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THB7  8/15/03  1:58 PM  Page 179

                                GEOMETRY OF PLANAR CAM PROFILES            179

               Sometimes the cam contour is given in polar coordinates r and q in the form r = r(q)
            or, equivalently, in parametric form as r = r(y) and q = q(y), where y denotes the angular
            displacement of the cam plate. In this case, the position vector of a point on the cam profile
            is given in terms of two mutually orthogonal unit vectors, e r and e q, pointing in the direc-
            tions in which r and q increase, respectively. Thus,
                                         p = () e r                        (7.5)
                                            rq
               Differentiating both sides of Eq. (7.5) with respect to the arc length l yields
                                                   r
                                 dp         dq     d dq
                                         rq
                                    ∫  e = ()  e +      e r                (7.6)
                                                q
                                 dl   t     dl    d dl
                                                   q
            where the chain rule and the relation de r /dq = e q have been used. On dot-multiplying both
            sides of Eq. (7.5) by e q and e r, one obtains
                                          d q             d d
                                                           r q
                          e ◊  e ∫ sin g = ()  ,  e ◊ e = cos g =  .       (7.7)
                                      r q
                           q
                                                  t
                              t
                                               r
                                                           q l
                                          d l             d d
               Furthermore, combining Eq. (7.7) yields
                                         rq ()       d r
                                   tang =    , rq ∫    .                   (7.8)
                                                ¢()
                                          ¢()
                                         r q         d q
               These results are used in deriving the synthesis equations of the cam profile (Angeles
            and Lopez-Cajún, 1991).
            7.2.2 Pressure Angle
            The direction of the tangent to the cam profile, given by the angle g between the tangent
            and the radius vector r, is used to determine the pressure angle of the profile. This vari-
            able can be defined as the angle between the normal to the cam profile and the velocity
            of the point of the follower at which the force exerted by the cam is applied. Let us assume
            in this discussion that the velocity of the point of interest is parallel to line L, which is
            shown in Fig. 7.2, the normal to the cam profile being denoted by N.
               Since we assume L to be known, we know the angle between L and line OP, called m
            here. With this information, we should be able to calculate the pressure angle a in terms
            of g and m. In fact, from Fig. 7.2, if the difference D between m and a is subtracted from
            g, the angle between N and T is obtained, which is p/2, i.e.,
                                         g - D  =  p 2
            where
                                          D= ma
                                              -
            and hence,

                                           p
                                       a =  - ( g -  m)
                                           2
            which is the desired relation. Expressions for m vary from case to case, but the principle
            is the same. In particular, m depends on the angle of rotation of the cam, y, and the posi-
            tion of the follower, whether translational or angular.
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