Trim Equations of Motion for Aircraft Design: Turning Flight, Pull-Up and Push-Over

Abstract

NOMENCLATURE The development of performance-optimal stability and control design solutions for advanced conventional and unconventional aircraft configuration shapes pose a particular challenge during the conceptual design phase. This design challenge can be attributed to a lack of design methods available, chronic data shortage when dealing with novel configuration shapes, and permanent time pressure during the early conceptual design segment. The situation is particularly apt when evaluating symmetric or asymmetric aircraft shapes in asymmetric flight conditions. n a v = normal acceleration b = span c = mean aerodynamic chord (m.a.c.) D C = drag coefficient (aircraft) 0 D C = drag coefficient (aircraft) for zero angle-of-attack q D C = variation of aircraft drag coefficient with pitch rate r D C = variation of aircraft drag coefficient with yaw rate α D C = variation of aircraft drag coefficient with angle of attack l C = rolling moment coefficient (aircraft) When evaluating the stability and control characteristics of a flight vehicle, all design decision-making needs to be based on a trimmed flight vehicle, ideally in six degrees-of-freedom. The task of trimming the vehicle in symmetric and asymmetric flight conditions represents, however, a non-trivial task during the conceptual design phase. This paper presents the development and discusses the application of the trim equations of motion (steady state equations of motion) for turning flight, pull-up and push-over flight applicable to conventional and unconventional aircraft shapes of symmetric and asymmetric layout. The fully developed six-degree-of-freedom formulation aims to trim those flight conditions key to sizing the vehicle’s control effectors during the conceptual design stage. The asymmetric-oblique-wing aircraft has been selected as the development ‘benchmark vehicle’, because of its unequaled development potential, the inherent inclusion of the range of symmetric aircraft types, and the fact that the majority of critical flight conditions for the design of controls are asymmetric flight conditions. Overall, the primary research aim is the development of a generic analytical framework with the capability to trim the aircraft in six degrees-of-freedom for the assessment of control power required/available. 0 l C = rolling moment coefficient (aircraft) for zero angleof-attack q l C = variation of aircraft rolling moment coefficient with pitch rate r l C = variation of aircraft rolling moment coefficient with yaw rate β l C = variation of aircraft rolling moment coefficient with angle of sideslip L C = lift coefficient (aircraft) 0 L C = lift coefficient (aircraft) for zero angle-of-attack q L C = variation of aircraft lift coefficient with pitch rate r L C = variation of aircraft lift coefficient with yaw rate α L C = variation of aircraft lift coefficient with angle of attack m C = pitching moment coefficient (aircraft) 0 m C = pitching moment coefficient (aircraft) for zero angleof-attack q m C = variation of aircraft pitching moment coefficient with pitch rate r m C = variation of aircraft pitching moment coefficient with yaw rate α m C = variation of aircraft pitching moment coefficient with angle of attack 1 AIAA Atmospheric Flight Mechanics Conference and Exhibit 11-14 August 2003, Austin, Texas AIAA 2003-5693 Copyright © 2003 by Copyright 2003 by Bernd Chudoba. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Z Y X , , = components of resultant force n C = yawing moment coefficient (aircraft) α , β = angle of attack, angle of sideslip 0 n C = yawing moment coefficient (aircraft) for zero angleof-attack δ = control effector deflection angle γ = flight path angle q n C = variation of aircraft yawing moment coefficient with pitch rate ∆ = increment (perturbation) of a parameter; non-zero reference value r n C = variation of aircraft yawing moment coefficient with yaw rate Γ = dihedral angle N M X τ τ τ , , = corrections for propulsive installation β n C = variation of aircraft yawing moment coefficient with angle of sideslip ψ θ φ , , = Euler angles v ω = angular velocity vector Subscripts Y C = side force coefficient (aircraft) A = aerodynamic 0 Y C = side force coefficient (aircraft) for zero angle-ofattack B, b = body frame FB D = drag increment due to inoperative engine DiCE = Directional CE q Y C = variation of aircraft sideforce coefficient with pitch rate i = variation with CE incidence angle i,j,k/l,m,n = variable indices r Y C = variation of aircraft sideforce coefficient with yaw rate LaCE = Lateral CE LoCE = Longitudinal CE s = stability axes β Y C = variation of aircraft side force coefficient with angle of sideslip T = thrust δ = variation with CE deflection Superscripts B F = body axes: FB (c.g., x, y, z) E = inertial system E F = frame of reference (inertial system) attached to the Earth: FE (OE, xE, yE, zE) INTRODUCTION z T y T x T F F F , , = scalar components of T v CONTEXT. A study has been conducted over a fouryear period to develop a generic stability and control (sc wing reference area T = thrust vector z y x T T T , , = scalar components of T v v DESIGN FOR STABILITY AND CONTROL. In AeroMech, all design decision-making is based on the evaluation of the trimmed aircraft in six degrees-offreedom (6-DOF), an exception being the take-off rotation maneuver. The trim (steady state) EOM are solved to determine control power available/required for a range of pre-defined critical flight conditions (DCFC). The dynamic (small perturbation) EOM are solved to estimate the gain constants required to restore w v u , , = scalar components of V in FB, perturbed values of U, V and W W V U , , = scalar velocity components of V r W r = weight (aircraft) V = aircraft velocity vector E E E z y x , , = coordinates of aircraft mass center relative to fixed axes (inertial system FE) T T T z y x , , = thrust line coordinates relative to the aircraft c.g.