弹箭非线性角运动周期解稳定性分析

Stability Analysis of Periodic Solution for Nonlinear Angle Motion of Projectiles and Rockets

为了分析和计算弹箭非线性角运动周期解的稳定性,推导了弹箭的非线性角运动方程组。以某型火箭弹高原试验为例,计算了立方马格努斯力矩系数取不同值时的角运动相图和庞加莱截面图;通过Poincare映射计算了线性马格努斯力矩系数作为分岔参数时的分岔图;利用推广的打靶法计算了角运动周期解的幅值和周期,结合Floquet理论分析了周期解的稳定性。结果表明,在高空低密度的情况下,考虑非线性马格努斯力矩系数后,当马格努斯力矩系数达到一定范围时,火箭弹角运动由零平衡位置分岔出稳定的周期运动。

In order to analyze and calculate the stability of the periodic solution for nonlinear angle motion of projectiles and rockets, the equations of the nonlinear angle motion were derived. Taking a rocket plateau test for an example, the phase portraits of angle motion and Poincare surface of section were calculated while the value of the cubic Magnus moment coefficient（MMC） varies. The bifurcation diagram was calculated based on Poincare map in which the linear MMC was selected as the bifurcation parameter. The amplitude and cycle of the periodic solution were calculated by generalized shooting method, and then the stability of the periodic solution was analyzed by using Floquet theory. In the case of low density at high altitude, the stable periodic motion is bifurcated from zero equilibrium position of the angle motion of the rocket after considering the nonlinear MMC while the MMC reaches a certain range.