Solid Mechanics and Its Applications

Volume 217

Series editors

J.R. Barber, Ann Arbor, USAAnders Klarbring, Linköping, Sweden

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The fundamental questions arising in mechanics are: Why?, How?, and Howmuch? The aim of this series is to provide lucid accounts written by authoritativeresearchers giving vision and insight in answering these questions on the subject ofmechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus itincludes the foundation of mechanics; variational formulations; computationalmechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrationsof solids and structures; dynamical systems and chaos; the theories of elasticity,plasticity and viscoelasticity; composite materials; rods, beams, shells andmembranes; structural control and stability; soils, rocks and geomechanics;fracture; tribology; experimental mechanics; biomechanics and machine design.

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More information about this series at http://www.springer.com/series/6557

Earl H. Dowell

A Modern Coursein AeroelasticityFifth Revised and Enlarged Edition

123

Earl H. DowellMechanical Engineeringand Materials Science

Duke UniversityDurham, NCUSA

ISSN 0925-0042 ISSN 2214-7764 (electronic)ISBN 978-3-319-09452-6 ISBN 978-3-319-09453-3 (eBook)DOI 10.1007/978-3-319-09453-3

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© Springer International Publishing Switzerland 2015This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publisher’s location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

The authors would liketo pay tribute toRobert H. Scanlan, asuperb aeroelastician,an inspiring teacher,and a consummatementor and friend. Heis greatly missed.

Preface to the Fifth Edition

In this fifth edition, a new chapter is added, Chap. 14, Some Recent Advances inNonlinear Aeroelasticity: Fluid-Structure Interaction in the Twenty First Century,with a discussion of some of the most recent research results that have beenobtained in the last decade. Also a new author and distinguished colleague,Dr. Deman Tang, has joined us. And the opportunity has been taken to correct allthe typographical errors that we and our readers have found.

With this edition, the first author is making available upon request video/audiorecordings of his semester long lectures that cover Chaps. 1–4 as well as selectedlectures on current research topics. It is planned to continually update these video/audio lectures and these updates will also be made available to those who purchasethe new edition. Also available are lecture notes and additional homework problemsand their solutions augmenting those already included in the text.

Earl H. Dowell

vii

Preface to the Fourth Edition

In this edition, several new chapters have been added and others substantially revisedand edited. Chapter 6 on Aeroelasticity in Civil Engineering originally authored byRobert Scanlan has been substantially revised by his close colleague, Emil Simiu.Chapter 9 on Modeling of Fluid-Structure Interaction by Earl Dowell and KennethHall is entirely new and discusses modern methods for treating linear and nonlinearunsteady aerodynamics based upon computational fluid dynamics models and theirsolution. Chapter 11 by Earl Dowell, John Edwards and Thomas Strganac onNoninearity Aeroelasticity is also new and provides a review of recent results.Chapter 12 by Robert Clark and David Cox on Aeroelastic Control is also new andprovides an authoritative account of recent developments. Finally Chapter 13by Kenneth Hall onModern Analysis for Complex and Nonlinear Unsteady Flows inTurbomachinery is also new and provides an insightful and unique account of thisimportant topic. Many other chapters have been edited for greater clarity as well, andauthor and subject indices are also provided.

Dr. Deman Tang has provided invaluable contributions to the production of thetext, and all of the authors would like to acknowledge his efforts with greatappreciation.

Earl H. Dowell

ix

Preface to the Third Edition

The authors would like to thank all those readers of the first and second editions whohave written with comments and suggestion. In the third edition, the opportunity hasbeen taken to revise and update Chapters 1 through 9. Also three new chapters havebeen added, i.e., Chapter 10, Experimental Aeroelasticity, Chapter 11, NonlinearAeroelasticity; and Chapter 12, Aeroelastic Control. Chapter 10 is a brief intro-duction to a vast subject: Chapter 11 is an overview of a frontier of research; andChapter 12 is the first connected, authoritative account of the feedback control ofaeroelastic systems. Chapter 12 meets a significant need in the literature. The authorsof the first and second editions welcome two new authors, David Peters who hasprovided a valuable revision of Chapter 7 on rotorcraft, and Edward Crawley whohas provided Chapter 10 on aeroelastic control. It is a privilege and a pleasure tohave them as members of the team. The author of Chapter 10 would also like toacknowledge the great help he has received over the year from his distinguishedcolleague, Wilmer H. “Bill” Reed, III, in the study of experimental aeroelasticity.Mr. Reed kindly provided the figures for Chapter 10. The author of Chapter 12would like to acknowledge the significant scholarly contribution of Charrissa Linand Ken Kazarus in preparing the chapter on aeroelastic control. Finally the readersof the first and second editions will note that the authors and subject indices havebeen omitted from this edition. If any reader finds this an inconvenience, pleasecontact the editor and we will reconsider the matter for the next edition.

Earl H. Dowell

xi

Preface to the Second Edition

The authors would like to thank all those readers who have written with commentsand errata for the first edition. Many of these have been incorporated into thesecond edition. They would like to thank Professor Holt Ashley of Stanford Uni-versity who has been most helpful in identifying and correcting various errata.

Also the opportunity has been taken in the second edition to bring up-to-dateseveral of the chapters as well as add a chapter on unsteady transonic aerodynamicsand aeroelasticity. Chapters 2, 5, 6 and 8 have been substantially revised. These coverthe topics of Static Aeroelasticity, Stall Flutter, Aeroelastic Problems of Civil Engi-neering Structures and Aeroelasticity in Turbomachines, respectively. Chapter 9,Unsteady Transonic Aerodynamics and Aeroelasticity, is new and covers this rapidlydeveloping subject in more breadth and depth than the first edition. Again, theemphasis is on fundamental concepts rather than, for example, computer codedevelopment per se. Unfortunately due to the press of other commitments, it has notbeen possible to revise Chapter 7, Aeroelastic Problems of Rotorcraft. However, theShort Bibliography has been expanded for this subject as well as for others. It is hopedthat the readers of the first edition and also new readers will find the second editionworthy of their study.

Earl H. Dowell

xiii

Preface to the First Edition

A reader who achieves a substantial command of the material contained in this bookshould be able to read with understanding most of the literature in the field. Possibleexceptions may be certain special aspects of the subject such as the aeroelasticity ofplates and shells or the use of electronic feedback control to modify aeroelasticbehavior. The first author has considered the former topic in a separate volume. Thelatter topic is also deserving of a separate volume.

In the first portion of the book, the basic physical phenomena of divergence,control surface effectiveness, flutter and gust response of aeronautical vehicles aretreated. As an indication of the expanding scope of the field, representativeexamples are also drawn from the nonaeronautical literature. To aid the student whois encountering these phenomena for the first time, each is introduced in the contextof a simple physical model and then reconsidered systematically in more compli-cated models using more sophisticated mathematics.

Beyond the introductory portion of the book, there are several special features ofthe text. One is the treatment of unsteady aerodynamics. This crucial part ofaeroelasticity is usually the most difficult for the experienced practitioner as well asthe student. The discussion is developed from the fundamental theory underlyingnumerical lifting surface analysis. Not only the well-known results for subsonic andsupersonic flow are covered; but also some of the recent developments for transonicflow, which hold promise of bringing effective solution techniques to this importantregime.

Professor Sisto’s chapter on Stall Flutter is an authoritative account of thisimportant topic. A difficult and still incompletely understood phenomenon, stallflutter is discussed in terms of its fundamental aspects as well as its significance inapplications. The reader will find this chapter particularly helpful as an introductionto this complex subject.

Another special feature is a series of chapters on three areas of advancedapplication of the fundamentals of aeroelasticity. The first of these is a discussion ofAeroelastic Problems of Civil Engineering Structures by Professor Scanlan. Thenext is a discussion on Aeroelasticity of Helicopters and V/STOL aircraft byProfessor Curtiss. The final chapter in this series treats Aeroelasticity in

xv

Turbomachines and is by Professor Sisto. This series of chapters is unique in theaeroelasticity literature, and the first author feels particularly fortunate to have thecontributions of these eminent experts.

The emphasis in this book in on fundamentals because no single volume canhope to be comprehensive in terms of applications. However, the above threechapters should give the reader an appreciation for the relationship between theoryand practice. One of the continual fascinations of aeroelasticity is this close inter-play between fundamentals and applications. If one is to deal successfully withapplications, a solid grounding in the fundamentals is essential.

For the beginning student, a first course in aeroelasticity could cover Chapters 1-3and selected portions of 4. For a second course and the advanced student or researchworker, the remaining chapters would be appropriate. In the latter portions of thebook, more comprehensive literature citations are given to permit ready access to thecurrent literature.

The reader familiar with the standard texts by Scanlan and Rosenbaum, Fung,Bisplinghoff, Ashley and Halfman and Bisplinghoff and Ashley will appreciatereadily the debt the authors owe to them. Recent books by Petre1 and Forsching2

should also be mentioned though these are less accessible to an English speakingaudience. It is hoped the reader will find this volume a worthy successor.

Earl H. Dowell

1 Petre, A., Theory of Aeroelasticity. Vol. I Statics, Vol. II Dynamics. In Romanian PublishingHouse of the Academy of the Socialist Republic of Romania, Bucharest, 1966.2 Forsching, H.W., Fundamentals of Aeroelasticity. In German. Springer-Verlag, Berlin, 1974.

xvi Preface to the First Edition

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Static Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Typical Section Model of an Airfoil . . . . . . . . . . . . . . . . . . . 3

2.1.1 Typical Section Model with Control Surface . . . . . . . 82.1.2 Typical Section Model—Nonlinear Effects. . . . . . . . . 13

2.2 One Dimensional Aeroelastic Model of Airfoils . . . . . . . . . . . 152.2.1 Beam-Rod Representation of Large Aspect

Ratio Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Eigenvalue and Eigenfunction Approach . . . . . . . . . . 192.2.3 Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Rolling of a Straight Wing. . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Integral Equation of Equilibrium. . . . . . . . . . . . . . . . 242.3.2 Derivation of Equation of Equilibrium. . . . . . . . . . . . 252.3.3 Calculation of Cαα . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.4 Sketch of Function Sðy1; ηÞ . . . . . . . . . . . . . . . . . . . 272.3.5 Aerodynamic Forces (Including

Spanwise Induction) . . . . . . . . . . . . . . . . . . . . . . . . 282.3.6 Aeroelastic Equations of Equilibrium and Lumped

Element Solution Method . . . . . . . . . . . . . . . . . . . . 312.3.7 Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.8 Reversal and Rolling Effectiveness . . . . . . . . . . . . . . 342.3.9 Integral Equation Eigenvalue Problem

and the Experimental Determination of InfluenceFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Two Dimensional Aeroelastic Model of Lifting Surfaces . . . . . 392.4.1 Two Dimensional Structures—Integral

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.2 Two Dimensional Aerodynamic Surfaces—Integral

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4.3 Solution by Matrix-Lumped Element Approach . . . . . 42

xvii

2.5 Other Physical Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . 432.5.1 Fluid Flow Through a Flexible Pipe . . . . . . . . . . . . . 432.5.2 (Low Speed) Fluid Flow Over a Flexible Wall . . . . . . 46

2.6 Sweptwing Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Dynamic Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.2 Many Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.3 Continuous Body . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.4 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.5 Nonpotential Forces . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Lagrange’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 Example—Typical Section Equations of Motion . . . . . 61

3.3 Dynamics of the Typical Section Model of An Airfoil. . . . . . . 643.3.1 Sinusoidal Motion. . . . . . . . . . . . . . . . . . . . . . . . . . 653.3.2 Periodic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.3 Arbitrary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 673.3.4 Random Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.5 Flutter—An Introduction to Dynamic Aeroelastic

Instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3.6 Quasi-Steady, Aerodynamic Theory . . . . . . . . . . . . . 85

3.4 Aerodynamic Forces for Airfoils—An Introductionand Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4.1 Aerodynamic Theories Available . . . . . . . . . . . . . . . 913.4.2 General Approximations . . . . . . . . . . . . . . . . . . . . . 943.4.3 Slender Body or Slender (Low Aspect Ratio)

Wing Approximation. . . . . . . . . . . . . . . . . . . . . . . . 963.5 Solutions to the Aeroelastic Equations of Motion . . . . . . . . . . 97

3.5.1 Time Domain Solutions . . . . . . . . . . . . . . . . . . . . . . 983.5.2 Frequency Domain Solutions . . . . . . . . . . . . . . . . . . 99

3.6 Representative Results and Computational Considerations . . . . 1023.6.1 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.6.2 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . 1043.6.3 Flutter and Gust Response Classification Including

Parameter Trends . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.6.4 Gust Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.7 Generalized Equations of Motion for Complex Structures . . . . 1243.7.1 Lagrange’s Equations and Modal Methods

(Rayleigh-Ritz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.7.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.7.3 Strain (Potential Elastic) Energy . . . . . . . . . . . . . . . . 127

xviii Contents

3.7.4 Natural Frequencies and Modes-Eigenvaluesand Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.7.5 Evaluation of Generalized Aerodynamic Forces . . . . . 1323.7.6 Equations of Motion and Solution Methods . . . . . . . . 1343.7.7 Integral Equations of Equilibrium . . . . . . . . . . . . . . . 1353.7.8 Natural Frequencies and Modes . . . . . . . . . . . . . . . . 1373.7.9 Forced Motion Including Aerodynamic Forces . . . . . . 140

3.8 Other Fluid-Structural Interaction Phenomena . . . . . . . . . . . . . 1513.8.1 Fluid Flow Through a Flexible Pipe:

“Firehose” Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.8.2 (High Speed) Fluid Flow Over a Flexible Wall—

A Simple Prototype for Plate or Panel Flutter . . . . . . . 154References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4 Nonsteady Aerodynamics of Lifting and Non-lifting Surfaces . . . . 1614.1 Basic Fluid Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . 163

4.1.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . 1634.1.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . 1644.1.3 Irrotational Flow, Kelvin’s Theorem and Bernoulli’s

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.1.4 Derivation of a Single Equation for Velocity

Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.1.5 Small Perturbation Theory . . . . . . . . . . . . . . . . . . . . 170

4.2 Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.2.1 Two-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . 1774.2.2 Simple Harmonic Motion of the Airfoil . . . . . . . . . . . 1774.2.3 Discussion of Inversion . . . . . . . . . . . . . . . . . . . . . . 1804.2.4 Discussion of Physical Significance of the Results . . . 1834.2.5 Gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.2.6 Transient Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.2.7 Lift, Due to Airfoil Motion . . . . . . . . . . . . . . . . . . . 1864.2.8 Lift, Due to Atmospheric Gust . . . . . . . . . . . . . . . . . 1874.2.9 Three Dimensional Flow . . . . . . . . . . . . . . . . . . . . . 191

4.3 Subsonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.3.1 Derivation of the Integral Equation by Transform

Methods and Solution by Collocation . . . . . . . . . . . . 1974.3.2 An Alternative Determination of the Kernel Function

Using Green’s Theorem. . . . . . . . . . . . . . . . . . . . . . 2004.3.3 Incompressible, Three-Dimensional Flow . . . . . . . . . . 2024.3.4 Compressible, Three-Dimensional Flow . . . . . . . . . . . 2064.3.5 Incompressible, Two-Dimensional Flow. . . . . . . . . . . 211

4.4 Representative Numerical Results . . . . . . . . . . . . . . . . . . . . . 230

Contents xix

4.5 Transonic Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5 Stall Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2655.2 Analytical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.3 Stability and Aerodynamic Work . . . . . . . . . . . . . . . . . . . . . 2685.4 Bending Stall Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695.5 Nonlinear Mechanics Description . . . . . . . . . . . . . . . . . . . . . 2715.6 Torsional Stall Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.7 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.8 Reduced Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.9 Computational Stalled Flow . . . . . . . . . . . . . . . . . . . . . . . . . 278References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

6 Aeroelasticity in Civil Engineering. . . . . . . . . . . . . . . . . . . . . . . . 2856.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

6.1.1 Vortex-Induced Oscillation. . . . . . . . . . . . . . . . . . . . 2876.1.2 Galloping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2986.1.3 Torsional Divergence . . . . . . . . . . . . . . . . . . . . . . . 3086.1.4 Flutter and Buffeting in the Presence

of Aeroelastic Effects . . . . . . . . . . . . . . . . . . . . . . . 3096.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

6.2.1 Suspension-Span Bridges . . . . . . . . . . . . . . . . . . . . . 3166.2.2 Tall Chimneys and Stacks, and Tall Buildings . . . . . . 335

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

7 Aeroelastic Response of Rotorcraft . . . . . . . . . . . . . . . . . . . . . . . 3497.1 Blade Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

7.1.1 Articulated, Rigid Blade Motion . . . . . . . . . . . . . . . . 3517.1.2 Elastic Motion of Hingeless Blades . . . . . . . . . . . . . . 360

7.2 Stall Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3717.3 Rotor-Body Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3757.4 Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

7.4.1 Dynamic Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 3947.4.2 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . 4007.4.3 Finite-State Wake Modelling . . . . . . . . . . . . . . . . . . 401

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

8 Aeroelasticity in Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . 4098.1 Aeroelastic Environment in Turbomachines . . . . . . . . . . . . . . 4108.2 The Compressor Performance Map . . . . . . . . . . . . . . . . . . . . 411

xx Contents

8.3 Blade Mode Shapes and Materials of Construction . . . . . . . . . 4158.4 Nonsteady Potential Flow in Cascades. . . . . . . . . . . . . . . . . . 4168.5 Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.6 Periodically Stalled Flow in Turbomachines . . . . . . . . . . . . . . 4258.7 Stall Flutter in Turbomachines . . . . . . . . . . . . . . . . . . . . . . . 4288.8 Choking Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4308.9 Aeroelastic Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4318.10 Recent Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

9 Modeling of Fluid-Structure Interaction. . . . . . . . . . . . . . . . . . . . 4399.1 The Range of Physical Models . . . . . . . . . . . . . . . . . . . . . . 439

9.1.1 The Classical Models . . . . . . . . . . . . . . . . . . . . . . . 4399.1.2 The Distinction Between Linear and Nonlinear

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.1.3 Computational Fluid Dynamics Models . . . . . . . . . . 4439.1.4 The Computational Challenge of Fluid Structure

Interaction Modeling . . . . . . . . . . . . . . . . . . . . . . . . 4439.2 Time-Linearized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

9.2.1 Classical Aerodynamic Theory . . . . . . . . . . . . . . . . . 4449.2.2 Classical Hydrodynamic Stability Theory. . . . . . . . . . 4449.2.3 Parallel Shear Flow with an Inviscid Dynamic

Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4459.2.4 General Time-Linearized Analysis. . . . . . . . . . . . . . . 4469.2.5 Some Numerical Examples . . . . . . . . . . . . . . . . . . . 447

9.3 Nonlinear Dynamical Models . . . . . . . . . . . . . . . . . . . . . . . . 4479.3.1 Harmonic Balance Method. . . . . . . . . . . . . . . . . . . . 4499.3.2 System Identification Methods . . . . . . . . . . . . . . . . . 4509.3.3 Nonlinear Reduced-Order Models . . . . . . . . . . . . . . . 4519.3.4 Reduced-Order Models . . . . . . . . . . . . . . . . . . . . . . 4519.3.5 Constructing Reduced Order Models . . . . . . . . . . . . . 4529.3.6 Linear and Nonlinear Fluid Models . . . . . . . . . . . . . . 4539.3.7 Eigenmode Computational Methodology . . . . . . . . . . 4539.3.8 Proper Orthogonal Decomposition Modes . . . . . . . . . 4549.3.9 Balanced Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4559.3.10 Synergy Among the Modal Methods . . . . . . . . . . . . . 4559.3.11 Input/Output Models . . . . . . . . . . . . . . . . . . . . . . . . 4559.3.12 Structural, Aerodynamic, and Aeroelastic Modes . . . . 4569.3.13 Representative Results . . . . . . . . . . . . . . . . . . . . . . . 458

9.4 Concluding Remarks and Directions for Future Research . . . . . 470References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

Contents xxi

10 Experimental Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47910.1 Review of Structural Dynamics Experiments . . . . . . . . . . . . . 47910.2 Wind Tunnel Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 480

10.2.1 Sub-critical Flutter Testing . . . . . . . . . . . . . . . . . . . . 48110.2.2 Approaching the Flutter Boundary . . . . . . . . . . . . . . 48110.2.3 Safety Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48110.2.4 Research Tests Versus Clearance Tests . . . . . . . . . . . 48210.2.5 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

10.3 Flight Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48210.3.1 Approaching the Flutter Boundary . . . . . . . . . . . . . . 48210.3.2 When Is Flight Flutter Testing Required?. . . . . . . . . . 48310.3.3 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48310.3.4 Examples of Recent Flight Flutter Test Programs . . . . 483

10.4 The Role of Experimentation and Theory in Design . . . . . . . . 483References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

11 Nonlinear Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48711.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48711.2 Generic Nonlinear Aeroelastic Behavior. . . . . . . . . . . . . . . . . 48811.3 Flight Experience with Nonlinear Aeroelastic Effects. . . . . . . . 490

11.3.1 Nonlinear Aerodynamic Effects . . . . . . . . . . . . . . . . 49111.3.2 Freeplay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49111.3.3 Geometric Structural Nonlinearities . . . . . . . . . . . . . . 492

11.4 Physical Sources of Nonlinearities. . . . . . . . . . . . . . . . . . . . . 49211.5 Efficient Computation of Unsteady Aerodynamic Forces:

Linear and Nonlinear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49311.6 Correlations of Experiment/Theory and Theory/Theory . . . . . . 495

11.6.1 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . 49511.7 Flutter Boundaries in Transonic Flow . . . . . . . . . . . . . . . . . . 499

11.7.1 AGARD 445.6 Wing . . . . . . . . . . . . . . . . . . . . . . . 49911.7.2 HSCT Rigid and Flexible Semispan Models. . . . . . . . 50011.7.3 Benchmark Active Control Technology Model . . . . . . 50111.7.4 Isogai Case A Model. . . . . . . . . . . . . . . . . . . . . . . . 503

11.8 Limit Cycle Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 50511.8.1 Airfoils with Stiffness Nonlinearities . . . . . . . . . . . . . 50511.8.2 Nonlinear Internal Resonance Behavior . . . . . . . . . . . 50611.8.3 Delta Wings with Geometrical Plate Nonlinearities . . . 50811.8.4 Very High Aspect Ratio Wings with Both Structural

and Aerodynamic Nonlinearities . . . . . . . . . . . . . . . . 50911.8.5 Nonlinear Structural Damping . . . . . . . . . . . . . . . . . 51111.8.6 Large Shock Motions and Flow Separation . . . . . . . . 51111.8.7 Abrupt Wing Stall. . . . . . . . . . . . . . . . . . . . . . . . . . 52111.8.8 Uncertainty due to Nonlinearity . . . . . . . . . . . . . . . . 522

xxii Contents

11.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

12 Aeroelastic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53112.2 Linear System Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

12.2.1 System Interconnections . . . . . . . . . . . . . . . . . . . . . 53212.2.2 Controllability and Observability. . . . . . . . . . . . . . . . 535

12.3 Aeroelasticity: Aerodynamic Feedback . . . . . . . . . . . . . . . . . 53612.3.1 Development of a Typical Section Model . . . . . . . . . 53712.3.2 Aerodynamic Model, 2D . . . . . . . . . . . . . . . . . . . . . 53912.3.3 Balanced Model Reduction . . . . . . . . . . . . . . . . . . . 54112.3.4 Combined Aeroelastic Model . . . . . . . . . . . . . . . . . . 54312.3.5 Development of a Delta Wing Model . . . . . . . . . . . . 54612.3.6 Transducer Effects . . . . . . . . . . . . . . . . . . . . . . . . . 54812.3.7 Aerodynamic Model, 3D . . . . . . . . . . . . . . . . . . . . . 55112.3.8 Coupled System . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

12.4 Open-Loop Design Considerations . . . . . . . . . . . . . . . . . . . . 55412.4.1 HSVs and the Modal Model. . . . . . . . . . . . . . . . . . . 55512.4.2 Optimization Strategy . . . . . . . . . . . . . . . . . . . . . . . 55612.4.3 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . 559

12.5 Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56012.5.1 Control of the Typical Section Model . . . . . . . . . . . . 56112.5.2 Control of the Delta Wing Model . . . . . . . . . . . . . . . 563

12.6 Parameter Varying Models . . . . . . . . . . . . . . . . . . . . . . . . . . 56412.6.1 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . 56512.6.2 LMI Controller Specifications. . . . . . . . . . . . . . . . . . 56612.6.3 An LMI Design for the Typical Section. . . . . . . . . . . 569

12.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57112.7.1 Typical Section Experiment . . . . . . . . . . . . . . . . . . . 57112.7.2 LPV System Identification . . . . . . . . . . . . . . . . . . . . 57212.7.3 Closed-Loop Results . . . . . . . . . . . . . . . . . . . . . . . . 57412.7.4 Delta Wing Experiment . . . . . . . . . . . . . . . . . . . . . . 575

12.8 Closing Comments on Aeroelastic Control . . . . . . . . . . . . . . . 581References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

13 Modern Analysis for Complex and Nonlinear UnsteadyFlows in Turbomachinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58513.1 Linearized Analysis of Unsteady Flows . . . . . . . . . . . . . . . . . 58613.2 Analysis of Unsteady Flows in Multistage Machines . . . . . . . . 59213.3 The Harmonic Balance Method for Nonlinear Unsteady

Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59613.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

Contents xxiii

14 Some Recent Advances in Nonlinear Aeroelasticity. . . . . . . . . . . . 60914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61014.2 Motivation and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61014.3 Current Examples of Recent Advances . . . . . . . . . . . . . . . . . 612

14.3.1 Transonic and Subsonic Panel Flutter . . . . . . . . . . . . 61214.3.2 Freeplay Induced Flutter and Limit Cycle

Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61514.3.3 Reduced Order Modeling of Unsteady

Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61914.4 Transonic Flutter and LCO of Lifting Surfaces . . . . . . . . . . . . 622

14.4.1 Generic Nonlinear Aeroelastic Behavior. . . . . . . . . . . 62214.4.2 Flight Experience with Nonlinear Aeroelastic

Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62414.4.3 Physical Sources of Nonlinearities. . . . . . . . . . . . . . . 62614.4.4 Efficient and Accurate Computation of Unsteady

Aerodynamic Forces: Linear and Nonlinear . . . . . . . . 62714.4.5 Experimental/Theoretical Correlations . . . . . . . . . . . . 627

14.5 Aerodynamic LCO: Buffet, AWS and NSV . . . . . . . . . . . . . . 63914.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Appendix A: A Primer for Structural Response to RandomPressure Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

Appendix B: Some Example Problems . . . . . . . . . . . . . . . . . . . . . . . . 659

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

xxiv Contents

Contributors

Robert Clark Mechanical Engineering and Materials Science, Duke University,Durham, NC, USA

David Cox Guidance and Control Branch, NASA Langley Research Center,Hampton, VA, USA

H.C. Curtiss Jr. Mechanical and Aerospace Engineering, Princeton University,Princeton, NJ, USA

Earl H. Dowell Mechanical Engineering and Materials Science, Duke University,Durham, NC, USA

John W. Edwards Aeroelasticity Branch, NASA Langley Research Center,Hampton, VA, USA

Kenneth C. Hall Mechanical Engineering and Materials Science, Duke Univer-sity, Durham, NC, USA

David A. Peters Mechanical Engineering, Washington University, St. Louis, MO,USA

Robert Scanlan Civil Engineering, Johns Hopkins University, Balimore, MD,USA

Emil Simiu National Institute for Standards and Technology, Gaithersburg, MD,USA

Fernando Sisto Mechanical Engineering, Stevens Institute of Technology,Hoboken, NJ, USA

Thomas W. Strganac Aerospace Engineering, Texas A&M University, CollegeStation, TX, USA

Deman Tang Research Scientist of Mechanical Engineering and MaterialsScience, Duke University, Durham, NC, USA

xxv

Short Bibliography

Books1. Bolotin VV (1963) Nonconservative Problems of the Elastic Theory of Stability. Pergamon

Press2. Bisplinghoff RL, Ashley H, Halfman RL (1955) Aeroelasticity. Addison-Wesley Publishing

Company, Cambridge, Mass., (BAH)3. Bisplinghoff RL, Ashley H (1962) Principles of Aeroelasticity. John Wiley and Sons Inc., NewYork, Also available in Dover Edition. (BA)

4. Fung YC (1955) An Introduction to the Theory of Aeroelasticity. John Wiley and Sons Inc.,New York, Also available in Dover Edition

5. Scanlan RH, Rosenbaum R (1951) Introduction to the Study of Aircraft Vibration and Flutter.The Macmillan Company, New York. Also available in Dover Edition

6. AGARD Manual on Aeroelasticity, Vols. I–VII, Beginning 1959 with continual updating.(AGARD)

7. Ashley H, Dugundji J, Rainey AG (1969) Notebook for Aeroelasticity. AIAA ProfessionalSeminar Series

8. Dowell EH (1975) Aeroelasticity of Plates and Shells. Noordhoff International Publishing,Leyden

9. Simiu E, Scanlan RH (1996) Wind Effects on Structures. John Wiley and Sons10. Johnson W, Helicopter Theory. Princeton University Press11. Dowell EH, Ilgamov M (1988) Studies in Nonlinear Aeroelasticity. Springer-Verlag12. Paidoussis MP (1988) Fluid-Structure Interactions: Slender Structures and Axial Flow, Vol. 1.

Academic Press

In parentheses, abbreviations for the above books are indicated which are used in the text.

Survey articles

1. Garrick IE (1976) Aeroelasticity-Frontiers and Beyond, 13th Von Karman Lecture. J. ofAircraft 13(9):641–657

2. Several Authors (1976) Unsteady Aerodynamics. Contribution of the Structures and MaterialsPanel to the Fluid Dynamics Panel Round Table Discussion on Unsteady Aerodynamics.Goettingen, AGARD Report R-645. March 1976

xxvii

3. Rodden WP (1976) A Comparison of Methods Used in Interfering Lifting Surface Theory.AGARD Report R-643. March 1976

4. Ashley H (1970) Aeroelasticity. Applied Mechanics Reviews. February 19705. Abramson HN (1969) Hydroelasticity: A Review of Hydrofoil Flutter. Applied Mechanics

Reviews. February 19696. Many Authors (1969) Aeroelastic Effects From a Flight Mechanics Standpoint. AGARD.

Conference Proceedings No. 467. Landhal MT, Stark VJE (1968) Numerical Lifting Surface Theory-Problems and Progress.

AIAA 6(11):2049–2060, November 19688. Many Authors (1967) Symposium on Fluid-Solid Interactions. ASME Annual Winter Meeting.November 1967

9. Kaza KRV (1988) Development of Aeroelastic Analysis Methods for Turborotors andPropfans-Including Mistuning. In: Lewis Structure Technology. Vol. 1. Proceedings, NASALewis Research Center

10. Ericsson LE, Reading JP (1988) Fluid Mechanics of Dynamic Stall. Part I, Unsteady FlowConcepts, and Part II, Prediction of Full Scale Characteristics. J. Fluids and Structures 2(1):1–33, (2):113–143

11. Mabey DG (1998) Some Aspects of Aircraft Dynamic Loads Due to Flow Separation.AGARD-R-750. February 1998

12. Yates EC. Jr., Whitlow W. Jr. (1987) Development of Computational Methods for UnsteadyAerodynamics at the NASA Langley Research Center. In: AGARD-R-749. Future Researchon Transonic Unsteady Aerodynamics and its Aeroelastic Applications. August 1987

13. Gad-el-Hak M (1987) Unsteady Separation on Lifting Surfaces. Applied Mechanics Reviews40(4):441–453

14. Hajela P (ed) (1987) Recent Trends in Aeroelasticity, Structures and Structural Dynamics.University of Florida Press, Gainesville

15. Jameson A (1983) The Evolution of Computational Methods in Aerodynamics. J. AppliedMechanics 50(4):1052–1070

16. Seebass R (1984) Advances in the Understanding and Computation of Unsteady TransonicFlows. In: Krothapalli A, Smith C (eds) Recent Advances on Aerodynamics, Springer-Verlag

17. McCroskey WJ (1982) Unsteady Airfoils. In: Annual Reviews of Fluid Mechanics. Vol. 14,pp. 285–311

18. Tijdeman H, Seebass R (1980) Transonic Flow Past Oscillating Airfoils. In: Annual Reviewsof Fluid Mechanics. Vol. 12, pp. 181–222

19. Ormiston R, Warmbrodt W, Hodges D et al. (1988) Survey of Army/NASA RotocraftAeroelastic Stability Research. NASA TM 101026 and USAASCOM TR 88-A-005

20. Dowell EH, Hall KC (2001) Modeling of Fluid-Structure Interaction. In: Annual Reviews ofFluid Mechanics. Vol. 33, pp. 445–490

21. Eastep, Franklin E (ed) (2003) Flight Vehicle Aeroelasticity. A series of invited articles byseveral authors in the J. of Aircraft 40(5):809–874

Journals

AHS JournalAIAA JournalASCE Transactions, Engineering Mechanics DivisionASME Transaction, Journal of Applied MechanicsInternational Journal of Solids and StructuresJournal of AircraftJournal of Fluids and StructuresJournal of Sound and Vibration

xxviii Short Bibliography

Other journals will have aeroelasticity articles, of course, but these are amongthose with the most consistent coverage.

The impact of aeroelasticity on design is not discussed in any detail in this book.For insight into this important area the reader may consult the following volumesprepared by the National Aeronautics and Space Administration in its series onSPACE VEHICLE DESIGN CRITERIA. Although these documents focus onspace vehicle application, much of the material is relevant to aircraft as well. Thedepth and breadth of coverage varies considerably from one volume to the next,but each contains at least a brief State-of-the-Art review of its topics as well as adiscussion of Recommended Design Practices. Further some important topics areincluded which have not been treated at all in the present book. These include, asalready mentioned in the Preface.

Aeroelasticity of plates and shells (panel flutter) (NASA SP-8004) andAeroelastic effects on control systems dynamics (NASA SP-8016, NASA SP-8036 NASA SP-8079) as well as Structural response to timedependent separatedfluid flows (buffeting) (NASA SP-8001) Fluid motions inside elastic containers(fuel sloshing) (NASA SP-8009, NASA SP- 8031) and Coupled structural—propulsion instability (POGO) (NASA SP-8055)

It was intended to revise these volumes periodically to keep them up-to-date.Unfortunately this has not yet been done.

1. NASA SP-8001 1970Buffeting During Atmospheric Ascent

2. NASA SP-8002 1964Flight Loads Measurements During Launch and Exit

3. NASA SP-8003 1964Flutter, Buzz and Divergence

4. NASA SP-8004 1972Panel Flutter

5. NASA SP-8006 1965Local Steady Aerodynamic Loads During Launch and Exit

6. NASA SP-8008 1965Prelaunch Ground Wind Loads

7. NASA SP-8012 1968Natural Vibration Wind Analysis

8. NASA SP-8016 1969Effect of Structural Flexibility on Spacecraft Control System

9. NASA SP-8009 1968Propellant Slosh Loads

10. NASA SP-8031 1969Slosh Suppression

11. NASA SP-8035 1970Wind Loads During Ascent

Short Bibliography xxix

12. NASA SP-8036 1970Effect of Structural Flexibility on Launch Vehicle Control System

13. NASA SP-8050 1970Structural Vibration Prediction

14. NASA SP-8055 1970Prevention of Coupled Structure-Propulsion Instability (POGO)

15. NASA SP-8079 1971Structural Interaction with Control Systems

xxx Short Bibliography