Basins of attraction of the bistable region of time-delayed cutting dynamics

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.

Figure 1

Physical model of the cutting process, with an SEM (scanning electron microscopy) image of the chips formation [29], which generates cutting forces.

Figure 2

Stability boundaries (solid black) and boundaries for the UZ obtained by DDEBIFTOOL (dot-dashed blue).

Figure 3

Outmost layer of the workpiece is mapped onto the $\theta -s\left(\theta \right)$ plane with a thick solid curve for the surface profile and the coordinate origin for the beginning of the tool-workpiece engagement. Thin dashed lines, thin solid curves, and a dot-dashed line are for nominal, real, and intended tool passes, respectively. Every time $\theta$ reaches $2\pi$, the tool position is mapped back to $\theta =0$.

Figure 4

(a) Tool deflection as it penetrates into the workpiece; the tool equilibrium is moved from $x=0$ ($y_1=-y_{10}$) to $y_1=0$. (b) Initial surface profile.

Figure 5

Basins of attraction and basin stability for (a, b) $\mathrm{\Omega }=0.230$ and ${\eta }_1/{\eta }_{\text{min}}=2.620$, (c, d) $\mathrm{\Omega }=0.240$ and ${\eta }_1/{\eta }_{\text{min}}=4.620$, and (e, f) $\mathrm{\Omega }=0.250$ and ${\eta }_1/{\eta }_{\text{min}}=6.920$, where the the peaks of $S_N$ always coincides with the nature frequency $\omega$.

Figure 6

(a) Overall basin stability $S_B$ as a function of $\eta /{\eta }_{\text{min}}$ for $\mathrm{\Omega }=0.24$, with the regions for a fast growth of $S_B$ been blown up in panel (b). (c–e) Basins of attraction and basin stability distributions for ${\eta }_1/{\eta }_{\text{min}}=4.60,4.63$, and 4.64.

Figure 7

(a) Relative thickness of the “boundary layer” (b) and its location in the $\mathrm{\Omega }-{\eta }_1/{\eta }_{\text{min}}$ plane.

Figure 8

Different points of penetration yield different tool passes and coordinates of $\theta -s$.

Figure 9

(a) Influence of geometry of tool engagement on the basin stability for $\mathrm{\Omega }=0.240$ and ${\eta }_1/{\eta }_{\text{min}}=4.630$, with basin stability distributions and basins of attraction for (b) $\pi /2$, (c) $\pi$, and (d) $3\pi /2$ added for demonstration.

Figure 10

(a) Basin stability $S_B$ is plotted as functions of ${\eta }_1/{\eta }_{\text{min}}$ for $N\varphi =0,\pi /2,\pi$, and $3\pi /2$, respectively. (b) Contour plot of $S_B$ as a function of ${\eta }_1/{\eta }_{\text{min}}$ and $N\varphi$.

Figure 11

Different geometry of tool engagement induce different cutting depth at the beginning of tool-workpiece interactions, where (a) favorable and (b) unfavorable geometry correspond to $N\varphi =\pi /2$ and $N\varphi =3\pi /2$, respectively.

Figure 12

(a) GIM, IF, and SL of the cutting vary as functions of the cutting geometry, with the basins of attraction for $N\varphi =0,\pi /4,\pi /2,3\pi /4,\pi ,3\pi /2$, and $5\pi /4$ plotted in panels (b–g). The rest of the parameters are selected as $\mathrm{\Omega }=0.24,{\eta }_1/{\eta }_{\text{min}}=4.60,N=5$, and $a_N=1.0$.

Figure 13

(a) Schematic illustration of hammer striking of the cutting tool. With different points of striking $\psi$ and hitting strength $\mathrm{\Delta }{y}_2$, the cutting chatter for $\mathrm{\Omega }=0.24$ and ${\eta }_1/{\eta }_{\text{min}}=4.55$ might (a, b) decay to zero or (c, d) return to the unstable cutting, respectively.

Figure 14

(a) Evolution of $S_B$ with respect to the increase of ${\eta }_1/{\eta }_{\text{min}}$ with typical basins of attraction for ${\eta }_1/{\eta }_{\text{min}}=4.30,4.40,4.49,4.50,4.51,4.53,4.54,4.60$, and 4.65 plotted in panels (b–j). The gray background for ${\eta }_1/{\eta }_{\text{min}}\in (4.49,4.54)$ [corresponding with panels (e–g)] is added to marked the fastest evolution of $S_B$, which connect a moderate increase of $S_B$ on its left side and a slow grow of $S_B$ on its right side.