Influence of Feed Velocity on Nonlinear Dynamics of Turning Process

Abstract

A more comprehensive orthogonal turning model is developed in order to further study the influence of feed velocity on frictional chatter. Nonlinear dynamic behavior of the cutting tool in two directions is presented by using bifurcation diagram, phase portrait, and Poincaré section. It can be found that the cutting tool has a variety of dynamic behaviors at different feed velocity and cutting velocity, such as periodic motion, quasi-periodic motion, and chaotic motion. Furthermore, the vibration displacement of the cutting tool is affected by the feed velocity, especially for relatively high feed velocity which will result in the cutting tool vibration displacement increase in the cutting direction but a decrease in the feed direction. In addition, it is clear that the stick–slip phenomenon only appears in the cutting direction in our work.

Appendix

Substituting Eqs. 2,3,4 and 5 into Eq. (1), we obtain the vibration response of the cutting tool motion in both directions

$$\begin{aligned} & m\ddot{x} + c_{x} \dot{x} + k_{x} x = K(2\pi v_{f} /\Omega - y)H(2\pi v_{f} /\Omega - y)\mu_{x} ({\text{sgn}} (\Omega R - \dot{x}) - a_{x} (\Omega R - \dot{x}) + \beta_{x} (\Omega R - \dot{x})^{3} ) + Q(2\pi v_{f} /\Omega - y)(c(\Omega R - \dot{x} - 1)^{2} + 1)H(2\pi v_{f} /\Omega - y)H(\Omega R - \dot{x}) \\ & m\ddot{y} + c_{y} \dot{y} + k_{y} y = Q(2\pi v_{f} /\Omega - y)(c(\Omega R - \dot{x} - 1)^{2} + 1)H(2\pi v_{f} /\Omega - y)H(\Omega R - \dot{x})\mu_{y} ({\text{sgn}} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y}) - a_{y} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y}) + \beta_{y} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y})^{3} ) + K(2\pi v_{f} /\Omega - y)H(2\pi v_{f} /\Omega - y) \\ \end{aligned}$$

(6)

In order to reduce the number of parameters, Eq. (6) is simplified as the following dimensionless equation.

$$\begin{aligned} & \ddot{x} + 2\xi_{x} \dot{x} + x = k(2\pi v_{f} /\Omega - y)H(2\pi v_{f} /\Omega - y)\mu_{x} ({\text{sgn}} (\Omega R - \dot{x}) - a_{x} (\Omega R - \dot{x}) + \beta_{x} (\Omega R - \dot{x})^{3} ) + q(2\pi v_{f} /\Omega - y)(c(\Omega R - \dot{x} - 1)^{2} + 1)H(2\pi v_{f} /\Omega - y)H(\Omega R - \dot{x}) \\ & \ddot{y} + 2\xi_{y} \sqrt \alpha \dot{y} + \alpha y = q(2\pi v_{f} /\Omega - y)(c(\Omega R - \dot{x} - 1)^{2} + 1)H(2\pi v_{f} /\Omega - y)H(\Omega R - \dot{x})\mu_{y} ({\text{sgn}} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y}) - a_{y} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y}) + \beta_{y} ((\Omega R - \dot{x})\tan \varphi + v_{f} - \dot{y})^{3} ) + k(2\pi v_{f} /\Omega - y)H(2\pi v_{f} /\Omega - y) \\ \end{aligned}$$

(7)