FLIGHT DYNAMICS OF THE FLEXIBLE AIRCRAFT INCLUDING UNSTEADY AERODYNAMIC EFFECTS

Grégori Pogorzelski(1,2), Roberto G. A. da Silva(1) and Pedro Paglione(1)

1Technological Institute of Aeronautics – ITA /IEA – S. J dos Campos – SP – Brazil2CENIC Eng. Ind. Com. Ltda – S. J dos Campos – SP – Brazil

Aeroelasticity DomainStructural elastic modes interaction with external forcesStructural stability analysesLinear aeroelastic tools used to predict frequency domain response

Flight Dynamics DomainSix degree of freedom nonlinear modelsAircraft response in the time domain when subjected to maneuversFlight stability and flight quality analyses

Need for a Coupled ModelStructural optimization payload increase and performance gainIncrease in aircraft flexibility and lowering of elastic frequenciesHigher level of interaction between flexible and rigid body modesClassical rigid body flight dynamics may become inaccurate

The modeling methodology and calculation procedure are organized in a straightforward manner in order to allow fast computations and parametric studies. The classical rigid body/quasi-steady aerodynamic model can also be simulated for comparison purposes. Some of the results that can be generated are:

•Calculation of trimmed flight conditions (straight and horizontal curve flight);•Stability analysis (rigid/elastic body and coupled conditions);•Calculation of V-g-f diagrams for parametric stability analysis;•Frequency response calculation associated to command inputs;•Time domain response for command inputs: rigid body state variables, aerodynamic lag terms and deformation history;•Evaluation of the rational function approximation of AIC coefficients.

A series of test cases and simulations were performed in order to test the cited model capabilities. The calculation procedure was applied to the high aspect ratio sailplane and two more flexible variants.

•Parametric studies indicated the reasonable number of elastic modes and aerodynamic lag terms to be incorporated;•The impact of different modeling approaches is evident: quasi-steady/unsteady aerodynamics and rigid/flexible body;•It was detected the need for a coupled model when the aircraft flexibility is increased, specially during maneuvers at high speeds.

AERODYNAMIC MODEL•Small perturbation unsteady subsonic strip theory for swept wings used as the aerodynamic representation basis;•The modified strip theory of Yates [1] is adapted in order to take into account both, the flexible and rigid body degrees o freedom contribution;•Flexible degrees of freedom are introduced by means of equivalent beam plunge and twist movement associated to each elastic mode shape;•Rigid body degrees of freedom (angle of attack and sideslip angle) are treated as equivalent swept, dihedral and twist angles at each spanwise station;•Forces and moments produced by each strip are summed up yielding the frequency domain Aerodynamic Influence Coefficient Matrix (AIC) for the entire aircraft;•The AIC is written as a Padé approximant in the Roger’s form allowing the subsequent representation of incremental aerodynamic forces in the time domain;•State variables and corresponding dynamics associated to aerodynamic lag phenomenon are produced;•Linearized force and moment expressions are written in terms of stability and control derivatives;•Calculated AIC terms can be updated in order to take into account aircraft’s known stability derivatives and drag polar parameters;

STRUCTURAL MODEL•A shell finite element model provides the necessary data for a modal analysis producing lifting surfaces deformed shapes associated to the normal elastic modes of interest;•Deformed and undeformed surfaces are interpolated from the data available at the nodal points by means of continuous surface spline functions (equation of a thin plate in bending);•A torsional moment at the wing tip is used, as load case, to calculate the elastic axis;•From the interpolation functions, deformed and undeformed sections can be obtained at each desired spanwise station;•Elastic axis coordinate is obtained by a least squares procedure in a way that leads to the best adjustment of the deformed section to the one undergoing a rigid body rotation;•Once the elastic axis is determined, a similar least squares scheme is adopted for calculating the beam equivalent displacement and rotation angle for each strip and elastic mode;

COUPLED EQUATIONS OF MOTION•Differential equations of motion are derived from Lagrangean principles according to the formulation of Waszak and Schmidt [2];•Basic assumptions are small perturbations and constant inertia tensor of a continuous elastic body;•The use of the mean axes reference frame allows the equations to be written in the inertially decoupled form;•State variables and corresponding dynamic equations associated to the aerodynamic lag effects are naturally incorporated in the formulation.

[1] Yates, E. C. J. (1958). Calculation of flutter characteristics for finite-span swept or unswept wings at subsonic and supersonic speeds by a modified strip analysis. Tech. Rep., Washington. NACA RM L57L10.[2] Waszak, M. R. and Schmidt, D. K. (1988). Flight dynamics of aeroelastic vehicles. Journal of Aircraft, 25(6), 563-571.[3] Pogorzelski, G., Zalmanovici, A., Silva, R. G. A., Paglione, P. Flight dynamics of the flexible aircraft including unsteady aerodynamic effects. In: International Forum on Aeroelasticity and Structural Dynamics - IFASD 2009, 2009, Seattle, WA, EUA. Proceedings of the International Forum on Aeroelasticity and Structural Dynamics - IFASD 2009, 2009.

•Aerodynamic model modifications in order to include gust response loads calculation;•Inclusion of unsteady panel method modules for aerodynamic calculation;•Possible consideration of inertial coupling terms, structural rotational degrees of freedom and variable inertia tensor;•Validation using flight test data;•The coupled dynamic model can be applied and tested in studies related to: aeroservoelasticity, gust-alleviation investigations and flight loads calculation;•The procedure is intended to evolve into a fast calculation tool to be used during aircraft early design phase.

The objective is to develop an integrated and general methodology for modeling the dynamic behavior of flexible aircrafts. The procedure is focused on the investigation of flexibility and subsonic unsteady aerodynamic effects on aircraft flight mechanics. Study cases are simulated using the data of a high aspect ratio sailplane made of composite material.

BACKGROUND

PURPOSE

MATERIALS AND METHODS

RESULTS

FUTURE DEVELOPMENTS

BIBLIOGRAPHY