Time Optimal Transfer Function of a Mechanism

Abstract

A variety of optimization problems in theoretical mechanics are related to the problem of finding the quickest operating mechanism among all possible mechanisms. It can be formulated in the following way. Let us consider a mechanical system with one degree of freedom consisting of leading and lagging links with masses M1 and M2 respectively. Source of mechanical energy is attached to the leading link and its potential energy is known as a function of the position of the leading link. Positions of the leading - x and lagging links - y are related through the position function. Now the optimization problem can be formulated in the following way: find the position function such that the lagging link is transferred from initial position to the final position in the shortest time. We have obtained an analytic solution to this problem using variational calculus methods in the case then dissipative forces acting on the system can be neglected. We have shown that this problem is analogous to the classical brachistochtone problem. The qualitative feature of this result is that the optical mechanical system tends to accelerate the leading link first in order to maximize the power being extracted from the mechanical energy source. Analytical solution is applied for the optimization of the fast acting electric switch design. Leading link load and operating time reductions with the use of the optimal transfer function is demonstrated. This approach can be generalized for a variety of mechanisms where the operating time is critical.