Optimal speed of response of a roller follower cam driven system

This dissertation analyzes the problem of determination of the maximal speed of response of a roller follower plate cam driven system. The solution to the problem of the lift of a roller from lower to upper dwelling conditions under the dual constraints of (1) a maximal allowed stress, and (2) no undercutting of the cam, was obtained using the analogy of the "Bang Bang Problem"; this problem has a very simple analytical solution, but its implications can be very usefully applied to the problem of cam lift. After applying this analogy in the present study, a technique of nonlinear programming was used to deal with the process of minimizing the angle of rotation of the cam under the constraint of a maximum allowable stress.

Using the stress constraint at its limiting value (i.e., at the border of the domain) an initial value problem of a second order non-linear differential equation is defined, which can be solved numerically. The study of the singular solution of this problem and the study of this ODE in the phase plane provide important information about the existence of restrictions in the phase plane as well as its singular points. The maximum stress constraint is found to provide a maximum rate of deceleration for a low angular velocity problem. Also for low angular velocity it was necessary to add an additional constraint of a maximum acceleration rate, because the dual constraint did not bound the acceleration.

In the region of high angular velocity it was found that the maximal stress constraint provides a maximum acceleration rate but a maximum deceleration rate is obtained using a low stress.

The geometrical constraint of no undercutting determines the area of the phase plane for a feasible solution.