Fundamentals of Applied Dynamics

Citation

Williams, J. H., Jr. (1996). Fundamentals of Applied Dynamics. John Wiley & Sons.

Chapter Summary

The book has nine chapters and a set of appendices, and runs to 854 pages. Its organising idea, set out in the author's preface, is a unification of Newtonian dynamics (the direct, or momentum, approach) with Lagrangian dynamics (the indirect, variational or work-energy approach). The momentum approach is treated mostly as perspective on material readers have met before; the longer presentations are on the Lagrangian approach. Equations of motion are formulated and then solved, mainly for linear mechanical systems, with electrical and electromechanical systems also included.

Chapter 1: Our Niche in the Cosmos

A historical chapter, which the preface calls a personal and unorthodox perspective. It looks at the sources from antiquity, ancient Egyptian and Mesopotamian astronomy and mathematics, mathematics of other early cultures, the Hellenic era, the unification of celestial and terrestrial motion, and variational principles in dynamics. The author suggests it be assigned for reading or skimming, and it can serve as a reference for readers interested in history.

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

The chapter has three main sections: introduction, design and modeling (the design process, the modeling process and the book's more modest goals), and the direct and indirect approaches to formulating equations of motion. The preface says it stresses that the technical goal of the analysis is engineering design, and that any formulation must satisfy three requirements: kinematical, force-dynamical and constitutive.

Chapter 3: Kinematics

The chapter covers position, velocity and acceleration, plane kinematics of rigid bodies (including types of plane motion and a cautionary note about finite rotations), the time rate of change of a vector in a rotating frame, kinematic analysis using intermediate frames, and generalizations of the kinematic expressions. The preface presents it as a three-dimensional treatment with an unusual approach.

Chapter 4: Momentum Formulation for Systems of Particles

The direct approach. It sets out Newton's laws, the particle, linear momentum and force, inertial frames and the universal law of gravitation; then torque and angular momentum for a particle; then worked formulations in two kinds of particle-dynamics problems. The preface says an instructor can spend a week or more on this chapter or skip it without losing continuity.

Chapter 5: Variational Formulation for Systems of Particles

The indirect approach. Sections cover work and state functions (including energy and coenergy), generalized coordinates, admissible variations, degrees of freedom, constraints and holonomicity, variational principles, equations of motion for holonomic systems, the work-energy relation and the nature of Lagrangian dynamics. The preface takes Hamilton's principle as the fundamental tenet, with Lagrange's equations as a consequence, and points to a four-step procedure summarised in Table 5-1.

Chapter 6: Dynamics of Systems Containing Rigid Bodies

Both formulations are extended to extended bodies. It covers momentum principles and the Newton-Euler equations, the inertia tensor, parallel-axes theorem, principal directions and moments of inertia, dynamics of rigid bodies by the direct approach, the Lagrangian for rigid bodies (kinetic coenergy and potential energy functions) and equations of motion for systems in plane motion.

Chapter 7: Dynamics of Electrical and Electromechanical Systems

The Lagrangian approach applied to circuits and transducers: formulation of equations of motion for electrical networks, constitutive relations for circuit elements, Hamilton's principle and Lagrange's equations with generalized charge and flux-linkage variables, constitutive relations for movable-plate capacitors and movable-core inductors, and electromechanical systems.

Chapter 8: Vibration of Linear Lumped-Parameter Systems

Responses of single-degree-of-freedom first-order and second-order systems (free, step, ramp and harmonic response; natural frequency via static deflection; logarithmic decrement; energy loss), two-degree-of-freedom systems (natural modes, response to initial conditions, harmonic response) and the stability of nonlinear systems. The preface notes that complex variables are introduced alongside real-variable methods.

Chapter 9: Dynamics of Continuous Systems

Equations of motion for one-dimensional continua: longitudinal motion of a rod, twisting of a shaft, an electric transmission line and flexural motion of a beam. Natural modes by separation of variables, eigenfunctions for second- and fourth-order systems, and responses to initial conditions and to harmonic excitation, each illustrated with examples.

Appendices

The appendices listed in the table of contents cover finite rotation, general kinematic analysis, momentum principles for systems of particles, elementary results of the calculus of variations, formulations of Hamilton's principle, Lagrange's form of d'Alembert's principle, a review of electromagnetic theory and approximations, complex numbers and useful formulas of complex variables and trigonometry, the temporal function for synchronous motion of two-degree-of-freedom systems, stability analyses of nonlinear systems, and strain energy functions. The preface says that few appendices are needed on a first exposure and that much of their material lies beyond what an undergraduate urgently needs.

Key Concepts

Chapters 1–3: Perspective, modeling and kinematics

• Three fundamental requirements: any formulation of the equations of motion must be kinematical, force-dynamical and constitutive.
• Direct and indirect approaches: momentum-based and variational or work-energy formulations of the same problem.
• Kinematics in rotating and intermediate frames: tracking how a vector changes when viewed from a rotating frame.

Chapters 4–6: Formulation of equations of motion

• Momentum principles and the Newton-Euler equations: the direct approach for particles and extended bodies.
• Energy and coenergy; generalized coordinates: the quantities used to formulate Lagrange's equations.
• Hamilton's principle and Lagrange's equations: the variational route to equations of motion for holonomic systems.
• Inertia tensor: the property that characterises how a rigid body's mass is distributed.

Chapters 7–9: Extensions and response

• Electromechanical systems: lumped-parameter circuits and transducers treated with the same variational framework.
• Free and forced response: first- and second-order, and two-degree-of-freedom, linear systems.
• Continuous systems: rods, shafts, transmission lines and beams, solved by natural modes.

Critical Analysis

Strengths

• A deliberate structure. The preface describes a consistent regime: three fundamental requirements, a four-step procedure for the indirect approach, and structured examples that become progressively harder, with summary tables and flow charts.
• Both approaches are developed side by side. Chapters 4 and 5 present the direct and the indirect approaches, and Chapter 6 extends both, so a reader can compare the formulations on the same problems.
• A wider scope than a standard dynamics course. The book includes electrical and electromechanical systems and continuous systems, and carries the formulation into vibration.
• Support for different levels. The preface gives course routes (a quarter course on Chapters 1–4 or 5; a semester on Chapters 1–6; a full year on Chapters 1–9) and describes three problem categories: elementary ones (about one-sixth), problems at sophomore engineering level (about half) and harder ones that often emphasise design (the rest).

Limitations

• Selective in topics. The author lists topics drafted but left out, among them three-dimensional rigid-body dynamics with Euler angles and gyroscopes, orbital dynamics, nonholonomic systems, electric motors and generators, Rayleigh's principle and Fourier analysis.
• Limited modeling. The preface says the book gives very little on quantitative modeling of physical systems, which was reduced from an earlier draft.
• Unconventional style. The author states that the repetition of phrasing and the de-emphasis of formal mathematics in the body are intentional. Readers who prefer a formula-driven, problem-type approach may find the emphasis on formulation unfamiliar.
• Mostly linear systems. The solutions concentrate on linear mechanical systems; nonlinear systems appear mainly in the stability analysis of Chapter 8.

Where It Fits in the Course

In this statics and dynamics course, Williams is the more theory-oriented dynamics text. Hibbeler's Engineering Mechanics: Statics and Meriam and Kraige's Engineering Mechanics: Dynamics cover the vectors, force balances, energy and momentum used to solve engineering problems; Williams does not replace them but approaches the same ground from the direct and Lagrangian formulations. Bird and Ross's Mechanical Engineering Principles is a student-friendly entry for readers starting engineering, and the OpenStax University Physics Volume 1 supplies the calculus-based mechanics that Williams assumes from a first physics course.

Rao's Mechanical Vibrations is a detailed vibration text covering single- and multi-degree-of-freedom systems, measurement and control. Williams' Chapters 8 and 9 give a shorter treatment of vibration and continuous systems, from the viewpoint of the equations of motion formulated earlier in the book, so Rao is the natural follow-up for a specialised vibrations course.

How to Use This Book

The preface describes the prerequisites as modest: calculus including vector analysis, a first physics course using calculus, and recommended (possibly concurrent) courses in differential equations and mechanics of solids. It is intended for undergraduate and graduate students and practising engineers who want an accessible introduction to applied dynamics or a coherent view of its theoretical foundations through the appendices.

A practical plan:

• First course: Chapters 1–4 or 5, with Chapter 3 and Chapter 5 given the most attention. Chapter 1 can be read as background and Chapter 4 reviewed quickly if momentum principles are familiar.
• Extended mechanics course: add Chapter 6 for rigid bodies, then selected parts of Chapters 7, 8 or 9 according to interest in electromechanical systems, vibration or continua.
• Theory-oriented readers: use the appendices on finite rotation, the calculus of variations and Hamilton's principle alongside Chapters 3 and 5.

Solve the same problem by the Newtonian and the Lagrangian route whenever you can, since the comparison is the point of the book. Take the end-of-chapter problems in the order of their categories, from the elementary to those that involve design.