Engineering Dynamics: A Comprehensive Introduction - Hardcover

Kasdin, N. Jeremy; Paley, Derek A.

Hardcover

ISBN 10: 0691135371 ISBN 13: 9780691135373

Publisher: Princeton University Press, 2011

Synopsis

An accessible yet rigorous introduction to engineering dynamics

This textbook introduces undergraduate students to engineering dynamics using an innovative approach that is at once accessible and comprehensive. Combining the strengths of both beginner and advanced dynamics texts, this book has students solving dynamics problems from the very start and gradually guides them from the basics to increasingly more challenging topics without ever sacrificing rigor.

Engineering Dynamics spans the full range of mechanics problems, from one-dimensional particle kinematics to three-dimensional rigid-body dynamics, including an introduction to Lagrange's and Kane's methods. It skillfully blends an easy-to-read, conversational style with careful attention to the physics and mathematics of engineering dynamics, and emphasizes the formal systematic notation students need to solve problems correctly and succeed in more advanced courses. This richly illustrated textbook features numerous real-world examples and problems, incorporating a wide range of difficulty; ample use of MATLAB for solving problems; helpful tutorials; suggestions for further reading; and detailed appendixes.

• Provides an accessible yet rigorous introduction to engineering dynamics
• Uses an explicit vector-based notation to facilitate understanding

Professors: A supplementary Instructor's Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: https://press.princeton.edu/class_use/solutions.html

About this title

About the Author

N. Jeremy Kasdin is professor of mechanical and aerospace engineering and lead investigator for the Terrestrial Planet Finder project at Princeton University. Derek A. Paley is assistant professor of aerospace engineering and director of the Collective Dynamics and Control Laboratory at the University of Maryland.

From the Back Cover

"There are few courses in the engineering curriculum that cause students more difficulty than rigid-body dynamics. By laying out the foundations of the subject with precision and clarity through unambiguous notation and rigorous definitions, Engineering Dynamics goes a long way toward remedying this situation. Numerous examples with motivating applications demonstrate the underlying ideas and solution techniques. This landmark text stands apart in the field, and will be welcomed by students and instructors alike."--Dennis S. Bernstein, University of Michigan

"Kasdin and Paley provide a thorough and rigorous introduction to engineering dynamics. They hit all the required topics, and also present material not normally addressed by an introductory text. This is an ambitious book and the authors carry it out well. It is in many ways better than almost all other comparable texts."--Geoffrey Shiflett, University of Southern California

From the Inside Flap

"There are few courses in the engineering curriculum that cause students more difficulty than rigid-body dynamics. By laying out the foundations of the subject with precision and clarity through unambiguous notation and rigorous definitions, Engineering Dynamics goes a long way toward remedying this situation. Numerous examples with motivating applications demonstrate the underlying ideas and solution techniques. This landmark text stands apart in the field, and will be welcomed by students and instructors alike."--Dennis S. Bernstein, University of Michigan

"Kasdin and Paley provide a thorough and rigorous introduction to engineering dynamics. They hit all the required topics, and also present material not normally addressed by an introductory text. This is an ambitious book and the authors carry it out well. It is in many ways better than almost all other comparable texts."--Geoffrey Shiflett, University of Southern California

Excerpt. © Reprinted by permission. All rights reserved.

ENGINEERING DYNAMICS

A Comprehensive IntroductionBy N. Jeremy Kasdin Derek A. Paley

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-13537-3

Contents

Preface...................................................................................xiChapter 1. Introduction...................................................................1Chapter 2. Newtonian Mechanics............................................................11Chapter 3. Planar Kinematics and Kinetics of a Particle...................................45Chapter 4. Linear and Angular Momentum of a Particle......................................113Chapter 5. Energy of a Particle...........................................................148Chapter 6. Linear Momentum of a Multiparticle System......................................189Chapter 7. Angular Momentum and Energy of a Multiparticle System..........................245Chapter 8. Relative Motion in a Rotating Frame............................................295Chapter 9. Dynamics of a Planar Rigid Body................................................337Chapter 10. Particle Kinematics and Kinetics in Three Dimensions..........................409Chapter 11. Multiparticle and Rigid-Body Dynamics in Three Dimensions.....................465Chapter 12. Some Important Examples.......................................................537Chapter 13. An Introduction to Analytical Mechanics.......................................580Appendix A. A Brief Review of Calculus....................................................623Appendix B. Vector Algebra and Useful Identities..........................................635Appendix C. Differential Equations........................................................645Appendix D. Moments of Inertia of Selected Bodies.........................................660Bibliography..............................................................................663Index.....................................................................................667

Chapter One

Introduction

1.1 What Is Dynamics?

Dynamics is the science that describes the motion of bodies. Also called mechanics (we use the terms interchangeably throughout the book), its development was the first great success of modern physics. Much notation has changed, and physics has grown more sophisticated, but we still use the same fundamental ideas that Isaac Newton developed more than 300 years ago (using the formulation provided by Leonhard Euler and Joseph Louis Lagrange). The basic mathematical formulation and physical principles have stood the test of time and are indispensable tools of the practicing engineer.

Let's be more precise in our definition. Dynamics is the discipline that determines the position and velocity of an object under the action of forces. Specifically, it is about finding a set of differential equations that can be solved (either exactly or numerically on a computer) to determine the trajectory of a body.

In only the second paragraph of the book we have already introduced a great number of terms that require careful, mathematical definitions to proceed with the physics and eventually solve problems (and, perhaps, understand our admittedly very qualitative definition): position, velocity, orientation, force, object, body, differential equation, and trajectory. Although you may have an intuitive idea of what some of these terms represent, all have rigorous meanings in the context of dynamics. This rigor—and careful notation—is an essential part of the way we approach the subject of dynamics in this book. If you find some of the notation to be rather burdensome and superfluous early on, trust us! By the time you reach Part Two, you will find it indispensable.

We begin in this chapter and the next by providing qualitative definitions of the important concepts that introduce you to our notation, using only relatively simple ideas from geometry and calculus. In Chapter 3, we are much more careful and present the precise mathematical definitions as well as the full vector formulation of dynamics.

1.1.1 Vectors

We live in a three-dimensional Euclidean universe; we can completely locate the position of a point P relative to a reference point O in space by its relative distance in three perpendicular directions. (In Part One we talk about points rather than extended bodies and, consequently, don't have to keep track of the orientation of a body, as is necessary when discussing rigid bodies in Parts Three and Four.) We often call the reference point O the origin. An abstract quantity, the vector, is defined to represent the position of P relative to O, both in distance and direction.

Qualitative Definition 1.1 A vector is a geometric entity that has both magnitude and direction in space.

A position vector is denoted by a boldface, roman-type letter with subscripts that indicate its head and tail. For example, the position r_P/O of point P relative to the origin O is a vector (Figure 1.1). An important geometric property of vectors is that they can be added to get a new vector, called the resultant vector. Figure 1.1b illustrates how two vectors are added to obtain a new vector of different magnitude and direction by placing the summed vectors "head to tail."

When the position of point P changes with time, the position at time t is denoted by r_P/O(t). In this case, the velocity of point P with respect to O is also a vector. However, to define the velocity correctly, we need to introduce the concept of a reference frame.

1.1.2 Reference Frames, Coordinates, and Velocity

We have all heard about reference frames since high school, and you may already have an idea of what one is. For example, on a moving train, objects that are stationary on the train—and thus with respect to a reference frame fixed to the train—move with respect to a reference frame fixed to the ground (as in Figure 1.2). To successfully use dynamics, such an intuitive understanding is essential. Later chapters discuss how reference frames fit into the physics and how to use them mathematically; for that reason, we revisit the topic again in Chapter 3. For now, we summarize our intuition in the following qualitative definition of a reference frame.

Qualitative Definition 1.2 A reference frame is a point of view from which observations and measurements are made regarding the motion of a system.

It is impossible to overemphasize the importance of this concept. Solving a problem in dynamics always starts with defining the necessary reference frames.

From basic geometry, you may be used to seeing a reference frame written as three perpendicular axes meeting at an origin O, as illustrated in Figure 1.3. This representation is standard, as it highlights the three orthogonal Euclidean directions. However, this recollection should not be confused with a coordinate system. The reference frame and the coordinate system are not the same concept, but rather complement one another. It is necessary to introduce the reference frame to define a coordinate system, which we do next.

Definition 1.3 A coordinate system is the set of scalars that locate the position of a point relative to another point in a reference frame.

In our three-dimensional Euclidean universe, it takes three scalars to specify the position of a point P in a reference frame. The most natural set of scalars (the three numbers usually labeled x, y, and z) are Cartesian coordinates. These coordinates represent the location of P in each of the three orthogonal directions of the reference frame. (Recall the discussion of vectors in the previous section stating that the position of P relative to O is specified in three perpendicular directions.) Cartesian coordinates, however, are only one possible set of the many different scalar coordinates, a number of which are discussed later in the book. Nevertheless, we begin the study of dynamics with Cartesian coordinates because they have a one-to-one correspondence with the directions of a reference frame. It is for this reason that the Cartesian-coordinate directions are often thought to define the reference frame (but don't let this lure you into forgetting the distinction between a coordinate system and a reference frame). We return to the concepts of reference frames and coordinate systems and discuss the relationship between a coordinate system and a vector in Chapter 3.

Throughout the book, reference frames are always labeled. Later we will be solving problems that employ many different frames, and these labels will become very important. Thus we often write the three Cartesian coordinates as (x, y, z)I , explicitly noting the reference frame—here labeled I—in which the coordinates are specified. (The reason for the letter I will become apparent later.)

Likewise, the change in time of a point's position (i.e., the velocity) only has meaning when referred to some reference frame (recall the train example). For that reason, we always explicitly point out the appropriate reference frame when writing the velocity. A superscript calligraphic letter is used to indicate the frame. Figure 1.4 shows a schematic picture of the velocity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the instantaneous rate of change in time of the position r_P/O with respect to the frame I.

We can also express the velocity of point P with respect to O as the rate of change [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (Appendix A reviews some basic rules of calculus if you are rusty.) Because the variables x, y, and z are scalars, their time derivatives do not need a frame identification. We maintain the notation (x, y, z)_I , however, to remind you that these three scalars are the rates of change of the three position coordinates in frame I. The rate of change of a scalar Cartesian coordinate is called the speed to distinguish it from the velocity. We return to this topic and discuss it in depth and more formally in Chapter 3.

1.1.3 Equations of Motion

We now return to the definition of dynamics. Trajectory signifies the complete specification of the three positions and three speeds of a point in a reference frame as a function of time. It takes six quantities in our three-dimensional universe to completely specify the motion of a point. This is not necessarily obvious. Why six quantities and not three? Isn't the position enough (since we can always find the velocity by differentiating)? The answer is no, because dynamics is about more than just specifying the position and velocity. It is about finding equations, based on Newton's laws, that allow us to predict the complete trajectory of an object given only its state at a single moment in time. By state we mean the three positions and three speeds of the point. These six quantities, defined at a single moment in time, are called the initial conditions. The tools of dynamics allow us to find a set of differential equations that can be solved—using these initial conditions—for the position and velocity at any later time. These differential equations are called equations of motion.

Definition 1.4 The equations of motion of a point are three second-order differential equations whose solution is the position and velocity of the point as a function of time.

To see this a bit more clearly, imagine that we know the three position variables x(t), y(t), and z(t) of a point in frame I at some time t and wish to know the position some short time later, t + Δt. Without the velocity at t we are lost; the point could move anywhere. However, with the three speeds x(t), y(t), and z(t), we know everything; the new position of the point in I is (x(t) + x(t) t, y(t) + y(t) Δt, z(t) + z(t) Δt)_I. The equations of motion allow us to find the speeds at time t + Δt. The six positions and speeds are sufficient to find the complete trajectory.

As an example, one of the simplest equations of motion is that for a mass on a spring. The position of the mass is given by the Cartesian coordinate x, and the force due to the spring is given by -kx (see Figure 2.7c). The position thus satisfies the following second-order differential equation, obtained by equating the force with the mass times acceleration and solving for the acceleration:

x = k/m x.

This differential equation is an equation of motion. Its solution gives x(t) and x(t), the trajectory of the mass point. Don't worry if you didn't follow how the equation was obtained; that is covered in Chapter 2.

Many equations of motion cannot be solved exactly; a computer is required to find numerical trajectories. You will have an opportunity to do this many times in this book. However, often we skip solving for the trajectory and find special solutions or conditions on the states by setting the time equal to a specific value, finding certain conditions on the forces, or setting the acceleration to a constant or zero (sometimes called a steady state). One particularly useful such solution is known as an equilibrium point. The mathematical details of equilibrium solutions are presented in Chapter 12, but it is useful to have a qualitative understanding now, as we will be finding equilibrium solutions of many systems here and there throughout the book.

Qualitative Definition 1.5 An equilibrium point of a dynamic system is a specific solution of the equations of motion in which the rates of change of the states are all zero.

In other words, an equilibrium point is a configuration in which the system is at rest. For the mass-spring system, for example, there is one equilibrium point, which corresponds to the mass situated at precisely the rest length of the spring. Mathematically, if x(t) is an equilibrium point, then x(t) = 0 and x(t) = 0. Thus x(t) = x(0), where x(0) is the initial condition at time t = 0. So an equilibrium point is a solution whose value over time remains equal to its initial value.

In summary, dynamics is about finding three second-order differential equations that can be solved for the complete trajectory of an object. The equations can be solved—using the six initial conditions—either analytically (by hand) or numerically (by a computer). It is true that other scalar quantities can be used to specify the position rather than Cartesian coordinates; we will begin to study alternate coordinate systems in detail in Chapter 3. However, we will always need six independent scalars. The remainder of this book describes methods for finding equations of motion—first for a point (particle) and later for extended (rigid) bodies—and presents various techniques for completely or partially solving them.

1.2 Organization of the Book

The next chapter reviews the physics of mechanics, covering Newton's laws in depth. We also start to solve simple problems. All the essential physical concepts that form the foundation for the rest of the book are presented in that chapter. Our approach is slightly unconventional in that we begin solving dynamics problems at the outset—in Chapter 2—to highlight the meaning of Newton's laws and how we incorporate the underlying postulates into our methodology.

The remainder of the book is divided into five parts plus a set of four appendices. We divide the book into parts to highlight the logical separation of main topics and show how rigid-body motion builds on the key concepts of particle motion. The material could be covered in one semester by leaving out certain topics or stretched over multiple semesters or quarters. In Part One we restrict ourselves to studying only the planar motion of single particles. Thus motion in only two dimensions is studied; we thus need only four scalars to specify a particle's state rather than six. We do this to simplify the mathematics and focus on the key physical concepts, allowing you to develop an understanding of the procedures used to solve dynamics problems. You will solve an amazing array of real and important problems in Part One.

Chapter 3 returns to first principles and lays out the mathematical framework for a full vector treatment of kinematics and dynamics in the plane. Our focus is on the use of various coordinate systems and approaches to treating velocity and acceleration. Throughout the chapter we return to the same example: the simple pendulum. While this example may seem a bit academic, our approach is to focus repeatedly on this relatively simple system to emphasize the various new techniques presented and explain how they interrelate and add value. At the end of the chapter these new concepts are used to solve a selection of more difficult problems.

Chapters 4 and 5 present the concepts of momentum and energy, respectively, for a particle. It is here that we begin to solve equations of motion for the characteristics of trajectories (also called integrals of the motion). These ideas will be useful throughout the remainder of your study of dynamics and form the foundation of modern physics.

Part Two presents an introduction to multiparticle systems (Chapters 6 and 7). The previous concepts are generalized to simultaneously study many, possibly interacting, particles. In Chapter 6 we introduce two important examples of multiparticle systems—collisions and variable-mass systems. Chapter 7 sets the stage for the rigid-body discussions in Parts Three and Four by analyzing angular momentum and energy for many particles.

Part Three introduces rigid-body dynamics in the plane. We show (Chapters 8 and 9) how to specialize our tools to study a rigid collection of particles (i.e., particles whose relative positions are fixed). In particular, the definition of equations of motion is expanded to include the differential equations that describe the orientation of a rigid body. We use these ideas to study a variety of important engineering systems. We still confine our study to motion in the plane, however, to focus on the physical concepts without being burdened by the complexity of three-dimensional kinematics. It is here that we introduce the moment of inertia and, most importantly, the separation of angular momentum.

(Continues...)

Excerpted from ENGINEERING DYNAMICSby N. Jeremy Kasdin Derek A. Paley Copyright © 2011 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Edition details

• Publisher: Princeton University Press
• Publication date: 2011
• Language: English
• ISBN 10: 0691135371
• ISBN 13: 9780691135373
• Binding: Hardcover
• Number of pages: 688