# Gear Geometry

The tooth profile of spur and helical gears is based on a truncated involute curve (Fig. 18.19), which is defined as a curve that connects a locus of points that are generated at the end of a taut string when it is unwound from the tangent of a base circle (known as the evolute). Litvin and Fuentes [58] involute gearing, first proposed by Euler, has many advantages in its use, including: (1) ease of manu-facturability, (2) lack of transmission errors when the gear center distance is changed, and (3) the tooth-to-tooth force is applied along a constant line of action throughout the time of meshing. Through geometrical analysis, it can be found that the geometry of contacting involute gear teeth can be represented by circular discs of varying radii. For this reason, the contact theory of Hertz [33] has the ability to provide a reasonable solution for the elastic deformation, pressure distribution, and real contact area between mating teeth, although it must be noted that Hertzian analysis is based on the assumption of static, dry (nonlubricated), and friction-less conditionsâ€”none of which are experienced between moving gear teeth. A more appropriate analysis involves the combined study of lubricant flow and pressurization along with deflection of tooth surfaces, as will be discussed in Sect. 18.5.1.3.

Both spur and helical gearing geometries are common within wind turbine gearboxes. In the case of spur gears, the contact region is a straight line across the depth of the tooth, such that at any time either one or two teeth are in contact. Helical gears, however, are skewed in the axial direction, causing each tooth to appear as a segment of a helix. Because the teeth are angled with respect to the axis of rotation, the contact region is composed of a series of slanted lines, with several teeth in contact at a given time. The angled teeth engage more gradually than do spur gear teeth, causing them to run more smoothly and quietly [60]. In regard to loading, spur gears impose only radial loads on their bearings.

Fig. 18.20 Equivalent radius of curvature for contacting gear teeth, from [62]

Single helical gears, meanwhile, impose both thrust and radial loads on their bearings. Double helical gears, which are side-to-side combinations of helical gears of opposing axial skewness, develop equal and opposite thrust reactions which serve to cancel out the thrust load [61].