2.1. Geometry of fluted shank
Surfaces Σ1 to Σ6 are the surface patches on the fluted shank. These surfaces are formed when a composite curve in XY plane is swept as per a sweeping rule. These are known as helicoidal
Fig. 2a. Modeling of an end mill tooth.
surface. The surface patches could better be visualized from the unfolded view of flat end mill (Figs. 2a and 2b). The position vectors of the end vertices of different sections of the composite profile curve (V1 . . . V7) and center points of the three circular arcs (c1, c2, c3) can be evaluated [7,8] to parametrically model the profile curve.
2.2. Modeling of fluted surfaces of end mill
The cross-sectional profile of a fluted section of an end mill consists of three parametric linear edges and three parametric circular arcs, namely, p1(s) to p6(s). Edges p1(s), p2(s) and p6(s) are straight, while p3(s), p4(s) and p5(s) are circular in two- dimensional space. The generic definition of the sectional profile in XY plane in terms of parameter s may be represented by,
pi(s) = [fi1(s) fi2(s) 0 1] .
The fluted surface is obtained by combined rotational and parallel sweeping and is parametrically described by,
p(s, φ) = p(s) · [TS ], where In the above equation, L1 is the length of fluted shank for flat end mills.
2.3. Sweeping rules
Fig. 3. Composite sectional curve.
surfaces. Fig. 3 shows the composite sectional curve (V1 . . . V7), consisting of six segments. Out of these six, three segments V1V2, V2V3 and V6V7 correspond to the three land widths, namely peripheral land, heel and face, and are shown as straight lines on a two-dimensional projective plane. The other three segments V3V4, V4V5 and V5V6 are circular arcs of radii r3, r2 and R respectively, corresponding to the fillet, the back of the tooth and the blending
The fluted section of an end mill can have a right helix or a left helix. If the flute’s spiral has a clockwise contour when looked along the cutter axis from either end, then it is a right helix or else the helix is left [13]. For a right helix cutter, the cross-section curve
rotates by an angle +φ about the axis in the right-hand sense.
Three different sweeping rules can be formulated for the fluted
shank and the end profile of the cutter. These rules are for
(i) Cylindrical Helical Path
(ii) Conical Helical Path
(iii) Hemispherical Helical Path
Cylindrical helical path —The path when the composite profile curve is swept helically along a cylinder is known as a cylindrical helical path. For a helical cutter let φ be the parameter denoting the angular movement, P the pitch of the helix, DC the cylindrical cutter diameter and L1 the length of the fluted part of cutter, then the mathematical definition of the helix is,
x = (DC /2) cos φ
y = (DC /2) sin φ
z = (Pφ)/(2π), where 0 ≤ φ ≤ (2π L1/P).
Similarly, one can formulate the equations of conical and hemispherical helical paths. The helical path along a frustum of cone of cutting end diameter DC and shank side diameter DS is defined by,
x = (D/2) cos φ y = (D/2) sin φ z = (Pφ)/(2π),
where D = DC + (DS − DC )z/L1 and 0 ≤ φ ≤ (2π L1/P).
This is valid for both types of frustum of cones i.e. when DC < DS and when DC > DS . The helical path along the hemispherical object of diameter DC is given as,
x = (D/2) cos ψ cos φ y = (D/2) cos ψ sin φ z = (DC /2)(1 − sin ψ), 通用立铣刀的三维建模英文文献和中文翻译(3):http://www.chuibin.com/fanyi/lunwen_206285.html