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CHAPTER Ⅰ FUNDAMENTAL DEFINITIONS AND ELEMENTARY PROPERTIES 1

1．1 Preliminary Remarks 1

1．2 Notation and Principal Types of Matrix 1

1．3 Summation of Matrices and Scalar Multipliers 4

1．4 Multiplication of Matrices 6

1．5 Continued Products of Matrices 9

1．6 Properties of Diagonal and Unit Matrices 12

1．7 Partitioning of Matrices into Submatrioes 13

1．8 Determinants of Square Matrices 16

1．9 Singular Matrices，Degeneracy，and Rank 18

1．10 Adjoint Matrices 21

1．11 Reciprocal Matrices and Division 22

1．12 Square Matrices with Null Product 23

1．13 Reversal of Order in Products when Matrices are Transposed or Reciprocated 25

1．14 Linear Substitutions 26

1．15 Bilinear and Quadratic Forms 28

1．16 Discriminants and One-Signed Quadratic Forms 30

1．17 Special Types of Square Matrix 33

CHAPTER Ⅱ POWERS OF MATRICES，SERIES，AND INFINITESIMAL CALCULUS 37

2．1 Introductory 37

2．2 Powers of Matrices 37

2．3 Polynomials of Matrices 39

2．4 Infinite Series of Matrices 40

2．5 The Exponential Function 41

2．6 Differentiation of Matrices 43

2．7 Differentiation of the Exponential Function 45

2．8 Matrices of Differential Operators 46

2．9 Change of the Independent Variables 48

2．10 Integration of Matrices 52

2．11 The Matrizant 53

CHAPTER Ⅲ LAMBDA-MATRICES AND CANONICAL FORMS 57

3．1 Preliminary Remarks 57

PART Ⅰ．Lambda-Matrices 57

3．2 Lambda-Matrices 57

3．3 Multiplication and Division of Lambda-Matrices 58

3．4 Remainder Theorems for Lambda-Matrices 60

3．5 The Determinantal Equation and the Adjoint of a Lambda-Matrix 61

3．6 The Characteristic Matrix of a Square Matrix and the Latent Roots 64

3．7 The Cayley-Hamilton Theorem 70

3．8 The Adjoint and Derived Adjoints of the Characteristic Matrix 73

3．9 Sylvester＇s Theorem 78

3．10 Confiuent Form of Sylvester＇s Theorem 83

PART Ⅱ． Canonical Forms 87

3．11 Elementary Operations on Matrices 87

3．12 Equivalent Matrices 89

3．13 A Canonical Form for Square Matrices of Rank r 89

3．14 Equivalent Lambda-Matrices 90

3．15 Smith＇s Canonical Form for Lambda-Matrices 91

3．16 Collineatory Transformation of a Numerical Matrix to a Canonical Form 93

CHAPTER Ⅳ MISCELLANEOUS NUMERICAL METHODS 96

4．1 Range of the Subjects Treated 96

PART Ⅰ．Determinants，Reciprocal and Adjoint Matrices，and Systems of Linear Algebraic Equations 96

4．2 Preliminary Remarks 96

4．3 Triangular and Related Matrices 97

4．4 Reduction of Triangular and Related Matrices to Diagonal Form 102

4．5 Reciprocals of Triangular and Related Matrices 103

4．6 Computation of Determinants 106

4．7 Computation of Reciprocal Matrices 108

4．8 Reciprocation by the Method of Postmultipliers 109

4．9 Reciprocation by the Method of Submatrices 112

4．10 Reciprocation by Direct Operations on Rows 119

4．11 Improvement of the Accuracy of an Approximate Reciprocal Matrix 120

4．12 Computation of the Adjoint of a Singular Matrix 121

4．13 Numerical Solution of Simultaneous Linear Algebraic Equations 125

PART Ⅱ．High Powers of a Matrix and the Latent Soots 133

4．14 Preliminary Summary of Sylvester＇s Theorem 133

4．15 Evaluation of the Dominant Latent Roots from the Limiting Form of a High Power of a Matrix 134

4．16 Evaluation of the Matrix Coefficients Z for the Dominant Roots 138

4．17 Simplified Iterative Methods 140

4．18 Computation of the Non-Dominant Latent Roots 143

4．19 Upper Bounds to the Powers of a Matrix 145

PART Ⅲ．Algebraic Equations of General Degree 148

4．20 Solution of Algebraic Equations and Adaptation of Aitken＇s Formulae 148

4．21 General Remarks on Iterative Methods 150

4．22 Situation of the Roots of an Algebraic Equation 151

CHAPTER Ⅴ LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 156

PART Ⅰ．General Properties 156

5．1 Systems of Simultaneous Differential Equations 156

5．2 Equivalent Systems 158

5．3 Transformation of the Dependent Variables 159

5．4 Triangular Systems and a Fundamental Theorem 160

5．5 Conversion of a System of General Order into a First-Order System 162

5．6 The Adjoint and Derived Adjoint Matrices 165

5．7 Construction of the Constituent Solutions 167

5．8 Numerical Evaluation of the Constituent Solutions 172

5．9 Expansions in Partial Fractions 176

PART Ⅱ．Construction of the Complementary Function and of a Particular Integral 178

5．10 The Complementary Function 178

5．11 Construction of a Particular Integral 183

CHAPTER Ⅵ LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS (continued) 186

PART Ⅰ．Boundary Problems 186

6．1 Preliminary Remarks 186

6．2 Characteristic Numbers 187

6．3 Notation for One-Point Boundary Problems 188

6．4 Direct Solution of the General One-Point Boundary Problem 191

6．5 Special Solution for Standard One-Point Boundary Problems 195

6．6 Confluent Form of the Special Solution 198

6．7 Notation and Direct Solution for Two-Point Boundary Problems 200

PART Ⅱ．Systems of First Order 202

6．8 Preliminary Remarks 202

6．9 Special Solution of the General First-Order System，and its Connection with Heaviside＇s Method 203

6．10 Determinantal Equation，Adjoint Matrices，and Modal Columns for the Simple First-Order System 205

6．11 General，Direct，and Special Solutions of the Simple First-Order System 206

6．12 Power Series Solution of Simple First-Order Systems 209

6．13 Power Series Solution of the Simple First-Order System for a Two-Point Boundary Problem 211

CHAPTER Ⅶ NUMERICAL SOLUTIONS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 212

7．1 Range of the Chapter 212

7．2 Existence Theorems and Singularities 212

7．3 Fundamental Solutions of a Single Linear Homogeneous Equation 214

7．4 Systems of Simultaneous Linear Differential Equations 215

7．5 The Peano-Baker Method of Integration 217

7．6 Various Properties of the Matrizant 218

7．7 A Continuation Formula 219

7．8 Solution of the Homogeneous First-Order System of Equations in Power Series 222

7．9 Collocation and Galerkin＇s Method 224

7．10 Examples of Numerical Solution by Collocation and Galerkin＇s Method 228

7．11 The Method of Mean Coefficients 232

7．12 Solution by Mean Coefficients：Example No．1 233

7．13 Example No．2 237

7．14 Example No．3 240

7．15 Example No．4 243

CHAPTER Ⅷ KINEMATICS AND DYNAMICS OF SYSTEMS 246

PART Ⅰ．Frames of Reference and Kinematics 246

8．1 Frames of Reference 246

8．2 Change of Reference Axes in Two Dimensions 247

8．3 Angular Coordinates of a Three-Dimensional Moving Frame of Reference 260

8．4 The Orthogonal Matrix of Transformation 261

8．5 Matrices Representing Finite Rotations of a Frame of Reference 261

8．6 Matrix of Transformation and Instantaneous Angular Velocities Expressed in Angular Coordinates 266

8．7 Components of Velocity and Acceleration 266

8．8 Kinematic Constraint of a Rigid Body 260

8．9 Systems of Rigid Bodies and Generalised Coordinates 260

PART Ⅱ．Statics and Dynamics of Systems 262

8．10 Virtual Work and the Conditions of Equilibrium 262

8．11 Conservative and Non-Conservative Fields of Force 263

8．12 Dynamical Systems 266

8．13 Equations of Motion of an Aeroplane 267

8．14 Lagrange＇s Equations Of Motion of a Holonomous System 269

8．15 Ignoration of Coordinates 272

8．16 The Generalised Components of Momentum and Hamilton＇s Equations 274

8．17 Lagrange＇s Equations with a Moving Frame of Reference 277

CHAPTER Ⅸ SYSTEMS WITH LINEAR DYNAMICAL EQUATIONS 280

9．1 Introductory Remarks 280

9．2 Disturbed Motions 280

9．3 Conservative System Disturbed from Equilibrium 281

9．4 Disturbed Steady Motion of a Conservative System with Ignorable Coordinates 282

9．5 Small Motions of Systems Subject to Aerodynamical Forces 283

9．6 Free Disturbed Steady Motion of an Aeroplane 284

9．7 Review of Notation and Terminology for General Linear Systems 288

9．8 General Nature of the Constituent Motions 289

9．9 Modal Columns for a Linear Conservative System 291

9．10 The Direct Solution for a Linear Conservative System and the Normal Coordinates 295

9．11 Orthogonal Properties of the Modal Columns and Rayleigh＇s Principle for Conservative Systems 299

9．12 Forced Oscillations of Aerodynamical Systems 302

CHAPTER Ⅹ ITERATIVE NUMERICAL SOLUTIONS OF LINEAR DYNAMICAL PROBLEMS 308

10．1 Introductory 308

PART Ⅰ．Systems with Damping Forces Absent 308

10．2 Remarks on the Underlying Theory 308

10．3 Example No．1：Oscillations of a Triple Pendulum 310

10．4 Example No．2：Torsional Oscillations of a Uniform Cantilever 314

10．5 Example No．3：Torsional Oscillations of a Multi-Cylinder Engine 316

10．6 Example No．4：Flexural Oscillations of a Tapered Beam 318

10．7 Example No．5：Symmetrical Vibrations of an Annular Membrane 320

10．8 Example No．6：A System with Two Equal Frequencies 322

10．9 Example No．7：The Static Twist of an Aeroplane Wing under Aerodynamical Load 325

PART Ⅱ．Systems with Damping Forces Present 327

10．10 Preliminary Remarks 327

10．11 Example：The Oscillations of a Wing in an Airstream 328

CHAPTER Ⅺ DYNAMICAL SYSTEMS WITH SOLID FRICTION 332

11．1 Introduction 332

11．2 The Dynamical Equations 336

11．3 Various Identities 336

11．4 Complete Motion when only One Coordinate is Frictionally Constrained 339

11．5 Illustrative Treatment for Ankylotic Motion 344

11．6 Steady Oscillations when only One Coordinate is Frictionally Constrained 345

11．7 Discussion of the Conditions for Steady Oscillations 348

11．8 Stability of the Steady Oscillations 350

11．9 A Graphical Method for the Complete Motion of Binary Systems 354

CHAPTER Ⅻ ILLUSTRATIVE APPLICATIONS OF FRICTION THEORY TO FLUTTER PROBLEMS 358

12．1 Introductory 358

PART Ⅰ．Aeroplane No．1 362

12．2 Numerical Data 362

12．3 Steady Oscillations on Aeroplane No．1 at V =260．(Rudder Frictionally Constrained) 363

12．4 Steady Oscillations on Aeroplane No．1 at Various Speeds．(Rudder Frictionally Constrained) 367

12．5 Steady Oscillations on Aeroplane No．1．(Fuselage Frictionally Constrained) 369

PART Ⅱ．Aeroplane No．2 369

12．6 Numerical Data 369

12．7 Steady Oscillations on Aeroplane No．2．(Rudder Frictionally Constrained) 370

12．8 Steady Oscillations on Aeroplane No．2．(Fuselage Frictionally Constrained) 372

12．9 Graphical Investigation of Complete Motion on Aeroplane No．2 at V = 230．(Rudder Frictionally Constrained) 372

PART Ⅲ．Aeroplane No．3 380

1210 Aeroplane No．3 380

CHAPTER ⅩⅢ PITCHING OSCILLATI0NS OF A FRICTIONALLY CONSTRAINED AEROFOIL 382

13．1 Preliminary Remarks 382

PART Ⅰ．The Test System and its Design 383

13．2 Description of the Aerofoil System 383

13．3 Data Relating to the Design of the Test System 384

13．4 Graphical Interpretation of the Criterion for Steady Oscillations 387

13．5 Alternative Treatment Based on the Use of Inertias as Parameters 389

13．6 Theoretical Behaviour of the Test System 392

PART Ⅱ．Experimental Investigation 395

13．7 Preliminary Calibrations of the Actual Test System 395

13．8 Observations of Frictional Oscillations 395

13．9 Other Oscillations Exhibited by the Test System 398

List of References 399

List of Authors Cited 403

Index 404