effect of structural nonlinearity on the dynamic response of a coupled acoustic-aeroelastic system

7/26/2019 Effect of Structural Nonlinearity on the Dynamic Response of a Coupled Acoustic-Aeroelastic System

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Australian Aerospace Congress, 23-24 February 2015, Melbourne

Effectof Structural Nonlinearity on the DynamicResponse of a Coupled Acoustic-Aeroelastic System

Nicholas F. Giannelis 1,*and Gareth A. Vio 1

1School of Aeronautical, Mechanical and Mechatronic Engineering, Building J11, The University of Sydney, NSW, 2006, Australia*[email protected]

Abstract

Aircraft are inherently exposed to acoustic fields during operation. Various studies in active flutter

suppression have shown that acoustic excitation of an aeroelastic system can be tuned to delay thepresence of the flutter instability. This paper aims to investigate the influence of an acoustic field on the

flutter stability of a simple aeroelastic model. The finite element method is used to couple the acousticand aeroelastic systems. Linear flutter analysis is conducted through eigenvalue analysis of a state spaceformulation of the aeroelastic equations of motion. Control surface freeplay is modelled and nonlinearflutter analysis is conducted through numerical integration. In both the linear and nonlinear analyses, thecoupled aero-acoustic system is found to exhibit a flutter velocity on the order of 9% in excess of the

aeroelastic system. Further, the inclusion of acoustic harmonics introduces interesting complexities to thenonlinear responses of the structural degrees-of-freedom.

Keywords:Nonlinear Aeroelasticity, Acoustics, Flutter

Introduction

Aircraft unavoidably operate in the presence of acoustic fields, and given structural frequencies ofcomparable order, aero-acoustic interactions may act to alter the inherent aeroelastic stability of a flightsystem. In [1], this phenomenon is examined experimentally, where acoustic excitation is successfully

employed to achieve active utter suppression. A number of studies on acoustic flutter suppression insupersonic cascades have also yielded promising results [2-4]. More recently, developments in theformulation of coupled aero-acoustic systems have been seen, with [5, 6] detailing a unified BoundaryElement/Finite Element coupling scheme. In [7], the Finite Element Method is used to couple a simpleaeroelastic system with an acoustic field, with linear flutter analysis revealing improved stability in thepresence of the acoustic medium. Linear aeroelastic analysis is however limited in its ability to modelcomplex phenomena, and [8] shows that even in simple aeroelastic systems, nonlinear Limit Cycle

Oscillations (LCOs) and bifurcations can present to provide insight to the underlying system stability.

In this paper, the nonlinear aeroelastic response of a simple coupled aero-acoustic system is investigated.The aeroelastic, acoustic and coupled systems are detailed, followed by the methodology in determiningthe linear and nonlinear stability boundaries. The results of the linear flutter analysis are then presented,

along with a discussion of the bifurcation behaviour, time response and restoring force characteristics ofthe nonlinear systems. To conclude, the key results are summarised.

ModelAeroelastic System

The structural model investigated in the present work is an extension of the Hancock wing proposed in[9]. The 3 degree-of-freedom (DoF) system, illustrated in Figure 1, consists of a rigid wing with flap

angle, twist angleand control surface deflection angle. Bending, torsion and control surface DoFsare constrained by three discrete springs of stiffness, and respectively. The particular wingconsidered exhibits a 2 m chord, 12 m span, aerodynamic centre (ac) at 25% chord, flexural axis () at40% chord and hinge axis () at 80% chord. The wind-off natural frequencies of the flap, pitch andcontrol modes are imposed at 7 Hz, 15 Hz and 22 Hz respectively.


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Fig. 1: 3 DoF Hancock Model

Although the inertial definitions of the structure pertain from a uniform flat plate, the aerodynamic model

assumes a conventional aerofoil section with an ideal lift curve slope of 2 . Modified strip theory isemployed to model the aerodynamics under a quasi-steady assumption, where the aerodynamics of asystem experiencing variable oscillatory motions are equal to the instantaneous values of an equivalent

system under constant motion. To better approximate the actual flutter mechanism under the quasi-steadyassumption, aerodynamic pitch damping ( ) and control damping ( ) derivatives are also included.The coupled aeroelastic system is represented by the second order ordinary differential equation:

(1)where , and are the aeroelastic inertial, damping and stiffness matrices respectively, are the physical displacements and and are the velocities and accelerations ofeach structural DoF respectively. See [10] for full derivation and definition of the component matrices inEquation 1, noting that in the present analysis no structural damping is considered.

Acoustic System

The acoustic field is modelled through the finite element method by a single four node quadrilateralisoparametric element encompassing the planform area of the wing. Analogous to the aeroelastic system,the acoustic field is represented by the second order differential equation:

(2)where and are the acoustic inertial and stiffness matrices respectively (see [11] for completedefinitions), is a vector of nodal pressures, is a vector of the second time derivatives of nodalpressures, is the fluid density (1.225 kg/m3) and is a spatial coupling matrix defined as:

(3)

where and are matrices of the structural and acoustic shape functions respectively, is theboundary between the fluid and the structure and is a transformation matrix between the acoustic andstructural degrees of freedom. The matrix of acoustic shape functions is derived from the definition of thefour node isoparametric element (see [12] for a complete derivation). The matrix of structural shape

functions is computed in physical coordinates via modal analysis. The displacement is determined ateach of the acoustic nodes by:

( ) ( ) (4)where

represents the node number of the acoustic element.


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Aeroelastic-Acoustic CouplingCoupling between the aeroelastic and acoustic systems proceeds following the method described in [11]and is achieved through the unsymmetric system of equations given by:

[

] {

}

{

} [

]

(5)

where is the speed of sound in air at sea level (343 m/s).Linear Flutter

To determine the linear utter stability of the aeroelastic and coupled aero-acoustic systems, a state spacemethod is employed to determine the frequency and damping of the systems described by Equations 1 and5 across a range of airspeeds. The systems are transformed to first order form:

(6)

where, and are the mass, damping and stiffness matrices respectively of the system underconsideration. For each case, the damping of the structural modes is tracked across the velocity range,with the flutter condition given by the first crossing of zero damping.

Freeplay Nonlinearity

The effect of a hinge stiffness dead-band on the flutter stability of the aeroelastic and aero-acoustic

systems is determined by including a nonlinear restoring force function () given by:

{ (7)

where is the spring stiffness and to defines the freeplay region. The time responses of theoriginal and acoustic systems are computed through numerical integration under an explicit Runge-Kutta(4, 5) scheme. To determine the flutter stability no external forcing function is applied, however an initialcondition of +5 control surface deflection is introduced to examine the systems response. The timeresponses are computed across an analogous range of airspeeds to the linear analysis, with a time step of

0.0001 s and points. From the final 20% of the response signal, the amplitude of control surfacedeflection is extracted at the zero velocity crossings to determine whether the system is decaying, unstableor exhibiting LCOs.

ResultsLinear Flutter

The frequency and damping for each of the structural degrees of freedom across a range of airspeeds isgiven in Figure 2. The results are in qualitative agreement with the findings of [7], where the inclusion ofan acoustic field delays the onset of flutter. A 9.3% increase in stability is observed in the acoustic case,

with flutter speed rising from 323 m/s to 353 m/s. In both the original and acoustic systems, instability isa product of coalescence in the flap and pitch modes, with the pitch DoF becoming unstable at the flutter

point. It is important to note that due to the inherent stiffness of the structural model, the linear flutterspeed is observed in the supersonic regime. This paper proceeds under the assumption that low speed

aerodynamics remain valid, as the principle concern is the effect of the acoustic field on the structuraldynamics.

The increase in stability observed in the acoustic system can be explained by the introduction of theacoustic harmonics, which act to shift the wind-off natural frequency of the pitch DoF from 14.3 Hz to15.9 Hz. The increase in frequency spacing between the fluttering modes necessitates a greater energy

input before the instability occurs. Further, the transition to flutter in the presence of the acoustic field ismore gradual, with the pitch damping of the acoustic system exhibiting a shallower gradient through the

zero damping condition than that of the original system.


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Fig. 2: V-g & V- Plots of Linear Systems

Bifurcation Behaviour

The linear flutter analysis is limited in scope due to its inability to capture LCOs. The presence of suchLCOs may act to delay the onset of flutter, and as such, a time domain analysis is performed to comparethe nonlinear responses of the original and acoustic systems. In Figure 3, the bifurcation plots of both

models are given. The presence of the acoustic field yields significant differences in the topology of thebifurcation branches when compared to the original system. For the original system represented in Figure3a, the structure exhibits a decaying response to the initial control surface deflection up to the point of thelinear flutter velocity. A single low amplitude LCO is then observed until 348 m/s, where a secondequilibrium branch appears. Beyond 355 m/s two stable low amplitude LCOs persist in the system up to

383 m/s, where apparent period doubling begins. Flutter in the nonlinear original system occurs at 397m/s, 22.9% greater than the flutter speed predicted through linear analysis.

(a) Original System (b) Acoustic System

Fig. 3: Bifurcation Diagrams of the Control Surface DoF

In Figure 3b, the bifurcation diagram of the coupled acoustic system is presented. The acoustic systemexhibits very low amplitude LCOs at velocities below the linear flutter speed of 353 m/s, with purely

decaying responses observed from 334 m/s. The amplitude of the acoustic LCOs increases marginally upto 387 m/s, where the system exhibits a purely decaying response. Above 387 m/s, four stable LCObranches exist centred about a positive control surface deflection, with a single branch centred about anegative control deflection equilibrium. The acoustic system further shifts to higher amplitude LCOs

above 405 m/s, with amplitudes increasing up to the flutter point of 435 m/s, 23.2% greater than the linearflutter speed.


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From the bifurcation diagrams of Figure 3, several interesting observations can be made. Where theoriginal system conveys symmetry about the equilibrium control surface deflection (0), this behaviour isnot evident in the acoustic responses. Under positive initial control deflection, the acoustic LCO orbitsappear to centre about a positive fixed point. This asymmetry is further investigated by performing anequivalent analysis with negative deflection as an initial condition. In Figure 3a, the topology of thebifurcation branching remains consistent regardless of the sense of the initial condition. Conversely, inFigure 3b, the stable orbits now centre about negative control surface deflection, highlighting an increasedsensitivity to initial conditions in the presence of an acoustic field. Nonetheless, the coupled acousticsystem exhibits significantly greater inherent stability, both in the absolute flutter speed and in theamplitude of oscillations sustainable prior to flutter. The nonlinear flutter velocity of the acoustic systemexceeds that of the original system by 9.6%, whilst prior to flutter a 30 LCO is sustained, 376% largerthan the 6.3 LCO of the aeroelastic model.

The ability of the acoustic system to sustain a variety of stable LCOs at a particular airspeed is furtherillustrated in Figure 4. With an initial control deflection of 5 at 390 m/s, the acoustic system is permittedto settle to a low amplitude LCO. An impulsive force of negative sense is then introduced, and followingthe decay of transients, the response is observed to settle to a higher amplitude LCO. The response

highlights the presence of multiple equilibrium orbits in the acoustic system, and the persistence ofstability at velocities well in excess of the flutter point predicted through linear analysis. The particularattractor to which an orbit is drawn is highly dependent on the initial conditions; however responses maybe forced to orbit about a different attractor, given appropriate excitation.

Fig. 4: Time Response of Acoustic System at 390 m/s

Restoring Force

The increased complexity introduced to the system through the acoustic field is evident in the restoringforce plots of Figure 5, which highlight the restoring force magnitude and characteristics in the nonlinearDoF of the systems. The dead-band zone is clearly visible in both systems, with no hysteresis arising fromthe presence of the acoustic coupling. There is an obvious shift in the centre of oscillation for the acoustic

signal, with attraction to the fixed point, and repulsion by the fixed point. The original systemoscillates symmetrically around the points, as also observed in Figure 3a.The original system presents a high restoring force for the same angle of oscillation, when compared tothe coupled acoustic system. This can be explained by the extra energy required to energise the higherorder frequency, clearly present in the response of the coupled system. Whereas the orbits of the originalsystem in Figure 5a are of type period-1, the presence of acoustic harmonics in the coupled systemincreases the periodicity of the orbits to period-6, hence producing the complex response of Figure 5b.

Conclusions

In this paper, the influence of an acoustic field on the aeroelastic behaviour of a simple 3 DoF system hasbeen investigated. Linear analysis finds an increase in flutter speed of 9.3% in the coupled system relativeto the original, a result of the acoustic harmonics increasing the wind-off frequency spacing of thefluttering modes. With the inclusion of a control hinge freeplay nonlinearity, interesting behaviour is


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observed in the acoustic system. A higher sensitivity to initial conditions is present, with apparentasymmetry in the bifurcation diagram. Multiple LCO branches are also seen to persist at any givenairspeed, and the acoustic system is found to capable of sustaining very high amplitude oscillations priorto instability. The flutter point of the acoustic system determined through nonlinear analysis is found toexceed the linear flutter speed by 23.2%, and remains 9.6% greater than the nonlinear flutter speed of theoriginal system. The presence of an acoustic medium is also shown to significantly increase theperiodicity and complexity of the control surface DoF orbits.

(a) Original System (b) Acoustic System

Fig. 5: Restoring Force in Control Surface DoF at 340 m/s

Acknowledgements

The authors would like to acknowledge Kok Hwa Yu and Harijono Djojohihardjo for their assistance inclarifying the foundations and methodology in their work.

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effect of structural nonlinearity on the dynamic response of a coupled acoustic-aeroelastic system

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