Page 105 - Cam Design Handbook

THB4 8/15/03 1:01 PM Page 93

POLYNOMIAL AND FOURIER SERIES CAM CURVES 93

TABLE 4.1 Zero Coefficient Terms

Powers in Zero derivatives Powers in
general polynomials (at 0 = 0) remaining terms
0 and 1 None 1
0 to 3 y 2, 3
0 to 5 y¢, y≤ 3, 4, 5
0 to 7 y¢, y≤, y 4, 5, 6, 7

These eight conditions establish the general polynomial
q
y = C + C + C q 2 + C q 3 + C q 4 + C q 5 + C q 6 + C q .
7
0 1 2 3 4 5 6 7
In dealing with high-degree polynomials, simplifications can be made if we realize that
dy
n
when q = 0, = 0 . Subsequently, the coefficient for that derivative is C n = 0. This is
dq n
summarized in Table 4.1 for polynomials thus far derived for the DRD event.
For the 4-5-6-7 polynomial, C 0 = C 1 = C 2 = C 3 = 0. The remaining conditions at q = 1
give the following equations:
C + C + C + C = 1
4 5 6 7
4C + 5C + 6C + 7C = 0
4 5 6 7
+
12C + 2030C + 42C = 0
C
4 5 6 7
24C + 60120C + 210C = 0.
+
C
4 5 6 7
Solving these simultaneously for the 4-5-6-7 polynomial yields
q
-
6
y = 35q 4 -84q 5 + 7020q 7
q
-
3
y ¢ =140420q 4 + 420q 5 -140q 6
-
q
2
y ¢¢ = 4201680q 3 + 2100q 4 -840q 5
4
y ¢¢¢ = 840q - 5040q 2 + 8400q 3 - 4200q . (4.5)
Figure 4.3 shows a plot of these curves.
If we compare this acceleration curve with the lower order 3-4-5 polynomial, we see
larger maximum acceleration and larger maximum jerk values. This indicates a possible
inferiority to the 3-4-5 polynomial. A zero value of jerk is not required since manufactur-
ing limits generally cannot meet this control. Next, let’s compare the physical significance
of the 4-5-6-7 polynomial. Table 4.2 shows the dynamic comparison with the 3-4-5 poly-
nomial, basic simple harmonic, and cycloidal curves.
The physical meaning of this controlled acceleration is best depicted by large-scale
plotting of the curves. Figure 4.4 compares these curves with an expanded view of the end
points. The data is based on 2 percent of follower rise and 14 percent of cam rotation.
Further control of fourth and higher derivatives raises the maximum acceleration and also
the time consumed for the initial and final displacement of the cam, and manufacturing
limits are often demanded.