Aeroelasticity

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AEROELASTICITY /

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Raymond L. Bisplinghoff. Holt Ashley and Robert L. Halfman

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AEROELASTICITY

AEROELASTICITY Raymond L. Bisplinghoff

Holt Ashley Robert L. Halfman

DOVER PUBLICATIONS, INC. MINEOLA, NEW YORK

Copyright Copyright ©1955 by Addison-Wesley Publishing Company, Inc. Copyright © renewed 1983 by Raymond L. Bisplinghoff , Holt Ashley and Robert L. Halfman. All rights reserved under Pan American and International Copyright Conventions.

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Bibliographical Note This Dover edition , first published in 1996, is a corrected republication of the work first published by Addison- Wesley Publishing Company, Cambridge, Mass., 1955. The authors have provided a number of corrections for the Dover edition.

Library of Congress Cataloging -in-Publication Data Bisplinghoff , Raymond L. Aeroelasticity / Raymond L. Bisplinghoff , Holt Ashley, Robert L. Halfman. cm. P Corrected republication of the work originally published: Cambridge, Mass. : Addison- Wesley, 1955. Includes bibliographical references and index . ISBN 0-486-69189-6 (pbk.) 1. Aeroelasticity. I. Ashley, Holt. II. Halfman, Robert L. III. Title. TL574.A37B5397 1996 629.132 ' 362 dc20 96-5412 CIP

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Manufactured in the United States of America Dover Publications, Inc. , 31 East 2nd Street, Mineola, N.Y. 11501

PREFACE The objective of the authors in writing Aeroelasticity has been to provide both a textbook for advanced engineering students and a reference book for practicing engineers. In selecting material for the book it was the authors’ conviction that not only the practical aspects of aeroelasticity should be treated, but also the aerodynamic and structural tools upon which these rest. Accordingly, the book divides roughly into two halves; the first deals with the tools and the second with applications of the tools to aeroelastic phenomena. The authors' convictions concerning a need for further treatment of the tools do not stem from a feeling that they are inadequately treated elsewhere but rather from the realization that they are not treated from the point of view of the aeroelastician . The first chapter emphasizes the role of aeroelasticity among the aeronautical sciences and its influence on modern design. Chapters 2, 3, and 4 are concerned with the deformation behavior of airplane structures under static and dynamic loads. These three chapters comprise the total treatment of the structural tools. The aerodynamic tools are treated in Chapters 5, 6, and 7. The reader will observe that although steady-state aerodynamics is discussed briefly, the primary emphasis is on unsteady phenomena. Chapter 8 brings together for the first time the aerodynamic and structural tools and treats the subject of static aeroelasticity. Problems of static aeroelasticity are characterized by the absence of the independent variable time, and they are introduced first because of their simplicity. Chapter 9 is concerned with flutter and Chapter 10 with dynamic response phenomena. Whereas the former entails essentially a harmonic dependence of the motion on time, the latter includes a class of problems in which the motion of the system may vary in a transient manner with time. Chapters 11 and 12 treat , respectively, the important subjects of aeroelastic model theory and model design and construction ; the final chapter is concerned with experimental techniques for studying aeroelastic phenomena. Although the space devoted to experimental methods is relatively small, it is not the authors’ intention to imply that experimental tools and techniques in aeroelasticity are of minor importance in the solution of practical problems. Indeed , aeroelastic phenomena encountered at the forefront of modern design often do not yield to analytical methods, and if solutions are to be obtained within a reasonable length of time the employment of experimental methods is absolutely necessary. The authors have endeavored to write each chapter by progressing from the easy to the hard. Thus the engineering instructor who seeks to use this book as an elementary text in aeroelasticity will find that his purpose is served by merely using the first parts of selected chapters. For example, the book may be used as an introductory text in aeroelasticity for senior or graduate students in aeronautical engineering by using Chapter 1 and the v

VI

PREFACE

first parts of Chapters 2, 3, 5, 8, and 9. The mathematical prerequisites for an understanding of these portions of the book are the mathematics courses included in the usual engineering curriculum, through differential equations. The latter course should have at least an introduction to the notions of partial derivatives and partial differential equations. An introductory laboratory course in experimental aeroelasticity can be based upon Chapters 11, 12, and 13. Advanced courses in aeroelasticity may be based upon the latter parts of Chapters 2, 3, 5, 8, and 9, as well as Chapters 4, 6, 7, and 10. In general, the mathematical prerequisites for an understanding of the complete book include a course in advanced calculus in addition to the courses mentioned above. The practicing engineer who uses the book for reference purposes will find that the authors have attempted to present applications of funda mentals instead of the compendium of standard tabular methods which may be in current favor. Although many numerical examples are included, it is unlikely that the practicing engineer will often find the particular problem that he is concerned with at the moment. However, it is hoped that the illustrative examples will always be of some value to the reader in perceiving how the fundamental tools may be applied to his case. The authors arranged the material content of the book and the outlines of each chapter in close cooperation. Then each author worked on certain chapters independently, with R. L. Bisplinghoff concentrating on the structural tools, H. Ashley on the aerodynamic tools, and R . L. Halfman on the experimental aspects. Finally, the applications to aeroelastic phenomena were prepared jointly and the entire manuscript was worked over by the. three authors to ensure continuity. Acknowledgement is due a great many people who aided in bringing the book to completion. Professor Eric Reissner’s counsel is gratefully acknowledged. A number of M.I.T. staff members and former students read portions of the manuscript and offered valuable advice and criticism. These include Professors Shatswell Ober, James Mar, Theodore Pian , Morton Finston, and Leon Trilling of the M.I.T. Department of Aeronautical Engineering ; Garabed Zartarian, Hua Lin, John McCarthy, Kenneth Foss, and Robert Staley of the Aeroelastic and Structures Re search Laboratory at M .I .T. ; Mr. M . J . Turner of the Boeing Airplane Co., Mr. H. C. Johnson of the Glenn L. Martin Company, Professor H. C. Martin of the University of Washington, and Professor K. Washizu of the University of Tokyo. The numerical examples were worked out by Mr. John Martuccelli and Mr . Yechiel Shulman of the Aeroelastic and Structures Research Laboratory . The authors express their sincere appreciation to all of these people, and to Miss Nancy Ladd for so ably performing the seemingly endless chore of typing and preparing the manuscript.

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Cambridge, Mass. February, 1955.

R.L.B. , H .A.

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R.L.H.

CONTENTS CHAPTER

1-1 1-2 1-3 1-4 CHAPTER

1. INTRODUCTION TO AEROELASTICITY Definitions Historical background Influence of aeroelastic phenomena on design Comparison of wing critical speeds

2. DEFORMATIONS

OF

1 1 3 7 13

AIRPLANE STRUCTURES UNDER STATIC

LOADS

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11 2-12 2-13 2-14 2-15

Introduction Elastic properties of structures Deformation due to several forces. Influence coefficients Properties of influence coefficients Strain energy in terms of influence coefficients Deformations under distributed forces. Influence functions Properties of influence functions The simplified elastic airplane Deformations of airplane wings Integration by weighting matrices Energy methods in deflection calculations Deformations of slender unswept wings Influence functions and coefficients of slender swept wings Deformations and influence coefficients of low aspect ratio wings Influence coefficients of complex built-up wings by the principle of minimum strain energy 2-16 Influence coefficients of complex built-up wings by the principle of

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minimum potential energy 2-17 Calculation of deformations of solid wings of variable thickness and complex built-up wings by the Rayleigh-Ritz method . . CHAPTER

3. DEFORMATIONS

17 17 21 23 23 26 28 30 33 38 47 49

51

57 60

OF AIRPLANE STRUCTURES UNDER DYNAMIC

67 67 67

LOADS

3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11

15 15

Introduction Differential equations of motion of a beam .

. . .

Integral equation of motion of a slender beam Dynamic equilibrium of slender rotating beams in bending

95 98 102 106 114 124 125 129

Dynamic equilibrium of slender beams in torsion . Dynamic equilibrium of restrained airplane wing . Dynamic equilibrium of the unrestrained elastic airplane

Energy methods Approximate methods of solution to practical problems Approximate solutions by the Rayleigh Ritz method . Approximate solutions by the lumped parameter method

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CHAPTER 4. APPROXIMATE

METHODS OF SHAPES AND FREQUENCIES

COMPUTING

NATURAL

MODE

132 132 4-1 Introduction . . 132 4-2 Natural modes and frequencies by energy methods integral the from frequencies shapes derived and 4 3 Natural mode 146 equation Vll

*«

*

CONTENTS

Vlll

4-4 Natural mode shapes and frequencies derived from the differential

. .159

equation

4-5 Solution of characteristic equations 4-6 . Natural modes and frequencies of complex airplane 4-7 Natural modes and frequencies of rotating beams

CHAPTER 5. AERODYNAMIC

TOOLS : TWO COMPRESSIBLE FLOW

5-1 5-2 5-3 5-4 5-5 5-6 5-7

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.

164

structures . 172 .

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184

AND THREE DIMENSIONAL IN

188 188 200 204 208 221

Fundamentals : the concept of small disturbances Properties of incompressible flow with and without circulation . Vortex flow Thin airfoils in steady motion Finite wings in steady motion Thin airfoils oscillating in incompressible flow 251 Arbitrary motion of thin airfoils in incompressible flow ; the gust problem 281

CHAPTER 6. AERODYNAMIC TOOLS: COMPRESSIBLE FLOW 6-1 Introduction 6-2 Wings and airfoils in steady subsonic flow ; the Prandtl-Glauert transformation 6-3 Airfoils and wings in steady supersonic flow 6-4 Oscillating airfoils in subsonic flow 6-5 Arbitrary small motions of airfoils in subsonic flow . . . . 6-6 Oscillating airfoils at supersonic speeds 6-7 Indicial airfoil motions in supersonic flow 6-8 Unsteady motion of airfoils at Mach number one

296 303 317 332 353 367 375

CHAPTER 7. WINGS AND BODIES IN THREE-DIMENSIONAL UNSTEADY FLOW 7-1 Introduction . . 7-2 Oscillating finite wings in incompressible flow . . . . 7-3 The influence of sweep 7-4 Wings of very low aspect ratio in unsteady motion . . . . 7-5 The influence of compressibility on oscillating wings of finite span 7-6 Unsteady motion of nonlifting bodies

380 380 381 394 400 405 414

CHAPTER 8. STATIC AEROELASTIC PHENOMENA . 8-1 Introduction . wing Twisting simple aileron of 8 2 two-dimensional with wings 8-3 Slender straight 8-4 Swept wings 8-5 Low aspect-ratio lifting surfaces of arbitrary planform and stiffness

421 421 421 427

294 294

474 516

527 CHAPTER 9. FLUTTER 527 9-1 Introduction. The nature of flutter 532 9-2 Flutter of a simple system with two degrees of freedom . 9-3 Exact treatment of the bending-torsion flutter of a uniform cantilever wing 9-4 Aeroelastic modes 9-5 Flutter analysis by assumed-mode methods 9-6 Inclusion of finite span effects in flutter calculations . 9-7 The effect of compressibility on flutter

.

.

.

545 551 555 590 595

CONTENTS

9-8 9-9 9-10 9-11

IX

Flutter of swept wings Wings of low aspect ratio Single-degree-of -freedom flutter Certain other interesting types of flutter

604 613 617 626

CHAPTER 10. DYNAMIC RESPONSE PHENOMENA 10-1 Introduction 10-2 Equations of disturbed motion of an elastic airplane . 10-3 Systems with prescribed time-dependent external forces . 10-4 Transient stresses during landing 10-5 Systems with external forces depending upon the motion . 10-6 Dynamic response to a discrete gust 10-7 Dynamic response to continuous atmospheric turbulence .

632 632 633

CHAPTER 11. AEROELASTIC MODEL THEORY 11-1 Introduction 11-2 Dimensional concepts 11-3 Equations of motion 11-4 Vibration model similarity laws 11-5 Similarity laws for systems under steady airloads . 11-6 Flutter model similarity laws 11-7 The unrestrained flutter model 11-8 The dynamic stability model

695

635 650 659 673 685 695 695

.

.

.

698 699 705 712 715

CHAPTER 12. MODEL DESIGN AND CONSTRUCTION 12-1 Introduction 12-2 Structural simulation 12-3 Elastic properties as functions of one variable . . . . . . . . 12-4 Elastic properties as functions of two variables 12 5 Shape simulation 12-6 Inertial simulation

717 717 718 724 735 741 745

CHAPTER 13. TESTING TECHNIQUES 13-1 Introduction 13-2 Measurement of structural flexibility 13-3 Measurement of natural frequencies and mode shapes 13-4 Steady-state aeroelastic testing 13-5 Dynamic aeroelastic testing full scale model scale 13-6 Dynamic aeroelastic testing

749 749 749 753 779 781 787

APPENDICES. MATHEMATICAL TOOLS A Matrices Integration by weighting numbers B

803 805 809 813

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C

Linear systems

REFERENCES

827

AUTHOR INDEX

851

SUBJECT INDEX

855

CHAPTER 1 INTRODUCTION TO AEROELASTICITY 1-1 Definitions. The term aeroelasticity has been applied by aeronautical engineers to an important class of problems in airplane design. It is often defined as a science which studies the mutual interaction between aerodynamic forces and elastic forces, and the influence of this interaction on airplane design . Aeroelastic problems would not exist if airplane structures were perfectly rigid . Modern airplane structures are very flexible , and this flexibility is fundamentally responsible for the various types of aeroelastic phenomena. Structural flexibility itself may not be objectionable ; however, aeroelastic phenomena arise when structural deformations induce additional aerodynamic forces. These additional aerodynamic forces may produce additional structural deformations which will induce still greater aerodynamic forces. Such interactions may tend to become smaller and smaller until a condition of stable equilibrium is reached , or they may tend to diverge and destroy the structure. The term aeroelasticity, however, is not completely descriptive, since many important aeroelastic phenomena involve inertial forces as well as aerodynamic and elastic forces. We shall apply a definition in which the term aeroelasticity includes phenomena involving interactions among inertial, aerodynamic, and elastic forces, and other phenomena involving interactions between aerodynamic and elastic forces. The former will be referred to as dynamic and the latter as static aeroelastic phenomena. Collar ( Ref . 1-1) has ingeniously classified problems in aeroelasticity by means of a triangle of forces. Referring to Fig. 1-1, the three types of forces, aerodynamic, elastic, and inertial, represented by the symbols A , E , and 7, respectively, are placed at the vertices of a triangle. Each aeroelastic phenomenon can be located on the diagram according to its relation to the three vertices. For example, dynamic aeroelastic phenomena such as flutter, F, lie within the triangle, since they involve all three types of forces and must be bonded to all three vertices. Static aeroelastic phenomena such as wing divergence, D , lie outside the triangle on the upper left side, since they involve only aerodynamic and elastic forces. Although it is difficult to define precise limits on the field of aeroelasticity, the classes of problems connected by solid lines to the vertices in Fig. 1-1 are usually accepted as the principal ones. Of course , other borderline fields can be placed on the diagram. For example, the fields of mechanical vibrations, V , and rigid-body aerodynamic stability, DS , are connected to the vertices by dotted lines . It is very likely that in certain cases the dynamic stability 1

Aeroelastic phenomena As Aerodynamic force E: Elastic force I: Inertial force

Related fields V: Mechanical vibrations DS: Dynamic stability

F: B: Z: L: D: C: R: DSA: SSA:

Flutter Buffeting Dynamic response Load distribution Divergence

Control effectiveness Control system reversal Aeroelastic effects on dynamic stability Aeroelastic effects on static stability

Fig . 1-1. The aeroelastic triangle of forces.

problem is influenced by airplane flexibility and it would therefore be where it would be moved within the triangle to correspond with regarded as a dynamic aeroelastic problem . It will be convenient to state concise definitions of each aeroelastic phenomenon which appears on the diagram in Fig . 1-1. Flutter , F . A dynamic instability occurring in an aircraft in flight , at a speed called the flutter speed , where the elasticity of the structure plays an essential part in the instability . Buffeting , B . Transient vibrations of aircraft structural components due to aerodynamic impulses produced by the wake behind wings, nacelles, fuselage pods, or other components of the airplane. Dynamic response , Z . Transient response of aircraft structural components produced by rapidly applied loads due to gusts, landing, gun reactions, abrupt control motions, moving shock waves, or other dynamic loads. Aeroelastic effects on stability ,