This chapter discusses and analyzes whether hypoid gearsets, designed and manufactured today, are based on a precise theory or on approximation. As an opening statement, it can be affirmed that hypoid gearsets with a non-generated gear member and parallel depth teeth have a mathematically exact base geometry (refer to Figure 12). Also, hypoids with tapered depth teeth, which use helical motion during pinion generation, have this mathematically exact base geometry if the ring gear is non-generated. A generalized law of gearing was interpreted by Errichello and Stadtfeld (Ref. 1) and reads: “Conjugate gears transmit uniform rotary motion from one shaft to another by means of gear teeth. The common normal to the profiles of these teeth, at all points of contact, must pass through a fixed-point P in the common connecting line that intersects the two shaft axes and is normal to the pitch element.”

The topics of this chapter are structured accordingly in: 

The three fundamental laws of gearing 

Perfect conjugacy 

Real-world applications 

Transmission design 

Heat treatment, lapping and grinding

The Three Fundamental Laws of Gearing The first fundamental law of gearing ng • VR = 0, also implies |N1 × R1| = i • |N2 × R2|, where i is the constant transmission ratio. The three cases in Figure 2 visualize the problem of a non-constant ratio and are noncompliant with the first gearing law because of a ratio change from case to case. This problem led Leonard Euler to discover the involute tooth profile. A simplistic mathematical approach teaches that the effective radius vector R remains unchanged while the contacting point between two mating flanks moves from Rb2 to Rb1 as shown to the right in Figure 3 (movement along the line of action).

The line of action in parallel axes cylindrical gearing is straight, connecting the two base circles. If the surface normal vectors N1 and N2 are within the line of action, then the vector product |Ni ×Ri| remains constant during a complete mesh cycle. The consequent application of this principle leads to the construction of an involute, as shown to the right in Figure 3. The line that forms the tooth surface elements while traveling from position “a” to “f” along the line of engagement (line of action) is always perpendicular to the line of action. This principle implies that a tool, simply with straight cutting edges as shown in Figure 4, can be used to form the complex involute profile. Figure 4 also demonstrates the principle of profile shift, while maintaining the first fundamental law of gearing.

The second fundamental law of gearing, which was proposed in 2017 (Ref. 2), pln × Vm + ng = 0 (Ref. 1), is a redundant relationship to the first gearing law and it is limited to cylindrical gears with parallel axes and straight bevel gears without hypoid offset. A third fundamental gearing law is proposed in Ref. 2 in two different notations. The first notation covers only the special case of ratio = 1: The above notation applies only for cases with equal base pitch diameters between pinion and gear because the circular pitch and not the angular pitch has to be equal between pinion, gear, and the operating base pitch. The second notation: is consistent with the requirement of equal circular pitch, which makes this notation more relevant. However, it does not add additional substance to the first law of gearing. As a conclusion, it can be stated that the first law of gearing is sufficient and applies without restrictions to all kinds of gearing.