Abstract
In this chapter of the book, special cases of generation of enveloping surfaces is disclosed. For this purpose, a concept of reversibly-enveloping surfaces is introduced. For generation of reversibly-enveloping surfaces a novel method is proposed. This method is illustrated by an example of generation of reversibly-enveloping surfaces in case tooth flanks for geometrically accurate (ideal or perfect) crossed-axis gear pairs. The performed analysis makes it possible a conclusion that two Olivier principles of generation of enveloping surfaces: (a) in general case are not valid, and (b) in a degenerate case these two principles are useless. Ultimately, there is no sense to apply Olivier principles for the purpose of generation of reversibly-enveloping smooth regular part surfaces. Part surfaces those allow for sliding over themselves are considered as a particular degenerated case of enveloping surfaces.
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Notes
- 1.
It is of importance to stress here that angular velocity is considered in this monograph as a vector directed along the axis of rotation in a direction defined by the right-hand screw rule. It is understood here and below that rotations are not vectors in nature. Therefore, special care is required when treating rotations as vectors.
- 2.
It should be stressed here that “\( R \)-gearing” is the only kind of crossed-axes gearing that ensures line contact of the worm threads with the tooth flanks of the worm gear. No other kind of gearing features line contact of this kind. Many efforts have been undertaken by Dr. J. Fillips to develop a kind of spatial gearing having line contact between the tooth flanks. In the design of spatial gearing proposed by Dr. J. Fillips (2003), both tooth flanks of the gear and of the pinion are generated by a plane that is traveling in relation to the axis of rotation of the gear, \( {\fancyscript{G}} \), and to the axis of rotation of the pinion (when the pinion tooth flank, \( {\fancyscript{P}} \), is generated). In \( R \)-gearing, neither tooth flanks of a gear, nor of a mating pinion can be generated by a plane. Therefore, it becomes evident that in the spatial gearing proposed by Dr. J. Fillips, the tooth flanks of the mating gears never make line contact. Research in this area was later carried out by Dr. Stachel (circa 2006, and others) to determine a special combination of the design parameters of the gearing under which the tooth flanks make line contact.
- 3.
In the case under consideration, the angular velocity vector, \( {\varvec{\upomega}}_{pl} \), can be used instead of the unit vector, \( {\mathbf{p}}_{ \ln }. \)
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Radzevich, S.P. (2020). Generation of Enveloping Surfaces: Special Cases. In: Geometry of Surfaces. Springer, Cham. https://doi.org/10.1007/978-3-030-22184-3_10
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DOI: https://doi.org/10.1007/978-3-030-22184-3_10
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