Optimum Path of an Airplane -- Minimum Time to Climb

Abstract

In this paper, the theory of variation is applied to the determination of the minimum time to climb for an airplane of constant weight. I t is evident that when sufficient boundary conditions are given—that is, for the beginning and the end point of the path—there is one and only one curve corresponding to the minimum time to climb. Because of the relative complexity of the resulting differential equation, it is convenient to generate such paths by a step-by-step procedure. The end point is then bracketed by proper choice of the higher derivatives at the origin. I t is evident that any decrease in weight during the time of the flight may also be properly bracketed by solving for the two limiting cases of initial and final weights of the airplane. I t will be observed, however, that the decrease in weight is actually too small to be of significance on the shape of the path. The solution is in a form which readily lends itself to routine machine calculation. Numerical calculations have been completed fcr a large number of cases pertaining to the Republic F-105 fighterbomber. Experimental flight tests are in the planning stage.