Abstract: One of the key issues in the modeling of milling processes is to determine the response of the cutting forces to the dynamic variations of the uncut chip thickness. The uncut chip thickness depends on the relative position between the milled surfaces left by the successive cutting teeth of the tool. A cutting tooth intersects the path of its previous tooth, and forms an angle with the previous tooth passing through the same point of intersection. Given the intersection angle, both the variable time delay and the instantaneous uncut chip thickness can be calculated directly. Through the analysis of trochoidal tooth paths and using the intersection angle as an auxiliary variable, a transcendental equation is developed to model of the geometry of chip formation. Then two new approaches are proposed to determine the intersection angle. The first approach converts the transcendental equation into an ordinary differential equation of the intersection angle, then solving it numerically without recursive root-finding algorithms; assuming the intersection angle is infinitesimal, another approach approximates the transcendental equation by replacing the sine function with a liner function, and then solve it analytically. Case studies with different process parameters show that the analytical approach can provide a high accuracy in practical milling operations, with a simpler expression compared with other models. The proposed numerical method is suitable for embedding into the milling process simulation or stability prediction algorithms which are sensitive to accuracy and efficiency.

Key words: approximate analytic solution, milling processes, numeric method, trochoidal tooth path, uncut chip thickness, variable delay