Gekeler Mathematical Methods for Mechanics

A Handbook with MATLAB Experiments
1. Auflage 2008
ISBN: 978-3-540-69279-9
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 624 Seiten

Mathematics is undoubtedly the key to state-of-the-art high technology. It is aninternationaltechnicallanguageandprovestobeaneternallyyoungscience to those who have learned its ways. Long an indispensable part of research thanks to modeling and simulation, mathematics is enjoying particular vit- ity now more than ever. Nevertheless, this stormy development is resulting in increasingly high requirements for students in technical disciplines, while general interest in mathematics continues to wane at the same time. This book and its appendices on the Internet seek to deal with this issue, helping students master the di?cult transition from the receptive to the productive phase of their education. The author has repeatedly held a three-semester introductory course - titled Higher Mathematics at the University of Stuttgart and used a series of 'handouts' to show further aspects, make the course contents more motiv- ing, and connect with the mechanics lectures taking place at the same time. One part of the book has more or less evolved from this on its own. True to the original objective, this part treats a variety of separate topics of varying degrees of di?culty; nevertheless, all these topics are oriented to mechanics. Anotherpartofthisbookseekstoo?eraselectionofunderstandablereal- ticmodelsthatcanbeimplementeddirectlyfromthemultitudeofmathema- calresources.TheauthordoesnotattempttohidehispreferenceofNumerical Mathematics and thus places importance on careful theoretical preparation.

Autoren/Hrsg.

Gekeler, Eckart W.

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;1 Mathematical Auxiliaries;17
3.1;1.1 Matrix Computations;17
3.2;1.2 Brief on Vector Analysis;33
3.3;1.3 Curves in R3;41
3.4;1.4 Linear Differential Equations;43
3.5;1.5 Linear Differential Systems of First Order;47
3.6;1.6 The Flux Integral and its Vector Field;54
3.7;1.7 Vector Spaces;61
3.8;1.8 Derivatives;68
3.9;1.9 Mappings in Banach Spaces;73
3.10;1.10 Convex Sets and Functions;77
3.11;1.11 Quadratic Functionals;86
4;2 Numerical Methods;94
4.1;2.1 Interpolation and Approximation;95
4.2;2.2 Orthogonal Polynomials;105
4.3;2.3 Numerical Integration;109
4.4;2.4 Initial Value Problems;120
4.5;2.5 Boundary Value Problems;143
4.6;2.6 Periodic Problems;148
4.7;2.7 Differential-Algebraic Problems;151
4.8;2.8 Hints to the MATLAB programs;156
5;3 Optimization;158
5.1;3.1 Minimization of a Function;159
5.2;3.2 Extrema with Constraints;164
5.3;3.3 Linear Programming;169
5.4;3.4 Linear-Quadratic Problems;179
5.5;3.5 Nonlinear Optimization;184
5.6;3.6 A Brief on Lagrange Theory;192
5.7;3.7 Hints to the MATLAB Programs;206
6;4 Variation and Control;208
6.1;4.1 Variation;209
6.2;4.2 Control Problems without Constraints;226
6.3;4.3 Control Problems with Constraints;235
6.4;4.4 Examples;241
6.5;4.5 On the Reentry Problem;251
6.6;4.6 Hints to the MATLAB programs;255
7;5 The Road as Goal;256
7.1;5.1 Bifurcation Problems;257
7.2;5.2 Scaling;272
7.3;5.3 Calculation of Singular Points;279
7.4;5.4 Ordinary Differential Systems;283
7.5;5.5 Hopf Bifurcation;290
7.6;5.6 Numerical Bifurcation;303
7.7;5.7 Continuation;310
7.8;5.8 Hints to the MATLAB Programs;315
8;6 Mass Points and Rigid Bodies;316
8.1;6.1 The Force and its Moment;316
8.2;6.2 Dynamics of a Mass Point;318
8.3;6.3 Mass Point in Central Field;325
8.4;6.4 Systems of Mass Points;334
8.5;6.5 The Three-Body Problem;343
8.6;6.6 Rotating Frames;349
8.7;6.7 Inertia Tensor and Top;354
8.8;6.8 On Multibody Problems;364
8.9;6.9 On Some Principles of Mechanics;368
8.10;6.10 Hints to the MATLAB Programs;372
9;7 Rods and Beams;373
9.1;7.1 Bending Beam;373
9.2;7.2 Eigenvalue Problems;381
9.3;7.3 Numerical Approximation;387
9.4;7.4 Frameworks of Rods;390
9.5;7.5 Frameworks of Beams;396
9.6;7.6 Hints to the MATLAB Programs;400
10;8 Continuum Theory;401
10.1;8.1 Deformations;401
10.2;8.2 The Three Transport Theorems;407
10.3;8.3 Conservation Laws;410
10.4;8.4 Material Forms;417
10.5;8.5 Linear Elasticity Theory;422
10.6;8.6 Discs;427
10.7;8.7 Kirchhoff’s Plate;429
10.8;8.8 Von Karman’s Plate and the Membrane;435
10.9;8.9 On Fluids and Gases;438
10.10;8.10 Navier-Stokes Equations;441
11;9 Finite Elements;449
11.1;9.1 Elliptic Boundary Value Problems;449
11.2;9.2 From Formula to Figure, Example;453
11.3;9.3 Constructing Finite Elements;459
11.4;9.4 Further Topics;466
11.5;9.5 On Singular Elements;481
11.6;9.6 Navier-Stokes Equations;485
11.7;9.7 Mixed Applications;496
11.8;9.8 Examples;503
11.9;9.9 Hints to MATLAB Programs;512
12;10 A Survey on Tensor Calculus;517
12.1;10.1 Tensor Algebra;517
12.2;10.2 Algebra of Alternating Tensors;534
12.3;10.3 Differential Forms in Rn;539
12.4;10.4 Tensor Analysis;551
12.5;10.5 Examples;564
12.6;10.6 Transformation Groups;569
13;11 Case Studies;574
13.1;11.1 An Example of Gas Dynamics;574
13.2;11.2 The Reissner-Mindlin Plate;576
13.3;11.3 Examples of Multibody Problems;578
13.4;11.4 Dancing Discs;581
13.5;11.5 Buckling of a Circular Plate;587
14;12 Appendix;590
14.1;12.1 Notations and Tables;590
14.2;12.2 Matrix Zoo;594
14.3;12.3 Translation and Rotation;596
14.4;12.4 Trigonometric Interpolation;598
14.5;12.5 Further Properties of Vector Spaces;604
14.6;12.6 Cycloids;606
14.7;12.7 Quaternions and Rotations;609
15;References;612
16;Index;623