ADVANCES OF RESEARCH ON BIFURCATIONS AND CHAOS IN VIBRO-IMPACT SYSTEM
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摘要: 针对工程实际中普遍存在的碰撞振动系统这种典型的非光滑动力系统, 其研究具有重要的理论意义和工程实用价值. 碰撞振动系统动力学的分析与研究方法主要有理论分析、数值模拟以及应用与实验研究. 为了研究碰撞振动系统的周期运动稳定性、分岔及混沌, 采用的手段有建立Poincar\'{e}映射、中心流形和范式方法, 映射的分岔与混沌理论是碰撞振动系统研究的理论基础. 首先简述了碰撞振动系统的分析与研究方法, 光滑非线性系统动力学的分析方法部分可以推广到碰撞振动系统, 碰撞振动的不连续性导致一些方法的适用性和有效性问题. 进一步综述了碰撞振动系统周期运动稳定性、分岔、混沌及奇异性的理论研究和工程应用现状. 最后着重结合相关离散型映射系统的动力学发展, 对碰撞振动系统的分岔与混沌研究及存在的主要问题进行了讨论, 并展望了其发展趋势.Abstract: The vibro-impact system found in many practicalsystems, as a typical non-smooth dynamical system is importantin engineering. Its studies include theoretical analysis, numericalsimulations, applied and experimental studies. By establishing thePoincar\'{e} map, using the center manifold theorem and the theory of normalforms, the periodic motion stability, bifurcation and chaos of the vibro-impactsystem can be investigated. The theory of bifurcation and chaos of map isone of the bases in studying vibro-impact systems. The smooth nonlinear dynamics canbe partly extended to vibro-impact systems, but the discontinuity caused byimpact will affect applicability and effectiveness of some methods.The theoretical studies and engineering applications of stability,bifurcations, chaos and singularity of vibro-impact systems are surveyedin this paper. Finally, the discussion centers around some unsolved problemsof bifurcations and chaos of vibro-impact systems, and the advance in thecorresponding discrete map dynamics. The future research trends aresuggested.
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Key words:
- vibro-impact /
- stability /
- bifurcation /
- chaos
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