Optimum supersonic climb

Abstract

In this paper, a simple, non-sensitive, robust, and accurate computational procedure is presented and it does not require a prior knowledge of a reference trajectory for computing the optimum trajectory for minimum-time-toclimb to a prescribed final altitude of a supersonic aircraft. From the program, we fnst generate the possible range of initial Mach numbers, initial altitudes and climb times for a fastest climb to a specified final altitude and Mach number while the final flight path angle is left open. Then from this family of optimum trajectories we find the relation between the minimum climb time and the specified final flight path angle for a set of given initial and final altitudes and Mach numbers, and initial flight path angle. The program routinely converges to the solution of the fastest transfer between two sets of prescribed terminal condition on altitude, Mach number and flight path angle. This program is an efficient tool for analyzing climb performance of high speed aircraft equipped with a variety of propulsion systems. In particular, climb performances between jet engine aircraft and rocket engine aircraft are compared and illustrated. In atmospheric flight, a minimum-time-to-climb problem leads to solving a two-point boundary value problem for a set of nonlinear ordinary differential equations. Although this problem has received much attention in the ,past1 *, all the known methods for its solution exhibit some deficiencies, such as less in accuracy, poor convergence, lack of robustness in the numerical computation, and most notably excessive computational cost. Moreover, a good initial reference trajectory is usually necessary for computational convergence. In this study, we develop a non-sensitive, robust, and accurate computation scheme to solve the minimumtime-to-climb problem for a supersonic fighter aircraft from * Professor, Department of Aerospace Engineering. Member A I M t Presently, Associate Scientist, Chung Shan Institute of Science and Technology, P. 0. Box 90008-15-15, LungTan, Taiwan 32526, Republic of China a set of specified initial altitude, Mach number, and flight path angle to a set of specified final altitude, Mach number, and flight path angle or free final flight path angle. From the program, we first generate the possible range of initial Mach numbers, climb times and initial altitudes for a fastest climb to a specified final altitude and Mach number. Then from this family of optimum trajectories we find the relation between the minimum climb time and the specified final flight path angle for a set of given initial and final altitudes, Mach numbers, and initial flight path angle. The program routinely converges to the solution of the fastest transfer between two sets of prescribed terminal condition on altitude, Mach number and flight path angle. The solutions for various boundary conditions clearly display the influence of the initial Mach number and initial altitude on the optimum climb technique. In addition, the optimum control depends on the type of propulsion systems. In this respect, climb performances between jet engine aircraft and rocket engine aircraft are compared and illustrated.