School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu 611731, China.
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China.
Phys Rev E. 2017 Sep;96(3-1):032205. doi: 10.1103/PhysRevE.96.032205. Epub 2017 Sep 6.
This paper investigates the effects of bistability in a nonsmooth time-delayed dynamical system, which is often manifested in science and engineering. Previous studies on cutting dynamics have demonstrated persistent coexistence of chatter and chatter-free responses in a bistable region located in the linearly stable zone. As there is no widely accepted definition of basins of attraction for time-delayed systems, bistable regions are coined as unsafe zones (UZs). Hence, we have attempted to define the basins of attraction and stability basins for a typical delayed system to get insight into the bistability in systems with time delays. Special attention was paid to the influences of delayed initial conditions, starting points, and states at time zero on the long-term dynamics of time-delayed systems. By using this concept, it has been confirmed that the chatter is prone to occur when the waviness frequency in the workpiece surface coincides with the effective natural frequency of the cutting process. Further investigations unveil a thin "boundary layer" inside the UZ in the immediate vicinity of the stability boundary, in which we observe an extremely fast growth of the chatter basin stability. The results reveal that the system is more stable when the initial cutting depth is smaller. The physics of the tool deflection at the instant of the tool-workpiece engagement is used to evaluate the cutting safety, and the safe level could be zero when the geometry of tool engagement is unfavorable. Finally, the basins of attraction are used to quench the chatter by a single strike, where the resultant "islands" offer an opportunity to suppress the chatter even when the cutting is very close to the stability boundary.
本文研究了非光滑时滞动力系统中的双稳性效应，这种双稳性在科学和工程中经常出现。先前关于切削动力学的研究表明，在位于线性稳定区的双稳区内，存在颤振和无颤振响应的持续共存。由于对于时滞系统，没有被广泛接受的吸引域定义，因此将双稳区称为不安全区（UZs）。因此，我们试图定义典型时滞系统的吸引域和稳定域，以深入了解时滞系统中的双稳性。特别关注了时滞初始条件、起始点和零时状态对时滞系统长期动力学的影响。通过使用这个概念，已经证实了当工件表面的波纹频率与切削过程的有效固有频率相吻合时，颤振更容易发生。进一步的研究揭示了在稳定边界附近的 UZ 内存在一个很薄的“边界层”，在这个边界层内，我们观察到颤振域稳定性的极快增长。结果表明，初始切削深度较小时，系统更稳定。在刀具与工件接触的瞬间，刀具挠度的物理特性被用来评估切削安全性，当刀具啮合的几何形状不利时，安全水平可能为零。最后，通过单次打击来使用吸引域来消除颤振，其中产生的“岛屿”即使在切削非常接近稳定边界时也提供了抑制颤振的机会。
