Preface by James H. Williams, Jr.
from Fundamentals of Applied Dynamics

SPLASH! The only copy of the freshly edited version of several chapters of the fundamental dynamics textbook I am writing is thrust overboard into the Caribbean Sea, as the penero nearly capsizes along the Archipielago Los Roques. Believing that there is a Force in the universe that is forever prepared to hurt us if we do not proceed calmly and take all things in stride, I relax. I withdraw. I reflect. I come to understand that there is no such entity as the book, only a book among all those which are the culminations of countless hours of moods and thoughts, among all those which do not splash into the Sea. Like so many before it, perhaps this loss too, if probed deliberately and deeply enough, will be almost worth the enlightenment.

As the manuscript strikes the surface of the water, almost instantly there is calm. Time stops, sounds cease, breaking waves hang in midair. My vision is telescoped onto the sinking manuscript; deeper and deeper it sinks into the crystal clear Caribbean Sea. I am inspired by the clarity of the water. Believing as I do that students find books on dynamics either (i) too difficult to read or/and (ii) so encumbered by a series of special cases that little scope is developed, my mission becomes clearer. As I edit my next draft, I must seek

• Clarity to quicken readers who will pore over my words in their independent efforts to acquire details;
• Structure to empower readers to approach with confidence new problems; and
• Perspective to embolden readers to acquire a broad view of the world because without it, they will forever remain incomplete.

Repetition will be used as a technique in support of each of these goals. Learning is volatile and requires a layering process. After one layer is allowed to set but before it evaporates, another layer must be applied. And so it builds, layer upon layer, until the principles for a lifetime are established. I have been told that one can become an excellent golfer by developing ``muscle memory'' through repetition, though I have no personal evidence of this. On the other hand, I believe that superb engineering or scientific skills can be developed only through the process of repetition.

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Clarity

Most authors seek clarity. As I use it here, clarity is intended to emphasize that this book is written for students. Students, whether taking a formal course or engaging in self-study, should expect to be able to read this book and more quickly understand what they read, more so than they may have come to expect of many engineering textbooks. This does not eliminate the teacher but should elevate the level of student-teacher interaction. A textbook should be literally a manual of instruction, not simply a compilation of topics, often requiring intense engagement by its readers in order to fulfill their implicit agreement with the author.

No newly encountered concept is incipiently clear. Repetition serves the goal of clarity. Many questions students have asked over the years and my responses to them often drive both the character and the length of my presentation. Examples sometimes precede theory; and although it may not be obvious--that is, until you decide to read comparable material elsewhere--there is a de-emphasis on formal mathematics in the body (nine chapters) of the book.

Throughout the book, examples will focus on the analysis of highly idealized, though very useful, models. In this way, emphasis on the principles of dynamics can be maintained. To a very modest extent, end-of-chapter problems require some modelling of physical conditions.

Structure

A good textbook must be more than clear; it must also be structured. Students, as do all analysts, need to feel secure in order to confidently confront and engage new problems. A textbook which fails to empower the reader in this manner has itself failed. The structure of a sound regimen is a vital aspect of such confidence. Structure will be found throughout the book; most notably associated with the discipline itself, the deliberate style of the presentation, and the format of the text, all as discussed below.

Discipline: (i) The two principal approaches to analysis in dynamics are drawn. In mechanical problems, these are called the direct (or momentum) approach and the indirect (or work-energy) approach. (ii) Regardless of the approach used, three fundamental requirements must be satisfied. These points are the focus of Chapter 2 (eqn.\ (2-1)). (iii) Within the indirect approach, four steps to equation formulation are delineated. This point is emphasized in Chapter 5 (Table 5-1). These disciplinary aspects of structure are associated more with formulation of the equations of motion as opposed to the solution of the equations of motion.

Style of Presentation: The repetition which serves clarity also serves structure. The unorthodox style of repeating phraseology in deriving equations of motion for different systems is intentional and without apology. In so doing, I seek to emphasize a consistency of thought and technique. I want to reinforce a mode of thought as much as I want to deliver the results; indeed, a major part of the ``results'' is the mode of thought which I seek to inculcate.

Format: (i) The disciplinary structure outlined above is exploited in numerous examples. The principles are comparably few and easy to articulate; the applications are exceedingly broad and rich. The purpose of the examples is to expose the variety of nuances which arise time after time in a broad range of models. The examples are highly structured in a consistent regimen of steps, becoming increasingly difficult as the reader proceeds. (ii) Summary tables, often in the form of flow charts, are provided throughout the book. The flow charts which appear primarily in the solutions oriented--as opposed to the formulation oriented--chapters are provided to reinforce the appropriate structure. (iii) The twelve appendixes serve several purposes, including providing (1) relevant material which is new but extends beyond the goals or level of the book--Appendixes A and C are of this kind; (2) relevant material which is not new but not readily accessible in elementary or intermediate textbooks--Appendix B is of this kind; (3) material which simply shifts many detailed and repetitive mathematical manipulations away from the chapters of the book--Appendixes E, I, J, K and L are of this kind; (4) theoretical foundations, sometimes opinionated and unorthodox in their presentation, which are omitted from the body of the book and which represent the underpinnings of the discipline--Appendixes D, F, G and H are of this kind. Therefore, these appendixes provide underlying theoretical and mathematical support for the body, thus streamlining the main presentation and enhancing its structure. Furthermore, these appendixes will allow the teaching of the subject in a more theoretical format, particularly if the book is used for an introductory (post)graduate course. In an initial exposure to this material, few of the appendixes will be necessary, particularly for the undergraduate who will likely find much of the material to be beyond that which is urgent.

Perspective

The perspective which I seek to enhance is an appreciation for the history of dynamics as well as a framework for future learning within this discipline or any discipline. Historical references throughout the book serve this goal. Also, I am reaching out to students who will study dynamics in the future as well as those who will not. In some respects, I want students to view me as a somewhat knowledgeable colleague, but as a colleague nonetheless. I am seeking to generate interest in potential applications, intellectual interest in the underlying concepts, and the ability to read comparable and more advanced literature. Some of the materials--especially several of the appendixes--which are not used substantially in the book serve these goals of perspective for future study. Ultimately, as the book's title suggests, I am trying to probe and expose the underlying fundamentals of applied dynamics. There is every attempt not to ``sweep under the rug'' or circumvent any of the subtleties of the subject. I have seen enough obscurity in textbooks to last a lifetime.

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Chapter 1 is my own unorthodox and personalized historical perspective. It is at once as obvious to me as much as it may be debatable to others. It is virtually impossible to write anything in the history of science without incurring the ire of one or more ``professionals''. The history of any subject is invariably richer than any categorization or sequencing of its features can reveal. Here, I want to raise and refresh the spirit of several of the great contributors to the discipline of dynamics. My purpose is to provoke and to encourage readers who care about such matters to read broadly and thoughtfully and then to formulate their own perspectives. Furthermore, very few, if any, ideas, theories or inventions can be cited as having sprung exclusively from the mind of a single individual. The names of great individuals as cited here serve as mileposts in the evolution of civilization as much as for purposes of attribution and are not attempts at deification. The concept of the illuminating light bulb in the head of an isolated genius is frequently and substantially a fabrication--a convenience for various mythologies and comic strips, but little else. I recommend that professors who use this book in their teaching of dynamics not assign Chapter 1 as required reading since students will read it on their own ... Hold it; forget what you just read! I recommend that professors who use this book in their teaching of dynamics assign Chapter 1 as required reading, or perhaps just for skimming. Its contents may not be fully digested in the first reading. Some students may come to regard it like a symphony orchestra in the city which, though rarely attended, provides comfort simply in the knowledge that it's there; a journey available, but nevertheless a journey declined. Other students may return to it--including its bibliography--as a useful reference, even beyond the purview of dynamics.

Chapter 2 emphasizes two major points: (i) whether obvious or obscure, the ultimate technical, as opposed to philosophical, goal of all our analyses is engineering design; and (ii) the formulation of the equations of motion for mechanical systems necessitates the satisfaction of three fundamental requirements--kinematical, force-dynamical, and constitutive in character. A third major topic is skirted, namely, the quantitative modelling of engineering systems. An early draft of this book dealt with modelling to a much greater degree than here; Appendix A is a condensed compromise between that earlier presentation and the present one which attempts very little on the matter of quantitative modelling. (In passing, I note that it is a widely held tenet in some scientific circles that everything in the universe can be modelled. Some people would argue that aesthetics or beauty or even perhaps love can be quantitatively analyzed. My own view is that love simply represents the identical neural configuration in an individual as when he or she is eating fine chocolate. Appendix A makes the point that a goal to quantitatively model everything of importance in engineering is unachievable.) Ultimately, I concluded that the goal of emphasizing the fundamentals of applied dynamics would be diluted too much by the inclusion of the topic of quantitative modelling.

I want students to understand that throughout this book the goal of the analyses is design. We are clearly investigating dynamics, but we want to maintain a cognizance of design. Here's what I mean.

In most courses in dynamics the results of the analyses are the motions of bodies or the force accompanying those motions or simply the equations of motion. Indeed, these three types of ``results'' are the respective cores of Chapters 3, 4 and 5; and if students fully appreciate these three aspects of dynamics, they will be well along in their understanding of the scope of the discipline. However, a course in dynamics which terminates there and leaves students with the impression that these are the results has not fulfilled its potential. Equations of motion are obviously intermediate results, but frequently, so too are the motions and forces obtained from those equations. To be of practical engineering value, such forces and motions must be related to some design criteria or specifications, including, for example, stresses and strains which should be assessed to be acceptable or unacceptable. In this regard, important design features including the geometry and materials properties of components can be isolated to assess their impact on stresses and strains in individual members. By the term design, I intend to emphasize this point. While detailed design goals can not be explored here--after all, as discussed in Appendix A, design requires addressing a broad set of criteria--if students who use this book realize that the calculated motions and forces often represent intermediate results leading to a stress analysis, materials selection, or (and) the satisfaction of a manufacturing or operating specification,then my exhortations on this matter will have been adequately rewarded.

Chapter 3 is a presentation of three-dimensional kinematics, kinematics being the study of the geometry of motion without regard for forces. Chapter 3 generalizes kinematics in a manner which I believe the reader will find to be unusual in its approach and efficacy as well as highly accessible. Chapter 3 (and its cited appendixes which I do not recommend during an initial encounter) will richly reward all the careful attention which the reader chooses to devote to it.

Chapter 4 formally introduces the direct approach, variously called vectorial mechanics or the momentum approach. With Newton's laws as the foundation, several ``kinds'' of dynamical problems are defined. Then, following the re-emphasis of the three fundamental requirements (kinematical, force-dynamical, and constitutive), several examples reveal the structure and simplicity of many dynamical formulations. One feature of the flexibility of the book is that the reader (or instructor) may spend a week--as I generally do--or month(s) on this chapter, or skip it altogether without loss of continuity. (Indeed, the book is so constructed that the same could be said for several chapters.)

This textbook emphasizes the work-energy approach (indirect approach), although there is an attempt to contrast and compare it with the momentum approach (direct approach). This choice of emphasis is not because students fully understand the momentum principles which they have previously encountered in their physics courses. After all, as I rediscover every time I teach dynamics, a mature understanding of momentum principles is a lifelong pursuit. It's simply that to repeat the principles of dynamics in the same style with many of the same problems as previously encountered is a reliable prescription for boring most students, especially the more brilliant ones, including those who regularly watch MTV. My instructional philosophy is to remind students that they have already studied momentum principles, and as I revisit such principles (which they too must repeat along with me) I want to broaden the disciplinary and philosophical context of those familiar ideas to include work-energy principles, both approaches in some semblance of their respective proper historical placements.

On a separate note regarding Chapter 4, in problems where there is a constant force, there will be a constant acceleration; the model of gravity in the vicinity of the Earth's surface is the most frequently cited example of this kind. Such problems are frequently treated exclusively via kinematics, which is not strictly correct, notwithstanding Galileo Galilei. Accelerations imply the presence of forces, requiring some constitutive relation which connects the two. An appreciation of this requirement between forces and accelerations is important in understanding the underlying structure of dynamical formulations, and is thus important in reinforcing the strong analogy between statics and dynamics.

Chapter 5 formally introduces the indirect approach, variously called lagrangian dynamics or variational dynamics. The presentation is particularly deliberate. The frequently maligned and misunderstood concepts of energy and coenergy are introduced and delineated. Here we adopt Hamilton's principle as our fundamental tenet, and Lagrange's equations as a derivable consequence which is useful for lumped-parameter systems. Table 5-1 summarizes the structure of the formulation of the equations of motion for lumped-parameter systems. The four-step procedure in Table 5-1 is used repeatedly as it will unquestionably become a part of the student's analytical lexicon. With the approach exposed in Table 5-1, only the details of its application remain. Each of the examples which follow Table 5-1 contains one or more nuances which expose the details of the application of the lagrangian formulation. It is important to appreciate each of these subtleties, as indeed these are the fine points that will be repeatedly required in the end-of-chapter problems, and ultimately in professional practice.

Chapter 6 expands both the momentum formulation of Chapter 4 and the lagrangian formulation of Chapter 5 to extended bodies, but with an emphasis on the lagrangian approach. Prior to the principal business of the lagrangian formulation, a summary of momentum principles for particles and extended bodies is given in Table 6-1. This table and the accompanying analyses lay out clearly (i) the relationship between linear momentum and angular momentum for particles (ii) the relationship between linear momentum and angular momentum for extended bodies and (iii) the independence of linear momentum and angular momentum for extended bodies. The concept of a rigid body is defined. In extending the lagrangian formulation for particles to the lagrangian formulation for rigid bodies, the very important inertia tensor is derived, as are the kinetic coenergy function\/ and the potential energy function. Several nuance revealing examples complete the chapter.

Chapter 7 presents a lagrangian formulation of lumped-parameter electrical and electromechanical devices. In an accompanying appendix (Appendix H), Maxwell's equations are used to derive the required energy functions as well as Kirchhoff's rules which are used in the lagrangian formulation of such systems. To my knowledge, equations of motion for operational amplifiers are formulated via this approach for the first time. (I have foregone the introduction of transistors via this approach.) An illuminating summary table (Table 7-2: LOVE) is provided for electrical networks; the structure of the table should figuratively catapult itself toward the reader.

Chapter 8 contains the response analyses of one and two degree of freedom systems. Concepts of stability are introduced. Although the role of complex analysis is introduced, the chapter can be completed with little encounter of complex variables. For example, in obtaining the response of such systems to harmonic excitation, the particular solutions are obtained via complex variables as an alternative to working exclusively with real variables. Such a parallel presentation is designed to allay some of the discomfort to which students surrender which encountering the eerie little i = square root of -1.

Chapter 9 is a detailed presentation of the formulation of equations of motion for one-dimensional continuum models. In addition to the continuing emphasis on fundamentals and underlying disciplinary structure, the reader will find a renewed display of detail in both the formulation and the solution phases of the analyses. These details--although sometimes complicated and nearly always omitted in textbooks--are exhibited because they too possess a structure which I hope will be revealed. Readers who think the presentation is too detailed should do something more useful such as reading a different book, mowing the lawn or watching television; readers who want to learn about the formulation, solution, and the disciplinary structure of continuum analyses will find them introduced here.

It has been said variously that to teach is to choose. The number of topics which were drafted but not inducted is significant. For example, among these topics are the following: three-dimensional rigid body dynamics, especially Euler angles and related material on gyroscopes; orbital dynamics; nonholonomic systems; variational exposition on transistors; electric motors and generators; dynamic reciprocity; Rayleigh's principle; extended state-space analysis; and Fourier analysis and related spectral analysis. In omitting these topics, I was able to hold to the length of the volume in your hands, and still not compromise my expressed goals.

Several points may be made regarding the administrative aspects of the book: (i) the designations of the measures of the book, in decreasing order, are chapters, sections, subsections and subdivisions; (ii) concerning units, both the metric system and the British or U.S. customary system are used at will; and (iii) although a comprehensive list of references is given at the end of the book, in general, references within the text are held to a minimum by the mere expedient of ignoring them.

This is a textbook on the fundamentals of applied dynamics. As such, it attempts to present a coherent unification of newtonian (yet another name for the direct approach) and lagrangian dynamics. The presentation on newtonian dynamics, which the reader has encountered previously, is more in the context of perspective; the more extensive presentations are on lagrangian dynamics. Equations of motion are formulated and solutions are obtained, with an emphasis on solutions of linear mechanical systems although electrical and electromechanical systems are also analyzed. Throughout the book, there is reiteration of structure, both in the formulations and the solutions.

The book is constructed to appeal to students--undergraduate and (post)graduate--and practicing engineers who seek (i) an accessible comprehensive introduction to applied dynamics and/or (ii) a coherent perspective of the theoretical underpinnings of the discipline, particularly via more extensive use of the appendixes. It is the use of the appendixes, the choice of homework problems, and the pace which will distinguish the initial and intermediate level courses.

The prerequisites for a course using the book are quite modest. They include a first course in differential and integral calculus, including vector analysis, and a first course in physics in which calculus is used. In the United States, these are typically taken in the freshman year in most engineering and science curricula. Also, first courses in differential equations and the mechanics of solids are recommended although such courses may be studied contemporaneously.

The book can be used for a one quarter introductory course (Chapters 1 through 4 or 5; or Chapters 1 through 4 plus the direct approach in Chapter 6); a one semester course (Chapters 1 through 6 plus dabs of one or more of the other chapters); or a full year course (Chapters 1 through 9). Thus, as suggested by the one quarter course, the book can be used exclusively for the direct approach in a first quarter, the indirect approach in a second quarter, and vibration in a third quarter. Further, with the appropriate omissions, an intermediate course might cover essentially the entire book in a semester.

In the one semester undergraduate course in Mechanical Engineering at MIT, I use various sections throughout the book. I cover Chapters 1 through 6, with Chapters 1, 2 and 4 used primarily for perspective. Thus, great emphasis is placed on Chapters 3, 5 and 6. Then, I typically cover portions of Chapters 7 and/or 8 and/or 9.

End-of-chapter problems are cast in three categories. The first category (labelled I), consisting of approximately one-sixth of the problems, is of an elementary nature. In chapters containing kinematics (Chapter 3), momentum principles (Chapters 4 and 6), electrical networks (Chapter 7) and vibration (Chapter 8), students will find these problems comparable to those already encountered in freshman physics. Indeed, these problems have been extracted directly from the superb elementary Wiley physics textbook by Halliday, Resnick and Walker (HRW). It is these problems which should reinforce my cited level of entry for this textbook as first-level undergraduate. In HRW, readers will find alternative and often enlightening discussions of many of the topics presented in this textbook. Readers are strongly encouraged to return to HRW or any physics textbook for such enhancing discussions. The second category (labelled II), consisting of approximately one-half of the problems, begins at the level of the typical engineering sophomore; such problems are covered in books listed in Section 10-4 of the references. These problems further underscore the undergraduate level of entry for this textbook. The problems in category II, however, become increasingly difficult and subtle, and it is their solutions which should make clearer both the breadth and the efficacy of the approaches emphasized here. That is--and this is very important--problems which may be familiar to the reader will be addressed distinctly differently here, in accordance with the regimens which I advocate. The third category (labelled III), comprising the remainder of the problems, consists of exercises which are generally more difficult than those in category II, and frequently with the additional feature of emphasizing design concepts or issues.

Clearly, categories II and III--indeed all three categories--could have been mingled; they are in Chapter 7. There is an instructional pedagogy embedded in providing these categories which is somewhat counterbalanced by an intellectual drawback; after all, the ``real world'' does not present itself to us so neatly categorized. To teach is to choose.

The path through the subject which I cut is both a subtle and personalized one. It is a path which explores both elementary and intermediate level concepts in dynamics, and which is directed to the uninitiated readers for whom this book may represent their sole professional encounter with the discipline, as well as to readers who seek disciplinary foundations for advance study. Essentially every mathematical manipulation can be constructed by visual reference only; the remainder may require extremely modest operations. Keep in mind that this is neither dynamics for poets nor a formal treatise; it is a dynamics textbook--a manual of instruction.

I take seriously the statement often attributed to Albert Einstein that ``Everything should be made as simple as possible, but no simpler'', adding here that everything should also be repeated. When asked ``How long should a man's legs be?'' Abraham Lincoln is reputedly to have responded ``Long enough to reach the ground.'' In writing several of the sections which are indeed longer than the reader is likely to find on comparable material in other textbooks, I sought only to make the concepts as simple as possible but long enough--with enough detail--so that students, in completing the material, would feel that their feet were on solid ground. They should have acquired clarity of my exposition, a sense of its formal structure, and a perspective of both its historical development and its potential application.

Learning should be fun. Enjoy.