Virtual pole method applied at the profiling of the gear shaped cutter tool for generating a K-type bore

. The “virtual pole” method allows profiling of the cutting tools that generate by rolling. This mode of generation is advantageous in terms of the accuracy of the geometric shape obtained. The mentioned method combines the analytical accuracy of profiling with a simplified calculation mode, compared to the established algorithms. The paper presents an application made on the basis of the “virtual pole” method, for the profiling of the gear shaped tool, designed to generate the bore of a bush with a K-profile. Based on this methodology, in house software was developed for the type of profile mentioned above. The program allows the numerical and graphical representation of the profile to be generated, the contact curve and the profile of the generating tool.


Introduction
The processing of the bores profiled by generating, by the rolling method, has the advantage of a good precision of the shape of the surface to be obtained. For this type of generating, the primary peripheral surface of the generating tool and the generated surface are reciprocally enveloping surfaces. This determines that the surface to be obtained is generated by a large number of cuts and, moreover, between the contour of this surface and the profile of the generating tool, to be a relative sliding.
For these two reasons, the real surface obtained is less likely to be affected by the shape errors of the generating surface. One of the classic methods of obtaining the profiled bores is the processing with gear shaped cutter tools. In the case of these tools, a problem of major importance is the profiling of the cutting edge of the tool, so that it will obtain the desired shape for the piece profile [1].
The paper presents an application of the "virtual pole" method [2] made for the profiling of the gear shaped cutter tool, designed to generate the profiled bore of a bush with K-profile. Also known as polygonal sections, the K-profiles of the shafts and bores have the advantage of being self-centering, having no areas with tension concentrators and ensuring an easy assembly and disassembly [3].
Huang et. al [4] states that this type of assembly ensures a lower processing cost with 40÷50% compared to the splined shaft assembly and also a lower vibration and noise level compared to the key assembly.
Polygonal profiles can have a wide variety of shapes, given their number of lobes, but those with three lobes (P3G) and four lobes (P4G) are standardized [3,5].

The applying algorithm for virtual pole method
The "virtual pole" method, widely presented in the papers [2,6], is a variant of the Willis method, according to which: "the profiles that transmit rotational movement between two parallel axes admit, at the contact point, a common normal, which passes through the gearing pole". The "virtual pole" method is based on the observation that, in the case of tools that generate by rolling, the enwrapping condition can be defined by identifying a "virtual pole" of gearing. This virtual pole is actually the intersection point between the normal taken to the generated profile, in the current point and the centrode associated with the piece. Figure 1 shows the reference systems: xy is the fixed reference system; XY -the mobile system, joined with the profile to be generated, Σ; C1 -the centrode associated with the piece. In the current point T, belonging to the profile to be generated, Σ, the normal n-n has been taken. It intersects the centrode, C1 in the point Pv, representing the virtual pole.
If it is admitted that the profile to be generated, Σ, has the parametric equations: ; : , with u, independent variable parameter, then the position vector of the current point, T has the form: (2) In this point, the normal vector will have the form: λ -being a scalar dimension which represents the distance between the current point, T and the virtual pole, Pv.
The position vector of the point Pv is obtained by summing the vectors r  and  N  [2], obtaining the form: The size of the distance λ is determined from the belonging condition of the point Pv to the centrode C1.
The centrode equations are: ; : and thus the belonging condition becomes: ; The parameter λ can be eliminated from the system of equations (6): representing the enveloping condition, meaning the connection between the independent parameters u and φ1.
Once known the specific value 1 1u    , for which the point Pv coincides with the gearing pole, the position of the T point, in its absolute movement, can be determined: meaning the position which it occupies on the contact curve, in the rolling process: Considering the rolling condition, namely the condition that the two centrodes roll without sliding: it can be determine the position of the tool when the piece rotates with the angle φ1u, see also figure 2. The absolute movement of the tool is: x ω x (11) and the correlation between xo and x systems is given by: From the equations (9), in the fixed reference system, the TF-point coordinates are obtained -be these F T x and F T y .
If the coordinate transformation (12) is taken into consideration, the TF-point coordinates can be determined in the fixed system x0y0, namely Applying to this point the absolute movement of the tool, given by (11), its coordinates are obtained in the reference system ξη, associated with the tool: When the current point T go through the whole profile Σ, the TS-point completely crosses the profile S, reciprocally enveloping to profile Σ.

2.1
The profiling of the gear shaped cutter tool designed to process a K-type bore The K-type profile equations are given by: X ( u ) R e cos n u cos u n e sin n u sin u Y( u ) R e cos n u sin u n e sin n u cos u where n is the number of lobes, and e is the eccentricity. In figure 3 are represented the reference systems, the two centrodes, C1 -associated with the piece and C2 -associated with the tool, the profile Σ, on which the current point T was considered, as well as the normal to the profile, n-n. The reference systems used are: xy is a fixed reference system, having the origin in the centre of the centrode C1 of radius Rrp; xoyo -fixed reference system, having the origin in the centre of centrode C2, of radius Rrs; XY -mobile reference system, joined with the Σ profile and which initially has the axes superimposed on the xy system; the absolute movement parameter of this system is φ1; ξη -mobile reference system, joined with the S profile (of the tool), initially having the axes superimposed on the xoyo system. The movement parameter of this system is the angle φ2, related to the angle φ1, is given by the rolling condition (10).
Taking into account the partial derivatives of the Σ profile equations in relation with the variable u: X e 1 n cos n u R sin u, Y e 1 n cos n u R cos u According to equation (3), in T-point, the normal at the profile Σ is: e 1 n cos n u R cos u e 1 n cos n u R sin u The position vector of the T-point is given by: R e cos n u cos u n e sin n u sin u R e cos n u sin u n e cos n u cos u As result according to (4), the position vector of the virtual pole is: R e cos n u cos u ne sin n u sinu e 1 n cos n u R cos u R e cos n u sin u ne sin n u cos u e 1 n cos n u R sin u Intersecting this vector with the C1 centrode: : ; , is obtained: R cos R e cos nu cos u ne sin nu sin u e 1 n cos nu R cos u R sin R e cos nu sin u ne sin nu cos u e 1 n cos nu R sin u From (20), by removing the λ parameter, is obtained the specific form of the enwrapping condition: The value of the angle φ1u for which the condition (21) is accomplished, represents the value of the angular parameter for which the Pv point coincides with the gearing pole, P.
For a certain value of parameter u and a corresponding value φ1u, determined from the enveloping condition (21), the T-point, belonging to the Σ profile, will have, in the fixed reference system, the coordinates given by the absolute movement of the blank, (8), are: Note: In this position, T-point also belongs to the gearing curve, meaning F T T  . While the blank moved with the angular parameter φ1u, the tool rotates with the parameter φ2u, whose value is given by the rolling condition (10).
Therefore, the position of the tool reference system is given by its absolute movement, (11), by developing obtaining: T : x ( u ) cos y ( u ) sin x u sin y ( u ) cos For u, taking values from umin to umax, the TS-point go through the entire profile of the tool, S.

Numerical example:
For a profile defined by R=60 mm; e=5 mm; n=3 lobes (according to DIN 32711, [7]), the points coordinates on the generating tool profile are presented in table 1, and the profile to be generated, the gearing curve and the generating tool profile are shown in figure 4.

Conclusions
The algorithm based on the "virtual pole" method represents an improved method for profiling of tools that generate by enwrapping, by the rolling method. The application presented in the paper demonstrates the accuracy and simplicity of the "virtual pole" method and, at the same time, allows verifying the results obtained with this method.
Based on the presented methodology, a calculation program was created, which allows the profiling and graphical representation of the gear shaped cutter tool, designed to process a K-type bore.