Page 192 - Cam Design Handbook
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          180                      CAM DESIGN HANDBOOK


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                         FIGURE 7.2.  Notation for pressure-angle determination.




          7.2.3 Cam Curvature
          The shape of a planar curve G depends on the rate of change of the direction of its tangent
          with respect to the arc length, a measure that is called the curvature of G, and designated
          hereafter as k. This variable plays an important role in the design of cam mechanisms, for
          it is directly related to the occurrence of cusps and the already mentioned effect known as
          undercutting. The reciprocal of the curvature k is the radius of curvature r, i.e.,
                                            1
                                         r =  .                          (7.9)
                                            k
          The radius of curvature at a point of the cam profile is the radius of a circle tangent at that
          point to the cam profile, on the concave side, as shown in Fig. 7.3. The curvature of that
          circle is the same as that of the cam profile. For our purposes, the radius of curvature is
          positive if the center K of the circle is located between the center of rotation O and the
          point of tangency Q; otherwise, the radius of curvature is negative. In the discussion that
          follows, formulas for calculating the curvature at any point of a cam profile are derived.
             Let us consider a planar curve, as shown in Fig. 7.4. At any point P on the curve, whose
          position vector is denoted by p, a unique orthonormal pair of vectors is defined, namely,
          the tangent and the normal vectors indicated in that figure as e t and e n. Let l measure the
          arc length as in Fig. 7.4.
             Unit vectors e t and e n, their derivatives with respect to l, and the curvature k are related
          by the Frenet-Serret formulas, Brand (1965), namely,
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