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Session 1265

Breguet's Formulas for Aircraft Range & Endurance An Application of Integral Calculus

Colonel Kip P. Nygren, Major Robert R. Schulz United States Military Academy

Introduction

At the United States Military Academy, faculty attempt to expose cadets to highly integrated learning scenarios. In an effort to reinforce that the world is not compartmentalized similar to the academic environment, the Department of Mathematical Sciences conducts sessions known as Interactive Lively Applications Projects (ILAPs). In particular, the engineering curriculum provides an excellent base of support to highlight the applications of single and multivariable calculus.

This paper focuses on one such ILAP using the Breguet range and endurance equations as the foundation for some insight into the physical significance of integral calculus. In a recent semester, members of the Department of Civil & Mechanical Engineering sponsored an ILAP where cadets Figure 1. Typical US Air Force Airplane learned how integral calculus supports airplane design.

Aircraft Performance Parameters

Knowledge of the requisite aircraft velocity for maximum range and maximum endurance operations is essential for air crews to optimize flight performance. Engineers focus on maximum range and maximum endurance characteristics to optimize the aircraft design for a specific mission. This information is also pivotal in the marketing of the aircraft. The need to understand maximum range and maximum endurance characteristics of an aircraft cannot be understated. Specific examples of aircraft designed for these conditions include Burt Rutan’s Voyager (maximum range), Unmanned Aerial Vehicles (maximum endurance), U-2 reconnaissance aircraft (maximum range and altitude), and the P-3 Orion submarine hunter (maximum endurance).

At the most basic level, the range and endurance of an aircraft depend directly on the quantity of fuel available and the rate at which the fuel is consumed per distance traveled or per hour in the air. Given an unlimited amount of fuel, any aircraft could fly continuously until, of course, the aircraft experiences a failure. This is the concept of aerial refueling used by the U.S. Air Force. However, fuel availability for practical aircraft is limited. Therefore, the characteristics of the aircraft and the environment that affect the maximum range and endurance must be found. Such a situation involving several variables is a natural application of multivariable calculus. In addition, the total range and total endurance are functions of the aircraft weight, which is changing continuously as fuel is consumed.

1996 ASEE Annual Conference Proceedings

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Schulz, M. R. R., & Nygren, C. K. P. (1996, June), Breguet's Formulas For Aircraft Range & Endurance An Application Of Integral Calculus Paper presented at 1996 Annual Conference, Washington, District of Columbia. 10.18260/1-2--5901