19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 AN APPROACH FOR CHARACTERIZING THE AERODYNAMICS/STRUCTURE COUPLING AT THE AIRCRAFT DESIGN STAGE IN THE PRESENCE OF STRUCTURAL

1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 AN APPROACH FOR CHARACTERIZING THE AERODYNAMICS/STRUCTURE COUPLING AT THE AIRCRAFT DESIGN STAGE IN THE PRESENCE OF STRUCTURAL NONLINEARITIES PACS: Tb Arévalo Lozano, Félix 1 1 EADS/CASA Structural Dynamics and Aeroelasticity Department; Paseo John Lennon s/n, Getafe, Madrid; felix.arevalo@casa.eads.net. Associate Professor. Departamento de Fundamentos Matemáticos. Universidad Politécnica de Madrid. felix.arevalo@upm.es ABSTRACT Aeroelasticity is the discipline of the aeronautical engineering that studies the vibration of the aircraft flexible structure as affected by the surrounding air. The aeroelastic behaviour of an aircraft depends basically of four major inputs: structure, inertia, aerodynamics, and flight control systems (aeroservoelasticity). The classical approach considers linear models for the structure, and does not include the effect on structural nonlinearities like freeplay/hysteresis on control surfaces, hardening/softening of the structure with high deformations, and so on. These nonlinearities can modify the classical aeroelastic behaviour of the structure by introducing additional vibration-type instabilities like Limit Cycle Oscillations (LCOs) or Chaotic response. These undamped oscillations, although not catastrophic, have an important influence on the fatigue life of the structure. This report supplies a generic overview of the effect of the concentrated structural nonlinearities on the vibration of the aircraft flexible structure, and describes the current theoretical and experimental methodologies that the aeronautical industry performs to take into account these nonlinearities at the aircraft design stage. Besides this, a novel simulation technology that predicts the response in the presence of structural nonlinearities as a function flight speed, nonlinearity characteristics, and other parameters that could affect the response is presented. The advantages and limitations of this methodology are analysed and a way forward is proposed for future studies. INTRODUCTION The aircrafts are designed to avoid the so-called classical aeroelastic instabilities caused by coupling of inertial, structural, and aerodynamic forces. Divergence, control effectiveness, control system reversal, buffet, and flutter, are classical aeroelastic-type instabilities (see [1]) that are covered into the aircraft design stage. Besides this, the appearance of flight-by-wire aircrafts introduced the need of assessing the flight control laws (FCL), which govern the motion of the control surfaces, to avoid aeroelastic coupling (aeroservoelasticity). Flutter (w/o and with FCL) is the most critical aeroelastic instability: above a certain flight speed, the so-called Flutter Speed U F, the aerodynamic forces amplify the structural deformations and cause the inadvertent catastrophic failure of the structure. This is a dynamic effect that involves the motion and deformation of the aircraft caused by the unsteady aerodynamic forces acting on the deformed structure. Because it is easier mathematically to describe the aerodynamic loads due to simple harmonic motion, theoretical flutter analyses often consists of assuming in advance that all dependent variables are proportional to e iωt (ω real), and reduce the flutter timedomain dynamical equations to a frequency-domain complex eigenvalue problem. The hypothesis of harmonic solution is only valid in case of linearity in all the components that affect to the aeroelaticity: structure, inertia, aerodynamics and FCL. However, there are many instances where nonlinearities can be important, and where an understanding of their effects is crucial to an efficient and safe design:

2 Structural nonlinearities can be characterized as either distributed or concentrated (see [3]), according to their origin. Distributed nonlinearities arise from slippage in riveted joints or from buckling in a built-up structure, for example, whereas concentrated nonlinearities are associated with such localized phenomena as backlash (or freeplay), damping hysteresis, or saturation on non-powered and powered controls. These latter nonlinearities are generally the most important. Missile control surfaces thar are designed to be easily attached or removed, all-movable aircraft lifting surfaces such as horizontal tails, or rotatable pylons on variable-sweep aircraft all exhibit nonlinear behaviour that can be potentially dangerous from an aeroelastic standpoint. An excellent discussion of structural nonlinearities in aircraft is contained in [3], along with a summary of both experimental and theoretical techniques employed to evaluate their effects. The situation with regard to missiles is summarized in [4]. Inertia nonlinearities arise from time-dependent mass (fuel consumption) or inertia (fuel sloshing in high-performance fighters). Aerodynamic nonlinearities are currently associated to the high-speed transonic regime where shock-induced and trailing edge separation plays an important role (see [5]), vortex iteration, or separated flow due to high angle of attack. Among the aforementioned structural nonlinearities, the freeplay/hysteresis on the actuators of control surfaces is usually the critical one. The appearance of fly-by-wire aircrafts with unbalanced control surfaces has increased the number of in-flight incidences associated to undamped LCOs caused by freeplay/hysteresis in control surfaces. As examples, Ref [14] describes that the primary cause of excessive vibration in the aft cabin of Model A320 series airplanes was the excessive freeplay in the elevator attachments; the Model B737 (see Figure 1) also exhibited freeplay-induced vibrations of the aileron balance tab (see [13]), and the Delegate of the Australian Civil Aviation Safety Aurhority concludes: The potential for vibration of the control surface should be avoided because the point of transition from vibration to divergent flutter is unknown. Both Civil and Military Airworthiness Authorities are recently requiring more specific analyses and ground/flight tests to substantiate the avoidance of Figure 1 Boeing 737 these phenomena. Military specifications (Ref [#]) define a maximum freeplay of [deg] to avoid undesired LCO-type vibrations. However, this value seems to be excessively conservative: recent flight tests on the F-16 aircraft with a freeplay of 0.2 [deg] did not exhibit LCOs. The prediction of LCOs is usually not deeply studied at the preliminary design stage of the aircraft. Moreover, there are no standard simulation methodologies that allow predicting the type of aeroelastic response in the presence of structural nonlinearities depending on the freeplay deadband, flight speed, and other aircraft parameters. Although an exclusively theoretical tool that predicts the aeroelastic response could not be accurate enough (specially for transonic regime, aerodynamically-loaded flight controls, non-linear flight control laws, ), it allows the engineer to perform qualitative analyses and predict potential problems in disciplines like fatigue damage. In fact, the characterization of the response in terms of damped motion, undamped harmonic motion, undamped chaotic motion, supplies important information on the cycles that should be used for the subsequent fatigue analyses of the materials. STRUCTURE/AERODYNAMICS COUPLING Linear Systems The Doublet-Lattice method DLM is the most extended subsonic code (Mach number below 0.7 approx.) in the Aeronautical Industry for calculation of the unsteady aerodynamic forces in the frequency-domain. This method allows reliable computation times for massive flutter analyses that are common at the aircraft design stage. Figure 2 shows the structural (MSC.NASTRAN Finite Element Model), inertia (lumped-mass model) and aerodynamics model (DLM) that 2

3 compose the aeroelastic model for flutter simulations on a typical aircraft. The displacements/rotations at the structural points are interpolated into the aerodynamic mesh by using different spline-type methods. Figure 2 Aeroelastic model used in the design of a typical aircraft. Apart from the DLM method, other more-refined Computational Fluid Dynamics (CFD) codes are used for calculating the unsteady aerodynamic forces in the frequency-domain: Transonic Small Disturbance Method (TSD), Full Potential Method (FP), Euler, or Navier-Stokes. These methods should be used beyond Mach number 0.8 (approx.) to capture transonic effects and shock waves, although have the drawbacks of excessive time computation, non-robustness, and the need of wind-tunnel or flight tests to determine some parameters of the code. Nonlinear Systems In the presence of nonlinearities, the classical frequency-domain approach fails and it is necessary to solve the structure/aerodynamics coupling by other methodologies. The following methods are currently being used in solving nonlinear systems (see [8]): 1. Describing Function or Harmonic Balance Method (see [8], chapter 6): The nonlinear system is replaced by an equivalent linear system, and the response is assumed to be harmonic. 2. Point Transformation Method (PTM), only valid for piecewise linear systems (see [8], chapter 4): The state-space system is divided into a number of distinct regions, in each of which the system s dynamical behaviour can be analysed by linear techniques, with the solutions for different regions being matched together at the boundaries. 3. Numerical Continuation ([12]): once an initial set of solutions of the state-space equations are found by using a Runge-Kutta or linear methodology, the continuation method allows to find other possible solutions of the same period for a variation in any parameter. 4. Time-domain solvers: Direct time-marching integration (Runge-Kutta, ) The Describing Function and Numerical Continuation are only strictly applicable when searching for harmonic-type LCOs. The PTM can be applied for all types of responses (harmonic, chaotic, ), but it assumed a piecewise-linear nonlinearity. Finally, the time-domain solution is able to capture all the different motions (including the chaotic behaviour) and is valid for all type of nonlinearities (freeplay, hysteresis, bilinear, stiffness cubic hardening or softening, and so on), and therefore is the most general methodology although also the most time-expensive method from the computational standpoint. RESEARCH AND DEVELOPMENT ON CONCENTRATED STRUCTURAL NONLINEARITIES This section summarises the main difficulties when dealing with concentrated structural nonlinearities. The aeroelastic equations should be stated in the time-domain (State-Space form) and must be integrated by using a dedicated integration methodology that captures the corners of the freeplay/hysteresis nonlinearity. 3

4 1 Hinge Moment M h The modal formulation states the aircraft deformation as a linear combination of a set of normal modes (usually retaining normal modes up to Hz is enough), what reduces the number of degrees of freedom from thousands to hundred of normal modes. The freeplay/hysteresis nonlinearity can be understood as a piecewise linear system, and different set of normal modes are obtained depending on the region of the nonlinearity where the nonlinear DoF stays. The 2 3 Rotation Angle δ Figure 3 Freeplay nonlinearity on the control surface rotation. following lines will be centred on the freeplay nonlinearity, although the explanation is directly extrapolated to the hysteresis-type nonlinearity that can be understood as a superposition of two freeplay nonlinearities. Figure 2 shows the freeplay nonlinearity on the control surface rotation angle. If the rotation angle is outside the deadband zone (zones 1 or 3 in Figure 2), the control surface hinge moment is proportional to the rotation angle increments with the nominal stiffness of the system, i.e., M h =K nom δ. On the other hand, if the rotation angle is inside the deadband zone (zone 2 in Figure 2), the control surface is free and the hinge moment is zero, i.e., M h =0. The set of normal modes associated to the deadband zone 2 is different with respect to that of zones 1 and 3, and this implies that the system equations should be changed when passing through the freeplay corners. However, if the modal base is not the same for outside (zone 1 or 3) and inside the deadband zone (zone 2), the transition between the systems introduce discontinuities as a result of the non-perfect matching between the two different set of normal modes, what makes necessary the usage of an unique set of normal modes independently of the freeplay zone. This unique modal base is built by merging the set of modes and different methods are being implemented in the industry. Another difficulty when dealing with nonlinearities is the formulation of the unsteady aerodynamic forces in the time-domain. EADS/CASA performs this calculation by using the Rational Approximation of the frequency-domain Doublet-Lattice matrices ([ ]). This approach allows characterizing the response of the aircraft on the entire flight envelope by running hundred of time-domain simulations. Finally, the integration software is a critical point when dealing with freeplay/hysteresis nonlinearities. The system equations should be changed when passing from one zone to another and the switching points are the corners of the nonlinearity. Different integration methods exists in the literature, but the most effective from a computational standpoint is the Henon s method ([11]): as the nonlinear variable reaches the corner, the state-space equations are formulated to include the time as dependent variable and solve the exact time of passing through the corner with only one time step. The author of this paper has developed a software for solving nonlinear aeroelastic systems taking into account the aforementioned difficulties, and a missile-type configuration has been chosen as test to show its applicability. The following section summarises the main results. APPLICATION TO MISSILE CONFIGURATIONS: RESULTS AND DISCUSSION The previous methodology is applied to a missile-type configuration with hysteresis nonlinearity on the control surface rotation angle. The missile is installed into the aircraft supported by a pylon as sketched in Figure 3, and the lateral vibrations are simulated by considering the following three degrees of freedom: lateral DoF, yaw DoF, and control surface rotation. Lateral vibrations are probable to exist due to the low frequency of the lateral and yaw pylon/missile normal modes: around 5 Hz the lateral DoF, and 10 Hz the yaw DoF. The control surface rotation mode is assumed to be at 25 Hz approx. 4

The unsteady aerodynamic forces are directly calculated in the time-domain by using the Slender Body Theory formulation.

Some results are depicted below in Figure 5, which shows the number of characteristic frequencies of the time history associated to the control surface rotation angle as a function of the Flight

It can also be seen that for a non-dimensional deadband amplitude 0.006 (± 0.34 [deg]) and Flight Speed 260 KTAS, the number of characteristic frequencies is 93, what means no periodic motion, i.e., chaotictype response.

5 The unsteady aerodynamic forces are directly calculated in the time-domain by using the Slender Body Theory formulation. The aeroelastic equations are formulated in the State-Space form and are integrated by using an integration Fortran software based on the Henon s method (see [11]) to capture the hysteresis corners. Some results are depicted below in Figure 5, which shows the number of characteristic frequencies of the time history associated to the control surface rotation angle as a function of the Flight Speed (x-axis) and the hysteresis deadband amplitude (y-axis). The label D means damped motion, which occurs at low speeds, and the linear flutter speed is 500 KTAS approx. It can also be seen that for a non-dimensional deadband amplitude (± 0.34 [deg]) and Flight Speed 260 KTAS, the number of characteristic frequencies is 93, what means no periodic motion, i.e., chaotictype response. The time-history of the rotation angle and the LATERAL DoF YAW DoF phase plane (rotation speed vs rotation angle) corresponding to this point are also shown below in Figure 5. As the flight speed approaches the flutter speed, the responses tend to be a periodic with 2 or 1 characteristic frequencies. Figure 6 shows the bifurcation diagram (flight speed as parameter) of the control surface rotation angle for a Figure 4 Missile configuration nondimensional deandband amplitude of The bifurcation installed into a fighter diagram represents the MAX/min of the control surface rotation as a function of the flight speed. At low speeds, the motion is damped and the control surface tends to approaches one of the corners of the hysteresis nonlinearity. At flight speed 240 KTAS, the damped motion explodes and appears non-periodic oscillatory motion. This change from steady state (damped motion) to oscillatory response is called Hopf bifurcation with hard loss of stability ([9]). Figure 5 Number of characteristic frequencies as a function of Flight Speed (x-axis) and hysteresis non-dimensional deadband amplitude (y-axis) 5

6 Figure 6 Bifurcation diagram of the control surface rotation angle CONCLUSIONS This paper has introduced the different methodologies that are currently used for the structural/aerodynamics coupling in the aeroelastic domain, with special emphasis on the treatment of concentrated structural nonlinearities. There are several methods for solving nonlinear aeroelastic systems (Describing Function, Point Transformation Method, Piecewise Linear Systems, Numerical Continuation, ), but the direct time-domain integration is the only one that supplies all the possible responses, from harmonic to chaotic behaviour, and is applicable to all types of structural nonlinearities. A time-domain integrator has been developed for solving the aeroelastic state-space equations, and has been applied to a missile-type configuration with hysteresis on the control surface rotation. The main results when varying the hysteresis deadband amplitude are depicted, obtaining different types of motion: damped, harmonic LCO, and non-periodic (chaotic?) motion. Once the applicability of this software has been shown for missile-type configurations, further studies will be centred on applying it to aircraft-type configurations. References: [1] R. L. Bisplinghoff, H. Ashley, R.L. Halfman: Aeroelasticity. Dover Publications, Mineola, New York (1996). [2] M. Ilkowska, A. Miskiewicz: Sharpness versus brightness: A comparison of magnitude estimates. Acta Acustica united with Acustica 92, No.5 (2006) [3] E. Breitbach. Effect of Structural Nonlinearities on Aircraft Vibration and Flutter. AGARD Report 665 (1978). [4] G. Dailey, R.J. Oedy, W.J. Werback: A State-of-the-Art Review of Methods in Aeroelasticity and Structural Analyses for Guided Weapons. Paper 36, presented at the 10 th Naval Symposium on Aeroballistics, Frederiscksburg, Va. (1975). [5] J.J. Meijer, A.M. Cunningham (Jr.): A Prediction Method of Transonic Limit Cycle Oscillation Characteristics of Fighter Aircraft using Adapted Steady Wind Tunnel Data. NLR TP U. National Aerospace Laboratory NLR, Amsterdam, The Netherlands (1994). [6] Department of Transportation, FAA: Airworthiness Directives; Airbus Model A319, A320, and A321 Series Airplanes. Federal Register, Vol. 66, No. 161 (2001). [7] Commonwealth of Australia. Civil Aviation Safety Authority: Boeing 737 Series Aeroplanes, AD/B737/298 Aileron Balance Tab (2006). [8] P.A. Cook: Nonlinear Dynamical Systems. Prentice Hall (1994). [9] R. Seydel: Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos. Springer-Verlag (1994). [10] M. Karpel, and M. Newman: Accelerated Convergence for Vibration Modes Using the Substructure Coupling Method and Fictitious Coupling Masses. Israel Journal of Technology (1975). [11] M.D. Conner, L.N. Virgin, and E.H. Dowell: Accurate Numerical Integration of State-Space Models for Aeroelastic Systems with Freeplay. AIAA Journal, 34(10), (1996). [12] I. Roberts, D.P. Jones, N.A.J. Lieven, M. di Bernardo, and A.R. Champneys: Analysis of Piecewise Linear Aeroelastic Systems Using Numerical Continuation. (2001). [13] D. Villiers. Airworthiness Directive of Civil Aviation Safety Authority, part (2006). [14] USA Federal Register, Rules and Regulations, Vol. 66, No. 161 (2001) 6

PROGRESS IN THE PREDICTION OF AEROSERVOELASTIC INSTABILITIES ON LARGE CIVIL TRANSPORT AIRCRAFT

ICAS 2000 CONGRESS PROGRESS IN THE PREDICTION OF AEROSERVOELASTIC INSTABILITIES ON LARGE CIVIL TRANSPORT AIRCRAFT M.LACABANNE, A.LAPORTE AEROSPATIALE MATRA AIRBUS, 31060 Toulouse Cedex 03, France Abstract