Nonlinear region of attraction of angular motion for a controlled projectile with self-rotating wraparound fins
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摘要:
针对有控自旋卷弧翼弹箭在鸭舵控制力与非线性气动力作用下的稳定性问题，开展非线性角运动特性研究。根据该类弹箭的弹道和结构特点，建立七自由度刚体运动模型，据此推导状态空间形式的非线性角运动方程；选取弹道特征点，利用数值仿真和多项式拟合方法得到对应的非线性气动力系数，采用基于平方和规划的吸引域估计方法，定量分析鸭舵控制和非线性气动力系数对该类弹箭稳定域的影响。理论分析及仿真结果表明：鸭舵控制力大小对弹箭角运动的稳定吸引域影响较大，对于给定算例，当舵偏角为20°时，与无控状态相比，吸引域边界减小约16.9%，而控制力方位对其影响却很小；升力系数导数、静力矩系数导数和前、后体马格努斯力矩系数导数的非线性项是影响该类弹箭角运动稳定吸引域的主要因素，且后体非线性马格努斯力矩对吸引域的影响明显大于前体。
Abstract:A study on the nonlinear angular motion characteristics has been conducted with the goal of addressing the stability issue of a controlled projectile with self-rotating wraparound fins under the influence of nonlinear aerodynamic force and duck rudder control force. According to the ballistic and structural characteristics of this type of sling and arrows, a seven-degree-of-freedom rigid-body motion model is established, according to which the nonlinear angular equations of motion in the state-space form are deduced. The characteristic points of the ballistic trajectory are selected, and the corresponding nonlinear aerodynamic coefficients are obtained by using the numerical simulation and polynomial fitting methods, and the estimation of the domain of attraction based on the sum-of-squares planning is adopted to analyse the effects of the duck rudder control and nonlinear aerodynamic coefficients on the stability domain of this type of slings and arrows in a quantitative manner. Based on the results of the simulation and theoretical analysis, the size of the duck rudder control force has a significant impact on the stable attraction domain of the angular motion of the projectile and arrow. For example, in the paper, the attraction domain boundary decreases by approximately 16.9% when the rudder deflection angle
δ c is 20° compared to the uncontrolled state, while the control force azimuth has barely any effect. The lift coefficient derivative, the static moment coefficient derivative, and the nonlinear terms of the forebody and aft body Magnus moment coefficient derivative are the main factors affecting the stable attraction domain of the projectile and arrow in this type of motion. The nonlinear Magnus moment of the rear body has a significantly larger effect on the region of attraction of the angular motion of the projectile than that of the front body. -
图 1 有控自旋卷弧翼弹箭模型示意图
Figure 1. The model diagram of controlled projectile with self-rotating wraparound fins
图 4 前、后体马格努斯力矩系数随攻角变化
Figure 4. The Magnus moment coefficient of front and rear bodies varies with the angle of attack
图 5 吸引域估计结果的空间轨迹
Figure 5. The spatial trajectory of the estimated result of the region of attraction
图 6 吸引域估计结果的相平面轨迹
Figure 6. The phase plane trajectory of the estimated result of the region of attraction
图 7 不同初始攻角下的攻角运动轨迹
Figure 7. Angle of attack motion trajectories at different initial angles of attack
图 8 不同控制方位角下的吸引域估计结果
Figure 8. The estimation results of region of attraction under different control azimuth
图 9 不同舵偏角对应的吸引域估计结果
Figure 9. The estimation results of region of attraction corresponding to different rudder angles
图 10 吸引域边界随$ {\kappa _1} $和$ {\kappa _4} $变化
Figure 10. Region of attraction boundaries vary with $ {\kappa _1} $and $ {\kappa _4} $
图 11 吸引域边界随$ {\kappa _2} $和$ {\kappa _3} $变化
Figure 11. Region of attraction boundaries vary with $ {\kappa _2} $ and $ {\kappa _3} $
图 12 吸引域边界随$ {\tau _1} $和$ {\tau _2} $变化
Figure 12. Region of attraction boundaries vary with $ {\tau _1} $ and $ {\tau _2} $
图 13 吸引域边界随$ {\kappa _5} $和$ {\kappa _6} $变化
Figure 13. Region of attraction boundaries vary with $ {\kappa _5} $ and $ {\kappa _6} $
图 14 吸引域边界随$ {\tau _4} $和$ {\tau _5} $变化
Figure 14. Region of attraction boundaries vary with $ {\tau _4} $ and $ {\tau _5} $
表 1 弹箭物理参数
Table 1. Physical parameters of projectile
$ {m_{\text{F}}} $/kg $ {m_{\text{B}}} $/kg $ {r_{\text{F}}} $/m $ {r_{\text{B}}} $/m $ {A_{\text{F}}} $/(kg·m2) $ {A_{\rm B} } $/(kg·m2) $ {C_{\text{F}}} $/(kg·m2) $ {C_{\rm B} } $/(kg·m2) $ r $/m d/m l/m $ {\delta _{\rm c}} $/(°) 15.10 24.90 0.152 4 0.307 3 1.265 5.767 0.023 0.094 0.619 8 0.101 6 0.101 6 5 
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表 2 弹道顶点处的状态参数
Table 2. State parameters at the apex of the trajectory
t/s v/(m·s−1) Ma $ {\dot \gamma _{\rm F}} $/(rad·s−1) $ {\dot \gamma _{\rm B}} $/(rad·s−1) $ {\theta _{\mathrm{a}}} $/(°) 28.10 230.67 0.714 13.13 182.89 0
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表 3 各气动系数拟合决定系数
Table 3. The coefficient of determination of each aerodynamic coefficient parameter
参数 $ {R^2} $ $ {c_x} $ 0.999 $ c_y^\prime $ 0.995 $ m_{\textit{z}}^\prime $ 0.994 $ m_{{\textit{zz}}}^\prime $ 0.998 $ {m}_{\text{F},y}^{\prime \prime} $ 0.983 $ m_{{\rm B},y}^\prime $ 0.976
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表 4 不同阶次下Lyapunov函数所对应的水平集
Table 4. The level sets corresponding to Lyapunov functions of different orders
V的阶次 水平集 2 1.000 1 4 1.000 2 6 1.001 2
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