正规形法在弹箭非线性运动分析中的应用

非线性气动力对弹箭运动特性具有重要影响，而其复杂性和有效分析工具的缺乏往往制约了弹箭非线性运动理论的发展。为探索正规形方法在弹箭非线性运动分析中的应用，构造了考虑二次非线性阻尼和七次非线性静力矩下攻角方程的正规形，进而求得攻角的通用解析解，通过数值积分验证了其在较大攻角范围内的有效性，该解析解也同样适用于无阻尼角运动和更高或更低阶静力矩作用下的角运动分析。基于正规形方法导出的初始条件关系，给出了保守但简洁的稳定初始条件范围的计算方法，结合平衡点分析，可较为准确地预测弹箭在非线性气动力作用下形成的极限环及其稳定性。

Applying the method of normal forms to projectile nonlinear motion analysis

Nonlinearity especially from aerodynamic coefficients in high orders has a significant effect on projectile dynamics. Its investigation has been hindered in the conventional analysis by the complexity in nonlinear motion equations and the lack of appropriate analysis tools. Therefore, the widely used method of normal forms was introduced for the analysis of projectile angular motion. Considering the second order damping and the seventh order static moment terms, the normal form of the angular motion was derived and thus the universal analytical solution of the angle of attack is obtained, which is verified to show good agreement with the numerical integration results over a wide range of angle of attack and also demonstrates its being applicable to the undamped case and the cases with lower or higher order of static moment. In addition, the obtained relationship between initial conditions can give a conventional but simple determination of the region of attraction to the origin. Also, the amplitude equation combined with the equilibrium analysis provides a accurate prediction for the existence and stability of limit cycle in angular motion.