Preface
Chapter 1. Introduction to Aeroelasticity ....................... 1
1.1 The Scope of Aeroelasticity ................................ 3
1.2 The Terminology in Aeroelastic Analyses .................... 6
1.3 The Role of Aeroelasticity in Aerospace Vehicle Design ..... 9
1.4 Force and Coordinate Systems .............................. 14
Chapter 2. A Brief History of Aeroelasticity ................... 21
2.1 Early Examples of Aeroelastic Phenemena ................... 21
2.2 Aeroelasticity in Aeronautics ............................. 30
2.3 Insights Gained During World War I ........................ 32
2.4 Developments After World War I ............................ 36
2.5 Some Accident Statistics .................................. 36
2.6 Developments Since World War II ........................... 45
2.7 Aeroelasticity in Stability and Control ................... 46
2.8 Development of Unsteady Aerodynamic Theory ................ 47
2.9 Early Experimental Techniques ............................. 49
Chapter 3. Aerodynamics: Steady and Unsteady Theory ............ 59
3.1 Aerodynamic Characteristics of Airfoils ................... 60
3.2 The Unsteady Two-Dimensional Airfoil ...................... 66
3.3 Estimation of Dynamic Stability Derivatives ............... 92
3.4 Unsteady Potential Theory ................................. 97
3.5 Approximate Analytical Solutions for Wings and Bodies .... 112
3.6 Numerical Solutions for Lifting Surfaces in Steady Flow .. 127
3.7 Numerical Solutions for Lifting Surfaces in
Oscillatory Motion ....................................... 138
3.8 Numerical Solutions for Interfering Lifting Surfaces
and Bodies ............................................... 153
3.9 Aerodynamic Influence Coefficients ....................... 164
3.10 Computational Fluid Dynamics ............................. 168
Chapter 4. Structures: Deflection Theory ...................... 177
4.1 Elementary Bending and Torsion Theory .................... 177
4.2 Energy Theorems .......................................... 181
4.3 Shear Center and Torsional Stiffness of a Torque Box ..... 184
4.4 Deflections of an Elastic Axis with Variable Stiffness ... 191
4.5 Interpolation of Flexibility Influence Coefficients ...... 198
4.6 Thin Plate Theory and Surface Splines .................... 208
4.7 Finite Element Analysis .................................. 229
4.8 The Direct Stiffness Method .............................. 230
4.9 A Stiffness Matrix from Assumed Deflections .............. 243
4.10 Reciprocity of Flexibility and Stiffness ................. 246
4.11 A Stiffness Matrix from Assumed Stresses ................. 248
Chapter 5. Vibrations: Frequencies and Modes .................. 253
5.1 A Single Degree-of-Freedom System ........................ 253
5.2 Structural Damping ....................................... 260
5.3 A Two Degree-of-Freedom System ........................... 264
5.4 The Vibrating String ..................................... 268
5.5 Torsional Vibrations of a Rod ............................ 275
5.6 Bending Vibrations of a Beam ............................. 279
5.7 Vibrations of an Elastic Axis ............................ 283
5.8 The Coupled Mass Matrix .................................. 284
5.9 Matrix Formulation of the Vibration Problem .............. 293
5.10 Orthogonality of Calculated and Measured Vibration
Modes .................................................... 296
5.11 A Method for Deriving Flexibilty Matrices from Ground
Vibration Tests .......................................... 301
Chapter 6. Quasistatic Aeroelasticity: Maneuvering Flight ..... 311
6.1 The Two-Dimensional Airfoil .............................. 311
6.2 A Rigid Swept Wing on Flexible Supports .................. 319
6.3 The Slender Straight Wing ................................ 324
6.4 Influence Coefficient Formulation for General Planforms .. 331
6.5 Stiffness Matrix Formulation for General Configurations .. 344
6.6 Divergence of a Restrained Vehicle ....................... 349
6.7 Modal Formulation for Static Aeroelasticity .............. 350
6.8 Equations of Motion Including the Effects of the
Dynamic Structure ........................................ 351
6.9 Speed Derivatives and Aeroelastic Effects on Drag ........ 358
6.10 Dihedral Effect of a Flexible Wing ....................... 360
6.11 Corrections to Measurements on "Rigid" Wind Tunnel
Models ................................................... 364
Chapter 7. Flutter: The Dynamic Aeroelastic Instability ....... 367
7.1 Quasistatic Flutter Analysis ............................. 368
7.2 First Order Unsteady Flutter Analysis .................... 373
7.3 The American or K-Method of Flutter Analysis ............. 378
7.4 The British or PK-Method of Flutter Analysis ............. 394
7.5 The Generalized Aeroelastic Analysis Method (GAAM) ....... 400
7.6 The Similarity Parameters of Flutter ..................... 404
1.1 Control Surface Flutter .................................. 409
7.7 Panel Flutter ............................................ 411
7.8 Propeller/Nacelle Whirl Flutter .......................... 416
7.9 T-tail Flutter ........................................... 428
Chapter 8. Transient Response: Landing and Gust Loads ......... 437
8.1 Response to Arbitrary Time-Dependent External Forces ..... 438
8.2 Numerical Methods for Response Calculations .............. 448
8.3 Transient Lift on a Maneuvering Airfoil .................. 458
8.4 Response of a Two-Dimensional Airfoil to a Discrete
Gust ..................................................... 461
8.5 Methods for Calculating Transient Stresses ............... 469
8.6 Response of a Flight Vehicle to a Discrete Uniform
Atmospheric Gust ......................................... 472
Chapter 9. Random Response: Atmospheric Turbulence and
Runway Roughness .............................................. 477
9.1 Statistical Description of Random Loading ................ 478
9.2 Random Response in Terms of Covariance Functions ......... 481
9.3 Random Response in Terms of Power Spectra ................ 489
9.4 Frequency of Exceeding a Specified Value ................. 499
9.5 The Harmonic Gust Functions .............................. 501
9.6 Examples of Power Spectral Densities ..................... 505
9.7 Design Loads for Random Atmospheric Gusts ................ 511
9.8 Examples of Random Gust Response ......................... 520
Chapter 10. Aeroservoelasticity: Control System Interaction ... 531
10.1 Transfer Functions ....................................... 532
10.2 Equations of Motion in State-Space ....................... 536
10.3 Functions for Approximating Generalized Oscillatory
Aerodynamic Forces ....................................... 547
10.4 Presentation of Results of Stability Analyses ............ 550
10.5 Design of Active Controls ................................ 552
10.6 Design of a Structural Mode Control System ............... 553
10.7 Design of a Gust Alleviation System ...................... 556
10.8 Design of a Flutter Suppression System ................... 559
Chapter 11. Aerothermoelasticity: High Speed Atmospheric
Flight ........................................................ 563
11.1 Large Deflections of Heated Structures ................... 563
11.2 Aerodynamic Heating ...................................... 567
11.3 Example of Aerothermoelastic Flutter ..................... JUO
Chapter 12. Aeroelastic Design: Optimization .................. 575
by Erwin H. Johnson, Ph. D., and Garret N. Vanderplaats, Ph.
D., PE
12.1 Aeroelastic Optimization ................................. 578
12.2 Numerical Optimization Basics ............................ 580
12.3 Aeroelastic Responses .................................... 586
12.4 Design Sensitivities ..................................... 587
12.5 Aeroelastic Design Sensitivities ......................... 591
12.6 The Optimizer ............................................ 594
12.7 Example of an Aeroelastic Design ......................... 599
Appendix A. Matrix Algebra .................................... 607
A.l Definitions .............................................. 607
A.2 Addition and Subtraction ................................. 608
A.3 Multiplication ........................................... 608
A.4 Inversion ................................................ 609
A.5 Differentiation .......................................... 610
A.6 Integration .............................................. 611
A.7 Partitioning ............................................. 611
A.8 Eigenvalues and Eigenvectors ............................. 613
Appendix B. Laplace and Fourier Transforms .................... 619
B.1 The Laplace Transform .................................... 619
B.2 Some General Properties of the Laplace Transform ......... 620
B.3 Laplace Transforms of Simple Functions ................... 623
B.4 The Solution of a Differential Equation by Means
of a Laplace Transform Table ............................. 625
B.5 The Inverse Laplace Transform ............................ 627
B.6 The Fourier Transform .................................... 629
Appendix C. Probability and Its Distributions ................. 637
C.1 Basic Notions ............................................ 637
C.2 Random Variables and Their Probability Functions.......... 639
C.3 Expectation and Moments .................................. 643
C.4 The Gaussian Distribution ................................ 645
C.5 The Rayleigh Distribution ................................ 648
Appendix D. The Development of the Doublet-Lattice Method ..... 651
1 Introduction ............................................... 651
2 The Doublet-Lattice Method (DLM) ........................... 652
3 Later Developments ......................................... 654
4 Publicizing the DLM ........................................ 655
5 Continued Developments ..................................... 657
6 Recent Developments ........................................ 658
7 Concluding Remarks ......................................... 659
Appendix E. Flight Mechanics of a Rigid Vehicle ............... 661
Introduction .................................................. 661
Symbols ....................................................... 663
Vector Equations of Motion .................................... 665
Scalar Equations .............................................. 667
Concluding Remarks ............................................ 668
Appendix F. Some Correspondence with the Federal Aviation
Administration ................................................ 671
Appendix G. Galerkin - The Man and the Method ................. 699
The Man ....................................................... 699
The Method .................................................... 700
Appendix H. Autobiography ..................................... 747
1 Introduction ............................................... 748
2 Early Aerospace Employment ................................. 748
3 Graduate Study ............................................. 749
4 Flutter Analysis ........................................... 749
5 Teaching ................................................... 750
6 Consulting ................................................. 750
7 Retirement ................................................. 753
8 My Favorite Contributions to the Literature ................ 753
9 My Least Favorite Contribution to the Literature ........... 755
References and Abbreviations .................................. 757
Index of Authors .............................................. 795
Index of Subjects ............................................. 805
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