Ministerstvo ekonomichnogo rozvytku I torgivli. Derzhavnyi department intelektualnoji vlasnosti. Data reiestratsii 29.01.2019. (in Ukrainian)



ISSN 2707-451X (Online)

ISSN 2707-4501 (Print)

Previous  |  Full text (in Ukrainian)  |  Next




Copyright © 2019-2020 V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine,

National Academy of Sciences of Ukraine.

All rights reserved.