
№ 2 (20) – 2023


VERIFICATION OF THE MATHEMATICAL FLIGHT MODEL THE PROJECTILE AS A MODIFIED MODEL OF THE MATERIAL POINT


https://doi.org/10.37129/23137509.2023.20.3444




R. Bubenshchykov 


S. Bondarenko, Candidate of Technical Sciences, Associate Professor 


V. Hrabchak, Doctor of Technical Sciences, Professor 








Cite in the List of bibliographic references (DSTU 8302:2015)


Бубенщиков Р. В., Бондаренко С. В., Грабчак В. І. Верифікація математичної моделі польоту снаряда як модифікованої моделі матеріальної точки. Збірник наукових праць Військової академії (м. Одеса). 2023. № 2 (20). С. 3444. https://doi.org/10.37129/23137509.2023.20.3444




Abstract


The analysis of mathematical models of the flight of projectiles has been carried out, it is shown that the nature of their provision varies depending on the degree of reliability of the representation of the real physical process of its flight by a mathematical model, the adequate consideration of certain forces (moments) acting on the projectile, as well as the level of information about external flight conditions. In the simplest mathematical model with three degrees of freedom, the projectile is considered as a material point that moves in the atmosphere under the influence of gravity and full aerodynamic force. The dynamics of projectile flight is most completely described by the mathematical model of the movement of a solid body with six degrees of freedom. In the intermediate model with four degrees of freedom, all aerodynamic forces are taken into account, the orientation of the projectile is characterized by taking into account the angle of nutation, and the kinetic energy of the rotational movement is taken into account through the angular velocity of the projectile around its axis of symmetry. The process of restori ng the coefficients of aerodynamic forces using the material point model does not provide the necessary accuracy and adequacy of the description of the projectile flight process; of the modified material point model is much simpler (the least complex) compared to the model of the motion of a rigid body with six degrees of freedom due to the smaller number of coefficients and differential equations included in its composition. The differential equations of the modified material point model are provided  the equation of motion of the center of mass and the equation of nutational oscillations of the projectile in scalar form; an assessment of its accuracy (adequacy) was carried out on the example of the 155mm PF projectile M2000, one of the family of projectiles of the company Denel Naschem – Assegai M200X Series 155mm Projectiles. A comparison of the characteristics of the 155mm PF of the M2000 projectile calculated according to the 6DoF model and the modified material point model showed their high convergence; the error does not exceed 1%.


Keywords


aerodynamic forces (moments), projectile, mathematical models, modified model, aerodynamic coefficients, simulation, accuracy, flight parameters.




List of bibliographic references



Дмитриевский А. А., Лисенко Л. Н. Внешняя баллистика : учебник. Москва : Машиностроение, 2005. 607 с.

STANAG 4355 (Edition 3), The modified point mass and five degrees of freedom trajectory models: NSAl0454(2009)JAIS/4355, dated 17 April 2009. 95 p. (NATO Standardization Agency).

Carlucci D. E., Jacobson S. S. Ballistics, theory and design of guns and ammunition : book. London, New York : Taylor & Francis Group, 2007. 514 p.

McCoy R. L. Modern Exterior Ballistics. Atglen, PA. : Schiffer Military History, 2012. 328 p.

Baranowski, L., Gadomski, B., Majewski, P. & Szymonik, J. Explicit ”ballistic mmodel”: a refinement of the implicit ”modified point mass trajectory model”. Bulletin of the Polish Academy of sciences technical sciences. 2016. Vol. 64, No.1, pp. 8189. DOI: 10.1515/bpasts20160010.

Калиткин Н. Н. Численные методы. Москва : Наука, 1978. 512 с.

Kincaid D. Numerical analysis. Brooks : Cole Publishing Company. 1991. 690 p.

Baranowski L. Eﬀect of the mathematical model and integration step on the accuracy of the results of computation of artillery projectile ﬂight parameters. Bulletin of the Polish Academy of sciences technical sciences. 2013. Vol. 61, No. 2, pp. 475484. DOI: 10.2478/bpasts20130047.




References



Dmitriyevskiy, A. A., & Lisenko, L. N. (2005). External ballistics. Mashinostroyeniye Publ. [in Russian].

STANAG 4355 (2009). The modified point mass and five degrees of freedom trajectory models : NSAl0454(2009)JAIS/4355 (NATO Standardization Agency). (Edition 3).

Carlucci, D. E., & Jacobson, S. S. (2007). Ballistics, theory and design of guns and ammunition : book. London, New York : Taylor & Francis Group.

McCoy, R. L. (2012). Modern Exterior Ballistics. Atglen, PA. : Schiffer Military History.

Baranowski, L., Gadomski, B., Majewski, P. & Szymonik, J. (2016). Explicit ”ballistic mmodel”: a refinement of the implicit ”modified point mass trajectory model”. Bulletin of the Polish Academy of sciences technical sciences,64, 1, 81–89. DOI: 10.1515/bpasts20160010.

Kalitkin, N. N. (1978). Numerical Methods.Nauka Publ. [in Russian].

Kincaid, D.(1991). Numerical analysis. Brooks : Cole Publishing Company Publ.[in English].

Baranowski, L.(2013). Eﬀect of the mathematical model and integration step on the accuracy of the results of computation of artillery projectile ﬂight parameters. Bulletin of the Polish Academy of sciences technical sciences, 61, 2, 475–484. DOI: 10.2478/bpasts20130047.
