Statistical basin of attraction in time-delayed cutting dynamics: Modelling and computation

Abstract

This paper proposes a novel concept of the statistical basin of attraction to analyse the multiple stability in nonlinear time-delayed dynamical systems and shows how they can be computed. This concept has been applied to the cutting dynamics, which has been extensively investigated by the authors. Due to the nonlinearity and non-smoothness of tool-workpiece interactions, the cutting dynamics always exhibit large-amplitude chatter entering a linearly stable zone, making the area below stability boundaries unsafe for high material removal rates. Meanwhile, a thorough investigation of the multiple stability in the cutting dynamics is hampered by infinite-many dimensions introduced by time delays, which induce difficulties in computation and visualization of the conventional basin of attraction. To address this issue, infinite-many dimensional time-delayed states are approximated by a Fourier series aligned on a straight line, and the coefficients of the basis functions and the cutting process are used to construct the statistical basin of attraction. Inside the statistical basin of attraction, a safe basin with no probability of chatter occurrence exists. These findings are instrumental in designing a new state-dependent intermittent control to guide the cutting dynamics towards the safe basins. It is also seen that the state-dependent intermittent control is efficient in improving the cutting safety and shrinking the unsafe zones, even when the targeted basin for the control is larger than the real safe basin.