Equations of motion of a quasisteady flight vehicle utilizing restrained static aeroelastic characteristics

Abstract

Mean axes provide the usual reference in maneuvering and dynamic response analyses of flexible vehicles. Attached or structural axes have also been used because the flexibility characteristics of the structure are only determined for a restrained structure. If the structural axes are employed, a relationship is required for the orientations of the structural axes relative to the mean axes, since the equations of motion determine the orientations of the mean axes. If this relationship is not considered, as it has not been in a number of publications, the solution to the equations of motion of the structural axes is not invariant with the choice of support configuration used in the calculation of the structural flexibility influence coefficients and is, therefore, incorrect. This relationship and the correct equations of motion for the structural axes are presented, and the correctness of the formulation is demonstrated numerically in studies of longitudinal maneuvering of an example forward-swept-wing airplane at constant forward velocity. Although the paper assumes constant airspeed in the basic developments, it concludes with discussions of aeroelastic speed derivatives and aeroelastic effects on drag. The paper assumes quasisteady equilibrium of the structure with its applied loads throughout. Nomenclature A= aeroelastic deflection amplification factor a = flexibility, a F is free-body flexibility a s = amplitude of motion of support reference point #0 = airfoil two-dimensional lift curve slope B = aeroelastic load amplification factor b = reference span C = generalized aerodynamic force coefficient: C/ is inertial value, C q is angular rate value, C a is incidence and control surface value, C^ is incidence rate value, C 0 is initial value C e = experimental control point force coefficient C h = complex oscillatory aerodynamic influence coefficient C hs = static aerodynamic influence coefficient C m = aerodynamic pitching moment coefficient C z = aerodynamic normal force coefficient c =mean aerodynamic chord c f = local lift curve slope EI X

Procedures and Models for Aeroservoelastic Analysis and Design

“…The basic assumption implies that the elastic deformations are orthogonal to the rigid-body modes. While the classic discrete-coordinate approach [33] needed a series of operations to implement this condition, the basic modal approach got it automatically due to the orthogonality of free-free elastic modes to the rigid-body ones [8]. Mathematically, the basic assumption implies that the effects of the elastic velocities and accelerations, f _ x x e g and f € x x e g, on the static aeroelastic equilibrium are negligible.…”

Aeroservoelastic Design Optimization of a Flexible Wing

“…Some of the work that addressed the active aeroelastic wing design adopted quasi-steady flight models, where structural deflections are assumed to have reached their steady-state conditions [17,15,18]. Mathematical formulations of flexible aircraft flight dynamics are mainly based on one of two approaches: the mean-axes method, or the quasi-coordinate method.…”

Consistent Structural Linearization in Flexible Aircraft Dynamics with Large Rigid-Body Motion

“…The aerodynamic loads are obtained as the generalized forces corresponding to the mode shapes of the structure, including their rigid-body component. 5 The resulting equations of motion (EoM), however, neglect some gyroscopic coupling terms (which may have an important effect on both the flight dynamics and the maneuver loads) and are therefore only valid for perturbations about steady flight of relatively stiff vehicles. A more general solution has been proposed using quasi-coordinates, [6][7][8] which directly solve the fully-coupled EoM of a flexible body (in a body-attached reference frame) under the assumption of small structural deformations.…”