A Modern Course in Aeroelasticity

A Modern Course in Aeroelasticity
اسم المؤلف
5 أبريل 2015

A Modern Course in Aeroelasticity
Duke University,
Durham, NC, U.S.A.
Duke University,
Durham, NC, U.S.A.
NASA Langley Research Center,
Hampton, VA, U.S.A.
Princeton University,
Princeton, NJ, U.S.A.
NASA Langley Research Center,
Hampton, VA, U.S.A.
Duke University,
Durham, NC, U.S.A.
Washington University,
St. Louis, MO, U.S.A.
Johns Hopkins University,
Baltimore, MD, U.S.A.
National Institute for Standards and
Technology, Gaithersburg, MD, U.S.A.
Stevens Institute of Technology,
Hoboken, NJ, U.S.A.
Texas A&M University,
College Station, TX, U.S.A.
P r e f a c e 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
2.1 Typical Section Model of An Airfoil 5
Typical section model with control surface 10
Typical section model—nonlinear effects 16
2.2 One Dimensional Aeroelastic Model of Airfoils 18
Beam-rod representation of large aspect ratio wing 18
Eigenvalue and eigenfunction approach 22
Galerkin’s method 24
2.3 Rolling of a Straight Wing 26
Integral equation of equilibrium 26
Derivation of equation of equilibrium 27
Calculation of Cαα 28
Sketch of function S(y1, η) 28
Aerodynamic forces (including spanwise induction) 30
Aeroelastic equations of equilibrium and lumped
element solution method 32
Divergence 33
Reversal and rolling effectiveness 34
I nt e g r a l e q u a t i o n e i g e nva l u e p r o b l e m a n d t h e
e x p er i m e nt a l d e t e r m i n a t i o n o f i n fl u e n c e f u n c t i o n s 3 7
2 . 4 Two D i m e n s i o n a l A e r o el a s t i c M o d e l o f L i f t i n g S u r f a c e s 4 1
Two d imensional structures—integral representation 41
Two d imensional aero dynamic surfaces—integral
representation 42
Solution by matrix-lump ed element approach 43
2.5 O ther Physical Phenomena 44
F l u i d fl ow t h r o u g h a fl e x i b l e p i p e 4 4
(Low sp eed) fluid fl ow over a fl exible wall 47
2 . 6 S wep tw i n g D i ver g e n c e 4 7
References for Chapter 2 51
3 . DY N A M I C A E RO E L A S T I C I T Y ( D OW E L L ) 5 3
3 . 1 H a m i l t o n ’ s P r i n c i p l e 5 4
S i n g l e p a r t i c l e 5 4
M a ny p a r t i c l e s 5 6
C o nt i nu o u s b o d y 5 6
Potential energy 56
Nonp otential forces 59
3 . 2 L a g r a n g e ’ s E q u a t i o n s 6 0
E x a m p l e — ty p i c a l se c t i o n eq u a t i o n s o f m o t i o n 6 1
3.3 Dynamics of the Ty pical S ection M o d el of An Airfoil 64
S i nu s o i d a l m o t i o n 6 4
Per i o d i c m o t i o n 67
A r b i t r a r y mo t i o n 6 7
R a n d o m mo t i o n 7 3
F l u t t e r – a n i nt r o d u c t i o n t o d y n a m i c a e r o el a s t i c i n s t a b i l i ty 8 1
Quasi-steady, aero d ynamic theory 85
3.4 Aero d ynamic Forces 87
A e r o d y n a m i c t h e o r i e s avai l a b l e 91
G e n e r a l a p p r ox i m a t i o n s 9 5
‘ S t r i p t h e o r y ’ a p p r ox i m a t i o n 9 5
‘ Q u a s i s t e a d y ’ a p p r ox i m a t i o n 95
Slender b o d y or slender ( low asp ect ratio) wing
a p p r ox i m a t i o n 9 6
3 . 5 S o l u t i o n s t o t h e A e r o el a s t i c E q u a t i o n s o f M o t i o n 9 7
Time domain solutions 98
Fr e q u e n c y d o m a i n so l u t i o n s 1 0 0
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 3 165
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
Reduction to classical acoustics 177
Boundary conditions 178
Symmetry and anti-symmetry 180
4.2 Supersonic Flow 182
Two-dimensional flow 182
Simple harmonic motion of the airfoil 183
Discussion of inversion 185
Discussion of physical significance of the results 187
Gusts 189
Transient motion 190
Lift, due to airfoil motion 191
Lift, due to atmospheric gust 192
Three dimensional flow 195
4.3 Subsonic Flow 201
Derivation of the integral equation by transform methods
and solution by collocation 201
An alternative determination of the Kernel Function
using Green’s Theorem 204
Incompressible, three-dimensional flow 207
Compressible, three-dimensional flow 211
Incompressible, two-dimensional flow 215
Simple harmonic motion of an airfoil 218
Transient motion 224
Evaluation of integrals 229
4.4 Representative Numerical Results 232
4.5 Transonic Flow 238
References for Chapter 4 270
5.1 Background 275
5.2 Analytical formulation 276
5.3 Stability and aerodynamic work 278
5.4 Bending stall flutter 279
5.5 Nonlinear mechanics description 281
5.6 Torsional stall flutter 282
5.7 General comments 285
5.8 Reduced order models 288
Contents xi
5.9 Computational stalled flow 289
References for Chapter 5 294
6.1 Vortex-induced Oscillation 301
Vortex shedding 301
Modeling of vortex-induced oscillations 305
Coupled two-degree-of-freedom equations: wake oscillator
models 306
Single-degree-of- freedom model of vortex-induced response310
6.2 Galloping 314
Equation of motion of galloping bodies. The Glauert-Den
Hartog necessary condition for galloping instability 314
Description of galloping motion 320
Chaotic galloping of two elastically coupled square bars 321
Wake galloping : physical description and analysis 321
6.3 Torsional Divergence 327
6.4 Flutter and Buffeting in the Presence of Aeroelastic
Effects 328
Formulation and analytical solution of the twodimensional bridge flutter problem in smooth flow 330
Bridge section response to excitation by turbulent wind
in the presence of aeroelastic effects 334
6.5 Suspension-Span Bridges 336
Wind tunnel testing of suspended-span bridges 336
Torsional divergence analysis for a full bridge 338
Locked-in vortex-induced response 340
Flutter and buffeting of a full-span bridge 350
Reduction of bridge susceptibility to flutter 360
6.6 Tall Chimneys and Stacks, and Tall Buildings 361
Tall chimneys and stacks 361
Tall buildings 365
References for Chapter 6 367
7.1 Blade Dynamics 379
Articulated, rigid blade motion 379
Elastic motion of hingeless blades 390
7 . 2 S t a l l F l u t t e r 4 0 3
7 . 3 R o t o r – B o d y C o u p l i n g 4 0 9
7.4 Unsteady Aero d ynamics 433
D y n a m i c i n fl ow 4 3 4
Fr equency d omain 440
Finite-state wake mo delling 441
S u m m a r y 4 4 4
References for Chapter 7 444
8 . A E RO E L A S T I C I T Y I N T U R B O M AC H I N E S ( S I S T O ) 4 5 3
8 . 1 A e r o el a s t i c E nv i r o n m e nt i n Tu r b o m a ch i n e s 4 5 4
8.2 The Compressor Performance Map 455
8 . 3 Bl a d e M o d e S h a p es a n d M a t e r i a l s o f Co n s t r u c t i o n 4 6 0
8.4 Nonsteady Potential Flow i n Cascades 462
8.5 Compressible Flow 467
8 . 6 Pe r i o d i c a l l y S t a l l e d F l ow i n Tu r b o m a ch i n e s 4 7 1
8 . 7 S t a l l F l u t t e r i n Tu r b o m a ch i n e s 4 7 5
8 . 8 Ch o k i n g F l u t t e r 4 7 7
8.9 Aero elastic Eigenvalues 479
8.10 Recent Trends 481
References for Chapter 8 487
I N T E R AC T I O N ( D OW E L L A N D H A L L ) 4 9 1
9 . 1 T h e R a n g e O f P hy s i c a l M o d e l s 4 9 1
The classical mo d els 491
The d istinction b etween linear and nonlinear mo d els 494
C o m p u t a t i o n a l fl u i d d y n a m i c s m o d e l s 4 9 5
T h e co m p u t a t i o n a l ch a l l e n g e o f fl u i d st r u c t u r e i nt e r a c t i o n
mo deling 495
9.2 Time-Linearized Mo dels 496
Classical aero d ynamic theory 496
C l a s s i c a l hy d r o d y n a m i c st a b i l i ty t h e o r y 4 9 7
Pa r a l l e l sh e a r fl ow wi t h a n i nv i s c i d d y n a m i c p e r t u r b a t i o n 4 9 7
General time-linearized analysis 498
Some nu merical examples 500
9 . 3 N o n l i n e a r D y n a m i c a l M o d e l s 5 0 0
Harmonic b alance metho d 503
Contents xiii
System identification methods 503
Nonlinear reduced-order models 504
Reduced-order models 504
Constructing reduced order models 505
Linear and nonlinear fluid models 506
Eigenmode computational methodology 507
Proper orthogonal decomposition modes 508
Balanced modes 509
Synergy among the modal methods 509
Input/output models 509
Structural, aerodynamic, and aeroelastic modes 511
Representative results 512
The effects of spatial discretization and a finite
computational domain 512
The effects of mach number and steady angle of attack:
subsonic and transonic flows 516
The effects of viscosity 521
Nonlinear aeroelastic reduced-order models 522
9.4 Concluding Remarks 524
References for Chapter 9 529
Appendix: Singular-Value Decomposition, Proper Orthogonal
Decomposition, & Balanced Modes 538
10.1 Review of Structural Dynamics Experiments 541
10.2 Wind Tunnel Experiments 543
Sub-critical flutter testing 543
Approaching the flutter boundary 544
Safety devices 544
Research tests vs. clearance tests 544
Scaling laws 544
10.3 Flight Experiments 545
Approaching the flutter boundary 545
When is flight flutter testing required? 545
Excitation 545
Examples of recent flight flutter test programs 546
10.4 The Role of Experimentation and Theory in Design 546
References for Chapter 10 548
1 1 . N O N L I N E A R A E RO E L A S T I C I T Y ( D OW E L L ,
E DWA R D S A N D S T RG A N AC ) 551
11.1 Introduction 551
11.2 Generic Nonlinear Aeroelastic Behavior 552
11.3 Flight Experience with Nonlinear Aeroelastic Effects 554
Nonlinear aerodynamic effects 556
Freeplay 556
Geometric structural nonlinearities 557
11.4 Physical Sources of Nonlinearities 557
11.5 Efficient Computation of Unsteady Aerodynamic Forces:
Linear and Nonlinear 558
11.6 Correlations of Experiment/Theory and Theory/Theory 560
Aerodynamic forces 560
11.7 Flutter Boundaries in Transonic Flow 566
11.8 Limit Cycle Oscillations 573
Airfoils with stiffness nonlinearities 573
Nonlinear internal resonance behavior 575
Delta wings with geometrical plate nonlinearities 577
Very high aspect ratio wings with both structural and
aerodynamic nonlinearities 578
Nonlinear structural damping 581
Large shock motions and flow separation 581
Abrupt wing stall 594
Uncertainty due to nonlinearity 595
References for Chapter 11 598
12.1 Introduction 611
12.2 Linear System Theory 612
System interconnections 612
Controllability and observability 615
12.3 Aeroelasticity: Aerodynamic Feedback 617
Development of a typical section model 617
Aerodynamic model, 2D 619
Balanced model reduction 622
Combined aeroelastic model 623
Development of a delta wing model 627
Transducer effects 630
Contents xv
Aerodynamic model, 3D 633
Coupled system 634
12.4 Open-Loop Design Considerations 636
HSVs and the modal model 637
Optimization strategy 638
Optimization results 641
12.5 Control Law Design 642
Control of the typical section model 644
Control of the delta wing model 647
12.6 Parameter Varying Models 647
Linear matrix inequalities 648
LMI controller specifications 649
An LMI design for the typical section 652
12.7 Experimental Results 654
Typical section experiment 655
LPV system identification 656
Closed-loop results 658
Delta wing experiment 664
12.8 Closing Comments 667
References for Chapter 12 669
13.1 Linearized Analysis of Unsteady Flows 676
13.2 Analysis of Unsteady Flows 683
13.3 Harmonic Balance Method 688
13.4 Conclusions 699
References for Chapter 13 701
Appendices 705
Appendix A: A Primer For Structural Response To
Random Pressure Fluctuations
A.1 Introduction 705
A.2 Excitation-Response Relation For The Structure 705
A.3 Sharp Resonance or Low Damping Approximation 709
Nomenclature 710
References for Appendix A 710
A p p en d i x B: S o m e E x a m p l e P r o b l e m s 711
B.1 For Chapter 2 711
B.2 For Section 3.1 724
B.3 For Section 3.3 730
B.4 For Section 3.6 735
B.5 For Section 4.1 738
Index 743
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