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StaticAeroelasticity apuntes, vibraciones
- 1. Technische UniversitÀt MÌnchen Wind Energy Institute Static Aeroelasticity Carlo L. Bottasso, Stefano Cacciola October 2015
- 2. Static Aeroelasticity -2- Contents ⢠Divergence of the typical section ⢠Aileron effectiveness and reversal ⢠Operators ⢠Straight and swept cantilever wings: divergence and load redistribution
- 4. Static Aeroelasticity -4- Divergence Structural deflection under aerodynamic loading that enhances further the aerodynamic loading itself It is a static aeroelastic phenomenon (no presence of inertial or unsteady aerodynamics effects) Typical section (simplest model of blade torsional deformation): Zero lift line Aerodynamic center Elastic axis Structural deformation ðŸ ðð ðŒð ð ð ðŒð
- 5. Static Aeroelasticity -5- Reminder: Resultants on Airfoils Aerodynamic center: ⢠Point about which moment is constant wrt AoA ð¶ (typically ðŽðšðª < ð) ⢠Close to ð/ð and rather constant for a wide range of AoAs before stall Center of pressure: ððªð· = ððšðª â ΀ ðŽðšðª ð³, ΀ ððªð· ð = ΀ ððšðª ð â ΀ ðªðŽðšðª ðªð³ ⢠Point of application of the aerodynamic pressure field resultant (i.e. pressure field is reduced to a single force vector with null moment) ⢠Changes with lift (i.e. AoA). Pre-stall:ðªð³ = ðªð³ð¶ð¶ and therefore ൠððªð· ð = ൠððšðª ð â ΀ ðªðŽðšðª ðªð³ð¶ð¶ TE LE AC â c/4 Aerodynamic force resultant (due to pressure and shear) ൠððªð· ð ð ðšðª Non-dimensionalization (forces and moment per unit span): ð³ = ð ð ððœð ððªð³ ð« = ð ð ððœð ððªð« ðŽðšðª = ð ð ððœð ðð ðªðŽðšðª
- 6. Static Aeroelasticity -6- Reminder: Resultants on Airfoils Re dependence at high a Separation and Stall cD vs. a dependent on Re (source: I.H. Abbott, A.E. von Doenhoff, âTheory of Wing Sections: Including a Summary of Airfoil Dataâ) ðªð³ ððð ðªð« ððð Lift coefficient Moment coefficient wrt c/4 point Drag coefficient Moment coefficient wrt AC Example: NACA 2412 Airfoil
- 7. Static Aeroelasticity -7- tanâð ðªð³ð¶ ðªð³ = ðªð³ð¶ð¶ in the pre-stall region, where the flow is attached For inviscid flows over thin airfoils: ðªð³ð¶ = ðÏ In general: ðªð³ð¶ < ðÏ (depending on airfoil and operating condition) ðð³ ðð = ðð Angle of zero lift ðð LE TE Geometric angle of attack ðð³ ðð Aerodynamic angle of attack ðð = ð ðð = ð Reminder: Resultants on Airfoils ðªð³ = ðªð³ð + ðªð³ð¶ð¶ ðªð³ = ðªð³ð¶ð¶
- 8. Static Aeroelasticity -8- Divergence Lift (per unit span): ð¿ = ððð¶ð¿ = ððð¶ð¿ðŒ ðŒð + ð with ð = 1 2 ðð2 Aerodynamic moment: ððžðŽ ðŽ = ððŽð¶ + ð¿ðð = ðð2 ð¶ððŽð¶ + ðð¶ð¿ðŒ ðŒð + ð Structural moment: ððžðŽ ð = ðŸð Equilibrium: ððžðŽ ðŽ = ððžðŽ ð Structural torsional deflection: ð = ðð2 ð¶ððŽð¶ +ðð¶ð¿ðŒðŒð ðŸâððð2ð¶ð¿ðŒ
- 9. Static Aeroelasticity -9- Divergence Structural torsional deflection: ð = ðð2 ð¶ððŽð¶ +ðð¶ð¿ðŒðŒð ðŸâððð2ð¶ð¿ðŒ Divergence: ð > 0 ð â ðð· = ðŸ ðð2ð¶ð¿ðŒ ð â â ð¿ = 1 + ð/ðð· 1âð/ðð· ððð¶ð¿ðŒ ðŒð + ð/ðð· 1âð/ðð· ð ð ð ð¶ððŽð¶ The divergence speed could be increased by increasing torsional stiffness, but this must be traded against increased weight/cost Divergence dynamic pressure ð ð ðð· ð¿ ð ðð· Elastic lift (limited by stall) Rigid lift ð¿ð = ððð¶ð¿ðŒ ðŒð
- 10. Static Aeroelasticity -10- ðœ ð Why Divergence Matters It is a crucial design parameter for ensuring safety and structural integrity Flight envelope in terms of the ðœ-ð diagram: Structural damage or failure because of divergence or flutter ðœðµð¬
- 12. Static Aeroelasticity -12- Aileron Effectiveness and Reversal ⢠To increase lift, deflect aileron (down) to increase camber ⢠Resulting nose-down moment decreases AoA and hence lift, when wing is torsionally flexible ⢠If lift decrease due to AoA change is larger than lift increase due to increase in camber, then lift decreases instead of increasing (aileron reversal) ⢠Same for aileron up deflection Zero lift line Aerodynamic center Elastic axis Structural deformation ð¿ ðŸ ðð ðŒð ð ð ðŒð
- 13. Static Aeroelasticity -13- Aileron Effectiveness and Reversal Lift (per unit span): ð¿ = ððð¶ð¿ = ððð¶ð¿ðŒ ðŒð + ð + ððð¶ð¿ð¿ ð¿ with ð = 1 2 ðð2 Aerodynamic moment: ððžðŽ ðŽ = ððŽð¶ + ð¿ðð = ðð2 ð¶ððŽð¶ + ð¶ððŽð¶ ð¿ ð¿ + ð ð¶ð¿ðŒ ðŒð + ð + ð¶ð¿ð¿ ð¿ Structural moment: ððžðŽ ð = ðŸð Equilibrium: ððžðŽ ðŽ = ððžðŽ ð Structural torsional deflection: ð = ðð2 ð¶ððŽð¶ +ð¶ððŽð¶ ð¿ ð¿+ð ð¶ð¿ðŒðŒð+ð¶ð¿ð¿ ð¿ ðŸâððð2ð¶ð¿ðŒ
- 14. Static Aeroelasticity -14- Aileron Effectiveness and Reversal (assuming ð¶ððŽð¶ = 0 for simplicity) Lift: ð¿ = 1 + ðð2 ðð¶ð¿ðŒ ðŸ â ððð2ð¶ð¿ðŒ ððð¶ð¿ðŒ ðŒð + ð + ðð2 ΀ ð¶ððŽð¶ ð¿ ð¶ð¿ðŒ ð¶ð¿ð¿ ðŸ â ððð2ð¶ð¿ðŒ ððð¶ð¿ð¿ ð¿ Reordering: ð¿ = 1 + ΀ ð ðð· 1 â ΀ ð ðð· ððð¶ð¿ðŒ ðŒð + 1 â ΀ ð ðð 1 â ΀ ð ðð· ððð¶ð¿ð¿ ð¿ with: ⢠Divergence dynamic pressure: ðð· = ðŸ ðð2ð¶ð¿ðŒ where if ð > 0, ð â ðð·, ð â â ⢠Reversal dynamic pressure: ðð = â ðŸð¶ð¿ð¿ ð2ð¶ððŽð¶ ð¿ ð¶ð¿ðŒ (notice: ð¶ððŽð¶ ð¿ < 0) Remark: typically ðð < ðð· Lift due to elastic effect at null Rigid lift Lift due to aileron deflection
- 15. Static Aeroelasticity -15- Aileron Effectiveness and Reversal Aileron Effectiveness (ðŽðž): ðŽðž = ð¿ð¿ ð¿ð¿ð = 1 â ΀ ð ðð 1 â ΀ ð ðð· ððð¶ð¿ð¿ ððð¶ð¿ð¿ = 1 â ΀ ð ðð 1 â ΀ ð ðð· Structural torsional deflection: ð = ðð2 ð¶ððŽð¶ + ð¶ððŽð¶ ð¿ ð¿ + ð ð¶ð¿ðŒ ðŒð + ðð¶ð¿ð¿ ð¿ ðŸ â ððð2ð¶ð¿ðŒ = ΀ ð ðð· 1 â ΀ ð ðð· ðŒð â ð ðð 1 â ΀ ðð ðð· 1 â ΀ ð ðð· ð¶ð¿ð¿ ð¶ð¿ðŒ ð¿ >0 (nose up) <0 (nose down) for (reduces AoA) ðŽðž ΀ ð ðð· 1 1 Stick right, roll left (and viceversa)!
- 16. Static Aeroelasticity -16- Why Aileron Effectiveness Matters Increase torsional stiffness to increase ðð , but significant weight penalty Better solution: high speed ailerons, spoilerons, flaperons Additional benefit: no adverse yaw effect Roll by the use of spoilers ⶠ(notice left wing spoiler up, left wing down, positive direct yaw effect)
- 18. Static Aeroelasticity -18- Structural Operators Structural operator: relates elastic deflections (rotations) to forces (and moments) ð = ð® ð Simplest possible example: spring-force system ð = ð ð¥ Governing equilibrium equation Applied forces (and/or moments) Elastic deflections (and/or rotations) Structural operator ð ð¥ ð Applied force Displacement Structural operator ð® = ð
- 19. Static Aeroelasticity -19- Structural Operators Example: wing bending The wing is modeled as a beam subjected to a distributed load: ⢠Depending on the technique used to model the structure, the operator can be differential, integral or a matrix ⢠For real life problems, numerical methods are typically necessary and the matrix approach is often the most effective (see FEM) ð(ð¥) Deformed configuration Undeformed configuration Governing equation Structural operator ð2 ðð¥2 ðžðŒ(ð¥) ð2ð€ ð¥ ðð¥2 = ð(ð¥) ð® = ð2 ðð¥2 ðžðŒ(ð¥) ð2 â ðð¥2 ð¥ ð€ nur von einer variable abhÀngig (x) = ordinary equation
- 20. Static Aeroelasticity -20- Structural Operators Finite Element Method (FEM) Governing equation in âweak formâ (multiply both sides by arbitrary test function ð£(ð¥), then integrate over the span of the beam): න ð£ð ð2 ðð¥2 ðžðŒ(ð¥) ð2 ð€ ð¥ ðð¥2 dð¥ = න ð£ð ð ð¥ dð¥ , â ð£(ð¥) Integrate by parts (twice) the left hand side: න ð2 ð£(ð¥) ðð¥2 ð ðžðŒ(ð¥) ð2 ð€ ð¥ ðð¥2 dð¥ = න ð£ð ð ð¥ dð¥ , â ð£(ð¥) Boundary conditions: á ð£ ð¥=0 = 0 (null displacement) àž ð2 ð€ ðð¥2 ð¥=ð¿ = 0 (null bending moment) ဠðð£ ðð¥ ð¥=0 = 0 (null slope) àž ð3 ð€ ðð¥3 ð¥=ð¿ = 0 (null shear force)
- 21. Static Aeroelasticity -21- Structural Operators The beam is represented as a collection of finite elements: On each element, the elastic displacements and the test functions are approximated using suitable shape functions: element 1 element 2 ⊠element i ⊠Elemental DoFs: nodal displacements and rotations ð£ ð¥ = ðµð ð¥ ðe ð€ ð¥ = ðµð ð¥ ðe Nodal DoFs of test function Nodal DoFs of elastic displacement field ðµ ð¥ = ð¢1 ð1 ð¢2 ð2 ð¢1 = 1 ð1 = ð¢2 = ð2 = 0 ð1 = 1 ð¢1 = ð¢2 = ð2 = 0 ð¢2 = 1 ð¢1 = ð1 = ð2 = 0 ð2 = 1 ð1 = ð¢2 = ð¢2 = 0
- 22. Static Aeroelasticity -22- Structural Operators Insert interpolating approximations into weak form: ðð න ð2 ðµ(ð¥) ðð¥2 ðžðŒ(ð¥) ð2 ðµ ðð¥2 ð dð¥ ðe = ðð න ðµ ð¥ ð ð¥ dð¥ , â ð Due to arbitrarity of ð, the local equilibrium ð²eðe = ðe holds true for each element Global equilibrium is obtained by assembling the contributions over all elements: ð = ð² ð â ð = ð²â1 ð = ðªðð ð Element stiffness matrix ð²e Element load vector ðe Load vector Nodal displacements and rotations Structural operator ð® = ð² Influence coefficient (or compliance) matrix ðªðð = ð® â1
- 23. Static Aeroelasticity -23- Structural Operators Example: wing torsion The wing is modelled as a beam subjected to a torsional distributed load: Similarly to the bending case, use of the FEM leads to: ð = ð¯ ð â ð = ð¯â1ð = ðªðœðœð Deformed configuration Undeformed configuration Governing equation Structural operator ð ðð¥ ðºðŒ(ð¥) ðð ð¥ ðð¥ = ð(ð¥) ð® = ð ðð¥ ðºðŒ(ð¥) ð â ðð¥ ð(ð¥) ð¥ Moment vector Nodal rotations Structural operator ð® = ð¯ Matrix of influence coefficients (compliance matrix) ðªðœðœ = ð® â1 torsional stiffness
- 24. Static Aeroelasticity -24- Elastic Equilibrium of the Wing The elastic behavior of the structure is represented by matrices of influence coefficients: Similarly, an aerodynamic operator can be defined as well ÎðŠ1 ÎðŠ3 ÎðŠ2 ÎðŠð ⊠A.C. E.A. ðð ðEAð ðð 1 2 3 i ð = ðªðð ð ð = ðªðð ðð¬ðš
- 25. Static Aeroelasticity -25- Aerodynamic Operators Aerodynamic operator: relates deflections (and in turn angles of attack) to aerodynamic loads Simple example: single airfoil Depending on need, it may be more convenient to relate AoA and non- dimensional aerodynamic coefficients ð A. C. ðŒ ð¿ ð ð¿ = ðð2 2 ð ðð¶ð¿ ððŒ ðŒ Aerodynamic loads Angle of attack Aerodynamic operator ð = ð ðŒ ð ð0 ð = ððð0 Cl alpha
- 26. Static Aeroelasticity -26- Example: 2-D strip theory Assumption: each strip acts independently from the others Goal: describe the relationship between lift distribution and angle of attack Aerodynamic Operators ð¿ð = ðâðŠðððð0ð ðŒð ð¶ð¿ð = ð0ð ðŒð ðŒ1 ðŒ2 â® ðŒð = ൠ1 ð01 ൠ1 ð02 0 0 Ⱡൠ1 ð0ð ð¿1/ð1ð ð¿2/ð2ð â® ð¿ð/ððð âðŠð ðŠ 1 2 3 ⊠ð ⊠ð ð¶ = ð â1 ðªð¿ Non-dimensional lift distribution Angles of attack Inverse of the aerodynamic operator
- 27. Static Aeroelasticity -27- Aerodynamic Operators Example: 3D lifting line theory (Prandtl) for steady incompressible flow ðŒ ðŠ = ð¿(ðŠ) ð0 ðŠ ð ðŠ ð + 1 8ð න ൠâð 2 ൠð 2 d dð ð ðŠ ð¶ð¿ ðŠ dð ðŠ â ð Approximating the integral with a quadrature rule: ðŒ1 ðŒ2 â® ðŒð = ൠ1 ð01 ൠ1 ð02 0 0 Ⱡൠ1 ð0ð + 1 8ð ð11 ⯠ð1ð â® â± â® ðð1 ⯠ððð ð¿1/ð1ð ð¿2/ð2ð â® ð¿ð/ððð More sophisticated models can be used to account for compressibility/Mach number effects Correction for finite length wing Strip theory Correction for finite length wing Strip theory
- 28. Static Aeroelasticity -28- Inertial Operator Inertial operator: relates deflections (rotations) to inertial forces (and moments) ðð° = â ð Simplest possible example: spring-mass system ðð° = ð ð2ð¥ ðð¡2 Inertial forces (and/or moments) Elastic deflections (and/or rotations) Inertial operator ð¥ ð Inertial force Displacement Inertial operator â = ð ð2 â ðð¡2 ð
- 29. Static Aeroelasticity -29- Equilibrium Dynamic aeroelasticity: â ð = ð ð + ð® ð + ð¬ Static aeroelasticity: 0 = ð ð + ð® ð + ð¬ Vibration problems: â ð = ð® ð + ð¬ Flight mechanics: â ð = ð ð + ð¬ Inertial forces Aerodynamic forces Structural forces External forces vgl. collers triangle
- 31. Static Aeroelasticity -31- Straight Cantilever Wing Consider a straight wing clamped at the root: âðŠð ðŠ ðð ð E.A. (Elastic axis) ð ðŒF A. C. E. A. ðŒ â ðŠ ð ð¿ ðAC ðŒ = ðŒB ðŠ + ð ðŠ ðŒB ðŠ = ðŒF + ðœ ðŠ Rigid angle of attack Elastic twist Angle of attack at wing root (dof to trim the aircraft) Built-in twist (to delay tip stall) Reference plane through wing root Deflection of elastic axis unterschiedliche cord line
- 32. Static Aeroelasticity -32- Loads Acting on the Wing Goal: look for lift distribution over the wing The wing is divided into strips of width ð¥ðŠð Force and moment acting on strip ð: ðð = ðððð¥ðŠðð0ðŒð ðEAð = ðððÎðŠðððð0ðŒð + ððð 2 ÎðŠðð¶ðACð ð: dynamic pressure ðð: mean chord of the ð-th strip ð0: slope of the lift coefficient ð¶ðACð : moment coefficient about the aerodynamic center of the ð-th strip ÎðŠ1 ÎðŠ3 ÎðŠ2 ÎðŠð ⊠A.C. E.A. ðð ðEAð ðð 1 2 3 i Auf die aerodynamische Achse bezogen (structural Problem) local aoa
- 33. Static Aeroelasticity -33- Loads Acting on the Wing Lift coefficient: ð¶ð¿ð = ð0ðŒ ð = ð0ðŒB ð + ð0ð ð = = ð¶ð¿ð B + ð¶ð¿ð E Force and moment for the ð-th strip: ðð = ðð¥ðŠððð ð¶ð¿ð B + ð¶ð¿ð E ððžðŽð = ðÎðŠððððð ð¶ð¿ð B + ð¶ð¿ð E + ðÎðŠððð 2 ð¶ðACð Forces and moments for all strips (in vector/matrix notation): ð = ð diag ÎðŠððð ðð¿ B + ðð¿ E ðEA = ð diag ÎðŠððððð ðð¿ B + ðð¿ E + ð diag ÎðŠððð 2 ððAC Rigid lift coefficient (known) Elastic lift coefficient (unknown) ðð³ (â) = â® ðªð³ (â) â®
- 34. Static Aeroelasticity -34- Loads Acting on the Wing The elastic behavior of the structure is represented by matrices of influence coefficients: ð = ðªââ ð ð = ðªðð ðEA Similarly, aerodynamics can be expressed in matrix notation as: ð¶ = ðâ1 ðªð³ where ð = ðâ1 ðªð³ E ð¶ðµ = ðâ1ðªð³ B Elastic bending of the wing Elastic torsion of the wing
- 35. Static Aeroelasticity -35- Equilibrium ⊠combining and rearranging the torsional equilibrium ⊠ðâ1 â ðð¬ ðð³ ðž = ð where ð = ðð¬ðððŒð + ðð¬ðð· + ððððAC ð¬ = ðªðð diag ð¥ðŠððððð , ð = ðªðð diag ð¥ðŠððð 2 , ð = 1 1 â® 1 Aerodynamic loads ð = ð diag ÎðŠððð ðL B + ðL E ðEA = ð diag ÎðŠððððð ðL B + ðL E + +ð diag ÎðŠððð 2 ððAC Elastic deflections ð = ðªââ ð ð = ðªðð ðEA Aerodynamic angles ð = ðâ1 ðð³ E ð¶ðµ = ðâ1 ðð³ B Moment due to root wing angle of attack Moment due to wing built-in twist Aerodynamic moment System of n equations for the n unknowns ðªð³ E
- 36. Static Aeroelasticity -36- Equilibrium Given vehicle AoA ðŒð , use ðâ1 â ðð¬ ðð³ ðž = ð and solve for ðð³ ðž Then compute: - Load distribution: ð = ð diag ÎðŠððð ðL B + ðL E - Moment distribution: ðEA = ð diag ÎðŠððððð ðL B + ðL E + ð diag ÎðŠððð 2 ððAC - Elastic deflection: ð = ðªââ ð - Elastic twist: ð = ðªðð ðEA
- 37. Static Aeroelasticity -37- Divergence Speed The divergence dynamic pressure ðð· of the wing is found by solving the following eigenvalue problem: det ðâ1 â ðð·ð¬ = 0 If we consider compressibility effects, then ð = ð ð = ð ðD A similar analysis can be made for a free-flying vehicle (instead of a cantilever wing), by adding the flight mechanics equilibrium equations The solution yields the vehicle trim (AoA, elevator deflection), together with the wing deflections ðð· ð |eigs| Determinante ist Produkt der Eigenwerte. sobald ein Eigenwert 0 ist der divergence Druck erreicht
- 38. Static Aeroelasticity -38- Typical Results ⢠The total lift remains the same (to yield equilibrium), but elasticity causes an outward redistribution of lift, because of increased twist at the wing tip ⢠Hence, the bending moment at the root is increased Load ð¹ Wing span Elastic wing Rigid wing der getrimmte Zustand erfordert ein Gleichgewicht. daher muss FlÀche unter kurven und somit der gesamt Auftrieb gleich sein
- 40. Static Aeroelasticity -40- ð High aspect ratio wing with elastic axis ðŠ: Swept Wing Remark: ⢠Bending and torsion are decoupled when referring to ð but ⢠Better refer to ð when expressing aerodynamic loads ðŠ ð E.A. (Elastic axis) ðŠ ð¥ ð¬ ð§ ð¬ ð, ðEA ð, ðEA ð, ðð ðŠ â ð lÀngsstabilitÀt aufgrund unterschiedlicher Geschwindigkeiten senkrecht zur flÃŒgelvorderkante
- 41. Static Aeroelasticity -41- Swept Wing Structural behavior: Bending behavior: ൠð = ðª ââ ð + ðª âð ðð ð = ðª ðâ ð + ðª ðð ðð Torsional behavior: áð = ðª ðð ðð Change of reference: ðð = âsinð¬ ðEA ðð = cosð¬ ðEA ð = cosð¬ ð â sinð¬ ð Decoupled when referring to structural axis ðŠ
- 42. Static Aeroelasticity -42- Lift Distribution Vertical displacement of wing sections: ð = ðª ââ ð â sinð¬ ðª âð ðEA Rotation of wing sections: ð = cos2 ð¬ ðª ðð ððžðŽ â sinð¬ ðª ðâ ð â ðª ðð sinð¬ ððžðŽ = = âsinð¬ ðª ðâ ð + cos2 ð¬ðª ðð + sin2 ð¬ ðª ðð ððžðŽ and in turn ð = ðªââ ð + ðªâð ððžðŽ ð = ðªðâ ð + ðªðð ððžðŽ where ðªââ = ðª ââ , ðªðâ = âsinÎ ðª ðâ and ðªðð = cos2 Î ðª ðð + sin2 Î ðª ðð jetzt mit torsion-displacement coupling da nicht mehr auf elastic axis bezogen
- 43. Static Aeroelasticity -43- Lift Distribution All developments for the straight wing remain valid and the solution is formally identical (only a redefinition of ð¬ is needed) Equilibrium: ðâ1 â ðð¬ ðð³ ðž = ð where ð = ðð¬ðððŒð¹ + ðð¬ðð· + ðððªðŽAC and ð¬ = ðªðð diag(ð¥ðŠððððð) + ðªðâ diag(ð¥ðŠððððð), ð = ðªðð diag ð¥ðŠððð 2 , Solve for ðð³ ðž , then compute complete deflected configuration of the wing The divergence speed is found by solving the following eigenvalue problem: det ðâ1 â ðð·ð¬ = 0 Moment due to wing root angle of attack Moment due to wing built-in twist Aerodynamic moment New term from lift-torsion coupling due to sweep System of n equations for the n unknowns ðªð³ E
- 44. Static Aeroelasticity -44- Typical Results ⢠Bending induces a nose-down rotation of the aerodynamic section: ðªðâ = âsinÎ ðª ðâ ⢠The tip of the wing is easy to bend: the angle of attack decreases at the tip ⢠The total lift remains the same (to yield equilibrium), but elasticity causes an inward redistribution of lift Remark: for forward sweep, the effect is the opposite: bending increases the AOA Load ð¹ Wing span Elastic wing Rigid wing
- 45. Static Aeroelasticity -45- Divergence Speed Sweep angle, Î ðð· -30 deg 30 deg Notice the much lower divergence dynamic pressure for swept- forward wings Swept-forward Swept-back self unload durch mehr bending und dadurch weniger twist
- 46. Static Aeroelasticity -46- Divergence Speed Remark: forward sweeping has interesting possible advantages ⢠Configuration: better internal layout because of aft wing spar ⢠Flight mechanics: higher maneuverability (unstable in pitch and yaw); reduced drag; delayed tip stall (better high AoA behavior), ⊠X-29 solution to counteract divergence without excessive weight penalty: âaeroelastic tailoringâ (bend-twist coupling) by the use of anisotropic composite materials Similar solutions are used on wind turbine blades for passive load alleviation Grumman X-29 ⌠Angle of composite fibers Bending Resulting twisting ðªðððð ðâ < 0 Compression/extension due to bending aligned with fibers