Nonlinear Region of Attraction Estimation for Projectile’s Angular Motion

doi:10.12382/bgxb.2022.0417

Abstract

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

CLC Number:

TJ760.1

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

Figures/Tables 14

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

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

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

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

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

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

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

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

Fig.4 Spatial representation of regions of attraction estimation

Fig.5 Profile of regions of attraction estimation

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

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

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

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

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

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