《Elementary Matrices and Some Applications to Dynamics and Differential Equations》求取 ⇩

CHAPTER Ⅰ FUNDAMENTAL DEFINITIONS AND ELEMENTARY PROPERTIES1

1．1Preliminary Remarks1

1．2 Notation and Principal Types of Matrix1

1．3 Summation of Matrices and Scalar Multipliers4

1．4 Multiplication of Matrices6

1．5 Continued Products of Matrices9

1．6 Properties of Diagonal and Unit Matrices12

1．7 Partitioning of Matrices into Submatrioes13

1．8 Determinants of Square Matrices16

1．9 Singular Matrices，Degeneracy，and Rank18

1．10 Adjoint Matrices21

1．11 Reciprocal Matrices and Division22

1．12 Square Matrices with Null Product23

1．13 Reversal of Order in Products when Matrices are Transposed or Reciprocated25

1．14 Linear Substitutions26

1．15 Bilinear and Quadratic Forms28

1．16 Discriminants and One-Signed Quadratic Forms30

1．17 Special Types of Square Matrix33

CHAPTER Ⅱ POWERS OF MATRICES，SERIES，AND INFINITESIMAL CALCULUS37

2．1Introductory37

2．2 Powers of Matrices37

2．3 Polynomials of Matrices39

2．4 Infinite Series of Matrices40

2．5 The Exponential Function41

2．6 Differentiation of Matrices43

2．7 Differentiation of the Exponential Function45

2．8 Matrices of Differential Operators46

2．9 Change of the Independent Variables48

2．10 Integration of Matrices52

2．11 The Matrizant53

CHAPTER Ⅲ LAMBDA-MATRICES AND CANONICAL FORMS57

3．1Preliminary Remarks57

PART Ⅰ．Lambda-Matrices57

3．2Lambda-Matrices57

3．3 Multiplication and Division of Lambda-Matrices58

3．4 Remainder Theorems for Lambda-Matrices60

3．5 The Determinantal Equation and the Adjoint of a Lambda-Matrix61

3．6 The Characteristic Matrix of a Square Matrix and the Latent Roots64

3．7 The Cayley-Hamilton Theorem70

3．8 The Adjoint and Derived Adjoints of the Characteristic Matrix73

3．9 Sylvester＇s Theorem78

3．10 Confiuent Form of Sylvester＇s Theorem83

PART Ⅱ． Canonical Forms87

3．11Elementary Operations on Matrices87

3．12 Equivalent Matrices89

3．13 A Canonical Form for Square Matrices of Rank r89

3．14 Equivalent Lambda-Matrices90

3．15 Smith＇s Canonical Form for Lambda-Matrices91

3．16 Collineatory Transformation of a Numerical Matrix to a Canonical Form93

CHAPTER Ⅳ MISCELLANEOUS NUMERICAL METHODS96

4．1Range of the Subjects Treated96

PART Ⅰ．Determinants，Reciprocal and Adjoint Matrices，and Systems of Linear Algebraic Equations96

4．2Preliminary Remarks96

4．3 Triangular and Related Matrices97

4．4 Reduction of Triangular and Related Matrices to Diagonal Form102

4．5 Reciprocals of Triangular and Related Matrices103

4．6 Computation of Determinants106

4．7 Computation of Reciprocal Matrices108

4．8 Reciprocation by the Method of Postmultipliers109

4．9 Reciprocation by the Method of Submatrices112

4．10 Reciprocation by Direct Operations on Rows119

4．11 Improvement of the Accuracy of an Approximate Reciprocal Matrix120

4．12 Computation of the Adjoint of a Singular Matrix121

4．13 Numerical Solution of Simultaneous Linear Algebraic Equations125

PART Ⅱ．High Powers of a Matrix and the Latent Soots133

4．14Preliminary Summary of Sylvester＇s Theorem133

4．15 Evaluation of the Dominant Latent Roots from the Limiting Form of a High Power of a Matrix134

4．16 Evaluation of the Matrix Coefficients Z for the Dominant Roots138

4．17 Simplified Iterative Methods140

4．18 Computation of the Non-Dominant Latent Roots143

4．19 Upper Bounds to the Powers of a Matrix145

PART Ⅲ．Algebraic Equations of General Degree148

4．20Solution of Algebraic Equations and Adaptation of Aitken＇s Formulae148

4．21 General Remarks on Iterative Methods150

4．22 Situation of the Roots of an Algebraic Equation151

CHAPTER Ⅴ LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS156

PART Ⅰ．General Properties156

5．1Systems of Simultaneous Differential Equations156

5．2 Equivalent Systems158

5．3 Transformation of the Dependent Variables159

5．4 Triangular Systems and a Fundamental Theorem160

5．5 Conversion of a System of General Order into a First-Order System162

5．6 The Adjoint and Derived Adjoint Matrices165

5．7 Construction of the Constituent Solutions167

5．8 Numerical Evaluation of the Constituent Solutions172

5．9 Expansions in Partial Fractions176

PART Ⅱ．Construction of the Complementary Function and of a Particular Integral178

5．10The Complementary Function178

5．11 Construction of a Particular Integral183

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

PART Ⅰ．Boundary Problems186

6．1Preliminary Remarks186

6．2 Characteristic Numbers187

6．3 Notation for One-Point Boundary Problems188

6．4 Direct Solution of the General One-Point Boundary Problem191

6．5 Special Solution for Standard One-Point Boundary Problems195

6．6 Confluent Form of the Special Solution198

6．7 Notation and Direct Solution for Two-Point Boundary Problems200

PART Ⅱ．Systems of First Order202

6．8Preliminary Remarks202

6．9 Special Solution of the General First-Order System，and its Connection with Heaviside＇s Method203

6．10 Determinantal Equation，Adjoint Matrices，and Modal Columns for the Simple First-Order System205

6．11 General，Direct，and Special Solutions of the Simple First-Order System206

6．12 Power Series Solution of Simple First-Order Systems209

6．13 Power Series Solution of the Simple First-Order System for a Two-Point Boundary Problem211

CHAPTER Ⅶ NUMERICAL SOLUTIONS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS212

7．1Range of the Chapter212

7．2 Existence Theorems and Singularities212

7．3 Fundamental Solutions of a Single Linear Homogeneous Equation214

7．4 Systems of Simultaneous Linear Differential Equations215

7．5 The Peano-Baker Method of Integration217

7．6 Various Properties of the Matrizant218

7．7 A Continuation Formula219

7．8 Solution of the Homogeneous First-Order System of Equations in Power Series222

7．9 Collocation and Galerkin＇s Method224

7．10 Examples of Numerical Solution by Collocation and Galerkin＇s Method228

7．11 The Method of Mean Coefficients232

7．12 Solution by Mean Coefficients：Example No．1233

7．13 Example No．2237

7．14 Example No．3240

7．15 Example No．4243

CHAPTER Ⅷ KINEMATICS AND DYNAMICS OF SYSTEMS246

PART Ⅰ．Frames of Reference and Kinematics246

8．1Frames of Reference246

8．2 Change of Reference Axes in Two Dimensions247

8．3 Angular Coordinates of a Three-Dimensional Moving Frame of Reference260

8．4 The Orthogonal Matrix of Transformation261

8．5 Matrices Representing Finite Rotations of a Frame of Reference261

8．6 Matrix of Transformation and Instantaneous Angular Velocities Expressed in Angular Coordinates266

8．7 Components of Velocity and Acceleration266

8．8 Kinematic Constraint of a Rigid Body260

8．9 Systems of Rigid Bodies and Generalised Coordinates260

PART Ⅱ．Statics and Dynamics of Systems262

8．10Virtual Work and the Conditions of Equilibrium262

8．11 Conservative and Non-Conservative Fields of Force263

8．12 Dynamical Systems266

8．13 Equations of Motion of an Aeroplane267

8．14 Lagrange＇s Equations Of Motion of a Holonomous System269

8．15 Ignoration of Coordinates272

8．16 The Generalised Components of Momentum and Hamilton＇s Equations274

8．17 Lagrange＇s Equations with a Moving Frame of Reference277

CHAPTER Ⅸ SYSTEMS WITH LINEAR DYNAMICAL EQUATIONS280

9．1Introductory Remarks280

9．2 Disturbed Motions280

9．3 Conservative System Disturbed from Equilibrium281

9．4 Disturbed Steady Motion of a Conservative System with Ignorable Coordinates282

9．5 Small Motions of Systems Subject to Aerodynamical Forces283

9．6 Free Disturbed Steady Motion of an Aeroplane284

9．7 Review of Notation and Terminology for General Linear Systems288

9．8 General Nature of the Constituent Motions289

9．9 Modal Columns for a Linear Conservative System291

9．10 The Direct Solution for a Linear Conservative System and the Normal Coordinates295

9．11 Orthogonal Properties of the Modal Columns and Rayleigh＇s Principle for Conservative Systems299

9．12 Forced Oscillations of Aerodynamical Systems302

CHAPTER Ⅹ ITERATIVE NUMERICAL SOLUTIONS OF LINEAR DYNAMICAL PROBLEMS308

10．1Introductory308

PART Ⅰ．Systems with Damping Forces Absent308

10．2 Remarks on the Underlying Theory308

10．3Example No．1：Oscillations of a Triple Pendulum310

10．4 Example No．2：Torsional Oscillations of a Uniform Cantilever314

10．5 Example No．3：Torsional Oscillations of a Multi-Cylinder Engine316

10．6 Example No．4：Flexural Oscillations of a Tapered Beam318

10．7 Example No．5：Symmetrical Vibrations of an Annular Membrane320

10．8 Example No．6：A System with Two Equal Frequencies322

10．9 Example No．7：The Static Twist of an Aeroplane Wing under Aerodynamical Load325

PART Ⅱ．Systems with Damping Forces Present327

10．10Preliminary Remarks327

10．11 Example：The Oscillations of a Wing in an Airstream328

CHAPTER Ⅺ DYNAMICAL SYSTEMS WITH SOLID FRICTION332

11．1Introduction332

11．2 The Dynamical Equations336

11．3 Various Identities336

11．4 Complete Motion when only One Coordinate is Frictionally Constrained339

11．5 Illustrative Treatment for Ankylotic Motion344

11．6 Steady Oscillations when only One Coordinate is Frictionally Constrained345

11．7 Discussion of the Conditions for Steady Oscillations348

11．8 Stability of the Steady Oscillations350

11．9 A Graphical Method for the Complete Motion of Binary Systems354

CHAPTER Ⅻ ILLUSTRATIVE APPLICATIONS OF FRICTION THEORY TO FLUTTER PROBLEMS358

12．1Introductory358

PART Ⅰ．Aeroplane No．1362

12．2Numerical Data362

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．2369

12．6Numerical Data369

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．3380

1210 Aeroplane No．3380

CHAPTER ⅩⅢ PITCHING OSCILLATI0NS OF A FRICTIONALLY CONSTRAINED AEROFOIL382

13．1Preliminary Remarks382

PART Ⅰ．The Test System and its Design383

13．2Description of the Aerofoil System383

13．3 Data Relating to the Design of the Test System384

13．4 Graphical Interpretation of the Criterion for Steady Oscillations387

13．5 Alternative Treatment Based on the Use of Inertias as Parameters389

13．6 Theoretical Behaviour of the Test System392

PART Ⅱ．Experimental Investigation395

13．7Preliminary Calibrations of the Actual Test System395

13．8 Observations of Frictional Oscillations395

13．9 Other Oscillations Exhibited by the Test System398

List of References399

List of Authors Cited403

Index404