A graphical solution in CATIA for profiling end mill tool which generates a helical surface

The generation of a helical flute, which belongs to a helical cylindrical surface with constant pitch, can be made using end mill tools. The tools on this type are easiest to make than the side mills and represent a less expensive solution. The end mill profiling may be done using the classical theorems of surfaces enveloping, analytical expressed, as Olivier theorem or Nikolaev method. In this paper is proposed an algorithm, developed in the CATIA design environment, for profiling such tool’s type. The proposed solution is intuitive, rigorous and fast due to the utilization of the graphical design environment capabilities. Numerical examples are considered in order to validate the quality of this method.


Introduction
The complex helical surface generation of a helical flute (the case of helical teeth, cutting tool's flutes, compressor rotors, helical pumps rotors) may be realized with end mill tools.
The tool's construction is simple and less expensive. Usually, the end mill axis is perpendicular to the helical surface axis. The helical flute may be generated in a single position of the tool only that its normal section is symmetrical regarding the tool's axis. This is a limitation of this generating process.
A specific problem of the generation by enwrapping with end mill is the determination of the tool's profile. There a used the Olivier or Gohman theorems [1][2][3], in analytical form, if are known the parametrical equations which describe the helical surface.
More, the complementary theorems, as the "minimum distance" method, can be use [4]. Also, it is specific for this problem the Nikolaev theorem [6], as general solution for the issue of generation be enveloping.
Analytical solutions are applicable if the helical surface's flank is known in analytical form. The development of the graphical Auto-CAD or CATIA design environment allows approaching these problems using specifically capabilities [5], [8][9][10]. More, using CNC machines these problems can be addressed using this type of machine-tools [7], [11].
In this paper is proposed an algorithm, based on the specific theorem of the "minimum distance", developed in CATIA graphical design environment. The algorithm leads to rigorous solutions and very intuitive. The specific issues of tool's profile discontinuities may be approached using the proposed method.
According to the theorem, in a plane perpendicular to the tool's axis A , is determined onto the  surface a curve belong to this -C.  , which, together with the plane P (see (1) ; .
The distance from the C current point to the X2 axis ( A ) is: The condition that this distance be minimal is given by the equation: The assembly of equations (1) and (5) determines, for each value of the H parameter a value of the d distance, (4), representing the minimum distance to the A axis (X) -the axis of the end mill. The axial section of the end mill is: where uH is the value of the u parameter from the C curve depending to the H arbitrary variable.

The frontal profiles of the three-lobs rotor
It is presented, in figure 2, the crossing section of the three-lobs rotor from the construction of the Roots compressor [16]. The roots compressor is composed by two three-lobs rotors. The geometry of these rotors is presented in figure 2 and includes: circle's arcs, with radius r, on the zones AC and BD and an epicycloids' arc on the CB portion.
The reference systems are defined as follows: x1y1z1 is the reference system joined with the C1 centrode; x2y2z2 -reference system joined with the C2 centrode; XYZ -reference system joined with the helical flank (the X axis is the symmetry axis of the gap); X2Y2Z2 -reference system joined with the rotor.
-The circle arc AC : They are defined θImin = 0 and θImax from intersecting condition between the circle (7) and the circle with radius Rr, -The epicycloids arc CB is described in the relative motion of the two centrodes, C1 and C2, with radius Rr. The movement of the x2y2 reference system regarding the x1y1 reference system is described by the coordinates transformation where: ω3(φ1) and ω3(φ2) are the rotation matrix around the O1 and O2 origins; A12 -is the distance between the rotation axes.
The x2 matrix is given by  The relative motion (10) and the rolling condition of the C1 and C2 centrodes, determine the epicycloids CB in the x1y1 reference system: -The BD circle arc:

The helical flank's surfaces of the three-lobed rotor
The two rotors are cylindrical helical surfaces with constant pitch, described by transformations such as for clockwise worm with p helical parameter.
The flank with the generatrix AC , see (7): cos cos sin sin ; cos sin sin cos ; , with Ψ -angular parameter of rotation around the z1 axis. The helical flank with the generatrix an epicycloids arc CB , see (16): . .

The Nikolaev theorem
For a validation criterion of the graphical method, an analytical method is accepted as profiling method of the side mill or end mill, based on a fundamental method -the method of helical motion decomposition, known as the Nikolaev theorem [11], figure 3.a.
The reference systems are defined as follows: XYZ is the reference system associated with the helical flanks of the three-lobed rotor; the Z axis overlapped to the V axis of the helical surface; X2Y2Z2 -associated with the side mill, the A axis, revolution axis perpendicular to the helix with the maximum diameter of the composed surface.  The unfold line of the helix from the cylinder with the radius Re is presented in figure 3.b. The Nikolaev theorem assumes that the vectors N  , A and 1 r are on the same plane, where: N  is the normal to the Σ composed surface of the flute of three-lobed rotor; A -axis versor of the future side mill tool;  The reporting of the rotor's frontal profile, and also of the rotor's helical surfaces, at the XYZ reference system (with X axis as symmetry axis of the helical flute) assume the coordinates transformation, see figure 4: In this way, the helical flank's equations, in the new reference system, see (23) -for the helical flank generated by the epicycloids arc CB (see equations (24) with x1(θ2), y1(θ2) given by (25).
The directional cosines of A axis -the axis of further side mill, for a helical surface -clockwise worm, are: with X, Y, Z from (32), (34), (35). In this way, the enwrapping condition is defined in the XYZ reference system, joined with the axis of the helical rotor.
In principle, the condition (29) determined for each helical surface from the construction of the helical flute, see (32), (34), (35), is a function dependent on two variable parameters: for the surface with generatrix AC ; for the surface with the generatrix CB ; for the surface with generatrix BD .
By rotating the segments of characteristic curve around the A axis -the axis of side mill -it is generated the composed primary peripheral surface which constitutes the primary peripheral surface of end mill: SAB, SCB and SBD.
In order to simplify the drawing of the characteristic curve's segments, it is useful to represent the characteristic curves in the reference system joined with A axis -the axis of the side mill (see figure   3) by a coordinates' transformation under the form: In this way, the equations (32), (34), (35), with the transformation (53), will be defined in the X2Y2Z2 reference system, joined with the A axis -the axis of the further side mill tool.
Similarly, the characteristic curves (52), by means of the transformation of (53), will be reported to the X2Y2Z2 reference system of the further side mill, in principle, under the form ; .
The axial section of the revolution surface SA, is determined from the equations under the form:

The graphical method in CATIA, for profiling the end mill tool for generating the compressor worm
In figure 6 the axis position for the end mill tool's surfaces, A , is presented, regarding the composed helical surface axis, V .
The issue of tools profiling, for this current case -the flank of Roots compressor rotor, can be solved according to the previously presented methodology, see section 2.
In figure 6, there are presented the plane sections X2 = H together with planes perpendicular on the  The composed characteristic curve is presented in table 1. The axial section can be determined now, see figure 8 and table 2.  It is obvious the discontinuity onto the tool's profile. This is due to the presence of singular point, B, onto the rotor's crossing section.

Conclusions
The presented algorithm is based on the complementary theorem of the "minimum distance" and uses the capabilities of graphical modelling provided by CATIA design environment. The algorithm allows determining the contact points between the helical surface and the revolution surface. These points are points onto the characteristic curve.
The solid model of the end mill tool can be generated and, starting from this, the axial section of this tool.
The numerical example proves the method quality. The algorithm is simple, intuitive and leads to very rigorous results.