See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/338398356 Parametric Roll Maneuverability Analysis of a High-Aspect-Ratio-Wing Civil Transport Aircraft Conference Paper · January 2020 DOI: 10.2514/6.2020-1191 CITATIONS READS 23 767 5 authors, including: Cristina Riso Divya Sanghi Georgia Institute of Technology University of Michigan 35 PUBLICATIONS 339 CITATIONS 7 PUBLICATIONS 43 CITATIONS SEE PROFILE Carlos E. S. Cesnik University of Michigan 364 PUBLICATIONS 11,622 CITATIONS SEE PROFILE All content following this page was uploaded by Cristina Riso on 18 January 2020. The user has requested enhancement of the downloaded file. SEE PROFILE This is a preprint of the following article: Cristina Riso, Divya Sanghi, Carlos E. S. Cesnik, Fabio Vetrano, Patrick Teufel. Parametric Roll Maneuverability Analysis of a High-Aspect-Ratio-Wing Civil Transport Aircraft, 2020 AIAA SciTech Forum, Orlando, FL, January 6–10, 2020. The published article may differ from this preprint and is available at: https://arc.aiaa.org/doi/10.2514/6.2020-1191. Parametric Roll Maneuverability Analysis of a High-Aspect-Ratio-Wing Civil Transport Aircraft Cristina Riso1 , Divya Sanghi1 , Carlos E. S. Cesnik1 , Fabio Vetrano2 , and Patrick Teufel2 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109 2 Airbus Operations S.A.S. Abstract Next-generation civil transport aircraft may take advantage of high-aspect-ratio wings for lower induced drag. However, these high-aspect-ratio wings are very flexible and may degrade aircraft roll maneuverability. This paper analyzes the roll maneuverability of a high-aspect-ratio-wing civil transport aircraft derived from a contemporary configuration with regular wing aspect ratio. Roll maneuverability is analyzed using the University of Michigan’s Nonlinear Aeroelastic Simulation Toolbox for capturing geometrically nonlinear effects in the aircraft aeroelastic response and their influence on the flight dynamic response. Results for parametric variations of wing stiffness and mass distributions are presented to discuss design guidelines for improving roll maneuverability in aircraft with high-aspect-ratio wings. Contents 1 2 3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2.1 2.2 Fundamental Description . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled Flight Dynamic and Nonlinear Aeroelastic Equations of Motion . . 3 4 Flexible and Very Flexible Aircraft Numerical Models . . . . . . . . . . . . . . 4 3.1 3.2 3.3 4 5 High-Fidelity Structural Models . . . . . . . . . . . . . . . . . . . . . . . Low-Order Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . Low-Order Transonic Aerodynamic Models . . . . . . . . . . . . . . . . . 5 6 10 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1 4.2 Static Roll Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Roll Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 16 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 .1 .2 24 25 Equivalent Beam Mass Model . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Beam Stiffness Model . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction The next-generation of civil transport aircraft may take advantage of high-aspect-ratio wings (HARWs) that increase aerodynamic efficiency by developing lower induced drag. However, these HARWs undergo larger structural deflections than conventional wings with regular aspect ratios. Large structural deflections reduce the change in wing lift for given aileron input, so degrading aircraft roll maneuverability [1, 2] to the point it may be inadequate for certification standards. Therefore, it is desirable to investigate the impact of HARWs on aircraft roll maneuverability and the benefits that can be achieved by tailoring the wing stiffness and mass distributions. Despite the recent interest in HARW aeroelasticity [3] and in passive aeroelastic tailoring [4], most of previous research focused on aircraft performance or stability [5, 6, 7, 8] while only few studies investigated roll maneuverability. Gibson et al. [9] maximized the aileron reversal speed of a wingbox by reinforcing its upper skin using topology optimization. Kitson et al. [10] studied the static and dynamic roll response of a modified version of the Common Research Model (CRM) [11] having a wing aspect ratio of 13.5. More specifically, they observed an improved roll maneuverability when considering a tow-steered composite HARW wingbox in place of a more flexible metallic wingbox with the same aspect ratio. Motivated by the growing interest in HARW configurations for civil transport aviation, this paper investigates the roll maneuverability of a HARW aircraft model representing a hypothetical next-generation civil transport aircraft. To this aim, a very flexible aircraft (VFA) test case with a HARW is developed from the Airbus XRF1 flexible aircraft (FA) test case. The high-fidelity MSC Nastran global finite element model (GFEM) of each test case is converted to a fully flexible equivalent beam-type representation for analyzing roll maneuverability in the University of Michigan Nonlinear Aeroelastic Simulation Toolbox (UM/NAST) [12]. Firstly, the roll maneuverability of the FA and VFA test cases are compared to highlight the impact of introducing a HARW in an otherwise identical aircraft design. Next, results for parametric variations of wing stiffness and mass distributions are presented to identify design guidelines for improving roll maneuverability in the next generation of HARW aircraft. The paper is organized as follows. Section 2 summarizes the formulation implemented into the UM/NAST framework [12]. Section 3 introduces the UM/NAST numerical models of the FA and VFA test cases used in this study. Section 4 compares the FA and VFA roll maneuverability and investigates the impact of parametric variations in the wing stiffness and mass distributions. A section of concluding remarks summarizes the identified passive aeroelastic tailoring guidelines for improving roll maneuverability in HARW aircraft. 2 Theoretical Background This section summarizes the coupled flight dynamic and nonlinear aeroelastic formulation implemented into UM/NAST [12], which is used as the computational framework for the roll maneuverability studies of this paper. 2 wz wx p Bx Bz By wy pB Gz Gx Gy Figure 1: Global, body, and local frames used for describing aircraft motion. 2.1 Fundamental Description UM/NAST is a low-order computational framework to analyze aircraft in free flight by considering fully coupled rigid-body dynamics, nonlinear structural dynamics, and unsteady aerodynamics. Aircraft are described as systems of fully flexible, geometrically nonlinear beams following the strain-based formulation of Su and Cesnik [13]. The aircraft motion is described by introducing a global inertial frame G, a global body frame B, and local frames w along the flexible beam-type members representing the vehicle components. The origin and orientation of these frames are illustrated in Fig. 1. The position of the B frame in the G frame is given by the vector pB , which is resolved in the B frame. The B frame is defined such that Bx points downstream, By points towards the right wingtip, and Bz is the cross product of Bx and By . Each local frame w has origin at a point p along the reference beam axis of a vehicle member and is oriented such that w x is along that axis, w y points to the leading edge, and wz is the cross product of w x and w y . The position and orientation of a point along the reference axis of vehicle component with respect to the B frame are described by the following 12-component column vector: h(s)T = p(s)T w x (s)T w y (s)T wz (s)T (1) where s is the local abscissa and p, w x , w y , and wz are resolved in the B frame. The formulation of Su and Cesnik [13] solves for the structural dynamics using strains as the independent degrees of freedom (DOFs). A strain-based beam-type finite element has three nodes and four strain DOFs denoted by ε x , κ x , κ y , and κz . These are the local extensional strain and the 3 twist, out-of-plane, and in-plane bending curvatures of the reference beam axis, respectively, which are assumed to be constant throughout the element. Once strains are known, nodal positions and orientations are recovered by integrating kinematic relations [12, 13] from the start to the end of each beam-type member. 2.2 Coupled Flight Dynamic and Nonlinear Aeroelastic Equations of Motion The coupled equations of motion for the rigid-body and elastic DOFs can be cast as [12] MFF MF B εÜ CFF CF B εÛ KFF 0 ε R + + = F (2) MBF MBB βÛ CBF CBB β 0 0 b RB 1 ζÛ = − Ωζ ζ 2GB Û PB = C 0 β (3) where the independent DOFs and their time derivatives are listed in the column vectors ε ε q= = pB b θB εÛ εÛ qÛ = = vB β ωB εÜ εÜ qÜ = Û = vÛ B β ωÛ B (4) In Eqs. (2) and (3), ε is the strain vector for the entire model, vB and ωB are the linear and angular velocities of the B frame with respect to the G frame, the vector b consists of their integrals, and pB and the vector of quaternions ζ give the position and orientation of the B frame with respect to the G frame. The matrices MFF , MF B , MBF , and MBB in Eq. (2) are generalized mass matrices, CFF , CF B , CBF , and CBB are generalized damping matrices, and KFF = K is the constant stiffness matrix of the constitutive relation of the strain-based formulation [13]. The generalized load vectors RF and RB consist of loads due to initial strain, gravity, and point or distributed forces and moments. Aerodynamic loads are modeled as distributed loads and can be computed using various formulations available in UM/NAST such as the two-dimensional finite-state unsteady thin airfoil theory of Peters et al. [14], the unsteady vortex lattice method [15], and the method of segments [16]. When aerodynamic loads are computed using the Peters’ theory, Eqs. (2) and 3 are augmented with additional first-order equations for describing the dynamics of inflow states that account for unsteady wake effects. Finally, Ωζ is a matrix relating the quaternion vector with its derivative and C GB is a rotation matrix from the B to the G frame. Details on the derivation of Eq. (2) and (3) are reported in Refs. [12, 13]. 3 Flexible and Very Flexible Aircraft Numerical Models This section describes the FA and VFA test cases considered for the roll maneuverability studies of this paper. Starting from the high-fidelity GFEMs of these test cases, low-order beam-type representations are developed in UM/NAST to quantify roll maneuverability by means of fully coupled flight dynamic and nonlinear aeroelastic analyses. The beam-type 4 (a) Wingbox model (b) Full-vehicle model Figure 2: GFEMs of the FA (black) and VFA (blue) test cases. representations of the FA and VFA test cases are verified by comparing modal results and aeroelastic trim results from UM/NAST with reference solutions from MSC Nastran modal analyses and from MSC Nastran/CFD aeroelastic trim analyses. 3.1 High-Fidelity Structural Models The FA test case used in this paper is the Airbus XRF1, an industrial standard multidisciplinary research test case representing a typical contemporary configuration for a long-range widebody civil transport aircraft. A HARW derivative of the XRF1, named XRF1-HARW, is developed from the available XRF1 GFEM to obtain a VFA test case representing a nextgeneration HARW vehicle of the same category. The FA and VFA GFEMs are shown in Fig. 2 and differ only for the wingbox. The HARW wingbox in the VFA GFEM was designed in a previous aerostructural optimization [17] conducted at the University of Michigan. The inertial properties and natural frequencies of the VFA and FA test cases (in vacuum) are compared in Tables 1 and 2 for a typical half-loaded mass case1. Trends are similar for 1OOP = out-of-plane bending, IP = in-plane bending. 5 Table 1: VFA to FA test case inertial property comparison (half-loaded case). Property Property variation Mass Longitudinal CG position Lateral CG position Vertical CG position Roll moment of inertia Pitch moment of inertia Yaw inertia moment of inertia +3.19% 0.60% −3.10% 3.54% 44.37% 1.40% 10.13% Table 2: VFA to FA test case free-free natural frequency comparison (half-loaded case). Mode Frequency variation Wing OOP bending (sym) Wing OOP bending (asym) Wing/pylon OOP bending (sym) Wing/pylon OOP bending (asym) Wing IP bending (asym) Wing/pylon/fuselage OOP bending (sym) −41.42% −33.40% −20.43% −18.43% −18.20% −14.47% other mass cases. The frequency variations refer to the first six free-free elastic modes shown in Fig. 3. The VFA shows +44% and +10% increases in the roll and yaw moments of inertia, respectively, and significantly lower natural vibration frequencies. 3.2 Low-Order Structural Models Roll maneuverability is analyzed using UM/NAST [12] for capturing geometrically nonlinear effects in the aircraft aeroelastic response and their impact on the flight dynamic response. This requires converting the GFEM of each aircraft test case to an equivalent beam-type representation compatible with the UM/NAST strain-based formulation. The Fem2Stick (F2S) code was developed at the University of Michigan [18] for extracting spanwise stiffness and mass distributions from a GFEM for creating its beam-type counterpart in UM/NAST [18]. F2S allows to effectively model wing-type flexible members as equivalent beams [18, 10]. However, it assumes infinite axial stiffness and neglects couplings between axial and bending/torsion flexibility. These assumptions make F2S not accurate when applied to tube-type components like fuselage segments or pylons. For developing fully flexible models of the FA and VFA test cases in UM/NAST, an enhanced GFEM-to-beam model order reduction framework was developed [19] that also identifies the axial stiffness spanwise distribution along with the distributions of the axial/bending and axial/torsion coupling terms. Details on the GFEM-to-beam model order reduction process are provided in Appendix. The GFEM-to-beam model order reduction procedure is applied to the isolated GFEM of each FA member for obtaining the equivalent beam stiffness and mass properties. Next, 6 (a) Symmetric wing OOP bending (mode #1) (b) Antisymmetric wing OOP bending (mode #2) (c) Symmetric wing/pylon OOP bending (mode (d) Antisymmetric wing/pylon OOP bending #3) (mode #4) (f) Symmetric wing/pylon/fuselage OOP bending (e) Antisymmetric wing IP bending (mode #5) (mode #6) Figure 3: VFA test case free-free mode shapes (half-loaded case). Figure 4: Beam-type representation of the FA (black) and VFA (blue) test cases. 7 Table 3: MSC Nastran vs. UM/NAST normalized FA wing natural frequencies (half-loaded case). Mode # Mode type MSC Nastran UM/NAST Error 1 2 3 4 5 6 7 8 9 10 OOP bending OOP bending IP bending OOP bending Torsion IP bending OOP bending Torsion OOP bending IP bending 1.00 2.76 3.21 6.14 6.89 8.12 9.81 11.94 13.71 16.01 0.99 2.73 3.22 6.11 7.06 8.54 10.17 11.98 15.13 17.33 −1.38% −0.93% 0.37% −0.48% 2.52% 5.17% 3.63% 0.32% 10.32% 8.24% Table 4: MSC Nastran vs. UM/NAST normalized VFA wing natural frequencies (half-loaded case). Mode # Mode type MSC Nastran UM/NAST Error 1 2 3 4 5 6 7 8 9 10 OOP bending OOP bending IP bending OOP bending IP bending Torsion OOP bending Torsion IP bending OOP bending 0.56 1.38 1.59 2.78 3.95 4.76 4.79 6.93 8.45 9.29 0.57 1.42 1.61 2.94 4.06 4.76 5.05 7.00 7.70 8.81 0.76% 2.82% 1.22% 5.42% 2.67% 0.05% 5.52% 1.09% −8.78% −5.15% 8 Table 5: MSC Nastran vs. UM/NAST normalized FA full-vehicle natural frequencies (halfloaded case). Mode # Mode type MSC Nastran UM/NAST Error 1 2 3 4 5 6 7 8 9 10 Fuselage bending Wing OOP bending (sym) Wing OOP bending (asym) Fuselage bending Left wing/pylon OOP bending Right wing/pylon OOP bending Right pylon OOP bending Left pylon OOP bending Fuselage bending Fuselage bending 1.00 1.08 1.09 1.10 1.34 1.34 1.65 1.65 1.72 1.74 1.01 1.07 1.07 1.08 1.34 1.34 1.88 1.88 1.54 1.83 0.98% −0.34% −1.09% −1.97% 0.40% 0.33% 14.30% 14.16% −10.28% 4.91% Table 6: MSC Nastran vs. UM/NAST normalized VFA full-vehicle natural frequencies (half-loaded case). Mode # Mode type MSC Nastran UM/NAST Error 1 2 3 4 5 6 7 8 9 10 Wing OOP bending (asym) Wing OOP bending (sym) Fuselage bending Right wing/pylon IP bending Left wing/pylon IP bending Fuselage bending Right wing/pylon OOP bending Left wing/pylon OOP bending Wing OOP bending (asym) Wing OOP bending (sym) 0.62 0.62 1.00 1.06 1.07 1.10 1.33 1.33 1.54 1.54 0.62 0.62 1.01 0.99 0.99 1.08 1.21 1.21 1.54 1.54 −0.51% −0.51% 0.93% −6.59% −7.70% −1.21% −9.08% −9.36% 0.46% 0.53% 9 property distributions for the individual members are combined for forming the UM/NAST full-vehicle model. The VFA model is developed by replacing the FA wing beam model with the beam-type representation of the HARW, while the other vehicle members remain unvaried. The beam-type representations of the VFA and FA test cases are shown in Fig. 4. The mass models are verified by comparing the vehicle total mass, center of mass, and inertia tensors against the corresponding quantities evaluated from the GFEM. Results match, because the GFEM-to-beam mass model reduction is numerically exact (see Appendix). The stiffness models are verified by comparing the natural frequencies to the ones evaluated for the GFEM. The MSC Nastran and UM/NAST natural frequencies for the first ten wing modes, normalized by the first FA wing frequency, are compared in Tables 3 (FA) and 4 (VFA). Results are for a typical half-loaded mass case. The MSC Nastran and UM/NAST natural frequencies for the first ten full-vehicle modes, normalized by the first FA full-vehicle frequency, are compared in Tables 5 (FA) and 6 (VFA). Results also are for a typical half-loaded mass case. The MSC Nastran full-vehicle natural frequencies are computed by clamping the fuselage cross-section at the wing reference axis location, because that point is kept fixed in the UM/NAST modal analysis. The frequency error in the wing-dominated modes is about 15%, which increases to 10-15% for fuselage- and pylon-dominated modes. The larger errors for these modes are due to the less accurate equivalent beam stiffness properties evaluated for tube-type members due to their more complex geometries, features such as cut-outs, and lower stiffness resulting in cross-sectional deformations that are not captured by a beam representation. However, the full-vehicle natural frequency comparisons are acceptable for the analyses of this paper that focuses on roll maneuverability. 3.3 Low-Order Transonic Aerodynamic Models The FA and VFA aerodynamics is modeled using a low-order surrogate-based transonic aerodynamic model based on the method of segments [16]. This approach allows one to capture three-dimensional transonic-flow effects with sufficient accuracy for roll maneuverability analyses and low computational cost that facilitates parametric studies. The method of segments describes a flexible lifting member as a collection of twodimensional rigid cross-sections along the span which undergo arbitrary three-dimensional translations and rotations. The aerodynamic loads on each cross-section are provided by a kriging surrogate model that returns the lift, drag, and moment coefficients for given local instantaneous Mach number and effective angle of attack. The kriging surrogate model for each cross-section is created by fitting a database of aerodynamic coefficients computed from full-vehicle steady rigid CFD simulations for various flight Mach number M and body angle of attack α conditions. Although the kriging surrogates are based on steady CFD data, they are used with the local flow conditions during simulations. Therefore, this description captures the effect of rigid-body motion and structural deflections on the aerodynamic loads, although it lacks the modeling of unsteady lags due to compressibility and shed wake. The kriging surrogates for the FA and VFA test cases are based on coefficient values from steady rigid CFD solutions for M ∈ [0.6, 0.89] and α ∈ [−4, +5] deg and are created in the Matlab DACE toolbox [20] using a zero-th order regression model and a linear correlation model. The kriging surrogates for the lift, drag, and pitching moment coefficients at the 10 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0.01 0.03 0.04 0.05 0.06 0.07 0.08 0.1 Coefficient value 1 Coefficient value 0.02 0.5 0 -0.5 5 0.05 0 5 0.9 0 Local AoA (deg) 0.9 0 0.8 -5 0.7 -10 0.6 0.8 -5 0.7 -10 Local AoA (deg) Mach number (a) Lift coefficient 0.6 Mach number (b) Drag coefficient -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 Coefficient value -0.05 -0.1 -0.15 -0.2 5 0.9 0 0.8 -5 Local AoA (deg) 0.7 -10 0.6 Mach number (c) Moment coefficient Figure 5: Kriging surrogates for the aerodynamic coefficients at the FA wing midspan node. 11 0.15 Normalized z coordinate Normalized z coordinate 0.15 0.12 MSC Nastran/CFD 0.09 UM/NAST 0.06 Undeformed 0.03 0 0 0.2 0.4 0.6 0.8 0.12 0.09 MSC Nastran/CFD 0.06 Undeformed 0.03 0 1 UM/NAST 0 0.2 0.4 0.6 0.8 Normalized y coordinate Normalized y coordinate (a) Empty case (b) Fully loaded case 1 Figure 6: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim deflection. Table 7: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim parameters (empty case). Empty weight Parameter MSC Nastran/CFD UM/NAST Error Angle of attack (deg) Elevator (deg) Tip z-displ. (% semispan) 0.55 −2.11 2.36 0.51 −1.66 2.38 −7.22% −21.18% 0.56% FA wing midspan cross-section are shown in Fig. 5 as an example. The surrogates for each spanwise location are included into the FA and VFA UM/NAST models and the complete aeroelastic models are verified by comparing UM/NAST trim results with reference MSC Nastran/CFD results. Comparisons for the FA are shown in Fig. 6 and Table 7 for a typical cruise trim condition and two mass cases corresponding to the minimum and maximum weight. The aircraft wing deformation is well captured by the UM/NAST model. The difference in the trim parameters is motivated by the absence of fuselage aerodynamics in UM/NAST, but it is acceptable for the scope of this study. Table 8: MSC Nastran/CFD vs. UM/NAST FA aeroelastic trim parameters (fully loaded case). Fully loaded Parameter MSC Nastran/CFD UM/NAST Error Angle of attack (deg) Elevator (deg) Tip z-displ. (% semispan) 2.37 −3.79 5.58 2.78 −4.25 5.65 17.31% 12.33% 1.35% 12 -0.1 WR2 0 Normalized x coordinate Normalized x coordinate -0.1 WR3-1 0.1 WR3-2 WR1 0.2 WR3-3 WR3-4 WR3-5 0.3 0.4 0.5 0.6 WR3-6 0 0.2 0.4 0.6 0.8 WR3-1 0.1 WR3-2 WR1 0.2 WR3-3 WR3-4 0.3 WR3-5 0.4 0.5 0.6 1 WR2 0 0 0.2 0.4 0.6 WR3-6 0.8 1 Normalized y coordinate Normalized y coordinate (a) FA wing (b) VFA wing Figure 7: FA and VFA wing planform subdivision for parametric studies. 4 Results This section discusses static and dynamic roll maneuverability results for the FA and VFA (half-loaded cases) with baseline wing properties and for parametric variations of the wing stiffness and mass distributions. For the parametric studies, the FA and VFA wings are subdivided into three regions denoted by WR1, WR2, and WR3, respectively. These regions are shown in Fig. 7 (coordinates are normalized by the semispan of each wing). The WR1 and WR2 regions correspond to the wing segments from the vehicle centerline to the wing-fuselage interface (WR1) and from the wing-fuselage interface to the wing-pylon interface (WR2). The pylon is placed at the same dimensional spanwise location in the FA and VFA, and thus at a more inboard normalized location in the VFA. The WR3 region spans from the pylon to the wingtip and it is subdivided into six smaller subregions named WR3-1 to WR3-6. The inboard and outboard ailerons are located in WR3-4 and WR3-5, respectively, which occupy the same normalized spanwise locations in the FA and VFA. Wing stiffness and mass properties are varied in the entire WR3 and in its subregions individually while property variations in the WR1 and WR2 regions do not impact roll maneuverability. 4.1 Static Roll Response The static roll response is analyzed by quantifying the roll aerodynamic moment increment on the FA and VFA isolated wings with root-clamped boundary conditions for given antisymmetric inboard/outboard aileron deflection. Results are for Mach number M = 0.83, altitude h = 8484 m, and root angle of attack α = 2 deg. Results for the FA and VFA wings with baseline properties are shown in Fig. 8a. The VFA wing experiences a smaller roll aerodynamic moment increment for given aileron deflection even in the presence of a longer moment arm of the ailerons compared to the FA. This is because the VFA wing is more flexible and thus each half-wing undergoes larger positive/negative out-of-plane bending displacements due to downward/upward aileron 13 5 x10 VFA outboard aileron Normalized spanwise center of lift 2 Roll moment (N-m) 0 FA outboard aileron -2 FA inboard aileron -4 VFA inboard aileron -6 -8 -10 0 1 2 3 4 5 0.44 0.42 FA right half-wing FA left half-wing 0.4 0.38 VFA right half-wing 0.36 VFA left half-wing 0.34 0.32 0 1 Aileron deflection (deg) 2 3 4 5 Aileron deflection (deg) (a) Roll moment (inboard/outboard aileron deflec- (b) Center of lift position (inboard aileron deflection) tion) 50 Aileron effectivness variation (%) Aileron effectivness variation (%) Figure 8: Static roll response of the baseline FA and VFA wings. 40 30 20 10 Reversed out-of-plane bending/torsion coupling Zero out-of-plane bending/torsion coupling Torsion 0 -10 -20 Out-of-plane bending/torsion coupling Out-of-plane bending -30 -40 -50 0.85 0.9 0.95 1 1.05 Scaling factor 1.1 1.15 (a) FA 50 40 Out-of-plane bending 30 20 10 0 Out-of-plane bending/torsion coupling Torsion Zero out-of-plane bending/torsion coupling -10 -20 Reversed out-of-plane bending/torsion coupling -30 -40 -50 0.85 0.9 0.95 1 1.05 Scaling factor 1.1 1.15 (b) VFA Figure 9: Inboard aileron effectiveness variation for uniform stiffness scaling in the FA and VFA WR3 regions. 14 deflections. For an aft-wing, the twist of aerodynamic cross-sections due to out-of-plane bending displacements contrasts the lift variation commanded by the ailerons. Therefore, the VFA wing develops a smaller roll aerodynamic moment (in magnitude) for given aileron deflection compared to the FA. This trend is also shown by the variation in the center of lift position along the span reported in Fig. 8b. For both configurations, inboard ailerons are more effective than outboard ailerons due to their larger area and lower out-of-plane bending displacements. The VFA wing also shows a moderately nonlinear static roll response when deflecting either pair of ailerons which is not observed for the FA wing (see Fig. 8a). Parametric static analyses focus on variations in the FA and VFA wing stiffness properties. This is because mass properties only influence the static response through the self-weight, which does not significantly impact the results. The varied stiffness properties are the torsion and out-of-plane bending stiffness terms and the out-of-plane bending/torsion coupling term, which have the highest influence on the static roll response. The effect of stiffness property variations is quantified in terms of inboard aileron effectiveness, that is, the slope in the curves in Fig 8a. For the VFA, the aileron effectiveness is computed by averaging slopes at different values of the inboard aileron deflection due to the nonlinearity in the aileron response (see Fig. 8). The variation in aileron effectiveness for uniform stiffness scaling in the entire WR3 is shown in Fig. 9. Increasing the torsion or the out-of-plane bending stiffness improves aileron effectiveness for both the FA and VFA wings. However, increasing the out-of-plane bending stiffness results in a more significant improvement compared to increasing torsion, as expected for an aft-swept wing. Increasing the out-of-plane bending stiffness reduces structural displacements in the presence of aileron deflections, so mitigating the loss in aileron effectiveness due to the twist of aerodynamic cross-sections induced by bending. These trends are observed for both the FA and VFA wings. The VFA aileron effectiveness is more sensitive to variations in the out-of-plane bending and torsion stiffness, as expected due to higher wing flexibility. Varying the magnitude of the out-of-plane bending/torsion coupling does not have a significant impact, while reversing its sign improves the FA aileron effectiveness as shown in Fig. 9. This is because the out-of-plane bending/torsion coupling term is mainly negative in the FA WR3 region, resulting in a wash-out effect which is detrimental for aileron effectiveness. On the other hand, a positive coupling term causes a positive elastic twist for upward out-of-plane bending, which is beneficial for aileron effectiveness. The out-of-plane bending/torsion coupling term is already positive in most of the VFA WR3 region by design, although improving aileron effectiveness was not a design goal [17]. Therefore, reversing the coupling term sign reduces aileron effectiveness as shown in Fig. 9. As a further verification of these trends, results for zero out-of-plane bending/torsion coupling are also shown in Fig. 9. For the FA, zero coupling increases aileron effectiveness, although not as much as in the case of reversed coupling (meaning positive coupling for this configuration). The opposite occurs for the VFA where the coupling term is positive by design, and thus setting it to zero reduces aileron effectiveness. Results for fixed stiffness scaling factors applied to WR3 subregions are shown in Tables 9 and 10. Varying the torsion stiffness in one subregion at a time has a limited impact on aileron effectiveness, particularly for the FA wing. This is expected because varying the torsion stiffness in the entire WR3 does not have a large impact either. The largest aileron effectiveness improvement is achieved by increasing the out-of-plane bending stiffness in the 15 Table 9: Inboard aileron effectiveness variation for stiffness scaling in FA WR3 subregions. Aileron effectiveness variation (%) Torsion stiffness (10% increase) Out-of-plane bending stiffness (10% increase) Out-of-plane bending/torsion coupling (reversed sign) Uniform 1.93 6.76 13.70 WR3-1 WR3-2 WR3-3 WR3-4 WR3-5 WR3-6 0.55 0.87 −0.01 0.15 0.03 0.00 2.13 3.46 0.66 −0.13 −0.08 0.00 2.15 7.72 −0.02 −0.14 −0.16 0.00 WR3-1 and WR3-2 regions (around the midspan) for both the FA and VFA wings. Varying the properties in other WR3 subregions has a smaller or negligible effect. Considering that the FA wing has a negative out-of-plane bending/torsion coupling, reversing its sign in WR31 and WR3-2 also improves aileron effectiveness. In contrast, reversing the coupling term sign in these regions degrades aileron effectiveness for the VFA wing. 4.2 Dynamic Roll Response The dynamic roll response is analyzed by comparing the time histories of the roll rate and roll angle2 developed for an antisymmetric deflection of the inboard ailerons applied to the FA and VFA full-vehicle models trimmed for level flight at M = 0.83 and h = 8484 m. Results for FA and VFA with baseline wing properties are shown in Fig. 10. Roll rates and angles follow the flight dynamic convention (roll axis towards the vehicle nose). The FA develops a larger roll rate and roll angle (in magnitude) at a given simulation time compared to the VFA due to higher wing stiffness and smaller vehicle roll moment of inertia (see Table 1). Furthermore, the roll rate of the VFA slightly increases at t = 1 s when the aileron starts to deflect before becoming negative following the applied command, and it shows oscillations that are not observed for the FA. These oscillations are also present in the wingtip vertical displacement and can be explained by observing the in-plane tip displacement of the FA and VFA wings. The in-plane tip displacement of the VFA wing has a higher amplitude and smaller damping associated to it which causes the oscillations observed in the roll rate and vertical tip displacement. Dynamic parametric analyses for uniform scaling in the wing mass distribution are shown in Fig. 11. Results for uniform scaling in other inertial parameters such as mass offset components and moments of inertia are not reported because they have no significant impact on the results. Scaling the mass distribution uniformly also has only a slight impact on the 2The quantitative measures of roll rate and roll angle are based off an assumed value of Clδ and as such the results are for a qualitative comparison only. 16 Aileron deflection (deg) 6 5 4 3 2 1 0 0 1 2 3 4 5 Time (s) (a) Right inboard aileron input 2 0.5 0 0 Roll rate (deg/s) Roll angle (deg) VFA -0.5 -1 FA -1.5 -2 VFA FA -2 -4 -6 -2.5 -3 0 1 2 3 4 -8 5 2 3 (b) Roll rate (c) Roll angle 4 5 0.2 Tip displacement (% semi-span) Tip displacement (% semi-span) 1 Time (s) 8 6 VFA 4 2 FA 0 -2 0 Time (s) 0.15 0.1 FA 0.05 VFA 0 -0.05 0 1 2 3 4 -0.1 5 Time (s) 0 1 2 3 4 5 Time (s) (d) Out-of-plane wingtip displacement (with re- (e) In-plane wingtip displacement (with respect to spect to trim) trim) Figure 10: Dynamic roll response of the baseline FA and VFA in free flight. 17 Table 10: Inboard aileron effectiveness variation for stiffness scaling in VFA WR3 subregions. Aileron effectiveness variation (%) Torsion stiffness (10% increase) Out-of-plane bending stiffness (10% increase) Out-of-plane bending/torsion coupling (reversed sign) Uniform 10.56 20.88 −38.20 WR3-1 WR3-2 WR3-3 WR3-4 WR3-5 WR3-6 2.50 3.95 0.57 −1.70 0.21 −0.00 6.32 11.77 2.87 0.32 −0.57 0.00 −24.70 −10.74 −3.62 0.19 0.08 0.00 roll responses for both the FA and VFA. Therefore, it is concluded that the roll response is primarily influenced by the wing stiffness, and region studies with respect to mass property variations are not performed. Next, the dynamic roll response is analyzed for variations in the wing stiffness properties to confirm the static analysis trends. Results for uniform stiffness scaling in WR3 are shown in Figs. 12 to 14. As in the static studies, practical variations in the torsion stiffness have limited impact on the roll response. However, reducing the torsion stiffness causes the FA to become unstable at the assumed flight condition, as shown by the growing oscillations in the roll rate time history in Fig. 12a. Increasing the out-of-plane bending stiffness improves the roll dynamic response for both the FA and VFA, resulting in a larger roll rate and roll angle (in magnitude) developed at a given simulation time. Figure 14 compares the FA and VFA results with reversed sign of the out-of-plane bending/torsion coupling term. As observed in the static analyses, the FA roll response improves when reversing the coupling term sign, which is negative in the outer wing. In contrast, the VFA response degrades when reversing the coupling term sign since this is already positive in the baseline design. Finally, Fig. 15 compares the dynamic roll responses of the FA and VFA obtained considering the best-case stiffness properties from the previous studies. For the FA, the out-of-plane bending stiffness is increased by 15% uniformly in WR3 and the sign of the outof-plane bending/torsion coupling term is reversed. For the VFA, the out-of-plane bending stiffness is increased by 15% and with no sign change in the out-of-plane bending/torsion coupling term. The applied stiffness variations improve the roll dynamic response for both the FA and VFA, in line with the previous results. However, the VFA roll response remains significantly slower than the one of the FA. This suggests that passive aeroelastic tailoring alone is not sufficient to ensure adequate roll maneuverability in HARW aircraft for practically achievable variations in the stiffness properties, but it has to be complemented with other design modifications. For instance, this study assumed inboard and outboard ailerons to be at the same normalized spanwise locations in the FA and VFA wings. Since the static results showed that inboard ailerons are more effective for both configurations, a more inboard 18 0.5 0.5 Baseline Scaling factor = 0.85 Scaling factor = 1.15 0 0 -0.5 Roll rate (deg/s) Roll rate (deg/s) -0.5 -1 -1.5 -1.5 -2 -2.5 -2 -2.5 0 1 2 3 4 -3 5 1 2 3 Time (s) (a) FA roll rate (b) VFA roll rate 2 2 0 0 -2 -4 -6 -8 0 Time (s) Roll angle (deg) Roll angle (deg) -3 -1 4 5 4 5 -2 -4 -6 0 1 2 3 4 -8 5 0 1 2 3 Time (s) Time (s) (c) FA roll angle (d) VFA roll angle Figure 11: Dynamic roll response in free flight for uniform mass scaling in the FA and VFA WR3 regions. 19 0.5 0 0 -0.5 Roll rate (deg/s) -0.5 Roll rate (deg/s) 0.5 Baseline Scaling factor = 0.85 Scaling factor = 0.90 Scaling factor = 1.10 Scaling factor = 1.15 -1 -1.5 -1.5 -2 -2.5 -2 -2.5 0 1 2 3 4 -3 5 1 2 3 Time (s) (a) FA roll rate (b) VFA roll rate 2 2 0 0 -2 -4 -6 -8 0 Time (s) Roll angle (deg) Roll angle (deg) -3 -1 4 5 4 5 -2 -4 -6 0 1 2 3 4 -8 5 0 1 2 3 Time (s) Time (s) (c) FA roll angle (d) VFA roll angle Figure 12: Roll dynamic response in free flight for uniform torsion stiffness scaling in the FA and VFA WR3 regions. aileron placement in the VFA wing can help mitigating the roll maneuverability degradation due to a longer wing span. 5 Concluding Remarks This paper analyzed the roll maneuverability of a VFA civil transport aircraft model with a HARW derived from a contemporary FA configuration. Analyses were conducted in the UM/NAST framework to capture geometrically nonlinear effects in the aircraft aeroelastic response and their impact on the flight dynamic response. The aileron effectiveness and free-flight dynamic roll response of the FA and VFA test cases were first compared to study the impact of introducing a HARW in an otherwise identical aircraft design. Next, parametric variations of wing stiffness and mass distributions were applied to the FA and VFA wings to study their impact on roll maneuverability. The static analyses showed that the VFA wing has reduced aileron effectiveness compared 20 Baseline Scaling factor = 0.85 Scaling factor = 0.90 Scaling factor = 1.10 Scaling factor = 1.15 0.5 0 0.5 0 -0.5 Roll rate (deg/s) Roll rate (deg/s) -0.5 -1 -1.5 -1.5 -2 -2.5 -2 -2.5 0 1 2 3 4 -3 5 1 2 3 Time (s) (a) FA roll rate (b) VFA roll rate 2 2 0 0 -2 -4 -6 -8 0 Time (s) Roll angle (deg) Roll angle (deg) -3 -1 4 5 4 5 -2 -4 -6 0 1 2 3 4 -8 5 0 1 2 3 Time (s) Time (s) (c) FA roll angle (d) VFA roll angle Figure 13: Roll dynamic response in free flight for uniform out-of-plane bending stiffness scaling in the FA and VFA WR3 regions. 21 0.5 0.5 0 0 -0.5 -0.5 Roll rate (deg/s) Roll rate (deg/s) Reversed coupling -1 -1.5 Reversed coupling -2.5 -2 -2.5 0 1 2 3 4 -3 5 1 2 3 Time (s) (a) FA roll rate (b) VFA roll rate 2 2 0 0 -2 Baseline -4 Reversed coupling -6 -8 0 Time (s) Roll angle (deg) Roll angle (deg) -3 -1 -1.5 Baseline -2 Baseline 4 5 Reversed coupling Baseline -2 -4 -6 0 1 2 3 4 -8 5 0 1 2 3 4 Time (s) Time (s) (c) FA roll angle (d) VFA roll angle 5 Figure 14: Roll dynamic response in free flight for uniform out-of-plane bending/torsion coupling term reversal in the FA and VFA WR3 regions. 22 0 0 -0.5 -0.5 Roll rate (deg/s) 0.5 Roll rate (deg/s) 0.5 -1 -1.5 -2 -3 Best stiffness properties 0 1 2 3 4 -2 -2.5 -3 5 0 1 2 3 4 Time (s) Time (s) (a) FA roll rate (b) VFA roll rate 2 5 0.5 0 0 Baseline -0.5 Roll rate (deg/s) Roll angle (deg) Best stiffness properties -1 -1.5 Baseline -2.5 Baseline -2 Baseline -4 -1.5 Best stiffness properties -6 Best stiffness properties -1 -2 -2.5 -8 0 1 2 3 4 -3 5 0 1 2 3 4 Time (s) Time (s) (c) FA roll angle (d) VFA roll angle 5 Figure 15: Roll dynamic response in free flight for uniform out-of-plane stiffness scaling by 15% in the FA and VFA WR3 regions and reversed coupling term in the FA WR3 region only. 23 to the FA wing due to its higher flexibility, which results in larger out-of-plane bending displacements for a given aileron input. The aileron effectiveness of both the FA and VFA wings improved by increasing the out-of-plane bending stiffness primarily around the wing midspan. A smaller improvement was also achieved by increasing the torsion stiffness in the same region. A positive out-of-plane bending/torsion stiffness coupling term is beneficial for aileron effectiveness. For the FA where the out-of-plane bending/torsion stiffness coupling term is mainly negative along the span, aileron effectiveness improved by reversing the coupling term sign. In contrast, the VFA wing already has a mainly positive out-of-plane bending/torsion stiffness coupling term along the span by design with a beneficial effect on aileron effectiveness. The dynamic analyses showed that the VFA full vehicle develops smaller roll rate and angle (in magnitude) for a given simulation time when subject to the same aileron input as the FA. Uniformly scaling the wing mass properties did not significantly impact the roll response, while uniformly scaling the wing stiffness properties confirmed the trends observed in the static analyses. However, the VFA did not recover the roll maneuverability of the FA for practical stiffness property variations. Future work will investigate the benefits of combined wing passive aeroelastic tailoring and varying aileron placement for improving roll maneuverability in HARW aircraft. Acknowledgments The material of this paper is based upon work supported by Airbus in the frame of the AirbusMichigan Center for Aero-Servo-Elasticity of Very Flexible Aircraft. The authors would like to thank Airbus for providing the XRF1 test case as a mechanism for demonstrating the approaches presented in this paper. Appendix: GFEM-to-Beam Model Order Reduction This appendix details the procedure for converting high-fidelity GFEMs into equivalent strainbased beam-type representations compatible with the UM/NAST formulation. The procedure is implemented into a GFEM-to-beam model order reduction framework developed at the University of Michigan [19]. .1 Equivalent Beam Mass Model A GFEM can be represented as a collection of lumped masses mi with positions xi + δi in the global reference frame, where xi are the GFEM nodes positions and δi the mass offsets, and lumped inertia tensors Ji with respect to xi + δi also given in the global reference frame (i = 1, . . . , N, where N is the number of nodes). The equivalent mass mref , mass offset δref , and inertia tensor Jref at a generic beam reference axis node are obtained by summing the contributions from the Nnb GFEM nodes in the nearest neighborhood: mref = Nnb Õ i=1 mi δref Nnb 1 Õ mi (xi + δi − xref ) = mref i=1 24 Jref = Nnb Õ i=1 (Ji + Ji,tr ) (5) where xref is the position of the reference beam axis node (in the global reference frame) and Ji,tr is the transport moment of mi from xi + δi to xref . The information in Eq. (5) can be directly included in UM/NAST by defining rigid-bodies attached to the beam element nodes [12]. .2 Equivalent Beam Stiffness Model The stiffness module generates the strain-based cross-sectional stiffness matrix k that relates the internal load vector f at a reference axis node to the local strain vector : f = k (6) In Eq. (6), f and are 6-component vectors listing the internal force and moment resultants and the strain components, respectively, all resolved in the local reference beam at the current beam cross-section. Following the UM/NAST formulation [12], this has the x axis along the beam, the y axis towards the leading edge, and the z oriented as the cross-product of x and y. The 6×6 matrix k describes the axial, shear, torsion, and bending behaviors of the beam. For the strain-based formulation used in UM/NAST [12, 13], the k matrix is reduced to a 4×4 matrix that retains only the terms associated with extension, torsion, and bending, along with the related couplings. This improves the previous F2S implementation [18], which assumed infinite axial stiffness and neglected the related coupling terms. The equivalent beam stiffness matrix k for an aircraft component is computed as follows. The component GFEM is subject to six static independent tip load cases, each considering only one force or moment component in the global reference frame. A linear static analysis with root-clamped boundary conditions is solved for each load case and the 3D displacement field of the GFEM is output has a 1D field using interpolation elements. The 1D displacement fields are next transformed from the GFEM global reference frame into the local frames defined by each pair of consecutive reference beam axis nodes A and B. These frames have x axis aligned from A to B, y axis towards the leading edge, and z axis upward, consistently with the local frames of the UM/NAST formulation [12, 13]. Considering two consecutive points A and B, one can write f B = K∆u (7) In Eq. (7), f B is the load vector at node B due to the applied tip load and ∆u := u B −u A, where u A and u B are the displacements vectors of the nodes A and B (listing nodal translations and rotations). Both f B and ∆u are resolved in the local frame associated to the segment AB. Equation (8) gives 6 scalar equations per load case for a total of 36 equations that allow to solve for K. Once K is known, k is obtained by solving K −1 Q−1 = k −1 HQ−1 + E k −1 (8) for k −1 using the method of Ref. [21]. The matrices Q, E, and H in Eq. (8) are constant for each segment AB and function of the segment length. Once k −1 is evaluated, it is reduced to 4×4 and inverted again to obtain the cross-sectional stiffness matrix compatible with the UM/NAST strain-based formulation. The process is repeated for each pair of nodes along the reference beam axis for obtaining the equivalent beam stiffness distributions of the entire component. Next, the process is repeated for each component and the obtained stiffness (and 25 mass) distributions are assembled together for forming the UM/NAST model of the entire vehicle. References [1] T. A. Weisshaar, C. 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