Elementary Matrices And Some Applications To Dynamics And Differential Equations

Cover......Page 1
Elementary Matrices And Some Applications To Dynamics And Differential Equations......Page 2
Contents......Page 4
Preface......Page 16
1.2 Notation and Principal Types of Matrix......Page 18
1.3 Summation of Matrices and Scalar Multipliers......Page 21
1.4 Multiplication of Matrices......Page 23
1.5 Continued Products of Matrices......Page 26
1.6 Properties of Diagonal and Unit Matrices......Page 29
1.7 Partitioning of Matrices into Submatrices......Page 30
1.8 Determinants of Square Matrices......Page 33
1.9 Singular Matrices, Degeneracy, and Rank......Page 35
1.10 Adjoint Matrices......Page 38
1.11 Reciprocal Matrices and Division......Page 39
1.12 Square Matrices with Null Product......Page 40
1.13 Reversal of Order in Products when Matrices are Transposed or Reciprocated......Page 42
1.14 Linear Substitutions......Page 43
1.15 Bilinear and Quadratic Forms......Page 45
1.16 Discriminants and One-Signed Quadratic Forms......Page 47
1.17 Special Types of Square Matrix......Page 50
2.2 Powers of Matrices......Page 54
2.3 Polynomials of Matrices......Page 56
2.4 Infinite Series of Matrices......Page 57
2.5 The Exponential Function......Page 58
2.6 Differentiation of Matrices......Page 60
2.7 Differentiation of the Exponential Function......Page 62
2.8 Matrices of Differential Operators......Page 63
2.9 Change of the Independent Variables......Page 65
2.10 Integration of Matrices......Page 69
2.11 The Matrizant......Page 70
3.2 Lambda-Matrices......Page 74
3.3 Multiplication and Division of Lambda-Matrices......Page 75
3.4 Remainder Theorems for Lambda-Matrices......Page 77
3.5 The Determinantal Equation and the Adjoint of a Lambda-Matrix......Page 78
3.6 The Characteristic Matrix of a Square Matrix and the Latent Roots......Page 81
3.7 The Cayley-Hamilton Theorem......Page 87
3.8 The Adjoint and Derived Adjoints of the Characteristic Matrix......Page 90
3.9 Sylvester\'s Theorem......Page 95
3.10 Confluent Form of Sylvester\'s Theorem......Page 100
3.11 Elementary Operations on Matrices......Page 104
3.13 A Canonical Form for Square Matrices of Rank r......Page 106
3.14 Equivalent Lambda-Matrices......Page 107
3.15 Smith\'s Canonical Form for Lambda-Matrices......Page 108
3.16 Collineatory Transformation of a Numerical Matrix to a Canonical Form......Page 110
4.2 Preliminary Remarks......Page 113
4.3 Triangular and Related Matrices......Page 114
4.4 Reduction of Triangular and Related Matrices to Diagonal Form......Page 119
4.5 Reciprocals of Triangular and Related Matrices......Page 120
4.6 Computation of Determinants......Page 123
4.7 Computation of Reciprocal Matrices......Page 125
4.8 Reciprocation by the Method of Postmultipliers......Page 126
4.9 Reciprocation by the Method of Submatrices......Page 129
4.10 Reciprocation by Direct Operations on Rows......Page 136
4.11 Improvement of the Accuracy of an Approximate Reciprocal Matrix......Page 137
4.12 Computation of the Adjoint of a Singular Matrix......Page 138
4.13 Numerical Solution of Simultaneous Linear Algebraic Equations......Page 142
4.14 Preliminary Summary of Sylvester\'s Theorem......Page 150
4.15 Evaluation of the Dominant Latent Roots from the Limiting Form of a High Power of a Matrix......Page 151
4.16 Evaluation of the Matrix Coefficients Z for the Dominant Roots......Page 155
4.17 Simplified Iterative Methods......Page 157
4.18 Computation of the Non-Dominant Latent Roots......Page 160
4.19 Upper Bounds to the Powers of a Matrix......Page 162
4.20 Solution of Algebraic Equations and Adaptation of Aitken\'s Formulae......Page 165
4.21 General Remarks on Iterative Methods......Page 167
4.22 Situation of the Roots of an Algebraic Equation......Page 168
5.1 Systems of Simultaneous Differential Equations......Page 173
5.2 Equivalent Systems......Page 175
5.3 Transformation of the Dependent Variables......Page 176
5.4 Triangular Systems and a Fundamental Theorem......Page 177
5.5 Conversion of a System of General Order into a First-Order System......Page 179
5.6 The Adjoint and Derived Adjoint Matrices......Page 182
5.7 Construction of the Constituent Solutions......Page 184
5.8 Numerical Evaluation of the Constituent Solutions......Page 189
5.9 Expansions in Partial Fractions......Page 192
5.10 The Complementary Function......Page 195
5.11 Construction of a Particular Integral......Page 200
6.1 Preliminary Remarks......Page 203
6.2 Characteristic Numbers......Page 204
6.3 Notation for One-Point Boundary Problems......Page 205
6.4 Direct Solution of the General One-Point Boundary Problem......Page 208
6.5 Special Solution for Standard One-Point Boundary Problems......Page 212
6.6 Confluent Form of the Special Solution......Page 215
6.7 Notation and Direct Solution for Two-Point Boundary Problems......Page 217
6.8 Preliminary Remarks......Page 219
6.9 Special Solution of the General First-Order System, and its Connection with Heaviside\'s Method......Page 220
6.10 Determinantal Equation, Adjoint Matrices, and Modal Columns for the Simple First-Order System......Page 222
6.11 General, Direct, and Special Solutions of the Simple First-Order System......Page 223
6.12 Power Series Solution of Simple First-Order Systems......Page 226
6.13 Power Series Solution of the Simple First-Order System for a Two-Point Boundary Problem......Page 228
7.2 Existence Theorems and Singularities......Page 229
7.3 Fundamental Solutions of a Single Linear Homogeneous Equation......Page 231
7.4 Systems of Simultaneous Linear Differential Equations......Page 232
7.5 The Peano-Baker Method of Integration......Page 234
7.6 Various Properties of the Matrizant......Page 235
7.7 A Continuation Formula......Page 236
7.8 Solution of the Homogeneous First-Order System of Equations in Power Series......Page 239
7.9 Collocation and Galerkin\'s Method......Page 241
7.10 Examples of Numerical Solution by Collocation and Galerkin\'s Method......Page 245
7.11 The Method of Mean Coefficients......Page 249
7.12 Solution by Mean Coefficients: Example No. 1......Page 250
7.13 Example No. 2......Page 254
7.14 Example No. 3......Page 257
7.15 Example No. 4......Page 260
8.1 Frames of Reference......Page 263
8.2 Change of Reference Axes in Two Dimensions......Page 264
8.3 Angular Coordinates of a Three-Dimensional Moving Frame of Reference......Page 267
8.5 Matrices Representing Finite Rotations of a Frame of Reference......Page 268
8.6 Matrix of Transformation and Instantaneous Angular Velocities Expressed in Angular Coordinate......Page 272
8.7 Components of Velocity and Acceleration......Page 273
8.8 Kinematic Constraint of a Rigid Body......Page 276
8.9 Systems of Rigid Bodies and Generalised Coordinates......Page 277
8.10 Virtual Work and the Conditions of Equilibrium......Page 279
8.11 Conservative and Non-Conservative Fields of Force......Page 280
8.12 Dynamical Systems......Page 283
8.13 Equations of Motion of an Aeroplane......Page 284
8.14 Lagrange\'s Equations of Motion of a Holonomous System......Page 286
8.15 Ignoration of Coordinates......Page 289
8.16 The Generalised Components of Momentum and Hamilton\'s Equations......Page 291
8.17 Lagrange\'s Equations with a Moving Frame of Reference......Page 294
9.2 Disturbed Motions......Page 297
9.3 Conservative System Disturbed from Equilibrium......Page 298
9.4 Disturbed Steady Motion of a Conservative System with Ignorable Coordinates......Page 299
9.5 Small Motions of Systems Subject to Aerodynamical Forces......Page 300
9.6 Free Disturbed Steady Motion of an Aeroplane......Page 301
9.7 Review of Notation and Terminology for General Linear Systems......Page 305
9.8 General Nature of the Constituent Motions......Page 306
9.9 Modal Columns for a Linear Conservative System......Page 308
9.10 The Direct Solution for a Linear Conservative System and the Normal Coordinates......Page 312
9.11 Orthogonal Properties of the Modal Columns and Rayleigh\'s Principle for Conservative Systems......Page 316
9.12 Forced Oscillations of Aerodynamical Systems......Page 319
10.2 Remarks on the Underlying Theory......Page 325
10.3 Example No. 1: Oscillations of a Triple Pendulum......Page 327
10.4 Example No. 2: Torsional Oscillations of a Uniform Cantilever......Page 331
10.5 Example No. 3: Torsional Oscillations of a Multi-Cylinder Engine......Page 333
10.6 Example No. 4: Flexural Oscillations of a Tapered Beam......Page 335
10.7 Example No. 5: Symmetrical Vibrations of an Annular Membrane......Page 337
10.8 Example No. 6: A System with Two Equal Frequencies......Page 339
10.9 Example No. 7: The Static Twist of an Aeroplane Wing under Aerodynamical Load......Page 342
10.10 Preliminary Remarks......Page 344
10.11 Example: The Oscillations of a Wing in an Airstream......Page 345
11.1 Introduction......Page 349
11.2 The Dynamical Equations......Page 352
11.3 Various Identities......Page 353
11.4 Complete Motion when only One Coordinate is Frictionally Constrained......Page 356
11.5 Illustrative Treatment for Ankylotic Motion......Page 361
11.6 Steady Oscillations when only One Coordinate is Frictionally Constrained......Page 362
11.7 Discussion of the Conditions for Steady Oscillations......Page 365
11.8 Stability of the Steady Oscillations......Page 367
11.9 A Graphical Method for the Complete Motion of Binary Systems......Page 371
12.1 Introductory......Page 375
12.2 Numerical Data......Page 379
12.3 Steady Oscillations on Aeroplane No. 1 at V = 260. (Rudder Frictionally Constrained)......Page 380
12.4 Steady Oscillations on Aeroplane No. 1 at Various Speeds. (Rudder Frictionally Constrained)......Page 384
12.6 Numerical Data......Page 386
12.7 Steady Oscillations on Aeroplane No. 2. (Rudder Frictionally Constrained)......Page 387
12.9 Graphical Investigation of Complete Motion on Aeroplane No. 2 at V = 230. (Rudder Frictionally Constrained)......Page 389
12.10 Aeroplane No. 3......Page 397
13.1 Preliminary Remarks......Page 399
13.2 Description of the Aerofoil System......Page 400
13.3 Data Relating to the Design of the Test System......Page 401
13.4 Graphical Interpretation of the Criterion for Steady Oscillations......Page 404
13.5 Alternative Treatment Based on the Use of Inertias as Parameters......Page 406
13.6 Theoretical Behaviour of the Test System......Page 409
13.8 Observations of Frictional Oscillations......Page 412
13.9 Other Oscillations Exhibited by the Test System......Page 415
List of References......Page 416
List of Authors Cited......Page 420
Index......Page 421