Méthode numérique pour l'étude des surfaces enveloppées

Abstract

Analytical methods for wrapping up surfaces: Oliver's method, Gohman's kinematic method, Willis's normals method and the method of the ‘minimum distance’, suppose the use of the surface equations in an analytical and vectorial form, for the partial derivatives determination for the direct writing of the wrapping up conditions (Oliver and the method of the minimum distance) or for the determination of the surfaces normals (Gohman, Willis). To solve the wrapping up conditions which are usually transcendental equations on an analytical manner is difficult. The methods are deficient in the situations when the surfaces whose wrappings are to be determined are wrappings of other surfaces too and when the wrapping up conditions can be transcendental of two parameters. To avoid these problems, a numerical method is proposed for calculating wrapping up surfaces, which is based on a point by point representation of the surfaces to be generated, having in view that all the surfaces can be described point by point with an accuracy which can satisfy from a technical point of view. The work presents a method to divide the surfaces leading to a representation by matrices. By this method, the generation of the surfaces associated with axoids in rolling, with particularities for the generation with gear-rack tool, shaping tool and rotary cutter, is analysed. The specific theorems for the generation by wrapping up of the helical surfaces, for the work with disc and cylindrical-frontal tools are presented. The tools for the generation of the relieved surfaces (radial and radial on the helical line) are also analysed and the specific theorems for the profiling of these tools as conjugated surfaces of the relieved surfaces. The contact between a helical surface and a cylindrical surface for the determination of the algorithm for the helical tool profiling is treated like a special problem. Numerical examples for profiling of gear-rack tools and disc tools for radial relief and helical tool are presented.