An Extension of the Theory of the Optimum Burning Program for the Level Flight of a Rocket-Powered Aircraft

Abstract

The optimum burning program for the horizontal flight of a rocket-powered aircraft is analyzed and discussed. The results of previous theories are extended to cover the general problem of minimizing an arbitrary function of the final values of time, mass, distance, and velocity. By using the indirect methods of the calculus of variations, it is shown that the totality of extremal arcs is composed of zero thrust sub-arcs, sub-arcs to be flown with maximum engine output, and variable thrust sub-arcs. For the latter, a closed solution is obtained. Particular problems, such as maximum range, maximum endurance, minimum propellant consumption, and maximum velocity increase, are treated within the general frame of the present theory for various types of boundary conditions. Detailed attention is devoted to the thrust programing which maximizes the range for the case where the overall flying time is prescribed. A method for constructing extremal paths is supplied under the assumption of a parabolic drag polar having either constant coefficients or coefficients depending on the Mach Number. An important difficulty associated with the linear aspect of the present problem is constituted by the fact that the LegendreClebsch condition fails to yield any information on the minimum or maximum character of the Eulerian paths. Moreover, the Weierstrassian function is zero at all points of the variable-thrust sub-arc. This difficulty is overcome with a generalization of a previous method of the same author, based on the use of Green's theorem.