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The indexable ball end mill is a sophisticated cutting tool designed for machining complex, free-form surfaces. Compared to traditional integral ball end mills, it combines the advantages of indexable cutting tools with enhanced performance due to the overlapping of multiple blades during the cutting process. This overlapping technique helps reduce non-free cutting conditions, thus minimizing cutting forces [1]. While literature [2] has explored the geometric fundamentals of the end edge of indexable ball end mills, there remains a gap in research regarding the overlapping of peripheral and end edges. Building upon the theoretical foundation in [2], this paper presents a mathematical model to calculate the geometrical errors that occur near the overlap region of these edges. The model also provides control parameters for optimizing the peripheral shape.
As shown in Figure 1, a coordinate system is established where the radius of the indexable ball end mill is R, and the blade inclination is ls. The rake face of the blade is initially designed as a square to maximize utilization, with side length A. The tangent vector T0 at the point of overlap is given by T0 = (0, 1, m) / √(1 + m²), where m = cot(w1) and w1 is the helix angle. This vector is perpendicular to the x-axis. The cylindrical parameter equation is defined as r = (Rcosq, Rsinq, V), where q is the angular variable and V is the height variable. The midpoint N of the cutting edge lies on the cylindrical surface.
To ensure the tangent vector after rotation aligns with T0, the direction vector P of the cutting edge CD must satisfy T0 × P = 0. By setting P = T0, the straight line CD passing through N is derived, leading to the equation of a single-leaf hyperboloid. When the blade rotates around the z-axis, the axial section of the hyperboloid is calculated, revealing the maximum shape error at point C. Using the derived equations, the maximum geometric error Dmax is determined.
To reduce this error, the peripheral edge modification angle f is introduced. By chamfering the blade’s edge and replacing the elliptical curve with a straight line NC", the error caused by the C and D points is minimized. The calculation of f involves trigonometric relationships based on the geometry of the blade. After modifying the edge, the new maximum error D’max is significantly reduced, improving the surface quality.
Additionally, the impact of the cutting edge on the overlap between the end and peripheral edges is analyzed. Adjustments are made to minimize "under-cut" effects, ensuring smoother transitions. The angle a' between the revolution surface and the end edge is calculated, offering insights into the surface's curvature and alignment.
A numerical example is provided using specific values: R = 25 mm, A = 12 mm, w1 = 20°, and ls = 15°. Calculations show that the use of a polygonal cutting edge instead of a straight one significantly reduces geometric errors, enabling a smooth connection between the peripheral and end edges. This approach enhances blade utilization, simplifies manufacturing, and improves economic efficiency. The results demonstrate the practical value of this method in modern machining applications.
October 11, 2025