A Strain-Based Approach For Geometrically Nonlinear .
A Strain-Based Approach forGeometrically Nonlinear AeroelasticityWeihua SuDepartment of Aerospace Engineering and MechanicsUniversity of AlabamaTuscaloosa, ALPresented at the School of Aeronautic Science and EngineeringBeijing University of Aeronautics and AstronauticsBeijing, ChinaJuly 29, 2013Aeroelasticity and StructuralDynamics Research Laboratory
2About Me Assistant Professor, University of AlabamaEducation– B.S. 2000 (BUAA)– M.S. 2002 (BUAA)– Ph.D. 2008 (Univ. of Michigan) Research areas–––––– Nonlinear aeroelasticityStructural dynamicsActive/smart structuresFlight dynamicsActive controlUAV, MAV, Wind Turbine, etc.Research lab:– Aeroelasticity and Structural Dynamics Research Laboratory– www.bama.ua.edu/ wsu2/Aeroelasticity and StructuralDynamics Research Laboratory
3Overview Introduction– Motivation and background– Objective Nonlinear aeroelastic formulation– Geometrically nonlinear beam model– Aerodynamic and flight dynamic formulations Numerical studies– Flutter instability– Response to external disturbance (gust) UAV design and flight tests Concluding remarks Other research areasAeroelasticity and StructuralDynamics Research Laboratory
4High-Altitude Long-Endurance (HALE) Aircraft Aircraft for surveillance, target acquisition, and communications Desired features:– long operation range– long loiter time L W0 R or E ln D W1 High aerodynamic efficiencyLow structural weight fraction High-aspect-ratio wingsVery flexible aircraftAeroelasticity and StructuralDynamics Research Laboratory
5Wing Aspect Ratio and Aerodynamic EfficiencyF-22aAR 2.36B787-8AR 11.08ETA (Germany)AR 51.33Large aspect ratio for high aerodynamicefficiencyBut The large AR brings something interestingAeroelasticity and StructuralDynamics Research Laboratory
6U.S. Air Force Sensorcraft Studies HALE aircraft may adopt rather unconventional configurations:Unmanned vehiclesSensor platformVery high fuel fractions (up to 60%)ISR (intelligence, surveillance, reconnaissance)“Sensorcraft” Concepts (Lucia, 2005)Aeroelasticity and StructuralDynamics Research Laboratory
7Pushing the Flight Envelop AeroVironment’s Helios: 24 hrsAeroVironment’s Global Observer:One weekDARPA’s Vulture Program: 5 yearsAeroelasticity and StructuralHigh lift-to-drag ratio wings and low structural weight fractionDynamics Research Laboratory
8Background Aeroelastic response of vehicles with long, high-aspect-ratio wings isinherently nonlinear:– Large elastic deflections: structurally nonlinear– Large angles of attack: aerodynamic nonlinear– Local transonic effects: aerodynamic nonlinear Low frequency aeroelastic response couples with flight dynamics withnonlinearities possibly dominating the vehicle response– Trajectory and attitude– Stability (including body-freedom flutter)– Response to disturbances Combined nonlinear effects alter loads, stability, performanceAeroelasticity and StructuralDynamics Research Laboratory
9Mishap of Helios PrototypeThe number one root cause/recommendation from NASA[1]was:[That] more advanced, multidisciplinary (structures,aeroelastic, aerodynamics, atmospheric, materials,propulsion, controls, etc.) time-domain analysismethods appropriate to highly flexible, morphingvehicles [be developed].[1] Noll, T. E., Brown, J. M., Perez-Davis, M. E., Ishmael, S. D., Tiffany, G. C.,and Gaier, M., “Investigation of the Helios Prototype Aircraft Mishap,” Tech. rep.,NASA, January 2004.Aeroelasticity and StructuralDynamics Research Laboratory
10Focus Objective:– Understand the aeroelastic response of very flexible aircraft during normaland unusual flight conditions Structural integrity Stability Controllability– Generate parametric models for concept design of unconventionalconfigurations– Explore different electric/mechanical/aero mechanisms for vehicleaeroelastic control Approach:– Develop reduced-order aeroservoelastic formulation For preliminary vehicle and control design studies or more detailed analysis Able to simulate fully flexible vehicle with 6 rigid-body DoF’s– Numerically investigate aeroelastic response of different vehicleconfigurations under different nonlinear effectsAeroelasticity and StructuralDynamics Research Laboratory
11Reduced-Order Aeroelastic FrameworkA Multidisciplinary ApproachAerodynamicsIncompressible 2-D unsteadyaerodynamics, Prandtl-Glauertand tip loss corrections, rically nonlinearcomposite beamRigid-BodyDynamicsNonlinear 6-DoFvehicle dynamicsSimplified free-flight analysis and simulation for full aircraftAeroelasticity and StructuralDynamics Research Laboratory
Reduced-Order Aeroelastic Framework(Cont’d)12A Multidisciplinary ApproachSimplified free-flight analysis and simulation for full aircraftAeroelasticity and StructuralDynamics Research Laboratory
13Reduced-Order Structural Modeling From 3D elastic problem to 2D beam cross-sectional analysis and 1Dbeam modelDimensional reduction using the Variational-Asymptotic Method:– Active thin-walled solution (mid-line discretization)– VABS (finite-element discretization)– User defined stiffness constantsCross-Section Stiffness and Actuation Constants Fx K11 M x K 21 M y K31 M z K 41K12K 22K32K 42K13K 23K33K 43K14 x B11K 24 x B21 K34 y B31K 44 z B41B12B22B32B42B1m v1 B2 m v2 B3m vm B4 m Aeroelasticityand StructuralDynamics Research Laboratory
Highlight of Strain-Based GeometricallyNonlinear Beam Formulation14 Geometrically exact formulation – no approximation todeformation of beam reference line Reduced number of degrees of freedom Efficient in solving geometrically nonlinear static problem Beam strain (curvature) is directly measured by control sensor –facilitate control design and study Catch geometrically nonlinear behavior of flexible isotropic andcomposite wings Provide structural dynamic models for nonlinear aeroelastic andcontrol studies of very flexible slender structures Difficulty in solving statically indeterminate beams – with splitsand jointsAeroelasticity and StructuralDynamics Research Laboratory
15Basic Coordinate Systems Global frame (G) Body frame (B) – origin notnecessary to be C.G. of vehicle Body frame motion variables p b B B p B vB b B B p B vB b B B Local beam frame (w) Auxiliary local frame (b)Aeroelasticity and StructuralDynamics Research Laboratory
Strained-Based Finite Element BeamFormulation 16Geometrically nonlinear beam formulation[2]Four local strain degrees-of-freedom (ε): extension, twist, flatwisebending, and chordwise bendingConstant-strain elementsCapture large complex deformations with fewerelements – computationally efficientIsotropic and anisotropic constitutive relationsSample element deformationswith constant strainStrains ( ) and body velocities ( )are independent variables[2] Su, W., and Cesnik, C. E. S., “Strain-Based Geometrically Nonlinear Beam Formulation forModeling Very Flexible Aircraft,” International Journal of Solids and Structures, Vol.48, No. 16-17,Aeroelasticityand StructuralDynamics Research Laboratory2011, pp. 2349-2360. doi: 10.1016/j.ijsolstr.2011.04.012
17Nodal Position and Orientation Arbitrary point in a cross-section(a)pa p B pw xw x yw y zw zBody FrameMotion Elastic StrainContributionPosition and orientation ofnodes along beam reference linehT (s) pTB pTw (s), wTx (s), wTy ( s), wTz ( s) Differential and variation of h arerelated to and h J h J hb bdh J h d J hb dbh J h J hbb J h J hb h J h J h J hb J hb whereJ h h , J hb h bAeroelasticity and StructuralDynamics Research Laboratory
Virtual Work – Inertia of Flexible BeamMembers18 Inertia forcePosition: p aVelocity and acceleration:va, aaVirtual work: W int pa aa dAds V Consists of virtual displacement h and dependent variable hAeroelasticity and StructuralDynamics Research Laboratory
19Virtual Work of Flexible Beam Members Internal strain and strain rate W int (s)T k (s) (s) 0 (s) ( s)T c(s) (s) dss Virtual work of external load W ext uT ( x, y, z ) f ( x, y, z )dVV Assembly of total virtual workInternal VW External VWKinematics(Jacobians)EOMStrain-basedform h J h J hb bh J h J h J hb J hb Aeroelasticity and StructuralDynamics Research Laboratory
20Formulation Based on Principle of Virtual Work W 0iiEquations of Motion M FF ( ) M FB ( ) CFF ( , , ) CFB ( , , ) K FF M ( ) M ( ) C ( , , ) C ( , , ) 0 BFBB BFBB Generalized Mass0 RF 0 b RB Generalized DampingGeneralizedGeneralizedStiffnessForce J Tp F dist J T M dist J Tp pt J T pt RF K FF 0 J hT T Ng T B F T B M T F T M J pb J pb RB 0 J hb J b J b Aeroelasticity and StructuralDynamics Research Laboratory
Unsteady Aerodynamics – Finite-State InflowTheory21 2-D Theodorsen-like unsteady aerodynamics (Peters et al., 94, 95)Inflow velocity z 1 lmc b 2 z y d 2 by 2 b d 0 2 bc1 y 2 y y y 2 1 mmc b 2 b 2 yz dy y 0 2 b 2c4 y 2 8 Glauert expansion of inflow velocityas function of inflow states, λn 0 1bn n 2 n 1 Finite state differential equation is transformed to independentvariables and E1 E2 z E3 E4 F1 F2 F3 Aeroelasticity and StructuralDynamics Research Laboratory
22Finite-State Inflow Theory: Modifications Aerodynamic coefficient from XFoil (Re effects) Compressibility accounted for by Prandtl-Glauert correction Spanwise aerodynamic correctionsf Lcorr 1 e s Simplified stall modelAeroelasticity and StructuralDynamics Research Laboratory
23Flight Dynamics ModelingThe trajectory and orientation of a fixed body reference frame, B, at point O,which in general is not the aircraft’s center of massAeroelasticity and StructuralDynamics Research Laboratory
24Complete-Aircraft Dynamics Model Elastic equations of motionStrains (4 by N structural d.o.f.) M ( ) C ( , , ) K R ( , , , pB , , , , ,u ) b Control inputsBody velocities (6 flight dynamic d.o.f.) Finite-state 2-D unsteady aerodynamics F1 F2 F3 Inflow states (m by N aerodynamic d.o.f.) Body reference frame propagation1 2PB C GB 0 Frame orientation (4 quaternions)Inertial velocities (6 d.o.f.)Aeroelasticity and StructuralDynamics Research Laboratory
25NAST: Function Block DiagramImplemented in MatlabQuickly generate majoraircraft componentsthrough identification ofbasic geometric andmaterial parametersAeroelasticity and StructuralDynamics Research Laboratory
Nonlinear Aeroelastic Simulation Toolbox(NAST)NAST is implemented in MatlabAutomatic generation of majoraircraft components throughidentification of basic geometricand material parametersSolutions: Nonlinear aeroelastic steady statedeformation/trim solution Linearized aeroelastic response Fully nonlinear time-marching aeroelasticsimulation Recovery of ply stress/strain, evaluation ofply failure Evaluation of flutter instability boundary,LCO Simulation of free flight of fully flexiblevehicle Structure and aeroelastic modes andfrequencies Closed-loop aeroelastic simulation26steady aerodynamic solutionfailure analysislarge structure deformationply stress/strain distributionpre-twist and curvaturemaneuver characteristicscross-section modeldiscrete control surfacesAeroelasticity and StructuralDynamics Research Laboratory
27Numerical StudiesAeroelasticity and StructuralDynamics Research Laboratory
28Blended-Wing-Body (BWB) Model Properties inspired from HiLDA (High Lift over Drag Active Wing)wind-tunnel modelElevon: 25% chordAeroelasticity and StructuralDynamics Research Laboratory
29Flutter of Constrained Vehicle Similar to constrained wind-tunnel model (no body DOFs) Fixed root angle of attack (8 deg) Free stream velocity 1% higher than flutter speedCoupled out-of-plane bending/torsion/in-plane bending modeAeroelasticity and StructuralDynamics Research Laboratory
Comparison of Flutter Modes with Rigid-BodyConstraintsAll cases trimmed for 6,096 m(20,000 ft) altitude, same fuelconditionFlutter SpeedFrequencyFullyconstraineddof’s172.52 m/s7.30 Hz plunging164.17 m/s7.07 Hz pitching andplunging123.17 m/s3.32 HzFree flight123.20 m/s3.32 HzTraditional wind-tunnel setupmaybe non-conservative – needrigid-body DOFs in the aeroelasticanalyses, simulations, and tests30Fullyconstrainedrigid-body DOFsAdditionalplunge DOFWith pitch andplunge DOFs(“same” for freeflight – 6 DOFs)Aeroelasticity and StructuralDynamics Research Laboratory
31Highly Flexible Flying Wing Model Representative of Helios prototype[3]– Five engines and three pods– Payloads applied at center pod– Empty gross mass: 726 kg[3] Patil, M. J., and Hodges, D. H., “Flight Dynamics of Highly Flexible Flying Wings,”Journal of Aircraft, Vol. 43, No. 6, 2006, pp. 1790-1798.Aeroelasticity and StructuralDynamics Research Laboratory
32Trim Results and Flight Stability Speed: 12.2 m/s at sea level; Payload: 0 – 227 kg (at center pod)Linearization about each trimmed condition with increase of payloadsRoot locus for phugoid mode (left: flexible, right: rigid)Unstable phugoid mode for payload 152 kgFlexibleRigidZero payload:span-loadedPayloadPayload[3]Full payload:center-loadedNonlinear aeroelastic/flight dynamic characteristics dependent on trimAeroelasticity and StructuralconditionsDynamics Research Laboratory
33Discrete Non-uniform Gust Model Fixed region in space Amplitude distribution– Peak at center and zero at boundary– Possibly different distribution in Eastand North directions– Smooth transition 1 t A(r , , t ) Ac 1 cos 2 tg 2 r nE AE (r ) sin 1 , 2 r0 AE cos AN sin 22 r nN AN (r ) sin 1 , 0 r r0 2 r0 Time variation: 1-cosine withdifferent temporal durationsAeroelasticity and StructuralDynamics Research Laboratory
Non-symmetric Gust Input and Response –Fully-Loaded Configuration 34Payload: 227 kg; gust region radius: 40 m;maximum gust center amplitude: 10 m/sNon-symmetric gust distribution: gusts mainly appliedon right wing2 s gust duration4 s gust durationGust duration impacts after-gust flight path8 s gust durationAeroelasticity and StructuralDynamics Research Laboratory
Instantaneous Vehicle Positions andOrientations35 Positions and orientations at 0, 5, 12, 18, 24, and 30 s,respectivelyFlight Direction2-s gust4-s gust8-s gustIllustration of unstable Phugoid modeAeroelasticity and StructuralDynamics Research Laboratory
Animation of Vehicle Motion with GustPerturbations362-s gust4-s gust8-s gustAeroelasticity and StructuralDynamics Research Laboratory
37Experimental StudiesAeroelasticity and StructuralDynamics Research Laboratory
38Duke University’s Wind-Tunnel Test Tang and Dowell’s high-aspect-ratio wing with a tip slender bodyNonlinear aeroelastic tests, studying geometrically nonlinear effects onwing responseData available in public domain for code validations[3][3]Photo from Tang and Dowell,Duke UniversitySu, Zhang, and Cesnik, IFASD 2009No complete vehicle aeroelastic/flight dynamic data available to supportAeroelasticity and Structuralfull formulation validationDynamics Research Laboratory
39X-HALE Project Designed and manufactured in Umich, with support from AFITand AFRLMy previous institute Design, build, and test experimental platform to providecontrolled nonlinear aeroelastic / flight dynamic coupled data tobe used for code validationAeroelasticity and StructuralDynamics Research Laboratory
40X-HALE Concept4 Modular Fairings6 Meter Wing Span- Data acquisition and propulsion- 64 single ended A/D channels- 64 differential A/D channels- Fiberglass / graphite / foam constructionCentral Guidance Fairing- GPS, INS, comm- 3 axis accelerometer- 3 axis IMU- 3 axis gyroEMX07 - Reflexed Airfoil- Positive moment coefficient5 Motors- PJS 1200- 8.5 N max thrust each- 2 differential outboard motors- Yaw to turn controlAeroelasticity and StructuralDynamics Research Laboratory
41Basic DimensionsControl Surface - 100% Elevon0.475m0.12 m0.65m1m6mTOP VIEW0.32m10º0.51m0.18m0.2mFRONT VIEW CROPPED - WINGTIPAeroelasticity and StructuralDynamics Research Laboratory
42Primary Flight Controls Forward thrust – all motorsYaw to turn – differential outboard motorsPitch – Tails 1 and 2Roll – All differential or 2-4 and 1-3 combinationAilerons on dihedral wing membersWing Label35Fairing / Tail Label311420264Mixing channels before and after servo switched controller (SSC)Aeroelasticity and StructuralDynamics Research Laboratory
43Wing ManufacturingAeroelasticity and StructuralDynamics Research Laboratory
44Still Going On Details about X-HALE design/simulation is published[4]First “hobby” flight– Jan 2011 at UM Oosterbaan Fieldhouse First flight of X-HALE– Aug, 2012 at Camp Atterbury, IN More information to be released in the future[4] Cesnik, C.E.S., Senatore, P.J., Su, W., Atkins, E.M., and Shearer, C.M., “X-HALE: AVery Flexible UAV for Nonlinear Aeroelastic Tests,” AIAA Journal, Vol. 50, No. 12, 2012,pp. 2820–2833.Aeroelasticity and StructuralDynamics Research Laboratory
45Concluding Remarks Numerical framework for modeling and analyzing very flexibleaircraft (VFA)– Coupled nonlinear aeroelastic/flight dynamic simulation (open andclosed loop)– Strain-based geometrically-nonlinear beam model– Incompressible unsteady aerodynamics (with compressibilitycorrections and stall models)– Rigid-body flight dynamics– Non-symmetric, spatially-distributed, discrete gust model– Skin wrinkling effects modeled as bilinear torsional stiffnessAeroelasticity and StructuralDynamics Research Laboratory
46Concluding Remarks (Cont’d) VFA have radically different behavior than conventional aircraft– Coupling between aircraft deformation and rigid-body motionschanges flutter boundaries– Flutter boundary in free flight condition may not be impacted bywing in-plane bending stiffness– Finite amplitude gust can excite instabilities– High instantaneous angle of attack on some wing stations results install, resulting in different transient responses of the wing and mayalter the vehicle flight behaviorAeroelasticity and StructuralDynamics Research Laboratory
47Concluding Remarks (Cont’d) What did we learn from the physics of VFA?– Deformed aircraft geometry, which depends on the operating (trim)condition, should be the basis in weight, structural, and stabilityanalyses– Traditional linear solution to VFA aeroelasticity might not besufficient – Nonlinear solution is required– Coupling between aeroelasticity and flight dynamics needs to beconsidered– Aeroelastic models should incorporate the rigid-body motion, andvice versa. Individual solutions might not be appropriateResearch on different aspects of Very Flexible Aircraft & Structures areon-going at the University of Alabama!Aeroelasticity and StructuralDynamics Research Laboratory
48Ongoing and Future Research AreasAeroelasticity and StructuralDynamics Research Laboratory
Classic Aeroelasticity Aeroelasticity of large-scale very flexible aircraftDARPA’s Vulture Program– Next-generation HALE UAVs DARPA, Air Force, Boeing, NASA – Enhanced structural modeling capability– Aerodynamics for different flight conditions andaircraft configurationsUSAF X-56A– Other emerging technologies in aerospace structures– Efficient aeroelastic
Aeroelasticity and Structural Dynamics Research Laboratory Unsteady Aerodynamics – Finite-State Inflow Theory 2-D Theodorsen-like unsteady aerodynamics (Peters et al., 94, 95) Glauert expansion of inflow velocity as function of inflow states, λ n 1 Finite state differen
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