Numerical and Analytical Methods with MATLAB

488 Pages·2009·12.76 MB·English

by  William Bober

NUMERICAL AND
ANALYTICAL
METHODS WITH
MATLAB®
93568_Book.indb 1 7/22/09 10:27:59 AM
CRC Series in
COMPUTATIONAL MECHANICS
and APPLIED ANALYSIS
Series Editor: J.N. Reddy
Texas A&M University
Published Titles
ADVANCED THERMODYNAMICS ENGINEERING
Kalyan Annamalai and Ishwar K. Puri
APPLIED FUNCTIONAL ANALYSIS
J. Tinsley Oden and Leszek F. Demkowicz
COMBUSTION SCIENCE AND ENGINEERING
Kalyan Annamalai and Ishwar K. Puri
EXACT SOLUTIONS FOR BUCKLING OF STRUCTURAL MEMBERS
C.M. Wang, C.Y. Wang, and J.N. Reddy
THE FINITE ELEMENT METHOD IN HEAT TRANSFER AND FLUID DYNAMICS,
Second Edition
J.N. Reddy and D.K. Gartling
MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS: THEORY
AND ANALYSIS, Second Edition
J.N. Reddy
NUMERICAL AND ANALYTICAL METHODS WITH MATLAB®
William Bober, Chi-Tay Tsai, and Oren Masory
PRACTICAL ANALYSIS OF COMPOSITE LAMINATES
J.N. Reddy and Antonio Miravete
SOLVING ORDINARY and PARTIAL BOUNDARY VALUE PROBLEMS in
SCIENCE and ENGINEERING
Karel Rektorys
93568_Book.indb 2 7/22/09 10:27:59 AM
CRC Series in
COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
NUMERICAL AND
ANALYTICAL
METHODS WITH
MATLAB®
WILLIAM BOBER
CHI-TAY TSAI
OREN MASORY
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
93568_Book.indb 3 7/22/09 10:27:59 AM
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-
ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2009 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20131120
International Standard Book Number-13: 978-1-4200-9357-5 (eBook - PDF)
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Contents
Preface ............................................................................................................xi
Acknowledgments .........................................................................................xv
Authors ........................................................................................................xvii
1 Numerical Modeling for Engineering ....................................................1
1.1 Computer Usage in Engineering ........................................................1
1.1.1 Importance of the Computer ................................................1
1.1.2 Computer Usage ...................................................................1
1.2 The Mathematical Model ..................................................................2
1.3 Computer Programming ...................................................................2
1.4 Preparing a Computer Program .........................................................3
1.5 Recommended Procedures for Writing a Program .............................3
1.6 Building Blocks in Writing a Program ...............................................3
2 MATLAB Fundamentals ........................................................................5
2.1 Introduction ......................................................................................5
2.1.1 The MATLAB Windows ......................................................5
2.2 Constructing a Program in MATLAB...............................................8
2.3 The MATLAB Program ....................................................................8
2.4 Program Examples ...........................................................................18
2.5 Debugging a Program......................................................................26
2.6 3-D and Contour Plots ....................................................................28
Projects ......................................................................................................29
References .................................................................................................44
3 Matrices ................................................................................................45
3.1 Matrix Operations ...........................................................................45
3.2 System of Linear Equations .............................................................48
3.2.1 The inv Function ................................................................49
3.2.2 The Gauss Elimination Function ........................................49
3.2.3 Examples ............................................................................50
v
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vi ◾ Contents
3.3 Gauss Elimination ...........................................................................55
3.4 The Gauss–Jordan Method ..............................................................61
3.5 Number of Solutions .......................................................................63
3.6 Inverse Matrix ................................................................................64
3.7 The Eigenvalue Problem ..................................................................68
Exercises .....................................................................................................72
Projects ......................................................................................................72
Reference ...................................................................................................76
4 Roots of Algebraic and Transcendental Equations ...............................77
4.1 The Search Method ........................................................................77
4.2 Bisection Method ...........................................................................77
4.3 Newton–Raphson Method ..............................................................79
4.4 The fzero Function ..........................................................................80
4.4.1 Example Programs .............................................................84
Projects ......................................................................................................83
Reference ...................................................................................................93
5 Numerical Integration ..........................................................................95
5.1 Numerical Integration and Simpson’s Rule ......................................95
5.2 Improper Integrals ...........................................................................98
5.3 MATLAB’s Quad Function...........................................................100
5.4 MATLAB’s DBLQUAD Function ................................................101
Exercises ...................................................................................................102
Projects ....................................................................................................103
6 Numerical Integration of Ordinary Differential Equations ...............105
6.1 The Initial Value Problem ..............................................................105
6.2 The Fourth-Order Runge–Kutta Method ......................................107
6.3 System of Two First-Order Equations ............................................108
6.4 A Single Second-Order Equation ...................................................108
6.5 MATLAB’s ODE Function ...........................................................110
6.6 Ordinary Differential Equations That Are Not
Initial Value Problems ...................................................................112
6.7 Solution of a Tri-Diagonal System of Linear Equations .................112
6.8 Difference Formulas ......................................................................115
6.9 Deflection of a Beam .....................................................................118
Projects ....................................................................................................121
7 Simulink .............................................................................................141
7.1 Introduction ..................................................................................141
7.2 Creating a Model in Simulink .......................................................141
7.3 Typical Building Blocks in Constructing a Model .........................143
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Contents ◾ vii
7.4 Constructing and Running the Model ..........................................146
7.5 Constructing a Subsystem .............................................................146
7.6 Using the mux and fcn Blocks .......................................................146
7.7 The Relay Block .............................................................................148
7.8 The Switch Block ...........................................................................149
7.9 Trigonometric Function Blocks .....................................................150
Projects ....................................................................................................152
8 Curve Fitting ......................................................................................155
8.1 Curve-Fitting Objective .................................................................155
8.2 Method of Least Squares ...............................................................155
8.3 Curve Fitting with the Exponential Function ................................158
8.4 MATLAB’s Curve-Fitting Function ..............................................160
8.5 Cubic Splines .................................................................................162
8.6 The Function Interp1 for Cubic Spline Curve Fitting ....................164
8.7 Curve Fitting with Fourier Series ..................................................164
Projects ....................................................................................................167
9 Optimization ......................................................................................171
9.1 Introduction ..................................................................................171
9.2 Unconstrained Optimization Problems .........................................172
9.3 Method of Steepest Descent ..........................................................173
9.4 Optimization with Constraints .....................................................176
9.5 MATLAB’s Optimization Function ..............................................178
Exercises ...................................................................................................182
Projects ....................................................................................................182
Reference .................................................................................................186
10 Partial Differential Equations ............................................................187
10.1 The Classification of Partial Differential Equations .......................187
10.2 Solution by Separation of Variables................................................188
10.2.1 The Vibrating String .........................................................188
10.2.2 Unsteady Heat Transfer I (Bar) .........................................192
10.2.3 Unsteady Heat Transfer II (Cylinder) ...............................199
10.3 Unsteady Heat Transfer in 2-D ....................................................206
10.4 Perturbation Theory and Sound Waves ..........................................211
10.5 Review of Finite Difference Formulas ............................................217
10.6 Example of Applying Finite Difference Methods to Partial
Differential Equations ...................................................................217
10.6.1 The Explicit Method .........................................................218
10.6.2 The Implicit Method .......................................................220
Projects ....................................................................................................221
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viii ◾ Contents
11 Iteration Method ................................................................................233
11.1 Iteration in Pipe Flow Analysis ......................................................233
11.2 The Gauss–Seidel Method .............................................................235
11.3 The Hardy–Cross Method .............................................................238
Projects ....................................................................................................241
References ................................................................................................248
12 Laplace Transforms ............................................................................249
12.1 Laplace Transform and Inverse Transform ....................................249
12.2 Transforms of Derivatives ..............................................................256
12.3 Ordinary Differential Equations, Initial Value Problem ................257
12.4 A Shifting Theorem ......................................................................260
12.5 The Unit Step Function .................................................................263
12.6 Laplace Transform of the Unit Step Function ...............................264
12.7 Convolution .................................................................................268
12.8 Laplace Transforms Applied to Partial Differential Equations .......271
12.9 Laplace Transforms and Complex Variables ..................................276
12.9.1 Residues and Poles ............................................................278
Exercises ...................................................................................................281
References ................................................................................................282
13 An Introduction to the Finite Element Method .................................283
13.1 Finite Element Method for Stress Analysis ....................................283
13.2 Structural Mechanics Plane Stress Analysis ...................................283
13.3 The Shape Function for a Linear Triangle Element ........................288
13.3.1 3-Node Triangular Element for 2-D Stress Analysis .........290
13.3.2 Shape Function in Area Coordinates ................................291
13.3.3 Finite Element Formulation Using Triangular
Elements ...........................................................................292
13.4 Finite Element Analysis Using MATLAB’s PDE Toolbox .............299
13.5 Structural Mechanics Plane Strain Analysis ..................................320
13.6 Model Analysis of 2-D Structures .................................................321
13.7 Finite Element Analysis for Heat Transfer .....................................325
Projects ....................................................................................................339
References ................................................................................................341
14 Control Systems .................................................................................343
14.1 Introduction .................................................................................343
14.2 Representation of Systems in MATLAB .......................................346
14.2.1 Transfer Function Representation ....................................348
14.2.2 Zero-Pole-Gain Format of Transfer Function
Representation ..................................................................350
14.2.3 State Space Representation ...............................................352
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Contents ◾ ix
14.3 Closed-Loop Systems .....................................................................355
14.3.1 DC Motor Modeling ........................................................355
14.3.2 Block Diagrams ................................................................357
14.3.3 MATLAB Tools for Defining the Closed-Loop System ....358
14.4 MATLAB Tools for the Performance Analysis
of Closed-Loop Systems .................................................................361
14.4.1 Root Locus Plots...............................................................361
14.4.2 Nyquist Plots ....................................................................363
14.5 MATLAB’s SISOtool ....................................................................367
14.5.1 Example to Be Used with SISOtool ..................................367
14.5.2 SISOtool Main Features ...................................................370
14.5.3 Using SISOtool to Design the Controller for Example
at Beginning of This Section .............................................371
14.6 Application of Simulink in Controls and Dynamic Systems ..........377
14.6.1 Example of Control of the Fluid Level in Coupled
Tanks ................................................................................377
14.6.2 Design of a Feed-Forward Loop Using Optimality
Criteria .............................................................................385
14.6.3 Active Suspension .............................................................391
14.6.4 Sampled Data Control System ..........................................394
14.6.5 Implementation of ADC and DAC in Simulink ...............397
14.7 Simulink’s Data Acquisition Toolbox ............................................405
14.7.1 Analog Input ...................................................................406
14.7.2 Analog Output .................................................................407
14.7.3 Digital Input....................................................................408
14.7.4 Digital Output ................................................................408
Projects ....................................................................................................412
Endnotes ..................................................................................................417
Appendix A ..................................................................................................419
Appendix B ..................................................................................................423
Appendix C .................................................................................................429
Index ...........................................................................................................439
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