【正文】 on. A point of a surface is called singular if it is singular for every parameterization of the surface7. A point that is singular in one parameterization of a surface may not be singular in other parameterizations. For a fixed value of ζ, equations 1 and 2 represent, in general, a curve on the surface which corresponds to this value of the parameter. If this is not a line of singular points, the curve slso lies on the envelope. The surface and the envelope are tangent to each other along this curve. Such curves are called characteristic lines of the family7. they can be used to find the contact lines between the surfaces of the cylindrical cam and the follower. THEORY OF ENVELOPES FOR DETERMINATION OF CYLINDRICALCAM PROFILES On the basis of the theory of envelopes, the profile of a cylindrical cam can be regarded as the envelope of the family of follower surfaces in relative positions between the cylindrical cam and the follower while the motion of cam proceeds. In such a condition, the input parameters of the cylindrical cam serve as the family parameters. Because the cylindrical or conical follower surface can be expressed in parametric form without difficulty, the theory of envelopes for a 1parameter of surfaces represented in parametric form (see equations 1 and 2) is used in determining the analytical equations of cylindricalcam profiles. As stated in the last section, a check for singular points on the follower surface is always needed. Figure 1a shows a cylindricalcam mechanism with a translating conical follower. The axis which the follower translates along is parallel to the axis of rotation of the cylindrical cam. a is the offset, that is, the normal distance between the longitudinal axis of the cam and that of the follower. R and L are the radius and the axial length of the cam, respectively. The rotation angle of the cylindrical cam is Ф2 about its axis. The distance traveled by the follower is s1 , which is a function of parameter Ф2 ,as follows: ? ?1 1 2SS?? (4) The displacement relationship (see equation 4 ) for the translating follower is assumed to be given. In figure 1b, the relative position of the follower when the follower moves is shown. The follower is in the form of a frustum of a cone. The semicone angle is α, and the smallest radius is r. δ1 is the height, and μ is the normal distance from the xz plane to the base of the cone. The fixed coordinate system Oxyz is located in such a way that the z axis is along the rotation axis of the cam, and the y axis is parallel to the longitudinal axis of the conical follower. the unit vectors of the x axis, y axis and z axis are i , j and k, respectively. By the use of the envelope technique to generate the cylindricalcam profile, the cam is assumed to be stationary. The follower rotates about the dam axis in the opposite direction. It is assumed that the follower rotates through an angle Ф2 about the axis. At the same time, the follower is transmitted a linear displacement s1 by the cam, as shown in Figure 1b. Consequently using the technique, if we introduce θ and δ as two parameters for the follower surface, the family of the follower surfaces can be described as ? ?? ? ? ?? ? ? ?? ?22 2 22 2 2,c os si n t a n c os c ossi n c os t a n c os si n1 t a n si nrra r ia r js r k? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ??? ? ? ? ?????? ? ? ? ? ?? ? ? ????? (5) where 0≤θ＜ 2pi And ф2 is the independent parameter of the cam motion. Referring to theory of envelopes for surfaces represented in parametric form (see equations 1and 2), we proceed with the solving process by finding ? ? ? ?? ? ? ?2222ta n c os c os sin ta nc os sin c os ta n sinrr rijk? ? ? ? ? ???? ? ? ? ???? ? ? ???????? ? ? ????? (6) There are no singular points on the family of surfaces, since(r +δtanα) ＞ 0 in actual applications. The profile equation satisfies equation 5 and the following equation: ? ?? ? ? ?? ?239。1se c t a nc os si n t a n si n c os c os si n0r r r rs r a? ? ?? ? ?? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ?? ? ?? ? ? ? ? ??????(7) Where 139。1 2dss d?? or ? ?2 2 212 t a n E E F GGF? ???? ? ? ???? ??? (8) Where ? ? ? ?39。1 c osta n sin c ossinEsFrGa?? ? ? ? ? ???? ? ? ? ? ?? Substituting equation 8 into equation 5, and eliminating θ, we obtain the profile equation of the cylindrical cam with a translating conical follower, and denote it as ? ?2,ccrr??? (9) As shown in equation 8,θ is a function of the selected followermotion program and the dimensional parameters. As a consequence, the cylindrical profile can be controlled by the chosen followermotion curves and the dimensional parameters. Two values of θ correspond to the two groove walls of the cylindrical cam. Now the profile of the cylindrical cam with a translating conical follower is derived by the new proposed method. As stated above, Dhande et and Chakraborty and Dhande2 have derived the profile equation of the same type of cam by the me