弹箭角运动的非线性吸引域估计

doi:10.12382/bgxb.2022.0417

摘要/Abstract

摘要：

为探索平方和规划方法在弹箭非线性吸引域估计中的应用,对计及高阶静力矩的无控尾翼弹平面角运动进行多项式化处理,并开展了非线性吸引域估计,通过构造李雅普诺夫函数并利用V-s迭代算法,得到了精度较高的吸引域估计;考虑法向力系数、俯仰力矩系数、赤道阻尼力矩系数以及马格努斯力矩系数的非线性,利用平方和优化得到脉冲末端修正迫弹空间角运动的非线性吸引域估计,由此获得脉冲参数(脉冲冲量和轴向作用位置)的有效组合范围,据此形成脉冲控制弹箭的参数设计方法。研究结果表明,利用平方和规划方法估计弹箭角运动的非线性吸引域,具有较高准确性,将估计结果用于弹箭参数设计是可行和有效的。

关键词: 攻角运动, 非线性气动力, 吸引域, 平方和规划, 稳定性

Abstract:

This paper explores the application of the Sum of Squares Programming (SOSP) in the nonlinear region of attraction estimation for projectiles. Considering quantic static moment, the nonlinear planar angular motion for an unguided fin-stabilized projectile is expressed in polynomials and its region of attraction is estimated using Lyapunov functions and V-s iterative algorithm, leading to an accurate estimation. Considering the nonlinear items of normal force coefficient, pitching moment coefficient, equatorial damping moment coefficient, and Magnus moment coefficient, the nonlinear region of attraction of the spacial angular motion of a terminal trajectory corrected mortar projectile with impulses is also studied. Thus, the effective combination of impulse parameters (impulse magnitude and axial acting position) is obtained and then a method for designing impulse parameters is proposed. The findings show that the estimation of the nonlinear region of attraction for projectile’s angular motion obtained by SOSP is of relatively high accuracy. It is feasible and effective to apply the estimation to projectile design.

Key words: motion of angle of attack, nonlinear aerodynamics, region of attraction, sum-of-squares programming, stability

中图分类号:

TJ760.1

李东阳, 常思江, 王中原. 弹箭角运动的非线性吸引域估计[J]. 兵工学报, 2023, 44(8): 2329-2341.

LI Dongyang, CHANG Sijiang, WANG Zhongyuan. Nonlinear Region of Attraction Estimation for Projectile’s Angular Motion[J]. Acta Armamentarii, 2023, 44(8): 2329-2341.

图/表 14

表1 系统稳定性分析

Table 1 System stability analysis

气动阻尼项符号组合	原点稳定性	极限环δ=2 $- H 0 / H 2$ 稳定性
H₀<0,H₂>0	不稳定平衡点	稳定极限环
H₀>0,H₂<0	稳定平衡点	不稳定极限环
H₀<0,H₂<0	不稳定平衡点	不存在极限环
H₀>0,H₂>0	稳定平衡点	不存在极限环

表1 系统稳定性分析

Table 1 System stability analysis

气动阻尼项符号组合	原点稳定性	极限环δ=2 $- H 0 / H 2$ 稳定性
H₀<0,H₂>0	不稳定平衡点	稳定极限环
H₀>0,H₂<0	稳定平衡点	不稳定极限环
H₀<0,H₂<0	不稳定平衡点	不存在极限环
H₀>0,H₂>0	稳定平衡点	不存在极限环

图1 不同初始条件下攻角运动相平面图

Fig.1 Phase plane diagram of angular motion under various initial conditions

图2 不同阶次李雅普诺夫函数估计下的吸引域

Fig.2 Regions of attraction using Lyapunov functions with various orders

表2 某120mm迫弹结构参数[10]

Table 2 Structure parameters of a 120mm mortar projectile[10]

参数	数值
弹径/mm	120
弹质量/kg	13.585
弹长/m	0.7049
重心位置(距弹底)/m	0.4229
轴向转动惯量/(kg·m²)	0.02335
横向转动惯量/(kg·m²)	0.23187

表3 弹道末段起始位置的弹道参数

Table 3 Ballistic parameters at the beginning of the terminal trajectory

时间/ s	速度/ (m·s^-1)	马赫数	俯仰角/ (°)	攻角/ (°)	高度/ m
38.07	219.50	0.65	-47.51	0.01	588.3

表4 弹道末段起始位置的角运动方程参数

Table 4 Parameters of angular motion function at the beginning of the terminal trajectory

H	M	T	R_z	R_y
1.11×10^-3	8.53×10^-5	1.29×10^-4	1.63×10^-8	0

图3 原模型和多项式近似模型的相轨迹比较

Fig.3 Comparison of phase trajectory between original model and polynomial approximation model

图4 吸引域估计结果的空间表示

Fig.4 Spatial representation of regions of attraction estimation

图5 吸引域估计结果的剖面图

Fig.5 Profile of regions of attraction estimation

图6 交集估计Ω∩在α'-β'平面上的位置变化

Fig.6 Phase diagram of intersection estimation Ω∩ in the α'-β' plane

表5 吸引域半径计算结果

Table 5 Calculated radius of the region of attraction

吸引域水平集	V₂	V₄
优化估计γ_∩	0.3967	0.9002
γ_∩式(47)计算值	0.3981
优化估计γ_∀	0.0676	0.8081
基于优化估计γ_∩的γ_∀计算值	0.0674
γ_∀式(47)计算值	0.0686

图7 满足水平集γ∀=0.0686的脉冲参数组合

Fig.7 Combination of impulse parameters satisfying the level set of γ∀=0.0686

图8 不同γ∀下的有效脉冲参数组合

Fig.8 Effective combination of impulse parameters with various values of γ∀

图9 γ∀与Ω∀中攻角和攻角速度最大值的对应关系

Fig.9 Maximum angle of attack and its rate of change for various γ∀ and Ω∀

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