The
fundamentals of gearing are illustrated through the spur gear tooth, both
because it is the simplest, and hence most comprehensible, and because it
is the form most widely used, particularly for instruments and control
systems.
The basic geometry and nomenclature of a spur gear mesh
is shown in Figure 2-1. The essential features of a gear mesh are:
1. Center
distance.
2. The pitch circle diameters (or pitch diameters).
3. Size of teeth (or module).
4. Number of teeth.
5. Pressure angle of the contacting involutes.
Details of these items along with their interdependence and definitions are covered in subsequent paragraphs.
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,
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* Numbers in parenthesis refer to references at end of text.
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:
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 it imparts high accuracy to the cut gear tooth.
3. 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 sting, 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 m 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, Figure2-3b, 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. 2-3 Generation and Action of Gear Teeth
