An Engineer's Guide to Mathematica
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Table of Engineering Applications xvii
Part 1 Introduction
1 Mathematica Environment and Basic Syntax 3
1.1 Introduction 3
1.2 Selecting Notebook Characteristics 4
1.3 Notebook Cells 8
1.4 Delimiters 12
1.5 Basic Syntax 12
1.5.1 Introduction 12
1.5.2 Templates: Greek Symbols and Mathematical Notation 15
1.5.3 Variable Names and Global Variables 18
1.6 Mathematical Constants 19
1.7 Complex Numbers 21
1.8 Elementary, Trigonometric, Hyperbolic, and a Few Special Functions 22
1.9 Strings 25
1.9.1 String Creation: StringJoin[] and ToString[] 25
1.9.2 Labeled Output: Print[], NumberForm[], EngineeringForm[], and TraditionalForm[] 26
1.10 Conversions, Relational Operators, and Transformation Rule 28
1.11 Engineering Units and Unit Conversions: Quantity[] and UnitConvert[] 30
1.12 Creation of CDF Documents and Documents in Other Formats 33
1.13 Functions Introduced in Chapter 1 34
Exercises 35
2 List Creation and Manipulation: Vectors and Matrices 39
2.1 Introduction 39
2.2 Creating Lists and Vectors 39
2.2.1 Introduction 39
2.2.2 Creating a List with Table[] 45
2.2.3 Summing Elements of a List: Total[] 46
2.2.4 Selecting Elements of a List 47
2.2.5 Identifying List Elements Matching a Pattern: Position[] 49
2.3 Creating Matrices 51
2.3.1 Introduction 51
2.3.2 Matrix Generation Using Table[] 54
2.3.3 Accessing Elements of Arrays 55
2.4 Matrix Operations on Vectors and Arrays 56
2.4.1 Introduction 56
2.4.2 Matrix Inverse and Determinant: Inverse[] and Det[] 57
2.5 Solution of a Linear System of Equations: LinearSolve[] 58
2.6 Eigenvalues and Eigenvectors: EigenSystem[] 59
2.7 Functions Introduced in Chapter 2 61
References 61
Exercises 61
3 User-Created Functions, Repetitive Operations, and Conditionals 69
3.1 Introduction 69
3.2 Expressions and Procedures as Functions 69
3.2.1 Introduction 69
3.2.2 Pure Function: Function[] 74
3.2.3 Module[] 78
3.3 Find Elements of a List that Meet a Criterion: Select[] 80
3.4 Conditionals 82
3.4.1 If[] 82
3.4.2 Which[] 83
3.5 Repetitive Operations 83
3.5.1 Do[] 83
3.5.2 While[] 83
3.5.3 Nest[] 84
3.5.4 Map[] 84
3.6 Examples of Repetitive Operations and Conditionals 85
3.7 Functions Introduced in Chapter 3 92
Exercises 92
4 Symbolic Operations 95
4.1 Introduction 95
4.2 Assumption Options 101
4.3 Solutions of Equations: Solve[] 101
4.4 Limits: Limit[] 105
4.5 Power Series: Series[], Coefficient[], and CoefficientList[] 108
4.6 Optimization: Maximize[]/Minimize[] 112
4.7 Differentiation: D[] 114
4.8 Integration: Integrate[] 120
4.9 Solutions of Ordinary Differential Equations: DSolve[] 126
4.10 Solutions of Partial Differential Equations: DSolve[] 136
4.11 Laplace Transform: LaplaceTransform[] and InverseLaplaceTransform[] 138
4.12 Functions Introduced in Chapter 4 145
References 145
Exercises 146
5 Numerical Evaluations of Equations 151
5.1 Introduction 151
5.2 Numerical Integration: NIntegrate[] 151
5.3 Numerical Solutions of Differential Equations: NDSolveValue[] and ParametricNDSolveValue[] 154
5.4 Numerical Solutions of Equations: NSolve[] 178
5.5 Roots of Transcendental Equations: FindRoot[] 180
5.6 Minimum and Maximum: FindMinimum[] and FindMaximum[] 182
5.7 Fitting of Data: Interpolation[] and FindFit[] 186
5.8 Discrete Fourier Transforms and Correlation: Fourier[], InverseFourier[], and ListCorrelate[] 189
5.9 Functions Introduced in Chapter 5 194
References 195
Exercises 196
6 Graphics 209
6.1 Introduction 209
6.2 2D Graphics 209
6.2.1 Basic Plotting 209
6.2.2 Basic Graph Enhancements 213
6.2.3 Common 2D Shapes: Graphics[] 217
6.2.4 Additional Graph Enhancements 222
6.2.5 Combining Figures: Show[] and GraphicsGrid[] 238
6.2.6 Tooltip[] 241
6.2.7 Exporting Graphics 244
6.3 3D Graphics 244
6.4 Summary of Functions Introduced in Chapter 6 253
References 254
Exercises 254
7 Interactive Graphics 263
7.1 Interactive Graphics: Manipulate[] 263
References 287
Exercises 287
Part 2 Engineering Applications
8 Vibrations of Spring-Mass Systems and Thin Beams 293
8.1 Introduction 293
8.2 Single Degree-of-Freedom Systems 294
8.2.1 Periodic Force on a Single Degree-of-Freedom System 294
8.2.2 Squeeze Film Damping and Viscous Fluid Damping 298
8.2.3 Electrostatic Attraction 302
8.2.4 Single Degree-of-Freedom System Energy Harvester 304
8.3 Two Degrees-of-Freedom Systems 307
8.3.1 Governing Equations 307
8.3.2 Response to Harmonic Excitation: Amplitude Response Functions 307
8.3.3 Enhanced Energy Harvester 310
8.4 Thin Beams 315
8.4.1 Natural Frequencies and Mode Shapes of a Cantilever Beam with In-Span Attachments 315
8.4.2 Effects of Electrostatic Force on the Natural Frequency and Stability of a Beam 318
8.4.3 Response of a Cantilever Beam with an In-Span Attachment to an Impulse Force 323
References 326
9 Statistics 327
9.1 Descriptive Statistics 327
9.1.1 Introduction 327
9.1.2 Location Statistics: Mean[], StandardDeviation[], and Quartile[] 327
9.1.3 Continuous Distribution Functions: PDF[] and CDF[] 329
9.1.4 Histograms and Probability Plots: Histogram[] and ProbabilityScalePlot [] 331
9.1.5 Whisker Plot: BoxWhiskerChart[] 332
9.1.6 Creating Data with Specified Distributions: RandomVariate[] 334
9.2 Probability of Continuous Random Variables 334
9.2.1 Probability for Different Distributions: NProbability[] 334
9.2.2 Inverse Cumulative Distribution Function: InverseCDF[] 337
9.2.3 Distribution Parameter Estimation: EstimatedDistribution[] and FindDistributionParameters[] 337
9.2.4 Confidence Intervals: ¿CI[] 340
9.2.5 Hypothesis Testing: LocationTest[] and VarianceTest[] 342
9.3 Regression Analysis: LinearModelFit[] 343
9.3.1 Simple Linear Regression 343
9.3.2 Multiple Linear Regression 347
9.4 Nonlinear Regression Analysis: NonLinearModelFit[] 351
9.5 Analysis of Variance (ANOVA) and Factorial Designs: ANOVA[] 354
9.6 Functions Introduced in Chapter 9 358
10 Control Systems and Signal Processing 359
10.1 Introduction 359
10.2 Model Generation: State-Space and Transfer Function Representation 359
10.2.1 Introduction 359
10.2.2 State-Space Models: StateSpaceModel[] 360
10.2.3 Transfer Function Models: TransferFunctionModel[] 362
10.3 Model Connections - Closed-Loop Systems and System Response: SystemsModelFeedbackConnect[] and SystemsModelSeriesConnect[] 363
10.4 Design Methods 369
10.4.1 Root Locus: RootLocusPlot[] 369
10.4.2 Bode Plot: BodePlot[] 371
10.4.3 Nichols Plot: NicholsPlot[] 372
10.5 Signal Processing 374
10.5.1 Filter Models: ButterworthFilterModel[], EllipticFilterModel[], ... 374
10.5.2 Windows: HammingWindow[], HannWindow[], ... 381
10.5.3 Spectrum Averaging 385
10.6 Aliasing 388
10.7 Functions Introduced in Chapter 10 390
Reference 391
11 Heat Transfer and Fluid Mechanics 393
11.1 Introduction 393
11.2 Conduction Heat Transfer 394
11.2.1 One-Dimensional Transient Heat Diffusion in Solids 394
11.2.2 Heat Transfer in Concentric Spheres: Ablation of a Tumor 398
11.2.3 Heat Flow Through Fins 401
11.3 Natural Convection Along Heated Plates 405
11.4 View Factor Between Two Parallel Rectangular Surfaces 408
11.5 Internal Viscous Flow 411
11.5.1 Laminar Flow in Horizontal Cylindrical Pipes 411
11.5.2 Flow in Three Reservoirs 412
11.6 External Flow 416
11.6.1 Pressure Coefficient of a Joukowski Airfoil 416
11.6.2 Surface Profile in Nonuniform Flow in Open Channels 419
References 423
Index 425
Table of Engineering Applications
Topic Example or Section Controls State-Space Models Section 10.2.2 Transfer Function Models Section 10.2.3 Model Connections – Closed-Loop Systems and System Response Section 10.3 PID Control System Example 10.1 Root Locus Section 10.4.1 Bode Plot Section 10.4.2 Nichols Plot Section 10.4.3 Engineering Mathematics Evaluating a Fourier Series Example 2.1 Convergence of a Series Example 2.2 Summing a Double Series Example 2.3 Solution of a System of Equations Example 2.4 Secant Method Example 3.14 Solution of a System of Equations Example 4.2 Radius of Curvature Example 4.10 Euler–Lagrange Equation Example 4.11 Fourier Coefficients Example 4.13 Cauchy Integral Formula Example 4.18 System of First-Order Equations and the Matrix Exponential Example 4.23 Laplace Transform Solution of an Inhomogeneous Differential Equation #1 Example 4.29 Laplace Transform Solution of an Inhomogeneous Differential Equation #2 Example 4.31 Limit Using Assumptions Example 4.4 Perturbation Solution #1 Example 4.7 Perturbation Solution #2 Example 4.8 Poincare Plot Example 5.11 Nonlinear Ordinary Differential Equation Example 5.12 Second-Order Differential Equation: Periodic Inhomogeneous Term Example 5.15 Interpolation Function from Some Data Example 5.24 Function’s Parameters for a Fit to Some Data Example 5.25 Parametric Solution to a Nonlinear Differential Equation Example 5.26 Fluid Mechanics Flow Around a Cylinder Example 6.6 Air Entrainment by Liquid Jets Example 5.14 Flow Around an Ellipse Example 7.6 Laminar Flow in Horizontal Cylindrical Pipes Section 11.5.1 Flow in Three Reservoirs Section 11.5.2 Pressure Coefficient of a Joukowski Airfoil Section 11.6.1 Surface Profile in Nonuniform Flow in Open Channels Section 11.6.2 Heat Transfer Heat Conduction in a Slab Example 5.13 One-Dimensional Transient Heat Diffusion in Solids Section 11.2.1 Heat Transfer in Concentric Spheres: Ablation of a Tumor Section 11.2.2 Heat Flow Through Fins Section 11.2.3 Natural Convection Along Heated Plates Section 11.3 View Factor Between Two Parallel Rectangular Surfaces Section 11.4 Kinematics Four-Bar Linkage Example 7.7 Signal Processing Spectral Analysis of a Sine Wave Example 5.27 Spectral Analysis of a Sine Wave of Finite Duration Example 5.28 Cross-Correlation of a Signal with Noise Example 5.29 Sum of Two Sinusoidal Signals Example 7.2 Steerable Sonar/Radar Array Example 7.3 Effects of Filters on Sinusoidal Signals Example 10.2 Effects of Windows on Spectral Analysis Example 10.3 Spectrum Averaging Example 10.4 Aliasing Example 10.5 Statistics Histograms Example 9.1 Whisker Plot Section 9.1.5 Confidence Intervals Section 9.2.4 Hypothesis Testing Section 9.2.5 Simple Linear Regression Section 9.3.1 Multiple Linear Regression Section 9.3.2 Nonlinear Regression Analysis Section 9.4 Two-Factor ANOVA Example 9.2 Four-Factor Factorial Analysis Example 9.3 Structures Analysis of Beams Example 4.20 Deformation of a Timoshenko Beam Example 4.21 Beam with a Concentrated Load Example 5.4 Beam with an Overhang Example 5.5 Beam with Abrupt Change in Properties Example 5.6 Deflection of a Uniformly Loaded Solid Circular Plate Example 5.16 von Mises Stress in a Stretched Plate with a Hole Example 7.4 Analysis of Beams Example 7.5 Vibrations Natural Frequencies of a Three Degrees-of-Freedom System Example 2.5 Natural Frequency Coefficient of a Two Degrees-of-Freedom System Example 3.5 Natural Frequencies of Beams Example 4.5 Peak Amplitude Response of a Single Degree-of-Freedom System Example 4.9 Response of a Two Degrees-of-Freedom System Example 4.30 Two Degrees-of-Freedom System Revisited Example 5.7 Particle Impact Damper Example 5.8 Change in Period of a Nonlinear System Example 5.9 Single Degree-of-Freedom System Example 5.10 Natural Frequencies of a Beam Clamped at Both Ends Example...
