CONTENTS
1. Introduction = 1
1.1 MATLAB : A Tool for Engineering Analysis = 1
1.2 Use of MATLAB Commands and Related Reference Materials = 2
1.2.1 Example Program to Compute the Value of e = 9
1.3 Description of MATLAB Commands and Related Reference Materials = 17
2. Elementary Aspects of MATLAB Graphics = 31
2.1 Introduction = 31
2.2 Overview of Graphics = 32
2.3 Polynomial Interpolation Example = 34
2.4 Conformal Mapping Example = 40
2.5 String Vibration Example = 46
2.6 Example on Animation of a Rctating Cube = 53
3. Summary of Concepts From Linear Algebra = 63
3.1 Introduction = 63
3.2 Vectors, Norms, Linear Independence, and Rank = 63
3.3 Systems of Linear Equations, Consistency, and Least Square Approximation = 65
3.4 Application of Least Square Approximation = 68
3.4.1 A Membrane Deflection Problem = 68
3.4.2 Mixed Boundary Value Problem for a Function Harmonic Inside a Circular Disk = 73
3.4.3 Using Rationa Functions to Conformally Map a Circular Disk Onto a Square = 79
3.5 Eigenvalue Problems = 86
3.5.1 Statement of the Problem = 86
3.5.2 Application to Solution of Martix Differential Equations = 89
3.6 Column Space, Null Space, Orthonormal Bases, and SVD = 89
3.7 Program Comparing FLOP Counts for Various Martix Operations = 92
4. Methods for Interpolation and Numerical Differentiation = 99
4.1 Concepts of Interpolation = 99
4.1.1 Example : Newton Polynomial Interpolation = 103
4.2 Interpolation, Differentiation, and Integration By Cubic Splines = 105
4.2.1 Example : Spline Interpolation Applied to sin(X) = 113
4.2.2 Example : Plotting of General Plane Curves = 114
4.3 Numerical Differentiations Using Finite Difference Formulas = 115
4.3.1 Example : Deriving General Difference Formulas = 118
4.3.2 Example : Deriving Adams Type Integration Formulas = 123
5. Gaussian Integration withApplication to Geometric Properties = 127
5.1 Fundamental Concepts and Intrinsic Integration Tools Provided in MATLAB = 127
5.2 Concepts of Gauss Integration = 133
5.3 Examples Comparing Different Integration Methods = 135
5.3.1 Example : Computation of Base Points and Weight Factors = 137
5.4 Line Integrals for Geometric Properties of Plane Areas = 147
5.4.1 Geometry Example Using a Simple Spline Interpolated Boundary = 151
5.5 Spline Approximation of General Boundary Shapes = 157
5.5.1 Program for Exact Properties of Any Area Bounded by Straight Lines and Circular Arcs = 161
5.5.2 Program to Analyze Spline Interpolated Boundaries = 171
6. Fourier Series and the FFT = 177
6.1 Definitions and Computation of Fourier Coefficients = 177
6.1.1 Trigonometric Interpolation and the FFT = 179
6.2 Some Applications = 182
6.2.1 Using the FFT to Compute Integer Order Bessel Functions = 183
6.2.2 Dynamic Response of a Mass on an Oscillating Foundations = 187
6.2.3 General Program to Construct Fourier Expansions = 202
7. Dynamic Response of Linear Second Order Systems = 215
7.1 Solving the Structural Dynamics Equations for Periodic Applied Forces = 215
7.1.1 Application to Oscillations of a Vertically Suspended Cable = 217
7.2 Direct Integration Methods = 229
7.2.1 Example on Cable Response by Direct Integration = 231
8. Integration of Nonlinear Intial Value Problems = 241
8.1 General Concepts on Numerical Integration of Nonlinear Matrix Differential Equations = 241
8.2 Runge-Kutta Methods and the ODE23 and ODE45 Integrators Provided in MATLAB = 243
8.3 Step-size Limits Necessary to Maintain Numerical Stability = 245
8.4 Discussion of Procedures to Maintain Accuracy by Varying Integration Step-size = 251
8.5 Example on Forced Oscillations of an Inverted Pendulum = 252
8.6 Example on Dynamics of a Chain with Specified End Motion = 260
8.7 FORTRAN MEX Implementation : Dynamics of a Chain with Specified End Motion = 276
8.7.1 Introduction = 276
8.7.2 MEX-Routine Development = 277
8.7.3 Discussion of Results from Using a MEX-routine = 285
9. Boundary Value Problems for Linear Partial Differential Equations = 299
9.1 Several Important Partial Differential Equations = 299
9.2 Solving the Laplace Equation Inside a Rectangular Region = 300
9.3 Transient Heat Conduction in One-Dimensional Slab = 311
9.4 Wave Propagation in a Beam with an Impact Moment Applied to One End = 316
9.5 Torsional Stresses in a Beam Natural Frequencies Obtained by Finite Element and Finite Difference Methods = 332
9.6 Accuracy Comparison for Euler Beam Natural Frequencies Obtained by Finite Element and Finite Difference Methods = 341
9.6.1 Mathematical Formulation = 341
9.6.2 Discussion of the Code = 347
9.6.3 Numerical Results = 349
A References = 365
B List of MATLAB Routines with Descriptions = 375
C MATLAB Utility Functions = 389
Index = 401
