Khí đàn hồi trong aeronautical engineering, mối liên hệ giữa kết cấu và khí động, vận tốc tối đa để xoắn không bị divergence và hiện tượng vẩy. cơ sở dựa trên môn dao động cơ học. Sách này tái bản lần thứ 4
A Modern Course in Aeroelasticity SOLID MECHANICS AND ITS APPLICATIONS Volume 116 Series Editor: G.M.L GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids The scope of the series covers the entire spectrum of solid mechanics Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design The median level of presentation is the first year graduate student Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity For a list of related mechanics titles, see final pages A Modern Course in Aeroelasticity Fourth Revised and Enlarged Edition by EARL H DOWELL (Editor) DAVID A PETERS Duke University, Durham, NC, U.S.A Washington University, St Louis, MO, U.S.A ROBERT CLARK ROBERT SCANLAN Duke University, Durham, NC, U.S.A Johns Hopkins University, Baltimore, MD, U.S.A DAVID COX EMIL SIMIU NASA Langley Research Center, Hampton, VA, U.S.A National Institute for Standards and Technology, Gaithersburg, MD, U.S.A H.C CURTISS, JR FERNANDO SISTO Princeton University, Princeton, NJ, U.S.A Stevens Institute of Technology, Hoboken, NJ, U.S.A JOHN W EDWARDS and NASA Langley Research Center, Hampton, VA, U.S.A THOMAS W STRGANAC KENNETH C HALL Texas A&M University, College Station, TX, U.S.A Duke University, Durham, NC, U.S.A KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 1-4020-2106-2 1-4020-2039-2 ©2005 Springer Science + Business Media, Inc Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.kluweronline.com http://www.springeronline.com The authors would like to pay tribute to Robert H Scanlan, a superb aeroelastician, an inspiring teacher, and a consummate mentor and friend He is greatly missed This page intentionally left blank Contents Preface xvii Preface to the First Edition xvii Preface to the Second Edition xix Preface to the Third Edition xx Preface to the Fourth Edition xxi Short Bibliography xxiii INTRODUCTION (DOWELL) STATIC AEROELASTICITY (DOWELL) 2.1 Typical Section Model of An Airfoil Typical section model with control surface Typical section model—nonlinear effects 10 16 2.2 One Dimensional Aeroelastic Model of Airfoils Beam-rod representation of large aspect ratio wing Eigenvalue and eigenfunction approach Galerkin’s method 18 18 22 24 2.3 Rolling of a Straight Wing Integral equation of equilibrium Derivation of equation of equilibrium Calculation of C αα Sketch of function S(y1 , η) Aerodynamic forces (including spanwise induction) Aeroelastic equations of equilibrium and lumped element solution method Divergence Reversal and rolling effectiveness 26 26 27 28 28 30 vii 32 33 34 viii A MODERN COURSE IN AEROELASTICITY Integral equation eigenvalue problem and the experimental determination of influence functions 2.4 Two Dimensional Aeroelastic Model of Lifting Surfaces Two dimensional structures—integral representation Two dimensional aerodynamic surfaces—integral representation Solution by matrix-lumped element approach 2.5 Other Physical Phenomena Fluid flow through a flexible pipe (Low speed) fluid flow over a flexible wall 2.6 Sweptwing Divergence References for Chapter 37 41 41 42 43 44 44 47 47 51 DYNAMIC AEROELASTICITY (DOWELL) 53 3.1 Hamilton’s Principle 54 Single particle 54 Many particles 56 Continuous body 56 Potential energy 56 Nonpotential forces 59 3.2 Lagrange’s Equations 60 Example—typical section equations of motion 61 3.3 Dynamics of the Typical Section Model of An Airfoil 64 Sinusoidal motion 64 Periodic motion 67 Arbitrary motion 67 Random motion 73 Flutter - an introduction to dynamic aeroelastic instability 81 Quasi-steady, aerodynamic theory 85 3.4 Aerodynamic Forces 87 Aerodynamic theories available 91 General approximations 95 ‘Strip theory’ approximation 95 ‘Quasisteady’ approximation 95 Slender body or slender (low aspect ratio) wing approximation 96 3.5 Solutions to the Aeroelastic Equations of Motion 97 Time domain solutions 98 Frequency domain solutions 100 Contents ix 3.6 Representative Results and Computational Considerations 103 Time domain 103 Frequency domain 103 Flutter and gust response classification including parameter trends 105 Flutter 105 Gust response 121 3.7 Generalized Equations of Motion for Complex Structures 128 Lagrange’s equations and modal methods (Rayleigh-Ritz) 128 Kinetic energy 129 Strain (potential elastic) energy 130 Examples 133 (a) Torsional vibrations of a rod 133 (b) Bending-torsional motion of a beam-rod 134 Natural frequencies and modes-eigenvalues and eigenvectors135 Evaluation of generalized aerodynamic forces 136 Equations of motion and solution methods 137 Integral equations of equilibrium 139 Natural frequencies and modes 141 Proof of orthogonality 143 Forced motion including aerodynamic forces 144 Examples 147 (a) Rigid wing undergoing translation responding to a gust147 (b) Wing undergoing translation and spanwise bending 153 (c) Random gusts-solution in the frequency domain 155 3.8 Other Fluid-Structural Interaction Phenomena 156 Fluid flow through a flexible pipe: “firehose” flutter 156 (High speed) fluid flow over a flexible wall - a simple prototype for plate or panel flutter 158 References for Chapter 165 NONSTEADY AERODYNAMICS (DOWELL) 169 4.1 Basic Fluid Dynamic Equations 169 Conservation of mass 170 Conservation of momentum 171 Irrotational flow, Kelvin’s theorem and Bernoulli’s equation172 Derivation of a single equation for velocity potential 174 Small perturbation theory 175 738 A MODERN COURSE IN AEROELASTICITY Substituting the various numerical parameters previously specified into (B.47) gives ω ωα = 2.099 or = 0.69 (B.48) ω ωα Using (B.48) in (B.45) along with the other numerical parameters gives −0.133µ2 + 0.121 µ− =0 k2 12k2 Solving for µ, 0.45 ± k2 k µ1,2 = 0.21 − 0.63 k2 (B.49) (B.50) Note that there is a maximum values of k possible, kM AX = [0.21/0.63] Larger k gives complex µ which are physically meaningless Also note that µ → 0.67, ∞ as k → ω/ωα is evaluated in (B.48) and for these simple aerodynamics does not vary with µ or k From (B.48) and a knowledge of k, U/bωα is known The above results are tabulated below k µ1 µ2 U/bωα 0.0 0.1 0.2 0.3 0.4 0.5 0.57 0.67 0.69 0.72 0.75 0.81 0.937 1.39 ∞ 89.6 22 9.28 4.3 2.66 1.39 ∞ 6.9 3.45 2.3 1.73 1.38 1.21 From the above table (as well as equation (B.50)) one sees that for 4µ < 0.67, no flutter is possible This is similar to the flutter behavior of the typical section at incompressible speeds At these low speeds mass ratios of this magnitude may occur in hydrofoil applications Although no such applications exist at high supersonic speeds, it is of interest at least from a fundamental point of view that this somewhat surprising behavior at small µ occurs there as well B.5 For Section 4.1 Question Starting from Bermoulli’s equation, show that ˆ a ˆ u ∼ M∞ a∞ U∞ Previously we had shown that the boundary condition on a moving body is (within a linear approximation) ∂ φˆ ∂z = z=0 ∂za ∂za + U∞ ∂t ∂x 739 Appendix B: Some Example Problems What is the corresponding boundary condition in terms of pˆ? Derive approximate formulae for the perturbation pressure over a twodimensional airfoil at supersonic speeds for very low and very high frequencies Answer Bernoulli’s equation is ∇φ · ∇φ ∂φ + + ∂t p p∞ dp1 ρ1 (p1 ) = U∞ Since ∂p p = constant and ∂ρ ργ we may evaluate integral in the above to obtain a2 ≡ U∞ ∂φ u2 a2 − a2∞ + = , ∂t γ−1 u ≡ |∇φ| Assume ˆ a =a∞ + a u =U∞ + u ˆ φ =U∞ x + φˆ where a ˆ a∞ , etc To first order −M∞ u ˆ ∂ φˆ a ˆ − + terms (ˆ a2 , etc) = U∞ a∞ ∂t γ − a∞ This means that M∞ (ˆ u/U∞ ) and a ˆ/a∞ are quantities of the same order, at least for ˆ steady flow where ∂ φ/∂t ≡ ∂ ∂ ∂ φˆ = Dza ; D≡ + U∞ (B.51) ∂z z=0 ∂t ∂x By the linearized momentum equation −∂p u = ρ∞ Dˆ ∂x but u ˆ = ∇x φˆ therefore pˆ = −ρ∞ Dpˆ therefore − From (B.51) and the above ∂ pˆ ∂ ˆ = ρ∞ (Dφ) ∂z ∂z ∂ ˆ = ρ∞ D = ρ∞ (D(φ)) ∂z −∂p ∂z z=0 = ρ∞ D2 za ∂ ˆ φ ∂z 740 A MODERN COURSE IN AEROELASTICITY ∂p = −ρ∞ D2 za at z = ∂z ∇2 φ − where ∂ ∂ + U∞ ∂t ∂x a2∞ (B.52) φ=0 ∂ ∂ ∂φ Big|z=0 = za + U∞ za ≡ w ∂z ∂t ∂x off wing ∂φ =0 thickness case ∂z z=0 This does not matter here, because there are no disturbances ahead of wing in supersonic flow =0 lifting case, φ z=0 ¯ z)eiωt and w = we For a two dimensional solution, let φ(x, z, t) = φ(x, ¯ iωt Thus 2¯ ∂ φ¯ ∂ φ¯ ∂ φ¯ ∂ φ + − −ω φ¯ + 2iωU∞ + U∞ =0 2 ∂x ∂z a∞ ∂x ∂x2 Recall u, v, w = for x ≤ (leading edge) in supersonic flow Taking a Laplace transform (quiescent condition at x = 0) Φ≡ then p2 Φ + or Thus ¯ −px dx φe ∂2Φ − [−ω Φ + 2iωpU∞ Φ + p2 U∞ Φ] ∂z a∞ d2 Φ ω2 2iωpM + p2 M Φ ≡ µ2 Φ −p2 − + dz a∞ a∞ Thus Now ∞ Φ = Be−µz dΦ dz = W, z=0 dΦ dz z=0 W ≡ = −µB, Hence Φ=− ∞ we ¯ −px dx B=− W µ w −µz e µ so ¯ z=0 = φ| L−1 − µ w(ξ, ¯ ω)dξ For low frequencies , we can ignore the ω terms so µ2 ∼ = (M − 1) p + iM ω a∞ (M − 1) 741 Appendix B: Some Example Problems − −1 = √ µ M2 − −1 µ L−1 = √ p+ iM ω a∞ (M −1) −1 exp[−iM ωx/a∞ (M − 1)] M2 − and x ¯ z=0 = √ −1 exp[−iM ω(x − ξ)/a∞ (M − 1)]w(ξ, ¯ ω)dξ φ| M2 − and if we select our coordinate system such that w(0) = 0, for low frequencies the perturbation pressure, pˆ, is from Bernoulli’s equation iω exp[iω(t − M x/a∞ (M − 1))] ρ∞ eiωt − pˆ = √ (M − 1) M2 − × exp[iM ωξ/a∞ (M − 1)]w(ξ, ¯ ω)dξ + U∞ w(x, ¯ ω) ρ∞ eiωt ∼ U∞ w(x) ¯ =√ M −1 For high frequencies , −ω 2iωpM d2 Φ = + + (M − 1)p2 Φ dx2 a2∞ a∞ ∼ = iω + pM a∞ Φ when we ignore the (−p2 ) term compared to those involving ω Then, −1 ∼ −1 = µ pM + aiω ∞ and ¯ z=0 = φ| L−1 −1 µ w(ξ, ¯ ω)dξ x−ξ by the convolution theorem Now L−1 −1 pM + iω a = −1 exp(−iωx/a∞ M ) M so x ¯ z=0 = − exp[−iω(x − ξ)/a∞ M ]w(ξ, ¯ x)dξ φ| M and from Bernoulli’s equation therefore x ρ∞ pˆ =❅ iω exp[iω(x − x/a∞ M )] exp(iωξ/a∞ M )w(ξ, ¯ ω)dξ M ❅ ρ∞ U∞ + ¯ ω) exp[iω(t − x/a∞ M )] exp(iωx/a∞ M )w(x, M x ❳ ρ∞ U∞ iω − ❳❳❳❳ exp[iω(t − x/a∞ M )] exp(iωξ/a∞ M )w(ξ, ¯ ω)dξ ❳ M a∞ M pˆ ∼ = ρ∞ U∞ w(x, ¯ ω)eiωt M for high frequencies (B.53) 742 A MODERN COURSE IN AEROELASTICITY This is known as the (linearized, small perturbation) piston theory approximation It is a useful and interesting exercise to determine pressure distributions, lift and moment for translation and rotation of a flate plate using the piston theory.∗ The low frequency approximation considered earlier is also useful in this respect ∗ Ashley, H and Zartarian, G., ‘Piston Theory - A New Aerodynamic Tool for the Aeroelastician’, J Aero Sciences, 23 (December 1956) pp 1109-1118 Index shock waves, 172, 266, 467, 484, 485, 493, 499, 509, 512, 519, 522, 552, 556, 558, 560, 572 slender body, 97, 156, 237, 381 strip theory, 37, 95, 147, 154, 381, 405, 485 subsonic, 94, 164, 165, 182, 187, 201, 215, 233, 239, 244, 251, 260, 266, 288, 467, 468, 482, 485, 493, 498, 516, 520, 560, 562, 566, 568, 590, 633, 671 supersonic, 94, 112, 158, 161, 165, 182, 215, 233, 239, 244, 467, 468, 481, 493, 498, 561, 566, 567, 584 Theodorsen’s function, 215, 223, 224, 331, 440 Theodorsen’s theory, 406, 440 transfer function, 77, 92, 506, 509, 510, 559 transonic, 91, 116, 166, 215, 238, 240, 243, 467, 485, 516, 520, 560, 562, 678, 680, 682, 691, 699 velocity potential, 172, 176, 201, 207, 215, 245, 468, 552 vortex shedding, 266, 475 Wagner function, 93–95, 226, 623, 624 Aeroelastic control, 612, 615, 627 Aeroelasticity, 1–3, 9, 10, 37, 46, 51–53, 81, 92, 110, 165, 166, 169, 288, 289, 300, 344, 377, 378, 434, 444, 453, 455, 462, 471, 487, 541, 551, 552, 555, 560, 566, 580, 585, 611, 617, 675 Air resonance, 409, 428, 429 Articulated blade, 379, 384 Articulated rotor, 377, 378, 389, 392, 396, 428 Aerodynamics acoustics, 169, 176, 177, 471, 485, 528, 533, 553, 676, 683 aerodynamic center, 64, 108, 379 aspect ratio, 18, 89, 96, 236, 365, 552, 557, 570, 576, 578 boundary condition, 88 circulation, 216, 271, 621 damping, 77, 79, 86, 116, 150, 305, 308, 311, 319, 321, 333, 360, 361, 366, 367, 388, 581, 694 damping , 67 Euler equations, 494, 495, 503, 507, 512, 516, 677, 678, 680, 688, 690, 691 finiteness condition, 217, 263, 513 Green’s functions, 40 gusts, 53, 100, 155, 189, 192, 227, 465, 688 incompressible, 92, 93, 95, 147, 207, 214, 283, 289, 329, 334, 462, 468, 512, 620, 633 influence coefficients, 200, 201, 480 influence function, 37, 38, 40, 53, 264, 268 irrotational flow, 495, 620, 633 Kelvin’s theorem, 172 Kussner function, 93, 94, 226, 228 Kutta condition, 203, 204, 220 Piston theory, 92, 95, 117, 188, 192, 233, 234, 237, 257, 491 quasi-steady, 76, 80, 85, 95, 96, 109, 150, 153, 161, 289, 310, 314, 433, 435, 436, 438, 440, 479 separation, 116, 238, 252, 265, 276, 288, 291, 294, 300, 302, 331, 335, 337, 360, 425, 472, 478, 485, 552, 558, 560, 578, 581, 582, 585, 595, 596, 611, 691 743 744 A MODERN COURSE IN AEROELASTICITY Beam, 44, 45, 47, 49, 50, 57, 58, 153, 378, 460, 461, 486, 576, 578, 597, 628 Beam-rod, 18, 19, 41, 42, 46, 48, 49, 134 Blade, 2, 110, 116, 276, 291–295, 328, 377–379, 381, 383–388, 390– 399, 401, 403–406, 408–411, 416–422, 427, 428, 430–432, 440, 442, 443, 453–458, 460– 465, 467, 470–481, 483–487, 511, 675, 677, 678, 682–688, 691, 699 Bluff body, 300, 369 Boundary conditions, 177, 178, 181, 192, 195, 208, 215, 246, 247, 290, 291, 300, 305, 472, 492, 493, 528, 677 Bridge, 2, 3, 110, 276, 299–301, 313, 327– 338, 340, 341, 344–351, 353, 356–360, 366 Buckling, 10, 45–47, 327, 552 Buffeting, 276, 300, 328, 330, 335, 336, 350, 351, 353, 356–359, 365, 475, 675 Buildings, 2, 299, 300, 331, 361, 363–367 Cascades, 462, 467, 471, 472, 475, 480, 487, 503, 678 Civil engineering, 2, 299, 300, 329 Collocation, 201, 204, 620 Computational, 72, 103, 111, 113, 266, 276, 289, 291, 292, 294, 472, 474, 479, 486, 487, 495, 496, 500, 504, 505, 507, 511–513, 517, 528, 556, 558–560, 566, 583, 585, 587, 591, 593, 632, 633, 669, 677, 678, 684, 690, 691, 699 Control transfer function, 612, 614, 636, 642, 644, 645, 649, 654, 660, 665 Control surface reversal, 12, 13, 36, 42, 53 Correlation function, 74, 75, 78, 124 Damping, 67, 79, 86, 87, 109–112, 125, 150, 286, 305, 308, 334, 350, 351, 358, 359, 363–365, 367, 384, 388, 391, 396, 399, 401, 405, 408, 426–432, 434, 454, 462, 469, 476, 479–481, 511, 514, 524, 542, 543, 557, 559, 572, 576, 585, 593, 594, 623, 629, 635, 645, 651–654, 657, 658, 687 Describing function, 510, 559 115, 317, 361, 389, 406, 461, 485, 556, 590, 637, 665, Divergence, 6, 8–10, 12, 13, 21, 22, 25, 26, 33–38, 42–47, 50, 51, 53, 83, 110, 205, 206, 300, 327, 328, 336, 338, 340, 379, 401, 454, 617 Dynamic response, 71, 73, 334, 335, 374, 552, 553, 569, 570, 641 Eigenfunctions, 22, 24, 25, 38, 39, 142, 143, 480 Eigenvalues, 38, 51, 102, 108, 115, 135, 136, 142, 143, 326, 340, 352, 479–481, 510, 511, 513–515, 517, 538, 539, 623, 625, 628, 649, 651–653, 657 Eigenvectors, 135, 136, 414, 415, 437, 499, 506–508, 511, 513 Elastic axis, 6, 7, 9, 11, 18, 31, 42, 47, 48, 59, 63, 64, 70, 84, 283, 285, 287, 288, 379, 454, 618, 619, 621, 623, 626, 655 Energy kinetic, 55, 63, 129, 162, 511, 618 potential, 56, 57, 62, 63, 128, 130, 131, 161, 511, 618 Equations of motion, 53, 54, 61, 63, 64, 82, 97, 99, 100, 105, 121, 126, 128, 137, 140, 144, 161, 162, 169, 305, 308, 314, 317, 320, 324, 329, 330, 332, 333, 344, 350–352, 361, 379, 381, 384, 386, 389, 391, 394, 398, 405, 408, 410, 411, 413, 414, 416– 418, 420, 427, 544, 577, 619, 629 Experiments flutter boundary, 543–545, 619, 625, 644, 646, 647, 655 gust response, 547 Finite difference method, 240, 247 Flutter choking, 477–479, 485 gallopig, 321 galloping, 276, 299, 300, 314, 317– 321, 323, 326, 327, 329, 367, 675 hydrofoil, 738 stall, xv, xvii, 116, 117, 120, 266, 275, 276, 279, 399, 403–406, 408, 409, 453, 454, 471–477, 481, 483–485, 512, 570, 577– 579, 594, 595 types of, 105, 108, 116, 285, 543, 570, 572 typical section, 5, 6, 8, 10, 11, 16, 18, 21, 46, 53, 54, 61, 63–65, 70, 81, 82, 86, 87, 89, 103, 105, 745 INDEX 121, 128, 136, 137, 147, 148, 155, 160, 161, 276, 573, 612, 617, 618, 620, 623, 625–627, 644, 646, 647, 652–657, 711, 731, 735, 738 Fourier transform, 68, 75, 76, 90, 100, 102, 138, 186, 190, 191, 196, 216, 259, 269, 352, 353, 542, 690, 707, 708, 734 Frequencies, 228, 287, 291, 292, 310, 353, 357, 358, 367, 391–395, 401, 410, 414, 416, 420, 421, 423– 426, 428, 437–439, 460–462, 473, 474, 485, 486, 501, 509– 511, 517, 520, 541–543, 555, 575–577, 584, 623, 625, 629, 630, 644, 647, 664, 675, 706, 739–741 Frequencies natural, 64, 66, 67, 72, 99, 103, 130, 135, 140, 141, 144, 147, 158, 163, 164, 292, 293, 295, 305, 319, 330, 348, 350, 361, 366, 367, 384, 387, 388, 391–395, 411, 414, 420, 421, 435, 457, 460, 473, 475, 479, 480, 485, 495, 511, 541, 542, 575, 576, 628–630, 637, 675, 676, 682 Generalized coordinate, 60, 61, 63, 161, 707 Generalized force, 60, 136, 138, 146, 154, 350, 355, 442, 506, 687, 707, 708 Generalized mass, 130, 146, 153 Gust cosine, 121–123 excitation, 138 frozen, 190, 192, 194 power spectral density, 80 random, 78, 100, 102, 122, 124, 126, 156 response, 103, 105, 121, 150 sharp edged, 121, 122, 190, 193, 227 sinusodal, 190 Hamilton’s principle, 54, 55, 59, 60 Helicopters, 276, 377, 378, 395, 398, 404, 408, 409, 428 Impulse function, 53, 89, 92–94, 98, 137, 359 Instability, 67, 81, 110, 157, 167, 276, 291, 292, 314, 327, 334, 349, 370, 388, 389, 399, 404, 405, 408, 409, 424, 426, 432, 472, 474, 475, 477, 484, 486, 634, 647, 675 Integral equations, 98, 128, 132, 139, 204, 264, 288, 465–467, 471, 493 Lagrange’s equations, 54, 60–63, 128, 135–137, 139, 162 Laplace transform, 68, 99, 148, 152, 155 Leading edge, 203, 227, 234, 235, 250, 251, 268, 288, 454, 467, 479, 486, 573, 577, 642, 691 Mach number, 161, 174, 182, 198, 233, 235, 243, 249, 251, 265, 456, 468, 475, 478, 483, 485, 491– 493, 498, 507, 516, 517, 522, 526, 554, 555, 560–562, 584, 590, 691, 692 Modes aeroelastic, 125 flutter, 544 natural, 110, 135, 140, 460, 706 spinning, 685–687 structural, 130, 461, 479 Motion arbitrary, 67, 69, 100, 208, 224, 499 flutter, 103, 109 gust, 77, 104, 121 periodic, 67, 499 random, 73 sinusoidal, 64, 67, 69, 74, 90, 108, 232 transient, 68, 71, 92–94, 166, 188, 190, 195, 224, 265 Nonlinearities aerodynamic, 116, 286, 289, 499, 694 structural, 577 Orthogonality, 130, 140, 143, 144, 146 Pipes, 156, 158, 160, 167 Plates, 57–59, 156, 496, 497, 576, 577, 586, 597, 622, 623, 627 Power flow, 285, 286 Power spectra, 75, 80, 102, 106 Pressure, 59, 64, 81, 84, 88, 91, 92, 95, 106, 116, 119, 136–138, 161, 172, 176, 181, 182, 188, 194, 195, 201–204, 207, 220, 223, 233–235, 277, 283, 287, 290, 304, 337, 405, 442, 552, 561, 570–572, 577, 583, 586, 621, 653, 677, 679, 680, 682, 685, 691, 694 Rayleigh-Ritz method, 128 746 A MODERN COURSE IN AEROELASTICITY Reduced frequency, 90, 96, 112, 185, 287, 294, 330, 333, 522, 530, 531, 563, 572, 623, 680, 691 Rotorcraft, 276, 433 Shock impulse, 680, 682, 694 Small perturbation theory, 91, 175, 178, 180, 243, 496, 552, 554, 676 Solutions frequency domain, 72, 100, 101, 138, 479, 559, 620, 676, 688, 699 time domain, 98, 102, 138, 510, 559, 603, 620, 676 Stiffness spring, 625 aeroelastic, 107 bending, 59, 391 nonlinear, 552, 556, 573, 574, 625 spring, 62, 125, 394, 573 torsional, 120, 328, 360, 379 Structural beam, 158 damping, 71, 72, 86, 111, 113, 114, 286, 306, 324, 329, 361, 366, 461, 481, 552, 557, 581, 694 influence function, 139, 146 plate, 130 stiffness, 101 transfer function, 77, 101, 506, 510, 559 Trailing edge, 454, 466, 560, 561, 569, 573, 642, 644 Transonic buzz, 116 Turbomachinery aeromechanics, 676 Turbomachines, 453, 454, 461, 471, 472, 475, 479, 486, 487 Virtual displacement, 54 Virtual work, 54–57, 59, 136, 162 Wave equation, 177, 195, 211, 492, 494, 552 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids The scope of the series covers the entire spectrum of solid mechanics Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design 10 11 12 13 14 15 16 17 18 19 20 21 22 23 R.T Haftka, Z G¨urdal and M.P Kamat: Elements of Structural Optimization 2nd rev.ed., 1990 ISBN 0-7923-0608-2 J.J Kalker: Three-Dimensional Elastic Bodies in Rolling Contact 1990 ISBN 0-7923-0712-7 P Karasudhi: Foundations of Solid Mechanics 1991 ISBN 0-7923-0772-0 Not published Not published J.F Doyle: Static and Dynamic Analysis of Structures With an Emphasis on Mechanics and Computer Matrix Methods 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-2 O.O Ochoa and J.N Reddy: Finite Element Analysis of Composite Laminates ISBN 0-7923-1125-6 M.H Aliabadi and D.P Rooke: Numerical Fracture Mechanics ISBN 0-7923-1175-2 J Angeles and C.S L´opez-Caj´un: Optimization of Cam Mechanisms 1991 ISBN 0-7923-1355-0 D.E Grierson, A Franchi and P Riva (eds.): Progress in Structural Engineering 1991 ISBN 0-7923-1396-8 R.T Haftka and Z G¨urdal: Elements of Structural Optimization 3rd rev and exp ed 1992 ISBN 0-7923-1504-9; Pb 0-7923-1505-7 J.R Barber: Elasticity 1992 ISBN 0-7923-1609-6; Pb 0-7923-1610-X H.S Tzou and G.L Anderson (eds.): Intelligent Structural Systems 1992 ISBN 0-7923-1920-6 E.E Gdoutos: Fracture Mechanics An Introduction 1993 ISBN 0-7923-1932-X J.P Ward: Solid Mechanics An Introduction 1992 ISBN 0-7923-1949-4 M Farshad: Design and Analysis of Shell Structures 1992 ISBN 0-7923-1950-8 H.S Tzou and T Fukuda (eds.): Precision Sensors, Actuators and Systems 1992 ISBN 0-7923-2015-8 J.R Vinson: The Behavior of Shells Composed of Isotropic and Composite Materials 1993 ISBN 0-7923-2113-8 H.S Tzou: Piezoelectric Shells Distributed Sensing and Control of Continua 1993 ISBN 0-7923-2186-3 W Schiehlen (ed.): Advanced Multibody System Dynamics Simulation and Software Tools 1993 ISBN 0-7923-2192-8 C.-W Lee: Vibration Analysis of Rotors 1993 ISBN 0-7923-2300-9 D.R Smith: An Introduction to Continuum Mechanics 1993 ISBN 0-7923-2454-4 G.M.L Gladwell: Inverse Problems in Scattering An Introduction 1993 ISBN 0-7923-2478-1 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 G Prathap: The Finite Element Method in Structural Mechanics 1993 ISBN 0-7923-2492-7 J Herskovits (ed.): Advances in Structural Optimization 1995 ISBN 0-7923-2510-9 M.A Gonz´alez-Palacios and J Angeles: Cam Synthesis 1993 ISBN 0-7923-2536-2 W.S Hall: The Boundary Element Method 1993 ISBN 0-7923-2580-X J Angeles, G Hommel and P Kov´acs (eds.): Computational Kinematics 1993 ISBN 0-7923-2585-0 A Curnier: Computational Methods in Solid Mechanics 1994 ISBN 0-7923-2761-6 D.A Hills and D Nowell: Mechanics of Fretting Fatigue 1994 ISBN 0-7923-2866-3 B Tabarrok and F.P.J Rimrott: Variational Methods and Complementary Formulations in Dynamics 1994 ISBN 0-7923-2923-6 E.H Dowell (ed.), E.F Crawley, H.C Curtiss Jr., D.A Peters, R H Scanlan and F Sisto: A Modern Course in Aeroelasticity Third Revised and Enlarged Edition 1995 ISBN 0-7923-2788-8; Pb: 0-7923-2789-6 A Preumont: Random Vibration and Spectral Analysis 1994 ISBN 0-7923-3036-6 J.N Reddy (ed.): Mechanics of Composite Materials Selected works of Nicholas J Pagano 1994 ISBN 0-7923-3041-2 A.P.S Selvadurai (ed.): Mechanics of Poroelastic Media 1996 ISBN 0-7923-3329-2 Z Mr´oz, D Weichert, S Dorosz (eds.): Inelastic Behaviour of Structures under Variable Loads 1995 ISBN 0-7923-3397-7 R Pyrz (ed.): IUTAM Symposium on Microstructure-Property Interactions in Composite Materials Proceedings of the IUTAM Symposium held in Aalborg, Denmark 1995 ISBN 0-7923-3427-2 M.I Friswell and J.E Mottershead: Finite Element Model Updating in Structural Dynamics 1995 ISBN 0-7923-3431-0 D.F Parker and A.H England (eds.): IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics Proceedings of the IUTAM Symposium held in Nottingham, U.K 1995 ISBN 0-7923-3594-5 J.-P Merlet and B Ravani (eds.): Computational Kinematics ’95 1995 ISBN 0-7923-3673-9 L.P Lebedev, I.I Vorovich and G.M.L Gladwell: Functional Analysis Applications in Mechanics and Inverse Problems 1996 ISBN 0-7923-3849-9 J Menˇcik: Mechanics of Components with Treated or Coated Surfaces 1996 ISBN 0-7923-3700-X D Bestle and W Schiehlen (eds.): IUTAM Symposium on Optimization of Mechanical Systems Proceedings of the IUTAM Symposium held in Stuttgart, Germany 1996 ISBN 0-7923-3830-8 D.A Hills, P.A Kelly, D.N Dai and A.M Korsunsky: Solution of Crack Problems The Distributed Dislocation Technique 1996 ISBN 0-7923-3848-0 V.A Squire, R.J Hosking, A.D Kerr and P.J Langhorne: Moving Loads on Ice Plates 1996 ISBN 0-7923-3953-3 A Pineau and A Zaoui (eds.): IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials Proceedings of the IUTAM Symposium held in S`evres, Paris, France 1996 ISBN 0-7923-4188-0 A Naess and S Krenk (eds.): IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics Proceedings of the IUTAM Symposium held in Trondheim, Norway 1996 ISBN 0-7923-4193-7 D Ies¸an and A Scalia: Thermoelastic Deformations 1996 ISBN 0-7923-4230-5 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 J.R Willis (ed.): IUTAM Symposium on Nonlinear Analysis of Fracture Proceedings of the IUTAM Symposium held in Cambridge, U.K 1997 ISBN 0-7923-4378-6 A Preumont: Vibration Control of Active Structures An Introduction 1997 ISBN 0-7923-4392-1 G.P Cherepanov: Methods of Fracture Mechanics: Solid Matter Physics 1997 ISBN 0-7923-4408-1 D.H van Campen (ed.): IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems Proceedings of the IUTAM Symposium held in Eindhoven, The Netherlands 1997 ISBN 0-7923-4429-4 N.A Fleck and A.C.F Cocks (eds.): IUTAM Symposium on Mechanics of Granular and Porous Materials Proceedings of the IUTAM Symposium held in Cambridge, U.K 1997 ISBN 0-7923-4553-3 J Roorda and N.K Srivastava (eds.): Trends in Structural Mechanics Theory, Practice, Education 1997 ISBN 0-7923-4603-3 Yu.A Mitropolskii and N Van Dao: Applied Asymptotic Methods in Nonlinear Oscillations 1997 ISBN 0-7923-4605-X C Guedes Soares (ed.): Probabilistic Methods for Structural Design 1997 ISBN 0-7923-4670-X D Franc¸ois, A Pineau and A Zaoui: Mechanical Behaviour of Materials Volume I: Elasticity and Plasticity 1998 ISBN 0-7923-4894-X D Franc¸ois, A Pineau and A Zaoui: Mechanical Behaviour of Materials Volume II: Viscoplasticity, Damage, Fracture and Contact Mechanics 1998 ISBN 0-7923-4895-8 L.T Tenek and J Argyris: Finite Element Analysis for Composite Structures 1998 ISBN 0-7923-4899-0 Y.A Bahei-El-Din and G.J Dvorak (eds.): IUTAM Symposium on Transformation Problems in Composite and Active Materials Proceedings of the IUTAM Symposium held in Cairo, Egypt 1998 ISBN 0-7923-5122-3 I.G Goryacheva: Contact Mechanics in Tribology 1998 ISBN 0-7923-5257-2 O.T Bruhns and E Stein (eds.): IUTAM Symposium on Micro- and Macrostructural Aspects of Thermoplasticity Proceedings of the IUTAM Symposium held in Bochum, Germany 1999 ISBN 0-7923-5265-3 F.C Moon: IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics Proceedings of the IUTAM Symposium held in Ithaca, NY, USA 1998 ISBN 0-7923-5276-9 R Wang: IUTAM Symposium on Rheology of Bodies with Defects Proceedings of the IUTAM Symposium held in Beijing, China 1999 ISBN 0-7923-5297-1 Yu.I Dimitrienko: Thermomechanics of Composites under High Temperatures 1999 ISBN 0-7923-4899-0 P Argoul, M Fr´emond and Q.S Nguyen (eds.): IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics Proceedings of the IUTAM Symposium held in Paris, France 1999 ISBN 0-7923-5450-8 F.J Fahy and W.G Price (eds.): IUTAM Symposium on Statistical Energy Analysis Proceedings of the IUTAM Symposium held in Southampton, U.K 1999 ISBN 0-7923-5457-5 H.A Mang and F.G Rammerstorfer (eds.): IUTAM Symposium on Discretization Methods in Structural Mechanics Proceedings of the IUTAM Symposium held in Vienna, Austria 1999 ISBN 0-7923-5591-1 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 P Pedersen and M.P Bendsøe (eds.): IUTAM Symposium on Synthesis in Bio Solid Mechanics Proceedings of the IUTAM Symposium held in Copenhagen, Denmark 1999 ISBN 0-7923-5615-2 S.K Agrawal and B.C Fabien: Optimization of Dynamic Systems 1999 ISBN 0-7923-5681-0 A Carpinteri: Nonlinear Crack Models for Nonmetallic Materials 1999 ISBN 0-7923-5750-7 F Pfeifer (ed.): IUTAM Symposium on Unilateral Multibody Contacts Proceedings of the IUTAM Symposium held in Munich, Germany 1999 ISBN 0-7923-6030-3 E Lavendelis and M Zakrzhevsky (eds.): IUTAM/IFToMM Symposium on Synthesis of Nonlinear Dynamical Systems Proceedings of the IUTAM/IFToMM Symposium held in Riga, Latvia 2000 ISBN 0-7923-6106-7 J.-P Merlet: Parallel Robots 2000 ISBN 0-7923-6308-6 J.T Pindera: Techniques of Tomographic Isodyne Stress Analysis 2000 ISBN 0-7923-6388-4 G.A Maugin, R Drouot and F Sidoroff (eds.): Continuum Thermomechanics The Art and Science of Modelling Material Behaviour 2000 ISBN 0-7923-6407-4 N Van Dao and E.J Kreuzer (eds.): IUTAM Symposium on Recent Developments in Non-linear Oscillations of Mechanical Systems 2000 ISBN 0-7923-6470-8 S.D Akbarov and A.N Guz: Mechanics of Curved Composites 2000 ISBN 0-7923-6477-5 M.B Rubin: Cosserat Theories: Shells, Rods and Points 2000 ISBN 0-7923-6489-9 S Pellegrino and S.D Guest (eds.): IUTAM-IASS Symposium on Deployable Structures: Theory and Applications Proceedings of the IUTAM-IASS Symposium held in Cambridge, U.K., 6–9 September 1998 2000 ISBN 0-7923-6516-X A.D Rosato and D.L Blackmore (eds.): IUTAM Symposium on Segregation in Granular Flows Proceedings of the IUTAM Symposium held in Cape May, NJ, U.S.A., June 5–10, 1999 2000 ISBN 0-7923-6547-X A Lagarde (ed.): IUTAM Symposium on Advanced Optical Methods and Applications in Solid Mechanics Proceedings of the IUTAM Symposium held in Futuroscope, Poitiers, France, August 31–September 4, 1998 2000 ISBN 0-7923-6604-2 D Weichert and G Maier (eds.): Inelastic Analysis of Structures under Variable Loads Theory and Engineering Applications 2000 ISBN 0-7923-6645-X T.-J Chuang and J.W Rudnicki (eds.): Multiscale Deformation and Fracture in Materials and Structures The James R Rice 60th Anniversary Volume 2001 ISBN 0-7923-6718-9 S Narayanan and R.N Iyengar (eds.): IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics Proceedings of the IUTAM Symposium held in Madras, Chennai, India, 4–8 January 1999 ISBN 0-7923-6733-2 S Murakami and N Ohno (eds.): IUTAM Symposium on Creep in Structures Proceedings of the IUTAM Symposium held in Nagoya, Japan, 3-7 April 2000 2001 ISBN 0-7923-6737-5 W Ehlers (ed.): IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials Proceedings of the IUTAM Symposium held at the University of Stuttgart, Germany, September 5-10, 1999 2001 ISBN 0-7923-6766-9 D Durban, D Givoli and J.G Simmonds (eds.): Advances in the Mechanis of Plates and Shells The Avinoam Libai Anniversary Volume 2001 ISBN 0-7923-6785-5 U Gabbert and H.-S Tzou (eds.): IUTAM Symposium on Smart Structures and Structonic Systems Proceedings of the IUTAM Symposium held in Magdeburg, Germany, 26–29 September 2000 2001 ISBN 0-7923-6968-8 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 Y Ivanov, V Cheshkov and M Natova: Polymer Composite Materials – Interface Phenomena & Processes 2001 ISBN 0-7923-7008-2 R.C McPhedran, L.C Botten and N.A Nicorovici (eds.): IUTAM Symposium on Mechanical and Electromagnetic Waves in Structured Media Proceedings of the IUTAM Symposium held in Sydney, NSW, Australia, 18-22 Januari 1999 2001 ISBN 0-7923-7038-4 D.A Sotiropoulos (ed.): IUTAM Symposium on Mechanical Waves for Composite Structures Characterization Proceedings of the IUTAM Symposium held in Chania, Crete, Greece, June 14-17, 2000 2001 ISBN 0-7923-7164-X V.M Alexandrov and D.A Pozharskii: Three-Dimensional Contact Problems 2001 ISBN 0-7923-7165-8 J.P Dempsey and H.H Shen (eds.): IUTAM Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics Proceedings of the IUTAM Symposium held in Fairbanks, Alaska, U.S.A., 13-16 June 2000 2001 ISBN 1-4020-0171-1 U Kirsch: Design-Oriented Analysis of Structures A Unified Approach 2002 ISBN 1-4020-0443-5 A Preumont: Vibration Control of Active Structures An Introduction (2nd Edition) 2002 ISBN 1-4020-0496-6 B.L Karihaloo (ed.): IUTAM Symposium on Analytical and Computational Fracture Mechanics of Non-Homogeneous Materials Proceedings of the IUTAM Symposium held in Cardiff, U.K., 18-22 June 2001 2002 ISBN 1-4020-0510-5 S.M Han and H Benaroya: Nonlinear and Stochastic Dynamics of Compliant Offshore Structures 2002 ISBN 1-4020-0573-3 A.M Linkov: Boundary Integral Equations in Elasticity Theory 2002 ISBN 1-4020-0574-1 L.P Lebedev, I.I Vorovich and G.M.L Gladwell: Functional Analysis Applications in Mechanics and Inverse Problems (2nd Edition) 2002 ISBN 1-4020-0667-5; Pb: 1-4020-0756-6 Q.P Sun (ed.): IUTAM Symposium on Mechanics of Martensitic Phase Transformation in Solids Proceedings of the IUTAM Symposium held in Hong Kong, China, 11-15 June 2001 2002 ISBN 1-4020-0741-8 M.L Munjal (ed.): IUTAM Symposium on Designing for Quietness Proceedings of the IUTAM Symposium held in Bangkok, India, 12-14 December 2000 2002 ISBN 1-4020-0765-5 J.A.C Martins and M.D.P Monteiro Marques (eds.): Contact Mechanics Proceedings of the 3rd Contact Mechanics International Symposium, Praia da Consolac¸a˜ o, Peniche, Portugal, 17-21 June 2001 2002 ISBN 1-4020-0811-2 H.R Drew and S Pellegrino (eds.): New Approaches to Structural Mechanics, Shells and Biological Structures 2002 ISBN 1-4020-0862-7 J.R Vinson and R.L Sierakowski: The Behavior of Structures Composed of Composite Materials Second Edition 2002 ISBN 1-4020-0904-6 Not yet published J.R Barber: Elasticity Second Edition 2002 ISBN Hb 1-4020-0964-X; Pb 1-4020-0966-6 C Miehe (ed.): IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains Proceedings of the IUTAM Symposium held in Stuttgart, Germany, 20-24 August 2001 2003 ISBN 1-4020-1170-9 Mechanics SOLID MECHANICS AND ITS APPLICATIONS Series Editor: G.M.L Gladwell 109 P St˚ahle and K.G Sundin (eds.): IUTAM Symposium on Field Analyses for Determination of Material Parameters – Experimental and Numerical Aspects Proceedings of the IUTAM Symposium held in Abisko National Park, Kiruna, Sweden, July 31 – August 4, 2000 2003 ISBN 1-4020-1283-7 110 N Sri Namachchivaya and Y.K Lin (eds.): IUTAM Symposium on Nonlnear Stochastic Dynamics Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26 – 30 August, 2000 2003 ISBN 1-4020-1471-6 111 H Sobieckzky (ed.): IUTAM Symposium Transsonicum IV Proceedings of the IUTAM Symposium held in G¨ottingen, Germany, 2–6 September 2002, 2003 ISBN 1-4020-1608-5 112 J.-C Samin and P Fisette: Symbolic Modeling of Multibody Systems 2003 ISBN 1-4020-1629-8 113 A.B Movchan (ed.): IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics Proceedings of the IUTAM Symposium held in Liverpool, United Kingdom, 8-11 July 2002 2003 ISBN 1-4020-1780-4 114 S Ahzi, M Cherkaoui, M.A Khaleel, H.M Zbib, M.A Zikry and B LaMatina (eds.): IUTAM Symposium on Multiscale Modeling and Characterization of Elastic-Inelastic Behavior of Engineering Materials Proceedings of the IUTAM Symposium held in Marrakech, Morocco, 20-25 October 2002 2004 ISBN 1-4020-1861-4 115 H Kitagawa and Y Shibutani (eds.): IUTAM Symposium on Mesoscopic Dynamics of Fracture Process and Materials Strength Proceedings of the IUTAM Symposium held in Osaka, Japan, 6-11 July 2003 Volume in celebration of Professor Kitagawa’s retirement 2004 ISBN 1-4020-2037-6 116 E.H Dowell, R.L Clark, D Cox, H.C Curtiss, Jr., K.C Hall, D.A Peters, R.H Scanlan, E Simiu, F Sisto and D Tang: A Modern Course in Aeroelasticity 4th Edition, 2004 ISBN 1-4020-2039-2 117 T Burczy´nski and A Osyczka (eds.): IUTAM Symposium on Evolutionary Methods in Mechanics Proceedings of the IUTAM Symposium held in Cracow, Poland, 24-27 September 2002 2004 ISBN 1-4020-2266-2 118 D Ies¸an: Thermoelastic Models of Continua 2004 ISBN 1-4020-2309-X Kluwer Academic Publishers – Dordrecht / Boston / London ... 1951 Also available in Dover Edition AGARD Manual on Aeroelasticity, Vols I-VII, Beginning 1959 with continual updating (AGARD) Ashley, H., Dugundji, J and Rainey, A G., Notebook for Aeroelasticity, ... Journals AHS Journal AIAA Journal ASCE Transactions, Engineering Mechanics Division ASME Transaction, Journal of Applied Mechanics International Journal of Solids and Structures Journal of Aircraft... A MODERN COURSE IN AEROELASTICITY NONLINEAR AEROELASTICITY (DOWELL, EDWARDS AND STRGANAC) 11.1 Introduction 11.2 Generic Nonlinear Aeroelastic Behavior 11.3 Flight Experience with Nonlinear Aeroelastic