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THB7  8/15/03  1:58 PM  Page 177

                                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-
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