弹丸飞行数学模型中的小角度逼近问题分析

V. Мaidaniuk
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摘要

本文讨论了弹丸飞行数学模型的建立问题,该模型能准确地描述弹丸在空中的运动。结果表明,数学模型表示的性质取决于数学模型所要求的真实物理弹丸飞行过程表示的可靠性程度,是否充分考虑作用在弹丸上的某些力(力矩),以及包括弹丸运动的空气参数在内的外部飞行条件的信息水平。同时,在微分方程系统中使用形状系数-一致系数导致数学模型“粗糙”,不能充分描述弹丸的飞行轨迹及其各个要素。该问题的解决方案在制定和实施程序,实现与北约部队互操作性的必要水平的技术解决方案,逐步放弃空气阻力的标准函数,过渡到单个函数和弹丸运动的数学模型期间特别相关,目前在联盟成员国中被接受。对现代数学模型的分析表明,它们的建立是基于一种近似方法,称为小角近似,其中,对于轴对称旋转弹丸,认为章动角足够小,气动力(力矩)仅取决于其飞行速度和章动角,并且只使用泰勒级数展开的线性项进行计算。考虑了弹丸的章动-进动特性,揭示了弹丸气动力(力矩)系数与章动角的非线性关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Analysis of problems of small-angle approximation in mathematical models of projectile flight
The article deals with the topical issue of developing mathematical models of projectile flight, which accurately describe the projectile motion in the air. It is shown that the nature of the mathematical models presentation varies depending on the required reliability degree of the real physical projectile flight process representation by the mathematical model, the adequate consideration of certain forces (moments) acting on the projectile, as well as the level of information about the external flight conditions which include the parameters of the air in which the projectile moves.At the same time, the use of the shape coefficient - the agreement coefficient in the differential equation system leads to "rough" mathematical models, which does not allow to adequately describe the projectile flight trajectory and its individual elements. The solution to this problem is especially relevant during developing and implementing procedures, technical solutions in the interest of achieving the necessary level of interoperability with NATO forces, the gradual abandonment of the standard functions of air resistance, the transition to individual functions and mathematical models of projectile motion, which are currently accepted in the member states of the Alliance. The conducted analysis of modern mathematical models showed that their construction is based on an approximate approach, which was called the small-angle approximation, in which, for an axisymmetric rotating projectile, it is considered that the nutation angles are sufficiently small, the aerodynamic forces (moments) depend only on the speed of its flight and the nutation angle, and only the linear terms of their Taylor series expansion are used in the calculations. The nutation-precessional behavior of the projectile was considered and the nonlinear dependencies of the coefficients of the aerodynamic forces (moments) of the projectile on the angles of nutation were revealed.
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