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Wing Flutter Analysis With An Uncoupled Method PDF

27 Pages·2003·5.44 MB·English
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Wing Flutter Analysis With An Uncoupled Method (cid:1) Koray KAVUKCUO LU M. Sc. Thesis Presentation Supervisor Prof. Dr. Mehmet A. AKGÜN Co-Supervisor Dr. Erdal OKTAY Presentation Outline • Introduction – Flutter • Flutter Modeling – Simple Systems – Coupled and P-K Methods – Akgün’s Method • Modal Analysis of AGARD Wing 445.6 – Finite Element Model – Results • Aerodynamic Analysis of AGARD Wing 445.6 – USER3D – CFD Model – Solution Procedure – Grid Transformation – Force Calculation – Results • Numerical Results – Discussion August 18, 2003 Koray KAVUKCUO(cid:1)LU 2 Introduction Flutter Dynamic instability of an elastic body in an airstream caused by the unsteady aerodynamic forces generated from elastic deformations of the structure. August 18, 2003 Koray KAVUKCUO(cid:1)LU 3 Flutter Modeling 1 – DOF System K a – Rigid Airfoil U – Unit Span a – Oscillation around the Leading Edge 2b (cid:1)(cid:1) iwt I a+ K a = M a =a e a a y 0 2 – DOF System z b b – Rigid Airfoil U x a – Unit Span h Ka – Pitching and Plunging Motion My (cid:1)(cid:1) c.g. (cid:1)(cid:1) ba bxa mh + S a+ mw2h = Q a h h Kh (cid:1)(cid:1) (cid:1)(cid:1) 2 S h + I a+ I w a= Q a a a a a August 18, 2003 Koray KAVUKCUO(cid:1)LU 4 Flutter Modeling P-K Method – Forced Response Analysis – Eigenvalue Solution [ ]{(cid:1)(cid:1)} [ ]{ } { (cid:1) (cid:1)(cid:1) } M q + K q = f (q,q,q,t) EOM for the system – Harmonic displacement – Transform to Modal Coordinates (cid:1) 1 (cid:2) [M ]s2 +[K ]- rV 2 [Q ] {h } = 0 Basic EV Flutter Eqn. * (cid:3) (cid:4) h h h h (cid:5) 2 (cid:6) Vk g s = (g + i) p = k(g + i) * Valid when = 0 b August 18, 2003 Koray KAVUKCUO(cid:1)LU 5 Flutter Modeling Coupled Methods – Solution of Aeroelastic Equations every time step – Grid Transformation – Coupling Algorithm – Weakly Coupled (Separate CFD – CSD) – Strongly Coupled (Stability = Least Stable Code) – Fully Coupled (Iterative solution at each time step) (cid:1)(cid:1) (cid:1) 2 q + 2zwq +w q = Q i i i i i i i – 2 SDOF Equations – Decoupled by Diagonalization – Solved at each time step August 18, 2003 Koray KAVUKCUO(cid:1)LU 6 Flutter Modeling Present Approach – Forced Response Analysis – Eigenvalue Solution – Aim is to decouple CSD and CFD [ ]{(cid:1)(cid:1)} [ ]{ } { (cid:1) (cid:1)(cid:1) } M q + K q = f (q,q,q,t) EOM for the system – Harmonic displacement { } [ ]{ } iwt q = F h e – Transform to Modal Coordinates { } ([ ]{ } iwt ) f = Im A f e j j Unsteady Aerodynamic Force Definition { } { } [ ] r = A f j j • Substitute to EOM and get Eigenvalue flutter problem August 18, 2003 Koray KAVUKCUO(cid:1)LU 7 Flutter Modeling { } [K ]-w2[M ]- [F]T [R]{h}= 0 Basic EV Flutter Eqn. M M – Similar to P-K method Formulation – Generalized Aerodynamic Force (GAF) Matrix ( [R] ) is unsymmetric – [R] = f(M,k) k : reduced frequency [ ] [ ] [ ] R = R + i R R I – By time integration get real and P (cid:7) [ ] p (cid:1) (cid:2) R = c T imaginary parts of aerodynamic force (cid:5) (cid:6) R p p=0 – Fit Pth order polynomial to aerodynamic P [ ] (cid:7) p (cid:1) (cid:2) force R = c Z (cid:5) (cid:6) I p p=0 • Transform into a Polynomial Eigenvalue Problem August 18, 2003 Koray KAVUKCUO(cid:1)LU 8 Flutter Modeling – Polynomial curve fitting in terms of reduced frequency (k) (cid:8) (cid:9) P (cid:7) (cid:10) k pQ (cid:11){h} = {0} Polynomial Eigenvalue Flutter Equation p (cid:12) (cid:13) p=0 (cid:1){ }(cid:2) h Transform to a real augmented system { } R h = (cid:3) (cid:4) {h } [Q] is (8x8) (cid:5) (cid:6) I (cid:1)[K ] [0] (cid:2) (cid:8)U (cid:9)2 (cid:1)[M ] [0] (cid:2) Q = (cid:3) M (cid:4) Q = Q -(cid:10) ¥ (cid:11) (cid:3) M (cid:4) 0 [0] [K ] 2 2 (cid:12) b (cid:13) [0] [M ] (cid:5) (cid:6) (cid:5) (cid:6) M M (cid:1)-[F]T (cid:1)T (cid:2) [F]T (cid:1)Z (cid:2) (cid:2) (cid:5) (cid:6) (cid:5) (cid:6) (cid:3) p p (cid:4) Q = Q Q = for p = 1,3,4 … P p p p (cid:3) (cid:4) -[F]T (cid:1)Z (cid:2) -[F]T (cid:1)Z (cid:2) (cid:5) (cid:5) (cid:6) (cid:5) (cid:6)(cid:6) p p August 18, 2003 Koray KAVUKCUO(cid:1)LU 9 Modal Analysis of AGARD Wing 445.6 Finite Element Model – Shell elements – Distributed thickness – Rotated element coordinate system – In accordance with real wing 0.6 0.4 h) Kolonay Up c n 0.2 Kolonay Down s (i Present Up s 0.0 e Present Down kn 0.0 0.2 0.4 0.6 0.8 1.0 NACA65004 Up c-0.2 hi NACA65004 Down T -0.4 -0.6 x/c August 18, 2003 Koray KAVUKCUO(cid:1)LU 10

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August 18, 2003 Koray KAVUKCUO LU 3 Flutter Dynamic instability of an elastic body in an airstream caused by the unsteady aerodynamic forces generated from elastic
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