Advanced Mathematics and Mechanics Applications Using MATLAB, 3rd Edition.pdf

Advanced Mathematics and Mechanics Applications Using MATLAB, 3rd Edition

Contents
1 Introduction
1.1 MATLAB:A Tool for Engineering Analysis
1.2 MATLAB Commands and Related Reference Materials
1.3 Example Problemon Financial Analysis
1.4 Computer Code and Results
1.4.1 Computer Output
1.4.2 Discussion of the MATLAB Code
1.4.3 Code for Financial Problem
2 Elementary Aspects of MATLAB Graphics
2.1 Introduction
2.2 Overviewof Graphics
2.3 Example Comparing Polynomial and Spline Interpolation
2.4 Conformal Mapping Example
2.5 Nonlinear Motion of a Damped Pendulum
2.6 A Linear Vibration Model
2.7 Example of Waves in an Elastic String
2.8 Properties of Curves and Surfaces
2.8.1 Curve Properties
2.8.2 Surface Properties
2.8.3 ProgramOutput and Code
3 Summary of Concepts fromLinear Algebra
3.1 Introduction
3.2 Vectors, Norms, Linear Independence, and Rank
3.3 Systems of Linear Equations, Consistency, and Least Squares Ap-
proximation
3.4 Applications of Least Squares Approximation
3.4.1 A Membrane Deßection Problem
3.4.2 Mixed Boundary Value Problem for a Function Harmonic
Inside a Circular Disk
3.4.3 Using Rational Functions to Conformally Map a Circular
Disk onto a Square
3.5 Eigenvalue Problems
3.5.1 Statement of the Problem
3.5.2 Application to Solution of Matrix Differential Equations
© 2003 by Chapman & Hall/CRC
3.5.3 The Structural Dynamics Equation
3.6 Computing Natural Frequencies for a Rectangular Membrane
3.7 Column Space, Null Space, Orthonormal Bases, and SVD
3.8 Computation Time to Run a MATLAB Program
4 Methods for Interpolation and Numerical Differentiation
4.1 Concepts of Interpolation
4.2 Interpolation, Differentiation, and Integration by Cubic Splines
4.2.1 Computing the Length and Area Bounded by a Curve
4.2.2 Example: Length and Enclosed Area for a Spline Curve
4.2.3 Generalizing the Intrinsic Spline Function in MATLAB
4.2.4 Example: A Spline Curve with Several Parts and Corners
4.3 Numerical Differentiation Using Finite Differences
4.3.1 Example: Programto Derive Difference Formulas
5 Gauss Integration with Geometric Property Applications
5.1 Fundamental Concepts and Intrinsic Integration Tools in MATLAB
5.2 Concepts of Gauss Integration
5.3 Comparing Results fromGauss Integration and Function QUADL
5.4 Geometrical Properties of Areas and Volumes
5.4.1 Area Property Program
5.4.2 ProgramAnalyzing Volumes of Revolution
5.5 Computing Solid Properties Using Triangular Surface Elements and
Using Symbolic Math
5.6 Numerical and Symbolic Results for the Example
5.7 Geometrical Properties of a Polyhedron
5.8 Evaluating Integrals Having Square Root Type Singularities
5.8.1 ProgramListing
5.9 Gauss Integration of a Multiple Integral
5.9.1 Example: Evaluating a Multiple Integral
6 Fourier Series and the Fast Fourier Transform
6.1 DeÞnitions and Computation of Fourier CoefÞcients
6.1.1 Trigonometric Interpolation and the Fast Fourier Transform
6.2 Some Applications
6.2.1 Using the FFT to Compute Integer Order Bessel Functions
6.2.2 Dynamic Response of a Mass on an Oscillating Foundation
6.2.3 General Programto Plot Fourier Expansions
7 Dynamic Response of Linear Second Order Systems
7.1 Solving the Structural Dynamics Equations for Periodic Forces
7.1.1 Application to Oscillations of a Vertically Suspended Cable
7.2 Direct Integration Methods
7.2.1 Example on Cable Response by Direct Integration
© 2003 by Chapman & Hall/CRC
8 Integration of Nonlinear Initial Value Problems
8.1 General Concepts on Numerical Integration of Nonlinear Matrix Dif-
ferential Equations
8.2 Runge-Kutta Methods and the ODE45 Integrator Provided in MAT-
LAB
8.3 Step-size Limits Necessary to Maintain Numerical Stability
8.4 Discussion of Procedures to Maintain Accuracy by Varying Integra-
tion Step-size
8.5 Example on Forced Oscillations of an Inverted Pendulum
8.6 Dynamics of a Spinning Top
8.7 Motion of a Projectile
8.8 Example on Dynamics of a Chain with SpeciÞed End Motion
8.9 Dynamics of an Elastic Chain
9 Boundary Value Problems for Partial Differential Equations
9.1 Several Important Partial Differential Equations
9.2 Solving the Laplace Equation inside a Rectangular Region
9.3 The Vibrating String
9.4 Force Moving on an Elastic String
9.4.1 Computer Analysis
9.5 Waves in Rectangular or Circular Membranes
9.5.1 Computer Formulation
9.5.2 Input Data for Programmembwave
9.6 Wave Propagation in a Beam with an Impact Moment Applied to
One End
9.7 Forced Vibration of a Pile Embedded in an Elastic Medium
9.8 Transient Heat Conduction in a One-Dimensional Slab
9.9 Transient Heat Conduction in a Circular Cylinder with Spatially Vary-
ing Boundary Temperature
9.9.1 ProblemFormulation
9.9.2 Computer Formulation
9.10 Torsional Stresses in a Beamof Rectangular Cross Section
10 Eigenvalue Problems and Applications
10.1 Introduction
10.2 Approximation Accuracy in a Simple Eigenvalue Problem
10.3 Stress Transformation and Principal Coordinates
10.3.1 Principal Stress Program
10.3.2 Principal Axes of the Inertia Tensor
10.4 Vibration of Truss Structures
10.4.1 Truss Vibration Program
10.5 Buckling of Axially Loaded Columns
10.5.1 Example for a Linearly Tapered Circular Cross Section
10.5.2 Numerical Results
© 2003 by Chapman & Hall/CRC
10.6 Accuracy Comparison for Euler Beam Natural Frequencies by Finite
Element and Finite Difference Methods
10.6.1 Mathematical Formulation
10.6.2 Discussion of the Code
10.6.3 Numerical Results
10.7 Vibration Modes of an Elliptic Membrane
10.7.1 Analytical Formulation
10.7.2 Computer Formulation
11 Bending Analysis of Beams of General Cross Section
11.1 Introduction
11.1.1 Analytical Formulation
11.1.2 Programto Analyze Beams of General Cross Section
11.1.3 ProgramOutput and Code
12 Applications of Analytic Functions
12.1 Properties of Analytic Functions
12.2 DeÞnition of Analyticity
12.3 Series Expansions
12.4 Integral Properties
12.4.1 Cauchy Integral Formula
12.4.2 Residue Theorem
12.5 Physical Problems Leading to Analytic Functions
12.5.1 Steady-State Heat Conduction
12.5.2 Incompressible Inviscid Fluid Flow
12.5.3 Torsion and Flexure of Elastic Beams
12.5.4 Plane Elastostatics
12.5.5 Electric Field Intensity
12.6 Branch Points and Multivalued Behavior
12.7 Conformal Mapping and Harmonic Functions
12.8 Mapping onto the Exterior or the Interior of an Ellipse
12.8.1 ProgramOutput and Code
12.9 Linear Fractional Transformations
12.9.1 ProgramOutput and Code
12.10 Schwarz-Christoffel Mapping onto a Square
12.10.1 ProgramOutput and Code
12.11 Determining Harmonic Functions in a Circular Disk
12.11.1 Numerical Results
12.11.2 ProgramOutput and Code
12.12 Inviscid Fluid Flow around an Elliptic Cylinder
12.12.1 ProgramOutput and Code
12.13 Torsional Stresses in a BeamMapped onto a Unit Disk
12.13.1 ProgramOutput and Code
12.14 Stress Analysis by the Kolosov-Muskhelishvili Method
12.14.1 ProgramOutput and Code
© 2003 by Chapman & Hall/CRC
12.14.2 Stressed Plate with an Elliptic Hole
12.14.3 ProgramOutput and Code
13 Nonlinear Optimization Applications
13.1 Basic Concepts
13.2 Initial Angle for a Projectile
13.3 Fitting Nonlinear Equations to Data
13.4 Nonlinear Deßections of a Cable
13.5 Quickest Time Descent Curve (the Brachistochrone)
13.6 Determining the Closest Points on Two Surfaces
13.6.1 Discussion of the Computer Code
A List of MATLAB Routines with Descriptions
B Selected Utility and Application Functions
References

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