**A resource book applying mathematics to solve engineering problems**

*Applied Engineering Analysis* is a concise textbook which demonstrates how to apply mathematics to solve engineering problems. It begins with an overview of engineering analysis and an introduction to mathematical modeling, followed by vector calculus, matrices and linear algebra, and applications of first and second order differential equations. Fourier series and Laplace transform are also covered, along with partial differential equations, numerical solutions to nonlinear and differential equations and an introduction to finite element analysis. The book also covers statistics with applications to design and statistical process controls.

Drawing on the author′s extensive industry and teaching experience, spanning 40 years, the book takes a pedagogical approach and includes examples, case studies and end of chapter problems. It is also accompanied by a website hosting a solutions manual and PowerPoint slides for instructors.

**Key features:**

- Strong emphasis on deriving equations, not just solving given equations, for the solution of engineering problems.
- Examples and problems of a practical nature with illustrations to enhance student′s self–learning.
- Numerical methods and techniques, including finite element analysis.
- Includes coverage of statistical methods for probabilistic design analysis of structures and statistical process control (SPC).

*Applied Engineering Analysis* is a resource book for engineering students and professionals to learn how to apply the mathematics experience and skills that they have already acquired to their engineering profession for innovation, problem solving, and decision making.

Preface xvii

Suggestions to instructors xxi

About the companion website xxv

**1 Overview of Engineering Analysis 1**

Chapter Learning Objectives 1

1.1 Introduction 1

1.2 Engineering Analysis and Engineering Practices 2

1.2.1 Creation 2

1.2.2 Problem Solving 2

1.2.3 Decision Making 3

1.3 Toolbox for Engineering Analysis 5

1.4 The Four Stages in Engineering Analysis 8

1.5 Examples of the Application of Engineering Analysis in Design 10

1.6 The Safety Factor in Engineering Analysis of Structures 17

1.7 Problems 19

**2 Mathematical Modeling 21**

Chapter Learning Objectives 21

2.1 Introduction 21

2.2 MathematicalModeling Terminology 26

2.2.1 The Numbers 26

2.2.1.1 Real Numbers 26

2.2.1.2 Imaginary Numbers 26

2.2.1.3 Absolute Values 26

2.2.1.4 Constants 26

2.2.1.5 Parameters 26

2.2.2 Variables 26

2.2.3 Functions 27

2.2.3.1 Form 1. Functions with Discrete Values 27

2.2.3.2 Form 2. Continuous Functions 27

2.2.3.3 Form 3. Piecewise Continuous Functions 28

2.2.4 Curve Fitting Technique in Engineering Analysis 30

2.2.4.1 Curve Fitting Using Polynomial Functions 30

2.2.5 Derivative 31

2.2.5.1 The Physical Meaning of Derivatives 32

2.2.5.2 Mathematical Expression of Derivatives 33

2.2.5.3 Orders of Derivatives 35

2.2.5.4 Higher–order Derivatives in Engineering Analyses 35

2.2.5.5 The Partial Derivatives 36

2.2.6 Integration 36

2.2.6.1 The Concept of Integration 36

2.2.6.2 Mathematical Expression of Integrals 37

2.3 Applications of Integrals 38

2.3.1 Plane Area by Integration 38

2.3.1.1 Plane Area Bounded by Two Curves 41

2.3.2 Volumes of Solids of Revolution 42

2.3.3 Centroids of Plane Areas 47

2.3.3.1 Centroid of a Solid of Plane Geometry with Straight Edges 49

2.3.3.2 Centroid of a Solid with Plane Geometry Defined by Multiple Functions 50

2.3.4 Average Value of Continuous Functions 52

2.4 Special Functions for MathematicalModeling 54

2.4.1 Special Functions in Solutions in MathematicalModeling 55

2.4.1.1 The Error Function and Complementary Error Function 55

2.4.1.2 The Gamma Function 56

2.4.1.3 Bessel Functions 56

2.4.2 Special Functions for Particular Physical Phenomena 58

2.4.2.1 Step Functions 58

2.4.2.2 Impulsive Functions 60

2.5 Differential Equations 62

2.5.1 The Laws of Physics for Derivation of Differential Equations 62

2.6 Problems 65

**3 Vectors and Vector Calculus 73**

Chapter Learning Objectives 73

3.1 Vector and Scalar Quantities 73

3.2 Vectors in Rectangular and Cylindrical Coordinate Systems 75

3.2.1 Position Vectors 75

3.3 Vectors in 2D Planes and 3D Spaces 78

3.4 Vector Algebra 79

3.4.1 Addition of Vectors 79

3.4.2 Subtraction of Vectors 79

3.4.3 Addition and Subtraction of Vectors Using Unit Vectors in Rectangular Coordinate Systems 80

3.4.4 Multiplication of Vectors 81

3.4.4.1 Scalar Multiplier 81

3.4.4.2 Dot Product 82

3.4.4.3 Cross Product 84

3.4.4.4 Cross Product of Vectors for Plane Areas 86

3.4.4.5 Triple product 86

3.4.4.6 Additional Laws of Vector Algebra 87

3.4.4.7 Use of Triple Product of Vectors for Solid Volume 87

3.5 Vector Calculus 88

3.5.1 Vector Functions 88

3.5.2 Derivatives of Vector Functions 89

3.5.3 Gradient, Divergence, and Curl 91

3.5.3.1 Gradient 91

3.5.3.2 Divergence 91

3.5.3.3 Curl 91

3.6 Applications of Vector Calculus in Engineering Analysis 92

3.6.1 In Heat Transfer 93

3.6.2 In Fluid Mechanics 93

3.6.3 In Electromagnetism with Maxwell s Equations 94

3.7 Application of Vector Calculus in Rigid Body Dynamics 95

3.7.1 Rigid Body in RectilinearMotion 95

3.7.2 Plane CurvilinearMotion in Rectangular Coordinates 97

3.7.3 Application of Vector Calculus in the Kinematics of Projectiles 100

3.7.4 Plane CurvilinearMotion in Cylindrical Coordinates 103

3.7.5 Plane CurvilinearMotion with Normal and Tangential Components 109

3.8 Problems 114

**4 Linear Algebra and Matrices 119**

Chapter Learning Objectives 119

4.1 Introduction to Linear Algebra and Matrices 119

4.2 Determinants and Matrices 121

4.2.1 Evaluation of Determinants 121

4.2.2 Matrices in Engineering Analysis 123

4.3 Different Forms of Matrices 123

4.3.1 Rectangular Matrices 123

4.3.2 Square Matrices 124

4.3.3 Row Matrices 124

4.3.4 Column Matrices 124

4.3.5 Upper Triangular Matrices 124

4.3.6 Lower Triangular Matrices 125

4.3.7 Diagonal Matrices 125

4.3.8 Unit Matrices 125

4.4 Transposition of Matrices 125

4.5 Matrix Algebra 126

4.5.1 Addition and Subtraction of Matrices 126

4.5.2 Multiplication of a Matrix by a Scalar Quantity 127

4.5.3 Multiplication of Two Matrices 127

4.5.4 Matrix Representation of Simultaneous Linear Equations 128

4.5.5 Additional Rules for Multiplication of Matrices 129

4.6 Matrix Inversion, [A] 1 129

4.7 Solution of Simultaneous Linear Equations 131

4.7.1 The Need for Solving Large Numbers of Simultaneous Linear Equations 131

4.7.2 Solution of Large Numbers of Simultaneous Linear Equations Using the Inverse Matrix Technique 133

4.7.3 Solution of Simultaneous Equations Using the Gaussian Elimination Method 135

4.8 Eigenvalues and Eigenfunctions 141

4.8.1 Eigenvalues and Eigenvectors of Matrices 142

4.8.2 Mathematical Expressions of Eigenvalues and Eigenvectors of Square Matrices 142

4.8.3 Application of Eigenvalues and Eigenfunctions in Engineering Analysis 146

4.9 Problems 148

**5 Overview of Fourier Series 151**

Chapter Learning Objectives 151

5.1 Introduction 151

5.2 Representing Periodic Functions by Fourier Series 152

5.3 Mathematical Expression of Fourier Series 154

5.4 Convergence of Fourier Series 161

5.5 Convergence of Fourier Series at Discontinuities 164

5.6 Problems 169

**6 Introduction to the Laplace Transform and Applications 171**

Chapter Learning Objectives 171

6.1 Introduction 171

6.2 Mathematical Operator of Laplace Transform 172

6.3 Properties of the Laplace Transform 174

6.3.1 Linear Operator Property 174

6.3.2 Shifting Property 175

6.3.3 Change of Scale Property 175

6.4 Inverse Laplace Transform 176

6.4.1 Using the Laplace Transform Tables in Reverse 176

6.4.2 The Partial Fraction Method 176

6.4.3 The Convolution Theorem 178

6.5 Laplace Transform of Derivatives 180

6.5.1 Laplace Transform of Ordinary Derivatives 180

6.5.2 Laplace Transform of Partial Derivatives 181

6.6 Solution of Ordinary Differential Equations Using Laplace Transforms 184

6.6.1 Laplace Transform for Solving Nonhomogeneous Differential Equations 184

6.6.2 Differential Equation for the Bending of Beams 186

6.7 Solution of Partial Differential Equations Using Laplace Transforms 192

6.8 Problems 195

**7 Application of First–order Differential Equations in Engineering Analysis 199**

Chapter Learning Objectives 199

7.1 Introduction 199

7.2 Solution Methods for First–order Ordinary Differential Equations 200

7.2.1 Solution Methods for Separable Differential Equations 200

7.2.2 Solution of Linear, Homogeneous Equations 201

7.2.3 Solution of Linear, Nonhomogeneous Equations 202

7.3 Application of First–order Differential Equations in Fluid Mechanics Analysis 204

7.3.1 Fundamental Concepts 204

7.3.2 The Bernoulli Equation 205

7.3.3 The Continuity Equation 206

7.4 Liquid Flow in Reservoirs, Tanks, and Funnels 206

7.4.1 Derivation of Differential Equations 207

7.4.2 Solution of Differential Equations 208

7.4.3 Drainage of Tapered Funnels 209

7.5 Application of First–order Differential Equations in Heat Transfer Analysis 217

7.5.1 Fourier s Law of Heat Conduction in Solids 217

7.5.2 Mathematical Expression of Fourier s Law 218

7.5.3 Heat Flux in a Three–dimensional Space 221

7.5.4 Newton s Cooling Law for Heat Convection 227

7.5.5 Heat Transfer between Solids and Fluids 227

7.6 Rigid Body Dynamics under the Influence of Gravitation 233

7.7 Problems 237

**8 Application of Second–order Ordinary Differential Equations in Mechanical Vibration Analysis 243**

Chapter Learning Objectives 243

8.1 Introduction 243

8.2 Solution Method for Typical Homogeneous, Second–order Linear Differential Equations with Constant Coefficients 243

8.3 Applications in Mechanical Vibration Analyses 246

8.3.1 What Is Mechanical Vibration? 246

8.3.2 Common Sources for Vibration 247

8.3.3 Common Types of Vibration 247

8.3.4 Classification of Mechanical Vibration Analyses 247

8.3.4.1 Free Vibration 247

8.3.4.2 Damped Vibration 248

8.3.4.3 Forced Vibration 249

8.4 Mathematical Modeling of Free Mechanical Vibration: Simple Mass Spring Systems 249

8.4.1 Solution of the Differential Equation 251

8.5 Modeling of Damped FreeMechanical Vibration: Simple Mass Spring Systems 254

8.5.1 The Physical Model 254

8.5.2 The Differential Equation 255

8.5.3 Solution of the Differential Equation 256

8.6 Solution of Nonhomogeneous, Second–order Linear Differential Equations with Constant Coefficients 258

8.6.1 Typical Equation and Solutions 258

8.6.2 The Complementary and Particular Solutions 258

8.6.3 The Particular Solutions 259

8.6.4 Special Case for Solution of Nonhomogeneous Second–order Differential Equations 263

8.7 Application in Forced Vibration Analysis 264

8.7.1 Derivation of the Differential Equation 264

8.7.2 Resonant Vibration 266

8.8 Near Resonant Vibration 273

8.9 Natural Frequencies of Structures and Modal Analysis 277

8.10 Problems 280

**9 Applications of Partial Differential Equations in Mechanical Engineering Analysis 285**

Chapter Learning Objectives 285

9.1 Introduction 285

9.2 Partial Derivatives 285

9.3 Solution Methods for Partial Differential Equations 287

9.3.1 The Separation of VariablesMethod 287

9.3.2 Laplace Transform Method for Solution of Partial Differential Equations 288

9.3.3 Fourier Transform Method for Solution of Partial Differential Equations 288

9.4 Partial Differential Equations for Heat Conduction in Solids 291

9.4.1 Heat Conduction in Engineering Analysis 291

9.4.2 Derivation of Partial Differential Equations for Heat Conduction Analysis 291

9.4.3 Heat Conduction Equation in Rectangular Coordinate Systems 292

9.4.4 Heat Conduction Equation in a Cylindrical Polar Coordinate System 293

9.4.5 General Heat Conduction Equation 293

9.4.6 Initial and Boundary Conditions 293

9.5 Solution of Partial Differential Equations for Transient Heat Conduction Analysis 298

9.5.1 Transient Heat Conduction Analysis in Rectangular Coordinate System 298

9.5.2 Transient Heat Conduction Analysis in the Cylindrical Polar Coordinate System 303

9.6 Solution of Partial Differential Equations for Steady–state Heat Conduction Analysis 308

9.6.1 Steady–state Heat Conduction Analysis in the Rectangular Coordinate System 308

9.6.2 Steady–state Heat Conduction Analysis in the Cylindrical Polar Coordinate System 311

9.7 Partial Differential Equations for Transverse Vibration of Cable Structures 314

9.7.1 Derivation of Partial Differential Equations for Free Vibration of Cable Structures 314

9.7.2 Solution of Partial Differential Equation for Free Vibration of Cable Structures 318

9.7.3 Convergence of Series Solutions 322

9.7.4 Modes of Vibration of Cable Structures 323

9.8 Partial Differential Equations for Transverse Vibration of Membranes 328

9.8.1 Derivation of the Partial Differential Equation 328

9.8.2 Solution of the Partial Differential Equation for Plate Vibration 331

9.8.3 Numerical Solution of the Partial Differential Equation for Plate Vibration 334

9.9 Problems 336

**10 Numerical Solution Methods for Engineering Analysis 339**

Chapter Learning Objectives 339

10.1 Introduction 339

10.2 Engineering Analysis with Numerical Solutions 340

10.3 Solution of Nonlinear Equations 341

10.3.1 Solution Using Microsoft Excel Software 341

10.3.2 The Newton RaphsonMethod 342

10.4 Numerical Integration Methods 347

10.4.1 The Trapezoidal Rule for Numerical Integration 348

10.4.2 Numerical Integration by Simpson s One–third Rule 352

10.4.3 Numerical Integration by Gaussian Quadrature 356

10.5 Numerical Methods for Solving Differential Equations 361

10.5.1 The Principle of Finite Difference 362

10.5.2 TheThree Basic Finite–difference Schemes 363

10.5.3 Finite–difference Formulation for Partial Derivatives 366

10.5.4 Numerical Solution of Differential Equations 367

10.5.4.1 The Second–order Runge Kutta Method 367

10.5.4.2 The Fourth–order Runge Kutta Method 369

10.5.4.3 Runge Kutta Method for Higher–order Differential Equations 370

10.6 Introduction to Numerical Analysis Software Packages 375

10.6.1 Introduction to Mathematica 375

10.6.2 Introduction to MATLAB 376

10.7 Problems 377

**11 Introduction to Finite–element Analysis 381**

Chapter Learning Objectives 381

11.1 Introduction 381

11.2 The Principle of Finite–element Analysis 383

11.3 Steps in Finite–element Analysis 383

11.3.1 Derivation of Interpolation Function for Simplex Elements with Scalar Quantities at Nodes 388

11.3.2 Derivation of Interpolation Function for Simplex Elements with Vector Quantities at Nodes 390

11.4 Output of Finite–element Analysis 401

11.5 Elastic Stress Analysis of Solid Structures by the Finite–elementMethod 403

11.5.1 Stresses 404

11.5.2 Displacements 406

11.5.3 Strains 406

11.5.4 Fundamental Relationships 407

11.5.4.1 Strain Displacement Relations 407

11.5.4.2 Stress Strain Relations 408

11.5.4.3 Strain Energy in Deformed Elastic Solids 409

11.5.5 Finite–element Formulation 409

11.5.6 Finite–element Formulation for One–dimensional Solid Structures 413

11.6 General–purpose Finite–element Analysis Codes 417

11.6.1 Common Features in General–purpose Finite–element Codes 419

11.6.2 Simulation using general–purpose finite–element codes 420

11.7 Problems 422

**12 Statistics for Engineering Analysis 425**

Chapter Learning Objectives 425

12.1 Introduction 425

12.2 Statistics in Engineering Practice 427

12.3 The Scope of Statistics 428

12.4 Common Concepts and Terminology in Statistical Analysis 430

12.4.1 The Mode of a Dataset 430

12.4.2 The Histogram of a Statistical Dataset 430

12.4.3 The Mean 431

12.4.4 The Median 433

12.4.5 Variation and Deviation 433

12.5 Standard Deviation ( ) and Variance ( 2) 434

12.5.1 The Standard Deviation 434

12.5.2 The Variance 434

12.6 The Normal Distribution Curve and Normal Distribution Function 435

12.7 Weibull Distribution Function for Probabilistic Engineering Design 437

12.7.1 Statistical Approach to the Design of Structures Made of Ceramic and Brittle Materials 438

12.7.2 TheWeibull Distribution Function 439

12.7.3 Estimation ofWeibull Parameters 441

12.7.4 Probabilistic Design of Structures with Random Fracture Strength of Materials 443

12.8 Statistical Quality Control 447

12.9 Statistical Process Control 448

12.9.1 Quality Issues in Industrial Automation and Mass Production 448

12.9.2 The Statistical Process Control Method 449

12.10 The Control Charts 450

12.10.1 Three–Sigma Control Charts 451

12.10.2 Control Charts for Sample Ranges (the R–Chart) 453

12.11 Problems 456

Bibliography 459

A Table for the Laplace Transform 463

B Recommended Units for Engineering Analysis 465

C Conversion of Units 467

D Application of MATLAB Software for Numerical Solutions in Engineering

Analysis 469

Index 483

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**TAI–RAN HSU,** *San Jose State University, USA*

**TAI–RAN HSU** is currently a Professor of Mechanical Engineering at San Jose State University (SJSU), San Jose, California, USA. He joined the SJSU as the Chair of the department in 1990 and served two terms until 1998, and also from 2012 to 2015. He served in a similar capacity at the University of Manitoba, Winnipeg, Canada before joining SJSU. Prior to his academic career, he worked as a design engineer with heat exchangers, steam power plant equipment, large steam turbines, and nuclear reactor fuel systems for major industries in Canada and U.S.A. He has published six books and edited another two on a wide ranging topics on finite element method in thermomechanics, microelectronics packaging, CAD, and MEMS and microsystems design and packaging. Additionally, he published over one hundred technical papers in archive journals and conference proceedings.