Page 469 - Cam Design Handbook

THB14 9/19/03 7:58 PM Page 457

SPECIAL CAM MECHANISMS 457

It may be observed that engaging gear teeth do not have rolling action except where
the point of contact is on the line of centers.

14.3 ROLLING BODIES OF BASIC CONTOURS

Pure rolling action between bodies occurs with basic contours (special cams) such as log-
arithmic spirals, ellipses, parabolas, and hyperbolas (see App. A). Appendix Aalso includes
the involute and Archimedes spiral curves. Note the Archimedes spiral curve is the straight
line curve in Chap. 2. The proposition for pure rolling action of curves is
• Centers of oscillation should be located at the foci of the curves.
• Lengths of the arc of action of the contacting bodies must be equal.
• Point of contact must be on the line of centers.
Figure 14.4 shows these rolling bodies in contact at point Q, and having foci F 1 and
F 2. The center oscillation may be at either of the foci, if two are available for each curve.
We see that the ellipse is the only curve that is closed and therefore the only one that will
give continuous action. All other curves in contact will have limited action, i.e., oscilla-
tion. However, sectors of logarithmic spirals may be combined to give continuous rolling
action. These bodies are called lobe wheels.
Logarithmic spirals of equal obliquity, Figs. 14.4a and 14.4b, are shown in contact at
point Q, pivoting around their foci. In Fig. 14.4a we see the bodies rotating in opposite
directions, whereas in Fig. 14.4b rotation is in the same direction. The latter is obviously
a more compact mechanism.
A logarithmic spiral and translating straight-sided follower is shown in Fig. 14.4c. It
can be shown that with pure rolling the contacting curve of the straight-sided follower is
a logarithmic spiral oscillating about its focus. This is so because of the inherent quality
of the logarithmic spiral, i.e., a tangent at all points makes a constant pressure angle with
a radial line.
A pair of equal ellipses, Fig. 14.4d, has pure rolling action which may occur for com-
plete rotation of both bodies. For oscillation, sectors of ellipses are feasible.
A pair of equal parabolas is shown in Fig. 14.4e. One parabola is oscillating about its
axis. The other parabolic body is translating, having the same shape as the first. The point
of contact Q is on a line through the center of oscillation perpendicular to the axis of the
translating body.
A pair of equal hyperbolas, Fig 14.4f, may be utilized for pure rolling action if
they are properly located relative to each other. Note the location of the foci with
respect to the hyperbolas’ curves. Hyperbolas are an excellent choice where space is at a
premium.
One of the shortcomings of logarithmic spirals is that complete rotaion is not possible.
Figure 14.4g shows a lobe wheel made up of sectors of logarithmic spirals or ellipses
which will roll with continuous action. Although a trilobes area is shown, any number of
lobes may be employed.
EXAMPLE We are given a toe-and-wiper cam mechanism having a straight-sided trans-
lating follower at an acute angle of 20° (Fig. 15.5). By mathematical and graphical means,
plot the contour of the driver having pure rolling action. Note: the contour is a logarith-
mic spiral, since the tangent to the curve is at a constant angle.
Solution In App. A, we have the equation for the logarithmic contour