空气阻力对完全非弹性蹦球动力学行为的影响

Effect of air damping on dynemical behaviors of a completely inelastic bouncing ball

一个在振动台面上蹦跳的小球具有复杂的运动形式,如倍周期分岔和混沌.如果球与台面间的碰撞是完全非弹性的,则球的运动是倍周期的,不存在混沌.在分岔相图中,鞍-结不稳定性引入"平台"结构,同时存在倍周期轨道的密集区.这里将研究空气的黏滞阻力对完全非弹性蹦球动力学行为的影响.分析表明,空气阻力很弱时,分岔序列不受影响,但分岔点的数值变大,"平台"和密集区加宽.空气阻力较大时,"平台"与密集区重叠.重叠区内原有产生倍周期运动的机理被破坏,球的运动是混沌的.

A ball dropped on a vertically vibrating table exhibits intricate dynamical behaviors including period-doubling bifurcations and chaos. If the collision between the bail and the table is completely inelastic, the motion of the bail is always periodic, and the plateaus caused by saddle-node instability and clumping structures for periodic trajectories occur in the bifurcation diagram. Here the effect of air damping on the dynamics of the bail with zero elasticity is analyzed. The air damping is treated as linear viscous one. It is shown that a weak air damping does not change the sequence of bifurcations, but makes the bifurcation points shift to larger values and broadens the transverse dimensions of the plateaus and the clumping zones in the diagrams. However, when the air damping becomes larger, overlapping between the plateaus and clumping zones takes place. In the overlapping section, the mechanism originally leading to periodic motion is destroyed, and chaos is introduced.