Lagrange Multipliers and Quasi-Steady Flight Mechanics

Abstract

A general method is presented for investigating optimum conditions of the quasi-steady mechanics of flight. For motion in a vertical plane the problem of extremizing an arbitrarily specified function of altitude, Mach Number, lift, path inclination, engine control parameter, and thrust inclination is considered for an aircraft which has to satisfy the equations of motion and 3 additional arbitrary constraints. A generalized solution is obtained in a determinantal form. The characteristic of this solution is that it unifies into a single equation the results of a large segment of the previous contributions to the quasi-steady mechanics of flight. Particular problems such as maximum speed, maximum range, maximum endurance, ceiling, steepest ascent, best rate of climb, flattest descent, etc., are thus all covered by the same determinantal equation. The optimum ratio (R) of induced drag to zero-lift drag is evaluated for arbitrary relationships between zero-lift drag coefficient, induced drag coefficient, thrust, specific fuel consumption, and Mach Number. Design problems are also investigated, such as those associated with the selection of the best wing surface or the best aspect ratio for a given performance to be optimized. Finally, the paper is completed with the study of the optimum flight conditions for curvilinear motion in a horizontal plane. Another general determinantal equation is derived, from which, as a particular case, the turning flight with maximum angle of bank, maximum angular velocity, or minimum radius of curvature is investigated.