一类碰撞振动系统的余维二分叉和Hopf分叉<sup>*</sup>

一类碰撞振动系统的余维二分叉和Hopf分叉^*

基金项目: * 国家自然科学基金

摘要: 本文研究弹簧质量系统对无穷大平面碰撞振动的分叉问题。证明了在接近完全弹性碰撞和在一些特殊的频率比附近,存在余维二分叉现象。利用映射的正则型理论,将Poincaré映射变换成含两个参数的正则型,通过分析该正则型,我们得到周期倍化分叉、周期1点、2点的Hopf分叉。并进行了数值验证。
- 碰撞振动 /
- 稳定性 /
- 余维二分叉
Abstract: In this paper,a spring-mass system impacting against an infinite large plane is analyzed. We have proved that phenomena of codimension two bifurcations exist when coefficient of restitution is near one and ratios of frequency near the same special values.By method of normal forms, we reduced Poincaré maps into normal forms containing two parameters, After studying these normal forms, we obtained period doubling bifurcations and Hopf bifurcations of period-one and period-two.The results have been verified by numerical works.
- impacting vibration /
- stability /
- bifurcation

[1]	S. W. Shaw and P. L. Holmes, A periodically forced piece-wise linear oscillator, Journal of Sound and Vibration, 90,1 (1983), 129-155.
[2]	P. C. Tung and S. W. Shaw, The dynamics of an impact print hammer, Journal of Vibration, Acoustics, Stress and Reliability in Design, 110, 4(i988), 193-199.
[3]	J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, New York:Springer-Verlag(1983),102-116.
[4]	Shu Zhdngzhou, Motion types and chaos of multi-body systems vibrating with impacts, Acta Mechanica Sinica, 7. 4 (1991). 369-375.
[5]	H. Ishihana, Y. Takagi and S. Maezawa, Analysis of impact vibration to deltafunction method-care of one degree-of-freedom system, first report: prefectly elastic collision, 20th Japan Nat. Congress for Applied Mechanics, Tokyo (1970).
[6]	谢建华.冲击式振动落砂机分叉问题研究,全国第三届运动稳定性与振动学术会议论文集:K运动稳定性、振动、分叉与混沌的研究》,中国科学技术出版社,北京(1992).
[7]	O. E. Lanford, Bifurcation of periodic solution into invariant tori: The work of Ruelle and Takens, Lecture Notes in Math., 322, Berlin-Heidelberg, New York, Springer-Verlag (1973).
[8]	Y. H. Wan, Computation of the stabity condition for the Hopf bifurcation of diffeomorphisms on R², SIAM, J. Appl. Math., 34, 1 (1978).