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Gear Cutting Tools for Machining Bevel Gears
Published in Stephen P. Radzevich, Gear Cutting Tools, 2017
Figure 13.3 illustrates the shape of straight tooth of a bevel gear. Major design parameters of the bevel gear are also indicated in the figure. In addition to the conventional design parameters, the addendum angle γa and the dedendum angle γd in the design of a bevel gear tooth are shown in Figure 13.3. Generally speaking, the tooth flank G of a straight bevel gear is shaped in the form of a conical surface. The apex of the cone surface coincides with the base cone apex Ag. In the design of perfect bevel gears the tooth profile is viewed as a spherical involute. The tooth profile serves as the directing line of the conical surface G. This means that the tooth flank can be interpreted as a locus of straight lines through a point of the tooth profile and the base cone apex Ag. Therefore, the straight bevel gear tooth flank G is a type of developable surfaces.
Gear Types and Nomenclature
Published in Stephen P. Radzevich, Dudley's Handbook of Practical Gear Design and Manufacture, 2021
Base cone, gear – The gear base cone is a cone with apex coincident with the gear apex, Ag, the axis of the base cone is aligned with the gear axis of rotation, Og, and the gear base cone is tangent to the plane of action, PA.
Perfect Crossed-Axes Gear Pairs: R-Gearing
Published in Stephen P. Radzevich, Theory of Gearing, 2018
It is natural to assume that the concave tooth flank (primarily of the gear, G) is located outward from the boundary N-cone, while the convex tooth flank (primarily of the pinion, P) is located within the interior of the boundary N-cone. However, as the apex, Apa, is not coincident either with the gear base cone apex, Ag, or with the pinion base cone apex, Ap, the boundary N-cone is not the only constraint in the geometry of the tooth flanks, G and P. The envelopes to consecutive positions of the boundary N-cone in their instant screw motion are used for this purpose instead. For the gear tooth flank, the constraint is generated when the boundary N-cone is rotated about the gear axis of rotation, Og. Similarly, for the pinion tooth flank, the constraint is generated when the boundary N-cone is rotated about the pinion axis of rotation, Op. Ultimately, the convex tooth flank of one member of the gear pair must be entirely located with the interior of the corresponding enveloping surface, while the concave tooth flank of another member of the gear pair must be entirely located outside the interior of the corresponding enveloping surface.
Procedural and relational understanding of pre-service mathematics teachers regarding spatial perception of angles in pyramids
Published in International Journal of Mathematical Education in Science and Technology, 2019
The three characteristics of relational thinking areGenerating insights (interpretations): For example, a formula of a pyramid volume shows that in pyramids with a common base (or bases with equal area), the longer the height, the greater the pyramid volume. A way of thinking which integrates analytical analysis with visualization plays an important role in the generation of insights, particularly in the field of geometry.Asking questions and presenting hypotheses: For example, after learning the theorem of congruent triangles by three sides, one can ask questions such as Can a congruence theorem by four sides apply also to quadrilaterals? or What are the congruence theorems of quadrilaterals [39]?Generalization through an inductive process: For example, in a square, the diagonals are bisected and perpendicular and form four isosceles triangles. One can see the square in this division as a net of a ‘degenerate’1 pyramid without a height. If one wants to pull upwards the diagonals’ intersection point and create a square pyramid, one has to decrease the angle in each of the congruent triangles. Thus, the value of the apex angle of a regular square pyramid should be less than 90°. In the same way, when dealing with a regular pentagon, the apex angle in each of the five triangles in the ‘degenerate’ case is 72° (360°:5). Hence, the apex angle of each of the lateral faces of a right and regular pentagonal pyramid is smaller than 72°. These conclusions lead to the generalization: The apex angle of the lateral face of a regular pyramid is smaller than the size of the angle between the equal edges in each of those triangles with a common vertex which comprise the regular polyhedron at the base of the pyramid.