Simulations of Oscillatory Systems: with Award-Winning Software, Physics of Oscillations

367 Pages·2015·18.078 MB·\367

by  Eugene I. Butikov (Author)

Physics B Simulations of
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Simulations of Oscillatory Systems: with Award-Winning Software, i
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Physics of Oscillations provides a hands-on way of visualizing and un- o Oscillatory Systems
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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.
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with Award-Winning Software,
The book is conveniently structured according to mathematical complexity. i
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Each chapter in Part I contains activities, questions, exercises, and problems
Physics of Oscillations
of varying levels of difficulty, from straightforward to quite challenging. u
Part II presents more sophisticated, highly mathematical material that l
delves into the serious theoretical background for the computer-aided study a
of oscillations. t
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The software package allows you to observe the motion of linear and
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nonlinear mechanical oscillatory systems and to obtain plots of the
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variables that describe the systems along with phase diagrams and plots

of energy transformations. These computer simulations provide clear, vivid
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illustrations of oscillations in various physical systems, bringing to life many
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abstract concepts, developing your physical intuition, and complementing
the analytical study of the subject. O
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Features c
• Provides interactive software that serves as a desktop laboratory for i
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exploring simulated systems and replicating the experiments
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• Enables you to perform interesting mini-research projects involving
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the physics of oscillations
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• Develops your physical intuition and theoretical foundations
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• Includes various examples of the behavior of simulated systems

• Requires no previous knowledge of algorithmic languages or S
programming y
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K24904 s
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K24904 cvr mech.indd 1 1/19/15 11:32 AM
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Simulations of
Oscillatory Systems
with Award-Winning Software,
Physics of Oscillations
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Simulations of
Oscillatory Systems
with Award-Winning Software,
Physics of Oscillations
Eugene I. Butikov
Department of Physics
Saint Petersburg State University
Saint Petersburg, Russia
(cid:105) (cid:105)
(cid:105) (cid:105)
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2015 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
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Version Date: 20150121
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Contents
Preface xi
Introduction xiii
ClassificationofOscillations . . . . . . . . . . . . . . . . . . . . . . . xiv
SimulatedPhysicalSystems . . . . . . . . . . . . . . . . . . . . . . . xvi
HowtoUsetheSoftware . . . . . . . . . . . . . . . . . . . . . . . . . xvii
NotestotheInstructor. . . . . . . . . . . . . . . . . . . . . . . . . . . xxii
I OscillationsinSimpleSystems 1
1 FreeOscillationsofaLinearOscillator 3
1.1 SummaryoftheTheory . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 GeneralConcepts . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 DifferentialEquationofaLinearTorsionOscillator . . . 4
1.1.3 TheTimeofDampingandtheQualityFactorQ . . . . . 6
1.1.4 ThePhaseDiagramofaLinearOscillator . . . . . . . . 9
1.1.5 EnergyTransformations . . . . . . . . . . . . . . . . . 11
1.1.6 TheComputerSimulationofaLinearOscillator . . . . . 13
1.2 ReviewofthePrincipalFormulas . . . . . . . . . . . . . . . . . 13
1.3 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 15
1.3.1 FreeUndampedOscillations . . . . . . . . . . . . . . . 15
1.3.2 DampedFreeOscillations . . . . . . . . . . . . . . . . 16
1.3.3 Non-OscillatoryMotionoftheSystem . . . . . . . . . . 17
2 TorsionSpringOscillatorwithDryFriction 19
2.1 SummaryoftheTheory . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 GeneralConcepts . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 ThePhysicalSystem . . . . . . . . . . . . . . . . . . . 19
2.1.3 TheDifferentialEquationoftheOscillator . . . . . . . . 22
2.1.4 DampingCausedbyDryFriction. . . . . . . . . . . . . 23
2.1.5 ThePhaseTrajectory . . . . . . . . . . . . . . . . . . . 24
2.1.6 EnergyTransformations . . . . . . . . . . . . . . . . . 25
2.1.7 TheRoleofViscousFriction . . . . . . . . . . . . . . . 26
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vi CONTENTS
2.2 ReviewofthePrincipalFormulas . . . . . . . . . . . . . . . . . 28
2.3 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 28
2.3.1 DampingCausedbyDryFriction. . . . . . . . . . . . . 29
2.3.2 InfluenceofViscousFriction . . . . . . . . . . . . . . . 30
3 ForcedOscillationsinaLinearSystem 33
3.1 SummaryoftheTheory . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 BasicConcepts . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 DiscussionofthePhysicalSystem . . . . . . . . . . . . 34
3.1.3 TheDifferentialEquationforForcedOscillations . . . . 36
3.1.4 ThePrincipleofSuperposition . . . . . . . . . . . . . . 37
3.2 Steady-StateForcedOscillations . . . . . . . . . . . . . . . . . 38
3.2.1 ForcedOscillationsintheAbsenceofFriction . . . . . . 38
3.2.2 TheResonanceCurve . . . . . . . . . . . . . . . . . . . 40
3.2.3 ResonanceoftheAngularVelocity . . . . . . . . . . . . 42
3.2.4 EnergyTransformations . . . . . . . . . . . . . . . . . 42
3.3 TransientProcesses . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 InitialConditionsThatEliminateaTransient . . . . . . 46
3.3.2 ForcedOscillationsfromRestatResonance . . . . . . . 47
3.3.3 MechanicalAnalogueoftheStimulatedEmissionof
Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.4 TransientProcessesNearResonance . . . . . . . . . . . 53
3.3.5 TransientProcessesFarfromResonance . . . . . . . . . 54
3.3.6 TransientProcessesandthePhaseTrajectory . . . . . . 55
3.4 ReviewofthePrincipalFormulas . . . . . . . . . . . . . . . . . 57
3.5 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 58
3.5.1 Steady-StateForcedOscillations . . . . . . . . . . . . . 58
3.5.2 TransientProcesses . . . . . . . . . . . . . . . . . . . . 60
4 Square-WaveExcitationofaLinearOscillator 65
4.1 TheoreticalBackground . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 ModelofthePhysicalSystem . . . . . . . . . . . . . . 65
4.1.2 TheDifferentialEquationofForcedOscillations . . . . 67
4.2 Steady-StateForcedOscillationsundertheSquare-WaveTorque 68
4.2.1 HarmonicsoftheDrivingForceandoftheSteady-State
Response . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.2 ForcedOscillationsasNaturalOscillationsaboutthe
AlternatingEquilibriumPositions . . . . . . . . . . . . 73
4.3 TransientProcessesundertheSquare-WaveExternalTorque. . . 75
4.4 EstimationoftheAmplitudeofSteady-StateOscillations . . . . 77
4.4.1 ResonantAmplitudeofSteady-StateOscillations . . . . 78
4.4.2 AmplitudeofSteadyOscillationsatStrongFriction . . . 80
4.4.3 AmplitudeofSteadyOscillationsatT =2nT . . . . . 81
0
4.4.4 Steady-StateOscillationsatHighFrequenciesofthe
Square-WaveTorque . . . . . . . . . . . . . . . . . . . 82
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CONTENTS vii
4.5 EnergyTransformations . . . . . . . . . . . . . . . . . . . . . . 83
4.6 TheElectromagneticAnalogueoftheMechanicalSystem . . . . 85
4.7 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 86
4.8 ReviewofthePrincipalFormulas . . . . . . . . . . . . . . . . . 87
4.9 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 88
4.9.1 SwingingoftheOscillatoratResonance . . . . . . . . . 88
4.9.2 Non-ResonantForcedOscillations . . . . . . . . . . . . 89
5 ParametricExcitationofOscillations 93
5.1 SummaryoftheTheory.GeneralConcepts . . . . . . . . . . . . 93
5.1.1 ClassificationofOscillations . . . . . . . . . . . . . . . 93
5.1.2 TheSimulatedPhysicalSystem . . . . . . . . . . . . . 94
5.1.3 ElectromagneticAnalogueoftheMechanicalSystem . . 97
5.1.4 ConditionsforParametricResonance . . . . . . . . . . 97
5.1.5 TheThresholdofParametricExcitation . . . . . . . . . 98
5.1.6 DifferentialEquationforParametricOscillations . . . . 101
5.1.7 TheMeanNaturalPeriodatLargeModulation. . . . . . 102
5.2 FrequencyRangesofParametricExcitation. . . . . . . . . . . . 103
5.2.1 MainIntervalofParametricExcitation . . . . . . . . . . 104
5.2.2 Third-OrderIntervalofParametricInstability . . . . . . 107
5.2.3 FrequencyRangesforResonancesofEvenOrders. . . . 109
5.2.4 IntersectionsoftheBoundariesatLargeModulation . . 112
5.2.5 IntervalsofExcitationinthePresenceofFriction . . . . 113
5.3 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 118
5.4.1 PrincipalParametricResonance . . . . . . . . . . . . . 118
5.4.2 ManualControloftheParameter . . . . . . . . . . . . . 121
5.4.3 ParametricResonancesofHighOrders. . . . . . . . . . 122
6 SinusoidalModulationoftheParameter 125
6.1 SummaryoftheTheory:BasicConcepts . . . . . . . . . . . . . 125
6.1.1 ThePhysicalSystem . . . . . . . . . . . . . . . . . . . 125
6.1.2 PhysicalReasonsforParametricExcitationatSmooth
Modulation . . . . . . . . . . . . . . . . . . . . . . . . 126
6.1.3 ConditionsofParametricResonance . . . . . . . . . . . 126
6.1.4 EnergyTransformationsatParametricExcitation . . . . 128
6.1.5 TheThresholdofParametricExcitation . . . . . . . . . 131
6.1.6 DifferentialEquationforSinusoidalMotionofthe
WeightsalongtheRod . . . . . . . . . . . . . . . . . . 132
6.2 TheIntervalsofParametricInstability . . . . . . . . . . . . . . 134
6.2.1 ThePrincipalIntervalofInstability. . . . . . . . . . . . 134
6.2.2 ResonanceoftheSecondOrder . . . . . . . . . . . . . 139
6.2.3 ResonancesoftheThirdandHigherOrders . . . . . . . 144
6.3 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 147
6.4 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 148
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viii CONTENTS
6.4.1 PrincipalParametricResonance . . . . . . . . . . . . . 148
6.4.2 ThePrincipalIntervalofParametricResonance . . . . . 150
6.4.3 TheSecondParametricResonance . . . . . . . . . . . . 151
II NonlinearOscillations 153
7 FreeOscillationsoftheRigidPendulum 155
7.1 SummaryoftheTheory . . . . . . . . . . . . . . . . . . . . . . 155
7.1.1 ThePhysicalSystem . . . . . . . . . . . . . . . . . . . 155
7.1.2 TheDifferentialEquationofMotionforaPendulum . . 157
7.1.3 DependenceofthePeriodontheAmplitude . . . . . . . 158
7.1.4 ThePhasePortraitofthePendulum . . . . . . . . . . . 159
7.1.5 ThePhasePortraitintheSimulationProgram . . . . . . 164
7.1.6 TheLimitingMotionalongtheSeparatrix . . . . . . . . 165
7.2 OscillationsofthePendulumwithExtremelyLarge
Amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.2.1 OscillationswithAmplitudesApproaching180 . . . . 169
◦
7.2.2 AnotherDerivationoftheExpressionforthePeriod
ofLargeOscillations . . . . . . . . . . . . . . . . . . . 173
7.3 PeriodofRevolutionsandLargeOscillations . . . . . . . . . . . 174
7.3.1 ThePeriodofFastRevolutions . . . . . . . . . . . . . . 174
7.3.2 RelationshipbetweenthePeriodsofRevolutions
andLargeOscillations . . . . . . . . . . . . . . . . . . 174
7.3.3 MeanValuesofthePotentialandKineticEnergies . . . 178
7.4 TheInfluenceofFriction . . . . . . . . . . . . . . . . . . . . . 179
7.4.1 ThePhasePortraitofthePenduluminthePresence
ofFriction . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.4.2 RevolutionsFollowedbyOscillations . . . . . . . . . . 180
7.5 ReviewofthePrincipalFormulas . . . . . . . . . . . . . . . . . 181
7.6 Questions,Problems,Suggestions . . . . . . . . . . . . . . . . . 182
7.6.1 SmallOscillationsofthePendulum . . . . . . . . . . . 182
7.6.2 OscillationswithLargeAmplitudes . . . . . . . . . . . 184
7.6.3 TheRotatingPendulum . . . . . . . . . . . . . . . . . . 187
8 RigidPlanarPendulumunderSinusoidalForcing 189
8.1 RegularResponseofaHarmonicallyDrivenRigidPendulum . . 189
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 189
8.1.2 ThePhysicalModel . . . . . . . . . . . . . . . . . . . . 190
8.1.3 BehaviorofthePendulumundertheSlowVarying
SinusoidalTorqueWhoseAmplitudeisCloseto1 . . . . 191
8.2 Steady-StateResponse-FrequencyCurves . . . . . . . . . . . . 196
8.2.1 ApproximateTheoreticalResonanceCurve . . . . . . . 196
8.2.2 Autoresonance,Hysteresis,andBistability . . . . . . . . 198
8.2.3 NonlinearResonanceanda“Bell-RingerMode” . . . . 199
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