Page 101 - Cam Design Handbook

THB4 8/15/03 1:01 PM Page 89

CHAPTER 4

POLYNOMIAL AND FOURIER
SERIES CAM CURVES

Harold A. Rothbart, D.Eng.

4.1 INTRODUCTION 89 4.6 POLYNOMIAL CURVE MODES OF
4.2 THE 2-3 POLYNOMIAL CAM CURVE 90 CONTROL—DRRD CAM 96
4.3 THE 3-4-5 POLYNOMIAL CAM CURVE 91 4.7 POLYNOMIAL CURVE EXPONENT
4.4 THE 4-5-6-7 POLYNOMIAL CAM MANIPULATION 99
CURVE 92 4.8 FOURIER SERIES CURVES—
4.5 POLYNOMIAL CURVE GENERAL DRD CAM 103
DERIVATION—DRD CAM 94

SYMBOLS

C i , i = 0, 1, 2,... n constant coefficients
h = maximum follower rise normalized
y = follower displacement, dimensionless
y¢= follower velocity, dimensionless
y≤= follower acceleration, dimensionless
y = follower jerk, dimensionless
b = cam angle for rise, h, normalized
q = cam angle rotation normalized

4.1 INTRODUCTION

In Chap. 2, Basic Cam Curves, and Chapter 3, Modified Cam Curves, we presented
two approaches for the engineers’ selection of an acceptable curve design. Now, we will
include a third choice of cam curve, that is, the use of algebraic polynomials to specify
the follower motion. This approach has special versatility especially in the high-speed
automotive field with its DRRD action. Also included in this chapter is a selection of
important Fourier series curves that have been applied for high-speed cam system action.
The application of algebraic polynomials was developed by Dudley (1952) as an
element of “polydyne” cams, discussed in Chap. 12, in which the differential equations of
motion of the cam-follower system are solved using polynomial follower motion equa-
tions. Stoddart (1953) shows an application of these polynomial equations to cam action.
The polynomial equation is of the form

y = C + C + C q 2 + C q 3 + ... + C q n (4.1)
q
0 1 2 3 n
For convenience Eq. (4.1) is normalized such that the rise, h, and maximum cam angle,
b, will both be set equal to unity. Therefore, for the follower:

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