2.3 The Involute Curve
There are almost an infinite number of curves that can be developed to satisfy the law of gearing, and many different curve forms have been tried in the past. Modem gearing (except for clock gears) based on involute teeth. This is due to three major advantages of the involute curve:
1. Conjugate action is independent of changes in
center distance.
2. The form of the basic rack tooth is straight-sided, and therefore is relatively simple
and can be accurately made; as a generating tool ft imparts high accuracy to the cut
gear tooth.
3. One cutter can generate all gear tooth numbers of the same pitch.
The involute curve is most easily understood as the
trace of a point at the end of a taut string that unwinds from a cylinder. It is imagined
that a point on a string, which is pulled taut in a fixed direction, projects its trace
onto a plane that rotates with the base circle. See Figure 1.2. The base cylinder, or base
circle as referred to in gear literature, fully defines the form of the involute and in a
gear it is an inherent parameter, though invisible.
The development and action of
mating teeth can be visualized by imagining the taut string as being unwound from one base
circle and wound on to the other, as shown in Figure 1.3a Thus, a single point on the
string simultaneously traces an involute on each base circles rotating plane. This pair of
involutes is conjugate, since at all points of contact the common normal is the common
tangent which passes through a fixed point on the line-of-centers. It a second
winding/unwinding taut string is wound around the base circles in the opposite direction,
Figure 1 .3b, oppositely curved involutes are generted which can accommodate motion
reversal. When the involute pairs are properly spaced the result is the involute gear
tooth, Figure 1.3c.
2.4 Pitch Circles
Referring to Figure 1.4 the tangent to the two base
circles is the line of contact, or line-of-action in gear vernacular. Where this line
crosses the line-of-centers establishes the pitch point, P. This in turn sets the size of
the pitch circles, or as commonly called, the pitch diameters. The ratio of the pitch
diameters gives the velocity ratio:
Velocity ratio of gear 2 to gear 1 = Z = D1
(1)
D2
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