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42 Silviu Berbinschi et al
- analytical methods, based on fundamental methods of surfaces enwrapping
– first Olivier theorem, Gohman theorem, normals method, W i l l i s [1], [2];
- complementary analytical methods – “minimum distance” method, the
“substitutive circles family” method, the “in-plane generating trajectories”
method [3]-[5];
- graphical-analytical methods [6];
- graphical methods, using the capabilities of CAD software [7].
We mention that the methods proposed and used for the study of
reciprocally enveloping surfaces respect the enveloping fundamental theorem.
The proposed solutions leads to comparable results, in most cases
identically, for the crossing profile tool’s shape, which generated by rolling
ordered curls profiles associated with a couple of rolling centrodes.
Literature include multiples applications for various domains as methods of
tool’s profiling, revolution surfaces, for generating helical surfaces [8]; the
modelling of the cylindrical surfaces with disk tools [9]; generation of the
helical compressors rotors [10].
In this paper, is proposed a method for profiling of the rack-gear tools
reciprocally enveloping with ordered curl profiles, based on the capabilities of
the CATIA design environment.
2. Generation Kinematics
In principle, the tool’s profiling determination problem for rack-gear tool
reciprocally enveloping with an ordered curls profile, associated with a circular
centrode, assume to respect the kinematics of the generating process, see figure
1.
Fig. 1 – Couple of rolling centrodes.
The two rolling centrodes, C1 —circle, associated with the profiles ordered
curls and C2 —straight line, associated with the rack-gear tool, are in rolling
movement, so, is respected the condition: