Preface ........................................................ xv
I Introducing MATLAB® ......................................... 1
1 Introduction to MATLAB® ...................................... 3
1.1 Starting MATLAB ......................................... 3
1.2 Using MATLAB as a simple calculator ..................... 5
1.3 How to quit MATLAB ...................................... 9
1.4 Using MATLAB as a scientific calculator ................ 10
1.4.1 Trigonometric functions ......................... 10
1.4.2 Inverse trigonometric functions ................. 13
1.4.3 Other elementary functions ...................... 15
1.5 Arrays of numbers ...................................... 16
1.6 Using MATLAB for plotting .............................. 18
1.6.1 Annotating a graph .............................. 20
1.7 Format ................................................. 21
1.8 Arrays of numbers ...................................... 22
1.8.1 Array elements .................................. 22
1.8.2 Plotting resolution ............................. 23
1.8.3 Array operations ................................ 24
1.9 Writing simple functions in MATLAB ..................... 27
1.10 Summary ................................................ 31
1.11 Examples ............................................... 34
1.12 More exercises ......................................... 42
2 Vectors and matrices ........................................ 47
2.1 Vectors in geometry .................................... 48
2.1.1 Arrays of point coordinates in the plane ........ 48
2.1.2 The perimeter of a polygon - for loops .......... 52
2.1.3 Vectorization ................................... 55
2.1.4 Arrays of point coordinates in solid geometry ... 56
2.1.5 Geometrical interpretation of vectors ........... 61
2.1.6 Operating with vectors .......................... 63
2.1.7 Vector basis .................................... 65
2.1.8 The scalar product .............................. 66
2.2 Vectors in mechanics ................................... 69
What every engineer should know about MATLAB® and Simulink®
2.2.1 Forces. The resultant of two or more forces ..... 69
2.2.2 Work as a scalar product ........................ 72
2.2.3 Velocities. Composition of velocities ........... 72
2.3 Matrices ............................................... 73
2.3.1 Introduction - the matrix product ............... 73
2.3.2 Determinants .................................... 77
2.4 Matrices in geometry ................................... 78
2.4.1 The vector product. Parallelogram area .......... 78
2.4.2 The scalar triple product. Parallelepiped
volume .......................................... 80
2.5 Transformations ........................................ 82
2.5.1 Translation - Matrix addition and subtraction ... 82
2.5.2 Rotation ........................................ 83
2.5.3 Homogeneous coordinates ......................... 84
2.6 Matrices in Mechanics .................................. 88
2.6.1 Angular velocity ................................ 88
2.6.2 Center of mass .................................. 89
2.6.3 Moments as vector products ...................... 91
2.7 Summary ................................................ 93
2.8 More exercises ......................................... 98
3 Equations .................................................. 103
3.1 Introduction .......................................... 103
3.2 Linear equations in geometry .......................... 103
3.2.1 The intersection of two lines .................. 103
3.2.2 Cramer's rule .................................. 104
3.2.3 MATLAB's solution of linear equations .......... 105
3.2.4 An example of an ill-conditioned system ........ 107
3.2.5 The intersection of three planes ............... 109
3.3 Linear equations in statics ........................... 109
3.3.1 A simple beam .................................. 109
3.4 Linear equations in electricity ....................... 112
3.4.1 A DC circuit ................................... 112
3.4.2 The method of loop currents .................... 114
3.5 On the solution of linear equations ................... 116
3.5.1 Homogeneous linear equations ................... 116
3.5.2 Overdetermined systems - least-squares
solution ....................................... 119
3.5.3 Underdetermined system ......................... 123
3.5.4 A singular system .............................. 126
3.5.5 Another singular system ........................ 128
3.6 Summary 1 ............................................. 132
3.7 More exercises ........................................ 134
3.8 Polynomial equations .................................. 135
3.8.1 MATLAB representation of polynomials ........... 135
3.8.2 The MATLAB root function ....................... 135
3.8.3 The MATLAB function conv ....................... 137
3.9 Iterative solution of equations ....................... 143
3.9.1 The Newton-Raphson method ...................... 143
3.9.2 Solving an equation with the command f zero .... 147
3.10 Summary 2 ............................................. 148
3.11 More exercises ........................................ 149
4 Processing and publishing the results ...................... 151
4.1 Copy and paste ........................................ 151
4.2 Diary ................................................. 152
4.3 Exporting and processing figures ...................... 152
4.4 Interpolation ......................................... 153
4.4.1 Interactive plotting and curve fitting ......... 153
4.5 The MATLAB®spline function ............................ 157
4.6 Importing data from Excel® - histograms ............... 165
4.7 Summary ............................................... 167
4.8 Exercises ............................................. 169
II Programming in MATLAB® ..................................... 171
5 Some facts about numerical computing ....................... 173
5.1 Introduction .......................................... 173
5.2 Computer-aided mistakes ............................... 174
5.2.1 A loop that does not stop ...................... 175
5.2.2 Errors in trigonometric functions .............. 176
5.2.3 An unexpected root ............................. 176
5.2.4 Other unexpected roots ......................... 178
5.2.5 Accumulating errors ............................ 179
5.3 Computer representation of numbers .................... 180
5.4 The set of computer numbers ........................... 184
5.5 Roundoff .............................................. 186
5.6 Roundoff errors ....................................... 187
5.7 Computer arithmetic ................................... 191
5.8 Why the examples in Section 5.2 failed ................ 193
5.8.1 Absorbtion ..................................... 193
5.8.2 Correcting a non-terminating loop .............. 194
5.8.3 Second-degree equation ......................... 194
5.8.4 Unexpected polynomial roots .................... 196
5.9 Truncation error ...................................... 199
5.10 Complexity ............................................ 202
5.10.1 Definition, examples ........................... 202
5.11 Horner's scheme ....................................... 205
5.12 Problems that cannot be solved ........................ 206
5.13 Summary ............................................... 208
5.14 More examples ......................................... 209
5.15 More exercises ........................................ 211
6 Data types and object-oriented programming ................. 215
6.1 Structures ............................................ 216
6.1.1 Where structures can help ...................... 216
6.1.2 Working with structures ........................ 217
6.2 Cell arrays ........................................... 219
6.3 Classes and object-oriented programming ............... 221
6.3.1 What is object-oriented programming? ........... 221
6.3.2 Calculations with units ........................ 222
6.3.3 Denning a class ................................ 224
6.3.4 Defining a subclass ............................ 229
6.3.5 Calculating with electrical units .............. 233
6.4 Summary ............................................... 238
6.5 Exercises ............................................. 240
III Progressing in MATLAB® .................................. 243
7 Complex numbers ............................................ 245
7.1 The introduction of complex numbers ................... 245
7.2 Complex numbers in MATLAB ............................. 245
7.3 Geometric representation .............................. 248
7.4 Trigonometric representation .......................... 250
7.5 Exponential representation ............................ 250
7.6 Functions of complex variables ........................ 253
7.7 Conformal mapping ..................................... 255
7.8 Phasors ............................................... 259
7.8.1 Phasors ........................................ 259
7.8.2 Phasors in mechanics ........................... 261
7.8.3 Phasors in electricity ......................... 265
7.9 An application in mechanical engineering -
a mechanism ........................................... 271
7.9.1 A four-link mechanism .......................... 271
7.9.2 Displacement analysis of the four-link
mechanism ...................................... 272
7.9.3 A MATLAB function that simulates the motion
of the four-link mechanism ..................... 274
7.9.4 Animation ...................................... 277
7.9.5 A variant of the function FourLink ............. 278
7.10 Summary ............................................... 281
7.11 Exercises ............................................. 283
8 Numerical integration ...................................... 287
8.1 Introduction .......................................... 287
8.2 The trapezoidal rule .................................. 288
8.2.1 The formula .................................... 288
8.2.2 The MATLAB trapz function ...................... 289
8.3 Simpson's rule ........................................ 290
8.3.1 The formula .................................... 290
8.3.2 A function that implements Simpson's rule ...... 292
8.4 The MATLAB quadl function ............................. 293
8.5 Symbolic calculation of integrals ..................... 295
8.6 Summary ............................................... 297
8.7 Exercises ............................................. 298
9 Ordinary differential equations ............................ 301
9.1 Introduction .......................................... 301
9.2 Numerical solution of ordinary differential
equations ............................................. 301
9.2.1 Cauchy form .................................... 301
9.3 Numerical solution of ordinary differential
equations ............................................. 302
9.3.1 Specifying the times of the solution ........... 305
9.3.2 Using alternative odesolvers ................... 306
9.3.3 Passing parameters to the model ................ 306
9.4 Alternative strategies to solve ordinary
differential equations ................................ 310
9.4.1 Runge-Kutta methods ............................ 312
9.4.2 Predictor-corrector methods .................... 315
9.4.3 Stiff systems .................................. 316
9.5 Conclusion: How to choose the odesolver ............... 323
9.6 Exercises ............................................. 324
10 More graphics .............................................. 327
10.1 Introduction .......................................... 327
10.2 Drawing at scale ...................................... 327
10.3 The cone surface and conic sections ................... 330
10.3.1 The cone surface ............................... 330
10.3.2 Conic sections ................................. 332
10.3.3 Developing the cone surface .................... 336
10.3.4 A helicoidal curve on the cone surface ......... 337
10.3.5 The listing of functions developed in this
section ........................................ 338
10.4 GUIs - graphical user interfaces ...................... 343
10.5 Summary ............................................... 355
10.6 Exercises ............................................. 356
11 An introduction to Simulink® ............................... 359
11.1 What is simulation? ................................... 359
11.2 Beats ................................................. 360
11.3 A model of the momentum law ........................... 366
11.4 Capacitor discharge ................................... 370
11.5 A mass-spring-dashpot system .......................... 376
11.6 A series RLC circuit .................................. 380
11.7 The pendulum .......................................... 383
11.7.1 The mathematical and the physical pendulum ..... 383
11.7.2 The phase plane ................................ 387
11.7.3 Running the simulation from a script file ...... 391
11.8 Exercises ............................................. 393
12 Applications in the frequency domain ....................... 395
12.1 Introduction .......................................... 395
12.2 Signals ............................................... 395
12.3 A short introduction to the DFT ....................... 398
12.4 The power spectrum .................................... 400
12.5 Trigonometric expansion of a signal ................... 407
12.6 High frequency signals and aliasing ................... 410
12.7 Bode plot ............................................. 412
12.8 Summary ............................................... 414
12.9 Exercises ............................................. 415
Answers to selected exercises ................................. 417
Bibliography .................................................. 423
Index ......................................................... 427
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