In this paper, the flow switching theory of discontinuous dynamical systems is used as the main tool to study the dynamical behaviors of a two-degree-of-freedom friction oscillator with elastic impacts, where considering that the inequality of static and kinetic friction forces results in the existence of flow barriers at the relative velocity between the mass and conveyor belt tending to zero. The negative feedback is also taken into account, \(\mathrm {i.e.}\), when the direction of the relative velocity between the mass and conveyor belt changes, the constant force in the periodic excitation force will also change to adapt to the variation in speed. Due to the discontinuity resulted from the friction and non-smoothness resulted from impact, the motion of the two masses can be divided into the following kinds: non-sliding motion (or free motion), sliding motion, non-sliding–stick motion and sliding–stick motion, and the phase space in system is divided into different domains and boundaries, where the vector field is continuous in each region but discontinuous on the velocity boundary. The corresponding normal vectors and G-functions on the separation boundaries are introduced to acquire the switching conditions for all possible motions such as passable motion, sliding motion, grazing motion and stick motion. The mapping theory provides an effective method for the discussion of periodic motion in discontinuous dynamical systems. The four-dimensional switching sets are defined by the means of the form of direct product of two-dimensional switching sets, and the mapping structures are introduced based on the switching sets to describe periodic motions. Eventually, in order to explain more intuitively the above motions in such a friction–impact system, the velocity, displacement, G-functions time histories and the trajectories in phase plane for the masses are illustrated by simulation numerically.

本文以不连续动力系统的流动切换理论为主要工具，研究了具有弹性冲击的两自由度摩擦振荡器的动力学行为，其中考虑了静，动摩擦力的不等式。在质量和传送带之间的相对速度趋于零时，存在流动障碍。负面反馈也被考虑为\（\ mathrm {ie} \），当质量和传送带之间的相对速度方向改变时，周期性激励力中的恒定力也将改变以适应速度的变化。由于摩擦产生的不连续性和冲击产生的非光滑性，两个质量块的运动可分为以下几种：非滑动运动（或自由运动），滑动运动，非滑动粘滞运动和滑动运动，系统中的相空间分为不同的域和边界，其中矢量场在每个区域中是连续的，但在速度边界上是不连续的。在分离边界上引入相应的法向矢量和G函数，以获取所有可能运动（如通过运动，滑动运动，放牧运动和坚持运动。映射理论为讨论不连续动力系统中的周期性运动提供了一种有效的方法。借助于二维开关集的直接积的形式来定义四维开关集，并且基于开关集引入映射结构以描述周期性运动。最终，为了更直观地解释在这样的摩擦-冲击系统中的上述运动，通过数值模拟来说明质量的速度，位移，G函数时间历史和相平面中的轨迹。借助于二维开关集的直接积的形式来定义四维开关集，并且基于开关集引入映射结构以描述周期性运动。最终，为了更直观地解释在这样的摩擦-冲击系统中的上述运动，通过数值模拟来说明质量的速度，位移，G函数的时间历史和相平面中的轨迹。借助于二维开关集的直接积的形式来定义四维开关集，并且基于开关集引入映射结构以描述周期性运动。最终，为了更直观地解释在这样的摩擦-冲击系统中的上述运动，通过数值模拟来说明质量的速度，位移，G函数的时间历史和相平面中的轨迹。