Simulations of oscillatory systems: with award-winning software, Physics of Oscillations
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press, Taylor & Francis Group
2015
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| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | XXIV, 340 S. |
| ISBN: | 9781498707688 |
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MARC
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| 245 | 1 | 0 | |a Simulations of oscillatory systems |b with award-winning software, Physics of Oscillations |c Eugene I. Butikov |
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| adam_text | I’hvsics
Simulations of Oscillatory Systems: with Award-Winning Software,
Physics of Oscillations provides a hands-on way of visualizing and un-
derstanding the fundamental concepts of the physics of oscillations. Both
the book and software are designed as exploration-oriented supplements for
courses in general physics and the theory of oscillations.
The book is conveniently structured according to mathematical complexity.
Each chapter in Part I contains activities, questions, exercises, and problems
of varying levels of difficulty, from straightforward to quite challenging.
Part II presents more sophisticated, highly mathematical material that
delves into the serious theoretical background for the computer-aided study
of oscillations.
The software package allows you to observe the motion of linear and
nonlinear mechanical oscillatory systems and to obtain plots of the
variables that describe the systems along with phase diagrams and plots
of energy transformations. These computer simulations provide clear, vivid
illustrations of oscillations in various physical systems, bringing to life many
abstract concepts, developing your physical intuition, and complementing
the analytical study of the subject.
Features
• Provides interactive software that serves as a desktop laboratory for
exploring simulated systems and replicating the experiments
• Enables you to perform interesting mini-research projects involving
the physics of oscillations
• Develops your physical intuition and theoretical foundations
• includes various examples of the behavior of simulated systems
• Requires no previous knowledge of algorithmic languages or
programming
Contents
11·
Preface xi
Introduction xiii
Classification of Oscillations.......................................xiv
Simulated Physical Systems ..........................................xvi
How to Use the Software.............................................xvii
Notes to the Instructor.............................................xxii
I Oscillations in Simple Systems 1
1 Free Oscillations of a Linear Oscillator 3
1.1 Summary of the Theory........................................... 3
1.1.1 General Concepts........................................ 3
1.1.2 Differential Equation of a Linear Torsion Oscillator ... 4
1.1.3 The Time of Damping and the Quality Factor Q............ 6
1.1.4 The Phase Diagram of a Linear Oscillator................ 9
1.1.5 Energy Transformations ................................ 11
1.1.6 The Computer Simulation of a Linear Oscillator......... 13
1.2 Review of the Principal Formulas............................... 13
1.3 Questions, Problems, Suggestions............................... 15
1.3.1 Free Undamped Oscillations............................. 15
1.3.2 Damped Free Oscillations .............................. 16
1.3.3 Non-Oscillatory Motion of the System................... 17
2 Torsion Spring Oscillator with Dry Friction 19
2.1 Summary of the Theory.......................................... 19
2.1.1 General Concepts....................................... 19
2.1.2 The Physical System.................................... 19
2.1.3 The Differential Equation of the Oscillator............ 22
2.1.4 Damping Caused by Dry Friction......................... 23
2.1.5 The Phase Trajectory................................... 24
2.1.6 Energy Transformations ................................ 25
2.1.7 The Role of Viscous Friction........................... 26
v
vi CONTENTS
2.2 Review of the Principal Formulas................................ 28
2.3 Questions, Problems, Suggestions................................ 28
2.3.1 Damping Caused by Dry Friction.......................... 29
2.3.2 Influence of Viscous Friction........................... 30
3 Forced Oscillations in a Linear System 33
3.1 Summary of the Theory........................................... 33
3.1.1 Basic Concepts.......................................... 34
3.1.2 Discussion of the Physical System....................... 34
3.1.3 The Differential Equation for Forced Oscillations .... 36
3.1.4 The Principle of Superposition.......................... 37
3.2 Steady-State Forced Oscillations................................ 38
3.2.1 Forced Oscillations in the Absence of Friction.......... 38
3.2.2 The Resonance Curve..................................... 40
3.2.3 Resonance of the Angular Velocity....................... 42
3.2.4 Energy Transformations ................................. 42
3.3 Transient Processes............................................. 45
3.3.1 Initial Conditions That Eliminate a Transient .......... 46
3.3.2 Forced Oscillations from Rest at Resonance.............. 47
3.3.3 Mechanical Analogue of the Stimulated Emission of
Radiation............................................... 50
3.3.4 Transient Processes Near Resonance...................... 53
3.3.5 Transient Processes Far from Resonance.................. 54
3.3.6 Transient Processes and the Phase Trajectory ........... 55
3.4 Review of the Principal Formulas................................ 57
3.5 Questions, Problems, Suggestions................................ 58
3.5.1 Steady-State Forced Oscillations........................ 58
3.5.2 Transient Processes..................................... 60
4 Square-Wave Excitation of a Linear Oscillator 65
4.1 Theoretical Background.......................................... 65
4.1.1 Model of the Physical System............................ 65
4.1.2 The Differential Equation of Forced Oscillations .... 67
4.2 Steady-State Forced Oscillations under the Square-Wave Torque 68
4.2.1 Harmonics of the Driving Force and of the Steady-State
Response................................................ 68
4.2.2 Forced Oscillations as Natural Oscillations about the
Alternating Equilibrium Positions....................... 73
4.3 Transient Processes under the Square-Wave External Torque ... 75
4.4 Estimation of the Amplitude of Steady-State Oscillations .... 77
4.4.1 Resonant Amplitude of Steady-State Oscillations .... 78
4.4.2 Amplitude of Steady Oscillations at Strong Friction ... 80
4.4.3 Amplitude of Steady Oscillations at T = 2nT0............ 81
4.4.4 Steady-State Oscillations at High Frequencies of the
Square-Wave Torque................................ 82
CONTENTS vii
4.5 Energy Transformations........................................... 83
4.6 The Electromagnetic Analogue of the Mechanical System .... 85
4.7 Concluding Remarks .............................................. 86
4.8 Review of the Principal Formulas................................. 87
4.9 Questions, Problems, Suggestions................................. 88
4.9.1 Swinging of the Oscillator at Resonance............ 88
4.9.2 Non-Resonant Forced Oscillations................... 89
5 Parametric Excitation of Oscillations 93
5.1 Summary of the Theory. General Concepts.......................... 93
5.1.1 Classification of Oscillations..................... 93
5.1.2 The Simulated Physical System ........................... 94
5.1.3 Electromagnetic Analogue of the Mechanical System . . 97
5.1.4 Conditions for Parametric Resonance ..................... 97
5.1.5 The Threshold of Parametric Excitation............. 98
5.1.6 Differential Equation for Parametric Oscillations .... 101
5.1.7 The Mean Natural Period at Large Modulation........102
5.2 Frequency Ranges of Parametric Excitation........................103
5.2.1 Main Interval of Parametric Excitation.............104
5.2.2 Third-Order Interval of Parametric Instability.....107
5.2.3 Frequency Ranges for Resonances of Even Orders .... 109
5.2.4 Intersections of the Boundaries at Large Modulation ..112
5.2.5 Intervals of Excitation in the Presence of Friction .... 113
5.3 Concluding Remarks ..............................................118
5.4 Questions, Problems, Suggestions.................................118
5.4.1 Principal Parametric Resonance.....................118
5.4.2 Manual Control of the Parameter....................121
5.4.3 Parametric Resonances of High Orders...............122
6 Sinusoidal Modulation of the Parameter 125
6.1 Summary of the Theory: Basic Concepts............................125
6.1.1 The Physical System................................125
6.1.2 Physical Reasons for Parametric Excitation at Smooth
Modulation.........................................126
6.1.3 Conditions of Parametric Resonance.................126
6.1.4 Energy Transformations at Parametric Excitation .... 128
6.1.5 The Threshold of Parametric Excitation.............131
6.1.6 Differential Equation for Sinusoidal Motion of the
Weights along the Rod..............................132
6.2 The Intervals of Parametric Instability .........................134
6.2.1 The Principal Interval of Instability..............134
6.2.2 Resonance of the Second Order ...........................139
6.2.3 Resonances of the Third and Higher Orders..........144
6.3 Concluding Remarks ..............................................147
6.4 Questions, Problems, Suggestions.................................148
viii CONTENTS
6.4.1 Principal Parametric Resonance..........................148
6.4.2 The Principal Interval of Parametric Resonance.......150
6.4.3 The Second Parametric Resonance.........................151
II Nonlinear Oscillations 153
7 Free Oscillations of the Rigid Pendulum 155
7.1 Summary of the Theory...........................................155
7.1.1 The Physical System.....................................155
7.1.2 The Differential Equation of Motion for a Pendulum . . 157
7.1.3 Dependence of the Period on the Amplitude...............158
7.1.4 The Phase Portrait of the Pendulum......................159
7.1.5 The Phase Portrait in the Simulation Program............164
7.1.6 The Limiting Motion along the Separatrix................165
7.2 Oscillations of the Pendulum with Extremely Large
Amplitudes.....................................................168
7.2.1 Oscillations with Amplitudes Approaching 180° .... 169
7.2.2 Another Derivation of the Expression for the Period
of Large Oscillations...................................173
7.3 Period of Revolutions and Large Oscillations.................174
7.3.1 The Period of Fast Revolutions..........................174
7.3.2 Relationship between the Periods of Revolutions
and Large Oscillations..............................174
7.3.3 Mean Values of the Potential and Kinetic Energies . . . 178
7.4 The Influence of Friction ......................................179
7.4.1 The Phase Portrait of the Pendulum in the Presence
of Friction.........................................179
7.4.2 Revolutions Followed by Oscillations................180
7.5 Review of the Principal Formulas............................181
7.6 Questions, Problems, Suggestions................................182
7.6.1 Small Oscillations of the Pendulum..................182
7.6.2 Oscillations with Large Amplitudes..................184
7.6.3 The Rotating Pendulum...............................187
8 Rigid Planar Pendulum under Sinusoidal Forcing 189
8.1 Regular Response of a Harmonically Driven Rigid Pendulum ..189
8.1.1 Introduction ...........................................189
8.1.2 The Physical Model..................................190
8.1.3 Behavior of the Pendulum under the Slow Varying
Sinusoidal Torque Whose Amplitude is Close to 1 .... 191
8.2 Steady-State Response-Frequency Curves .........................196
8.2.1 Approximate Theoretical Resonance Curve.............196
8.2.2 Autoresonance, Hysteresis, and Bistability..........198
8.2.3 Nonlinear Resonance and a “Bell-Ringer Mode” .... 199
CONTENTS ix
8.2.4 Symmetry-Breaking and Period-Doubling Bifurcations,
Chaos and the Crisis ....................................201
8.3 Subharmonic and Superharmonic Resonances.........................203
8.4 Other Extraordinary Regular Forced Oscillations..................206
8.5 Concluding Remarks ............................................ 211
9 Pendulum with a Square-Wave Modulated Length 213
9.1 The Investigated Physical System.................................213
9.1.1 The Square-Wave Modulation of the Pendulum Length . 214
9.1.2 Conditions and Peculiarities of Parametric Resonance . . 217
9.2 The Threshold of Parametric Excitation...........................218
9.2.1 The Energy Supplied by the Square-Wave Modulation . 218
9.2.2 Regime of Parametric Regeneration...................219
9.2.3 Transients over the Threshold ...........................221
9.2.4 Parametric Swinging with the Feedback...............222
9.3 Autoresonance, Bifurcations, Multi stability.....................223
9.3.1 Autoresonance.......................................223
9.3.2 Bifurcations of Symmetry Braking and Period Doubling 223
9.4 Quantitative Theory of Parametric Excitation.....................225
9.4.1 Differential Equation for Parametric Oscillations .... 225
9.4.2 The Mean Natural Period at Large Depth of Modulation 226
9.5 Frequency Ranges for Parametric Resonance........................227
9.5.1 Resonances of Odd Orders............................227
9.5.2 Main Interval of Parametric Instability.............228
9.5.3 Third-Order Interval of Parametric Instability......232
9.5.4 Parametric Resonances of Even Orders................233
9.5.5 Intersections of the Boundaries at Large Modulation . . 236
9.6 Intervals of Parametric Excitation in the Presence of Friction . . 237
9.6.1 Boundaries of Instability for Resonances of Odd Orders . 238
9.6.2 Resonances of Even Orders...........................242
9.7 Concluding Remarks ..............................................243
10 Rigid Pendulum with Oscillating Pivot 245
10.1 Introductory Notes...............................................245
10.2 Kapitza’s Pendulum — Dynamic Stabilization.......................246
10.3 The Physical Model of the Investigated System....................248
10.4 Parametric Resonance.............................................249
10.5 Physical Reasons for Stability of the Inverted Pendulum..........251
10.6 An Approximate Quantitative Theory of the Inverted Pendulum . 253
10.7 Exact Differential Equation for Pendulum with Oscillating Pivot 257
10.8 Effective Potential Function for a Pendulum......................258
10.9 Subharmonic Resonances of High Orders............................262
10.9.1 Multiple-Nodding Oscillations of the Parametrically
Driven Pendulum..........................................263
10.9.2 Spectrum of Small-Amplitude n-Periodic Oscillations . . 267
X
CONTENTS
10.9.3 Lower Boundary of the Dynamic Stabilization...........269
10.9.4 Subharmonic Resonances of Fractional Orders ..........273
10.9.5 Coexistence of Subharmonic Resonances of Different
Orders n..............................................275
10.10 Upper Boundary of Dynamic Stability..........................277
10.10.1 The “Flutter” Mode and Ordinary Parametric Resonance 278
10.10.2 Boundaries of the Second-Order Subharmonic
Resonance.............................................281
10.10.3 The Influence of Friction.............................283
10.11 Enhanced Criterion for Kapitza’s Pendulum Stability........286
10.11.1 Subharmonic Resonances at Arbitrary Frequencies
and Amplitudes of the Pivot Vibration.................287
10.11.2 Improved Lower Boundary of Dynamic Stabilization . . 290
10.11.3 Improved Upper Boundary of Dynamic Stabilization . . 294
10.12 Complicated Regular Motions of the Parametrically Driven
Pendulum......................................................298
10.13 Chaotic Motions of the Pendulum..............................300
10.14 Concluding Remarks...........................................303
11 Torsion Pendulum with Dry and Viscous Damping 307
11.1 Basics of the Theory.........................................307
11.1.1 Introduction..........................................307
11.1.2 The Physical System...................................309
11.1.3 The Differential Equation of the Oscillator...........311
11.1.4 Damping of Free Oscillations under Dry Friction .... 312
11.2 Sinusoidally Driven Oscillator with Dry Friction.............313
11.2.1 Resonance in the Oscillator with Dry Friction under
Sinusoidal Excitation.................................313
11.2.2 Analytical Solution for the Second Half-Cycle of the
Resonant Excitation...................................314
11.2.3 The Threshold of the Resonant Growth .................316
11.2.4 Resonance in the Presence of Dry and Viscous Friction . 317
11.2.5 Non-Resonant Forced Oscillations......................319
11.3 Excitation at Sub-Resonant Frequencies ......................322
11.4 Concluding Remarks ..........................................330
Bibliography 333
Index
339
|
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| author | Butikov, Evgenij Ivanovič 1940- |
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| id | DE-604.BV042555646 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:24:50Z |
| institution | BVB |
| isbn | 9781498707688 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027989448 |
| oclc_num | 955255100 |
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| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XXIV, 340 S. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| spelling | Butikov, Evgenij Ivanovič 1940- Verfasser (DE-588)1061192083 aut Simulations of oscillatory systems with award-winning software, Physics of Oscillations Eugene I. Butikov Boca Raton [u.a.] CRC Press, Taylor & Francis Group 2015 XXIV, 340 S. txt rdacontent n rdamedia nc rdacarrier Dynamisches System (DE-588)4013396-5 gnd rswk-swf Oszillator (DE-588)4132814-0 gnd rswk-swf Oszillator (DE-588)4132814-0 s Dynamisches System (DE-588)4013396-5 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027989448&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027989448&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Butikov, Evgenij Ivanovič 1940- Simulations of oscillatory systems with award-winning software, Physics of Oscillations Dynamisches System (DE-588)4013396-5 gnd Oszillator (DE-588)4132814-0 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4132814-0 |
| title | Simulations of oscillatory systems with award-winning software, Physics of Oscillations |
| title_auth | Simulations of oscillatory systems with award-winning software, Physics of Oscillations |
| title_exact_search | Simulations of oscillatory systems with award-winning software, Physics of Oscillations |
| title_full | Simulations of oscillatory systems with award-winning software, Physics of Oscillations Eugene I. Butikov |
| title_fullStr | Simulations of oscillatory systems with award-winning software, Physics of Oscillations Eugene I. Butikov |
| title_full_unstemmed | Simulations of oscillatory systems with award-winning software, Physics of Oscillations Eugene I. Butikov |
| title_short | Simulations of oscillatory systems |
| title_sort | simulations of oscillatory systems with award winning software physics of oscillations |
| title_sub | with award-winning software, Physics of Oscillations |
| topic | Dynamisches System (DE-588)4013396-5 gnd Oszillator (DE-588)4132814-0 gnd |
| topic_facet | Dynamisches System Oszillator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027989448&sequence=000001&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=027989448&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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