T-19
2.2 The Law Of Gearing
A primary requirement of gears is the constancy of angular velocities or proportionality of
position transmission. Precision instruments require positioning fidelity. High-speed and/or high-power
gear trains also require transmission at constant angular velocities in order to avoid severe dynamic
problems.
Constant velocity (i.e., constant ratio) motion transmission is defined as "conjugate action" of the
gear tooth profiles. A geometric relationship can be derived (2, 12)* for the form of the tooth profiles to
provide conjugate action, which is summarized as the Law of Gearing as follows:
"A common normal to the tooth profiles at their point of contact must, in all positions of the
contacting teeth, pass through a fixed point on the line-of-centers called the pitch point."
Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate
curves.
2.3 The Involute Curve
There is 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. Modern gearing (except for clock gears)
is based on involute teeth. This is due to three major advantages of the involute curve:
Conjugate action is independent of changes in center distance.
The form of the basic rack tooth is straight-sided, and therefore is relatively simple and can
be accurately made; as a generating tool it imparts high accuracy to the cut gear tooth.
One cutter can generate all gear teeth 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 2-2. 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 2-3a. Thus, a
single point on the string simultaneously traces an involute on each base circle's 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. If a second winding/unwinding taut string is
wound around the base circles in the opposite direction, Figure 2-3b, oppositely curved involutes are
generated which can accommodate motion reversal. When the involute pairs are properly spaced, the
result is the involute gear tooth, Figure 2-3c.
Fig. 2-2 Generation of an
Fig. 2-3 Generation and
Involute by a Taut String
Action of Gear Teeth
* Numbers in parenthesis refer to references at end of text.
1.
2.
3.
Trace Point
Involute
Curve
Base Cylinder
Unwinding
Taut String
Involute
Generating
Point on
Taut String
Base Circle
Base
Circle
Taut String
(a) Left-Hand
Involutes
(b) Right-Hand
Involutes
(c) Complete Teeth Generated
by Two Crossed Generating
Taut Strings
