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In Figure 5-2 the gear train has a difference of numbers of teeth of only 1; z1 = 30 and z2 = 31. This results in a reduction ratio of 1/30. Fig. 5-2 The Meshing of Internal Gear and          External Gear in which the Numbers of Teeth Difference is 1                     (z2 - z1 = 1)  SECTION 6 HELICAL GEARS The helical gear differs from the spur gear in that its teeth are twisted along a helical path in the axial direction. It resembles the spur gear in the plane of rotation, but in the axial direction it is as if there were a series of staggered spur gears. See Figure 6-1. This design brings forth a number of different features relative to the spur gear, two of the most important being as follows: Fig. 6-1  Helical Gear 1. Tooth strength is improved because of    the elongated helical wraparound tooth    base support. 2. Contact ratio is increased due to the    axial tooth overlap. Helical gears thus    tend to have greater load carrying    capacity than spur gears of the same    size. Spur gears, on the other hand,    have a somewhat higher efficiency. Helical gears are used in two forms: 1. Parallel shaft applications, which is    the largest usage. 2. Crossed-helicals (also called spiral or    screw gears) for connecting skew    shafts, usually at right angles. 6.1 Generation Of The Helical Tooth    The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear. However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in three dimensions to show changing axial features.    Referring to Figure 6-2, there is a base cylinder from which a Fig. 6-2  Generation of the Helical Tooth Profile

taut plane is unwrapped, analogous to the unwinding taut string of the spur gear in Figure 2-2. On the plane there is a straight line AB, which when wrapped on the base cylinder has a helical trace A0B0. As the taut plane is unwrapped, any point on the line AB can be visualized as tracing an involute from the base cylinder. Thus, there is an infinite series of involutes generated by line AB, all alike, but displaced in phase along a helix on the base cylinder.    Again, a concept analogous to the spur gear tooth development is to imagine the taut plane being wound from one base cylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugate helical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, a complete involute helicoid tooth is formed. 6.2 Fundamentals Of Helical Teeth    In the plane of rotation, the helical gear tooth is involute and all of the relationships governing spur gears apply to the helical. However, the axial twist of the teeth introduces a helix angle. Since the helix angle varies from the base of the tooth to the outside radius, the helix angle b is defined as the angle between the tangent to the helicoidal tooth at the intersection of the pitch cylinder and the tooth profile, and an element of the pitch cylinder. See Figure 6-3.    The direction of the helical twist is designated as either left or right. The direction is defined by the right-hand rule.    For helical gears, there are two related pitches - one in the plane of rotation and the other in a plane normal to the tooth. In addition, there is an axial pitch.    Referring to Figure 6-4, the two circular pitches are defined and related as follows:    pn = ptcosb = normal circular pitch    The normal circular pitch is less than the transverse radial pitch, pt in the plane of rotation; the ratio between the two being equal to the cosine of the helix angle.    Consistent with this, the normal module is less than the transverse (radial) module.    The axial pitch of a helical gear, px, is the distance between corresponding points of adjacent teeth measured parallel to the gear's axis - see Figure 6-5.           349