Abstract
Cam, a mechanism usually used to transform a rotary motion into the desired output motion, has been commonly used in the modern industry. In practice, various practical methods have been discussed to analyze and synthesize the cam mechanism. However, the mobility of the cam mechanism is seldom mathematically addressed. This article mathematically discusses the necessary and sufficient design conditions for four kinds of overconstrained cam mechanisms (the one with a translating flat-faced follower, a translating roller follower, an oscillating flat-faced follower, and an oscillating roller follower) to be mobile. For the first two overconstrained cam mechanisms, the relation between the design conditions and the mobility of the cam mechanism is derived by proving that the identical cam contour can be enveloped by the top and bottom follower faces. For the third overconstrained cam mechanism, the relation is derived by transforming the cam contour into a geometric layout associated with an orthoptic curve. For the fourth overconstrained cam mechanism, it is shown that the cam mechanism cannot be theoretically designed due to the variable length of the follower's arm, which does not obey the rigid body assumption. In conclusion, by means of these geometric methods, the first three kinds of cam mechanisms are proved to be mobile if and only if they satisfy the design conditions, and the last cam mechanism is proved to be theoretically infeasible.
