engineering differential equations
TRANSCRIPT
Engineering Differential Equations
Engineering Differential Equations
Bill Goodwine
Theory and Applications
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
ISBN 978-1-4419-7918-6 e-ISBN 978-1-4419-7919-3
subject to proprietary rights.
DOI 10.1007/978-1-4419-7919-3
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
All rights reserved. This work may not be translated or copied in whole or in part without the
Use in connection with any form of information storage and retrieval, electronic adaptation, computer
© Springer Science+Business Media, LLC 2011
Mathematics Subject Classification Codes (2010): 34-01, 34H05, 35B05, 34B24, 93-01, 65-01, 70J10, 70J25, 80-01.
MATLAB®
Fax: 508-647-7001E-mail: [email protected]
Tel: 508-647-7000
Springer New York Dordrecht Heidelberg London
Web: www.mathworks.com
Natick, MA, 01760-2098 USA
For MATLAB product information, please contact:
3 Apple Hill Drive
is a registered trademark of The MathWorks, Inc.
The MathWorks Inc.
Department of Aerospace and
Bill Goodwine
University of Notre Dame Mechanical Engineering
Notre Dame, Indiana 46556
To Amy, Bridget, Carolyn and Katie.
Preface
This book is intended for engineering undergraduate students, particularly aerospaceand mechanical engineers and students in other disciplines concerned with systemmodeling, analysis, and control. It is intended to be a relatively comprehensive treat-ment of engineering undergraduate differential equations as well as two primary ap-plications thereof: linear vibrations and classical feedback control. This material istraditionally separated into different courses in undergraduate engineering curric-ula, however, consistent with the theme of this book, the current trend to optimizeand streamline curricula results in many programs combining courses where there isa common underlying theoretical basis. Specifically this book was developed fromthe materials presented in a two-course, required, junior-level sequence of coursesthat I developed and have taught in the Department of Aerospace and MechanicalEngineering at the University of Notre Dame over the past several years. The ratio-nale behind the selection, arrangement, and relationship of the content of book hasfour primary facets.
The first facet relates to the role of mathematical analysis in modern engineer-ing. The modern reality, especially in industry but also in academic settings, is thatsophisticated software packages enabled by fast computing are starting to play adominant role in engineering analysis. Hence, to some extent, the skill of being ableto solve problems “by hand” is being displaced by computer simulation. This is notan argument for not covering what has been traditionally been the subject of en-gineering analysis courses, however, it can be taken as a justification for a slightlyaltered focus toward fundamental understanding versus problem solving by hand.Because the algorithmic aspects of many problem-solving methods are increasingly“hidden” in software, a fundamental understanding of the relationship between theattributes of a differential equation and the nature of its solution is critical; for fewthings are as dangerous as an engineer who places complete faith in the output of acomputer.
This point may be best represented in the language of educational objectives. Per-haps the most common is due to Bloom [7], which categorizes cognitive processesin a hierarchical manner. From the lowest to highest process there are: knowledge,comprehension, application, analysis, synthesis, and evaluation. Despite the label of
vii
viii Preface
“engineering analysis” many homework problems given to students fall within theapplication process; that is, they are asked to apply a particular solution or analysismethod to a given problem. Then, through repeated exposure to a subject on the ap-plication level, it is hoped that most students develop the ability (inductively throughexperience) to become competent at the higher levels, at which point higher-levelsynthesis or evaluation problems may be addressed.
With the application level being more and more automated, an increased focusin courses on the higher cognitive levels is necessary for the students to remaincompetent. I cannot argue that being able to “do” problems is not a necessary skill,and it is one that certainly has not been removed from this book. However, focusingon higher-level cognitive processes is what is going to serve students best. At aminimum, it will allow them to be distinguishable from a computer, which is alsoable to solve differential equations [27, 55]. By combining the mathematics and theapplication in the same course, the full range of the theoretical mathematics can beexercised through the engineering applications for which the students will ultimatelybe accountable.
The second facet is pedagogical. There are several nonstandard features of thecontent and presentation in this book that should be highlighted. First, there is anabundance of detailed examples. These are present, not to serve as a template fromwhich students can copy the procedure to solve a problem, but rather a recognitionof the fact that, although traditionally mathematics and related application fields aretaught in a deductive manner, inductive learning actually “promotes deeper learn-ing and longer retention of information” [15, 16, 33]. Thus, one way to considerthis abundance of examples is that they replace, to some extent, the more directapplication-oriented homework problems. Second, material is sometimes coveredor named in a nonstandard manner purely to promote a deeper understanding of theultimate result, but which otherwise does not directly help one “use” the result or isotherwise nonstandard. Examples of this are found throughout the text. A superficialexample would be naming the procedure normally referred to as “integrating fac-tors” for first-order equations “variation of parameters” because it shares a commonderivation with that method as typically applied to higher-order equations. A deeperexample would be in the study of frequency response methods for feedback control,where emphasis is placed on the fact that what is plotted in Bode plots relates to theharmonically forced steady-state solution of the system, when the usual use of theplot for stability under unity feedback is essentially unrelated to that. Finally, an ex-ample of including a whole nonstandard section is related to Taylor series methodsfor numerical methods. It is included purely as a setup for the Runge–Kutta methodto facilitate developing a deeper understanding by the students. This is particularlyimportant in such a case. When the end result is just a formula to be used, withoutthe proper development, many students would be inclined perhaps simply to be con-tent to “use” the formula, in which case the more modern approach would be simplyto bring the ode45() function in MATLAB� to the attention of the students.
The third facet is that this text provides a means for a streamlined and efficienttreatment of material normally covered in three courses. In the author’s programit resulted in combining three courses into two courses with little lost in terms of
Preface ix
content coverage. With the ever-broadening scope of engineering programs to in-clude more, for example, biological sciences and design content, this would providea means to allocate credit hours to such content without overly substantially cuttinginto the traditional engineering science material.
The fourth facet relates to the motivation engineering students have for studyingmathematics. Ultimately engineering students study mathematics in order to be ableto solve problems that are of importance to them. Although it is certainly legitimatefor engineering students to have courses solely focused on the mathematics followedby the applications, it has been my experience that engineering students approachthe mathematical subjects with much greater interest and enthusiasm when theyhave an application immediately at hand. Assessment from the sequence of coursesI teach verifies this conclusion.
Content
This book covers what is normally covered in undergraduate engineering differentialequations, vibrations, and controls courses. Less emphasis is placed on “recipes” orenumerated “procedures” to solve problems than is usual, although such content isnot completely missing. There are plenty of problems that ask the student to simplysolve some differential equations, however, quite a few of them are, for the reasonsoutlined above, deeper.
In addition to the combination of subjects unified by the content of the book,there are a couple of additional unique features related to content. The final chapteron nonlinear systems is perhaps longer than what is typically covered in engineeringcourses. Also, in the appendix there are many computer programs that were usedto solve the example problems. These are presented in both the C programminglanguage as well as in FORTRAN. The former is included because it is still widelyused. The latter is, perhaps somewhat uniquely, still used in the aerospace industrybut, more important, is fairly transparent in syntax, so that a student who is notproficient in programming can still easily determine what the program is doing.
It is important to note that the chapters are not of approximately uniform length.This is particularly important for an instructor making a course syllabus from thetable of contents to note.
A Web page has been created for this book that contains:
• Some media content such as movies illustrating problem solutions that areamenable to such a presentation
• Source code for computer programs• Additional exercises• Errata
The URL is: http://controls.ame.nd.edu/engdiffeq/
x Preface
Prerequisites
The student is assumed to have a good background in calculus (at least throughmultivariable calculus) and linear algebra. A dynamics course would be useful, butthe basic mechanics from the typical undergraduate engineering physics sequenceseems to suffice. A basic exposure to circuit analysis along the lines of the con-tent typical in introductory physics courses would also be helpful. Finally, a goodintroduction to computer programming would be very useful, but not necessarilyrequired.
Chapter Dependencies
The book is organized in what I consider the most logical order for a fundamentaltreatment of the subject matter. Even if a chapter does not explicitly depend on a pre-vious one, the general progression of understanding and sophistication that wouldbe developed when the chapters are covered in order was carefully considered.
However, curricular realities may prevent covering the chapters in order. Al-though it is not ideal, it would be possible to treat some of the material out of order.Specifically, the order of the following chapters may be altered without an extremedisruption in the logical flow of the material.
• Chapter 5 considers variable-coefficient ordinary differential equations. Themethod used, assuming a power series, is sufficiently different from the meth-ods that precede the chapter that it could really be considered at any point.
• Chapters 8 through 10 cover Laplace transforms and control applications. Itwould be possible to treat these chapters as an independent unit.
• Chapter 11 considers the simplest linear partial differential equations using theseparation of variables method. Hence, as long as Chapter 3 or the equivalent hasbeen covered, it should be possible to cover this material.
• Chapter 12 considers numerical methods, and hence really only requires an un-derstanding of Taylor series.
In the text, these chapters do occasionally refer back to earlier chapters, however,the dependence is typically one of pedagogy, rather than of theoretical necessity.These references could typically be treated in a lecture with a relatively quick aside.In my own case, for example, because of the structure of our curriculum, I coverpartial differential equations and numerical methods in the first semester of the two-semester sequence.
Acknowledgments
I would like to acknowledge the valuable help I have received from several of mycolleagues. Michael M. Stanisic provided many insightful comments and correc-tions to several chapters of a draft of this book. I would also like to thank Mary
Preface xi
Frandsen for reviewing and suggesting improvements to the section on music inChapter 11 and Robert A. Howland for reviewing and suggesting improvements tosome of the mechanics material in Chapter 1. I would also like to thank the approxi-mately 300 undergraduate students who used draft versions of this text in the coursesI teach and pointed out many typographical errors, substantive errors, confusinglywritten parts, and so on.
Finally, I would like to thank my wife, Amy, for her patience and supportthroughout the process of writing this book. Especially during the very long timeI thought this project was “95% finished,” she was unwaveringly supportive.
And now on to business.
Notre Dame, Indiana Bill GoodwineJune 2010
Contents
1 Introduction and Preliminaries 11.1 The Engineering Utility of Differential Equations . . . . . . . . . . 11.2 Sets, Relations, and Functions . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Relations and Functions . . . . . . . . . . . . . . . . . . . 41.2.3 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Implicit Functions . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Types of Differential Equations . . . . . . . . . . . . . . . . . . . 81.3.1 Ordinary Versus Partial Differential Equations . . . . . . . 91.3.2 The Order of a Differential Equation . . . . . . . . . . . . 101.3.3 Linear Versus Nonlinear Differential Equations . . . . . . . 111.3.4 Homogeneous Versus Inhomogeneous Linear Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.5 Constant-Coefficient Versus Variable-Coefficient Linear Dif-
ferential Equations . . . . . . . . . . . . . . . . . . . . . . 131.3.6 Types of Linear Second-Order Partial Differential Equations 14
1.4 Solutions of Differential Equations . . . . . . . . . . . . . . . . . 141.4.1 Types of Solutions to Differential Equations . . . . . . . . 151.4.2 Existence and Uniqueness of Solutions . . . . . . . . . . . 18
1.5 A Few Fundamental Principles from Science . . . . . . . . . . . . 191.5.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5.2 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . 211.5.3 Mechanical Components . . . . . . . . . . . . . . . . . . . 321.5.4 Kirchhoff’s Laws . . . . . . . . . . . . . . . . . . . . . . . 341.5.5 Electronic Components . . . . . . . . . . . . . . . . . . . 35
1.6 Introduction to Numerical Methods . . . . . . . . . . . . . . . . . 391.6.1 Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . 391.6.2 Determining an Appropriate Step Size . . . . . . . . . . . 431.6.3 Numerical Methods for Higher-Order Differential Equations 441.6.4 Numerical Packages . . . . . . . . . . . . . . . . . . . . . 45
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 First-Order Ordinary Differential Equations 572.1 Motivational Examples . . . . . . . . . . . . . . . . . . . . . . . . 57
xiii
2.2 Homogeneous Constant-Coefficient Linear First-Order OrdinaryDifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3 Inhomogeneous Constant-Coefficient Linear First-Order OrdinaryDifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . 632.3.1 Undetermined Coefficients . . . . . . . . . . . . . . . . . . 632.3.2 Complication: When the Assumed Solution Contains a Ho-
mogeneous Solution . . . . . . . . . . . . . . . . . . . . . 692.3.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . 74
2.4 Variable-Coefficient Linear First-Order Ordinary Differential Equa-tions: Variation of Parameters . . . . . . . . . . . . . . . . . . . . 75
2.5 Ordinary First-Order Nonlinear Differential Equations . . . . . . . 772.5.1 Separable Equations . . . . . . . . . . . . . . . . . . . . . 772.5.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . 792.5.3 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . 81
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3 Second-Order Linear Constant-Coefficient Ordinary Differential Equa-tions 913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Theory of Linear Homogeneous Equations . . . . . . . . . . . . . 93
3.2.1 The Principle of Superposition . . . . . . . . . . . . . . . 953.2.2 Linear Independence . . . . . . . . . . . . . . . . . . . . . 96
3.3 Constant-Coefficient Homogeneous Equations . . . . . . . . . . . 983.3.1 Distinct Real Roots . . . . . . . . . . . . . . . . . . . . . 1013.3.2 Complex Roots . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.3 Repeated Roots . . . . . . . . . . . . . . . . . . . . . . . 104
3.4 Inhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . 1063.4.1 The Method of Undetermined Coefficients Constant-Coefficient
Differential Equations . . . . . . . . . . . . . . . . . . . . 1063.4.2 Method of Variation of Parameters for Constant or Variable-
Coefficient Equations . . . . . . . . . . . . . . . . . . . . 1093.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Single Degree of Freedom Vibrations 1234.1 Free Undamped Oscillations . . . . . . . . . . . . . . . . . . . . . 1244.2 Harmonically Forced Undamped Vibrations . . . . . . . . . . . . . 126
4.2.1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.2.2 Near Resonance . . . . . . . . . . . . . . . . . . . . . . . 1344.2.3 Vibrating Base . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3 Free Damped Vibrations . . . . . . . . . . . . . . . . . . . . . . . 1374.3.1 Damping Ratio Greater than One . . . . . . . . . . . . . . 1374.3.2 Damping Ratio Equal to One . . . . . . . . . . . . . . . . 1394.3.3 Damping Ratio Less than One . . . . . . . . . . . . . . . . 139
Contentsxiv
4.4 Harmonically Forced Damped Vibrations . . . . . . . . . . . . . . 1394.4.1 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.4.2 Vibrating Base . . . . . . . . . . . . . . . . . . . . . . . . 143
4.5 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . 1474.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5 Variable-Coefficient Linear Ordinary Differential Equations 1615.1 Motivational Examples . . . . . . . . . . . . . . . . . . . . . . . . 1625.2 Convergence: Real Rational Functions . . . . . . . . . . . . . . . . 1665.3 Series Solutions About an Ordinary Point . . . . . . . . . . . . . . 1715.4 Series Solutions About a Singular Point . . . . . . . . . . . . . . . 1745.5 A Collection of Famous Series Solutions . . . . . . . . . . . . . . 182
5.5.1 Airy Equation . . . . . . . . . . . . . . . . . . . . . . . . 1825.5.2 Chebychev Equation . . . . . . . . . . . . . . . . . . . . . 1835.5.3 Hermite Equation . . . . . . . . . . . . . . . . . . . . . . 1855.5.4 Legendre Equation . . . . . . . . . . . . . . . . . . . . . . 1865.5.5 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . 186
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6 Systems of First-Order Linear Constant-Coefficient Ordinary Differ-ential Equations 1936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.2 Converting to Systems of First-Order Differential Equations . . . . 1976.3 Linearly Independent Full Set of Real Eigenvectors . . . . . . . . . 197
6.3.1 Some Useful Results from Linear Algebra . . . . . . . . . 1986.3.2 Solution Technique for ξ = Aξ . . . . . . . . . . . . . . . 200
6.4 Complex Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 2046.5 Generalized Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 211
6.5.1 Geometric and Algebraic Multiplicities . . . . . . . . . . . 2126.5.2 Homogeneous Solutions with Repeated Eigenvalues . . . . 216
6.6 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.7 The Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . 2306.8 Nonhomogeneous Systems of First-Order Equations . . . . . . . . 233
6.8.1 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . 2346.8.2 Undetermined Coefficients . . . . . . . . . . . . . . . . . . 2376.8.3 Variation of Parameters . . . . . . . . . . . . . . . . . . . 240
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7 Applications of Systems of First-Order Equations 2497.1 Multidegree of Freedom Vibrations . . . . . . . . . . . . . . . . . 249
7.1.1 Classical Approach . . . . . . . . . . . . . . . . . . . . . 2507.1.2 Eigenvalue and Eigenvector Approach . . . . . . . . . . . 2547.1.3 Forced Undamped Multidegree of Freedom Systems . . . . 2577.1.4 Vibration Absorbers . . . . . . . . . . . . . . . . . . . . . 259
7.2 Introduction to “Modern” Control . . . . . . . . . . . . . . . . . . 262
xvContents
7.2.1 State-Space Control Systems . . . . . . . . . . . . . . . . 2637.2.2 Pole Placement . . . . . . . . . . . . . . . . . . . . . . . . 2657.2.3 The Linear Quadratic Regulator . . . . . . . . . . . . . . . 267
7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8 The Laplace Transform 2798.1 Motivational Example . . . . . . . . . . . . . . . . . . . . . . . . 2798.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.3 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.3.1 The Laplace Transform of Some Common Functions . . . . 2848.3.2 Properties of the Laplace Transform . . . . . . . . . . . . . 288
8.4 Initial Value Problems and Discontinuous Forcing . . . . . . . . . 2928.5 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3048.6 Block Diagram Representation and Algebra . . . . . . . . . . . . . 3128.7 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . 3208.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
9 Classical Control Theory: Analysis 3299.1 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
9.1.1 Proportional Control . . . . . . . . . . . . . . . . . . . . . 3319.1.2 Proportional plus Derivative Control . . . . . . . . . . . . 3349.1.3 Proportional plus Integral plus Derivative Control . . . . . 337
9.2 Time Domain Specifications . . . . . . . . . . . . . . . . . . . . . 3399.3 Response Versus Pole Location . . . . . . . . . . . . . . . . . . . 341
9.3.1 Real Poles . . . . . . . . . . . . . . . . . . . . . . . . . . 3429.3.2 Poles at the Origin . . . . . . . . . . . . . . . . . . . . . . 3439.3.3 Purely Imaginary Poles . . . . . . . . . . . . . . . . . . . 3469.3.4 Complex Conjugate Poles . . . . . . . . . . . . . . . . . . 346
9.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3519.5 Response of a Second-Order System . . . . . . . . . . . . . . . . . 355
9.5.1 Second-Order System Step Response . . . . . . . . . . . . 3579.5.2 Additional Poles and Zeros . . . . . . . . . . . . . . . . . 364
9.6 Root Locus Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3739.6.1 Motivational Example . . . . . . . . . . . . . . . . . . . . 3739.6.2 A Quick Review of Functions of a Complex Variable . . . . 3789.6.3 Root Locus Plotting Rules . . . . . . . . . . . . . . . . . . 3819.6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 3939.6.5 Determining the Gain . . . . . . . . . . . . . . . . . . . . 3999.6.6 Computational Tools . . . . . . . . . . . . . . . . . . . . . 400
9.7 Frequency Response Analysis . . . . . . . . . . . . . . . . . . . . 4019.7.1 Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 4049.7.2 Nyquist Plots . . . . . . . . . . . . . . . . . . . . . . . . . 421
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
10 Classical Control Theory: Design 449
Contentsxvi
10.1 System Type and Steady-State Error . . . . . . . . . . . . . . . . . 45010.2 Controller Design Using a Root Locus Plot . . . . . . . . . . . . . 452
10.2.1 Proportional Control . . . . . . . . . . . . . . . . . . . . . 45210.2.2 Lead–Lag Compensation . . . . . . . . . . . . . . . . . . 460
10.3 Frequency Response Design . . . . . . . . . . . . . . . . . . . . . 46810.3.1 Lead Compensation . . . . . . . . . . . . . . . . . . . . . 47010.3.2 Lag Compensation . . . . . . . . . . . . . . . . . . . . . . 477
10.4 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47710.4.1 Lowpass Filters . . . . . . . . . . . . . . . . . . . . . . . 47710.4.2 Highpass Filters . . . . . . . . . . . . . . . . . . . . . . . 479
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
11 Partial Differential Equations 48511.1 The One-Dimensional Wave Equation . . . . . . . . . . . . . . . . 485
11.1.1 Derivation of the Wave Equation . . . . . . . . . . . . . . 48511.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 48711.1.3 Separation of Variables . . . . . . . . . . . . . . . . . . . 48811.1.4 Summary and Examples of the Solution to the Wave Equation494
11.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49711.2.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . 49811.2.2 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . 49911.2.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 50211.2.4 The General Fourier Series . . . . . . . . . . . . . . . . . 50411.2.5 Examples of Fourier Series . . . . . . . . . . . . . . . . . 50511.2.6 Summary Remarks and Theory . . . . . . . . . . . . . . . 511
11.3 The One-Dimensional Heat Equation . . . . . . . . . . . . . . . . 51111.3.1 Solution to the Heat Conduction Equation with Homoge-
neous Boundary Conditions . . . . . . . . . . . . . . . . . 51411.3.2 Solution to the Heat Equation with Inhomogeneous Bound-
ary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 51811.3.3 Solution to the Heat Equation with an Insulated End . . . . 520
11.4 Heat Conduction in Two Dimensions . . . . . . . . . . . . . . . . 52211.4.1 Laplace’s Equation: Steady-State Temperature Distribution 52211.4.2 Unsteady Two-Dimensional Heat Conduction with Homo-
geneous Boundary Conditions . . . . . . . . . . . . . . . . 52511.4.3 Unsteady Two-Dimensional Heat Conduction with Inho-
mogeneous Boundary Conditions . . . . . . . . . . . . . . 53011.5 Vibrating Membranes . . . . . . . . . . . . . . . . . . . . . . . . . 534
11.5.1 The Two-Dimensional Wave Equation in Rectangular Co-ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
11.5.2 The Two-Dimensional Wave Equation in Polar Coordinates 53511.5.3 Summary of the Solution to the Wave Equation in Polar
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 54211.5.4 Modes of Vibration of a Circular Drum Head . . . . . . . . 543
11.6 Sturm–Liouville Theory . . . . . . . . . . . . . . . . . . . . . . . 544
Contents xvii
11.7 The Euler–Bernoulli Beam Equation . . . . . . . . . . . . . . . . . 54711.7.1 Derivation of the Beam Equation . . . . . . . . . . . . . . 54711.7.2 Boundary Conditions for the Beam Equation . . . . . . . . 55211.7.3 Solutions to the Beam Equation . . . . . . . . . . . . . . . 554
11.8 A Musical Interlude . . . . . . . . . . . . . . . . . . . . . . . . . 55911.8.1 Defining Scales . . . . . . . . . . . . . . . . . . . . . . . 56111.8.2 A Simple Model for Dissonance . . . . . . . . . . . . . . . 56411.8.3 Percussion Instruments . . . . . . . . . . . . . . . . . . . 567
11.9 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 56811.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
12 Numerical Methods 57712.1 Another Look at Euler’s Method . . . . . . . . . . . . . . . . . . . 57712.2 Taylor Series Methods . . . . . . . . . . . . . . . . . . . . . . . . 580
12.2.1 Second-Order Taylor Series Expansion . . . . . . . . . . . 58212.2.2 Third-Order Taylor Series Expansion . . . . . . . . . . . . 584
12.3 The Runge–Kutta Method . . . . . . . . . . . . . . . . . . . . . . 58712.3.1 The First-Order Runge–Kutta Method . . . . . . . . . . . . 58912.3.2 The Second-Order Runge–Kutta Method . . . . . . . . . . 59012.3.3 The Third-Order Runge–Kutta Method . . . . . . . . . . . 59412.3.4 The Fourth-Order Runge–Kutta Method . . . . . . . . . . 596
12.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59712.5 Numerical Methods for Higher-Order Systems . . . . . . . . . . . 604
12.5.1 Systems of First-Order Ordinary Differential Equations . . 60412.5.2 Higher-Order Ordinary Differential Equations . . . . . . . 60412.5.3 Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . 60512.5.4 Second-Order Taylor Series . . . . . . . . . . . . . . . . . 60812.5.5 Fourth-Order Runge–Kutta . . . . . . . . . . . . . . . . . 608
12.6 Numerical Methods for Partial Differential Equations . . . . . . . . 61112.6.1 Finite Difference Approximation . . . . . . . . . . . . . . 61212.6.2 Finite Differences: Laplace’s Equation . . . . . . . . . . . 61412.6.3 Finite Differences: Heat Equation . . . . . . . . . . . . . . 61712.6.4 Finite Differences: The Wave Equation . . . . . . . . . . . 622
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
13 Introduction to Nonlinear Systems 63113.1 Motivation and Introduction . . . . . . . . . . . . . . . . . . . . . 631
13.1.1 Multiple Equilibria and Chaos . . . . . . . . . . . . . . . . 63113.1.2 The Phase Plane . . . . . . . . . . . . . . . . . . . . . . . 63313.1.3 Poincare Sections . . . . . . . . . . . . . . . . . . . . . . 63513.1.4 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . 635
13.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63713.3 Jacobian Linearization . . . . . . . . . . . . . . . . . . . . . . . . 64313.4 Geometry and Stability of Equilibrium Points in the Phase Plane . . 64813.5 Harmonic Balance and Describing Functions . . . . . . . . . . . . 655
xviii Contents
13.5.1 Harmonic Balance . . . . . . . . . . . . . . . . . . . . . . 65513.5.2 Describing Functions . . . . . . . . . . . . . . . . . . . . 658
13.6 Introduction to Local Bifurcation Analysis . . . . . . . . . . . . . 66613.6.1 Saddle-Node Bifurcations . . . . . . . . . . . . . . . . . . 66713.6.2 Pitchfork Bifurcations . . . . . . . . . . . . . . . . . . . . 67213.6.3 Concluding Comments . . . . . . . . . . . . . . . . . . . . 675
13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
A Some Complex Variable Theory 683A.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 683A.2 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . 685A.3 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . 686
B Linear Algebra Review 691B.1 Linear Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 691
B.1.1 Properties of Vector Operations in the Euclidean Plane . . . 691B.1.2 Definition and Examples of Vector Spaces . . . . . . . . . 694B.1.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . 699B.1.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 700
B.2 Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . 704B.2.1 Computing Determinants . . . . . . . . . . . . . . . . . . 704B.2.2 Computing a Matrix Inverse . . . . . . . . . . . . . . . . . 706
C Detailed Computations 707C.1 Computations Related to Fourier Series . . . . . . . . . . . . . . . 707
D Some Theorems 709D.1 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . 709D.2 Series Solutions About a Singular Point . . . . . . . . . . . . . . . 710
E Example Programs 713E.1 C Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713E.2 FORTRAN Programs . . . . . . . . . . . . . . . . . . . . . . . . . 725References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739
Index 741
Contents xix
