考虑单级直齿轮副，构建Poincaré映射，应用初值胞映射法、延拓打靶法以及数值仿真求解并追踪系统的共存吸引子及其演化，计算Jacobi矩阵的特征值，根据Floquet理论确定周期吸引子的稳定性与分岔类型，应用胞映射法计算吸引域，研究共存吸引子的分岔特征，揭示系统在极小参数区间存在的容易隐藏的吸引子信息以及鞍结型擦边分岔和混沌激变等不连续分岔行为。在一定参数条件下，单级直齿轮副存在大量的多吸引子共存现象以及周期倍化和鞍结分岔。由于非光滑因素的影响，擦边诱导的鞍结分岔引起系统终态的跳跃和迟滞。研究结果可为直齿轮副动力学行为的评价以及系统参数设计与优化提供指导。
Abstract
A single-stage spur gear pair is considered, and the Poincaré mapping is constructed. The coexisting attractors of the system and their evolutions are calculated and traced by applying initial value cell mapping method, continuation shooting method and numerical simulation. The eigenvalues of Jacobi matrix are calculated, and the stability and the type of bifurcation are determined by means of the Floquet theory. The basins of attraction are calculated by using cell mapping method. The bifurcation characteristics of coexisting attractors are studied, and the attractors existing in the minimum interval of system parameters, which are not easy to be found, are revealed as well as discontinuous bifurcation behaviors including saddle-node type of grazing bifurcation and chaos crisis. There are a large number of coexistence phenomena of multiple attractors and period-doubling and saddle-node bifurcations in the single-stage spur gear pair under certain parameter conditions. Due to the influence of non-smooth factor, the grazing-induced saddle-node bifurcation leads to the jump and hysteresis of final-state response of the system. The results can provide guidance for the evaluation of dynamic behavior of spur gear pair and the design and optimization of system parameters.
关键词
齿轮 /
共存吸引子 /
稳定性 /
分岔 /
激变
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Key words
Gear /
coexisting attractors /
stability /
bifurcation /
crisis
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