Page 102 - Cam Design Handbook

THB4 8/15/03 1:01 PM Page 90

90 CAM DESIGN HANDBOOK

Displacement, y, dimensionless

dy
Velocity, ¢ = , dimensionless
y
dq
dy ¢
Acceleration, ¢¢ = , dimensionless
y
dq
dy ¢¢
y
Jerk, ¢¢¢ = , dimensionless
dq
q = cam angle rotation, dimensionless
C i , i = 0, 1, 2,... n constant coefficients
Note that C i are chosen so that the displacement, y, and its derivatives satisfy the bound-
ary conditions of the motion. Eq. (4.1) is utilized by establishing as many boundary con-
ditions as necessary to define the mechanism motion. Polynomials can be employed to
produce acceptable cam profiles, especially at high speeds. The number of terms to Eq.
(4.1) should be properly limited to the desired design. Too many terms may add compli-
cations to the machine performance. Furthermore, the higher the order of terms, the slower
will be the initial and final displacements and the more accurately the cam profile must
be machined at these end points.
h
The simplest polynomial is the constant velocity polynomial, y = q , Eq. (2.14) of
b
Chap. 2, where the normalized displacement is
y =q (4.2)
This primitive cam has control only at the ends,
q = 0, y = 0

q =1, y =1.

4.2 THE 2-3 POLYNOMIAL CAM CURVE

For the cubic 2-3 polynomial follower motion, the boundary conditions are:

when q = 0, y = 0, y ¢ = 0
q =1, y =1, y ¢ = 0.
The cubic polynomial is employed
q
y = C + C + C q 2 + C q 3
0 1 2 3
The first derivative is the velocity
2
y ¢ = C + C q + C q 2
3
1 2 3
Subsituting the four boundary conditions into the polynomial equation and solving these
equations, we obtain the coefficients: