Analytical Prediction of Interrupted Cutting Periodic Motions in a Machine Tool

Abstract

The methodology for prediction of interrupted cutting periodic motions in a machining system is developed. The interrupted cutting mappings in the vicinity of the system constraints are defined. The criteria for the interrupted cutting periodic motions are developed through the state variables and mapping forms. The periodic interrupted cutting motions in a two-degree-of-freedom model are predicted numerically and analytically via closed form solutions. The chip and tool-piece seizure in the machine-tool system is also discussed. The bifurcations are caused by interactions of continuous dynamical systems in the neighborhood of the boundary.

Appendix

The dynamical system parameters for the machine-tool system, in the case the tool does not contact the work-piece, domain Ω₁ are

(A.1)

and

$$C^{(1)}_x=C^{(1)}_y=A^{(1)}_x=A^{(1)}_y=0.$$

(A.3)

The dynamical system parameters for this machine-tool system, in the case the tool contacts the work-piece where no cutting occurs, domain Ω₂ are

(A.4)

(A.5)

and

$$\everymath{\displaystyle}\left.{\begin{array}{l}C_x^{(2)}={1\over {m\Omega ^2}}\{k_1[x_1^\ast \sin \beta -y_1^\ast \cos \beta ]\sin \beta\},\\[4mm]C_y^{(2)}={1\over {m\Omega ^2}}\{-k_1[x_1^\ast \sin \beta -y_1^\ast \cos \beta ]\cos \beta\},\\[4mm]A_x^{(2)}={A\over {m\Omega ^2}}\sin \eta,\qquad A_y^{(2)}=\mathrm{A\over {m\Omega ^2}}\cos \eta.\end{array}}\right \}$$

(A.6)

The dynamical system parameters for this machine-tool system, in the case the tool contacts the work-piece where cutting occurs where $\dot{z}<0$ and D ₄>0, domain Ω₃ and $\dot{z}>0$, domain Ω₄;

(A.7)

(A.8)

and

(A.9)

(A.10)

$$A_{x}^{(j)}=\frac{A}{m{{\Omega }^{2}}}\sin \eta ,\,\,\,\,\,\,A_{y}^{(j)}=\frac{A}{m{{\Omega }^{2}}}\cos \eta \ $$

(A.11)

for j=3,4; respectively. The parameters for the machine-tool where the chip adheres to the tool-piece rake face $(\dot{z}\equiv0)$ are

$$d=\frac{1}{2m\Omega }[{{d}_{2}}+{{d}_{1}}{{\sin }^{2}}(\alpha +\beta )+{{d}_{x}}{{\cos }^{2}}\alpha +{{d}_{y}}{{\sin }^{2}}\alpha ],\ $$

(A.12)

$${{\omega }^{2}}=\frac{1}{m{{\Omega }^{2}}}[{{k}_{1}}{{\sin }^{2}}(\alpha +\beta )+{{k}_{2}}+{{k}_{x}}{{\cos }^{2}}\alpha +{{k}_{y}}{{\sin }^{2}}\alpha ],\ $$

(A.13)

$${{A}_{0}}=\frac{A}{m{{\Omega }^{2}}}\sin (\eta -\alpha ),\ $$

(A.14)

$${{B}_{0}}=\frac{V}{m{{\Omega }^{2}}}[{{k}_{1}}\cos (\alpha +\beta )\sin (\alpha +\beta )+({{k}_{x}}-{{k}_{y}})\cos \alpha \sin \alpha ],\ $$

(A.15)

(A.16)