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578 Dynamics of Mechanical Systems
N
P
Pitch Circles Pitch Circles P "Pulleys" (base
circles)
"Belt" (line of
action)
Pressure Line
(line of action)
FIGURE 17.4.1 FIGURE 17.4.2
Rolling wheels, pitch point, and pressure line. Line of action viewed as a belt connecting pulleys
(base circles) of rolling wheels.
17.4 Preliminary and Fundamental Concepts: Involute Curve Geometry
The law of conjugate action places restrictions on the gear tooth profile shape. Of all tooth
profiles producing conjugate action, the most widely used has the form of an involute of
a circle. In this section, we consider the geometry of such involute curves.
Consider Figure 17.4.1 showing a sketch of rolling wheels (pitch circles). In view of the
law of conjugate action, let the inclined line N represent the normal of the gear tooth
surfaces at their points of contact. Thus, N passes through the pitch point P.
For smooth gear teeth, the forces transmitted between the teeth will be directed along
N. For this reason N is often called the pressure line or line of action of the gear tooth.
The line of action N may be thought of as describing a crossed belt connecting two
pulleys attached to the rolling wheels, as depicted in Figure 17.4.2. In Figure 17.4.3, we
see that the radii of these pulleys are related to the radii of the rolling wheels through the
inclination angle θ of the line of action (pressure line). The profile circles of the pulleys
are called base circles, and the pressure line inclination angle θ is called the pressure angle.
Specifically the base and pitch circle radii r and r are related by the expressions:
b
p
r = r cosθ and r = r cosθ (17.4.1)
Ab Ap Bb Bp
or
rr = cosθ (17.4.2)
b p
Viewed still another way, the line of action as shown in Figures 17.3.4 and 17.4.3 is a
crossed tangent to the base circles. Because the line of action is normal to the gear tooth
surfaces, we can construct the gear tooth profile as the locus of points on the gear wheel
occupied by a typical point on the line of action. Specifically, suppose we consider again
the line of action modeled as a cable or belt wrapped around the base circle pulley. Then,
instead of thinking of the belt as moving along a fixed line away from the rotating pulley,
let us consider the pulley to be fixed with the belt being unwrapped around the pulley.

