METHODS OF ENGINEERING MATHEMATICS

Transcription

1 METHODS OF ENGINEERING MATHEMATICS Edward J. Hang Kyung K. Choi Department of Mechanical Engineering College of Engineering The University of Iowa Iowa City, Iowa 52242

2 METHODS OF ENGINEERING MATHEMATICS Edward.J. Haug Kyung K. Choi Department of Mechanical Engineering College of Engineering The University of Iowa Iowa City, Iowa Copyright 1992

3 CONTENTS 1 PREFACE MATRICES 1.1 Linear Equations 1 Examples of Linear Equations, 1 Solution of Linear Equations, 4 Exercises, Matrix Algebra 7 Matrix Equations, 7 Matrix Operations, 8 Exercises, 12 xiii Matrix Inverse and Elementary Matrix Operations Matrix Inverse, 14 Elementary Matrices, 16 Exercises, Determinants 20 Determinant of a Matrix, 20 Properties of Determinants, 21 Matrix Inverse and Cramer's Rule, Exercises, LINEAR ALGEBRA 2.1 Vector Algebra 30 Geometric Vectors, 30 Algebraic Vectors, 36 Exercises, Vector Spaces 39 n-dimensional Real Space (Rn). 39 Linear Independence of Vectors, Ill

4 IV Contents 3 Basis and Dimension of Vector Spaces, 43 Sub spaces, 46 Exercises, Rank of Matrices and Solution of Matrix Equations 48 Rank of a Matrix, 49 Elementary Row Operations, 50 Solution of Matrix Equations, 52 Exercises, Scalar Product and Norm 60 Scalar Product on a Vector Space, 60 Norm and Distance in a Vector Space, 60 Ortlwnormal Bases, 64 Gram-Schmidt Orthonormalization, 69 Exercises, 72 EIGENVALUE PROBLEMS AND QUADRATIC FORMS 3.1 Eigenvalue Problems 7 4 Eigenvalues and Eigenvectors, 75 Characteristic Equations, 76 Eigenvector Orthogonality for Symmetric Matrices, 80 Eigenvector Expansion, 83 Exercises, Quadratic Forms 88 Matrix Quadratic Forms, 88 Strain Energy, 89 Canonical (Diagonal Matrix) Forms, 93 Modal Matrices, 94 Exercises, Positive Definite Quadratic Forms 100 Positive Definite and Semidefinite Matrices, 100 Energy Scalar Product and Norm, 104 Exercises, Generalized Eigenvalue Problems 107 Generalized Eigenvalue Equations, 108 B-Orthogonality of Eigenvectors, 109 Diagonalization of a Pair of Quadratic Forms, 111 Second Order Linear Differential Equations of Vibration, 114 Exercises, Minimum Principles for Matrix Equations 119 Minimum Functional Theorem,

5 Contents v Ritz Method, 121 Minimization of the Rayleigh Quotient, 122 Exercises, INFINITE SERIES 4.1 Infinite Series Whose Terms Are Constants 125 Convergence of Series of Nonnegative Constants, 125 Alternating Series, 130 Absolute Convergence, 131 Exercises, Infinite Series Whose Terms Are Functions 133 Uniform Convergence, 134 Differentiation and Integration of Series, 138 Taylor Series, 141 Power Series, 145 Exercises, Power Series Solution of Ordinary Differential Equations 149 Expansions About Ordinary Points, 149 Expansions About Singular Points (Frobenius Method), 155 Exercises, Introduction to Special Functions 162 Legendre Polynomials, 163 Bessel Functions, 164 Exercises, FUNCTION SPACES AND FOURIER SERIES 5.1 The Algebra of Function Spaces 168 Representation of Functions, 168 Algebraic Operations with Functions, 169 Vector Spaces of Functions, 172 Exercises, Scalar Product and Norm in Function Spaces 174 Scalar Product and Norm for Integrable Functions, 174 The Space L2(0) of Square Integrable Functions, 179 Linear Independence and Orthogonality of Functions, 180 Gram-Schmidt Orthonormalization in L2(0), 182 Exercises, Bases and Completeness of Function Spaces 184 The Space..t2 of Square Summable Sequences,

6 vi Contents 6 Convergence of Sequences in Normed Vector Spaces, 188 Cauchy Sequences and Complete Vector Spaces, 190 Least Square Approximation and Complete Sets of Functions, 193 Exercises, Construction of Fourier Series 200 Piecewise Continuous Functions, 200 Even and Odd Functions, 202 Fourier Series for Periodic Functions, 204 Even and Odd Extensions of Functions, 208 Exercises, Pointwise Convergence of Fourier Series 212 Pointwise Convergence, 212 UniformConvergence, 216 Exercises, Differentiation and Integration of Fourier Series 220 Termwise Differentiation of Fourier Series, 220 Termwise Integration of Fourier Series, 222 Exercises, Fourier Series In Two Variables 226 Exercises, 229 MATHEMATICAL MODELS IN MECHANICS 6.1 Multiple Integral Theorems 230 Properties of Multiple Integrals, 230 The Mean Value Theorem and Leibniz's Rule, 232 Line and Surface Integrals, 234 Green and Divergence Theorems, 235 The Implicit Function Theorem, 240 Exercises, Material Derivative and Differential Forms 243 Description of a Continuum, 243 Material Derivative, 245 Differential Forms, 247 Exercises, Vibrating Strings and Membranes 251 The Transversely Vibrating String, 251 The Transversely Vibrating Membrane, 255 Steady State String and Membrane Problems, 259 Exercises,

7 Contents vii 6.4 Heat and Diffusion Problems 262 Conduction of Heat in Solids, 262 Diffusion of Gas in Porous Materials, 265 Exercises, Fluid Dynamics 268 Dynamics of Compressible Fluids, 268 I"otational, Incompressible, and Steady Isentropic Flows, 272 Propagation of Sound in Gas, 276 Exercises, Analogies (Physical Classification) INTRODUCTION TO THE THEORY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS First Order Partial Differential Equations 280 Reduction to Ordinary Differential Equations, 280 General Solutions, 281 Exercises, General Theory of Second Order Partial Differential Equations 284 Second Order Equations in Two Variables, 284 Cauchy Data, 286 The Cauchy-Kowalewski Existence Theorem, 289 Exercises, Characteristic Curves 291 Initial Curves, 291 Prototype Cauchy Problems, 293 Exercises, Stability of Solutions and Well-Posed Problems Classification of Linear Equations in Two Independent Variables 297 Canonical Forms of Linear Equations, 298 Hyperbolic Equations, 301 Parabolic Equations, 302 Elliptic Equations, Characteristic Variables 306 Exercises, Linear Equations in More Than Two Independent Variables 311 Canonical Form of Linear Equations, 311 Classification of Linear Equations, 313 Exercises, 314

8 viii 8 9 METHODS OF SOLVING SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS 8.1 The Wave Equation 316 General Solution of the Wave Equation, 316 Propagation ofwaves, 319 D'Alemberts Formula, 321 Uniqueness of Solution, 323 Exercises, The Heat Equation 327 Separation of Variables, 328 Existence of Solution, 330 Uniqueness of Solution, 331 Exercises, Laplace's Equation 333 Separation of Variables, 334 Existence of Solution, Elliptic Differential Equations 337 Elliptic Boundary-Value Problems, 337 General Properties of Harmonic Functions, 340 Separation ofvariablesfor the Laplace Equation on a Circle, 344 Well Posed Boundary-Value Problem, 347 Green's Function, 347 Exercises, Parabolic Differential Equations 351 Parabolic Initial-Boundary-Value Problems, 351 Separation ofvariables, 352 Maximum Principle, 355 Well Posed Initial-Boundary-Value Problem, 358 Exercises, Hyperbolic Differential Equations 360 Fundamental Solutions of the Cauchy Problem, 360 Domains of Dependence and Influence, 362 Initial-Boundary-Value Problem, 365 Wave Propagation, 373 Separation of Variables, 376 Exercises, 382 LINEAR OPERATOR THEORY IN MECHANICS 9.1 Linear Functionals 383 Bounded Functionals, 383 Contents

9 Contents IX Linear Functionals, 385 Riesz Representation Theorem, 387 Exercises, Linear Operator Equations 389 Linear Operators, 390 Bounded Linear Operators, 390 Symmetric Operators, 392 Positive Operators, 393 Matrices as Linear Operators, 396 Operator Eigenvalue Problems, 400 Exercises, Sturm-Liouville Problems 402 Eigenfunctions and Eigenvalues, 402 Completeness of Eigenfunctions in L2, 405 Exercises, Separation of Variables and Eigenfunction Expansions 407 Exercises, A Formal Treatment of the Eigenfunction Expansion Method 414 Separation of Variables, 415 Completeness of Eigenfunctions, 415 Green's Function, Green's Function for Ordinary Boundary-Value Problems 419 Influence Function, 420 Properties of Green's Functions, 421 Exercises, Completeness of Eigenfunctions 425 Green's Function for the Sturm-Liouville Operator, 425 Completeness of Sturm-Liouville Equations, 427 Completeness of Eigenfunctions of General Operators, VARIATIONAL METHODS FOR BOUNDARY-VALUE PROBLEMS Energy Convergence 433 Energy Scalar Product and Norm, 433 Energy Convergence, 434 Exercises, The Minimum Functional Theorem 438 Minimum Principles for Operator Equations, 438 The Space of Functions with Finite Energy, 440

10 X Contents Generalized Solutions of Operator Equations, 447 Exercises, Calculus of Variations 448 Variations of Functionals, 448 Minimization of Functionals, 450 Necessary Conditions for Minimization of Functionals, 451 Exercises, Natural Boundary Conditions 463 Exercises, Nonhomogeneous Boundary Conditions 466 Exercises, Eigenvalue Problems 471 Completeness of Eigenfunctions in Energy, 471 Minimization Principles for Eigenvalue Problems, 472 Exercises, Minimizing Sequences and the Ritz Method 477 Minimizing Sequences, 477 The Ritz Method of Generating Minimizing Sequences, 478 Exercises, Ritz Method for Equilibrium Problems 481 Selection of Coordinate Functions, 481 Ritz Approximate Solutions, 481 Exercises, Ritz Method for Eigenvalue Problems 486 Selection of Coordinate Functions, 487 Ritz Approximate Solutions, 487 Exercises, Other Variational Methods 494 The Method of Least Squares, 494 The Galerkin Method, 495 Exercises, APPLICATIONS OF VARIATIONAL METHODS Boundary-Value Problems for Ordinary Differential Equations 498 Theoretical Results, 498 Bending of a Beam of Variable Cross Section on an Elastic Foundation, 503 Eigenvalues of a Second-Order Ordinary Differential Equation, 504 Stability of a Column, 505 Vibration of a Beam of Variable Cross Section, 507

11 Contents xi 11.2 Second Order Partial Differential Equations 511 Boundary Value Problems for Poisson and Laplace Equations, 511 Torsion of a Rod of Rectangular Cross Section, Higher Order Equations and Systems of Partial Differential Equations 520 Bending of Thin Plates, 520 Bending of a Clamped Rectangular Plate, 524 Computer Solution of a Clamped Plate Problem, Solution of Beam and Plate Problems by the Galerkin Method 530 Static Analysis of Beams, 530 Simply Supported Beam, 531 Clamped Beam, 534 Static Analysis of Plates, 537 Simply Supported Plate, 538 Clamped Plate, 541 Vibration of Beams and Plates, AN INTRODUCTION TO FINITE ELEMENT METHODS Finite Elements for One Dimensional Boundary-Value Problems 547 Linear Elements, 548 Quadratic Elements, 552 Cubic Elements, Finite Elements for Two Dimensional Boundary-Value Problems 558 Linear Triangular Elements, 558 Quadratic Triangular Elements, 560 Cubic Triangular Elements, 561 Quintic Triangular Elements, 562 Rectangular Elements, Finite Element Solution of a System of Second Order Partial Differential Equations 564 Second Order Plate Boundary-Value Problems, 564 Galerkin Approximate Solution, Finite Element Methods in Dynamics 568 The Heat Equation, 569 Hyperbolic Equations, 571 REFERENCES INDEX

12 PREFACE Purpose and Uses of Text The purpose of this text is to develop and illustrate the use of mathematical methods that are essential for advanced study in a number of engineering disciplines. The fundamentals of linear algebra, matrices, infinite series, function spaces, ordinary differential equations, and partial differential equations are developed at an inte1mediate level of mathematical sophistication. Statements of definitions and presentations of theorems and their proofs are at the level of mathematical rigor that is required for advanced applications and research in disciplines such as Mechanical and Civil Engineering and related engineering specializations that employ modem methods of mechanics. Model engineering applications are employed throughout the text to illustrate the use of methods presented and to provide the engineer with a concrete framework in which mathematical methods may be related. This text has evolved from a two course sequence that has been taught at the advanced undergraduate and first year graduate level in the College of Engineering at the University of Iowa over the past twenty years. The first course, which constitutes Chapters 1 through 8 of this text, is a corequisi te for intermediate level courses in mechanics of materials, structural mechanics, finite element methods, dynamics, 11uid mechanics, and thermal sciences. This material and the contents of Chapters 9 through 12 are pre-requisites for advanced courses in continuum mechanics, elasticity, structural mechanics, dynamics of machines, finite element methods, fluid mechanics, and design optimization. The level of rigor of the text is the minimum essential to support modem engineering applications and research. Topical coverage in this text has been selected to support basic graduate level courses in engineering. Every attempt has been made to avoid making the text a cookbook that covers a plethora of methods, but without an adequate level of ligor. As a result, coverage has been limited to methods that are essential in numerous engineering graduate courses and in enginee1ing applications. Also, breadth of coverage has been sacrificed in favor of a reasonably thorough and rigorous treatment of central topics and mathematical methods that support the engineedng sciences. Organization of Text Linear algebraic equations occur in all branches of applied science and engineeling. Chapter 1 develops basic tools of matrix algebra that play an impmtant role in the solution of linear equations and in high speed digital computation. For most engineers, the mate1ial in xiii

13 xiv Preface Chapter 1 should be a review of techniques they learned in their undergraduate matrix algebra course. Chapter 2 presents concepts of linear algebra and matrix theory in a vector space setting, by first introducing the algebraic representation of geometric vectors. Vector algebra in any finite dimensional vector space is represented in tenus of matrix operations, which reduces vector operations to a form that can be easily implemented on a digital computer. Vector space concepts are also used to determine existence and uniqueness of solutions of linear equations, without necessadly solving them. Introduction of a norm extends the idea of distance between vectors and introduction of a scalar product extends the concept of orthogonality of vectors. These generalizations of conventional vector properties provide the engineer with a geometric viewpoint that takes advantage of basic ideas of three dimensional vector analysis, even in analysis of partial differential equations. Eigenvalue problems that arise in engineering applications such as vibration and stability and in mathematical techniques such as separation of variables and eigenvector expansion methods are introduced in Chapter 3. Another concept introduced in Chapter 3 is quadratic forms that are associated with energy in mechanical systems. Under certain physically meaningful conditions, it is shown that the solution of equilibrium equations in mechanics is equivalent to minimization of a quadratic form, which may be the total potential energy of the system. To support series solution techniques for ordinary and partial differential equations, infinite series whose terms are constants and functions are studied in Chapter 4. The important concept of uniform convergence of infinite series whose terms are functions is introduced and used to establish the convergence of power series solutions of ordinary differential equations. The Frobenius method is presented and special functions, such as Bessel functions, are introduced. Many engineering applications deal with continuum behavior of a solid or fluid medium, which may be described by a function u(x, t) of a spatial variable x and possibly time t. To assist the engineer to gain insight in continuum applications, the idea of function spaces of candidate solutions is introduced in Chapter 5. The concept of a collection of functions that are candidate solutions is shown to be a natural extension of the finite dimensional vector space ideas developed in Chapters 1 through 3. Much as in Chapter 2, function space algebra is defined and the concepts of scalar product and norm are used to establish a natural algebra and geometry of function spaces. With a norm and scalar product, concepts of closeness of approximation and orthogonality of functions are developed. One of the most important function spaces introduced in this chapter is the space L 2 (Q) of square integrable functions on a domain Q in the physical Euclidean space Rn. Similadties in basic properties and concepts of vector analysis in the finite dimensional space Rn and in the infinite dimensional function space L 2 (Q) are established to aid the engineer's intuition. The infinite dimensionality of function spaces requires that limit concepts be introduced in the function space setting. As a tool to describe a vadety of functions, methods of constructing Fourier sine and cosine series and the theory of their convergence are developed. In Chapter 6, differential equations of model continuum problems that are encountered in fluid dynamics, elasticity, and heat transfer are derived, using conservation laws

14 Preface XV of mechanics, multiple integral theorems, and the concept of the material derivative. While many engineering applications have fundamentally different physical properties, it is shown that the governing equations for each of these fields fall into one of four different forms of second order partial differential equation; Laplace (equilibrium), parabolic (heat), hyperbolic (wave), and eigenvalue equations. Second order linear partial differential equations are classified in Chapter 7, from a mathematical point of view, using characteristic variables. It is shown that the same three basic forms of equations obtained in Chapter 6, from an applications point of view, are obtained using a mathematical criteria. The Cauchy-Kowalewski Theorem that gives a theoretical method for analyzing second order partial differential equations is used in this chapter. The concepts of stability of solutions and well-posed problems are also introduced and related to the physical behavior of engineering systems. Methods of solving second order partial differential equations derived in Chapter 6 and classified in Chapter 7 are developed and illustrated in Chapter 8. Solution methods such as the method of separation of variables and D'Alembert's formula are presented and shown to be broadly applicable. Partial differential equations that govern the mechanics of a broad range of engineering systems are formulated in Chapter 9, in terms of linear operators. Basic concepts of linear operators are introduced and properties such as symmetry, boundedness, and positive definiteness are defined and related to physical characteristics of applications. Sturm Liouville problems that mise in separation of variables and associated eigenvalue problems are studied in detail and completeness of their eigenfunctions is established. Separation of variables and eigenfunction expansion methods introduced in Chapter 8 for second-order partial differential equations are extended to general problems of engineering analysis. Green's functions are introduced for boundary:-value problems and used to establish completeness of eigenfunctions of broad classes of operators. It is shown that the methods developed for model problems in Chapters 1 through 8 are in fact broadly applicable in engineering analysis, with mathematical properties that support both theoretical and computational methods. Variational methods for solving boundary-value problems that have become standard tools in the applied mathematics community are introduced and developed in Chapter 10. Concepts of convergence in energy are shown to be both physically natural and, in the case of positive-bounded below operators of mechanics, to yield stronger results than conventional convergence in the mean. The equivalence between solving boundary-value problems and minimizing energy functionals is established, as the foundation for modern computational methods in engineering mechanics. Basic ideas of the calculus of variations are introduced and used to establish criteria for identifying natural boundary conditions that may be ignored during va1iational solution of boundary-value problems. The Ritz method for creating approximate solutions of boundary-value problems is developed and its convergence properties are established for both equilibrium and eigenvalue problems. Finally, the Galerkin method is introduced and shown to be an extension of the Ritz method for broad classes of applications. In order to provide the reader with experience in formulation and numerical application of variational methods, moderate scale applications are presented in Chapter 11. The

15 xvi Preface governing equations in these applications are shown to satisfy criteria for application of variational methods and approximating functions are selected to be both theoretically correct and practical for computation. Analytical and numerical results for second order and higher-order ordinary and partial differential equations are presented and analyzed, to illustrate convergence characteristics of the Ritz and Galerkin methods. Chapter 12 introduces basic ideas of modem finite element methods, as a means to systematically construct approximating functions that are well-suited for application of variational methods. Finite element approximations are presented and illustrated with applications to beams and plates. Numerical comparisons with applications developed in Chapter 11 are presented to provide the reader with confidence that systematic finite element methods can be used to obtain sound approximate solutions of realistic boundaryvalue problems.