强共振情况下冲击成型机的亚谐与Hopf分岔

通过理论分析与数值仿真研究了双质体冲击振动成型机的周期运动在强共振条件下的亚谐分岔与Hopf分岔，证实了此系统的1／1周期运动在强共振(λ0^4=1)条件下可以分岔为稳定的4／4周期运动及概周期运动．讨论了冲击映射的奇异性，分析了冲击振动系统的“擦边”运动对强共振条件下周期运动及全局分岔的影响。

SUBHARMONIC AND HOPF BIFURCATIONS OF AN IMPACT-FORMING MACHINERY IN A STRONG RESONANCE CASE

An impact-forming machinery with double masses is considered. Dynamics of the system are studied with special attention to subharmonic and Hopf bifurcations of period 1 single-impact motion in a strong resonance case. The Poincare map of period 1 single-impact motion of the vibro-impact system is established. Bifurcation values and intersecting conditions of the period motion with one impact, in the strong resonance case, are determined. A center manifold theorem technique is applied to reduce the Poincare map to a two-dimensional one, which is put into normal form by theory of normal forms. By theory of subharmonic and Hopf bifurcations of fixed points in .R2-strong resonance, local dynamical behavior of the vibro-impact system, near by the points of resonance, may be analyzed. The theoretical analyses are verified by the results from simulation. The singularity of the Poincare map of the vibro-impact system, caused by the motion with grazing contact, is analyzed by numerical simulation. The influence of the motions with grazing contact on global bifurcations of period 1 single-impact motion, in the strong resonance case, is elucidated.