An analytical formulation of ZOA-based machining stability for complex tool geometries: application to 5-axis ball-end milling

Abstract

The paper describes a computationally convenient analytical formulation of the stability of the cutting process with respect to self-excited vibrations in the case of five-axis milling based on the commonly used zero-order approximation. In the case of five-axis milling with general milling cutters, it is difficult to calculate stable machining process conditions for two main reasons. The first reason is the difficulty of calculating the mean value of the cutting force Jacobian with respect to the regenerative displacement (closely related to a milling directional matrix) for a generally inclined tool, and the second reason is the nonlinearity of this Jacobian with respect to the process parameters, which means that the problem cannot be reduced to a linear eigenvalue problem as is usual for linear cases (e.g. cylindrical milling with respect to the axial depth of cut). In the first part, this paper presents a modification of the calculation of the machining stability limits for a nonlinearly dependent cutting force Jacobian. A new formulation of this Jacobian for a general tool based on the surface integral over the tool and workpiece engagement region is presented which leads directly to the mean value of the Jacobian of the cutting force (direction matrix) without the need to calculate it as a function of time and then calculate the mean value over one revolution. The advantage is that if we can analytically describe the engagement area, we also obtain an analytical relation for the cutting force Jacobian. This is presented with the practical example of a generally inclined ball-end mill. This analytical formulation of the force Jacobian allows the calculation of its derivatives with respect to the technological parameters (depth of cut, step over, tilt and lead angles), which is useful both for the calculation of stability diagrams and for solving optimization problems related to machining stability.