Abstract
This section of the book deals with gears and gear pairs. At the beginning, a concept of the “gear vector diagram” is introduced. This concept is extensively employed below aiming the development of a scientific classification of gearing. A concept of a “favorable line of contact” in a gear pair is another mean that is extensively used when developing a scientific classification of gear pairs. These two concepts along with the newly introduced concept of the “generic gear surface”, all together, make the scientific classification of gears and gear pairs possible.
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Notes
- 1.
The term “rotary-zero crossed-axes gear pair” is due to that in parallel-axes gearing of this kind the rotation of the gear is zero (i.e., ωg = 0). “Gear-to-rack gear pair” of a conventional design features a zero rotation of the gear: for a certain rotation of the pinion, ωp, the rotation of the gear, ωg, is always zero, ωg = 0. In a more general case of zero crossed-axes gearing, the gear rotation is not zero (i.e., ωg ≠ 0). By convention, the term “rotary-zero crossed-axes gear pair” is applied to gearing of all types, that is, to parallel-axes, intersected-axes, and crossed-axes gearing as well.
- 2.
The term “spherical gear pair” is incorrect as gears of other types, for example, crossed-axes gear pairs, are also engaged in mesh on a sphere. Therefore, replacement of the obsolete and extensively used term “conical gear pair” with the term “spherical gear pair” is not valid. In order to avoid ambiguities in further discussions, gearing of this type is referred to as “intersected-axes gearing,” or just as “Ia−gearing,” for simplicity.
- 3.
More accurately, the radius of the gear “sector,” and not of the gear, approaches infinity.
- 4.
The vector diagrams 1.1.2.1 and 1.3.2.1 correspond to the deeply degenerate designs of gear pairs. Because of these, significant features could be observed when developing tooth flanks for gearing that correspond to the gear vector diagrams 1.1.2.1 and 1.3.2.1. When friction between the interacting tooth flanks of the gear, \( \mathcal{G} \), and the pinion, \( \mathcal{P} \), is ignored, the tangential force by means of which the torque is transmitted from the driving shaft to the driven shaft acts along the common perpendicular, ng, to the interacting tooth flanks, \( \mathcal{G} \) and \( \mathcal{P} \). The common perpendicular, ng, intersect the pitch line, Pln, that is, it intersects the line of action of the vector of instant rotation, ωpl. In cases of gear pairs that correspond to the vector diagrams 1.1.2.1 and 1.3.2.1, all three rotation vectors, that is, ωg, ωp, and ωpl, are along a common straight line, Pln. Once the line of action of the vector ng intersects the line of action of the velocity vector, ωpl, the arm of tangential force in the gear pair becomes zero. This means that no torque can be transmitted by a gear pair of these particular kinds of gearing. Gear coupling is not a kind of gearing (no contact ratio can be defined). This discrepancy needs to be thoroughly investigated.
In reality, a gear axis and its mating pinion axis always are slightly misaligned. Under such a scenario no discrepancy is observed, and gear pairs can be designed in accordance to the vector diagrams 1.1.2.1 and 1.3.2.1.
- 5.
Jean-Gaston Darboux (August 14, 1842–February 23, 1917), a French mathematician.
- 6.
Remember that the algebraic values of the radii of principal curvatures, \( {R}_{1.g}^a \) and \( {R}_{2.g}^a \), relate to each other as \( {R}_{2.g}^a>{R}_{1.g}^a \). In the case of umbilical points, all radii of normal curvature are equal. Therefore, the principal directions, \( {\mathbf{t}}_{1.g}^a \) and \( {\mathbf{t}}_{2.g}^a \) (and, consequently, the principal radii of curvature, \( {R}_{1.g}^a \) and \( {R}_{2.g}^a \)), are not identified for umbilical points on gear generic surface.
- 7.
Jean Baptiste Marie Charles de la Place Meusnier (June 19, 1754–June 17, 1793), a French mathematician.
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Radzevich, S.P. (2022). Kinematic Foundations of Scientific Classification of Gearing. In: Radzevich, S.P. (eds) Recent Advances in Gearing. Springer, Cham. https://doi.org/10.1007/978-3-030-64638-7_1
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