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8.3 Velocity Ratio
    The velocity ratio, i, can be derived from the ratio of several parameters:
     i = z1 = d1 = sind1                         (8-1)
          z2    d2    sind2
   where: d = pitch angle (see Figure 8-5)
8.4 Forms Of Bevel Teeth *
   In the simplest design, the tooth elements are straight radial. converging at the cone apex. However, it is possible to have the teeth curve along a spiral as they converge on the cone apex. resulting in greater tooth overlap, analogous to the overlapping action of helical teeth. The result is a spiral bevel tooth. In addition, there are other possible variations. One is the zerol bevel, which is a curved tooth having elements that start and end on the same radial line.
   Straight bevel gears come in two variations depending upon the fabrication equipment. All current Gleason straight bevel generators are of the Coniflex form which gives an almost imperceptible convexity to the tooth surfaces. Older machines produce true straight elements. See Figure 8-6a.
   Straight bevel gears are the simplest and most widely used type of bevel gears for the transmission of power and/or motion between intersecting shafts. Straight bevel gears are recommended:
 1. When speeds are less than 300 meters/mm (1000 feet/ min - at higher speeds, straight bevel gears may be noisy.
 2. When loads are light, or for high static loads when surface wear is not a critical factor.
 3. When space, gear weight, and mountings are a premium. This includes planetary gear sets, where space does not permit the inclusion of rolling-element bearings.

Other forms of bevel gearing include the following:
     ¥ Coniflex gears (Figure 8-6b) are produced by current Gleason straight bevel gear generating machines that crown the sides of the teeth in their lengthwise direction. The teeth, therefore, tolerate small amounts of misalignment in the assembly of the gears and some displacement of the gears under load without concentrating the tooth contact at the ends of the teeth. Thus, for the operating conditions Coniflex gears are capable of transmitting larger loads than the predecessor Gleason straight bevel gears.
     ¥ Spiral bevels (Figure 8-6c) have curved oblique teeth which contact each other gradually and smoothly from one end to the other. Imagine cutting a straight bevel into an infinite number of
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The material in this section has been reprinted with the permission of McGraw Hill Book Co., Inc., New York, N.Y. from "Design of Bevel Gears by W. Coleman, Gear Design and Applications, N. Chironis, Editor, McGraw Hill, New York, N.Y. 1967, p. 57.

 short face width sections, angularly displace one relative to the other, and one has a spiral bevel gear. Well-designed spiral bevels have two or more teeth in contact at all times. The overlapping tooth action transmits motion more smoothly and quietly than with straight bevel gears.
   ¥ Zerol bevels (Figure 8-6d) have curved teeth similar to those of the spiral bevels, but with zero spiral angle at the middle of the face width; and they have little end thrust.
   Both spiral and Zerol gears can be cut on the same machines with the same circular face-mill cutters or ground on the same grinding machines. Both are produced with localized tooth contact which can be controlled for length, width, and shape.
   Functionally, however. Zerol bevels are similar to the straight bevels and thus carry the same ratings. In fact, Zerols can be used in the place of straight bevels without mounting changes.
   Zerol bevels are widely employed in the aircraft industry, where ground-tooth precision gears are generally required. Most hypoid cutting machines can cut spiral bevel, Zerol or hypoid gears.

8.5 Bevel Gear Calculations

   Let z1 and z2 be pinion and gear tooth numbers; shaft angle S and pitch cone angles d1 and d2 then:

     (8-2)

Generally, shaft angle S = 90º is most used. Other angles (Figure 8-7) are sometimes used. Then, it is called "bevel gear in nonright angle drive". The 90º case is called "bevel gear in right angle drive".
   When d1 = 90º,
Equation (8-2) becomes:

     (8-3)

   Miter gears are bevel gears with S = 90º and z1 = z2. Their speed ratio z1 / z2 = 1. They only change the direction of the shaft, but do not change the speed.
   Figure 8-8 depicts the meshing of bevel gears. The meshing must be considered in pairs. It is because the pitch cone angles d1 and d2 are restricted by the gear ratio z1 / z2 In the facial view, which is normal to the contact line of pitch cones, the meshing of bevel gears appears to be similar to the meshing of spur gears.

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