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GEOMETRY OF PLANAR CAM PROFILES 177
D= difference angle
D = incremental area
g = angle between tangent to cam and radius
G= cam surface or curve
k = cam surface curvature
l = arc length
m = angle between velocity vector and cam radius
m = ramp function from 0 to 1
r = radius to point Q on cam profile
r = mass density
s(y) = positive definite follower displacement function
q = angle to point Q on cam profile
j(y) = follower oscillation function
w = camshaft speed
7.1 INTRODUCTION
The geometric properties of the profile of disk cams is studied here. To produce accurate
m
values of both local and global properties of the contour, a suitable number of points {P i} 1
on the cam profile, with Cartesian coordinates (x i, y i), for i = 1,..., m, is required. Local
properties pertain to slope and curvature values throughout the contour; global properties
involve area or, correspondingly, volume, centroid location, and moments and products of
inertia of the cam.
Thus, global geometric properties require the calculation of integrals, which, broadly
speaking, involve numerically stable computations. That is, truncation errors induced by
the discretization of the contour are attenuated due to the filtering effect of integrations,
although care must be taken to produce results that are accurate enough. Various tech-
niques are currently available for the systematic evaluation of the integrals that arise in
this context, those yielding the highest accuracy for a fixed value of m being techniques
based on spline representations of the contour.
However, regarding the calculation of local properties, the truncation errors due to the
contour discretization are amplified by virtue of the differentiations involved. Various tech-
niques are available to cope with these problems; the techniques favored in this chapter
are those based on a spline representation of the contour.
Thus, a methodology for handling the discrete points via nonparametric and para-
metric cubic splines is described. In Sec. 7.2 various concepts of differential geometry are
outlined; these involve local properties such as slope, which is needed in the calculation
of the pressure angle, and cam curvature, which arises in avoiding cusps and undercut-
ting. In Sec. 7.3, the computation of the global properties of the cam is studied.
The computational issues around the calculation of the local and global geometric pro-
perties of planar cam plates are addressed in Sec. 7.4. An outline of relevant commercial
software is included in Sec. 7.5, and Sec. 7.6 closes with some case studies.
7.2 LOCAL PROPERTIES OF THE CAM PROFILE
The differential-geometry relationships pertaining to cam profiles are discussed in this
section. These relationships will be used later for determining the cam profile and some
related variables, such as the tangent orientation, required to evaluate the pressure angle,
and the curvature of the cam profile, required to verify the occurrence of cusps and under-