where 0 ≤ φ ≤ π DC /P and 0 ≤ ψ ≤ (π/2).
Here, φ is the angle about Z axis and ψ is the angle with XY plane and the relation between them is,
Table 1Forward mapping relations for flat end mill.
2.4. End surface geometry
The end surface geometry of a flat end mill consists of three planes and one blending surface. (i) Face Land (Σ7) (ii) Minor Flank (Σ8) and (iii) Rake Face Extension (Σ9) are the three planes whereas the blending surface, 2Σ9 blends surface patch Σ8 of the first tooth with surface patch Σ9 of the second tooth of the end mill. Face land (Σ7) is formed when an XY plane given by [u7 v7 0 1]
is transformed through rotation by a 3D rotational angle α7 about X
axis [Rx,α7 ], followed by rotation by an angle γ1 about Z axis [Rz,γ1 ]
and is defined as in Box I.
Minor Flank (Σ8) is formed when the XY plane is rotated by an angle α8 about the X axis [Rx,α8 ], followed by an angle γ1 about the Z axis [Rz,γ1 ], and then translated by a distance d82 = l1 cos γ2 along the Y axis and d83 = l1 cos γ2 sin α7 along the Z direction [Tyz ] and is given as in Box II.
Rake Face Extension (Σ9) is formed when a ZX plane ([u9 0 w9 1]) is rotated by an angle α9 about the X axis [Rx,α9 ] and by an angle γ1 about the Z axis [Rz,γ1 ]. Here, the helix angle λ = tan−1(P/π DC ), α9 = 90◦ − λ∗ and λ∗ = λ + (15◦ − 25◦) but
≤ 90◦. The surface Σ9 satisfies the relation in Box III.
3. Modeling of shank
The shank surface of a flat end mill may be modeled as a combination of two surface patches (i) cylindrical surface of revolution Σ50 and (ii) planar end surface Σ51. The cylindrical surface of shank in terms of parameter w, may be parametrically defined by,
Table 3b
Geometric parameters and input data for an end mill.
for 0 ≤ φ ≤ 2π , 0 ≤ w ≤ 1. The term L2 stands for overall length of end mill and diameter Dr may be given by,.DS /2, for straight shank(DC /2) + w(DS − DC )/2, for tapered shank.The planar end surface forming the end opposite to cutting end is parametrically modeled as,
p51(u, v) = [ u v L2 1 ] (2)for −∞ ≤ u, v ≤ ∞. A unit width, 45◦ chamfer between shank surface patches Σ50 and Σ51 is the only blending surface on the shank body of the cutter. The chamfer σ50,51 is modeled by revolving the edge having end point coordinates as (((DS /2) − 0.707), 0, L2) and (DS /2, 0, (L2 − 0.707)) about the Z axis and is given in Box IV,
4. Mapping relations
In this work, relations to map the proposed 3D rotational angles, used to define the geometry of the flat end mills, into conventional 2D projective angles and vice versa are developed. The former is called forward mapping while the latter is known as inverse mapping. The conventional angles are formed by projecting the surface patches of the end mill on planes of projection and the traditional geometry of end mill can be referred from [12– 14]. Table 1 shows the forward mapping relations for a flat end mill. The solution of these forward mapping relations establishes inverse mapping that helps to evaluate the 3D rotational angles if tool angles specified by conventional nomenclatures are known. Table 2 presents the inverse mapping for an end mill. 通用立铣刀的三维建模英文文献和中文翻译(4):http://www.chuibin.com/fanyi/lunwen_206285.html