Aerospace
, Volume 7 (11) – Nov 12, 2020

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aerospace Article Multiﬁdelity Sensitivity Study of Subsonic Wing Flutter for Hybrid Approaches in Aircraft Multidisciplinary Design and Optimisation 1 2, ,† Marco Berci and Francesco Torrigiani * Pilatus Aircraft Ltd, 6371 Stans, Switzerland; m.berci07@members.leeds.ac.uk Deutsches Zentrum für Luft- und Raumfahrt, 21129 Hamburg, Germany * Correspondence: Francesco.Torrigiani@dlr.de † Current address: DLR, Institute of System Architectures in Aeronautics, Hein-Saß-Weg 22, 21129 Hamburg, Germany. Received: 31 August 2020; Accepted: 3 November 2020; Published: 12 November 2020 Abstract: A comparative sensitivity study for the ﬂutter instability of aircraft wings in subsonic ﬂow is presented, using analytical models and numerical tools with different multidisciplinary approaches. The analyses build on previous elegant works and encompass parametric variations of aero-structural properties, quantifying their effect on the aeroelastic stability boundary. Differences in the multiﬁdelity results are critically assessed from both theoretical and computational perspectives, in view of possible practical applications within airplane preliminary design and optimisation. A robust hybrid strategy is then recommended, wherein the ﬂutter boundary is obtained using a higher-ﬁdelity approach while the ﬂutter sensitivity is computed adopting a lower-ﬁdelity approach. Keywords: aircraft design; aeroelastic stability; sensitivity analysis; ﬂexible wing; subsonic ﬂow 1. Introduction Within aircraft multidisciplinary design and optimisation (MDO) [1,2], efﬁcient methods and robust tools are highly sought after as effective reduced-order models (ROMs) [3–5] for fast parametric aeroelastic analyses [6–8], especially for sensitivity and uncertainty evaluation purposes at the preliminary design stage [9–11]. Simpliﬁed semi-analytical formulations for the bending–torsion instabilities of ﬂexible wings have been available for a long time but have inherent limitations [12–14], whereas complex ﬂuid-structure interaction (FSI) [15] procedures based on the ﬁnite element method (FEM) [16] and computational ﬂuid dynamics (CFD) [17] have recently been used to improve accuracy and generality, but remain computationally expensive and require special care [18,19] with signiﬁcant effort to pre- and post-process the numerical simulations. Continuing previous studies on multilevel techniques for practical aeronautical applications [20,21], this conceptual work investigates a robust hybrid strategy where the ﬂutter boundary is accurately obtained using a higher-ﬁdelity approach while the ﬂutter sensitivity is efﬁciently computed by adopting a (tuned) lower-ﬁdelity approach as an effective ROM. A comparative sensitivity study is hence presented for the aeroelastic stability boundary of a uniform cantilever wing in subsonic ﬂow as the standard benchmark [22]. The Typical Section idealisation [23,24] is employed as the analytical lower-ﬁdelity model, whereas the numerical higher-ﬁdelity model couples a beam-based FEM with panel-based CFD. Considering Loring’s wing [25] with a NACA0002 airfoil [26], the ﬂutter analyses build on earlier publications [27,28] and encompass parametric variations of wing properties, such as its structural inertia, stiffness and aspect ratio. The effects of the latter on the aeroelastic instability are quantiﬁed and differences between lower-ﬁdelity and higher-ﬁdelity results are critically assessed from Aerospace 2020, 7, 161; doi:10.3390/aerospace7110161 www.mdpi.com/journal/aerospace Aerospace 2020, 7, 161 2 of 22 both theoretical and computational perspectives, in order to study the necessary trade-off between complexity and costs of model ﬁdelity for intensive industrial MDO activities [29]. 2. Problem Formulation Uniform cantilever wings have long been used as the standard benchmark for fundamental studies on aeroelastic analyses and parametric sensitivities for the bending–torsion instabilities of ﬂexible wings, both numerically and experimentally [12,22]. They feature constant material properties (i.e., structural density r , Young’s elastic modulus E and shear modulus G), chord c, mass m and moment of inertia m per unit length at the inertial axis x (i.e., the line of sectional centres of gravity); CG bending stiffness EI and torsion stiffness GJ at the elastic axis x 0 (i.e., the line of sectional shear EA centers) along the semi-span l; structural damping is typically small and conservatively neglected [30]. With q(y, t) and h(y, t) the pitch and plunge motion of the ﬂexural axis at the time t, respectively, the wing vertical displacement w(x, y, t) is given as w = h + xq directly, x and y being the chordwise and spanwise directions. Assuming an Euler–Bernoulli beam model [31], the linear system of coupled partial differential equations (PDEs) for the dynamic aeroelastic response of wing bending and torsion undergoing small deformations reads: 0000 00 ¨ ¨ ¨ ¨ ¨ m h x q + EIh = DL, mq mx h x q GJh = DM, (1) CG CG CG with DL(y, t) and DM(y, t) being the sectional lift and pitching moment, respectively; the governing equations are then completed by both geometrical and natural boundary conditions, namely: 0 0 00 000 q (0, t) = GJq (l, t) = 0, h (0, t) = h (0, t) = EIh (l, t) = EIh (l, t) = 0. (2) As for the case of Loring’s wing, the problem formulation assumes an isotropic material without loss of generality; an anisotropic material (e.g., for the case of composite wings [32]) would feature elastic coupling between bending and torsion (due to different mechanical characteristics in respective directions [12]) but would not introduce conceptual changes to the overall methodology and main conclusion of this work. 2.1. Uncoupled Natural Vibration Modes Assuming h f c and q f c , where c(w, t) are the generalised coordinates of the uncoupled h h q q vibration mode shapes f(y) with natural frequencies w, the latter for the cantilever wing’s bending and torsion dynamics can independently be obtained by solving the relative homogeneous PDEs via separation of variables with the respective boundary conditions, namely [12]: g EI n GJ w = , w = , cosh g cos g + 1 = 0, cos n = 0, (3) l m l m h i y y cosh g + cos g y y y f = cosh g cos g sinh g sin g , f = sin n , (4) h q l l sinh g + sin g l l l which form a complete set of modal bases for the generalised solution of the aeroelastic equations; yet, note that these uncoupled bending and torsion modes are inherently orthogonal within their own bases but not to one another. 2.2. Unsteady Aerodynamic Model According to thin aerofoil theory for incompressible potential ﬂow, the sectional unsteady aerodynamic loads due to the wing motion may be written as [22]: h i c p ˙ ¨ ¨ ˙ ˙ DL = r c Uq h + x q + kUC (C V) , V = Uq h + x q, (5) MC T MC l/a 2 2 Aerospace 2020, 7, 161 3 of 22 c p c ¨ ˙ ¨ DM = r c q + x Uq x h x q + kUx C (C V) , (6) CP MC MC AC l/a T 2 2 32 where r and U are the reference air density and horizontal ﬂow speed; the aerodynamic centre x AC (where the circulatory load acts) and the control point x (where the non-penetration boundary CP condition for the vertical ﬂow speed V(y, t) is imposed) lay at the ﬁrst- and third-quarter chords, respectively; x refers to the mid-chord (where the non-circulatory load acts); C is the derivative MC l/a of the sectional lift coefﬁcient C with respect to the nominal angle of attack a. Within tuned strip theory [21], the scaling factor k introduces steady three-dimensional downwash effects due to the tip vortices [33] and takes part in the derivative of the wing lift coefﬁcient C = kC [34], while the complex lift-deﬁciency function C (k) is deﬁned in the reduced-frequency L/a l/a T k domain and embeds the lag due to two-dimensional inﬂow effects from the travelling ﬂat wake [35], namely: 4e ph H cw C = 2p 1 + p , k = , C = , k = , (7) l/a T 2 2 phe + C 2U 3 3 H + iH l/a 1 0 with e being the aerofoil thickness ratio, h the wing aspect ratio, e(h) Jones’ edge-velocity factor (i.e., the ratio between wing semi-perimeter and span [36]) and H (k) Hankel’s functions of the second type and n-th order. Note that the proposed approximate correction for the wing lift coefﬁcient is analogous to applying Oswald’s efﬁciency factor [37], whereas k 1 for standard strip theory. In summary, unsteady aerodynamics account for both circulatory lag and non-circulatory inertia effects [38], whereas quasi-unsteady aerodynamics neglect circulatory lag terms [39] within a quasi-static approach (i.e., C 1 with k << 1). Quasi-steady and steady aerodynamics then disregard non-circulatory inertia terms too, within a static approach (i.e., C = 1 and k 0) where the control point coincides with the elastic axis in the former case (in order to avoid unrealistic aerodynamic damping [23]) and all time derivatives are eventually discarded in the latter case. Note that the structural motion represents a time-dependent boundary condition for the incompressible airﬂow in all cases, but a synchronous proportional variation of the circulatory airload occurs within quasi-static approaches only [40]. 2.3. Modal Approach and Stability Analysis Given the natural vibration modes of the wing, the relative equations of motion can be transformed into ordinary differential equations (ODEs) by employing Ritz’s method [41], where generalised coordinates fc(t)g multiply the natural vibration mode shapes f. Recasting the whole aero-structural model in vector-matrix form, the governing modal equations then read: [M]fc ¨g + [C]fc ˙g + [K]fcg = f0g , fF g = [M ]fc ¨g + [C ]fc ˙g + [K ]fcg , (8) A A A A where the aeroelastic mass [M], damping [C] and stiffness [K] matrices are given by: [M] = [M ] [M ] , [C] = [C ] [C ] , [K] = [K ] [K ] , (9) S A S A S A and enforce a monolithic FSI, with fF (t)g the aerodynamic load; as anticipated, [C ] = [0] is hereby A S assumed for convenience without loss of generality. Due to linearity, the parametric aeroelastic stability of the wing is then monitored from the root locus of the characteristic equation for the ﬂutter determinant as the reference airspeed increases; in particular, the system becomes metastable when the real part of at least one eigenvalue l vanishes and leaves the response undamped, namely: lt 2 fcg fug e , l [M] + l [C] + [K] fug = f0g , (10) Aerospace 2020, 7, 161 4 of 22 with l = iw in the case of dynamic ﬂutter where a coupled resonant harmonic motion excites the eigenmode fug (Hopf’s bifurcation occurring when a couple of complex conjugate eigenvalues cross the imaginary axis), whereas l = 0 in the case of static divergence (occurring when the aeroelastic stiffness matrix becomes singular). Note that the eigenvalues and eigenvectors are suitably assumed to form a complete distinct set, as per most of the practical cases [42]; of course, a sufﬁcient number of natural vibration modes shall be employed in order to grant proper convergence of the modal analysis [43]. 2.4. Sensitivity Analysis The aeroelastic stability boundary acts as a typical constraint in aircraft MDO problems, where parametric sensitivities of both ﬂutter and divergence speeds with respect to variations in the design variables are required for gradient-based optimisation algorithms [44,45]. In general, such derivatives may numerically be obtained by performing many simulations with small alterations of each and every design variable; nevertheless, the aircraft’s conceptual and preliminary design stages would require extensive robust computations for the trends of both objective function and constraints while changing the design parameters [46,47]. Effective analytical solutions (with explicit expressions in a few special cases) have hence been derived by differentiating the governing equations, taking advantage of the self-adjoint nature of complex eigenproblems and then separating real and imaginary parts [48,49]. Especially when the sensitivity of the critical mode shape is also required, the continuation method may efﬁciently be employed to calculate the critical point and obtain all desired derivatives at the same time [50]. The eigenproblem arising in ﬂutter and divergence analysis may indeed be considered as a system of nonlinear equations with a normalisation condition for the eigenvectors; the solution is then numerically sought by the Newton–Raphson method, where the aeroelastic equations are differentiated with respect to a design parameter of interest p and the system is linearised at each iteration [51]. Within a single uniﬁed procedure, the eigenvalues’ and (right) eigenvectors’ derivatives may then automatically be determined along with the eigensolution itself at once, with no information about the transposed (left) problem being required; higher-order derivatives may be obtained by further differentiation. In particular, with U = U (p), w = w (p) and u = u (p) in the critical f g f g c c c ﬂight condition, the nonlinear eigenproblem for the ﬂutter mode reads [9]: [S]fu g = 0, fu g fu g = 1, [S] = w [M] + iw [C] + [K] , (11) c c c c and is differentiated to give the eigensolution sensitivity with respect to the design parameter as: 8 9 8 9 2 3 Re [S] Im [S] Refug RefVg > Re u > > Re i > f g f g > > > > > > > > 6 7 < = < = Im S Re S Im u Im V [ ] [ ] f g f g ¶ Imfu g Imfig 6 7 c 6 7 = , (12) T T 4Re u Im u 0 0 5 ¶p > U > > 0 > f g f g c c c > > > > > > > > : ; : ; T T w 0 Imfu g Refu g 0 0 c c c where the submatrices and subvectors involve the derivatives of the aeroelastic model, namely: ¶ [S] ¶ [S] ¶ [M] ¶ [C] ¶ [K] fVg = fu g , = w + iw + 2w [M] + i [C] , (13) c c c ¶w ¶w ¶w ¶w ¶w c c c c c ¶ [S] ¶ [S] ¶ [M] ¶ [C] ¶ [K] fug = fu g , = w + iw + , (14) c c ¶U ¶U ¶U ¶U ¶U c c c c c ¶ S ¶ S ¶ M ¶ C ¶ K [ ] [ ] [ ] [ ] [ ] fig = fu g , = w + iw + ; (15) c c ¶p ¶p ¶p ¶p ¶p in the degenerate case of static divergence with Imfu g = 0 and w = 0, all quantities are inherently c c real and the matrix of algebraic equations above specialises without the second and fourth rows and columns, respectively. Following previous studies [27], the derivatives may ﬁnally be normalised with Aerospace 2020, 7, 161 5 of 22 respect to the reference values of the involved quantities, thereby comparing all sensitivities within a coherent representation. 3. Lower-Fidelity Model The Typical Section abstraction [52] is hereby employed as the analytical lower-ﬁdelity model, as it is conceptually illuminating, inherently robust and computationally efﬁcient within its limitations. It resembles the section of the uniform wing and its fundamental aeroelastic behaviour for the bending–torsion coupling mechanism; matching the wing inertia and natural frequencies, such an abstraction, may be regarded as a condensed ROM. The structural arrangement is a classic mass-spring system for the wing section, where the coupled pitch and plunge motion is restrained by vertical 2 2 and angular linear springs with equivalent stiffnesses k = mw and k = mw (see Appendix A), h q h q respectively; it is representative of the inertial and elastic properties per unit length of the wing at about 75% of its span, where the structural arrangement becomes progressively ﬂexible and the aerodynamic load is still high. The model is hereby extended to include all bending modes having natural frequency below that of the ﬁrst torsion mode, as the instability may involve any combination of them in general. Considering the ﬁrst two uncoupled bending modes and the ﬁrst uncoupled torsion mode of Euler–Bernoulli’s beam model (with g 1.875, g 4.694 and n 1.571) [12], the aeroelastic 1 2 1 equations of motion for the wing’s Typical Section in (time-varying) steady incompressible ﬂow are: 2 3 2 3 m 0 mx f k 0 0 CG 6 7 6 7 [M ] = 0 m mx g , [K ] = 0 k r 0 , (16) 4 5 4 5 S CG S h mx f mx g m + mx 0 0 k CG CG q CG 2 3 0 0 f 6 7 [M ] = [0] , [C ] = [0] , [K ] = U cC 0 0 g , (17) 4 5 A A A L/a 0 0 x AC where the cross-projections of the non-orthogonal modes scale the off-diagonal coupling terms: Z Z l l 1 1 f = f (y, g ) f (y, n ) dy, g = f (y, g ) f (y, n ) dy, r = ; (18) h 1 q 1 h 2 q 1 l l 0 0 g in particular, all the latter are constant and read f 0.959, g 0.274 and r 39.275 for homogeneous aero-structural properties. Note that assuming a static aerodynamic approach is consistent with the rigorous application of the scaling factor k, which is hereby based on steady lifting-line theory; no damping is hence provided to the aeroelastic response. It is also worth stressing that the equations for the ﬁrst and second bending/plunge modes are not coupled directly, but the equation for the torsion/pitch mode couples them indirectly. Aero-Structural Parametric Derivatives The sensitivity of the aeroelastic matrices with respect to aero-structural parameters can readily be obtained in explicit analytical form, where the chain rule applies [28]. In particular, the derivatives with respect to the material density r and elastic modulus E are given by: ¶ [M] 1 ¶ [K] ¶ [M] ¶ [K] 1 = [M ] , = [0] , = [0] , = [K ] , (19) S S ¶r r ¶r ¶E ¶E E m m m whereas those with respect to reference ﬂow speed U and perturbation frequency w read: ¶ [M] ¶ [K] 2 ¶ [M] ¶ [K] = [0] , = [K ] , = [0] , = [0] , (20) ¶U ¶U U ¶w ¶w Aerospace 2020, 7, 161 6 of 22 and with respect to the wing semispan l it is: 2 3 2 0 0 ¶ [M] ¶ [K] 2 1 ¶C 6 7 L/a = [0] , = [D] [K ] + [K ] , [D] = 0 2 0 , (21) 4 5 S A ¶l ¶l l C ¶l L/a 0 0 1 ¶C 2 ¶C ¶C 1 k L/a L/a L/a = , = C . (22) L/a ¶l c ¶h ¶h h 4. Higher-Fidelity Model The high-ﬁdelity model is based on a FEM of a slender beam solved with the commercial code Nastran [53] for the structural analysis; the latter is coupled with either the doublet lattice method (DLM) available in the same software [54] or an in-house panel code based on Morino’s boundary element method (BEM) for the steady and unsteady aerodynamic analysis [55,56], thereby enforcing a numerical FSI. Both Nastran-based structural FEM and in-house aerodynamic BEM are generated with automatised routines, in order to ease parametric studies; this capability is exploited to perform numerical convergence studies (see Appendix B) and compare the analytical ﬂutter derivatives with their numerical counterparts obtained via ﬁnite difference. 4.1. Structural Model Following previous works [21], the wing structure is modelled with CBEAM elements and accounts for the distance between inertial and elastic axes; the node lying at the wing root is then clamped (see Figure 1). Further FEM nodes are placed at the leading and trailing edges of the wing and connected to the beam nodes with the rigid elements RBE2, in order to support mapping with the aerodynamic grid (in terms of both structural deformation and aerodynamic load: the latter modiﬁes the former, which changes the airﬂow boundary condition in turn) according to the closely-spaced rigid diaphragm (CSRD) assumption [57]. For the natural vibration analysis, shear deformation is neglected in order to obtain the Euler–Bernoulli beam model, with PBEAM deﬁning the properties (i.e., inertia and stiffness) of the beam element and SPC1 deﬁning the single-point constraint for the clamped root. Using Nastran’s SOL103 to obtain the structural eigenvalues and eigenvectors, the vibration analysis is performed while selecting Lanczos’ method [58] as available in EIGRL and normalising the modes to unit values of the generalised mass. Figure 1. Structural FEM of Loring’s wing. Aerospace 2020, 7, 161 7 of 22 4.2. Aerodynamic Model Standard ﬂutter prediction methods in the industrial environment are based on aerodynamic panel codes (mostly DLM [59,60]) that idealise wing and empennage as lifting surfaces; if considered, the aircraft fuselage is treated with dedicated elements for non-lifting bodies. The lower computational cost compensates for the lower ﬁdelity of the model; however, the reliability may be increased by correcting the aerodynamic inﬂuence coefﬁcients with higher-ﬁdelity data (which are generally not available at the preliminary stage of an MDO process though). Considering small perturbations of an unsteady irrotational ﬂow, the BEM proposed by Morino [55,56] is hereby adopted and consists of an integral representation of the velocity potential at any point of the computational domain in terms of the values on the surface surrounding the aircraft body and wake; the principle of superposition applies. The linear equation of acoustic waves propagation governs the unsteady ﬂow for the case of isentropic compressible ﬂuid and accounts for the ﬁnite speed of sound [61], whereas Laplace’s equation is resumed for the case of steady ﬂow by taking advantage of Prandtl–Glauert’s transformation [62]. Green’s function representing a unit-impulsive point source, the theoretical formulation of the boundary value problem for the perturbation potential is then based on Green’s formula [33], with (Neumann-type, generally instationary) non-penetration boundary conditions on the aircraft surface and wake as well as (Dirichlet-type, generally stationary) unperturbed asymptotic conditions far from the latter. Bernoulli’s theorem is linearised to calculate the pressure coefﬁcient and Kutta’s condition is imposed at the trailing edge of lifting surfaces, where the wake detaches and is shed back with the reference airspeed, trailing circulation variations without sustaining any pressure load [37]; its trajectory then represents a streamline at any time and may generally become part of a nonlinear iterative solution where roll-up occurs due to downwash effects, if it is not prescribed a priori (as is in fact typical for most practical applications [53,63], especially when characterised by moderately unsteady ﬂow). Thus, Morino’s BEM is able to deal with arbitrary 3D geometries in a uniﬁed manner, reducing the effort of the abstraction process and easing the integration with complex structural models. Although the original theory is valid for arbitrary motion in time or frequency domain, only harmonic oscillations in the frequency domain are here considered as suitable for ﬂutter analysis. The current implementation of Morino’s BEM is described in previous works [64] and is equivalent to an appropriate combination of doublets (for lifting bodies and wake) and point sources/sinks (for thickness effects and non-lifting bodies). The 3D geometry is approximated with ﬂat quadrilateral panels that follow the local wing surface and the aerodynamic potential is assumed constant over them, with an analytic expression for their mutual induction. Note that similar ﬂow conditions can be modelled by either vortex or doublet distributions and a quadrilateral doublet element is equivalent to a vortex ring placed at the panel edges [65]; higher-order methods have also been formulated in order to improve accuracy and computational efﬁciency for complex geometries and conﬁgurations, but they are more prone to becoming ill-posed and typically not necessary as long as enough quadrilateral panels reﬁne the aerodynamic grid for lower-order methods to capture high pressure gradients while still staying away from ill-conditioning [66]. Through the wake being assumed as ﬂat, rigid and parallel to the free-stream velocity in order to perform rigorous comparisons with Nastran’s DLM and Theodorsen’s theory without loss of generality (in the absence of aircraft fuselage and empennages [67]), a linear system is obtained: k k T s = T J , (23) s J where frequency-dependent T and T are the matrices of aerodynamic inﬂuence coefﬁcients (AIC), s J k k whereas s and J are the aerodynamic potential and normal wash associated with the k-th mode shape of the structural displacement [68], having mapped the latter onto the aerodynamic grid with an effective implementation of the inﬁnite plate spline (IPS) method [69]. The elements Q of the hk generalised aerodynamic forces (GAF) matrix Q(s) are then calculated from the work done by the Aerospace 2020, 7, 161 8 of 22 aerodynamic pressure C due to the k-th mode on the displacement deﬁned by the h-th mode f for a prescribed range of reduced frequencies [40], namely: h i k Q = f n A C (s ), (24) hk å i i p i=1 where n and A are the normal vector and area of the i-th aerodynamic panel, respectively. i i The NACA0002 aerofoil [26] is suitably adopted to obtain a baseline 3D representation of Loring’s wing surface (see Figure 2); this particular choice of thin symmetric aerofoil is consistent with the original work with no loss of generality, as the parametric sensitivity of the aeroelastic stability boundary with respect to aerofoil thickness was already found to be poor for small perturbations of subsonic potential ﬂow (i.e., in the absence of strong shock waves and/or signiﬁcant ﬂow separation) [70]. Figure 2. Aerodynamic panels model of Loring’s wing. 4.3. Aeroelastic Model After obtaining the natural vibration modes, the GAF are computed with Morino’s BEM, and the continuation method [50] is used to solve the aeroelastic stability eigenproblem and trace the root locus. In order to avoid its costly re-computation during the stability analysis, the GAF matrix is approximated by a rational expression in which the non-linear dependency on the complex reduced frequency s appears explicitly. Here the matrix fraction approach (MFA) [71] is used and preferred to the rational function approximation (RFA) [72], since the former exhibits higher accuracy than the latter for the same number of poles [73]; yet, both methods provide the analytical continuation of the GAF for complex reduced frequency, increasing the accuracy of damped solutions with respect to the p k method. According to MFA, the GAF matrix is expressed as fraction of matrix polynomials: ! ! M+2 M i i Q N s D s , (25) i i å å i=0 i=0 and the accuracy of the approximation increases with increasing the number of poles M, which is equal to the size of the state-space system [74]; in particular, three poles have been used in this proof-of-concept work. The approximation matrices D and N are both obtained by solving i i a least-square problem in which the distance from the GAF samples Q(ik) is minimised. Aerospace 2020, 7, 161 9 of 22 The continuation method hereby adopted provides a straightforward and efﬁcient tracking technique without using any correlation function, such as the modal assurance criterion (MAC) [75]. For this reason, the method is rather insensitive to the number of poles used for the aerodynamic ﬁnite-state approximation and is able to distinguish between actual and artiﬁcial aerodynamic states by construction. In the continuation method, the aeroelastic equations are differentiated with respect to the free-stream airspeed, thereby resulting in a system of ODEs which are solved with a predictor–corrector integration schema starting from an initial airspeed value for which a true solution is known (e.g., the natural vibration frequencies for U = 0). For validation purposes, ﬂutter analysis is also carried out with Nastran’s DLM directly: a ﬂat plate aligned with the free-stream is used as lifting surface and the aerodynamic panels are deﬁned with CAERO1 and PAERO1 cards; the distribution and number of such panels follow established guidelines and convergence studies (see Appendix B). The mode shapes are mapped onto the aerodynamic grid with the IPS method [76] using the SPLINE1 card and the GAF matrix is then generated for sixteen reduced frequencies speciﬁed in the MKAERO1 card, suitably ranging from k = 0 to k = 1 (with logarithmic-like spacing) as per literature studies and common practice [53,63]; a cubic spline is then exploited to interpolate the GAF therein. For the dynamic aeroelastic stability analysis, the p k method as available in the FLUTTER card is used: the aerodynamic damping is approximated as the imaginary part of the GAF matrix computed for (undamped) harmonic motion, limiting results’ accuracy to the case of lightly damped aeroelastic solutions [12]. The p k approximation of the aeroelastic eigenproblem [77] is then solved for all the free-stream speed values deﬁned in the FLFACT card, according to the reference length and air density speciﬁed in the AERO card. After damping ratios are obtained in the given speed range for all modes, the (undamped) ﬂutter point is determined by linear interpolation. 4.4. Aero-Structural Parametric Derivatives To calculate the sensitivity of the ﬂutter boundary with respect to any aero-structural parameter, the derivatives of the aeroelastic matrices are also necessary in the ﬁrst place. In particular, the derivatives of structural eigenvalues and eigenvectors with respect to structural parameters, such as wing material properties (e.g., density or elastic modulus) and element properties (e.g., moments of inertia or distance between elastic and inertia axis), are obtained with Nastran SOL200. The design variable label and initial value are deﬁned with the DESVAR card, which is connected to the relative bulk data entry by DVMREL1 or DVPREL1 cards for material or element properties, respectively; derivatives are hence computed for the structural responses deﬁned by DRESP1 cards, which are the ﬁrst n structural eigenvalues and eigenvectors. The GAF matrix is differentiated with respect to both reduced frequency and design variables: according to MFA, the ﬁnite-state approximation allows the analytic differentiation with respect to the complex reduced frequency s, whereas the method presented in previous works [73] was used for the derivatives with respect to the design variables p. In particular, by deﬁning the functions P s , p and R s , p as: k k k P = Q s , p , R = T s T J , (26) hk s J a sub-differentiation process is set up, and depending on the number of design variables and mode shapes to be considered [11], the derivatives may be obtained by either a direct approach: dP ¶P ¶P ds ¶R ds ¶R = + , = , (27) dp ¶p ¶s dp ¶s dp ¶p or an adjoint approach: T T dP ¶P ¶R ¶R ¶P = + L , L = . (28) dp ¶p ¶p ¶s ¶s Aerospace 2020, 7, 161 10 of 22 In computing the partial derivatives, it is important to note that the imaginary component is introduced only by the normal wash J and the wake’s inﬂuence coefﬁcients inside matrix T ; therefore, the partial derivatives of the steady (real) contribution to the GAF are computed via complex step [78] and the partial derivatives of the unsteady (complex) contribution are then analytically assembled. 5. Results and Discussion Loring’s uniform slender thin wing [25] is hereby considered, as both experimental wind-tunnel data and numerical results assuming two-dimensional incompressible potential ﬂow are available and all results can hence be explained from both physical and mathematical standpoints; moreover, this fundamental benchmark embeds the full complexity of the aeroelastic problem without introducing detrimental uncertainties. The wing’s chord is 0.305 m and the semi-span is 2.057 m, giving an aspect ratio 13.5. The inertial axis lays at 42.3% of the chord, with mass 8.05 kg/m and mass moment of inertia 0.0471 kgm. The wing root is clamped at the elastic axis, with ﬂexural stiffness 2 2 677.3 Nm and torsional stiffness 1018.9 Nm placed at 30% of the chord. The coupled natural vibration frequencies of the ﬁrst bending and torsion modes were observed at 1.29 and 18.1 Hz, respectively; that of the second bending mode was detected in between at 7.7 Hz. With a ﬂuid density 1.11 kg/m , the ﬂutter speed and frequency were experienced at 90.3 m/s and 10.2 Hz, which give a reduced frequency around 0.11; the subsonic ﬂow may then be considered as moderately unsteady. A Euler–Bernoulli beam model calculated the ﬁrst three coupled vibration frequencies as 1.29, 7.65 and 17.98 Hz; once coupled with two-dimensional unsteady aerodynamics for a ﬂat plate using standard strip theory, ﬂutter was predicted at 90.7 m/s and 9.2 Hz with good accuracy. In order to elaborate on the literature results and visualise the ﬂutter mechanism, the aeroelastic stability analysis of Loring’s wings is ﬁrst performed with the same assumptions as in the original publication and the approach here described in the Problem Formulation; the p k method (with Theodorsen’s exact function [35]) has consistently been used and results have been cross-veriﬁed against the state-space representation (with a common two-term RFA of Theodorsen’s function) [21]. By employing the ﬁrst two bending modes and the ﬁrst torsion mode, the respective coupled vibration frequencies are calculated as 1.21, 7.59 and 17.91 Hz; once still coupled with two-dimensional unsteady aerodynamics for a ﬂat plate using standard strip theory, ﬂutter is consistently predicted at 91.15 m/s and 9.2 Hz (which is indeed an excellent approximation, as conﬁrmed by a modal convergence study up to the ﬁrst three bending and torsion modes). The evolution of the aeroelastic system’s eigenvalues is presented in Figure 3 and conﬁrms static divergence to arise well beyond dynamic ﬂutter (i.e., U >> U ). d f Figure 3. Real (left) and imaginary (right) parts of the aeroelastic eigenvalues for Loring’s wing in incompressible ﬂow. Aerospace 2020, 7, 161 11 of 22 Due to the remarkable agreement between measurements and simulations, the assumptions of slender beam structure and two-dimensional incompressible potential ﬂow in the latter are hence deemed justiﬁed. Note that the instability mechanism involves second bending and ﬁrst torsion modes directly, but the ﬁrst bending mode is indirectly essential for their coupling to occur before that between ﬁrst bending and ﬁrst torsion modes (see Appendix A). 5.1. Aeroelastic Analyses The aeroelastic stability of Loring’s wing is investigated and compared using the proposed lower and higher-ﬁdelity models, while still assuming incompressible ﬂow. The analyses encompass parametric variations of the aero-structural properties, quantifying their effects on the aeroelastic stability boundary and critically assessing the differences in the multiﬁdelity results from both theoretical and computational perspectives, for possible practical applications in airplane design and optimisation adopting hybrid strategies. Figure 4 shows the aeroelastic stability analysis from the higher-ﬁdelity model, focusing on the same ﬂutter mechanism as found earlier: the ﬁrst torsion mode becoming unstable and extracting energy from the airﬂow through the coupled second bending mode, ﬂutter is found at 94.77 m/s and 10.04 Hz in excellent agreement with the experimental results and exhibits a slightly higher vibration frequency than the theoretical one. The aeroelastic stability analysis performed with Nastran and its embedded DLM for the ﬂat wing is also presented and conﬁrms the higher-ﬁdelity results based on Morino’s BEM with NACA0002 aerofoil, ﬂutter being found at 94.44 m/s and 10.38 Hz. Note that both sets of code share the same structural model, and the coupled natural vibration frequencies in the void are 1.21, 7.55 and 21.03 Hz for the ﬁrst three bending modes, whereas it is 17.88 Hz for the ﬁrst torsion mode; these ﬁrst four modes have been used in all analyses (see Appendix B). As expected from the wing aspect ratio being relatively large, three-dimensional downwash effects are actually moderate and the (beneﬁcial) steady ones on the (attenuated) airload distribution are roughly compensated by the (detrimental) unsteady ones on the (accelerated) airload evolution [21]; thus, the ﬂutter speed is just slightly higher than that predicted assuming unsteady two-dimensional ﬂow. Figure 4. Higher-ﬁdelity real (left) and imaginary (right) parts of the aeroelastic eigenvalues for Loring’s wing in incompressible ﬂow. Still focusing on the ﬂutter mechanism, Figure 5 then shows the aeroelastic stability analysis from the lower-ﬁdelity model: despite its inherent simplicity, the latter predicts ﬂutter quite accurately at 92.1 m/s and 9.09 Hz, thereby proving the effectiveness of such an idealisation (analogous stability diagrams have indeed been obtained by retaining only steady aerodynamic terms in the higher-ﬁdelity model, for cross-veriﬁcation purposes). The coupled natural vibration frequencies in the void are 1.21 and 7.59 Hz for the two bending modes, and 17.91 Hz for the torsion mode, in excellent agreement with the higher-ﬁdelity ones. The aeroelastic response is then metastable until the ﬁrst instability Aerospace 2020, 7, 161 12 of 22 occurs, as no damping is provided by either the elastic structure or static aerodynamics; to this respect, it is worth mentioning that the lift-derivative correction for the steady three-dimensional downwash (which tends to increase the ﬂutter speed, by providing lower airload) incidentally compensates the lack of aerodynamic lag (which tends to decrease the ﬂutter speed, by neglecting wake inﬂow). Note that, although the ﬂutter reduced-frequency is rather high for steady aerodynamics to be rigorously applicable, the latter was still adopted in order to exacerbate the difference between lower and higher-ﬁdelity models and stress the multilevel approach for more general conclusions. Figure 5. Lower-ﬁdelity real (left) and imaginary (right) parts of the aeroelastic eigenvalues for Loring’s wing in incompressible ﬂow. 5.2. Sensitivity Study Following previous studies on the aeroelastic stability boundary of slender cantilever wings and its sensitivity [27,28], linear FSI models give nonlinear ﬂutter trends as relevant aero-structural parameters are perturbed. The reciprocal positions of aerodynamic, elastic and inertial axes being often constrained by the available space inside the wing as well as the chosen structural layout and systems arrangement, the ﬂutter point’s sensitivity to the wing’s structural properties is individually explored by changing its material density r (which scales m and m) and elastic modulus E (which scales EI and GJ), whereas varying the wing’s semi-span l (which modiﬁes the lift coefﬁcient derivative C ) l/a alters its geometry and affects both structural and aerodynamic properties at the same time. From Figure 6, it can be seen that changing the wing’s material density alters all natural frequencies through the inertial properties and hence the ﬂutter frequency, but has marginal/no inﬂuence on the ﬂutter speed; all symbols give the actual nonlinear percentage variation of the ﬂutter point, whereas all lines with corresponding colour draw its normalised linear prediction. In particular, a 1% increment in the material density causes about a 0.5% decrement in the ﬂutter frequency; the negligible variation of the ﬂutter speed is conﬁrmed by previous works [27] as being due to the (small) beneﬁcial effect of an increase in sectional mass being compensated by an (almost) equal and opposite detrimental effect of an increase in mass moment of inertia. This outcome may indeed help while minimising the aeroplane weight, as reducing the material density does not signiﬁcantly affect the present ﬂutter boundary due to wing bending–torsion instability (especially in the absence of lumped masses). However, it shall be recalled that varying the wing inertia might cause other types of aero-structural instabilities to arise at the aircraft level due to modal coalescence, and monitoring the variation of the ﬂutter frequency is then important to preventing potential resonance. It is also worth mentioning that the small higher-ﬁdelity effect on the ﬂutter speed is due to the unsteady airload being frequency dependent through the lift-deﬁciency function and the related aerodynamic lag. Aerospace 2020, 7, 161 13 of 22 Figure 6. Flutter speed (left) and frequency (right) parametric variation and sensitivity to changes in the wing’s material density. As per Figure 7, changing the wing’s material elastic modulus alters all natural frequencies through the stiffness properties and hence the ﬂutter speed and frequency; in particular, a 1% increment in the elastic modulus induces about a 0.5% increment in all the latter, leaving the ﬂutter reduced frequency practically unchanged. This outcome is quantitatively conﬁrmed by previous works [27] as being due to the detrimental effect of an increase in ﬂexural stiffness being much lower than the beneﬁcial effect of an increase in torsional stiffness. The striking agreement between lower and higher-ﬁdelity results is mostly driven by the respective structural models being equivalent (as seen from the close agreement between the natural vibration frequencies), while the different aerodynamics play a minor role (as the airﬂow is moderately unsteady). The quasi-linear trend of the percentage variations reveals a rather large conﬁdence interval for the normalised sensitivity; however, the dimensional counterpart of the latter to be used by optimisation routines follows a highly nonlinear pattern (note that the explicit lower-ﬁdelity expression given in Appendix A for the torsional static divergence speed provides a straightforward theoretical check). Figure 7. Flutter speed (left) and frequency (right) parametric variation and sensitivity to changes in the wing’s material elastic modulus. Finally, Figure 8 shows that changing the wing semispan has a large effect on the stability boundary through the stiffness properties as well as the aerodynamic loads and hence on the ﬂutter speed and frequency. In particular, a 1% increment in the wing semispan induces about a 0.8% decrement in the critical speeds and about a 1.6% decrement in the ﬂutter frequency; note that analogous trends for the variations and orders of magnitude for the sensitivities were obtained in previous studies on similar slender wings [28], as a qualitative means of validation. Lower and higher- ﬁdelity results are still in very good agreement and differences are mainly due to unsteady downwash effect, as the aspect ratio of the wing changes considerably with the span and so does the sectional airload. Aerospace 2020, 7, 161 14 of 22 Figure 8. Flutter speed (left) and frequency (right) parametric variation and sensitivity to changes in the wing semispan. For the sake of completeness, it is worth mentioning that variations and sensitivities of the divergence speed (see Appendix A) with respect to the same aero-structural parameters considered above were consistently found to have trends and orders of magnitude very close to those pertaining the ﬂutter speed. In particular, it is observed that changing the wing’s geometry has a much larger impact on the ﬂutter boundary than changing the material properties (especially density); this is particularly true for the ﬂutter frequency, the variations of which exhibit a more signiﬁcant nonlinear behaviour and local curvature. Due to the sufﬁcient mutual separation of the natural vibration modes, it is also worth stressing that the instability mechanism did not change across the parametric variations, and the derivatives of the ﬂutter stability boundary have accurately been calculated by the lower-ﬁdelity model at a (marginal) fraction of the computational complexity and costs. In particular, semi-analytical solutions drastically reduced the latter, being almost instantaneous and demanding minimal pre- and post-processing (if any at all) while granting an enhanced theoretical understanding. Thus, the healthy combination of lower and higher-ﬁdelity models enables efﬁcient multidisciplinary exploration of a large design variable space for innovative aeroplane concepts and conﬁgurations; further numerical savings may still be obtained by exploiting reliable surrogate models for the higher-ﬁdelity solutions, with an effective synthesis of the underlying complexity [20]. 6. Conclusions Within the context of aircraft multidisciplinary design and optimisation, a comparative sensitivity study for the bending–torsion ﬂutter instability of ﬂexible aircraft wings in subsonic ﬂow has been presented. Analytical models and numerical tools with different complexities and ﬁdelities have been used, in view of possible practical applications exploiting multilevel approaches within the conceptual and preliminary MDO phases. Parametric studies have been performed where the effects of varying the wing’s aero-structural properties on the aeroelastic stability boundary have been quantiﬁed and critically assessed from both theoretical and computational perspectives. When the natural vibration modes of the wing are well separated from all other natural vibration modes of the aircraft and the aeroelastic instability mechanism does not change in nature, an efﬁcient hybrid strategy is then recommended where the ﬂutter analysis is performed using higher-ﬁdelity approaches, whereas the sensitivity analysis of the ﬂutter boundary is performed using lower-ﬁdelity approaches, thereby improving theoretical understanding and reducing computational costs while retaining accuracy. Future works are encouraged to investigate additional effects (e.g., wing sweep and ﬂow compressibility) and increased complexity (e.g., presence of a control surface) or perform the full MDO of a ﬂexible aircraft wing, effectively exploiting the proposed multiﬁdelity strategy. Aerospace 2020, 7, 161 15 of 22 Author Contributions: M.B. derived the analytical model and results; F.T. performed the numerical simulations; the authors then wrote the respective parts of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors would like to thank Ranjan Banerjee at City University of London and Rakesh Kapania at Virginia Polytechnic Institute and State University for the precious insights into their previous works. Conﬂicts of Interest: The authors declare no conﬂict of interest. Abbreviations Symbols A aerodynamic panel area c section chord C section lift C section lift derivative l/a C wing lift derivative L/a C pressure coefﬁcient C Theodorsen’s function C generalised damping matrix [ ] D aerodynamic approximation matrix (fraction denominator) e semiperimeter-to-span ratio E section Young’s elastic modulus f cross-projection of ﬁrst bending and ﬁrst torsion modes F generalised load vector f g g cross-projection of second bending and ﬁrst torsion modes G section shear elastic modulus h section ﬂexural (plunge) displacement H Hankel’s functions of the second type and n-th order I section ﬂexural area moments of inertia J section torsional area moments of inertia k reduced frequency k equivalent spring stiffness [K] generalised stiffness matrix l wing semi-span DL section aerodynamic force m section mass DM section aerodynamic moment [M] generalised mass matrix n aerodynamic panel normal vector N aerodynamic approximation matrix (fraction numerator) p design parameter Q generalised aerodynamic forces matrix r squared ratio of second and ﬁrst ﬂexural vibration frequencies s complex reduced frequency [S] system matrix t time T aerodynamic inﬂuence coefﬁcients matrix fug eigenvector U horizontal airspeed V vertical airspeed x chordwise coordinate y spanwise coordinate w section vertical displacement Aerospace 2020, 7, 161 16 of 22 Greek a angle of attack fcg generalised coordinates e aerofoil thickness ratio f natural vibration mode shape g ﬂexural natural vibration constant h wing aspect ratio k aerodynamic load scaling function l eigenvalue m section mass moment of inertia n torsional natural vibration constant q section torsional (pitch) displacement J normal wash matrix r air density r material density s aerodynamic potential matrix t reduced time w natural vibration frequency Subscripts A aerodynamic c critical f ﬂutter d divergence h ﬂexural S structural q torsional Acronyms AC aerodynamic centre AIC aerodynamic inﬂuence coefﬁcient BEM boundary element method CFD computational ﬂuid dynamics CG centre of gravity CP control point CSRD closely-spaced rigid diaphragm DLM doublet lattice method EA elastic axis FEM ﬁnite element method FSI ﬂuid-structure interaction GAF generalised aerodynamic forces IPS inﬁnite plate spline MAC modal assurance criterion MC mid-chord MDO multidisciplinary design and optimisation MFA matrix fraction approach MST modiﬁed strip theory ODE ordinary differential equation PDE partial differential equation QST quasi-steady theory RFA rational function approximation ROM reduced order model SST standard strip theory TST tuned strip theory Aerospace 2020, 7, 161 17 of 22 Appendix A. Aeroelastic Stability of the Typical Section with Steady Aerodynamics Providing full control on the results with respect to the speciﬁc assumptions of the methods and tools employed for the analysis, theoretical formulations grant a clear and complete overview of the problem which is essential for any engineering application. Although inherently limited in their general capabilities, analytical solutions then offer a wealth of qualitative information and quantitative details as well as fundamental insights and rigorous validation for both educational and practical purposes. Considering a single mode for the wing bending and torsion, respectively, the aeroelastic equations for the Typical Section with steady aerodynamics read [79]: ¨ ¨ m h x f q + k h = U cC f q, (A1) CG h L/a ¨ ¨ ¨ mq mx h x f q + k q = U x cC q, (A2) CG CG q AC L/a where f 1 for two-dimensional aerofoils [12], with f 1 and f 1. By neglecting the aerodynamic h q load, the coupled bending and torsion natural vibration frequencies are explicitly obtained as: 4 2 F w F w + F = 0, w = F F 4F F , (A3) 4 2 0 v 2 4 0 v v 2F 2 2 mw w m m 2 2 2 h q F = 1, F = w + 1 + x w , F = , (A4) 0 2 4 CG q h 2 2 2 2 m + mx (1 f ) m + mx (1 f ) CG CG where the uncoupled counterparts are resumed whenever inertial and elastic axes coincide (i.e., with x = 0). Otherwise, the characteristic equation for the ﬂutter determinant provides with the CG metastable boundary and gives the ﬂutter frequency as: 4 2 P w P w + P = 0, w = P P 4P P , (A5) 4 2 0 2 4 0 f f 2 2P and setting the inner radical discriminant equal to zero then gives the ﬂutter speed as: 4 2 D U + D U + D = 0, U = D D 4D D , (A6) 4 2 0 f 2 4 0 f f rD where the sign before the inner radical shall provide with the smallest positive real value; ﬁnally, note that setting both w and P equal to zero provides with the torsional static divergence f 0 speed. In particular, when the aerodynamic centre is ahead of the elastic axis (i.e., with x < 0), AC the latter explicitly reads: 2k U = , (A7) rx cC AC L/a regardless the bending stiffness; when the elastic axis is ahead of the inertial axis (i.e., with x > 0), CG the ﬂutter condition is given as: s s 1 P U = D 4D D D , w = , (A8) f 4 0 2 f rD 2P 4 4 where all coefﬁcients are analytical functions of the aero-structural parameters, namely: P = k k + U x cC , (A9) 0 h q AC L/a 2 Aerospace 2020, 7, 161 18 of 22 h i 2 2 2 P = m k + U x x f cC + k m + mx , (A10) 2 q AC CG L/a h CG h i 2 2 P = m m + mx 1 f , (A11) CG h i h i 2 2 2 D = mk + k m + mx 4m m + mx 1 f k k , (A12) 0 q h h q CG CG n h i h i o 2 2 2 2 D = 2m x x f k m + k m + mx 2 m + mx 1 f x k cC , (A13) AC CG q h AC h L/a CG CG h i D = m x x f cC . (A14) 4 AC CG L/a The formulas above were implemented and compared with literature results for the stability boundary of a Typical Section assuming standard strip theory (SST) with steady aerodynamics [12,79]: exact agreement was always found and provided rigorous validation, also with respect to the relative root locus. The same formulas have then been used to investigate the stability boundary of Loring’s wing Typical Section when either the ﬁrst or the second bending mode is coupled with the ﬁrst torsion mode, employing tuned strip theory (TST) with steady aerodynamics [21]. Without accounting for the modal cross-projection (i.e., f 1 for a “pitch & plunge” apparatus), ﬂutter is calculated at 106.5 m/s and 4.32 Hz in the former case whereas at 73.9 m/s and 11.28 Hz in the latter case; when the modal cross-projection is considered (i.e., f < 1 for a slender beam), ﬂutter is calculated at 109.7 m/s and 4.28 Hz in the former case whereas at 139.2 m/s and 9.60 Hz in the latter case. The static divergence speed pertains the torsion mode only and is found at 210.2 m/s, regardless the bending modes and their cross-projections. A direct comparison with the low-ﬁdelity results hence reveals that the interaction between bending and torsion modes as well as their cross-projections are essential to reproduce the correct behaviour of the ﬂutter mechanism, where the coalescence between ﬁrst bending mode and torsion mode drives that between second bending mode and the latter (which have closer natural vibration frequencies) to occur earlier. Appendix B. Higher-Fidelity Model Results Convergence Study The number of elements for the structural and aerodynamic models was deﬁned according to rigorous convergence studies. Following Nastran’s best practice [80] with k = 1 the maximum max reduced frequency employed in the aeroelastic analysis, ﬁfteen DLM panels have uniformly been distributed along the wing chord. When adopting Morino’s BEM, thirty aerodynamic panels were symmetrically placed along both upper and lower aerofoil surfaces according to the convergence study in Figure A1 (left), with a suitable reﬁnement around the leading edge in order to capture high pressure gradients. The number of aerodynamic panels in the span-wise direction was determined according to another convergence study shown in Figure A1 (right) and all higher-ﬁdelity results presented in this work were obtained with ﬁfteen panels strips uniformly distributed along the wing span, since sufﬁcient to grant a relative error below 1% for both ﬂutter speed and ﬂutter frequency. Following convergence studies in previous works [37,64], the wake extends for a hundred chords behind the wing and is modelled with a hundred rows of shed panels, the length of which is cubically increased with increasing their distance from the trailing edge. Figure A2 (left) shows the indicial lift-deﬁciency function F(t) (which represents the equivalent of the complex lift-deﬁciency function C in the reduced-time domain t) from a unit step in angle of attack, whereas Figure A2 (right) depicts the normalised distribution of the steady circulation k(y) (which represents the ratio between the sectional circulation in three- and two-dimensional ﬂow within the framework of modiﬁed strip theory, MST [13]); note that the unsteady lift development approaches quasi-steady theory (QST) asymptotically. These results also justify the assumption of two-dimensional ﬂow for the lower-ﬁdelity model, within the framework of TST [21] (to this respect, it is worth mentioning the higher-ﬁdelity Aerospace 2020, 7, 161 19 of 22 lift coefﬁcient derivative of the wing is obtained as C = 5.25, in excellent agreement with the L/a lower-ﬁdelity analytical estimation C = 5.21). L/a Figure A1. Flutter speed and frequency varying the number of aerodynamic panels along the wing chord (left) and span (right). Figure A2. Indicial lift-evolution function from a step-change in the angle of attack (left), spanwise lift-decay function (right) for Loring’s wing in incompressible ﬂow. Figure A3 (left) then shows the evolution of the natural vibration frequencies percentage error with varying the number of FEM nodes of the structural beam; the value obtained with the most dense grid is used as reference to compute the plotted error. The ﬁrst four natural vibration frequencies exhibit a good convergence behaviour, with the third one (i.e., the ﬁrst torsional mode) almost insensitive to the number of nodes. All higher-ﬁdelity results presented in this work were hence obtained with 20 nodes uniformly distributed along the wing span, since sufﬁcient to grant a relative error below 2%. 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In Proceedings of the 3rd MSC Worldwide Aerospace Conference and Technology Showcase, Toulouse, France, 16–24 September 2001. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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