Page 44 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 32

32 CAM DESIGN HANDBOOK

SIMPLE POLYNOMIAL CURVES The displacement equations of simple polynomial curves
are of the form

y = Cq 2 (2.12)
where n = any number
C = a constant
In this polynomial family, we have the following popular curves with integer powers:
straight line, n = 1; parabolic or constant acceleration, n = 2; cubic or constant jerk,
n = 3. High degree polynomial curves are shown in Chapter 3.
TRIGONOMETRIC CURVES The curves of trigonometric form are: simple harmonic
motion (SHM) or crank curve, which has a cosine acceleration curve; cycloidal, which
has a sine acceleration curve; double harmonic; and elliptical. (Appendix B presents tab-
ulated values of the simple harmonic motion curve and the cycloidal curve as a direct
approach to follower characteristics.)

OTHER CURVES In addition to these two families are the miscellaneous, little-used
curves: modified straight-line circular arc and the circular arc curves (see Chapter 14).
These are employed primarily as an improvement over the characteristics of the straight-
line curve and for special design requirements.
EXAMPLE A cam rotating at 120rpm has the positive acceleration part of its rise of 3/8
in. in 40 degrees of cam rotation. A simple polynomial curve having n = 2.4 is used. Find
the velocity and acceleration values at the end of 30 degrees of cam rotation.
2 p
7
.
Solution The angular velocity of cam = w = 120 ¥ = 12 5 rad sec. The total cam
60
angle = 40p/180 = 0.698 radian. Substituting into Eq. (2.12) gives the displacement
3 24 .
.
y = Cq n = = C(0698 )
8
Solving yields C = 0.888. Therefore, the basic equation is
.
y = 0888q 24 .
The velocity by differentiating Eq. (2.12) is
˙ y = wq n-1 ips
C n
Thus, the velocity after 30 degrees of rotation is
Ê p ˆ . 14
. ips
. )(2 4 30
.
˙ y = 0888 (12 57 . ) ¥ =108
Ë 180 ¯
The acceleration, differentiating again is
(
˙˙ y = Cw 2 n n - )q1 n-2
The acceleration after 30 degrees of rotation is
Ê
. )
. ) (2 4
˙˙ y = 0888 (12 57 2 . )(1 4 30 ¥ p ˆ . 04 = 364 in sec 2
.
Ë 180 ¯