Общий интеграл для одного класса нестационарных атмосферных летательных аппаратов и приложения для анализа траекторий

D.M Azimov,R.G Mukharlyamov

doi:10.22363/2312-9735-2017-25-2-161-169

Abstract

A complete analytical integration of the aircraft kinematic and dynamic equations of motion is presented. Different applications of defined integrals to trajectory analysis are considered. The dynamic equations are obtained under the assumptions, that acceleration due to aerodynamic lift, the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed, the aircraft body rate about the velocity axis is zero and the sideslip angle is zero. The general integral of these equations consists of six independent first integrals of motion and describes a class of non-steady flight trajectories in a maneuver plane. It will be shown that the dynamic equations can be derived and completely integrated in a closed-form for more general assumptions. The problem of computing thrust for a given trajectory has been considered. The trajectory is defined by constraint equation. Constraints stabilization equations, which have asymptotically stable trivial solution, are constructed. Explicitness can make the integrals applicable to modeling the trajectories of spacecraft, re-entry vehicles and missiles, and to the design of on-board targeting and guidance. An illustrative example is presented.

Highlights

This paper presents a complete analytical integration of the aircraft kinematic and dynamic equations obtained under the following assumptions: (a) acceleration due to aerodynamic lift, and the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed; (b) the aircraft body rate about the velocity axis is zero; (c) the sideslip angle is zero
It will be shown that the general integral of these equations consists of six independent first integrals which lead to the closed-form analytical solutions
Consider the F-frame formed by the triad of orthogonal unit vectors eF1, eF2, eF3 and with the origin O at the aircraft center of mass (COM): the unit vector eF1 is aligned with the velocity vector, eF3 forms the angle φ with lift and eF2 completes the right handed system

Summary

Introduction

This paper presents a complete analytical integration of the aircraft kinematic and dynamic equations obtained under the following assumptions: (a) acceleration due to aerodynamic lift, and the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed; (b) the aircraft body rate about the velocity axis is zero; (c) the sideslip angle is zero. The studies of the existing literature show that the aircraft equations can be integrated in a closed-form for some specific cases of quasi-steady and non-steady flights, including the cases of climb and cruise with constant altitude, velocity or lift acceleration, negligible flight path angle or small angle of attack [1, 2]. It will be shown that Eq (1) can be derived and completely integrated in a closed-form for a more general assumptions (a-c) with non-zero and variable angles of attack. Explicitness can make the integrals applicable to modeling the trajectories of spacecraft, re-entry vehicles and missiles, and to the design of on-board targeting and guidance [4]

Equations and Integrals for Non-steady Flight

Expressions for Thrust and Angle of Attack

Illustrative Example

Computation of Thrust with Stabilization of Constraints

Conclusions