IFASD-2013-16B REDUCED-ORDER AEROELASTIC MODELS FOR THE DYNAMICS OF MANOEUVRING FLEXIBLE AIRCRAFT Henrik Hesse 1 and Rafael Palacios *1 1 Department of Aeronautics Imperial College, London, United Kingdom Abstract: We investigate model reduction, using balancing methods, of the unsteady aerodynamics of flexible aircraft subject to asymmetric gust excitations. The aeroelas- tic response of the vehicle, with possibly large wing deformations at trim, is captured by coupling a geometrically-exact beam formulation with the three-dimensional unsteady vortex lattice method. Consistent linearisation of the structural degrees of freedom al- lows the projection on a few vibration modes of the unconstrained aircraft and permits the vehicle flight dynamics to have arbitrarily-large angular velocities. The linearised aerodynamic system, which defines the mapping between the small number of generalised coordinates and the aerodynamic loads, is then reduced through balanced truncation. Nu- merical studies on a representative high-altitude, long-endurance aircraft demonstrate the reduced-order modelling approach for spanwise non-uniform discrete gusts over a range of gust lengths and lateral placement. Keywords: reduced-order modelling, aeroservoelasticity, flexible-body dynamics, un- steady aerodynamics, very flexible aircraft NOMENCLATURE Symbols C global tangent damping matrix K global tangent stiffness matrix M global tangent mass matrix q generalised displacements in modal basis Q ext global vector of external forces, N t physical time, s u input vector v inertial translational velocity of the body-fixed frame, m/s W c controllability Gramian W o observability Gramian x state vector of a linear system identified by subscripts y output vector β vector of global translational and rotational velocities Γ circulation strength of a vortex ring, m 2 /s η vector of nodal displacements and rotations Θ Euler angles, rad ν global displacements and rotations as time integral of β Φ matrix of mode shapes χ coordinates of the aerodynamic lattice, m ω inertial angular velocity of the body-fixed frame, rad/s * Contact author: [email protected] (R. Palacios) 1
Transcript
IFASD-2013-16B
REDUCED-ORDER AEROELASTIC MODELS FOR THEDYNAMICS OF MANOEUVRING FLEXIBLE AIRCRAFT
Henrik Hesse1 and Rafael Palacios∗1
1Department of Aeronautics
Imperial College, London, United Kingdom
Abstract: We investigate model reduction, using balancing methods, of the unsteadyaerodynamics of flexible aircraft subject to asymmetric gust excitations. The aeroelas-tic response of the vehicle, with possibly large wing deformations at trim, is capturedby coupling a geometrically-exact beam formulation with the three-dimensional unsteadyvortex lattice method. Consistent linearisation of the structural degrees of freedom al-lows the projection on a few vibration modes of the unconstrained aircraft and permitsthe vehicle flight dynamics to have arbitrarily-large angular velocities. The linearisedaerodynamic system, which defines the mapping between the small number of generalisedcoordinates and the aerodynamic loads, is then reduced through balanced truncation. Nu-merical studies on a representative high-altitude, long-endurance aircraft demonstrate thereduced-order modelling approach for spanwise non-uniform discrete gusts over a rangeof gust lengths and lateral placement.
FSaeroelastic, includingrigid-body and structural states
R rigid-body degrees of freedomS structural degrees of freedomw wake
Superscripts
˙(•) derivatives with respect totime, t
¯(•) small perturbations around anequilibrium
(•)∗ conjugate transpose of a matrix
1 INTRODUCTION
Low-speed unsteady aerodynamic loads have traditionally been obtained in aeroelasticityusing the Doublet Lattice Method (DLM) [1], which computes the aerodynamic influencecoefficients for a few structural vibration modes over a range of reduced frequencies andflight conditions. Time-domain state-space models can then be obtained by means ofrational-function approximations on sampled aerodynamic data [2–4]. This approach hasbeen successfully applied to obtain transient responses and in controller design, but itassumes small normal displacements of the lifting surfaces, which may be no longer validin high-load situations on very flexible wings.
Time-domain aerodynamic models, such as the Unsteady Vortex Lattice Method (UVLM),provide a more flexible alternative to frequency-domain methods. They evaluate loadson the instantaneous deformed geometry and therefore can be coupled directly togeometrically-exact flexible-body dynamic models of the manoeuvring aircraft [5]. Thisimproved kinematic characterisation in the structural dynamics also removes the need fora priori assumptions on the inertial coupling between the vehicle structural and flight-dynamic response, as is done in the mean-axes approximation [6]. It was previouslyshown [7, 8] that such decoupling of the system variables may have a significant impacton the predicted vehicle dynamic gust and manoeuvre loads.
This is critical in the design of vehicles with higher-aspect-ratio wings, which tend to bemore vulnerable to gust excitations. The process to design against critical gust loads is acrucial part in the certification of large transport aircraft for airworthiness and requires alarge number of simulations for each configuration to demonstrate its resilience in terms offatigue and critical gust loads against continuous turbulence and vertical/lateral discretegusts. Efficient linear solutions based on DLM aerodynamics are accepted for certificationto obtain the coupled gust response for relatively stiff vehicles subject to spanwise uni-form gusts [4, 6, 10]. However, as aircraft become larger and more flexible, higher-fidelitytime-domain aeroelastic tools are required to accurately capture large wing deformationscoupled with the nonlinear flight dynamic response due to spanwise non-uniform gust ex-citations. This is particularly relevant for novel unmanned high-altitude, long-endurance(HALE) concepts of very high aspect-ratio [11–13]. In particular, Dillsaver et al. [14] us-ing 2D aerodynamics coupled with a geometrically-nonlinear beam formulation exploredthe gust response of a flexible flying wing subject to non-uniform gusts, as proposed inthe DARPA guidelines for the Vulture II programme [11]. Opposite to uniform discretegust responses, where shorter gusts lengths usually result in critical gust loads, they foundthat the longer non-uniform gusts lead to a larger aeroelastic response. Murua et al. [15]
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investigated the open- and closed-loop gust response of a full HALE configuration basedon 3D aerodynamics linearised around a geometrically-nonlinear trim equilibrium. Theresulting linear time-domain descriptions of the flexible aircraft dynamics has also beendemonstrated for spanwise non-uniform symmetric gusts.
Time-domain aeroelastic solutions expressed in terms of the instantaneous circulation inall aerodynamic panels are still computationally expensive for controller design and flightdynamic simulation, where lower-order models would be desired. This can be achievedthrough model reduction of the linearised unsteady aerodynamic system (around a refer-ence nonlinear equilibrium condition). Hall [16] demonstrated that the governing equa-tions of a time domain unsteady vortex lattice model can be formulated as a generalisedeigenvalue problem to produce a very compact, reduced-order aerodynamic model. Com-puting eigenvalues becomes however computationally expensive for large systems. Thisproblem has been addressed in the fluid dynamics community using the proper orthogonaldecomposition (POD), also known as the Karhunen-Loeve expansion [17], to produce abasis function of the system based on sample data of the flow [18, 19]. Dowell et al. [20]and Lucia et al. [21] both provide in-depth reviews of the application of POD to high-orderlinear and nonlinear computational fluid dynamic problems.
The resulting basis functions, obtained through either eigenanalysis or POD, accuratelycapture the internal dynamics of the system, but a large number of modes (not onlylightly-damped ones) may be required to effectively represent the particular input/outputmapping in a typical aeroelastic problem. This has been demonstrated by Rule et al. [22]for the unsteady vortex lattice method. Those authors propose an alternative modelreduction method, adopted from the controls community, to obtain a reduced-order aero-dynamic model that is optimal in terms of the particular transmission path between thesystem inputs and outputs. Balanced realisations [23,24] are especially useful in unsteadyaerodynamics, where a large number of states is used to transmit information from acomparably small number of inputs (geometry of the structure) to a small number ofoutputs (unsteady aerodynamic forces). Balanced states that have little contribution toa particular input/output mapping can be efficiently discarded [20]. The balanced modelreduction approach has already been demonstrated for aeroelastic systems on a 2D pitch-plunge problem [22] and on a full linear aircraft model to obtain the flight loads followingwake vortex encounters [25].
The performance of a balanced realisation is strongly linked to the number of system in-puts and outputs [26], e.g. the number of elastic and rigid-body degrees of freedom (DoF)in an aeroelastic problem. Hence, this work will propose a model reduction approach foran integrated flexible-aircraft formulation which allows us to control the number of in-puts/outputs of the aerodynamic system by projecting the flexible-body dynamics part ofthe coupled system on a small modal basis to reduce the number of generalised coordinatesand aerodynamic loads. This has been preferred to a direct system identification throughstep responses, as done recently by Haghighat et al. [27], since the balancing approachuses the state-space structure of the original model in the reduction process. To model thedynamics of manoeuvring flexible aircraft, previous work by the authors [7] demonstrateda consistent linearisation of the structural DoF in nonlinear flexible-aircraft dynamicsproblems. This provides a formulation which allows projection of the nonlinear flexible-body dynamics subsystem onto the vibration modes of the unconstrained structure at anyreference condition while keeping the nonlinearity in the rigid-body dynamics equations.
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The linearisation of the structural DoF is consistent such that all couplings between rigid-body and structural dynamics due to gyroscopic motions are preserved, which was foundto have a significant impact on the transient loads of the structure [7]. Coupled with a3D UVLM [5], this nonlinear modal approach has been demonstrated for the dynamicanalysis of manoeuvring flexible aircraft with possibly large, geometrically-nonlinear trimdeformations [8].
Continuing on those studies, this work will investigate the application of balanced trun-cation on the aerodynamics of the transient dynamic analysis of manoeuvring flexibleaircraft. Under the assumption of small elastic deformations, the unsteady aerodynamicloads are computed using the linearised version of the UVLM, which has been alreadyshown to be suitable for the stability analysis of flexible aircraft [5, 8] and control de-sign [15]. For manoeuvring aircraft subject to spanwise non-uniform or lateral gusts,however, the underlying rigid-body dynamics equations of motion (EoM) are nonlinearand we will focus in this work on balanced truncation of the (linear) unsteady aerody-namic subsystem only, whereas the nonlinear flexible-body dynamics are projected onto asmall number of dominant vibration modes of the unconstrained structure. This reducesthe number of generalised structural coordinates and aerodynamic loads – the inputs andoutputs of the aerodynamic system – which is necessary for an effective reduction of thebalanced aerodynamic subsystem [26]. The resulting integrated aeroelastic framework isof very low order, but captures the 3D unsteady features of the flow and the tight couplingbetween the nonlinear flight dynamics of the aircraft and the structural wing deforma-tions around a geometrically-nonlinear static aeroelastic equilibrium. To demonstrate thereduced-order modelling approach over a range of excitation frequencies, numerical stud-ies in this work will focus on gust responses of a flexible HALE configuration subject todiscrete gusts with spanwise variations of the vertical gust velocities.
2 FLEXIBLE-AIRCRAFT DYNAMICS MODEL
This section describes the aerodynamic and flexible-body dynamic models used to com-pute the dynamics of flexible aircraft. The underlying geometrically nonlinear descriptionof the flexible-aircraft dynamics can be found in our previous work [7, 28] and here weonly focus on aspects of the linearisation in both models, which will form the basis formodel reduction of the coupled system in the next section.
2.1 Unsteady Vortex Lattice Method
The unsteady aerodynamic loads in this work are obtained using the linearised form of theunsteady vortex lattice method developed by Murua et al. [5] and based on the descriptionby Hall [16]. In the UVLM, vortex ring quadrilateral elements are used to discretise bothlifting surfaces and their wakes, as illustrated in Figure 1. Each surface (bound) vortexring has an associated circulation strength, Γk, and a collocation point, at which theimpermeability boundary condition is satisfied. To obtain the state-space form of theUVLM, the governing equations are linearised on a frozen aerodynamic geometry [5, 16]around the aircraft trim condition with possibly large wing deformations and non-planarwake.
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Wake Vortex
Ring
Bound Vortex Ring
Lifting Surface
Wake Vortex Ring
Corner PointsV∞Δt
A
x
yz
Collocation Point
Trailing Edge
Γbk
nk→
wt-ΔtΓ
wt-2ΔtΓ
Figure 1: Unsteady vortex lattice method with vortex ring discretisation of lifting surface and wake forfree-stream flow in positive x direction.
The change in circulation strengths of the vortex rings around this steady-state solutionis then given in discrete time as
∆Γn+1 = AF∆Γn +BF∆unF , (1)
where superscripts n and n+ 1 refer to the current and next time steps and the subscriptF refers to the linear time-invariant (LTI) aerodynamics. The state vector, Γ, and inputs,uF , are given by
Γ =[Γ>b Γ>w Γ>b
]>, and uF =
[χ>b χ>b w>g
]>, (2)
with the bound (surface) and wake circulation strengths, Γb and Γw, respectively. Thecoordinates and time-derivatives of the bound aerodynamic grid, χb and χb, respectively,can account for small elastic deformations of the lifting surfaces as well as the rigid-bodyvelocities and orientation of the vehicle. The time-derivative of the aerodynamic grid,χb, is also included in Eq. (2) to account for deployment of control surfaces. Finally, wg
refers to gust inputs.
The unsteady aerodynamic loads, which result from the vorticity distribution at each timestep and are written in general as ∆yF , can be computed using the unsteady Bernoulliequation [5]. This will be written in symbolic form as
∆ynF = CF∆Γn +DF∆unF . (3)
where the system feedthrough accounts for small changes in orientation of the aerodynamicloads. The resulting aerodynamic state-space system can be tightly-coupled with theflexible-body dynamics model, which is discussed next.
2.2 Flexible-Body Dynamics Model
The flexible vehicle will be modelled using composite beam elements on a moving (body-attached) frame of reference. Starting from a geometrically nonlinear displacement-based
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formulation [29, 30], the elastic DoF are the displacements and rotations at the elementnodes, which have been linearised using perturbation methods [7]. This consistent lin-earisation is done around a static equilibrium (trimmed aircraft in forward flight) withpossibly large elastic deformations, which will be referred to as η. A detailed derivationof the linearisation can be found in previous work [7,8]. The resulting set of perturbationequations
M(η)
{¨η
β
}+ C(η, β)
{˙ηβ
}+ K(η, β)
{η0
}= Qext (η, ˙η, β, ζ) , (4)
accurately describe the structural dynamics of relatively-stiff aircraft undergoing arbi-trary manoeuvres with (not-necessarily small) rigid-body velocities, β(t), resulting insmall elastic deformations, η(t). The damping and stiffness matrices, C and K, respec-tively, originate from the perturbation of the discrete gyroscopic and elastic forces of thegeometrically nonlinear description and together with the mass matrix,M, account for allcoupling between the structural and rigid-body dynamics of the vehicle. It is important tonote that the mass matrix M only depends on the static deformations, η, to account fora change in rotational inertia, but the damping and stiffness matrices are also functionsof the instantaneous rigid-body velocity, β. See previous work by the authors [7] for adetailed derivation of each of these matrices. The input from the aerodynamic model isaccounted for in the external forcing term, Qext, which can also include thrust and gravityloads.
Equation (4) is solved together with the propagation equations that determine the positionand orientation of the body-fixed reference frame [31]. The equations are time-marchedusing an implicit, constant-acceleration Newmark integration scheme which was modifiedas in Geradin and Rixen [32] to introduce controlled positive algorithmic damping.
2.2.1 Modal Reduction of the Nonlinear System Equations
For the efficient representation of the dynamics of the flexible aircraft, we write the pertur-bation equations, Eq. (4), in terms of the natural modes of the unconstrained structureat a reference condition. These vibration modes are obtained from the unforced fully-linearised version of Eq. (4), and are used to define the following transformation:{
ην
}= Φ
{qν
}, (5)
where we have introduced the new variable ν with ˙ν = β. The vector of the projectedmodal coordinates is q and Φ is the matrix of the corresponding mode shapes, whichhas been transformed to separate the rigid-body motion of the body from the elasticdeformations due to the excitation of the structure [8]. A required assumption for thisseparation is that the body-fixed frame initially coincides with the center of mass (CM)and its axes to be aligned with the principal axes of the structure. The resulting basis isfound to be very convenient for the modal projection of the coupled system equations,
Φ>M(η)Φ
{q
β
}+ Φ>C(η, β)Φ
{qβ
}+ Φ>K(η, β)Φ
{q0
}= Φ>Qext(η, ˙η, β, ζ), (6)
which describes the arbitrarily large rigid-body motion of the flexible aircraft. This ap-proach is limited to small elastic deformations which are captured using the modified
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vibration modes of the unconstrained structure. The modal damping and stiffness matri-ces remain functions of the rigid-body DoF, β. However, it is easy to see that they have,respectively, linear and quadratic dependencies with β, and that it is possible to writethem in terms of third- and fourth-order tensors [7],
Φij CjkΦkl = cilrβr (t) ,
ΦijKjkΦkl = kstifil + kgyrilrsβr (t) βs (t) ,(7)
where we sum over repeated indices and have identified the contributions to the modalstiffness matrix from elastic and gyroscopic forces. The tensors c and k are constantin time and their dimensions are i, l = {1, ...,m} and r, s = {1, ..., 6} for m number ofmodes used in the expansion. These tensors are typically very sparse, and this approachgenerates efficient numerical solutions that keep the nonlinearities in the rigid-body DoFand all couplings with the linear structure at a low computational cost.
2.3 Coupled Aeroelastic and Flight Dynamic Framework
The presented models (and the geometrically nonlinear counterparts) have been imple-mented in the framework for Simulation of High Aspect Ratio Planes (SHARP) to studythe behaviour of flexible aircraft, including static aeroelastic analyses, trim, linear stabil-ity analyses, and fully nonlinear time-marching simulations [5,7,15,28,33]. In this sectionwe will focus on coupling the linearised models which form the basis for subsequent modelreduction.
2.3.1 Monolithic Framework for Linearised Flexible-Aircraft Dynamics
Murua et al. [5] demonstrated that the linearised aerodynamic formulation, Eq. (1), en-ables rigid-body motions and elastic deformations (both small) to be incorporated in aunified monolithic framework. This provides a powerful formulation to determine theaircraft dynamic stability and it can be very useful for efficient control design and opti-misation. As the linear UVLM is written in discrete time, temporal discretisation of theflexible-body system, linear form of Eq. (4), is also required before the fluid/structure cou-pling. A standard Newmark-β discretisation is used, which can also include algorithmicdamping in the structural dynamics response. After introducing the modal transformationof Eq. (5), this leads to the following aeroelastic state-space system [15]
∆xn+1 = AFS∆xn +BFS∆unFS
∆ynFS = CFS∆xn,(8)
where the state vector that completely determines the linear system is
x =[x>F | x>S
]>=[Γ>b Γ>w Γ>b | q> q> β> Θ>
]>, (9)
Note that the orientation of the aircraft is now given in terms of the Euler angles, Θ.The (aerodynamic) inputs to the system, uFS, account for gusts and control inputs in a(closed-loop) time-marching solution. The subscript FS is introduced here to separate theaeroelastic system variables from the linear equations of the UVLM in Eq. (1). Dependingon the nature of the problem, the system output vector yFS can include aerodynamicloads, elastic deformations, aircraft position, attitude, etc. From the homogeneous formof Eq. (8) one also obtains a discrete-time generalised eigenvalue problem to determinethe dynamic stability of the vehicle which includes aeroelastic and flight dynamic modes.
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2.3.2 Dynamic Response of Manoeuvring Flexible Aircraft
The linear equations above may be enough for stability augmentation in forward flight,but they cannot describe arbitrary vehicle manoeuvres. In this section we present theequations of motion of flexible aircraft undergoing large manoeuvres, e.g. due to gustsor control surface inputs. The resulting system is nonlinear because of gyroscopic forcesdominating the flight dynamics, however, under the assumption of small elastic deforma-tions of lifting surfaces during the transient dynamics (the reference condition may stillhave large displacements), we can assume that the unsteady aerodynamics are linear. Un-like aeroelastic approaches based on frequency-domain aerodynamics, the flexible-aircraftdynamics model in this work is developed by tightly coupling the modal form of theflexible-body equations, Eq. (6), with the linear unsteady vortex lattice model, Eq. (1),as illustrated in Figure 2.
The resulting description is based on the vibration modes of the unconstrained flexiblebody, Φ, to reduce (a) the size of the nonlinear flexible-body dynamics problem, and (b)the number of inputs and outputs of the aerodynamic system, Eq. (1). The resultingaerodynamic system can be projected such that
∆Γn+1 = A∆Γn +BSΦ∆uΦn +BF∆uF
n
∆ynF = Φ>C∆Γn + Φ>DSΦ∆uΦn + Φ>DF∆uF
n,(10)
where the inputs to the aerodynamic model are now the generalised coordinates of the
flexible-body equations, i.e. ∆uΦ =[q> q> β>
]>, while the aerodynamic inputs, uF , still
account for gust induced velocities and the deployment of control surfaces. The outputvector yF contains the unsteady loads due to the excitation of a certain vibration modeto obtain the generalised forces. Other applied loads, such as thrust and gravity, areassembled in the flexible-body dynamics subsystem and finally complete the generalisedforce vector Φ>Qext in Eq. (6).
Figure 2: Time-marching solution for manoeuvring flexible aircraft.
Even though the global coupled system equations are still nonlinear, we have isolated thelinear aerodynamic subsystem, which is to be solved at every subiteration in the solutionprocess. Similar to the monolithic system, the aerodynamic system requires a large num-ber of states to transfer the information from a small number of inputs (structural modeshapes) to a small number of outputs (generalised forces). Model reduction of the linear
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system will be the focus of the next section, where we can control the number of systeminputs and outputs by selecting only a few vibration modes for the structural problem.
Note that this approach does not require transformation to frequency domain to obtainthe generalised forces, but the linearisation of the unsteady aerodynamics is limited to aspecific aeroelastic reference condition with the dynamic pressure as a parameter. Thisrequires interpolation of the aerodynamic (reduced-order) model between different pa-rameter values, such as flight speed or altitude, to enlarge the flight envelope of veryflexible aircraft configurations [34]. However, if the linearisation is done around the un-deformed reference, as in DLM solutions, it is easy to show that the aerodynamic loadsin the resulting linear system, Eq. (10), are proportional to the forward-flight dynamicpressure. This would generate reduced-order aerodynamic models which are independentof the flight conditions, but it also constrains the admissible wing kinematics and will notbe further investigated in this paper.
The size of the linear aerodynamic system presented in Eq. (10) is dominated by thenumber of vortex panels used in the discretisation of lifting surfaces and the wake ofthe aircraft. Wake panels are needed to capture the unsteady effects, but they are alsoan excellent target for model reduction. A direct approach to reduce the size of theaeroelastic system is modal projection (and truncation) of the system states on a fewgeneralised coordinates [16]. However, as we discussed in the introduction, this approachmay require many states in the reduced-order model to capture the internal dynamics ofthe flow [22,26]. A more efficient basis function can be found by balancing the system sothat the input-output mapping drives the reduction process [22–24].
By balancing the states of the linear aerodynamic subsystem, the transmission path be-tween the system aerodynamic and structural inputs and unsteady aerodynamic outputsis captured in an optimal way. This approach was first discussed by Moore [23], whilea detailed description is provided by Glover [24]. Information about the past input andfuture output energy, expressed in terms of the system controllability and observabilityGramians, is used to identify a linear similarity transformation T that balances the stablesystem {A,B,C,D}, such that each balanced state is equally controllable and observable.The balanced representation of the aerodynamic system defined in Eq. (10) can then bewritten in terms of the transformed aerodynamic states, Γ = T−1Γ, as
∆Γn+1 = T−1AT∆Γn + T−1 (BSΦ∆uΦn +BF∆uF
n)
∆ynF = CT∆Γn + Φ>DSΦ∆uΦn + Φ>DF∆uF
n.(11)
Note that expressions for the discrete-time counter part exist, but we will follow thestandard implementation of these methods, which involves a bilinear transformation tocontinuous time prior to any model reduction [35].
The similarity transformation, T , is obtained using the controllability and observabil-ity Gramians, Wc and Wo, respectively, which are given for stable LTI systems by thealgebraic Lyapunov equations,
AWc −WcA> +BB> = 0,
A>Wo −WoA+ C>C = 0.(12)
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with B =[B>S B>F
]>. The controllability Gramian, Wc, is correlated inversely to the input
energy required by an actuator to reach a certain state and the observability Gramian,Wo, on the contrary, is related directly to the energy required by a sensor to observe aninitial state [36].
The square roots of the eigenvalues of the Gramian product, WcWo, are the Hankel singu-lar values (HSV) of the system, denoted as σ1 ≥ σ2 ≥ · · · ≥ σn > 0 [37]. The HSV are in-variant to the transformation and the Gramians of the balanced system, Wc = T−1WcT
−>
and Wo = T>WoT , are equal and diagonal, Wc = Wo = Σ = diag (σ1, ..., σn). Hence thestates with the largest HSV are the ones most involved in the energy transfer between thesystem inputs and outputs and the least controllable/observable states can be truncated.
The balancing transformation, T , is obtained here using the Square Root Method [38],which involves the Choleski factorisation of the system controllability and observabilityGramians, Wc and Wo. Reduction of the balanced aerodynamic system, obtained fromEq. (11), is finally achieved by truncating the corresponding rows and columns of thebalancing transformation, T , according to the HSV of each balanced state, σi.
4 OPEN-LOOP GUST RESPONSE OF A HIGH-ASPECT-RATIOAIRCRAFT
This section will demonstrate the reduced-order modelling approach on a representativeHALE aircraft configuration modelled in SHARP. The vehicle characteristics will be firstdefined in Subsection 4.1. To explore the use of linear aeroelastic/flight dynamic toolsin the certification process of large vehicle configurations [9], the HALE aircraft will befirst exposed to a uniform discrete gust (Subsection 4.2). More efficient, higher-aspect-ratio platforms are more susceptible to spanwise non-uniform gust distributions. Hence,in the second study (Subsection 4.3) aircraft responses to non-uniform discrete gusts willbe explored to expose the system nonlinearities and to demonstrate the model reductionprocess over a range of excitation frequencies and amplitudes.
4.1 Definition of HALE aircraft geometry
The flexible HALE configuration used for the numerical studies in this work is shown inFigure 3. The vehicle was introduced in previous work [8, 39] with a detailed descriptionof the aircraft and its material properties which are repeated in Table 1 for completeness.The aircraft is similar to those by Patil et al. [40] and Murua et al. [5] but it includeswinglets to increase resistance against spiral divergence and the point mass of 50 kg wasmoved to the centre of the main wing to guarantee static pitch stability [5]. The dihedralmembers are rigidly linked to the main wing at both ends at an angle of 20 deg. Thevehicle is powered by two massless propellers, which are modelled as point forces alsorigidly linked to the main wing, as defined in Figure 3. The mass per unit length of thefuselage is the same as that of the horizontal and vertical tail planes, and thus the totalmass of this aircraft, including payload and structural mass, is 75.4 kg.
Following the definition in Table 1, the aircraft consists of a large-aspect-ratio flexiblemain wing and the fuselage and T-tail are assumed to be rigid. The flexible vehicleis trimmed at a free-stream velocity of V∞ = 30 m/s and an altitude of 20 km (airdensity ρ = 0.0889 kg/m3) to obtain the geometrically-nonlinear, deformed configuration
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Payload, 50 kg1 m
5 mElevator, 0.25 m
10 m
1 m
Aileron, 4 m x 0.25 m Aileron, 4 m x 0.25 m
1.25 m 3.75 m20 deg
2 m 12 m 4 m
x
y
z
y
Figure 3: Undeformed HALE aircraft geometry (not to scale). [8]
at steady level flight. Trim results for this flexible configuration are presented in Hesseet al. [8]. A trim solution was obtained using 4 bound aerodynamic panels per chordwisemeter and 1 panel per spanwise meter of lifting surfaces, leading to 4218 aeroelastic states(3525 aerodynamic, 684 structural and 9 rigid-body dynamic states) in the subsequenttransient analyses. Figure 4 shows the trim deformations of the HALE configurationwith wing tip deflections of 13% compared to the semispan of the main wing, B = 16m. The resulting large effective wing dihedral at steady state has a significant effect onthe stability characteristics of the vehicle [8], which is evident in Table 2 comparing the(flexible) flight dynamics modes of the flexible configuration to the rigid aircraft modes.Note in particular the effect of flexibility on the lightly-damped phugoid and spiral modes.
Table 1: HALE aircraft properties. [8, 40]
Main wing Tail plane
Chord, c 1 m 0.5 m
Semi-span, B 16 m 2.5 m
Elastic axis (from l.e.) 0.5 m 0.25 m
Center of gravity (from l.e.) 0.5 m 0.25 m
Mass per unit length 0.75 kg/m 0.08 kg/m
Moment of inertia 0.1 kg·m 0.01 kg·mTorsional stiffness 2×104 N·m2 ∞Bending stiffness 4×104 N·m2 ∞In-plane bending stiffness 8×106 N·m2 ∞
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Table 2: Stability characteristics of the flexible HALE aircraft at trimmed flight compared to a rigidconfiguration.
Figure 4: Trim shape of HALE aircraft compared to rigid configuration. (Figure shows actual deforma-tions.)
4.2 Response to a Uniform Discrete Gust
Firstly, the HALE aircraft is subjected to a uniform discrete “1-cos” vertical gust to es-tablish design limit loads of the current configuration and identify critical gust lengths.This simple exercise follows the methods required for FAR Part 25 certification [9] oflarge (manned) aircraft subject to discrete or continuous gusts. Here, we will focuson the discrete vertical gust event to demonstrate the application of the state-spaceaeroelastic/flight-dynamics formulation (Eq. (8)) as an effective time-domain tool to ob-tain the large amount of data required for gust certification. The formulation can accountfor large trim deformations which is relevant for the expansion of the FAR Part 25 cer-tification procedures [9] to vehicles exhibiting geometric-nonlinear behaviour, as the oneanalysed in this work.
Following Murua et al. [15], the discrete gust event is modelled as a stationary disturbancewith a “1-cos” spatial chordwise variation while the spanwise distribution is constant inthis case. As the aircraft travels through the gust, each collocation point, k, of the boundaerodynamics panels experiences a different gust induced velocity. The spatial variationof gust velocity, Ug, at the penetration length into the gust, xg, is defined as
Ug = Uds
[1− cos
(πxgH
)]for 0 ≤ xg ≤ 2H, (13)
where H is the gust gradient length along the vehicle flight path. The design gust veloc-ity, Uds, is defined in terms of the altitude dependent reference velocity, Uref , and gustgradient, H, as
Uds = UrefFg
(H
106.7 m
)1/6
. (14)
Note that unlike the original certification all equations in this work are given in SI units.The reference velocity, Uref , is defined in Ref [9] for manned aircraft to altitudes of 18.3 km
12
and will be extrapolated to the current flight conditions at 20 km leading to Uref = 7 m/sas a reference velocity for all numerical studies presented in this work. Due to the lowstiffness properties, higher gust intensities at lower altitudes, as required for FAR Part 25certification [9], results in infeasible wing bending [15,41,42]. Hence, launching of such avehicle should be limited to relatively calm conditions. However, once the target altitudeis reached, the certification requirements remain necessary, as the aircraft is expectedto loiter at these conditions for months or even years. Hence, it is very likely that theaircraft experiences the limit reference gust velocities during operation and the flightprofile alleviation factor, Fg, in Eq. (14) is therefore assumed to be one.
−250 −200 −150 −100 −50 00
2
4
6
8
Ve
rtic
al g
ust
ve
locity [
m/s
]
Distance in x [m]
Figure 5: Variation of vertical gust velocity with the gust gradient, H, in consecutive steps from H = 5 m(black) to H = 110 m (grey) in increments of 5 m.
To investigate the dependency of the root bending moment with gust length, a series ofopen-loop gust responses will be computed for varying gust gradient lengths, H, with5 m ≤ H ≤ 110 m in increments of 5 m. Figure 5 presents the spatial distribution of thediscrete gusts with changing H and highlights the variation of the gust reference velocity,Uref , with the gust length according to Eq. (14). Note that, following the definition ofthe reference frames in Figure 3, the aircraft travels in the negative x direction. As thethe gust is modelled as a spatial event rather than a function of time, the state-spaceformulation allows to include penetration effects as the vehicle enters the gust. Thishighlights one advantage of the more general time-domain description over frequency-domain approaches commonly used for FAR Part 25 certification [9]. Note that this resultsin a large number of 188 gust inputs (the number of bound panels in the aerodynamiclattice) to the aeroelastic system.
0 1 2 3 4 5 6 7 8−200
−150
−100
−50
0
50
100
To
rsio
n m
om
en
t [N
m]
Time [s]
(a) Root torsion moment.
0 1 2 3 4 5 6 7 8−1500
−1000
−500
0
500
1000
1500
Be
nd
ing
mo
me
nt
[Nm
]
Time [s]
(b) Root bending moment around y
Figure 6: Incremental internal moments at the root of the main wing for HALE aircraft subject to discretegusts with varying gust gradients from H = 5 m (black) to H = 110 m (grey) in increments of5 m. Critical gust length is H = 30 m (dashed).
13
The vehicle gust response is first solved using the monolithic state-space formulation givenby Eq. (8) which couples the structural response with the flight dynamics of the uncon-strained aircraft. Figure 6 first presents the structural response of the HALE configurationin terms of the variation of the root torsion and bending moments of the main wing aboutthe steady-state equilibrium moments to identify the worst case gust. Due to the largebending deformations at trim leading to −1810 Nm root bending moment (compared to70 Nm in torsion), the bending moment around the y axis in addition to the trim valuesdetermines the critical gust length. A gust gradient length of H = 30 m was found toproduce the maximum root bending moment, as highlighted in Figure 6(b). The variationof the vehicle flight dynamic response with the gust gradient length, H, is presented inFigure 7. The pitch response in Figure 7(b) clearly demonstrates the penetration effectfor smaller gust gradients (H ≤ 10) as the tail of the aircraft enters the gust after 0.33 s(for flight velocity of 30 m/s and fuselage length of 10 m). The gust tuning was also exer-cised for downward (negative) discrete gusts, as required for FAR Part 25 certification [9],but due to the large trim deformations the worst-case gust occurs for upward gusts aspresented above.
0 2 4 6 8
0
2
4
6
8
Aircra
ft v
ert
ica
l ve
locity [
m/s
]
Time [s]
(a) Aircraft vertical velocity
0 2 4 6 8
−10
−5
0
5
Aircra
ft p
itch
ra
te [
de
g/s
]
Time [s]
(b) Aircraft pitch rate
Figure 7: Variation of HALE aircraft flight dynamic gust response with gust gradients from H = 5 m(black) to H = 110 m (grey) in increments of 5 m. Critical gust length is H = 30 m (dashed).
Even at these relatively high gust velocities, the transient wing deformations remain below11% of the wing semispan. This ensures that linear methods provide a good approxima-tion of the vehicle gust response [8]. So far, the gust tuning exercise was computed usingthe fully linearised EoM of the flexible aircraft including linearised flight dynamics. Theresulting state-space form of the coupled system in time domain facilitates the direct syn-thesis of load alleviation controllers [15], but is limited to small flight dynamic amplitudesand longitudinal motion. To explore the implications of the first assumption for thislongitudinal problem, Figure 8 compares the gust response for the worst-case gust usingthe linear state-space approach to the solution of the perturbed EoM which preserve thenonlinearity in the vehicle flight dynamics. Both the structural and the flight dynamicresults suggest that a fully linear approach (around a geometrically-nonlinear equilibriumpoint) can capture the dominant dynamics of flexible HALE aircraft subject to uniformdiscrete gusts.
At last, Figure 8 also superimposes the variation of the vertical gust velocities at the rootsof the main wing and T-tail to illustrate the penetration effect typical for tailed aircraftconfigurations. As the main wing enters the gust, the aerodynamic loads contribute tofurther wing bending which is catalysed when the tail enters the gust due to pitchingdown of the aircraft.
14
0 1 2 3 4 5 6 7 8
−1000
0
1000
Time [s]
Bendin
gm
om
ent [N
m] perturbed
state−space
(a) Root bending moment around y axis
0 1 2 3 4 5 6 7 8
0
2
4
6
Time [s]
Aircraftvertical
velocity[m/s]
perturbed
state−space
gust at main
gust at tail
Ver
tical
gus
tve
loci
ty [m
/s]
0
2
4
6
(b) Aircraft vertical velocity
0 1 2 3 4 5 6 7 8−2
0
2
Time [s]
Aircra
ft p
itch
rate
[deg/s
]
perturbed
state−space
(c) Aircraft pitch rate
Figure 8: Comparison of discrete gust response for the critical case with H = 30 m using the state-space(Eq. (8)) and the (nonlinear) perturbation (Eq. (4)) approach.
4.3 Response to Non-Uniform Gust Excitations
The standard uniform gust tuning provides a first starting point to determine the criticalgust loads in the design of high-aspect-ration aircraft. However, the increased span andrelatively low mass of such novel configurations increase the likelihood of a non-uniformgust event which can lead to induced bending moments that exceed the critical loadsdetermined for FAR Part 25 certification [9]. Hence, the guidelines for DARPA’s VultureII concepts [11] propose a simplified discrete gust model with a non-uniform, symmetricspanwise variation of gust intensities in addition to the “1-cos” chordwise distribution.
Following the implementation by Murua et al. [15] and DARPA’s guidelines [11], the non-uniform discrete gust event is modelled by extending Eq. (13) to account for a sinusoidalspanwise variation. Hence, the spatial variation of gust velocity, Ug, at the longitudinalpenetration length into the gust, xg, and the spanwise location, y, is now defined as
Ug = UAS
[1− cos
(πxgH
)]cos
(2πy
λg− ϕg
)for 0 ≤ xg ≤ 2H, ∀y (15)
where H is the gust gradient length along the vehicle flight path and λg is the wavelengthof the spanwise sine distribution which can include the the phase angle ϕg. The gustintensity is adjusted for the non-uniform case, such that [11]
UAS =Uds
2
(b
1524 m
)1/3
, (16)
15
where Uds is the design (uniform) gust velocity defined in Eq. (14) and the denominatorin Eq. (16) originates from the characteristic length of the reference gust from the vonKarman model.
Figure 9: Non-uniform DARPA gust profile for H = 40 m, λg = 32 m and ϕg = π/2 at the operatingaltitude of 20 km. Figure also includes trimmed aircraft to show relative scale.
Figure 9 shows an example of the resulting gust profile for a gust gradient of H = 40 mand the spanwise wavelength of λg = 32 m. Although the largest root bending momentcan be expected for a symmetric non-uniform gust [11], this work also includes the phaseangle ϕg to study the dynamic response of asymmetric gust events, as shown in Figure 9for the antisymmetric case with ϕg = π/2. Note that the spanwise variation is assumedto be continuous, which is important as the aircraft will exhibit a lateral response due tothe asymmetric gust excitation. Due to the lateral rigid-body motions, we will first solvethe vehicle gust response using the perturbation approach, which captures the nonlinearrigid-body dynamics of the aircraft as presented in Section 2.2. The effect of linearisingalso the vehicle flight dynamics in the prediction of the non-uniform gust responses willbe shown in Figure 12.
Firstly, we identify the dependency of the root bending momement with the non-uniformdiscrete gust. Figure 10 presents the gust response for varying longitudinal gust gradientlengths, H, with 10 m ≤ H ≤ 100 m in increments of 10 m, and spanwise gust length,λg, with 8 m ≤ H ≤ 40 m in increments of 4 m. To also include the effect of theplacement of the spanwise distribution, the phase angle ϕg is also varied where ϕg = 0corresponds to the symmetric non-uniform case, as proposed in Ref. [11], and ϕg = π/2is the antisymmetric case.
Figure 10 clearly shows that longer gusts along the vehicle flight path lead to largerbending moments for each ϕg, which is consistent with the results by Dillsaver et al. [14].Hence, the gust tuning was stopped at a maximum gradient length of H = 100 m,which is assumed to the worst realistic gust case, but a future comprehensive studyshould include even longer gusts. The gust tuning in the lateral direction, on the other
16
(a) ϕg = 0 (b) ϕg = π/4
(c) ϕg = π/2 (d) ϕg = 3π/4
Figure 10: Variation of maximum root bending moment of main wing due to non-uniform DARPA gustwith chordwise gradient H and spanwise gust length λg and phase ϕg. Magnitudes are givenin Nm and the markers highlight the worst gust case for each ϕg.
hand, reveals that the worst gust cases occur for spanwise wavelengths between 75 and87.5% of the overall span of the main wing (32 m), which demonstrates the importantdynamic effects at shorter spanwise excitations in the gust response of slender high-aspect-ratio configurations. This is less intuitive since wavelengths equal to the aircraft wingspan minimise the alleviation effect due to vehicle plunging motion [15]. Because of this“clamping ”effect, the symmetric case with ϕg = 0 in Figure 10(a) exhibits the largestbending deformations and constitutes the worst gust event for the HALE configurationanalysed in this work. This is consistent with the DARPA guidelines [11] and in thefindings in Refs. [14,15] for very flexible aircraft.
Figure 11(a) shows the elastic and flight dynamic response of the HALE aircraft sub-ject to symmetric non-uniform discrete DARPA gust excitations for the critical spanwisewavelength of λg = 24 m. The variation of the longitudinal gust gradient, H, from 10 to100 m demonstrates the quasi-steady nature of the non-uniform symmetric gust responseleading to bounded vehicle flight dynamics during the gust event. The worst gust case forthe range of longitudinal gust gradients studied in this work is highlighted with dashedcurves.
Although the spanwise symmetric gust leads to much higher bending moments and hencecritical gust loads, it is worthwhile to further investigate the asymmetric gust responsesalso presented in Figures 10-11 to demonstrate the effect of non-uniform gusts on thevehicle lateral dynamics. This is especially relevant for high-aspect-ratio platforms with
17
0 2 4 6 8−500
0
500
1000B
en
din
g
m
om
en
t [N
m]
0 2 4 6 8
0
0.5
1
Aircra
ft v
ert
ica
lve
locity [
m/s
]
0 2 4 6 8−6−4−2
024
Time [s]
Aircra
ft p
itch
rate
[d
eg
/s]
(a) ϕg = 0 and λg = 24 m
0 2 4 6 8
−300
−200
−100
0
100
Be
nd
ing
mo
me
nt
[Nm
]
0 2 4 6 8−0.4
−0.2
0
0.2
Aircra
ft v
ert
ica
lve
locity [
m/s
]
0 2 4 6 8
−2
−1
0
1
Aircra
ft p
itch
rate
[d
eg
/s]
0 2 4 6 8
0
5
10
Aircra
ft r
oll
rate
[d
eg
/s]
Time [s]
(b) ϕg = π/2 and λg = 28 m
Figure 11: Variation of HALE aircraft response to (a) symmetric and (b) anti-symmetric non-uniformDARPA gust with varying longitudinal gust gradients from H = 10 m (black) to H = 100 m(grey) in increments of 10 m. Worst gust length is H = 100 m in both cases (dashed).
lightly-damped or even unstable lateral stability characteristics, as demonstrated in pre-vious work [8]. Hence, Figure 11(b) shows the elastic and flight dynamic response due toantisymmetric non-uniform gust excitations for λg = 28 m, which is the critical spanwisewavelength for the worst gust case with H = 100 m (highlighted with dashed curves).The variation of the gust gradient length, H, shows that the vehicle rigid-body motionsincrease with increasing H which results in maximum root bending moment long after theantisymmetric gust disturbance. This dynamic behaviour is opposite to the symmetriccases in Figure 11(a) and highlights the importance of also accounting for asymmetricgust cases in the design of highly efficient aircraft.
Due to the possible lateral motions in the flight dynamic response of non-uniform gusts,the above gust tuning exercise was done using the perturbation approach which capturesthe possibly large and/or lateral rigid-body motions of the aircraft. To demonstrate the ef-fect of flight dynamic nonlinearities, Figure 12 compares the non-uniform gust response forthe critical chord- and spanwise gust lengths obtained using the (nonlinear) perturbationapproach, Eq. (4), to the fully-linearised solution of the state-space aeroelastic/flight-dynamics system defined in Eq. (8). Similar to the uniform case presented in Subsec-tion 4.2, the response to the symmetric non-uniform gust shown in Figure 12(a) remainspredominantly linear even for large gust velocities. This suggests that more efficient lin-ear methods provide a suitable tool to determine gust loads even for high-aspect-ratioHALE configurations subject to critical uniform and spanwise non-uniform symmetricgusts. Note that for more flexible configurations geometric nonlinearity due to extensivewing bending may contribute another nonlinear effect. However, for the symmetric casesstudied here, with the gust velocities given by the DARPA guidelines [11], transient wing
18
tip deformations never exceeded 10% of the wing semi-span (in addition to the aircrafttrim deformations).
0 2 4 6 8−1000
−500
0
500
1000
1500
Be
nd
ing
mo
me
nt
[Nm
]
Time [s]
perturbed
state−space
(a) ϕg = 0 and λg = 24 m
0 2 4 6 8−1000
−800
−600
−400
−200
0
200
400
Time [s]
Be
nd
ing
mo
me
nt
[Nm
]
perturbed
state−space
(b) ϕg = π/2 and λg = 28 m
Figure 12: Comparison of (a) symmetric and (b) anti-symmetric non-uniform gust responses for thecritical cases with H = 100 m using the state-space (Eq. (8)) and the (nonlinear) perturbation(Eq. (4)) approach.
For the spanwise non-uniform antisymmetric gust, Figure 12(b) demonstrates that linearmethods fail to accurately capture the vehicle response. The state-space solution dramat-ically overpredicts the structural response producing root bending moments close to thesymmetric case, as the gyroscopic effects due to the induced roll response are not captured.This highlights the importance of nonlinear, time-domain methods in the prediction ofcritical gust loads for HALE aircraft subject to non-uniform possibly-asymmetric discretegusts.
1 3 5 7 9 11 13 15 17 190
1
2
3
x 104
Ha
nke
l sin
gu
lar
va
lue
Balanced state
Figure 13: Hankel singular values of the balanced aerodynamic system for the HALE aircraft. Only thefirst 20 states are plotted.
The added numerical burden due to the higher fidelity makes time-domain methods lesssuitable for efficient aircraft design. This will be tackled next using the reduced-ordermodelling approach presented in Section 3 to reduce the relatively large aerodynamicsystem which consists of 3525 states. As introduced in Section 2.3.2, the (generalised)coordinates and velocities of the aircraft structural and flight dynamics response in addi-tion to the vertical gust velocities at each bound vortex panel constitute the inputs to theaerodynamic system which maps these inputs to the (generalised) aerodynamic loads. Toobtain an a prior criteria for the subsequent reduction, Figure 13 first presents the Hankelsingular values of the balanced aerodynamic system which indicate the contribution ofeach balanced state to the system input-output mapping and provide an error bound ofthe model reduction. Only the first 20 HSV (of the original 3525) are plotted here, as theerror in the model reduction decays significantly over this range.
19
0 2 4 6 8
−300
−200
−100
0
100B
en
din
gm
om
en
t [N
m]
0 2 4 6 8−0.4
−0.2
0
0.2
Aircra
ft v
ert
ica
lve
locity [
m/s
]
0 2 4 6 8
−2
−1
0
1
Aircra
ft p
itch
rate
[d
eg
/s]
0 2 4 6 8
0
5
10
Aircra
ft r
oll
rate
[d
eg
/s]
Time [s]
full−order 1 state 5 states 17 states
(a) Long (critical) gust with H = 100 m
0 0.5 1 1.5 2−60
−30
0
30
60
Be
nd
ing
mo
me
nt
[Nm
]
0 0.5 1 1.5 2−2
0
2
4x 10
−3
Aircra
ft v
ert
ica
lve
locity [
m/s
]
0 0.5 1 1.5 2−0.04
−0.02
0
0.02
0.04
Aircra
ft p
itch
rate
[d
eg
/s]
0 0.5 1 1.5 2−3
0
3
6
9A
ircra
ft r
oll
rate
[d
eg
/s]
Time [s]
full−order 1 state 5 states 17 states
(b) Short gust with H = 10 m
Figure 14: Comparison of different reduced-order responses following spanwise non-uniform antisymmet-ric gust excitations with λg = 28 m and ϕg = π/2.
Next, the unsteady aerodynamic system is reduced from 3525 to one, five and 17 states,with the corresponding HSV highlighted in Figure 13, following the balanced truncationapproach introduced in Section 3. Figure 14(a) shows the elastic and flight dynamicsresponse of the HALE aircraft subject to the worst-case antisymmetric DARPA gust withH = 100 m, λg = 28 m and ϕg = π/2 for the different reduced-order systems. Due to thequasi-steady nature of the response to the long gust case, only one aerodynamic state isneeded to accurately capture the 3D aerodynamic loads of the flexible HALE configura-tion. This confirms previous findings based on the open-loop response of the same vehiclesubject to commanded control surface inputs [39] and highlights the potential of balancingmethods, even for relatively large systems, to provide significant model reduction.
To also demonstrate the model reduction approach for higher frequency dynamics withpossible unsteady aerodynamic effects, Figure 14(b) presents the vehicle response for ashort antisymmetric gust with H = 10 m. Comparison of the flight dynamic responsesshows that the reduced-order models fail to predict the longitudinal response accurately,which appears to have a negligible effect on the root bending moment of the main wing.In fact, as few as five aerodynamic states where found to be sufficient to accurately predictthe elastic response even for this lateral problem with 3D unsteady aerodynamic effects.
Because of the general description of the gust inputs in terms of the vertical gust velocitiesat each aerodynamic panel, the reduced-order model needs to be computed only once foreach vehicle configuration and flight condition. The resulting reduced-order aeroelastic
20
and flight dynamics model can then be used to perform a large number of load studies in-cluding uniform and spanwise non-uniform asymmetric discrete gusts at a computationalburden similar to current aeroelastic methods based on frequency doublet lattice aero-dynamics and the mean-axes approximation. The proposed formulation, however, canaccount for very flexible configurations with geometrically-nonlinear wing deformationsat trimmed flight and incorporates important 3D unsteady aerodynamic effects [33] andall couplings between the vehicle structural and flight dynamics [8].
5 CONCLUSION
This paper has presented a model reduction approach on the coupled flight dynamics andaeroelastic response of flexible aircraft with large static trim deformations. Under theassumption of small wing deformations in the subsequent system dynamics, the unsteadyaerodynamic loads were obtained in time domain using the three-dimensional unsteadyvortex lattice method, which was linearised around the trimmed flight condition witha deformed aerodynamic lattice and a force-free wake. Consistent linearisation of thestructural degrees of freedom allowed the nonlinear flexible-body dynamics to be projectedonto a small number of dominant vibration modes of the unconstrained structure andpreserves all gyroscopic couplings between rigid-body and structural dynamics.
This aeroelastic modelling approach has been exercised on a representative HALE con-figuration first subject to discrete uniform gusts, as required for certification of largetransport aircraft, to explore the validity of standard linear tools in the computation ofcritical gust loads. For the longitudinal gust case explored first, a fully linear approach– including linear flight dynamics – was able to capture the aeroelastic/flight-dynamicresponse of the very flexible vehicle around a geometrically-nonlinear trim equilibriumincluding penetration effects of the T-tailed HALE configuration. This is relevant forthe certification of very-flexible-wing transport aircraft and may also be important in thedefinition of airworthiness requirements for unmanned vehicles.
The latter, however, can also be subject to spanwise non-uniform gusts due to the in-creased wing span of novel unmanned aircraft concepts. Hence, in the second study thereduced-order modelling approach was demonstrated for non-uniform asymmetric discretegusts. Based on a general description of the gust inputs, which accounts for the inducedgust velocities at each aerodynamic panel along the lifting surfaces, the spatial and tem-poral variation of the vertical gust velocities in the longitudinal and spanwise directionwas tuned to identify the critical gust case. Variation of the spanwise wavelength clearlyaffected the dynamic response of the vehicle with critical lateral wavelengths around 25%below the overall span of the aircraft main wing. Tuning of the longitudinal gust length,however, suggests a quasi-steady behaviour whereby the elastic response appears to fol-low the gust length for the symmetric case. Variation also of the spanwise placement ofthe gust resulted in nonlinear flight dynamics which were accurately captured using theproposed consistent linearisation. Subsequent balanced truncation of the linear aerody-namic system was demonstrated on the antisymmetric gust response of the vehicle, whichhighlighted the potential of balancing methods with dramatic model reductions whilethe inherent three-dimensional features of the unsteady flow and the nonlinearity in thevehicle flight dynamics are preserved.
21
In summary, the proposed balanced model reduction approach of the aeroelastic frame-work provides a powerful tool to simulate the dynamics of highly-efficient very flexibleaircraft with a similar computational cost of linear frequency approaches based on thedoublet lattice method. The proposed method however removes kinematic restrictionsand obtains the aerodynamic loads directly in time domain, which enables the analysisof very flexible aircraft subject to spanwise non-uniform gust excitations and makes italso ideal for control synthesis and optimisation. Moreover, even though the aeroelasticframework is projected onto a few vibration modes of the unconstrained structure, it ac-counts for the inertial coupling between the vehicle structural response and the nonlinearrigid-body dynamics.
ACKNOWLEDGMENTS
The authors would like to acknowledge the contribution from Joseba Murua in the gustmodelling of the state-space aeroelastic system. The work of Henrik Hesse is sponsoredby the UK Engineering and Physical Sciences Research Council (EPSRC). This supportis also gratefully acknowledged.
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