A systematic development for the theory of envelopes with proof is presented. It offers a simple and general procedure to determine the planar and spatial cam profiles. Based on the theory, a planar or spatial cam profile is defined as the envelope of its follower surfaces represented in a parametric form in different relative positions of the cam and the follower. The procedure is illustrated by determining the 2D and 3D cam profiles with analytical formulations. To demonstrate its accuracy and effectiveness, the results are then compared to those obtained by an earlier approach using the screw theory.

A systematic development for the theory of envelopes with proof is presented. It offers a simple and general procedure to determine the planar and spatial cam profiles. Based on the theory, a planar or spatial cam profile is defined as the envelope of its follower surfaces represented in parametric form in different relative positions of the cam and the follower. The procedure is illustrated by determining the two-dimensional and three-dimensional cam profiles with analytical expressions. To show its accuracy and effectiveness, the results are then compared to those obtained by an earlier approach using the screw theory.

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A simple, integrated procedure for the profile determination and kinematic analysis of cylindrical cams with oscillating conical roller-followers is presented. Based on the theory of envelopes for a one-parameter family of surfaces, the new method can be easily utilized to determine cylindrical cam profiles once the appropriate follower motion curves are given. In addition, the kinematic characteristics such as the contact lines between the cam and the roller surfaces, the pressure angles, and the principal curvatures of the cam surface are analyzed in the design process. To illustrate the accuracy and effectiveness of the proposed design tool, the analytical expressions derived in this study are compared to those obtained by using the principle of contact point. Numerical examples are also given to show the application of this integrated approach.