A modern course in aeroelasticity MECHANICS: DYNAMICAL SYSTEMS Editors: L. Meirovitch and G. /E Oravas E. H. Dowell, Aeroelasticity of Plates and Shells. 1974. ISBN 90-286-0404-9. D. G. B. Edelcn, Lagrangian Mechanics of Nonconservativc Non­ holonomic Systems. 1977. ISBN 90-286-0077-9. J. L. Junkins, An Introduction to Optical Estimation of Dynamical Systems. )978. ISBN 90-286-0067-1. E. H. Dowell et al., A Modern Course in Aeroelasticity. 1978. ISBN 90-286-0057-4. L. Meirovitch, Computational Methods in Structural Dynamics. 1980. ISBN 90-286-0580-0. B. Skalmierski and A. Tylikowski, Stochastic Processes in Dynamics. 1982. ISBN 90-247-2686-7. P. C. Müller and W. O. Schiehlen, Linear Vibrations. 1985. ISBN 90-247-2983-1. Gh. Buzdugan, E. Mihäilescu and M. Rade§, Vibration Measurement. 1986. ISBN 90-247-3111-9. G. M. L. Gladwell, Inverse Problems in Vibration. 1986. ISBN 90-247-3408-8. G. I. Schueller and M. Shinozuka (eds.), Stochastic Methods in Structural Dynamies. 1987. ISBN 90-247-3611-0. E. H. Dowell, H. C. Curtiss, Jr., R. H. Scanlan and F. Sisto, A Modern Course in Aeroelasticity. Second revised and enlarged edition. 1989. ISBN 0-7923-0062-9. A modern course in aeroelasticity Second revised and enlarged edition Earl H. Dowell, editor, Professor of Mechanical Engineering and Materials Science Duke University, Durharn, North Carolina, U.S.A. Howard C. Curtiss, Jr. Professor of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey, U.S.A. Robert H. Scanlan Professor of Civil Engineering Johns Hopkins University, Baltimore, Maryland, U.S.A. and Fernando Sisto Professor of Mechanical Engineering Stevens Institute of Technology, Hoboken, New Jersey, U.S.A. Springer-Science+Business Media, B.V. Ubrary of eoDlress Catalolilll iD hblkatioll Data A Modp.rn course in aeroelasticity. (Mecbanics, dynallical syste_ ; Includes bibliographies and indexes. 1. Aeroelasticity. 1. Dowell, E. H. H. Series. TL574.A37M62 1989 629.132'362 88-32051 ISBN 978-0-7923-0185-1 ISBN 978-94-015-7858-5 (eBook) DOI 10.1007/978-94-015-7858-5 This is the revised and enlarged edition 01 a book published in 1978 by Sijthof! & Noordhof! International Publishers, in the series •Mechanics: Dynamical Systems' (Volume 3) printed on acid/ree paper All Rights Reserved © 1989 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1989 Softcover reprint ofthe hardcover 2nd edition 1989 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Contents Preface xiii Prelace to the second edition xv Short bibliography xvii 1. Introduction . 1 2. Static aeroelastidty 3 2.1 Typical section model 01 an airfoil . 3 Typical section with control surface 8 Typical section-nonlinear effects . 12 2.2 One dimensional aeroelastic model 01 airloils 15 Beam-rod representation of large aspect ratio wing 15 Eigenvalue and eigenfunction approach 18 Galerkin's method .............. 20 2.3 Rolling 01 a straight wing. 22 Integral equation of equilibrium 22 Derivation of equation of equilibrium 24 Calculation of C"'. 25 Sketch of function S(Yl' 1/). 25 Aerodynamic forces (including spanwise induction) 27 Aeroelastic equations of equilibrium and lumped element solution method ..... 29 Divergence . 31 Reversal and rolling effectiveness 31 v Contents Integral equation eigenvalue problem and the experi- mental determination of influence functions . 35 2.4 Two dimensional aeroelastic model o{ lifting surfaces . .. 38 Two dimensional structures-integral representation . .. 38 Two dimensional aerodynamic surfaces-integral represen- tation .. 40 Solution by matrix-Iumped element approach. 40 2.5 Nonairfoil physical problems . 42 Fluid flow through a flexible pipe . 42 (Low speed) fluid flow over a flexible wall 45 2.6 Sweptwing divergence. 46 Re{erences tor Chapter 2 49 3. Dynamic aeroelasticity 51 3.1 Hamilton' s principle 52 Single particle 52 Many particles . 54 Continuous body 54 Potential energy 54 Nonpotential forces 57 3.2 Lagrange's equations 58 Example-typical section equations of motion 59 3.3 Dynamics ot the typical section modelot an airfoil 62 Sinusoidal motion 63 Periodic motion 65 Arbitrary motion 66 Random motion 72 Flutter-an introduction to dynamic aeroelastic insta- bility ........ 80 Quasi-steady, aerodynamic theory ....... 83 3.4 Aerodynamic torces tor airfoils-an introduction and summary ........ 85 Aerodynamic theories available 89 General approximations . 93 'Strip theory' approximation 93 vi Contents 'Quasi-steady' approximation . 93 Slender body or slender (low aspect ratio) wing approximation. 94 3.5 Solutions to the aeroelastic equations of motion 95 Time domain solutions .......... 96 Frequency domain solutions. 98 3.6 Representative results and computational considerations . I () I Time domain ................ 101 Frequency domain .............. 103 Flutter and gust response classification including para- meter trends 105 Flutter . 105 Gust response. 118 3.7 Generalized equations of motion for complex structures . 124 Lagrange's equations and modal methods 124 Kinetic energy . 126 Strain (potential, elastic) energy 126 Examples .......... 129 (a) Torsional vibrations of a rod. 129 (b) Bending-torsional motion of a beam-rod 130 Natural frequencies and modes-eigenvalues and eigen- vectors ............... 13 1 Evaluation of generalized aerodynamic forces 132 Equations of motion and solution methods . 133 Integral equations of equilibrium 135 Natural frequencies and modes . 137 Proof of orthogonality . 139 Forced motion including aerodynamic forces 140 Examples ............... 143 (a) Rigid wing undergoing translation responding to a gust . .. 143 (b) Wing undergoing translation and spanwise bending 149 (c) Random gusts-solution in the frequency domain. 151 3.8 Nonairfoil physical problems . .. 152 Fluid flow through a flexible pipe . .. 152 (High speed) fluid flow over a flexible wall-a simple prototype for plate flutter 155 References for Chapter 3 . .. 160 vii Contents 4. Nonsteady aerodynamics 01 liftinl ud non-liftinl snrfaces . 162 4.1 Basic fluid dynamic equations 162 Conservation of mass . 163 Conservation of momentum 164 Irrotational ftow, Kelvin's theorem and Bernoulli's equation ........... 165 Derivation of single equation for velo city potential 168 Small perturbation theory 170 Reduction to acousties 171 Boundary conditions. 172 Symmetry and anti-symmetry . 174 4.2 Supersonic flow . 177 Two-dimensional ftow . 177 Simple harmonie motion of the airfoil 178 Discussion of inversion ..... 180 Discussion of physieal significance of results 183 Gusts ........ 184 Transient motion . 185 Lift, due to airfoil motion 186 Lift. due to atmospheric gusts 187 Three-dimensional ftow 190 4.3 Subsonic flow .... 196 Derivation of the integral equation by transform methods and solution by collocation . 197 An alternative determination of the Kernel Function using Green's theorem ....... 200 Incompressible, three-dimensional ftow 202 Compressible, three-dimensional ftow . 207 Incompressible, two-dimensional ftow . 212 Simple harmonie motion of an airfoil 215 Transient motion . 222 Evaluation of integrals 226 4.4 Representative numerical results 230 4.5 Transonic flow . 237 References for Chapter 4 267 Vlll Contents s. StaU lutter 270 5.1 Background . .... 270 5.2 Analytical formulation 271 5.3 Stability and work flow 273 5.4 Bending stall fluuer . 274 5.5 Nonlinear mechanics description 275 5.6 Torsional stall fluuer . 277 5.7 General comments 280 5.8 Computational stalled flow 283 References for Chapter 5 . 288 6. Aeroelastic problems of civil eugineering strudures . 290 6.1 Vortex shedding .......... 292 Introduction . 292 Aspects of response to vortex sheddmg 295 Empirical models of vortex-induced oscillation 298 Commentary on vortex excitation models. 308 6.2 Galloping. 314 Across-wind galloping 314 Wake galloping 320 6.3 Divergence . 323 6.4 Flutter and buffeting 327 Basic concepts 327 Three-dimensional flutter and buffeting . 335 Single-mode flutter and buffeting 343 Indicial Function Formulations 345 References for Chapter 6 348 7. Aeroelastic problems of rotorcraft 355 7.1 Blade dynamies ..... 356 Articulated, rigid bl ade motion 358 Elastic motion of hingeless blades . 368 7.2 Stall flutter . 380 7.3 Blade motionl body coupling . 385 Two bladed rotors. 407 References for Chapter 7 . 408 ix Contents 8. Aeroelasticity in turbomachines 411 8.1 Aeroelastic environment in turbomachines 412 8.2 The compressor performance map . 414 8.3 Blade mode shapes and materials 01 construction 417 8.4 Nonsteady potential flow in cascades. 419 8.5 Compressible flow . 425 8.6 Periodically stalled flow in turbomachines 428 8.7 Stall flutter in turbomachines 432 8.8 Choking flutter . 434 8.9 Aeroelastic eigenvalues 435 8.10 Recent trends 438 References for Chapter 8 . 441 9. Unsteady transonie aerodynamies and aeroelasticity 443 Summary 443 Nomenclature 444 9.1 lntroduction 445 9.2 Linear/ nonlinear behavior in unsteady transonic aerody­ namies . 446 Motivation and general background 446 NACA 64A006 airfoil 448 Mach number trends 453 Conclusions 457 9.3 Viable alternative solution procedures to finite difference methods 459 Hounjet .... 459 Cockey .... 460 A possible synthesis 461 9.4 Nonuniqueness 462 Early work 462 Recent work . 462 Studies of Williams and Salas 469 Aileron buzz . 470 9.5 Effective,