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伏江龙, 吕小红, 刘一凡, 罗冠炜. 间隙约束悬臂梁系统的不连续分岔. 力学学报, 2024, 56(6): 1-13. DOI: 10.6052/0459-1879-23-581
引用本文: 伏江龙, 吕小红, 刘一凡, 罗冠炜. 间隙约束悬臂梁系统的不连续分岔. 力学学报, 2024, 56(6): 1-13. DOI: 10.6052/0459-1879-23-581
Fu Jianglong, Lyu Xiaohong, Liu Yifan, Luo Guanwei. Discontinuous bifurcations of a cantilever beam system with clearance restrictions. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(6): 1-13. DOI: 10.6052/0459-1879-23-581
Citation: Fu Jianglong, Lyu Xiaohong, Liu Yifan, Luo Guanwei. Discontinuous bifurcations of a cantilever beam system with clearance restrictions. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(6): 1-13. DOI: 10.6052/0459-1879-23-581

间隙约束悬臂梁系统的不连续分岔

DISCONTINUOUS BIFURCATIONS OF A CANTILEVER BEAM SYSTEM WITH CLEARANCE RESTRICTIONS

  • 摘要: 间隙约束悬臂梁广泛应用在机械和工程设计中, 其动态力学特性的优劣直接反映主机运行品质的高低, 是决定主机能否安全和高效运行的关键因素. 从多参数角度研究悬臂梁碰撞系统的动力学可实现系统的动态特性与功能目标的优化设计. 考虑间隙约束悬臂梁系统, 构建由光滑流映射复合的全局Poincaré映射及其Jacobi矩阵. 应用数值方法获得各类吸引子在两参数平面的存在区域, 结合数值仿真、延拓打靶法和胞映射研究相邻对称型周期1吸引子经混合运动域的转迁特征及共存吸引子的形成机理, 揭示不稳定吸引子在系统动力学演化过程中的重要作用, 以及不同类型的激变、迟滞、鞍结型周期倍化分岔和亚临界叉式分岔等不连续分岔行为. 结果表明, 在含对称间隙弹性碰撞系统中, 周期吸引子的不连续分岔呈现9种不同的类型. 叉式分岔使多吸引子共存更加普遍. 两个反对称的不稳定周期轨道可引起两个共存的混沌吸引子同时发生内部激变. 叉式型擦边分岔和9种不连续分岔的定义将进一步丰富非光滑系统动力学.

     

    Abstract: A cantilever beam with clearance restrictions is widely used in mechanical and engineering design. the quality of its dynamic properties directly reflects the operation quality of the overall system, and it is a key factor in determining whether the overall system can operate safely and efficiently. Study on the multi-parameter dynamics of cantilever beam collision system can achieve the optimal design of the dynamic characteristics and function target. Considering the cantilever beam system with clearance restrictions. The global Poincaré map composed of smooth flow maps and the Jacobian matrix are constructed. Based on Floquet theory and grazing conditions, the existence regions of various types of periodic and chaotic attractors in the two-parameter plane are obtained by applying numerical method. The transition characteristics of adjacent symmetric period-1 attractors through the beat motion zones and the formation mechanism of coexisting attractors are revealed by combining the numerical simulation, continuation shooting approach and cell mapping method. The important roles of unstable periodic attractors in the dynamic evolutions of the system are revealed, as well as different types of discontinuous bifurcations, such as crises, hysteresis, saddle-node-type period-doubling bifurcations and subcritical pitchfork bifurcations. The results indicate that there are nine types of discontinuous bifurcations in the evolutions of periodic attractors in the elastic impact system with symmetric clearances. The coexistences of multiple attractors are more common due to the occurrence of pitchfork bifurcations. The simultaneous interior crises of two coexisting chaotic attractors can be induced by a pair of asymmetric unstable periodic orbits. The definitions of pitchfork-type grazing bifurcation and nine types of discontinuous bifurcations will further enrich the dynamics of non-smooth systems.

     

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