Page 59 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 47

BASIC CURVES 47

2.13 ELLIPTICAL CURVE

This member of the trigonometric family is developed from projections of a semiellipse
(Fig. 2.13). The contour of the elliptical curve and its characteristics depend on the
assumed proportions of the major and minor axes. As the major axis increases the cam
becomes larger with the velocities of start and stop slower. In other words, the curve is
flatter at the top and the bottom as the ratio of the major axis to the minor axis is made
larger. If the major axis of the ellipse is zero in length, the contour in the displacement
diagram is a straight-line curve. A ratio of 2:4 gives a small cam for a given pressure
angle. Increasing the ratio further to 11:8 makes the curve approach a parabolic curve.
Further increase in the ratio is not practical, since velocity, acceleration, and cam size
become prohibitive. Equations for the ellipse are in Appendix A.
The layout consists of the projections of equal arcs on an ellipse of assumed major to
minor axis ratio (see Fig. 2.13):

1. Plot ordinate and abscissa axes with the total rise of the follower h equal to either the
major or the minor axis of the ellipse.
2. Describe two circles whose diameters are equal to major and minor axes.
3. Draw any radius AE cutting circles at E and F.
4. Draw lines through E parallel to one axis and through F parallel to the other. The inter-
section G of these two lines is a point on the ellipse. Continue in this manner and draw
the ellipse.
5. Divide both the abscissa of the displacement diagram and the arc of the ellipse into the
same number of equal parts, usually 4, 6, 8, 10, 12, or 16. Note, no relation exists
between the lengths of divisions on the displacement diagram and the divisions on the
ellipse except that they must be the same in number.
6. Project these intercepts to their respective cam angle division, and connect points to
yield the curve.
Next, we will establish the mathematical relationship for the elliptical curve of
different major and minor axes. From descriptive geometry, the basic equation for the
ellipse is

x e 2 + y e 2 = 1 (2.65)
a e 2 b e 2

r f
P

FIGURE 2.13. Elliptical curve construction.