一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉

通过理论分析和数值仿真,研究了一类二维冲击振动系统在一种强共振条件下的Hopf分叉与亚谐分叉。分析并证实了该类系统在此共振条件下可由稳定的周期1 1振动分叉为周期4 4振动或概周期振动,讨论了亚谐振动和概周期振动向混沌运动的演化过程。     

冲击渐进振动系统相邻基本振动的转迁规律

冲击振动现象广泛存在于动力机械系统中,使得系统表现出复杂的动力学响应.目前对冲击振动系统的p/1类基本振动的稳定性及分岔研究报道较少,而且已有的对冲击振动系统动力学的研究基本都是基于单参数分岔进行分析的.研究以小型振动冲击式打桩机为工程背景,建立了冲击渐进振动系统的力学模型.分析了激振器和缓冲垫发生碰撞的类型,以及滑块渐进运动的条件.给出了系统可能呈现的四种运动状态的判断条件和运动微分方程.通过二维参数分岔分析得到系统在(ω,l)参数平面内存在的各类周期振动的参数域和分布规律.详细分析了相邻p/1类基本振动的转迁规律.在5/1基本振动的参数域的右边区域,相邻p/1基本振动的参数域临界线上存在一个奇异点X_p,相邻p/1类基本振动的分岔特点以奇异点X_p为临界点.在l小于l_X_p的区域内,相邻p/1基本振动经实擦边分岔和鞍结分岔相互转迁,实擦边分岔线和鞍结分岔线之间存在迟滞域,迟滞域内,系统存在两个周期吸引子共存的现象.在l大于l_X_p的区域内,相邻p/l类基本振动的参数域之间存在一个中间过渡区域.中间过渡区域内,系统呈现(2p+2)/2和(2p+1)/2周期振动等.在5/1基本振动的参数域的左边区域,p/1基本振动经多重滑移分岔产生(P+1)/1基本振动.     

冲击消振器的概周期碰振运动分析

建立了冲击消振器对称周期运动的Poincar啨映射方程 ,讨论了对称周期运动的稳定性与局部分岔。通过数值仿真研究了冲击消振器在非共振、弱共振和强共振条件下的概周期碰振运动及其向混沌的转迁过程。     

碰撞振动系统强共振下的两参数动力学分析

建立了两自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在一对共轭复特征值在单位圆上并满足强共振(λ40=1)条件时,通过中心流型-范式方法将四维映射转变为二维范式映射。理论分析了系统两参数开折的局部动力学行为,扩展了单参数分岔理论,给出了n-1周期运动产生Hopf分岔和次谐分岔的条件。数值仿真验证了所得出的理论,证明系统在共振点附近存在稳定的Hopf分岔不变环面和次谐分岔4-4周期运动。     

具有单侧刚性约束的两自由度振动系统在强共振条件下的拟周期运动与混沌

采用理论分析和数值仿真相结合的方法,研究了一类两自由度碰撞振动系统在一种强共振条件下的Hopf分叉问题.分析并证实了碰撞振动系统在此共振条件下可由稳定的周期1-1振动分叉为不稳定的周期3-3振动,讨论了亚谐振动向混沌运动的演化过程.     

具有单侧刚性约束的两自由度振动系统在强共振条件下的拟周 …

采用理论分析和数值仿真相结合的方法,研究了一类两自由度碰撞振动系统在一种强共振条件下的Ｈｏｐｆ分叉问题,分析并证实了碰撞振动系统在此共振条件下可由稳定的周期１－１振动分叉为不稳定的周期３－３振动,讨论了亚谐振动向混沌运动的演化过程。     

一类分段光滑隔振系统的非线性动力学设计方法

分段光滑隔振系统是一类具备分段刚度或阻尼的非线性动力学系统,在振动控制领域中具有广泛代表性,诸如限位隔振系统、分级汽车悬挂等. 分段光滑的刚度或阻尼特性能够实现隔振系统的特定动力学性能及提升隔振性能,如抑制共振响应、提升共振区隔振性能等,但是亦会给隔振系统的动力学行为带来诸多不利影响. 以分段双线性分段光滑隔振系统为理论模型,系统研究了摒除不利于隔振的非线性动力学现象设计方法,包括幅值跳跃、周期运动的倍周期分岔等. 首先,利用平均法与奇异性理论给出了主共振频响曲线拓扑特征的完整拼图. 研究结果表明,参数空间分为4 个区域,其中2 个区域存在幅值跳跃,而其产生跳跃原因分别由鞍结分岔与擦边分岔所导致;基于此提出避免主共振跳跃的设计方法. 其次,建立了隔振有效区内周期运动的庞加莱映射,通过特征值分析给出了避免倍周期分岔发生的条件,证实增大阻尼可以抑制倍周期分岔的发生. 最后通过数值仿真分析了噪声对多稳态运动的影响. 研究结果发现在噪声影响下,分段光滑隔振系统的响应会在不同稳态间跃迁,非常不利于隔振. 因此,在完成跳跃与倍周期分岔的防治设计后,应采用数值仿真校验系统是否存在多稳态运动.      

非线性转子——密封系统低频失稳机理研究：不平衡系统的亚谐共振

本文采用Ｍｕｓｚｙｎｓｋａ密封力模型分析单圆盘转子－－密封系统的低频自激振动。文（１）研究了平衡转子的稳定性和分岔，本文研究不平衡转子在临界平衡点附近自激振动（周期扰动Ｈｏｐｆ分岔）的亚谐共振，给出了不同参数条件下的振动性态，为识别转子的亚谐共振故障及预防提供了一些新理论依据。     

黏弹性传动带1:3内共振时的周期和混沌运动

研究了参数激励作用下黏弹性传动带在1:3内共振时的周期解分岔和混沌动力学. 同时考虑传动带的线性外阻尼因素和材料内阻尼因素. 首先建立了具有线性外阻尼情况下的黏弹性传动带平面运动时的非线性动力学方程, 黏弹性材料的本构关系用Kelvin模型描述. 然后考虑黏弹性传动带的横向振动问题, 利用多尺度法和Galerkin离散法得到黏弹性传动带系统在1:3内共振时的平均方程. 最后利用数值模拟方法研究了黏弹性传动带系统的周期振动和混沌动力学, 得到了系统在不同参数下的混沌运动. 数值模拟结果说明黏弹性传动带系统存在周期分岔, 概周期运动及混沌运动.     

碰撞振动系统分岔与混沌的研究进展

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.      

线性碰振系统周期解擦边分岔的一类映射分析方法

擦边分岔是碰振机械系统的一种重要分岔行为. 以固定相位面作为Poincaré截面， 建立了线性碰振系统单碰周期$n$运动的Poincaré映射. 通过分析该映射，得到了系统 发生擦边分岔的条件和分岔方程，并以单自由度碰振系统为实例验证了分析结果的正确性. 该方法不仅可以计算线性碰振系统擦边分岔的参数值，还可以计算系统的任意周 期$n$解的分岔参数值.     

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T _in, T _on and T _out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions. The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.     

Codimension two bifurcation of a vibro-bounce system

A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.The project supported by the National Natural Science Foundation of China (10172042, 50475109) and the Natural Science Foundation of Gansu Province Government of China (ZS-031-A25-007-Z (key item))     

两自由度塑性碰撞振动系统的动力学研究

用三维映射表示具有单侧刚性约束的两自由度振动系统在塑性碰撞时的动力学方程。借助理论分析与数值方法研究了系统周期n-1振动的存在性与稳定性,描述了系统周期n-1振动的特点,讨论了碰撞振子与约束擦边引起的Poincare映射奇异性对系统全局分岔的影响。     

Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider

A two-degree-of-freedom plastic impact oscillator with a frictional slider is considered. Dynamics of the plastic impact oscillator are analyzed by a three-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the impact Poincaré map. The piecewise property of the map is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of the map is generated via the grazing contact of two masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularity and parameter variation on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams of before-impact velocity as a function of the excitation frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-N single-impact motions of the plastic impact oscillator are found to exhibit extensive and systematic characteristics. Dynamics of the impact oscillator, in the elastic impact case, is also analyzed. This type of impact is modelled by using the conditions of conservation of momentum and an instantaneous coefficient of restitution rule. The differences in periodic-impact motions and bifurcations are found by making a comparison between dynamic behaviors of the plastic and elastic impact oscillators with a frictional slider. The best progression of the plastic impact oscillator is found to occur in period-1 single-impact sticking motion with large impact velocity. The largest progression of the elastic impact oscillator occurs in period-1 multi-impact motion. The simulative results show that the plastic impact feature for the impact-progressive oscillator is of a considerable importance in minimizing adverse effects such as high noise levels, wear and tear caused by impacts.     

Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1：2 resonance case

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed. The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang.     

Dynamics of a plastic-impact system with oscillatory and progressive motions

A mathematical model is developed to describe oscillatory and progressive motions in dynamics of a plastic impact oscillator with a frictional slider. Dynamics of the impact oscillator is analyzed by a five-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of impacting masses immediately after the impact, and the singularity of the map is generated via the grazing contact of impacting masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and various parameters on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams for before-impact velocity versus forcing frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-n single-impact motions of the plastic-impact oscillator are found to exhibit extensive and systematic characteristics.     

索-梁耦合结构Hopf分岔的反控制

本文研究了索-梁耦合结构的Hopf分岔的反控制,动态窗口滤波反馈控制器在反控制领域有着很广泛的应用。本文通过使用这种控制器,可以使得受控系统在指定的平衡点处产生Hopf分岔。最后,根据庞加莱截面和级数展开法,证明了上述方法的有效性及可行性。