Table of Contents
for Fundamentals of Applied Dynamics
by James H. Williams, Jr.

Dedication

About the Author

Acknowledgments

Preface

Chapter 1   Our Niche in the Cosmos

1-1 Introduction

1-2 Why History?

1-3 Importance of Mathematics in the Development of Mechanics

1-4 Our Sources From Antiquity: Getting the Message from There to Here

1-4.1 Invention of Writing

1-4.2 Hieroglyphics

1-4.3 Cuneiform

1-4.4 Ancient Egyptian Papyri

1-4.5 Mesopotamian Clay Tablets

1-5 Ancient Egyptian Astronomy and Mathematics

1-5.1 Ancient Egyptian Astronomy

1-5.2 Ancient Egyptian Mathematics

1-6 Mesopotamian Astronomy and Mathematics

1-6.1 Mesopotamian Astronomy

1-6.2 Mesopotamian Mathematics

1-7 Mathematics of the Mayans, Indians, Arabs and Chinese

1-8 The First Great Engineering Society

1-9 Adverse Criticism of Ancient Egyptian and Mesopotamian Mathematics

1-10 Evolution Through Hellenic Era

1-11 The Unification of Celestial and Terrestrial Motion

1-11.1 Celestial Motion

1-11.2 Terrestrial Motion

1-11.3 Unification

1-12 Variational Principles in Dynamics

1-13 The Internationalism of Dynamics

1-14 Our Niche in the Cosmos

Chapter 2   Design, Modeling, and Formulation of Equations of Motion

2-1 Introduction

2-2 Design and Modeling

2-2.1 The Design Process

2-2.2 The Modeling Process

2-2.3 Our More Modest Goals

2-3 Direct and Indirect Approaches for Formulation of Equations of Motion

Chapter 3   Kinematics

3-1 Introduction

3-2 Position, Velocity and Acceleration

3-3 Plane Kinematics of Rigid Bodies

3-3.1 The General Motion of a Rigid Body

3-3.2 Types of Plane Motion of a Rigid Body

3-3.3 Angular Displacement, Angular Velocity and Angular Acceleration

3-3.4 A Cautionary Note About Finite Rotations

3-4 Time Rate of Change of Vector in Rotating Frame

3-5 Kinematic Analysis Utilizing Intermediate Frames

3-6 Generalizations of Kinematic Expressions

Chapter 4   Momentum Formulation for Systems of Particles

4-1 Introduction

4-2 The Fundamental Physics

4-2.1 Newton's Laws of Motion

4-2.2 A Particle

4-2.3 Linear Momentum and Force

4-2.4 Inertial Reference Frames

4-2.5 The Universal Law of Gravitation

4-3 Torque and Angular Momentum for a Particle

4-4 Formulation of Equations of Motion: Examples

4-4.1 Problems of Particle Dynamics of the First Kind

4-4.2 Problems of Particle Dynamics of the Second Kind

Chapter 5   Variational Formulation for Systems of Particles

5-1 Introduction

5-2 Formulation of Equations of Motion

5-3 Work and State Functions

5-3.1 Work

5-3.2 Kinetic State Functions

5-3.3 Potential State Functions

5-3.4 Energy and Coenergy

5-4 Generalized Variables and Variational Concepts

5-4.1 Generalized Coordinates

5-4.2 Admissible Variations, Degrees of Freedom, Geometric Constraints and Holonomicity

5-4.3 Variational Principles in Mechanics

5-4.4 Generalized Velocities and Generalized Forces for Holonomic Systems

5-5 Equations of Motion for Holonomic Mechanical Systems via Variational Principles

5-6 Work-Energy Relation

5-7 Nature of Lagrangian Dynamics

Chapter 6   Dynamics of Systems Containing Rigid Bodies

6-1 Introduction

6-2 Momentum Principles for Rigid Bodies

6-2.1 Review of Solids in Equilibrium and Particle Dynamics

6-2.2 Models of Rigid Bodies

6-2.3 Momentum Principles for Extended Bodies: The Newton-Euler Equations

6-2.4 Momentum Principles for Rigid Bodies Modeled as Systems of Particles

6-2.5 Momentum Principles for Rigid Bodies Modeled as Continua

6-3 Dynamic Properties of Rigid Bodies

6-3.1 The Inertia Tensor

6-3.2 Parallel-Axes Theorem

6-3.3 Principal Directions and Principal Moments of Inertia

6-3.4 Uses of Mass Symmetry

6-4 Dynamics of Rigid Bodies via Direct Approach

6-5 Lagrangian for Rigid Bodies

6-5.1 Kinetic Coenergy Function for Rigid Body

6-5.2 Potential Energy Function for Rigid Body

6-6 Equations of Motion for Systems Containing Rigid Bodies in Plane Motion

Chapter 7   Dynamics of Electrical and Electromechanical Systems

7-1 Introduction

7-2 Formulation of Equations of Motion for Electrical Networks

7-3 Constitutive Relations for Circuit Elements

7-3.1 Passive Elements

7-3.2 Active Electrical Elements

7-4 Hamilton's Principle and Lagrange's Equations for Electrical Networks

7-4.1 Generalized Charge Variables

7-4.2 Generalized Flux Linkage Variables

7-4.3 Work Expressions

7-4.4 Summary of Lumped-Parameter Offering of Variational Electricity

7-4.5 Examples

7-5 Constitutive Relations for Transducers

7-5.1 Ideal Movable-Plate Capacitor

7-5.2 Electrically Linear Movable-Plate Capacitor

7-5.3 Ideal Movable-Core Inductor

7-5.4 Magnetically Linear Movable-Core Inductor

7-6 Hamilton's Principle and Lagrange's Equations for Electromechanical Systems

7-6.1 Displacement-Charge Variables Formulation

7-6.2 Displacement-Flux Linkage Variables Formulation

7-7 Another Look at Lagrangian Dynamics

Chapter 8   Vibration of Linear Lumped-Parameter Systems

8-1 Introduction

8-2 Single Degree of Freedom First Order Systems

8-2.1 Free Response

8-2.2 Step Response

8-2.3 Ramp Response

8-2.4 Harmonic Response

8-2.5 Summary of Responses for Single Degree of Freedom First Order Systems

8-3 Single Degree of Freedom Second Order Systems

8-3.1 Free Response

8-3.2 Natural Frequency via Static Deflection

8-3.3 Logarithmic Decrement

8-3.4 Energy Loss of Free Vibration

8-3.5 Harmonic Response

8-3.6 Summary of Responses for Single Degree of Freedom Second Order Systems

8-4 Two Degree of Freedom Second Order Systems

8-4.1 Natural Modes of Vibration

8-4.2 Response to Initial Conditions

8-4.3 Harmonic Response

8-5 Stability of Nonlinear Systems

Chapter 9   Dynamics of Continuous Systems

9-1 Introduction

9-2 Equations of Motion

9-2.1 Longitudinal Motion of System Containing Rod

9-2.2 Twisting Motion of System Containing Shaft

9-2.3 Electric Transmission Line

9-2.4 Flexural Motion of System Containing Beam

9-2.5 Summaries

9-3 Natural Modes of Vibration

9-3.1 Method of Separation of Variables

9-3.2 Time Response

9-3.3 Eigenfunctions for Second-Order Systems

9-3.4 Eigenfunctions for Fourth-Order Systems

9-3.5 General Solutions for Free Undamped Vibration

9-4 Response to Initial Conditions

9-4.1 An Example: Release of Compressed Rod

9-4.2 An Example: Shaft Stopped after Rotation

9-4.3 An Example: Sliding Free Beam Initially Bent

9-5 Responses to Harmonic Excitations

9-5.1 An Example: Specified Harmonic Motion of Boundary

9-5.2 An Example: Distributed Harmonic Force

9-5.3 An Example: Harmonic Force on Boundary

9-6 Summaries

Bibliography

1 Historical

2 Astronomy

3 Design, Systems and Modeling

4 Elementary Dynamics

5 Intermediate/Advanced Dynamics

6 Hamilton's Law of Varying Action and Hamilton's Principle

7 Electrical and Electromechanical Systems

8 Vibration

Appendix A   Finite Rotation

A-1 Change in Position Vector Due to Finite Rotation

A-2 Finite Rotations Are Not Vectors

A-3 Do Rotations Ever Behave as Vectors?

A-3.1 Infinitesimal Rotations are Vectors

A-3.2 Consecutive Finite Rotations About a Common Axis are Vectors

Appendix B   General Kinematic Analysis

B-1 All Angular Velocities Defined With Respect to Fixed Reference Frame

B-2 Each Angular Velocity Defined With Respect to Immediately Preceding Frame

Appendix C   Momentum Principles for Systems of Particles

C-1 Asserted Momentum Principles

C-2 Principles for Single Particle

C-3 Principles for System of Particles

C-3.1 Asserted System Momentum Principles

C-3.2 System Momentum Principles Derived from Particle Momentum Principles

C-3.3 Conditions on Internal Forces

C-3.4 Relationships between Momentum Principles and Conditions on Internal Forces

C-3.5 Linear Momentum Principle in Terms of Centroidal Motion

C-3.6 Angular Momentum Principle about Arbitrary Point

C-3.7 System of Particle Model in Continuum Limit

C-4 Angular Momentum Principle in Noninertial Intermediate Frame

Appendix D   Elementary Results of the Calculus of Variations

D-1 Introduction

D-2 Summary of Elementary Results

D-3 Euler Equation: Necessary Condition for a Variational Indicator to Vanish

Appendix E   Some Formulations of the Principles of Hamilton

E-1 Mechanical Formulations

E-1.1 Hamilton's Law of Varying Action

E-1.2 Hamilton's Principle

E-1.3 Lagrange's Equations

E-1.4 Discussion

E-2 Hamilton's Principle for Electromechanical Systems Using a Displacement Charge Formulation

E-3 Hamilton's Principle for Electromechanical Systems Using a Displacement Flux Linkage Formulation

E-4 Work-Energy Relation Derived from Lagrange's Equations

Appendix F   Lagrange's Form of d'Alembert's Principle

F-1 Fundamental Concepts and Derivations

F-2 Examples

Appendix G   A Brief Review of Electromagnetic (EM) Theory and Approximations

G-1 Maxwell's Equations: Complete Form

G-1.1 Integral Form

G-1.2 Differential Form

G-2 Maxwell's Equations: Electrostatics and Magnetostatics

G-3 Maxwell's Equations: Electroquasistatics and Magnetoquasistatics

G-3.1 Electroquasistatics

G-3.2 Magnetoquasistatics

G-4 Energy Storage in Electroquasistatics and Magnetoquasistatics

G-4.1 Energy Storage in Electroquasistatics

G-4.2 Energy Storage in Magnetoquasistatics

G-5 Kirchhoff's ``Laws''

G-5.1 Kirchhoff's Current ``Law''

G-5.2 Kirchhoff's Voltage ``Law''

G-5.3 Summary

Appendix H   Complex Numbers and Some Useful Formulas of Complex Variables and Trigonometry

H-1 Introduction

H-2 Elementary Algebraic Operations of Complex Numbers

H-3 Complex Conjugates

H-4 A Useful Formula of Complex Variables

H-5 Use of Complex Variables in Harmonic Response Analyses

H-6 Useful Formulas of Trigonometry

Appendix I   Temporal Function for Synchronous Motion of Two Degree of Freedom Systems

I-1 Free Undamped Equations of Motion

I-2 Synchronous Motion

I-3 General Temporal Solution

I-4 Special (Semidefinite) Temporal Solution

I-5 Generalization to Systems Having More Degrees of Freedom

Appendix J   Stability Analyses of Nonlinear Systems

J-1 State-Space Stability Formulation

J-1.1 State-Space Representation of Equations of Motion

J-1.2 Equilibrium States

J-1.3 Linearization about Equilibrium States

J-1.4 Concept and Types of Stability

J-1.5 Stability of Linearized Systems

J-1.6 Local Stability of Nonlinear Systems

J-1.7 Nonlinear Stability Analyses

J-1.8 Summary of State-Space Stability Analysis

J-2 Nonlinear Stability Analysis for Conservative Systems

Appendix K   Strain Energy Functions

K-1 Concept

K-2 Strain Energy Density Function

K-3 Strain Energy Function

K-4 Examples