Advanced Mathematics and Mechanics Applications Using MATLAB

c262x_fm.pdf......Page 1
Table of Contents......Page 0
Preface......Page 4
Contents......Page 6
1.1 MATLAB: A T ool for Engineering Analysis......Page 11
1.2 MATLAB Commands and Related Reference Materials......Page 12
1.3 Example Problem on Financial Analysis......Page 13
1.4.1 Computer Output......Page 15
1.4.2 Discussion of the MA TLAB Code......Page 18
1.4.3 Code for Financial Problem......Page 19
2.1 Introduction......Page 24
2.2 Overview of Graphics......Page 25
2.3 Example Comparing Polynomial and Spline Interpolation......Page 26
2.4 Conformal Mapping Example......Page 31
MA TLAB Example......Page 34
2.5 Nonlinear Motion of a Damped Pendulum......Page 37
2.6 A Linear V ibration Model......Page 46
2.7 Example of W aves in an Elastic String......Page 52
MA TLAB Example......Page 56
2.8.1 Curve Properties......Page 60
Program Output and Code......Page 64
2.8.2 Surface Properties......Page 69
2.8.3 Program Output and Code......Page 74
3.2 Vectors, Norms, Linear Independence, and Rank......Page 81
3.3 Systems of Linear Equations, Consistency, and Least Squares Approximation......Page 82
3.4.1 A Membrane Deflection Problem......Page 85
MA TLAB Example......Page 88
3.4.2 Mixed Boundary V alue Problem for a Function Harmonic Inside a Circular Disk......Page 90
MA TLAB Example......Page 93
3.4.3 Using Rational Functions to Conformally Map a Circular Disk on to a Square......Page 95
MA TLAB Example......Page 96
3.5.1 Statement of the Problem......Page 100
3.5.2 Application to Solution of Matrix Differential Equations......Page 102
3.5.3 The Structural Dynamics Equation......Page 103
MATLAB Example......Page 105
3.6 Computing Natural Frequencies for a Rectangular Membrane......Page 111
3.7 Column Space, Null Space, Orthonormal Bases, and SVD......Page 115
3.8 Computation Time to Run a MA TLAB Program......Page 117
4.1 Concepts of Interpolation......Page 124
4.2 Interpolation, Differentiation, and Integration by Cubic Splines......Page 126
4.2.2 Example: Length and Enclosed Area for a Spline Curve......Page 128
4.2.3 Generalizing the Intrinsic Spline Function in MA TLAB......Page 133
Example: Spline Interpolation Applied to Sin(x)......Page 135
Example: Spline Curve Drawing the W ord MA TLAB......Page 138
4.3 Numerical Differentiation Using Finite Differences......Page 141
4.3.1 Example: Program to Derive Difference Formulas......Page 143
5.1 Fundamental Concepts and Intrinsic Integration Tools in MATLAB......Page 147
5.2 Concepts of Gauss Integration......Page 151
5.3 Comparing Results from Gauss Integration and Function QUADL......Page 154
Output from Program quadtest......Page 155
5.4 Geometrical Properties of Areas and V olumes......Page 159
5.4.1 Area Property Program......Page 163
5.4.2 Program Analyzing V olumes of Revolution......Page 169
5.5 Computing Solid Properties Using Triangular Surface Elements and Using Symbolic Math......Page 175
5.6 Numerical and Symbolic Results for the Example......Page 178
5.7 Geometrical Properties of a Polyhedron......Page 186
5.8.1 Program Listing......Page 196
5.9 Gauss Integration of a Multiple Integral......Page 201
5.9.1 Example: Evaluating a Multiple Integral......Page 203
6.1 Defnitions and Computation of Fourier Coeffcients......Page 207
6.1.1 Trigonometric Interpolation and the Fast Fourier Transform......Page 209
6.2.1 Using the FFT to Compute Integer Order Bessel Functions......Page 211
6.2.2 Dynamic Response of a Mass on an Oscillating Foundation......Page 214
6.2.3 General Program to Plot Fourier Expansions......Page 226
7.1 Solving the Structural Dynamics Equations for Periodic Applied Forces......Page 239
7.1.1 Application to Oscillations of a V ertically S u s p e n d e d C a b l e......Page 241
MA TLAB Example......Page 246
7.2 Direct Integration Methods......Page 251
7.2.1 Example on Cable Response by Direct Integration......Page 253
MA TLAB Example......Page 256
8.1 General Concepts on Numerical Integration of Nonlinear Ma- trix Differential Equations......Page 264
8.2 Runge-Kutta Methods and the ODE45 Integrator Provided in MATLAB......Page 266
8.3 Step-size Limits Necessary to Maintain Numerical Stability......Page 267
MATLAB Example......Page 271
8.4 Discussion of Procedures to Maintain Accuracy by Varying Integration Step-size......Page 272
8.5 Example on Forced Oscillations of an Inverted Pendulum......Page 273
MATLAB Example......Page 279
8.6 Dynamics of a Spinning Top......Page 282
Program Output and Code......Page 286
8.7 Motion of a Projectile......Page 289
Program Output and Code......Page 290
8.8 Example on Dynamics of a Chain with SpeciÞed End Motion......Page 293
Example on Nonlinear Cable Motion......Page 300
8.9 Dynamics of an Elastic Chain......Page 305
Program for Elastic Chain Dynamics......Page 308
9.1 Several Important Partial Differential Equations......Page 314
9.2 Solving the Laplace Equation inside a Rectangular Region......Page 315
9.3 The Vibrating String......Page 326
9.4 Force Moving on an Elastic String......Page 336
9.4.1 Computer Analysis......Page 337
9.5 Waves in Rectangular or Circular Membranes......Page 341
9.5.2 Input Data for Program membwave......Page 344
9.6 Wave Propagation in a Beam with an Impact Moment Applied to One End......Page 355
9.7 Forced V ibration of a Pile Embedded in an Elastic Medium......Page 369
9.8 Transient Heat Conduction in a One-Dimensional Slab......Page 378
9.9.1 Problem Formulation......Page 382
9.9.2 Computer Formulation......Page 384
9.10 Torsional Stresses in a Beam of Rectangular Cross Section......Page 393
10.2 Approximation Accuracy in a Simple Eigen value Problem......Page 402
10.3.1 Principal Stress Program......Page 407
10.3.2 Principal Axes of the Inertia T ensor......Page 408
10.4 V ibration of T russ Structures......Page 409
10.4.1 T russ V ibration Program......Page 415
10.5 Buckling of Axially Loaded Columns......Page 421
10.5.1 Example for a Linearly T apered Circular Cross Section......Page 426
10.5.2 Numerical Results......Page 429
Program Output and Code......Page 430
10.6.1 Mathematical Formulation......Page 436
10.6.2 Discussion of the Code......Page 440
10.6.3 Numerical Results......Page 441
MA TLAB Example......Page 445
10.7.1 Analytical Formulation......Page 455
10.7.2 Computer Formulation......Page 457
11.1 Introduction......Page 471
11.1.1 Analytical Formulation......Page 472
11.1.3 Program Output and Code......Page 476
12.2 Defnition of Analyticity......Page 493
12.4 Integral Properties......Page 495
12.4.2 Residue Theorem......Page 496
12.5.1 Steady-State Heat Conduction......Page 497
12.5.4 Plane Elastostatics......Page 498
12.6 Branch Points and Multivalued Behavior......Page 499
12.7 Conformal Mapping and Harmonic Functions......Page 501
12.8 Mapping onto the Exterior or the Interior of an Ellipse......Page 503
12.8.1 Program Output and Code......Page 507
12.9 Linear Fractional T ransformations......Page 512
12.9.1 Program Output and Code......Page 515
12.10 Schwarz-Christoffel Mapping onto a Square......Page 519
12.10.1 Program Output and Code......Page 522
12.11 Determining Harmonic Functions in a Circular Disk......Page 525
12.11.1 Numerical Results......Page 527
12.11.2 Program Output and Code......Page 531
12.12 In viscid Fluid Flow around an Elliptic Cylinder......Page 538
12.12.1 Program Output and Code......Page 541
12.13 Torsional Stresses in a Beam Mapped onto a Unit Disk......Page 543
12.13.1 Program Output and Code......Page 545
12.14 Stress Analysis by the Kolosov-Muskhelishvili Method......Page 549
12.14.1 Program Output and Code......Page 554
12.14.2 Stressed Plate with an Elliptic Hole......Page 560
12.14.3 Program Output and Code......Page 564
13.1 Basic Concepts......Page 567
13.2 Initial Angle for a Projectile......Page 569
Program Output and Code......Page 570
13.3 Fitting Nonlinear Equations to Data......Page 575
Program Output and Code......Page 577
13.4 Nonlinear Deßections of a Cable......Page 580
Program Output and Code......Page 581
13.5 Quickest Time Descent Curve (the Brachistochrone)......Page 585
Program Output and Code......Page 588
13.6 Determining the Closest Points on T wo Surfaces......Page 592
13.6.1 Discussion of the Computer Code......Page 595
List of MATLAB Routines with Descriptions......Page 610
Selected Utility and Application Functions......Page 620
References......Page 658