Paper n. 117

FULLY COUPLED STRUCTURAL-UNSTEADY AERODYNAMICS MODELLING

FOR AEROELASTIC RESPONSE OF ROTORCRAFT

G. Bernardini, J. Serafini, M. Molica Colella, M. Gennaretti

University Roma TreDept. of Mechanical and Industrial EngineeringVia della Vasca Navale 79 - 00146 Rome, Italy

e-mail: g.bernardini@uniroma3.it

Abstract

This paper deals with a computational aeroelastic toolfor the analysis of rotorcraft. It has been developed bycoupling a nonlinear beam model for blades (and wing)structural dynamics description with a boundary inte-gral equation solver for the prediction of aerodynamicloads. This solver is based on three-dimensional, un-steady, potential aerodynamic formulation. The Galerkinmethod is used for the spatial integration, whereas theperiodic blade (and wing) response is determined bya harmonic balance approach. This aeroelastic modelyields a unified approach for aeroelastic response andblade pressure prediction that may conveniently be usedfor aeroacoustic purposes and, in addition, is able toexamine configurations where blade-vortex interactionsand multiple-body aerodynamic interactions occur. Nu-merical results show the capability of the aeroelastic toolto evaluate blade response and vibratory hub loads fora helicopter main rotor in level flight conditions, andexamine the efficiency and robustness of the differentcomputational algorithms that might be applied in thepresented aeroelastic solver. A sensitivity analysis ofthe predictions on the aerodynamics model used will bealso discussed.

Introduction

The aim of this work is to present some features andrecent advances of a computational tool for the analysisof the aeroelastic response of helicopters and tiltrotors.Rotorcraft are affected by strong couplings between elas-tic deformations and aerodynamic loads, which signif-icantly contribute to the vibrations transmitted to thefuselage, as well as, to the emitted noise. A great varietyof flight conditions may be experienced by rotary-wingaircraft, each yielding specific aerodynamic environmentand corresponding elastic response. For instance, in thedescent flights of helicopters strong BVI (Blade-VortexInteraction) characterizes both vibrations and noise lev-els, while for tiltrotors in airplane mode the impact of

proprotor wake on the wing is a major contribution tovibrations which requires an accurate simulation for pro-viding a reliable aeroelastic response.

The computational aeroelastic tool for rotary-wing air-craft presented is the result of the research work doneby the authors during the last years [1]. It uses a three-dimensional, unsteady, aerodynamic solver based on theBoundary Element Method (BEM) for potential flowsintroduced in Ref. [2] as a development of the formula-tion presented in Ref. [3], that is able to predict strongBVI effects, with inclusion of the wake-wing impactsarising on tiltrotor configurations. This solver can beapplied both for direct evaluation of the aerodynamicloads and for the calculation of the wake inflow to beused within a section-load aerodynamic model. Thestructural model for the helicopter rotor blades (andwing-proprotor systems, as well) is based on the beamtheory developed in Ref. [4]. It consists of a nonlinearbending-torsional model (with inclusion of the gimbalequations, if it is present). The resulting set of integro-differential equations is discretized spatially through theGalerkin approach based on the natural modes of vi-bration of a cantilever, nonrotating beam (with addi-tional rigid-body modes, if necessary). The aerody-namic and the structural solvers are fully coupled byforcing the structural equations with the aerodynamicgeneralized loads computed by the BEM approach andevaluating the aerodynamic boundary conditions fromthe deformations given by the structural model. Con-sidering arbitrary steady flight conditions, the aeroelas-tic responses given in terms of blade deflections and vi-bratory loads (in rotating and nonrotating frames) areobtained through a harmonic balance procedure thatiteratively updates periodic deformations and aerody-namic loads until convergence. When the aerodynamicsolver is applied for the direct evaluation of loads (fully-coupled mode solution), the blade pressure distributionis among the outputs of the solution procedure which,in turn, may be applied in aeroacoustic formulations forthe identification of the noise field emitted.

Aerodynamic and structural models, as well as the tech-niques applied for the numerical integration of the fullycoupled aeroelastic system will be described with detailsin the following. In addition, a numerical investigationof the efficiency and robustness of different computa-tional algorithms that might be applied in the fully-coupled mode aeroelastic solution will be examined, alongwith the sensitivity of the predictions on the aerody-namic model used.

Structural dynamics modelling

Beam-like models are applied to describe the structuraldynamics of helicopter rotor blades, as well as that ofwing and proprotor blades of tiltrotors. These are basedon the nonlinear bending-torsion formulation presentedin Ref. [4], that is valid for straight, slender, homoge-neous, isotropic, nonuniform, twisted blades, undergo-ing moderate displacements.

For rotor blade analysis, second order terms are re-tained in the equations after application of an order-ing scheme dropping third-order terms (with respect tobending slope) not contributing to damping. The radialdisplacement is eliminated from the set of equations bysolving it in terms of local tension, and thus the resultingstructural operator consists of a set of coupled nonlineardifferential equations governing the bending of the elas-tic axis and the blade torsion [5]. If present, the effectsof blade precone angle, hinge offset, torque offset andmass offset are included in the model.

For tiltrotor analysis, the interactions between wing andpropellers are such that the wing deformation affectsproprotor blades kinetics and, in turn, proprotor hubloads force wing dynamics [6].

Aerodynamics modelling

The aerodynamics of helicopters is significantly affectedby the interactions among main rotor, fuselage and tailrotor, as well as the aerodynamics of tiltrotors is sig-nificantly affected by the interactions between propro-tors and wing. In particular, a crucial role is playedby the interactions between bodies and wakes (body-wake interactions, BVI). These include blade-wake andwing-wake impacts that produce impulsive pressures onthe body surfaces, and then strongly contribute to thevibratory loads transmitted to the airframe and gener-ate extremely annoying acoustic effects. Therefore, asimulation tool applied for the optimal design of newgeneration helicopters and tiltrotors has to be able topredict BVI occurrence and corresponding aeroelasticand aeroacoustic effects. In view of this, a boundaryintegral formulation for potential flows suited for theprediction of strong aerodynamic interaction effects hasbeen presented in Ref. [2]. It is a development of theformulation introduced in Ref. [3] and further extendedto rotors in forward flight in Ref. [7]. A brief outline ofthe potential-flow formulation suited for BVI analysis isgiven in the following.

It introduces the decomposition of the potential fieldinto an incident field, ϕ

I, and a scattered field, ϕ

S. The

scattered potential is generated by sources and doubletsover the body surfaces, S

B, and by doublets over the

wake portion that is very close to the trailing edge from

which emanated (near wake, SN

W). The incident poten-

tial is generated by the doublets over the complemen-

tary wake region that compose the far wake, SF

W.This is

the wake portion that may come in contact with otherblades. The scattered potential is discontinuous across

SN

W, whereas the incident potential is discontinuous across

SF

W. Hence, as demonstrated in Ref. [2], for ϕ = ϕ

I+ϕ

S

the scattered potential is obtained by

ϕS(x, t) =

∫SB

[G (χ− χ

I)− ϕ

S

∂G

∂n

]dS(y)

−∫SN

W

∆ϕS

∂G

∂ndS(y) (1)

where G = −1/4π r is the unit-source solution of the 3DLaplace equation, with r = ‖y − x‖, while ∆ϕ

Sis the

potential jump across the wake surface, that is knownfrom the past history of the potential discontinuity atthe trailing edge of the corresponding body through theKutta-Joukowski condition [2]. In addition, χ = v · n,with v representing the body velocity and n its outwardunit normal, whereas χ

I= u

I· n, with the velocity

induced by the far wake, uI, given by

uI(x, t) = ∇ϕ

I(x, t)

= −∇∫SF

W

∆ϕS(y

TE

W, t− ϑ)

∂G

∂ndS(y) (2)

The incident potential affects the scattered potentialthrough the induced-velocity term, χ

I, and, in turn, the

scattered potential affects the incident potential by itstrailing-edge discontinuity that is convected along thewake and yields the intensity of the doublet distributionover the far wake.

Obtaining the discrete form of Eq. (2) by using N pan-els over the far wake and recalling the vortex-doubletequivalence, the incident velocity field may be evaluatedthrough the following expression

uI(x, t)≈−

N∑k=1

∆ϕS(y

TE

Wk, t− ϑk)

∫Ck

∇xG× dy (3)

where yTE

Wkis the trailing edge position where the wake

material point currently in yWk

emanated at time t−ϑk,Ck denotes the contour line of the k-th far wake panel,and ∇x denotes gradient operator with respect to thevariable x. This equation represents the velocity fieldgiven by the Biot-Savart law applied to the vorticeshaving the shape of the far wake panel contours and

intensity ∆ϕS(y

TE

Wk, t − ϑk). Equation (3) is applied to

evaluate both the term χI

in equation (1) and, once ex-tended to the whole wake, the velocity field from whichthe wake shape evolution is determined (free-wake anal-ysis). Note that for an accurate prediction of BVI phe-nomena it is usually essential the accurate evaluation ofthe wake distortion in that a crucial role is played bythe relative positions between body and wake.

The final step of the formulation consists of introducingin Eq. (3) a finite-thickness vortex model that assures

a regular distribution of the induced velocity within thevortex core, and thus a stable and regular solution evenin body-vortex impact conditions [2].

Equation (1) is solved numerically by boundary ele-

ments, i.e., by dividing SB

and SN

Winto quadrilateral

panels, assuming ϕS, χ, χ

Iand ∆ϕ

Sto be piecewise con-

stant (zero-th order, boundary element method - BEM),and imposing that the equation be satisfied at the centerof each body element (collocation method) [2].

Once the potential field is known, the Bernoulli theoremyields the pressure distribution (see Appendix A) andhence blade loads may be evaluated through integration.

Aerodynamic loads from airfoil theories

Akin to the solution procedures applied in several aeroe-lastic prediction tools commonly used in rotorcraft ap-plications, in the present solver non fully-coupled solu-tions may be obtained. These are based on the evalua-tion of the aerodynamic forcing terms obtained as span-wise integration of the loads given by sectional aero-dynamics models derived from airfoil unsteady aerody-namics theories, and corrected with wake inflow (fromthe BEM solver) to take into account three-dimensionaleffects due to wake vortices.

For a thin, straight airfoil moving in an incompress-ible flow, following the Greenberg theory [8] (which isthe extension of the Theodorsen theory [9] to pulsatingairstream) it is possible to determine the aerodynamicforce acting on it by combining the non-circulatory lift,Lnc, orthogonal to the chord with the circulatory lift,Lc, directed along the normal to the relative wind (see,for instance, Ref. [5]). Specifically, for ρ denoting theair density and b denoting the airfoil semi-chord length,the non-circulatory lift is expressed as

Lnc(t) = π ρ b2 v n1/2

(4)

where v n1/2

denotes the time derivative of the normalcomponent of the relative wind evaluated at the airfoilmid point (positive upwards). The circulatory lift, thatin practical applications is the most relevant load con-tribution, is given by the following expression

Lc(t) = 2π ρ b V F−1[C(k) v n

3/4

](5)

where V denotes the relative wind velocity, F denotesFourier transformation and v n

3/4= F(v n

3/4), with v n

3/4

representing the normal component of the relative windevaluated at the airfoil 3/4-chord point. In addition,C(k) is the lift deficiency (complex) function definedby Theodorsen [9] in terms of the reduced frequency,k = ω b/V , where ω is the variable in the Fourier do-main and V is the mean (or reference) value of V whichis responsible for the shed vorticity convection (see Ref.[10] for a detailed analysis of the effects of non-uniformconvection of the shed vorticity along the wake). For aharmonic v n

3/4input, C(k) defines gain and phase shift

of the Lc/V harmonic response. The set of the aerody-namic loads acting on the airfoil includes also the pitch-ing moment about the 1/4-chord point (positive clock-wise for the relative wind directed from left to right),

which is given by [8]

M1/4(t) = − b2Lnc − π ρ

b3

2

(V ωa +

b

4ωa

)(6)

where ωa is the angular velocity of the airfoil. Oncev n

1/2, v n

3/4, ωa and V are expressed in terms of the blade

degrees of freedom (a major elastic contribution fromblade lag may appear also in V ), Eqs. (4), (5) and (6),combined with a model to take into account drag effects,may be applied to develop aeroelastic formulations forwings and rotors.

Harmonic balance for aeroelasticresponse analysis

The equations governing the blade structural dynam-ics coupled with the unsteady aerodynamic load mod-elling yield the aeroelastic integro-differential equationsto be integrated. Here, the space integration is per-formed through the Galerkin approach, starting fromthe description of the elastic axis deformation as a lin-ear combination of shape functions that satisfy homo-geneous boundary conditions. The resulting aeroelasticsystem consists of a set of nonlinear ordinary differentialequations of the type

M(t) q + C(t) q + K(t)q = f nlstr(t,q) + faer(t,q) (7)

where q denotes the vector of the Lagrangian coordi-nates, whereas M,C, and K are time-periodic, mass,damping, and stiffness structural matrices representingthe linear structural terms (note that these matrices aretime-variant because of the cyclic pitch contribution).Nonlinear structural contributions are collected in theforcing vector f nl

str(t,q), whereas vector faer(t,q) collectsthe generalized aerodynamic forces. The aerodynamicloads may be obtained either by integration along bladeand wing surfaces of the pressure distribution given bythe BEM solver outlined above (fully-coupled solution),or by spanwise integration of the expressions in Eqs. (4)-(6) derived from the airfoil theory, combined with thewake inflow predicted by the BEM solver (non fully-coupled solution).

Since the aim is the prediction of the aeroelastic steadyperiodic response during forward flight, the aeroelasticsystem in Eq. 7 is solved by using the harmonic balanceapproach. It is a methodology suitable for the analysisof the asymptotic solution (as time goes to infinity) ofdifferential equations forced by periodic terms, as in thepresent case. The harmonic balance solution consists of:(i) express LHS and RHS of the aeroelastic system (Eq.7) in terms of their Fourier series; (ii) equate the result-ing coefficients; (iii) solve the corresponding algebraicsystem in terms of the unknown Fourier coefficients ofthe Lagrangian coordinates of the problem. Specifically,expressing q and f = f nl

str + faer in terms of the followingFourier series

q(t) = q0 +

N∑n=1

[qcn cos(Ωnt) + qs

n sin(Ωnt)]

f(t) = f0 +

N∑n=1

[f cn cos(Ωnt) + f s

n sin(Ωnt)]

(where qcn,q

sn, f

cn and f s

n denote cosine and sine compo-nents of the n-th harmonic of q and f , whereas Ωn = nΩ,with Ω representing the fundamental frequency i.e., therotor angular velocity) and then combining with Eq. 7yields the following representation of the aeroelastic sys-tem harmonic components[

M + C + K]q = f (8)

whereqT =

qT0 qc

1T qs

1 qc2T qs

2T · · ·

and

fT =fT0 f c1

T fs1T f c2

T fs2T · · ·

Matrices M, C and K in Eq. 8 come out from Eq. 7 bycombining the harmonics of the q, q and q terms withthe harmonics of the matrices M,C, and K (see Ref.[1] for details). In particular, if M,C, and K were con-stant matrices, in Eq. 8 one would have block-diagonalmatrices and each harmonic of q would depend onlyon the same harmonic of f (the q-harmonics equationswould be uncoupled). Instead, in the problem underexamination the structural matrices are periodic andhence, once expressed in terms of the Fourier series andcombined with the harmonics of q, q and q, they yieldfully-populated M, C and K matrices, thus coupling thealgebraic equations for the unknown harmonics in q.

Because of the presence of nonlinear structural termsand of aerodynamic contributions in f , Eq. 8 has to besolved using an iterative procedure. To this aim, theNewton-Raphson method is applied. Specifically, theharmonic aeroelastic solution is obtained from conver-gence of the following iterative procedure (with n indi-cating the iteration step number)

qn =[M + C + K− Jaer

n−1

]−1

[fn−1 − Jaern−1 qn−1] (9)

where Jaern−1 is the aerodynamic Jacobian evaluated at

q = qn−1, while fn−1 denotes the sum of nonlinearstructural terms and aerodynamic loads evaluated atq = qn−1 (i.e., through the Lagrangian coordinatesgiven by the previous iteration step). Under this as-sumption, the aerodynamic portion of the forcing termis equivalent to faer0 + f nl

aer, with faer0 denoting the aero-dynamic load portion that is not influenced by the struc-tural deformation. For the sake of computational timesaving, the approximation of maintaining constant theaerodynamic Jacobian might be applied (indeed, thiswould avoid the evaluation of the inversion of the globalJacobian matrix at each step of the iterative process).

Evaluation of aerodynamic Jacobian

The aerodynamic Jacobian is evaluated numerically. Ob-serving that, at a given iteration step, it relates the har-monics of each perturbation structural Lagrangian vari-able to the harmonics of the corresponding generalizedaerodynamic forces, it is identified through a sequence oftime-marching harmonic responses of the BEM solver.Specifically, for a given Lagrangian variable, the pertur-bation harmonic responses to each of the 2N + 1 (small

amplitude) perturbation Fourier components consideredare evaluated. The (input and output) perturbations aredefined about the Fourier components corresponding tothe previous step solution. The Fourier components ofeach (multi-harmonic) set of output perturbation forcesdivided by the corresponding input perturbation yieldone column of the aerodynamic Jacobian matrix to beused in Eq. 9, i.e.:

Jaerij =

faeri (qj + ∆qj)− faeri (qj)

∆qj(10)

Numerical results

Here, some results from the aeroelastic formulation out-lined above are presented, with the aim of assessing theefficiency and robustness of different computational al-gorithms that might be used in the fully-coupled aeroe-lastic tool, along with analyzing the sensitivity of theaeroelastic response (blade tip deflections and vibratoryhub loads) on the used aerodynamic mode.

For these analyses, the four-bladed rotor examined inRef. [12], at which the reader is referred to for detailsabout structural and inertia properties, is considered.It has radius R = 4.93m, constant chord c = 0.395m,NACA 0012 section profile and a linear twist angle of−8. The rotor has been examined in level flight con-ditions with rotational speed Ω = 40rad/s, at advanceratios µ = 0.15 and µ = 0.3. For the two advance ratiosinvestigated, shaft angle and blade collective and cyclicpitch control angles are those determined in Ref. [12]with linear inflow.

Advance Ratio µ = 0.30

First, a convergence analysis of the iterative aeroelasticsolution algorithm is presented in Fig.1. Specifically,

Figure 1. Effect of relaxation factor on convergence

history. µ = 0.3.

it shows the effect of the relaxation factor used in theNewton-Raphson method in the error decay (note thathere the relaxation factor is defined as a coefficient thatmultiplies the vector fn−1 in Eq. 9). The relative quadraticerror is evaluated in terms of the difference between the

vibratory hub loads computed at two iteration steps.This choice is motivated by the fact that from the prac-tical point of view the interesting outcomes of the aeroe-lastic response analysis are vibratory hub loads and bladedeflections, and that the convergence of the former im-plies the convergence of the latter. For computationaltime saving, the aerodynamic Jacobian is kept constantduring the iterative process (thus yielding the so-calledfixed slope iteration procedure), equal to that evalu-ated numerically about the undeformed configuration(q = 0). The solution histories show that this approachis effective and leads to solution convergence for anyvalue of the relaxation factor, although a relaxation fac-tor close to 1 seems to increases slightly the convergencespeed.

In addition, note that a similar convergence history hasbeen achieved by using the constant aerodynamic Jaco-bian evaluated about the aeroelastic solution predictedby a simpler model based in quasi-steady aerodynamics(this means that the aerodynamic Jacobian is not sub-ject to significant variations in the neighborhood of thestate q = 0, and thus the most convenient procedure ofevaluating it about the undeformed configuration seemsto be appropriate).

Figure 2. Effect of aerodynamic Jacobian modelling

on convergence history. µ = 0.3.

Next, Fig. 2 shows the effect of the use of aerodynamicJacobian from different aerodynamic models on conver-gence. Specifically, the results obtained by using theJacobian evaluated through the BEM solver are com-pared with those obtained by using the Jacobian eval-uated through the quasi-steady aerodynamics, as wellas with those determined from the elimination of theaerodynamic Jacobian. As expected, no convergenceis achieved when the aerodynamic Jacobian is not in-cluded in the solution algorithm, but it is interesting toobserve that a very similar negative result is obtainedwhen the Jacobian from the quasi-steady aerodynamicmodel is applied. This suggests the necessity of usingthe same aerodynamic solver for solution and Jacobianevaluation.

Then, in order to examine the possibility of reducing thecomputational costs, a further survey was conducted onthe impact of the BEM mesh used for the evaluationof the Jacobian matrix on the solution convergence (in-

deed, as explained above, the numerical identification ofthe aerodynamic Jacobian might require the evaluationof a high number of aeroelastic responses, thus resultingtoo computationally expensive). Three different mesheshave been considered, as given in Table 1 (note that‘Solution’ mesh indicates the mesh applied also for theevaluation of the aerodynamic loads forcing the aeroe-lastic equations).

Time CalculationNB NW NT Step [s] Time [min]

Coarse 192 720 1 0.0031 4Finer 384 1920 2 0.0017 15Solution 1536 8640 3 0.0011 200

Table 1. BEM mesh and solution data for Jacobian

evaluation. NB =body panels, NW =wake panels,

NT =wake turns.

The results of this analysis are depicted in Fig. 3 thatdemonstrates that the application of a very high fidelity(and computationally expensive, see Table 1) Jacobiandoes not yield a convergence rate significantly fasterthan those provided by the use of less accurate (andless computationally expensive) Jacobians. Indeed, eventhe solution from a coarse (and computationally cheap)mesh for Jacobian evaluation is acceptable.

Figure 3. Effect of BEM mesh used for Jacobian

evaluation on convergence history. µ = 0.3.

Finally aeroelastic response obtained by using the pro-posed fully-coupled aeroelastic model is compared withthat obtained through the application of a quasi-steadyaerodynamic model in the solution procedure. The re-sults are given in terms of blade tip displacements (Fig.4) and vibratory hub loads (Figs. 5). As flap and lagtime histories show a similar behaviour, torsion fromthe BEM aerodynamic solution presents a higher har-monic content, probably due to the more accurate eval-uation of the wake inflow that plays a significant roleon this class of problems. Similar considerations can bemade about the evaluated 4/rev vibratory hub loads,for which the most relevant differences between the twoaeroelastic solutions appear in the out-of-plane loads,Fz and Mz, that are more directly related to the bladeelastic torsion deformation.

Figure 4. Blade tip deflections. µ = 0.3.

Figure 5. 4/rev vibratory hub loads. µ = 0.3.

Advance Ratio µ = 0.15

Next, the above analyses have been repeated for a ro-tor flight at lower advance ratio. In this condition thestrong influence of the velocity field induced by the wakeremaining closer to the rotor disk might negatively affectthe convergence of the numerical scheme.

Figure 6. Effect of relaxation factor on convergence

history. µ = 0.15.

Akin to the investigation presented for µ = 0.3, first

results concerning the effects of the relaxation parame-ter on the solution convergence history are presented inFig. 6. It shows that also in this case the error decayis sped up when higher values of the relaxation factor(close to 1) are used. In the overall, the convergenceerror threshold tends to be reached more slowly withrespect to the case with µ = 0.3. Although not shownhere, the analysis on the effect of aerodynamic Jaco-bian determined from different aerodynamics confirmsthat solution convergence is achieved neither when noJacobian is included in the iterative process, nor whena quasi-steady solver is used.

Figure 7. Effect of BEM mesh used for Jacobian

evaluation on convergence history. µ = 0.15.

Figure 7 presents the effect of the accuracy of the BEMsolver used to determine the Jacobian on the aeroe-lastic convergence. For the present advance ratio thefaster solution is clearly obtained using the finest ‘Solu-tion’ mesh. However, the trade-off between convergencespeed and computational costs would suggest the use ofthe coarse mesh.

Figure 8. Blade tip deflections. µ = 0.15.

The aeroelastic response results, given in terms of blade-tip deflections and vibratory hub loads, are presented inFigs. 8 and 9. The observation of the tip deflectionsreveals that, differently from what examined at µ = 0.3,a phase shift of about 180 exists between the flap timehistory predicted by the BEM approach and that given

Figure 9. 4/rev vibratory hub loads. µ = 0.15.

by quasi-steady aerodynamics, while two amplitudes arein agreement. Akin to what observed at µ = 0.3, ahigher harmonic content on the torsion deformation ispresent in the simulation from the BEM aerodynam-ics. In the overall, BEM and quasi-steady aerodynamicsblade tip predictions are closer for µ = 0.15 rather thanfor µ = 0.30. Concerning the 4/rev vibratory hub loads,in this case the differences between quasi-steady andBEM aerodynamics predictions are significantly greaterthan those calculated at µ = 0.3. This result is par-ticularly evident for the out-of-plane load components.Note that the vibratory loads at low speed are about oneorder of magnitude smaller than those at µ = 0.30, andthis fact could be one important reason for the slowerconvergence rate observed.

Conclusions

A fully-coupled aeroelastic solver for the analysis of ro-torcraft in steady flight conditions has been presented.Structural dynamics is based upon a non-linear beammodel, while a BEM solver for potential flows is usedfor the evaluation of the airloads. The numerical spaceintegration is performed through the Galerkin approach,while the time solution is evaluated by harmonic-balancetechnique. Performance and robustness of the proposednumerical solution algorithm has been analysed for afour-bladed rotor moving at two advance ratios. Par-ticular attention has been paid on the effect of relax-ation and Jacobian fidelity on the convergence rate ofthe solution. When using Jacobian numerically evalu-ated through the BEM solver, the convergence rate hasshown to be satisfactory, however depending on the re-laxation factor applied and the accuracy of BEM solverfor Jacobian identification. No convergence is achievedif the aerodynamic Jacobian is evaluated by a simpleraerodynamic solver (a quasi-steady model has been con-sidered in this work) or is even neglected. It has beenshown that a coarse BEM mesh yields a Jacobian thatguarantees quite fast convergence of the aeroelastic it-eration process, at much lower computational costs.

Finally, the aeroelastic responses (in terms of blade tipdisplacements and vibratory hub loads) obtained by theproposed fully coupled aeroelastic solver have been com-

pared with those from quasi-steady aerodynamics. Ma-jor differences have been observed on the prediction oftorsion deflection and out-of-plane vibratory loads.

Acknowledgments

This work has been partially supported by the Euro-pean Union Project ARISTOTEL (Grant agreement N.266073), financed under the Seventh Framework Pro-gramme (FP7/2007-2013).

References

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2. Gennaretti, M., Bernardini, G., ‘Novel BoundaryIntegral Formulation for Blade-Vortex InteractionAerodynamics of Helicopter Rotors,’ AIAA Jour-nal, Vol. 45, No. 6, 2007, pp. 1169-1176.

3. Morino, L., ‘A General Theory of unsteady com-pressible potential aaerodynamics,’ NASA CR-2464, 1974.

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Appendix A

From the decomposition of the potential field into a scat-tered component, ϕ

S, and an incident component, ϕ

I

(see section ‘Aerodynamic modelling’), in a body-fixedframe of reference the Bernoulli theorem reads

ϕS

+ ϕI− v

B· (∇ϕ

S+ u

I)

+‖∇ϕ

S+ u

I‖2

2+p

ρ=p0ρ

(11)

where p0 is the pressure of the undisturbed medium. Inorder to evaluate the pressure distribution, the expres-sion above requires the determination of ϕ

I, which is the

only term not directly available from the aerodynamicformulation presented in the section ‘Aerodynamic mod-elling’. The incident potential (and the correspondingtime derivative) could be obtained from the doublet dis-

tribution over the far wake, SF

W, using the following in-

tegral expression (see also Eq. (2))

ϕI(x, t) = −

∫SF

W

∆ϕS(y

TE

W, t− ϑ)

∂G

∂ndS(y) (12)

that, dividing the wake into panels, SF

Wk, is approxi-

mated as

ϕI(x, t)≈−

N∑k=1

−∆ϕS(y

TE

Wk, t− ϑk)

∫SF

Wk

∂G

∂ndS(y) (13)

However, as already mentioned in section ‘Aerodynamicmodelling’, the above expression cannot be applied tothose wake panels coming in contact with the body orpassing very close to it (BVI occurrence), in that theybecome numerically unstable and yield unrealistic po-tential distributions.

In order to avoid this problem, for the far wake pan-els that might experience BVI, the contribution to ϕ

I

is obtained in the following robust and accurate way.Consider the closed vortex at the contour of the k-thpanel of the far wake that might risk to come in con-

tact with the body, Ck = ∂SF

Wk, and introduce a frame

of reference rigidly connected with it such that, underthe assumption of undeformed wake panel, the incidentpotential induced by the vortex is constant in time oneach point of it. Next consider a body surface pointand a vortex frame point that at a given time coincide.Observing that for a generic function, f = f(x, t),

∂f

∂t

∣∣∣∣V

=∂f

∂t

∣∣∣∣B

+ vB−V· ∇f

where ∂f/∂t|V

and ∂f/∂t|B

denote time derivative ob-served, respectively, in the vortex frame and in the bodyframe, while v

B−V= v

V−v

Bis the relative velocity be-

tween the two frames at the considered point, for f = ϕkI

(with ϕkI

denoting the incident potential due to the k-thfar wake vortex) and reminding that ϕk

I= 0 in the vor-

tex frame, it is possible to obtain for the time derivativeof the incident potential in the body frame

∂ϕkI

∂t

∣∣∣∣∣B

= −vB−V· uk

I(14)

with ukI

known from the (regularized) k-th contributionin Eq. (3).

If the evaluation of the pressure is carried out withina free-wake solution, each far wake vortex is subjectto deformation. In this case, the related (doublet) po-tential induced in the corresponding vortex frame is nolonger stationary, although the intensity of the doubletis time independent (indeed, it is given by the potentialjump across the wake that is constant following a mate-rial point [3, 7]). Nevertheless, this contribution is nottaken into account for the vortices that are close to theevaluation point, in that negligible with respect to thecontribution from the term in Eq. (14), due to the veryhigh induced velocity arising.

A procedure similar to that described above for the eval-uation of the ϕk

Ifrom a wake vortex has been applied

in Ref. [11] for the evaluation of the pressure on a bodyinduced by vortex filaments.