Advanced Mathematics and Mechanics …

1 CHAPMAN & HALL/CRCA CRC Press CompanyBoca Raton London New York Washington, EditionAdvancedMathematicsand MechanicsApplications UsingHoward B. WilsonUniversity of AlabamaLouis H. TurcotteRose-Hulman Institute of TechnologyDavid HalpernUniversity of AlabamaMATLAB 2003 by Chapman & Hall/CRC This book contains information obtained from authentic and highly regarded sources. Reprinted materialis quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonableefforts have been made to publish reliable data and information, but the author and the publisher cannotassume responsibility for the validity of all materials or for the consequences of their this book nor any part may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, microfilming, and recording, or by any information storage orretrieval system, without prior permission in writing from the consent of CRC Press LLC does not extend to copying for general distribution, for promotion, forcreating new works, or for resale.

2 Specific permission must be obtained in writing from CRC Press LLCfor such all inquiries to CRC Press LLC, 2000 Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and areused only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at 2003 by Chapman & Hall/CRC No claim to original Government worksInternational Standard Book Number 1-58488-262-XLibrary of Congress Card Number 2002071267 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Wilson, Mathematics and Mechanics applications using MATLAB / Howard B. Wilson, Louis H. Turcotte, David Halpern. 3rd ed. p. 1-58488-262-X 1. MATLAB.

3 2. Engineering Mathematics Data processing. 3. Mechanics , Applied Data processing. I. Turcotte, Louis H. II. Halpern, David. III. Title. TA345 . W55 2002 620 .00151 dc212002071267 C262X disclaimer Page 1 Friday, August 2, 2002 11:45 AM 2003 by Chapman & Hall/CRCFor my dear wife, B. WilsonFor my loving wife, Evelyn, our departed cat, Patches, and my H. Turcotte 2003 by Chapman & Hall/CRCP refaceThis book uses MATLABR to analyze various applications in Mathematics and me-chanics. The authors hope to encourage engineers and scientists to consider thismodern programming environment as an excellent alternative to languages such asFORTRAN or C++. MATLAB1embodies an interactive environment with a highlevel programming language supporting both numerical and graphical commands fortwo- and three-dimensional data analysis and presentation.

4 The wealth of intrinsicmathematical commands to handle matrix algebra, Fourier series, differential equa-tions, and complex-valued functions makes simple calculator operations of manytasks previously requiring subroutine libraries with cumbersome argument analyze problems, drawn from our teaching and research interests, empha-sizing linear and nonlinear differential equation methods. Linear partial differentialequations and linear matrix differential equations are analyzed using eigenfunctionsand series solutions. Several types of physical problems are considered. Amongthese are heat conduction, harmonic response of strings, membranes, beams, andtrusses, geometrical properties of areas and volumes,flexure and buckling of inde-terminate beams, elastostatic stress analysis, and multi-dimensional integration of matrix differential equations is used in several examplesillustrating the utility of such methods as well as essential aspects of numerical ap-proximation.

5 Attention is restricted to the Runge-Kutta method which is adequate tohandle most situations. Space limitation led us to omit some interesting MATLAB features concerning predictor-corrector methods, stiff systems, and event book is not an introductory numerical analysis text. It is most useful as a ref-erence or a supplementary text in computationally oriented courses emphasizing ap-plications. The authors have previously solved many of the examples in MATLAB solutions consume over three hundred pages (over twelve thousandlines). Although few books published recently present this much code, comparableFORTRAN versions would probably be signifcantly longer. In fact, the concisenessof MATLAB was a primary motivation for writing the programs contain many comments and are intended for study as separate en-tities without an additional reference.

6 Consequently, some deliberate redundancy1 MATLAB is a registered trademark of The MathWorks, Inc. For additional information contact:The MathWorks, Apple Hill DriveNatick, MA 01760-1500(508) 647-7000, Fax: (508) 647-7001 Email: 2003 by Chapman & Hall/CRCexists between program comments and text discussions. We also list programs in astyle we feel will be helpful to most readers. The source listings show line numbersadjacent to the MATLAB code. MATLAB code does not use line numbers or permitgotostatements. We have numbered the lines to aid discussions of particular pro-gram segments. To conserve space, we often place multiple MATLAB statements onthe same line when this does not interrupt the of the programs presented are designed to operate under the version ofMATLAB and Microsoft Windows. Both the text and graphics windows should besimultaneously visible.

7 A windowed environment is essential for using capabilitieslike animation and interactive manipulation of three dimensionalfigures. The sourcecode for all of the programs in the book is available from the CRC Press website The program collection is organized using anindependent subdirectory for each of the thirteen third edition incorporates much new material on time dependent solutions oflinear partial differential equations. Animation is used whenever seeing the solutionevolve in time is helpful. Animation illustrates quite well phenomena like wavepropagation in strings and membranes. The interactive zoom and rotation features inMATLAB are also valuable tools for interpreting graphical programs in the book are academic examples, but some problem solutionsare useful as stand-alone analysis tools. Examples include geometrical property cal-culation, differentiation or integration of splines, Gauss integration of arbitrary order,and frequency analysis of trusses and chapter on eigenvalue problems presents applications in stress analysis, elasticstability, and linear system dynamics.

8 A chapter on analytic functions shows theefficiency of MATLAB for applying complex valued functions and the Fast FourierTransform (FFT) to harmonic and biharmonic functions. Finally, the book concludeswith a chapter applying multidimensional search to several nonlinear emphasize that this book is primarily for those concerned with physical appli-cations. A thorough grasp of Euclidean geometry, Newtonian Mechanics , and somemathematics beyond calculus is essential to understand most of the topics. Finally,the authors enjoy interacting with students, teachers, and researchers applying ad-vanced Mathematics to real world availability of economical computerhardware and the friendly software interface in MATLAB makes computing increas-ingly attractive to the entire technical community. If we manage to cultivate interestin MATLAB among engineers who only spend part of their time using computers,our primary goal will have been B.

9 H. 2003 by Chapman & ,Norms,LinearIndependence, Systems of Linear Equations, Consistency, and Least Squares Mixed Boundary Value Problem for a Function using Rational Functions to Conformally Map a 2003 by Chapman & ,NullSpace,OrthonormalBases, ,Differentiation, Computing Solid Properties using Triangular Surface Elements 2003 by Chapman & General Concepts on Numerical Integration of Nonlinear Matrix Runge-Kutta Methods and the ODE45 Integrator Provided in Discussion of Procedures to Maintain Accuracy by Varying Wave Propagation in a Beam with an Impact Moment Applied Transient Heat Conduction in a Circular Cylinder with Spatially 2003 by Chapman & Accuracy Comparison for Euler Beam Natural Frequencies by 2003 by Chapman & (theBrachistochrone) 2003 by Chapman & Hall/CRCC hapter MATLAB: A Tool for Engineering AnalysisThis book presents various MATLAB applications in Mechanics and applied math-ematics.

10 Our objective is to employ numerical methods in examples emphasizing theappeal of MATLAB as a programming tool. The programs are intended for study asa primary component of the text. The numerical methods used include interpola-tion, numerical integration,finite differences, linear algebra, Fourier analysis, rootsof nonlinear equations, linear differential equations, nonlinear differential equations,linear partial differential equations, analytic functions, and optimization intrinsic MATLAB functions are used along with some utility functions devel-oped by the authors. The physical applications vary widely from solution of linearand nonlinear differential equations in mechanical system dynamics to geometricalproperty calculations for areas and many years FORTRAN has been the favorite programming language for solv-ing mathematical and engineering problems on digital computers.