一类对称约束碰振系统的余维二擦边分岔的存在条件

PDF(679 KB)

振动与冲击 ›› 2022, Vol. 41 ›› Issue (5) : 273-279.

论文

徐洁琼，陈慧莹，伍帅，袁泉

作者信息 +

Existence conditions of co-dimension-2 grazing bifurcation for a class of symmetrically constrained vibro-impact systems

XU Jieqiong, CHEN Huiying, WU Shuai, YUAN Quan

Author information +

文章历史 +

摘要

类似光滑系统的余维二分岔的分类方法，余维二擦边分岔被划分为三种类型，分别是擦边点退化、退化环擦边（非双曲）以及两个擦边事件同时发生。本文分析了一个二自由度对称约束的碰撞振动系统，得到了该系统第二类余维二擦边分岔的存在条件。首先，考虑双侧擦边周期运动，理论推导出双侧擦边周期运动的存在性条件。利用不连续映射方法，得出1/1/n碰撞周期运动发生鞍结分岔和倍周期分岔的解析表达式。然后，结合双擦边周期运动的存在性条件和1/1/n碰撞周期运动的分岔条件，推导出发生余维二擦边分岔时满足的解析表达式，并以周期1运动为例，给出了余维二擦边分岔点的分布。

Abstract

similar to the approach used to classify co-dimension-two bifurcations in smooth systems, co-dimension-two grazing bifurcations can be put into one of the following three types: degenerate gazing point, grazing of degenerate cycles, simultaneous occurrence of two grazings. This paper obtained the existence condition of the second type co-dimension-two grazing bifurcation for a two-degree-of-freedom vibro-impact system with symmetric constraints. First, considering the periodic motion of the double-sided grazing, the existence condition of the periodic motion of the double-sided grazing was deduced theoretically. Using the discontinuous mapping method, the analytical expressions of saddle-node bifurcation and period doubling bifurcation for 1/1/n impact periodic motion are derived. Then, combining the existence condition of the grazing periodic motion and the bifurcation condition of the 1/1 /n impact periodic motion, the analytical expression of co-dimensional-two grazing bifurcation was obtained, and The distribution of co-dimensional-two bifurcation points is analyzed for period one motion.

导出引用

徐洁琼，陈慧莹，伍帅，袁泉. 一类对称约束碰振系统的余维二擦边分岔的存在条件[J]. 振动与冲击, 2022, 41(5): 273-279

XU Jieqiong, CHEN Huiying, WU Shuai, YUAN Quan. Existence conditions of co-dimension-2 grazing bifurcation for a class of symmetrically constrained vibro-impact systems[J]. Journal of Vibration and Shock, 2022, 41(5): 273-279

参考文献

[1] Wen G, Chen S, Jin Q. A new criterion of period-doubling bifurcation in maps and its application to an inertial impact shaker[J]. Journal of sound and vibration, 2008, 311(1-2): 212-223.
[2] Luo G, Zhang Y, Zhang J, et al. Periodic motions and bifurcations of vibro-impact systems near a strong resonance point[J]. Nonlinear Science And Complexity. 2007: 193-203..
[3] Nordmark A B. Non-periodic motion caused by grazing incidence in an impact oscillator[J]. Journal of Sound and Vibration, 1991, 145(2): 279-297.
[4] Nordmark A B. Universal limit mapping in grazing bifurcations[J]. Physical review E, 1997, 55(1): 266.
[5] Bernardo M, Budd C, Champneys A R, et al. Piecewise-smooth dynamical systems: theory and applications[M]. Springer Science & Business Media, 2008.
[6] Di Bernardo M, Budd C J, Champneys A R. Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems[J]. Physica D: Nonlinear Phenomena, 2001, 160(3-4): 222-254..
[7] Kowalczyk P, di Bernardo M, Champneys A R, et al. Two-parameter discontinuity-induced bifurcations of limit cycles: classification and open problems[J]. International Journal of bifurcation and chaos, 2006, 16(03): 601-629.
[8] Yin S, Shen Y, Wen G, et al. Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator[J]. Nonlinear Dynamics, 2017, 90(1): 443-456.
[9] Thota P, Zhao X, Dankowicz H. Co-dimension-two grazing bifurcations in single-degree-of-freedom impact oscillators[J]. Journal of Computational and Nonlinear Dynamics, 2006, 1(4): 328-335.
[10] Zhao X, Dankowicz H. Unfolding degenerate grazing dynamics in impact actuators[J]. Nonlinearity, 2005, 19(2): 399.
[11] Jiang H, Wiercigroch M. Geometrical insight into non-smooth bifurcations of a soft impact oscillator[J]. IMA Journal of Applied Mathematics, 2016, 81(4):662-678.
[12] Jiang H, Chong A S E, Ueda Y, et al. Grazing-induced bifurcations in impact oscillators with elastic and rigid constraints[J]. International Journal of Mechanical Sciences, 2017, 127: 204-214.
[13] Yin S, Wen G, Xu H, et al. Higher order zero time discontinuity mapping for analysis of degenerate grazing bifurcations of impacting oscillators[J]. Journal of Sound and Vibration, 2018, 437: 209-222.
[14] Xu H, Ji J. Analytical-numerical studies on the stability and bifurcations of periodic motion in the vibro-impact systems with clearances[J]. International Journal of Non-Linear Mechanics, 2019, 109: 155-165.
[15] 吕小红, 朱喜锋, 罗冠炜. 含双侧约束碰撞振动系统的 OGY 混沌控制[J]. 机械科学与技术, 2016, 35(4): 531-534..
LÜ Xiaohong, ZHU Xifeng, Luo Guanwei. Chaos control of vibro-impact system with two-sided constraints based on OGY[J].Mechanical Science and Technology for Aerospace Engineering, 2016, 35(4): 531-534.
[16] 井红岩. 双侧碰撞振动系统的n-p-q周期运动[D].哈尔滨工程大学,2016.
[17] Xu J , Chen P , Li Q . Theoretical analysis of co-dimension-two grazing bifurcations in n-degree-of-freedom impact oscillator with symmetrical constrains[J]. Nonlinear Dynamics, 2015, 82(4):1641-1657.