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The indexable ball end mill is a sophisticated tool designed for machining complex, free-form surfaces. Compared to traditional integral ball end mills, it offers the advantages of indexable cutting tools while significantly enhancing the cutting edge performance through overlapping blade processing methods. This approach helps reduce non-free cutting coefficients, thereby minimizing cutting forces [1]. While literature [2] has explored the geometric theory of the end edge of indexable ball end mills, there are no existing studies on the method of overlapping the peripheral and end edges. Based on the theoretical foundation in [2], this paper presents a mathematical model for the lapping of peripheral and end edges. The model can be used to calculate the geometric error of the machined surface near the overlap and optimize the control parameters for the peripheral shape.
Figure 1 shows the coordinate system used in this study. The indexable ball end mill has a radius R, and the blade is inclined at an angle ls. The rake face of the blade is initially set as a square shape, which allows all four sides to act as cutting edges, maximizing blade utilization. The side length of the square is A (as shown in Figure 2). The tangent vector T0 at the point of overlap between the end edge and the peripheral edge is given by T0 = (0, 1, m)T / √(1 + m²), where m = cot(w1), with w1 being the helix angle. From this equation, T0 is perpendicular to the x-axis.
The cylindrical parameter equation is defined as r = (Rcosq, Rsinq, V)T, where q is the angular variable and V is the cylinder height variable. The midpoint N of the blade’s cutting edge lies on the cylindrical surface. The direction vector CD is denoted as P. Assuming that the tangent after the rotation of the peripheral edge is parallel to T0, the condition T0 × P = 0 must be satisfied. By setting P equal to T0, the line CD passing through point N (Rcosq, Rsinq, V) is expressed as:
x = Rcosq
y = Rsinq + zV * m
Substituting into the equation of a single-leaf hyperboloid, we obtain:
y² - (z + mRsinq + 0.5Acosls)² = (Rcosq)²m
To simplify calculations, point C is assumed to lie on the xoy plane, and its projection on the z-axis is -2V, with -2V = Acosls, so V = -(Acosls)/2. Substituting into the equation gives:
y² - (z + mRsinq + 0.5Acosls)² = (Rcosq)²m
As shown in Figure 3, points C and D are not on the cylindrical surface but instead lie at z = 0 or z = 2V, causing maximum shape errors. The maximum positive y-coordinate (ymax) is calculated when z = 0:
ymax = [(Rsinq + A cosls)² + R²cos²q]¹â„² / 2m
Thus, the maximum geometric error at point C is:
Dmax = ymax - R
To reduce the geometric error, the peripheral edge modification angle f is introduced. The ideal scenario is when points D, N, and C intersect the rake face and the cylindrical surface, forming an ellipse (where Dmax = 0). However, due to the elliptical shape of the blade’s edge, a straight line NC'' is used instead. Chamfering the CND segment from both sides of N reduces the error caused by points C and D. The chamfer angle f is determined using the right triangle NCC:
tanf = 2{[(Rsinq + A cosls)² + R²sin²q]¹â„² - RA} / 2m
After calculating f, the new maximum geometric error D’max is computed by replacing R with (R - Dmax) in the previous equations. Additionally, the "less cut" phenomenon occurs when the blade is modified, and the negative error Dc0 is calculated by adjusting the blade dimensions. These modifications help achieve a smoother connection between the peripheral and end edges, improving surface quality and reducing manufacturing complexity.
Finally, the angle a’ between the axial profiles of the rotated surfaces is calculated. Using the coordinates of E’, F, and G, the slope of the hyperbola is derived, leading to the formula for a’:
a’ = arctan[Acosls(R + D’max)] / [4(D’max - Dc0)(2R + D’max + Dc0)]
This angle indicates the smoothness of the transition between the two edges. Smaller values of a’ and D’max indicate better surface accuracy and fit at the lap joint.
In a calculation example, with R = 25 mm, A = 12 mm, w1 = 20°, and ls = 15°, the values of Dmax, D’max, f, Dc0, and a’ were found to be 1.5335 mm, 1.45×10â»â´ mm, 14.34°, -0.00984 mm, and 0.395°, respectively. These results demonstrate that using a symmetrical polygonal cutting edge instead of a straight one significantly reduces geometric errors and enables a nearly smooth connection at the overlap. The blade can be rotated and reused, increasing efficiency and reducing costs. This method offers great practical and economic value in modern machining processes.
October 11, 2025