DISCONTINUOUS BIFURCATIONS OF COEXISTING ATTRACTORS FOR A GEAR TRANSMISSION SYSTEM
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摘要: 大量的多吸引子共存是引起齿轮传动系统具有丰富动力学行为的一个重要因素. 多吸引子共存时, 运动工况的变化以及不可避免的扰动都可能导致齿轮传动系统在不同运动行为之间跳跃变换, 对整个机器产生不良的影响. 目前, 一些隐藏的吸引子没有被发现, 共存吸引子的分岔演化规律没有被完全揭示. 考虑单自由度直齿圆柱齿轮传动系统, 构建由局部映射复合的Poincaré映射, 给出Jacobi矩阵特征值计算的半解析法. 应用数值仿真、延拓打靶法和Floquet特征乘子求解共存吸引子的稳定性与分岔, 应用胞映射法计算共存吸引子的吸引域, 讨论啮合频率、阻尼比和时变激励幅值对系统动力学的影响, 揭示齿轮传动系统倍周期型擦边分岔、亚临界倍周期分岔诱导的鞍结分岔和边界激变等不连续分岔行为. 倍周期分岔诱导的鞍结分岔引起相邻周期吸引子相互转迁的跳跃与迟滞, 使倍周期分岔呈现亚临界特性. 鞍结分岔是共存周期吸引子出现或消失的主要原因. 边界激变引起混沌吸引子及其吸引域突然消失, 对应周期吸引子的分岔终止.Abstract: An important factor of rich dynamics in the gear transmission system is that there are a large number of various types of co-existing attractors. When multiple attractors coexist, the change of motion conditions and the inevitable disturbance may cause the gear transmission system to jump between different motion behaviors. As a result, the whole machine is adversely affected, and sometimes, the system structure will be destroyed. At present, some hidden attractors have not been found, and the bifurcation evolution characteristics of coexisting attractors have not been fully revealed. A single-degree-of-freedom spur gear system is considered. The Poincaré mapping compounded by local maps is constructed, and semi-analytic calculation method of eigenvalues of Jacobi matrix is presented. The stability and bifurcations of coexisting attractors are studied by applying numerical simulation, continuation shooting method and Floquet multipliers, and the basins of attraction of coexisting attractors are calculated by using cell mapping method. The influence of the meshing frequency, damping ratio and amplitude of time-varying excitation on the system dynamics is analyzed, and the discontinuous bifurcation behaviors including PD-type grazing bifurcation, saddle-node bifurcation induced by subcritical period-doubling bifurcation and boundary crisis are revealed in the gear transmission system. The saddle-node bifurcation induced by period-doubling bifurcation leads to the jump and hysteresis in the transition between adjacent periodic attractors, resulting in that the period-doubling bifurcation presents subcritical feature. The saddle-node bifurcation is a major factor for the appearance and disappearance of coexisting periodic attractors. The boundary crisis leads the chaotic attractor and its basin of attraction to disappear, and the bifurcation of corresponding periodic attractor terminates.
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Key words:
- non-smooth system /
- gear /
- continuation shooting method /
- coexisting attractors /
- bifurcation
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图 2 延拓打靶法计算的分岔图
Figure 2. Bifurcation diagram calculated by continuation shooting method
图 6 随Pa变化的分岔图和相图
Figure 6. Bifurcation diagram of displacement as a function of Pa and trajectories
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