Stabilization of turning processes using spindle feedback with state-dependent delay

Figure 1.  Turning model

Figure 2.  The graphs of $y = f(\beta) = \frac{\sin\beta}{\cos\beta(1-\cos\beta)}$ and $y = g(\beta) = -\frac{2q}{\beta}$

Figure 3.  Graphs of $y = \delta(\beta) = \frac{\xi\beta^2(1-\cos\beta)+\beta^3\sin\beta}{2q(1-\cos\beta)+\beta\sin\beta}$ and $y = h_2(\beta) = -\frac{q\beta(1-\cos\beta)-\xi\beta-q\xi\sin\beta}{\xi\beta(1-\cos\beta)+\beta^2\sin\beta}$ with $\xi<2q$

Figure 4.  The graphs of $y = F(\beta) = \beta\cot\frac{\beta}{2}$, and those of $y = G(\beta)$, $y = H(\beta)$ which are the upper and lower part of the curve $\frac{(y+q)^2}{q^2}+\frac{\beta^2}{\left(\frac{q^2}{1-\frac{2q}{\xi}}\right)} = 1$, $\beta>0$, respectively

Figure 5.  The curves of $(\delta, \, h_1)$, $\delta>0$ where $\beta_1\frac{\sqrt{2\sqrt{5}-2}}{4\sqrt{5}-8} = 8.955929<q$

Figure 6.  The shaded region shows the details of the stability region of Figure 5 near the origin $(0, \, 0)$

Figure 7.  The curves of $(\delta, \, h_1)$, $\delta>0$ where $\beta_1\frac{\sqrt{2\sqrt{5}-2}}{4\sqrt{5}-8} = 8.955929>q$. The shaded region near the origin $(0, \, 0)$ is the stability region

Figure 8.  The curves of $(\delta, \, h_2)$, $\delta>0$, $h_2>0$ where $\xi<2q$. The shaded region near the origin $(0, \, 0)$ is the stability region

Figure 9.  The curves of $(\delta, \, h_2)$, $\delta>0, \, h_2>0$ with $\xi>2q$. The connected region near the origin $(0, \, 0)$ without crossing the lobes, or the vertical line $\delta = 0$ or the horizontal line $h_2 = 0$ is the stability region