Chapter
1.15 Bilinear and Quadratic Forms
1.16 Discriminants and One-Signed Quadratic Forms
1.17 Special Types of Square Matrix
CHAPTER II : POWERS OF MATRICES, SERIES, AND INFINITESIMAL CALCULUS
2.3 Polynomials of Matrices
2.4 Infinite Series of Matrices
2.5 The Exponential Function
2.6 Differentiation of Matrices
2.7 Differentiation of the Exponential Function
2.8 Matrices of Differential Operators
2.9 Change of the Independent Variables
2.10 Integration of Matrices
CHAPTER III : LAMBDA.MATRICES AND CANONICAL FORMS
3.3 Multiplication and Division of Lambda Matrices
3.4 Remainder Theorems for Lambda Matrices
3.5 The Determinantal Equation and the Adjoint of a Lambda Matrix
3.6 The Characteristic Matrix of a Square Matrix and the Latent Roots
3.7 The Cayley.Hamilton Theorem
3.8 The Adj oint and Derived Adjoints of the Characteristic Matrix
3.10 Confluent Form of Sylvester's Theorem
PART II : Canonical Forms
3.11 Elementary Operations on Matrices
3.13 A Canonical Form for Square Matrices of Rank r
3.14 Equivalent Lambda Matrices
3.15 Smith's Canonical Form for Lambda-Matrices
3.16 Collineatory Transformation of a Numerical Matrix to a Canonical Form
CHAPTER IV : MISCELLANEOUS NUMERICAL METHODS
4.1 Range of the Subjects Treated
PART I : Determinants, Reciprocal and Adjoint Matrices, and Systems of Linear Algebraic Equations
4.3 Triangular and Related Matrices
4.4 Reduction of Triangular and Related Matrices to Diagonal Form
4.5 Reciprocals of Triangular and Related Matrices
4.6 Computation of Determinants
4.7 Computation of Reciprocal Matrices
4.8 Reciprocation by the Method of Postmultipliers
4.9 Reciprocation by the Method of Submatrices
4.10 Reciprocation by Direct Operations on Rows
4.11 Improvement of the Accuracy of an Approximate Reciprocal Matrix
4.12 Computation of the Adjoint of a Singular Matrix
4.13 Numerical Solution of Simultaneous Linear Algebraic Equations
PART II : High Powers of a Matrix and the Latent Boots
4.14 Preliminary Summary of Sylvester's Theorem
4.15 Evaluation of the Dominant Latent Roots from the Limiting Form of a High Power of a Matrix
4.16 Evaluation of the Matrix Coefficients Z for the Dominant Roots
4.17 Simplified Iterative Methods
4.18 Computation of the Non Dominant Latent Roots
4.19 Upper Bounds to the Powers of a Matrix
PART III : Algebraic Equations of General Degree
4.20 Solution of Algebraic Equations and Adaptation of Aitken's Formulae
4.21 General Remarks on Iterative Methods
4.22 Situation of the Roots of an Algebraic Equation
CHAPTER V : LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
PART I : General Properties
5.1 Systems of Simultaneous Differential Equations
5.3 Transformation of the Dependent Variables
5.4 Triangular Systems and a Fundamental Theorem
5.5 Conversion of a System of General Order into a First Order System
5.6 The Adjoint and Derived Adjoint Matrices
5.7 Construction of the Constituent Solutions
5.8 Numerical Evaluation of the Constituent Solutions
5.9 Expansions in Partial Fractions
PART II : Construction of the Complementary Function and of a Particular Integral
5.10 The Complementary Function
5.11 Construction of a Particular Integral
CHAPTER VI : LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS (continued)
PART I : Boundary Problems
6.2 Characteristic Numbers
6.3 Notation for One Point Boundary Problems
6.4 Direct Solution of the General One Point Boundary Problem
6.5 Special Solution for Standard One Point Boundary Problems
6.6 Confluent Form of the Special Solution
6.7 Notation and Direct Solution for Two-Point Boundary Problems
PART II : Systems of First Order
6.9 Special Solution of the General First-Order System, and its Connection with Heaviside's Method
6.10 Determinantal Equation, Adjoint Matrices, and Modal Columns for the Simple First-Order System
6.11 General, Direct, and Special Solutions of the Simple First-Order System
6.12 Power Series Solution of Simple First-Order Systems
6.13 Power Series Solution of the Simple First-Order System for a Two-Point Boundary Problem
CHAPTER VII : NUMERICAL SOLUTIONS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS
7.2 Existence Theorems and Singularities
7.3 Fundamental Solutions of a Single Linear Homogeneous Equation
7.4 Systems of Simultaneous Linear Differential Equations
7.5 The Peano-Baker Method of Integration
7.6 Various Properties of the Matrizant
7.7 A Continuation Formula
7.8 Solution of the Homogeneous First-Order System of Equations in Power Series
7.9 Collocation and Galerkin's Method
7.10 Examples of Numerical Solution by Collocation and Galerkin's Method
7.11 The Method of Mean Coefficients
7.12 Solution by Mean Coefficients: Example No. 1
CHAPTER VIII : KINEMATICS AND DYNAMICS OF SYSTEMS
PART I : Frames of Reference and Kinematics
8.2 Change of Reference Axes in Two Dimensions
8.3 Angular Coordinates of a Three-Dimensional Moving Frame of Reference
8.4 The Orthogonal Matrix of Transformation
8.5 Matrices Representing Finite Rotations of a Frame of Reference
8.6 Matrix of Transformation and Instantaneous Angular Velocities Expressed in Angular Coordinates
8.7 Components of Velocity and Acceleration
8.8 Kinematic Constraint of a Rigid Body
8.9 Systems of Rigid Bodies and Generalised Coordinates
PART II : Statics and Dynamics of Systems
8.10 Virtual Work and the Conditions of Equilibrium
8.11 Conservative and Non-Conservative Fields of Force
8.13 Equations of Motion of an Aeroplane
8.14 Lagrange's Equations of Motion of a Holonomous System
8.15 Ignoration of Coordinates
8.16 The Generalised Components of Momentum and Hamilton's Equations
8.17 Lagrange's Equations with a Moving Frame of Reference
CHAPTER IX : SYSTEMS WITH LINEAR DYNAMICAL EQUATIONS
9.3 Conservative System Disturbed from Equilibrium
9.4 Disturbed Steady Motion of a Conservative System with Ignorable Coordinates
9.5 Small Motions of Systems Subject to Aerodynamical Forces
9.6 Free Disturbed Steady Motion of an Aeroplane
9.7 Review of Notation and Terminology for General Linear Systems
9.8 General Nature of the Constituent Motions
9.9 Modal Columns for a Linear Conservative System
9.10 The Direct Solution for a Linear Conservative System and the Normal Coordinates
9.11 Orthogonal Properties of the Modal Columns and Rayleigh's Principle for Conservative Systems
9.12 Forced Oscillations of Aerodynamical Systems
CHAPTER X : ITERATIVE NUMERICAL SOLUTIONS OF LINEAR DYNAMICAL PROBLEMS
PART I : Systems with Damping Forces Absent
10.2 Remarks on the Underlying Theory
10.3 Example No. 1: Oscillations of a Triple Pendulum
10.4 Example No. 2: Torsional Oscillations of a Uniform Cantilever
10.5 Example No. 3: Torsional Oscillations of a Multi-Cylinder Engine
10.6 Example No. 4: Flexural Oscillations of a Tapered Beam
10.7 Example No. 5: Symmetrical Vibrations of an Annular Membrane
10.8 Example No. 6: A System with Two Equal Frequencies
10.9 Example No. 7: The Static Twist of an Aeroplane Wing under Aerodynamical Load
PART II : Systems with Damping Forces Present
10.10 Preliminary Remarks
10.11 Example: The Oscillations of a Wing in an Airstream
CHAPTER XI : DYNAMICAL SYSTEMS WITH SOLID FRICTION
11.2 The Dynamical Equations
11.4 Complete Motion when only One Coordinate is Frictionally Constrained
11.5 Illustrative Treatment for Ankylotic Motion
11.6 Steady Oscillations when only One Coordinate is Frictionally Constrained
11.7 Discussion of the Conditions for Steady Oscillations
11.8 Stability of the Steady Oscillations
11.9 A Graphical Method for the Complete Motion of Binary Systems
CHAPTER XII : ILLUSTRATIVE APPLICATIONS OF FRICTION THEORY TO FLUTTER PROBLEMS
12.3 Steady Oscillations on Aeroplane No. 1 at V =260. (Rudder Frictionally Constrained)
12.4 Steady Oscillations on Aeroplane No. 1 at Various Speeds. (Rudder Frictionally Constrained)
12.5 Steady Oscillations on Aeroplane No. 1. (Fuselage Frictionally Constrained)
PART II : Aeroplane No. 2
12.7 Steady Oscillations on Aeroplane No. 2. (Rudder Frictionally Constrained)
12.8 Steady Oscillations on Aeroplane No. 2. (Fuselage Frictionally Constrained)
12.9 Graphical Investigation of Complete Motion on Aeroplane No. 2 at V = 230. (Rudder Frictionally Constrained)
PART III : Aeroplane No. 3
CHAPTER XIII : PITCHING OSCILLATIONS OF A FRICTIONALLY CONSTRAINED AEROFOIL
PART I : The Test System and its Design
13.2 Description of the Aerofoil System
13.3 Data Relating to the Design of the Test System
13.4 Graphical Interpretation of the Criterion for Steady Oscillations
13.5 Alternative Treatment Based on the Use of Inertias as Parameters
13.6 Theoretical Behaviour of the Test System
PART II : Experimental Investigation
13.7 Preliminary Calibrations of the Actual Test System
13.8 Observations of Frictional Oscillations
13.9 Other Oscillations Exhibited by the Test System
Elementary Matrices
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