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THB4 8/15/03 1:01 PM Page 99

POLYNOMIAL AND FOURIER SERIES CAM CURVES 99

4.7 POLYNOMIAL CURVE
EXPONENT MANIPULATION

Curve shape adjustment can be accomplished by changing the exponents of the polyno-
mial. This procedure at best is one of trial and error with some degree of predictability.
For example, the fifth-order polynomial can have the powers 3-4-5, 3-5-7, 3-6-9, etc. The
seventh-order polynomial can have powers of 4-5-6-7, 4-6-8-10, 4-8-12-16, 4-7-10-13,
and so forth. Figures 4.7 and 4.8 show the progressive changes of the polynomials. Note
that the higher the powers, the greater is the maximum acceleration that shifts toward the
start of the displacement curve.
Polynomial curves having exponents as high as 50 have been used experimentally for
automotive cams (see Thoren et al., 1952). Figure 4.9 shows some high-order DRRD
polynomial curves. The rise boundary conditions are
4
dy
when q = 0, y = 0, y ¢ = 0, y ¢¢ = 0, y ¢¢¢ = 0, = 0
d q 4
q =1, y = 0350, y ¢ = 0, y ¢¢ = 0, y ¢¢¢ = 0. (4.9)
.

The general polynomial equation is of the form
y = C q 2 + C q p + C q q + C q r + C q ,
w
q
p
0
w
r
where the exponents p, q, r, and w may have values such that
< < < £
8 £ pq r s 50.

Now let us indicate some polynomial curves that have been developed. The following are
four curves with excellent dynamic characteristics.

3-4-5

3-6-9
y"

Cam angle q

FIGURE 4.7. Acceleration of a family of fifth-order polynomial curves.