ESDU 86026:2011

INTRODUCTION

This Item presents an introduction to the use of polynomial
equations to model the motion of a cam and its follower. The Item
forms part of a series (References 1 to 16) that offers guidance on
the design and analysis of cam and follower mechanisms. It is
assumed that the user is familiar with other Items in this series
and particular reference is made to ESDU 82006^[2],
Selection of DRD Cam Laws, and ESDU 83027^[6], ESDU
92014^[10] and ESDU 93002^[12] relating to the
blending of cam profiles.

This Item presents three approaches by which a polynomial can be
used to satisfy a set of cam and follower motion requirements.

For the first approach a minimum-order
polynomial is chosen, using successive power terms whose
number is sufficient to satisfy the prescribed motion requirements.
These will consist of the boundary conditions specifying the start
and end of a segment or part of a segment and, possibly, precision
point^† conditions within a segment. Such motion
conditions may include displacement, velocity, acceleration and
other higher derivative values. The order of the polynomial is one
less than the number of imposed conditions. The coefficients of the
terms of the polynomial are determined from a set of simultaneous
equations, equal in number to the number of coefficients. A
computer program incorporating a routine for the solution of this
set of linear simultaneous equations is described in Appendix B and
has been used throughout this Item, when applicable, to obtain
displacement, velocity, acceleration and jerk equations for the
examples in this Item. Computer input and output files for these
cases are also provided.

A process called exponent manipulation provides
the second approach. As before the number of terms is determined by
the boundary conditions, giving a unique polynomial of higher-order
than the minimum-order equation derived by the use of the first
method. The result is a non-successive power series, the
lower-order terms of the minimum-order polynomial being replaced by
an equal number of higher-order terms. The polynomial is still
unique but the follower lift in comparison with results from the
minimum-order polynomial is more gradual at the start of the rise
motion and more rapid at the motion end. This "skewness" of the
displacement and higher derivative curves increases with the order
of the polynomial and this characteristic may be used to satisfy
precision points.

Blending provides the third approach. Parts of
a motion segment satisfied by individual polynomials are coupled to
form a complete segment. Blending points are necessarily precision
points in the complete segment. Smooth transition from one
part-segment to another requires a common displacement and velocity
at each blending point and may also require common accelerations
and higher derivatives. A polynomial law part-segment may also be
blended with part-segments defined by other cam laws. When blending
is used with minimum-order polynomials for parts of a motion
segment, the computer program can be used to obtain polynomial
solutions. Computer input and output files for these examples are
provided.

Appendix A provides a list of the coefficients and polynomial
equations of the standard cam laws used in this Item.

Comprehensive examples are included in the Item to illustrate
and compare the approaches that are presented.

^† In this Item "precision point" is taken to mean a
point on the follower path and/or a specific follower velocity and
/or acceleration that is to be reached at a specific cam angle
within that motion segment.