考虑到碰撞振动系统的Poincaré映射的隐式特点,在不改变原碰撞系统平衡解结构的前提下,采用线性反馈控制方法研究了一类三自由度含间隙双面碰撞振动系统Poincaré映射的叉式分岔的反控制问题。首先,建立闭环控制系统的六维Poincaré映射,针对由特征值特性描述的传统叉式分岔临界准则在六维的高维映射中只能通过数值试算来确定控制增益的困难,利用不直接依赖于特征值计算的显式临界准则获得了系统出现叉式分岔的控制参数区域。然后应用中心流形-范式方法进一步分析叉式分岔解的稳定性。最终数值仿真验证了在任意指定的系统参数点通过控制能实现稳定的叉式分岔解。
Abstract
In the premise of no change of periodic solutions of the original system and with consideration of the difficulties that given by the implicit Poincaré map of the vibro-impact system, anti-control of Pitchfork bifurcation on Poincaré map of a three-degree-of-freedom vibro-impact system is studied by using linear feedback control method. Firstly, the six-dimensional Poincaré map of close-loop system is established, to overcome the difficulty that the numerical computing method can be only used to determine control gains on basis of the classical critical criteria of Pitchfork bifurcation described by the properties of eigenvalues in six-dimensional map, an explicit Pitchfork critical criterion without using eigenvalues is used to obtain the controlling parameters area of two parameters. Then, the stability of the Pitchfork bifurcation is further analyzed by utilizing the center manifold and normal formal theory. Finally, the numerical experiments verify that the stable Pitchfork bifurcation solutions can be generated at an arbitrary specified parameters point by controlling.
关键词
叉式分岔 /
分岔反控制 /
稳定性 /
碰撞振动系统
{{custom_keyword}} /
Key words
Pitchfork bifurcation /
anti-controlling bifurcation /
stability /
vibro-impact system
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Shaw S W,Holmes P J. A periodically forced impact oscillator with large dissipation[J]. Journal of Applied Mechanics, 1983, 50(4a): 849-857.
[2] Holmes P J. The dynamics of repeated impacts with a sinusoidally vibrating table[J]. Journal of Sound and Vibration, 1982, 84(2): 173-189.
[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] Bureau E, Schilder F, Elmegård M, et al. Experimental bifurcation analysis of an impact oscillator-Determining stability[J]. Journal of Sound and Vibration, 2014, 333(21): 5464-5474.
[5] 金栋平,胡海岩,吴志强. 基于Hertz 接触模型的柔性梁碰撞振动分析[J]. 振动工程学报,1998,11(1):46-51.
JIN Dong-ping, HU Hai-yan, WU Zhi-qiang. Analysis of vibro-impacting flexible beams based on Hertzian contact model [J]. Journal of vibration engineering, 1998, 11(1): 46-51.
[6] Luo G W, Lv X H, Shi Y Q. Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: Diversity and parameter matching of periodic-impact motions[J]. International Journal of Non-Linear Mechanics, 2014, 65: 173-195.
[7] Ding W C, Li G F, Luo G W, et al. Torus T2 and its locking, doubling, chaos of a vibro-impact system[J]. Journal of the Franklin Institute, 2012, 349(1): 337-348.
[8] 盛冬平,朱如鹏,陆凤霞,等. 多间隙弯扭耦合齿轮非线性振动的分岔特性研究[J]. 振动与冲击, 2014,33(19): 116-122.
SHENG Dong-ping, ZHU Ru-peng, LU Feng-xia, et al. Bifurcation characteristics of bending-torsional coupled gear nonlinear vibration with multi-clearance [J]. Journal of vibration and shock, 2014, 33(19): 116-122.
[9] 金俐,陆启韶. 非光滑动力系统 Lyapunov 指数谱的计算方法[J]. 力学学报,2005,37(1):40-47.
JIN Li, LU Qi-shao. A method for calculating the spectrum of Lyapunov exponents of non-smooth dynamical systems[J]. Acta mechanica sinica, 2005, 37(1):40-47.
[10] Yue Y, Xie J H. Neimark–Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincaré map in a three-degree-of-freedom vibro-impact system[J]. International Journal of Non-Linear Mechanics, 2013, 48: 51-58.
[11] Chen G R, Lu J L, Nicholas B, et al. Bifurcation dynamics in discrete-time delay-feedback control systems[J]. International Journal of Bifurcation and Chaos, 1999, 9(1): 287-293.
[12] Alonso D M, Paolini E E, Moiola J L. An experimental application of the anticontrol of Hopf bifurcations[J]. International Journal of Bifurcation and Chaos, 2001, 11(7): 1977-1987.
[13] 刘素华,唐驾时. 四维Qi系统零平衡点的Hopf分岔反控制[J]. 物理学报,2008,57(10): 6162-6168.
LIU Su-hua, TANG Jia-shi. Anti-control of Hopf bifurcation at zero equilibrium of 4D Qi system [J]. Acta physica sinica, 2008, 57(10): 6162-6168.
[14] Wang X D, Deng L W, Zhang W L. Hopf bifurcation analysis and amplitude control of the modified Lorenz system[J]. Applied Mathematics and Computation, 2013, 225: 333-344.
[15] Wei Z C, Yang Q G. Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci[J]. Applied Mathematics and Computation, 2010, 217(1): 422-429.
[16] 罗冠炜,谢建华. 碰撞振动系统的周期运动和分岔[M]. 北京: 科学出版社,2004.
[17] Xu H D, Wen G L. Alternative criterion for investigation of pitchfork bifurcations of limit cycle in relay feedback systems[J]. Journal of Computational and Nonlinear Dynamics, 2014, 9(3): 031004-1 -031004-7.
[18] Kuznetsov Y A. Elements of applied bifurcation theory [M]. 2nd ed. New York: Springer-Verlag, 1998.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}