In order to analyze the angular motion stability of projectile, the equations of the nonlinear angular motion are derived, and the Hopf bifurcation analysis method of the nonlinear angular motion of projectile is given. Taking a rocket plateau test as an example, the air density is selected as the bifurcation parameter, and the Hurwitz criterion is used to judge the stability of the system. The bifurcation point is determined. Center manifold theory is proposed to reduce the system dimension, and then a three-order normal form of Hopf bifurcation is calculated by plotting the bifurcation diagram. In addition, the effect of the bifurcation parameter on the swing of the limit cycle is analyzed. The numerical simulations show that the bifurcation analysis method can be used to judge the stability of the system correctly and analyze the motion of limit cycle in the system accurately.
ZHONG Yang-wei, WANG Liang-ming, FU Jian, CHANG Si-jiang.
Hopf Bifurcation Analysis of Nonlinear Angular Motion Stability of Projectile. Acta Armamentarii. 2015, 36(7): 1195-1202 https://doi.org/10.3969/j.issn.1000-1093.2015.07.007
基金
国家自然科学基金项目（11402117）
{{custom_fund}}
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
［1］ 韩子鹏.弹箭外弹道学［M］.北京：北京理工大学出版社，2008. HAN Zi-Peng．Exterior ballistics of shells and rockets［M］．Beijing：Beijing Institute of Technology Press，2008.(in Chinese) ［2］ 李臣明，刘怡昕.大长径比远程弹箭的极限平面摆动及其抑制［J］.系统与仿真学报，2009，21(23)：7390-7393. LI Chen-ming，LIU Yi-xin． Limit plane swing motion and its restraining measure of un-rotary long-range missile with large ratio of length to diameter［J］．Journal of System Simulation，2009，21(23)： 7390-7393. (in Chinese) ［3］ 徐明友.高等外弹道学［M］.北京：高等教育出版社，2003. XU Ming-you．Advanced exterior ballistics［M］．Beijing：Higher Education Press，2003. (in Chinese) ［4］ 王华毕，吴甲生.火箭弹锥形运动的数学仿真与抑制措施 ［J］.北京理工大学学报，2007，27(3)：196-199. WANG Hua-bi，WU Jia-sheng．Coning motion of rockets, its numerical simulation and restraint［J］．Transactions of Beijing Institute of Technology，2007，27(3)：196-199. (in Chinese) ［5］ 王华毕，吴甲生.火箭弹锥形运动稳定性分析［J］.兵工学报，2008，29(5)：562-566. WANG Hua-bi，WU Jia-sheng．The coning motion stability analysis of rocket ［J］．Acta Armamentarii，2008，29(5)：562-566. (in Chinese) ［6］ 闫晓勇，杨树兴，张成.基于章动运动理论的火箭弹锥形运动稳定性分析［J］.兵工学报，2009，30(10)：1291-1296. YAN Xiao-yong，YANG Shu-xing，ZHANG Cheng．Analysis of stability for coning motion of rockets based on theory of nutation movement［J］．Acta Armamentarii，2009，30(10)：1291-1296. (in Chinese) ［7］ 李 克勇，赵良玉，周伟.一类旋转弹在高空中的锥形运动稳定性［J］. 动力学与控制学报，2012，10(3)：239-243. LI Ke-yong，ZHAO Liang-yu，ZHOU Wei．Stability for coning motion of a spinning projectile［J］．Journal of Dynamic and Control，2012，10(3)：239~243. (in Chinese) ［8］ MaoX R，Yang S X，Xu Y. Research on the coning motion of wrap around fin projectiles ［J］. Canadian Aeronautics and Space Journal，2006，52(3)：119-125. ［9］ 赵良玉，杨树兴.卷弧翼火箭弹圆锥运动收敛速度计算方法 ［J］.固体火箭技术，2009，32(1)：15-19. ZHAO Liang-yu，YANG Shu-xing．Research on convergence speed of coning motion of wrap around fin rockets［J］．Journal of Solid Rocket Technology，2009，32(1)：15-19. (in Chinese) ［10］ 赵 良玉，杨树兴，焦清介.提高卷弧翼火箭弹圆锥运动渐进稳定性 的几个方法［J］.固体火箭技术，2010，33(4)：369-372. ZHAO Liang-yu，YANG Shu-xing，JIAO Qing-jie．Several methods for improving asymptotic stability of coning motion of wrap around fin rockets ［J］．Journal of Solid Rocket Technology，2010，33(4)： 369-372. (in Chinese) ［11］ 任天荣，马建敏.基于陀螺力学的旋转导弹锥形运动分析［J］.宇航学报，2010，31(9)：2082-2087. REN Tian-rong，MA Jian-min．Coning motion analysis of spinning missile based on gyro dynamics ［J］．Journal of Astronautics，2010，31(9)：2082-2087. (in Chinese) ［12］ 任天荣，马建敏.旋转弹锥形运动发生区间及频率特性研究［J］.固体火箭技术，2014，37(3)：295-300. REN Tian-rong，MA Jian-min．Research on activating region and frequency characteristics of coning motion for spinning missiles ［J］．Journal of Solid Rocket Technology，2014，37(3)：295- 300. (in Chinese) ［13］ McCoyR L．Modern exterior ballistics ［M］. Atglen, Pennsylvania: Schiffer Publishing Ltd，1999. ［14］ 张琪昌，王洪礼，竺致文，等.分岔与混沌理论及应用［M］.天津： 天津大学出版社，2005. ZHANG Qi-chang，WANG Hong-li，ZHU Zhi-wen, et al. Bifurcation and chaos theory and its application ［M］．Tianjin：Tianjin University Press，2005. (in Chinese) ［15］ 陆启韶.常微分方程与动力系统［M］.北京：北京航空航天大学 出版社，2010. LU Qi-shao．Ordinary differential equations and dynamical systems ［M］．Beijing：Beijing University of Aeronautics and Astronautics Press, 2010.(in Chinese)
