Engineering Analysis

444 Pages·2018·22.033 MB·English

by  Merle C. Potter

Engineering Analysis
Merle C. Potter
Engineering Analysis
123
Merle C. Potter
Department ofMechanical Engineering
Michigan State University
EastLansing, MI
USA
ISBN978-3-319-91682-8 ISBN978-3-319-91683-5 (eBook)
https://doi.org/10.1007/978-3-319-91683-5
LibraryofCongressControlNumber:2018941988
©SpringerInternationalPublishingAG,partofSpringerNature2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeor
part of the material is concerned, specifically the rights of translation, reprinting, reuse of
illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,
andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis
publication does not imply, even in the absence of a specific statement, that such names are
exemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationin
thisbookarebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernor
the authors or the editors give a warranty, express or implied, with respect to the material
containedhereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremains
neutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternational
PublishingAGpartofSpringerNature
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To my wife, Gloria, who lets me cook and always loves what i make!
Contents
Preface xiii
1 Ordinary Differential Equations 1
1.1 Introduction 1
1.2 Definitions 2
1.3 Differential Equations of First Order 3
1.3.1 separable equations, 3
1.3.2 exact equations, 6
1.3.3 integrating Factors, 8
1.4 Physical Applications 9
1.4.1 simple electrical circuits, 9
1.4.2 The rate equation, 11
1.4.3 Fluid Flow, 13
1.5 Linear Differential Equations 14
1.6 H omogeneous Second-Order Linear Equations
with Constant Coefficients 16
1.7 Spring–Mass System—Free Motion 20
1.7.1 undamped Motion, 21
1.7.2 damped Motion, 24
1.7.3 The electrical circuit analog, 29
1.8 N onhomogeneous Second-Order Linear Equations
with Constant Coefficients 31
1.9 S pring–Mass System—Forced Motion 34
1.9.1 resonance, 37
1.9.2 near resonance, 38
1.9.3 F orced oscillations with damping, 40
1.10 Periodic Input Functions—Fourier Series 44
1.10.1 even and odd Functions, 48
1.10.2 Half-range expansions, 52
1.10.3 Forced oscillations, 54
vii
viii / Contents
1.11 The Cauchy Equation 56
1.12 Variation of Parameters 59
1.13 Miscellaneous Information 61
2 Power-Series Methods 70
2.1 Power Series 70
2.2 L inear Differential Equations with Variable Coefficients 74
2.3 Legendre’s Equation 83
2.4 The Method of Frobenius 87
2.4.1 distinct roots not differing by an integer, 88
2.4.2 double roots, 90
2.4.3 roots differing by an integer, 94
2.5 Bessel’s Equation 97
3 Laplace Transforms 111
3.1 Introduction 111
3.2 T he Laplace Transform 112
3.3 L aplace Transforms of Derivatives and Integrals 122
3.4 D erivatives and Integrals of Laplace Transforms 126
3.5 L aplace Transforms of Periodic Functions 129
3.6 I nverse Transforms—Partial Fractions 133
3.6.1 unrepeated linear Factor, 133
3.6.2 repeated linear Factor, 133
3.6.3 unrepeated Quadratic Factor, 134
3.6.4 repeated Quadratic Factor, 134
3.7 Solution of Differential Equations 138
4 Matrices and Determinants 153
4.1 Introduction 153
4.2 Matrices 154
4.3 Addition of Matrices 155
4.4 The Transpose and Some Special Matrices 157
4.5 Matrix Multiplication—Definition 161
4.6 Matrix Multiplication—Additional Properties 163
4.7 Determinants 165
Contents / ix
4.8 The Adjoint and the Inverse Matrices 171
4.9 Eigenvalues and Eigenvectors 178
5 Vector Analysis 190
5.1 Introduction 190
5.2 Vector Algebra 190
5.2.1 definitions, 190
5.2.2 addition and subtraction, 192
5.2.3 components of a Vector, 192
5.2.4 Multiplication, 194
5.3 Vector Differentiation 203
5.3.1 ordinary differentiation, 203
5.3.2 Partial differentiation, 208
5.4 The Gradient 211
5.5 Cylindrical and Spherical Coordinates 221
5.5.1 cylindrical coordinates, 221
5.5.2 spherical coordinates, 225
5.6 Integral Theorems 229
5.6.1 The divergence Theorem, 229
5.6.2 stokes’s Theorem, 232
6 Partial Differential Equations 244
6.1 Introduction 244
6.2 Wave Motion 246
6.2.1 V ibration of a stretched, Flexible string, 246
6.2.2 The Vibrating Membrane, 248
6.2.3 longitudinal Vibrations of an elastic Bar, 250
6.2.4 Transmission-line equations, 251
6.3 The d’Alembert Solution of the Wave Equation 254
6.4 Separation of Variables 258
6.5 Diffusion 270
6.6 Solution of the Diffusion Equation 273
6.6.1 a long, insulated rod with ends at Fixed Temperatures, 273
6.6.2 a long, Totally insulated rod, 277
6.6.3 T wo-dimensional Heat conduction in a long, rectangular Bar, 281
6.7 Electric Potential About a Spherical Surface 286
6.8 Heat Transfer in a Cylindrical Body 288
6.9 Gravitational Potential 292
x / Contents
7 Complex Variables 299
7.1 Introduction 299
7.2 Complex Numbers 299
7.3 Elementary Functions 305
7.4 Analytic Functions 311
7.5 Complex Integration 315
7.5.1 Green’s Theorem, 315
7.5.2 cauchy’s integral Theorem, 318
7.5.3 cauchy’s integral Formula, 322
7.6 Series 327
7.6.1 Taylor series, 327
7.6.2 laurent series, 328
7.7 Residues 335
8 Numerical Methods 348
8.1 Introduction 348
8.2 Finite-Difference Operators 350
8.3 T he Differential Operator Related to the Difference Operators 354
8.4 Truncation Error 359
8.5 Numerical Integration 362
8.6 Numerical Interpolation 367
8.7 Roots of Equations 369
8.8 I nitial-Value Problems—Ordinary Differential Equations 372
8.8.1 Taylor’s Method, 373
8.8.2 euler’s Method, 374
8.8.3 adams’ Method, 374
8.8.4 runge–Kutta Methods, 375
8.8.5 direct Method, 378
8.9 Higher-Order Equations 381
8.10 B oundary-Value Problems—Ordinary Differential Equations 388
8.10.1 iterative Method, 388
8.10.2 superposition, 389
8.10.3 simultaneous equations, 389
8.11 Numerical Stability 392
Contents / xi
8.12 Numerical Solution of Partial Differential Equations 392
8.12.1 The diffusion equation, 393
8.12.2 The Wave equation, 394
8.12.3 laplace’s equation, 395
Bibliography 406
Appendix 407
Table A1 U nited States Engineering Units, SI Units,
and Their Conversion Factors 407
Table A2 Gamma Function 408
Table A3 Error Function 409
Table A4 Bessel Functions 410
Answers to Selected Problems 414
Index 427