Machinery's Handbook, 31st Edition
2368
CAMS AND CAM DESIGN
y
P( x 0 , y 0 )
x
O
Q
2 x 0
2 x 0
Fig. 8. The Tangent at P Bisects OQ When Curve is a Parabola
Example: A cam follower is to rise 1 ∕ velocity, over an angle of 50 degrees; and then 1 ∕ In Fig. 9 the three rise distances are laid out, y 1 = 1 ∕
4 in. with constant acceleration; 1 1 ∕
4 in. with constant
2 in. with constant deceleration.
4 in., y 2 = 1 2 in., and horizontals drawn. Next, an arbitrary horizontal distance φ 2 proportional to 50 degrees is measured off and points A and B are located. The line AB is extended to M 1 and M 2 . By remembering that a tangent to a parabola, Fig. 8, will cut the abscissa axis at point ( X 0 /2, 0) where X 0 is the abscissa of the point of tangency, the two values φ 1 = 20 ° and φ 3 = 40 ° will be found. Analytically, 1 ∕ 4 in., y 3 = 1 ∕
50 2 1 50 2 1 φ φ
y y
M E 1
. . 125 025 125 050 . .
1
2 1
20
1 ∴ φ
°
=
=
=
°
φ
2
y y
FM
3
2 3
2
40
2 = In Fig. 9, the portions of the parabola have been drawn in; the details of this operation are as follows: Assume that accuracy to the nearest thousandth of one inch is desired, and it is decided to plot values for every 5 degrees of cam rotation. The formula for the acceleration portion of the parabolic curve is: (5) 3 ∴ φ ° ° φ = = Two different parabolas are involved in this example; one for accelerating the follower during a cam rotation of 20 degrees, the other for decelerating it in 40 degrees, these two being tangent, to opposite ends of the same line AB . In Fig. 9 only the first half of a complete acceleration-deceleration parabolic curve is used to blend with the left end of the straight line AB . Therefore, in using the Formula (5) substitute 2 y 1 for h and 2 φ 1 for β so that y h y 2 2 2 2 2 2 1 2 1 2 β φ φ φ = = ^ ^ h^ h h y T h t h 2 2 2 2 2 β φ = = c m
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