Introduction to Mechanics for Engineersby Andy Ruina & Rudra PratapCopyright and acknowledgements To the student

Preface

General issues about content, level, organization, style and motivation. Study advice starts on page To the student.

This is an engineering statics and dynamics text. It is an introduction, aimed primarily at engineering students, particularly mechanical, civil and biomedical engineers. The book emphasizes use of vectors, free-body diagrams, and computation. But, it’s not just cold recipes; the book is meant to help students build an intuition for mechanics.

The first four chapters cover the basic tools used in all of mechanics: vectors, free body diagrams, matrix math and ordinary differential equations (for dynamics). Then there are 5 chapters on engineering statics and 11 on engineering dynamics.

Prerequisite and co-requisite skills.

We assume students start with some math skills.

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Freshman calculus. Readers are assumed to have facility with the basic geometry, algebra, trigonometry, differentiation and integration used in elementary calculus. Some of these topics are briefly reviewed in this book, but not as ab initio tutorials.
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This book shows how to set up algebraic and differential equations for computer solution. You need to know, or be simultaneously learning, a computer language or package which can solve sets of linear algebraic equations, numerically integrate simple ordinary differential equations and make decent plots.

Exposure to other useful subjects might be useful, but is not assumed. For example:

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Completion of freshman physics may help but is not needed.
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Vector topics, especially dot and cross products, are introduced here from scratch in the context of mechanics.
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A background in linear algebra wouldn’t hurt, but the reduction of linear equations to matrix form is taught here. A key fact from linear algebra, also presented here, is that linear algebraic equations are usually easy to solve on a computer.
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A course in differential equations would also add perspective. But the basic concepts of differential equations are presented here as needed.

Coverage for courses.

The sections have been divided so that the homework problems selected from one section are usually about half of a typical weekly homework assignment; the theory and examples from one section might be adequately covered in about one lecture, plus or minus.

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Statics. A leisurely one-semester statics course, or a more fast-paced half-semester prelude to strength of materials should use chapters 1-9, excluding topics of less interest.
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Dynamics. A typical one-semester dynamics course will cover most of chapters 10-19, reviewing chapters 1-4 as needed.
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One-semester combined. A lower-level one-semester statics and dynamics course can cover the less advanced parts of chapters 1-7 and 10-17.
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Full-year combined. An advanced full-year statics and dynamics course could cover most of the book.
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Upper-level dynamics. The later parts of the book can serve as a start for more advanced dynamics courses.

A student who has learned the statics part of this book is well-prepared for using statics in engineering practice, for learning Strength of Materials and for going on to Dynamics. A student who has learned the dynamics portion is well prepared to go on to learn Vibrations, Systems Dynamics or more advanced Multi-object Dynamics.

Organization and formatting

Each subject is covered in various ways.

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Every section starts with descriptive text and short examples motivating and describing the theory;
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More detailed explanations of the theory are in boxes interspersed in the text. For example, one box explains the common derivation of angular momentum balance from ${\accentset{\rightharpoonup}{\boldsymbol{F}}}=m{\accentset{\rightharpoonup}{% \boldsymbol{a}}}$ (page LABEL:SystAngDeriv), one explains the genius of the wheel (page 5.3), and another connects ${\accentset{\rightharpoonup}{\boldsymbol{\omega}}}$ -based kinematics to $\hat{\boldsymbol{e}}_{r}$ - and $\hat{\boldsymbol{e}}_{\scriptscriptstyle\!\theta}$ -based kinematics (page 18.3);
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Sample problems (marked with a gray border) at the end of each section show how to do homework-like calculations. These set an example by their consistent use of free-body diagrams, systematic application of basic principles, vector notation, units, and checks against both intuition and special cases;
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Homework problems at the end of each chapter give students a chance to practice mechanics calculations. The first problems for each section build a student’s confidence with the basic ideas. The problems are ranked in approximate order of difficulty, with theoretical problems coming later. Problems marked with a * have an answer at the back of the book;
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Reference tables on the inside covers and end pages concisely summarize much of the content in the book. These tables can save students the time of hunting for formulas and definitions.

Notation

Clear vector notation helps students do problems. One common class of student errors comes from copying a textbook’s printed bold vector $\boldsymbol{F}\$ the same way as a plain-text scalar $F$ . We help reduce this error by use a redundant vector notation, e.g., a bold and harpooned ${\accentset{\rightharpoonup}{\boldsymbol{F}}}$ .

As for all authors and teachers concerned with motion in two and three dimensions (kinematics) we have struggled with the tradeoffs between a precise notation and a simple notation.

Perfectly precise notations are complex and intimidating. Simple notations can be ambiguous or hide key information.

For example, this expression for relative velocity, ${}^{\mathcal{B}}{\rm d}{\accentset{\rightharpoonup}{\boldsymbol{r}}}_{\!% \mathrm{P^{\prime}/0^{\prime}}}/{\rm d}t$ is probably intimidating to some, while this supposedly equivalent expression, ${\accentset{\rightharpoonup}{\boldsymbol{v}}}_{\rm rel}$ , is a bit too vague for others. Our attempt at clarity without too much clutter is summarized in the box on page 1.1.

Relation to other mechanics books

Although there are a few new details and approaches in this book, we admit that most of what is here can be found in other places, including freshman physics texts, other modern engineering texts, and hundreds of classic books.

Good freshman physics texts easily encompass half of this book’s contents. However, this book is a bit deeper, is more rigorous than many elementary physics texts, and is more oriented to engineering. Unfortunately, after freshman physics, too many students have only a vague notion of what mechanics is, and how it can be used. For example, some students leave freshman physics with the sense that a free-body diagram (or ‘force diagram’) is a vague conceptual picture with arrows for various forces and motions drawn on it this way and that. In the worst cases, freshman physics text illustrations sometimes do not make clear which force is acting on which object. Also, because freshman physics tends to avoid use of college math, many students leave freshman physics with little sense of how to use vectors or calculus to solve mechanics problems. In this book we try to lead students, students who may start with these fuzzy freshman-physics notions, into a world of precise, yet still intuitive, mechanics.

Various statics and dynamics textbooks cover much of the same material as this one. These textbooks have modern applications, ample samples, lots of pictures, and lots of homework problems. Many are excellent in some ways. Most of today’s engineering professors learned from one of these books. Nonetheless we wrote this book hoping to do still better.

Between about 1689 and 1980 hundreds of classic books were written with titles like Statics, Engineering mechanics, Dynamics, Machines, Mechanisms, Kinematics, or Elementary physics. Many thoughtfully cover most of the material here and sometimes much more. But, none are good modern textbooks; they lack an appropriate pace, style and organization; they are too reliant on geometry skills and not enough on vectors and numerics; and they don’t have enough modern applications, sample calculations, illustrations, or homework problems. There is much that can be learned from these older books

^†^†margin: Here are three good and universally respected classics:

J.P. Den Hartog, Mechanics, originally published in 1948 but still available as an inexpensive reprint (well written and insightful);

J.L. Synge and B.A. Griffith, Principles of Mechanics, through page 408. Originally published in 1942, reprinted in 1959 (good pedagogy but dry); and

E.J. Routh, Dynamics of a System of rigid bodies, Vol 1 (the “elementary” part through chapter 7. Originally published in 1905, but reprinted in 1960). Routh also has five other idea-packed statics and dynamics books. Routh shared college graduation honors with the now-more-famous physicist James Clerk Maxwell (of ‘Maxwell’s equations’).

. If you are really into mechanics you can find golden nuggets in these books.

This book is somewhat different in organization and approach than the others listed above. Some of our goals include

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showing the unity of the subject, while also
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presenting a complete description of the subject,
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using clear notation in figures and equations,
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using computers,
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using units consistently throughout,
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developing insights into how various things work, and
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using a friendly writing style.

This book also uses some important but not well-enough known concepts

^†^†margin: For example, we use angular momentum balance (appropriately expressed) with respect to any possibly-accelerating point, not just points selected from an arcane list.

What’s in these books?

The core equations in mechanics, including statics and dynamics, are balances of force, momentum and energy^†^†margin: To be precise, we also pay attention to balances of torque (moment) and both linear and angular momentum. . All of these balances are usually best represented using vectors and free body diagrams.

Vectors and Free Body Diagrams are the basic tools for setting up the equations for most mechanics problems.

Hence we have almost a whole short book (Book 1, Mechanics Toolset), on vectors and free body diagrams. There are other tools that are useful (or needed). So, we include here some tips on computer use, some things you should know about units and estimating sizes, and a brief introduction to ordinary differential equations.

Organization

At the highest level, there are three books (or, parts): 1) Mechanics Toolset (basic skills), 2) Statics, and 3) Dynamics. Within these, mechanics, the combination of Statics and Dynamics, progresses from elementary to advanced in four general ways:

1.
Number of spatial dimensions.
1. (a)
  
  One dimensional mechanics is the easiest. All forces are in one direction, say the $x$ direction (or the opposite), and there is no consideration of 2- or 3-dimensional geometry. Some people call this ‘scalar’ mechanics because vectors have minimal utility in 1D.
2. (b)
  
  Planar, or 2D mechanics is next easiest. 2D mechanics is most emphasized in simple applications. Geometry is important, but not too difficult.
3. (c)
  
  3D mechanics is most difficult. The geometry of three dimensions is surprisingly more difficult than 2 dimensions.
2.
Complexity of the motion.
1. (a)
  
  Statics assuming no motion (or, more precisely, negligible acceleration), is easiest.
2. (b)
  
  Straight-line motion is next easiest, assuming all points move parallel to a single given line, say the $x$ axis.
3. (c)
  
  Circular motion concerns systems where all points go in circles around a given point in a plane or, in 3D, around a given fixed axis.
4. (d)
  
  General motion, where points and objects can move any which way, is most general, and most difficult.
3.
Complexity of the system. In approximate order of difficulty, systems one can study in particle and rigid-object mechanics can be:
1. (a)
  
  A single particle.
2. (b)
  
  A system of particles.
3. (c)
  
  A rigid object.
4. (d)
  
  A collection of particles and rigid objects.
4.
The type of interaction. Parts of a system interact with each other with forces. In Dynamics, some interactions are easier to deal with than others.
1. (a)
  
  Forces determined by positions and velocities. The simplest interactions are when forces are from springs, dashpots and gravity. That is, when the forces can be found directly if you know the positions and velocities of all of the objects.
2. (b)
  
  Forces and accelerations are coupled. These are systems that have parts that interact with constraints like hinges and sliding joints. These ‘kinematic’ constraints are used to describe mechanisms.

Leaving aside the fourth category above (types of forces) a mechanics table of contents might have one chunk of text for each of the $2\times 3\times 3=18$ combinations:
I. Statics A. particle * 1D, 2D, 3D B. rigid object * 1D, 2D, 3D C. many objects * 1D, 2D, 3D II. Dynamics A. particle * 1D, 2D, 3D B. rigid object * 1D, 2D, 3D C. many objects * 1D, 2D, 3D ^†^†margin:

Filename:tfigure-wholecourse

However, these

2\times 3\times 3=18

chunks vary greatly in difficulty.

Simplest, the top sub-item from each list above, the statics of one particle, is almost trivial. In contrast, if one mastered the 3D multi-object dynamics of mechanisms, all of the other classifications are special, easier cases. But, 3D multi-object dynamics is generally considered an advanced graduate topic and is not part of this series of books.

So, in these books, we work our way partially down the lists above. After we cover the basic tools (Toolset: vectors, free body diagrams, and computation), we consider the simplest motion, that is no motion. This is called Statics. Then, in Dynamics we work our way from simpler to harder cases. The main emphasis, in this first course, is with 2-dimensional statics and dynamics. 3D statics is not wildly harder than 2D Statics, so it is reasonably covered. But 3D dynamics is, for most students, genuinely harder than 2D dynamics, so is only briefly introduced at the end.

What’s not in these books

These introductory books only go so far. Here are some things that are in some classical mechanics books but are not in these 3 books.

Relativity.: To physicists, classical mechanics also includes special and general relativity. To a physicist, the word ’classical’ means ‘deterministic’ or ‘non-quantum’, thus relativity, while coming hundreds of years after Newton, is still called classical. Here, classical mechanics means mechanics as understood by Newton and Euler. We have no discussion of relativity.
Lagrange’s equations.: A subset of mechanics problems^†^†margin: Advanced aside: Those with holonomic constraints and conservative forces are beautifully handled with Lagrange’s equations. Another related approach is Hamilton’s equations. A related idea is Hamilton’s principle. None of these are included here.
Multi-object formalisms.: Beyond not having Lagrange equations, we do not have any formal algorithms for dealing with multi-object dynamics systems.
Stress and strain tensors.: While we have slight discussion of stress and strain in the context of springs and bars, these books do not include 2- or 3-dimensional concepts of stress $\sigma_{ij}$ nor strain $\epsilon_{ij}$ . There are no tensors, and no Mohr’s circle here.
Beam and torsion theory.: While the book does include bending moment diagrams, we do not discuss the relation between curvature and moment in beams (i.e., no $M=EI/\rho$ ) nor the relation between twist and axial torque in rods (i.e., no $M=JG\phi/\ell$ ).

What do you think?

We have tried to make it as easy as possible for you to learn basic mechanics from this book. We present truth as we know it and as we think it is effectively communicated. Nonetheless we have surely made some technical and strategic errors. As you progress, please let us know your thoughts so that we can improve future editions.

Rudra Pratap,	pratap@iisc.ac.in
Andy Ruina,	andy.ruina@google.com