Ordinary differential equations and mechanical systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham ; Heidelberg ; New York ; Dordrecht ; London
Springer
2014
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xv, 614 Seiten Illustrationen, Diagramme |
| ISBN: | 9783319076584 |
Internformat
MARC
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| 020 | |a 9783319076584 |9 978-3-319-07658-4 | ||
| 035 | |a (OCoLC)897012806 | ||
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| 041 | 0 | |a eng | |
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| 100 | 1 | |a Awrejcewicz, Jan |d 1952- |0 (DE-588)120089122 |4 aut | |
| 245 | 1 | 0 | |a Ordinary differential equations and mechanical systems |c Jan Awrejcewicz |
| 264 | 1 | |a Cham ; Heidelberg ; New York ; Dordrecht ; London |b Springer |c 2014 | |
| 300 | |a xv, 614 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Mechanisches System |0 (DE-588)4132811-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-027615554 | ||
Datensatz im Suchindex
| _version_ | 1804152683383226368 |
|---|---|
| adam_text | Contents
1
Introduction
................................................................. 1
1.1
Existence
of
a Solution
............................................. 5
2
First-Order ODEs
.......................................................... 13
2.1
General Introduction
................................................ 14
2.2
Separable Equation
................................................. 16
2.3
Homogenous
Equations
............................................ 22
2.4
Linear
Equations
.................................................... 27
2.5
Exact
Differential
Equations
....................................... 39
2.6
Implicit Differential
Equations
Not Solved
with Respect to a Derivative
........................................ 45
3
Second-Order ODEs
....................................................... 51
3.1
Introduction
......................................................... 51
3.2
Linear ODEs
........................................................ 51
3.2.1
General Approaches
..................................... 51
3.2.2
Hypergeometric (Gauss) Equation
...................... 60
3.2.3
The Legendre Equation and Legendre Polynomials
... 63
3.2.4
The Bessel Equation
..................................... 69
3.2.5
ODEs with Periodic Coefficients
........................ 76
3.2.6
Modelling of Generalized Parametric Oscillator
....... 91
3.2.7
ODEs with Constant Coefficient
........................ 100
3.3 Lagrange
Equations and Variational Principle
..................... 107
3.4
Reduction to First-Order System of Equations
.................... 112
3.4.1
Mathematical Background
.............................. 113
3.4.2
The Lagrangian Function
................................ 116
3.5
Canonical Transformations
......................................... 118
3.6
Examples
............................................................ 120
Contents
3.7
Normal
Forms of Hamiltonian Systems
........................... 124
3.7.1
Parametric Form of Canonical Transformations
....... 125
3.7.2
Integration of Hamiltonian Equations
Perturbated by Damping
................................. 126
3.8
Geometrical Approach to the Swinging Pendulum
................ 130
3.8.1
The Analysed System and Geometrization
............. 133
3.9
Geometric Analysis of a Double Pendulum Dynamics
........... 138
3.9.1
The Pendulum and Geometrization
..................... 143
3.9.2
Numerical Simulations
.................................. 146
3.10
A Set of Linear Second-Order ODEs with Constant
Coefficients
.......................................................... 148
3.10.1
Conservative Systems
.................................... 149
3.10.2
Non-conservative Systems
.............................. 152
3.10.3
Modal Analysis and Identification
...................... 154
Linear ODEs
................................................................ 167
4.1
Introduction
......................................................... 167
4.2
Normal and Symmetric Forms
..................................... 168
4.3
Local Solutions (Existence, Extensions and Straightness)
........ 176
4.4
First-Order Linear Differential Equations
with Variable Coefficients
.......................................... 187
4.4.1
Introduction
.............................................. 187
4.4.2
Fundamental Matrix of Solutions
....................... 193
4.4.3
Homogeneous Differential Equations
................... 195
4.4.4
Examples of Homogeneous Linear
Differential Equations
................................... 196
4.4.5
Non-homogeneous Differential Equations
............. 206
4.4.6
Examples of Linear Non-homogenous
Differential Equations
................................... 207
4.4.7
Homogeneous Differential Equations with
Periodic Coefficients
..................................... 209
4.4.8
The Floquet Theory
...................................... 210
4.4.9
Reduction of Non-homogeneous Linear
Differential Equations with Periodic Coefficients
...... 212
4.4.10
Characteristic Multipliers
................................ 213
4.4.11
Characteristic Exponents
................................ 214
4.4.12
Structure of Solutions for Simple
Characteristic Multipliers
................................ 214
4.4.13
Solutions Structure for Case of Multiple Multipliers
.. 215
4.4.
1
4
Stability
.................................................. 217
4.4.15
Periodic Solutions to Homogenous
Differential Equations with Periodic Coefficients
...... 217
4.4.16
Periodic Solutions to Non-homogenous
Differential Equations with Periodic Coefficients
...... 218
Contents xiii
5
Higher-Order ODEs Polynomial Form
.................................. 221
5.1
Introduction
......................................................... 221
5.2 Linear
Homogeneous
Differential
Equations
...................... 223
5.3 Differential
Equations with Constant Coefficients
................ 226
5.4
Linear Non-homogeneous Differential Equations
with Constant Coefficients
......................................... 234
5.5
Differential Equations with Variable Coefficients
................. 240
6
Systems
...................................................................... 245
7
Theory and Criteria of Similarity
........................................ 253
8
Model and Modelling
....................................................... 263
8.1
Introduction
......................................................... 263
8.2
Mathematical Modelling
............................................ 266
8.3
Modelling in Mechanics
............................................ 267
8.4
General Characteristics of Mathematical Modelling
of Systems
........................................................... 273
8.5
Modelling Control Theory
.......................................... 273
8.5.1
Ordinary Automatic Control Systems
.................. 275
8.5.2
Distributed Automatic Control Systems
................ 279
8.5.3
Classification of Control Systems with
Respect to Another Criteria
............................. 280
8.5.4
Examples of Control Systems and Their
Block Diagrams
.......................................... 282
9
Phase Plane and Phase Space
............................................. 295
9.1
Introduction
......................................................... 295
9.2
Phase Plain and Singular Points
.................................... 297
9.3
Analysis of Singular Points
......................................... 304
9.3.
1 Unstable Node
........................................... 304
9.3.2
Stable Node
.............................................. 306
9.3.3
Critical Node
............................................. 308
9.3.4
Degenerate Node
......................................... 308
9.3.5
Saddle
.................................................... 310
9.3.6
Unstable Focus
........................................... 313
9.3.7
Stable Focus
.............................................. 313
9.3.8
Centre
..................................................... 314
9.4
Analysis of Singular Points Governed by Three
Differential Equations of First Order
.............................. 317
9.4.1
Theory Concerning the Solving a System
of Differential Equations and Method for
Determining Roots of a Polynomial of Third Degree
.. 317
9.4.2
Analysis of Singular Points Described by
Three First-Order Differential Equations
............... 318
Contents
10
Stability
......................................................................
329
10.1
Introduction
......................................................... 329
10.2
Lyapunov s Functions and Second Lyapunov s Method
.......... 348
10.3
Classical Theories of Stability and Chaotic Dynamics
............ 358
11
Modelling via Perturbation Methods
..................................... 363
1
1
.1
Introduction
......................................................... 363
11.2
Selected Classical Perturbative Methods
.......................... 364
11.2.1
Autonomous Systems
.................................... 364
11.2.2
Nonautonomous Systems
................................ 379
12
Continualization and Discretization
...................................... 395
12.1
Introduction
......................................................... 395
12.2
One-Dimensional Chain of Coupled Oscillators
.................. 395
12.3
Planar Hexagonal Net of Coupled Oscillators
..................... 400
12.4
Discretization
....................................................... 406
12.5
Modelling of Two-Dimensional
Anisotropie
Structures
.......... 410
13
Bifurcations
................................................................. 417
13.1
Introduction
......................................................... 417
13.2
Singular Points in
1
D
and 2D Vector Fields
....................... 421
13.2.1
ID Vector Fields
......................................... 421
13.2.2
Two-Dimensional Vector Fields
......................... 429
13.2.3
Local Bifurcation of Hyperbolic Fixed Points
.......... 436
13.2.4
Bifurcation of a Non-hyperbolic Fixed
Point
(Hopf
Bifurcation)
................................. 438
13.2.5
Double
Hopf
Bifurcation
................................ 451
13.3
Fixed Points of Maps
............................................... 458
13.4
Continuation Technique
............................................ 461
13.5
Global Bifurcations
................................................. 467
13.6
Piece-Wise Smooth Dynamical System
........................... 469
13.6.1
Introduction
.............................................. 469
13.6.2
Stability
.................................................. 471
13.6.3
Orbits Exhibiting Degenerated Contact
with Discontinuity Surfaces
............................. 477
13.6.4
Bifurcations in Filippov s Systems
...................... 480
13.6.5
Bifurcations of Stationary Points
....................... 480
13.6.6
Bifurcations of Periodic Orbits
.......................... 483
14
Optimization of Systems
................................................... 487
14.1
Introduction
......................................................... 487
14.2
Simple Examples of Optimization in Approximative
Problems
............................................................ 488
14.3
Conditional
Extrema
................................................ 490
14.4
Static Optimization
................................................. 494
14.4.1
Local Approximation of a Function
..................... 497
14.4.2
Stationary Points and Quadratic Forms
................. 498
Contents xv
14.5
Convexity of Sets of Functions
..................................... 499
14.6
Problems of Optimization Without Constraints
................... 502
14.7
Optimality Conditions of a Quadratic Form
....................... 504
14.8
Equivalence Constraints
............................................ 505
14.9
The
Lagrange
Function and Multipliers
........................... 507
14.10
Inequivalence Constraints
.......................................... 510
14.11
Shock Absorber Work Optimization
............................... 513
14.11.1
Introduction
.............................................. 513
14.11.2
Optimization Example
................................... 513
15
Chaos and Synchronization
............................................... 527
15.1
Introduction
......................................................... 527
15.2
Modelling and Identification of Chaos
............................. 531
15.3
Lyapunov Exponents
................................................ 539
15.4
Frequency Spectrum
................................................ 540
15.5
Function of Autocorrelation
........................................ 541
15.6
Modelling of Nonlinear Discrete Systems
......................... 542
15.6.1
Introduction
.............................................. 542
15.6.2
Bernoulli s Map
.......................................... 545
15.6.3
Logistic Map
............................................. 548
15.6.4
Map of a Circle into a Circle
............................ 553
15.6.5
Devil s Stairs, Farey Tree and Fibonacci Numbers
..... 557
15.6.6
Hénon Map
............................................... 560
15.6.7
Ikeda Map
................................................ 566
15.7
Modelling of Nonlinear Ordinary Differential Equations
......... 568
15.7.1
Introduction
.............................................. 568
15.7.2
Non-autonomous Oscillator with Different
Potentials
................................................. 569
15.7.3
Melnikov Function and Chaos
.......................... 571
1
5.7.4 Lorenz Attractor......................................... 575
15.8
Synchronization Phenomena of Coupled Triple Pendulums
...... 578
15.8.1
Mathematical Model
..................................... 578
15.8.2
Numerical Simulations
.................................. 584
15.9
Chaos and Synchronization Phenomena Exhibited
by Plates and Shells
................................................. 586
15.9.1
Introduction
.............................................. 586
15.9.2
One Layer Shell
.......................................... 588
15.9.3
Two-Layer Shell
......................................... 596
References
......................................................................... 605
Jan Awrejcewicz
Ordinary Differential Equations and Mechanical Systems
This book applies a step-by-step treatment of the current state-of-the-art of ordinary
differential equations used in modeling of engineering systems/processes and beyond.
It covers systematically ordered problems, beginning with first and second order ODEs,
linear and higher-order ODEs of polynomial form, theory and criteria of similarity,
modeling approaches, phase plane and phase space concepts, and stability optimization,
and ending on chaos and synchronization.
Presenting both an overview of the theory of the introductory differential equations
in the context of applicability and a systematic treatment of modeling of numerous
engineering and physical problems through linear and non-linear ODEs, the volume
is self-contained, yet serves both scientinc and engineering interests. The presentation
relies on a general treatment, analytical and numerical methods, concrete examples,
and engineering intuition.
The scientific background used is well balanced between elementary and advanced
level, making it a unique self-contained source for both theoretically and application-
oriented graduate and doctoral students, university teachers, researchers and engineers
of mechanical, civil, and mechatronic engineering.
|
| any_adam_object | 1 |
| author | Awrejcewicz, Jan 1952- |
| author_GND | (DE-588)120089122 |
| author_facet | Awrejcewicz, Jan 1952- |
| author_role | aut |
| author_sort | Awrejcewicz, Jan 1952- |
| author_variant | j a ja |
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| bvnumber | BV042176224 |
| classification_rvk | SK 520 |
| ctrlnum | (OCoLC)897012806 (DE-599)BVBBV042176224 |
| dewey-full | 515.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.352 |
| dewey-search | 515.352 |
| dewey-sort | 3515.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042176224 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:14:36Z |
| institution | BVB |
| isbn | 9783319076584 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027615554 |
| oclc_num | 897012806 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-83 |
| physical | xv, 614 Seiten Illustrationen, Diagramme |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| spelling | Awrejcewicz, Jan 1952- (DE-588)120089122 aut Ordinary differential equations and mechanical systems Jan Awrejcewicz Cham ; Heidelberg ; New York ; Dordrecht ; London Springer 2014 xv, 614 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Mechanisches System (DE-588)4132811-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Mechanisches System (DE-588)4132811-5 s DE-604 Erscheint auch als 978-3-319-07659-1 Online-Ausgabe Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615554&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615554&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Awrejcewicz, Jan 1952- Ordinary differential equations and mechanical systems Mechanisches System (DE-588)4132811-5 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4132811-5 (DE-588)4020929-5 |
| title | Ordinary differential equations and mechanical systems |
| title_auth | Ordinary differential equations and mechanical systems |
| title_exact_search | Ordinary differential equations and mechanical systems |
| title_full | Ordinary differential equations and mechanical systems Jan Awrejcewicz |
| title_fullStr | Ordinary differential equations and mechanical systems Jan Awrejcewicz |
| title_full_unstemmed | Ordinary differential equations and mechanical systems Jan Awrejcewicz |
| title_short | Ordinary differential equations and mechanical systems |
| title_sort | ordinary differential equations and mechanical systems |
| topic | Mechanisches System (DE-588)4132811-5 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Mechanisches System Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615554&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027615554&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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