Aspects of the Kinematic Theory of Spatial Transformations of Rotations: Analytic and Software Synthesis of Kinematic Pitch Configurations

Abstract

The present work deals with the kinematic theory of spatial rotations transformation in the context of defining a kinematic and geometrical essence of the basic building elements of the mathematical models for synthesis of hyperboloid gear sets, which are called by the authors of this study—pitch configurations: pitch circles and pitch surfaces. They configure one contemporary, as a content and terminology, direction from the Theory of Gearing, defining such basic characteristics of hyperboloid gears as: the structure and geometry of the gear system, the longitudinal and cross orientation of the active tooth surfaces of the gears, the values of the gears’ modules, the forces acting in the contact zone of the gears, on the shafts and on the bearing supports of the gear mechanism, coefficient of efficiency and etc. Hence, from a methodological and application view point, it is one of the reasons, that the authors of the current study have devoted a great part of their researches on the mentioned above thematic. Activities of such type are a potential opportunity for emerging of new ideas related to the creation of innovative hyperboloid gears. When these pitch configurations are synthesized in condition of a static contact of the active tooth surfaces of the hyperboloid gears, they define so-called geometric pitch configurations. These configurations are subject of study “Geometric Pitch Configurations—Basic Primitives of the Mathematical Models for the Synthesis of Hyperboloid Gear Drives”, published in Advance Gear Engineering, Mechanism and Machine Science 51, Springer. The research, oriented to the synthesis of hyperboloid transmissions, will be complete and effective, when it leads to the creation of an adequate mathematical model, describing the status of the pitch configurations in the process of spatial rotations transformation. In this work, the treated scientific concepts, are related to the content of the concepts of kinematic pitch configurations, when their geometry (shape and dimensions) and mutual position directly take into account the process (law) of spatial rotations transformation. These type characteristics of hyperboloid gears are of essential importance for their synthesis, for considering the magnitude and orientation of the sliding velocity vector in the mesh region of the active tooth surfaces.