May 16th, 2003 © 2002-2003, Team 02021 Nonlinear Dynamics Laboratory Equipment Andrew DickJoe Houtz...

May 16th, 2003May 16th, 2003 © 2002-2003, Team 02021© 2002-2003, Team 02021

Nonlinear Dynamics Nonlinear Dynamics Laboratory EquipmentLaboratory Equipment

Andrew DickAndrew Dick Joe HoutzJoe Houtz

Jeremy RedleckiJeremy Redlecki Ashley RiceAshley Rice

James StreeterJames Streeter

Customer: Mechanical Engineering Dept. at R•I•T Faculty Mentor: Dr. J.S. Török


OutlineOutline

Project GoalsProject GoalsPreliminary DesignPreliminary DesignRedesignRedesignFabricationFabricationEquipment TestingEquipment TestingExperiments and DemonstrationsExperiments and DemonstrationsConclusionConclusionQuestionsQuestions


Project GoalsProject Goals

Design, build, and test nonlinear dynamics Design, build, and test nonlinear dynamics laboratory equipment to aid in the teaching laboratory equipment to aid in the teaching of nonlinear dynamicsof nonlinear dynamics

Develop a set of procedures and manuals Develop a set of procedures and manuals for experiments and demonstrationsfor experiments and demonstrations


Preliminary DesignPreliminary Design

Chua’s CircuitChua’s Circuit Nonlinear Electrical SystemNonlinear Electrical System Chua’s DiodeChua’s Diode SpeakersSpeakers Acrylic CoverAcrylic Cover

Multi-well OscillatorMulti-well Oscillator Magneto-Elastic SystemMagneto-Elastic System MagnetsMagnets Motor & Eccentric MassMotor & Eccentric Mass Strain GaugesStrain Gauges


Preliminary DesignPreliminary Design

Chaotic WaterwheelChaotic Waterwheel Acrylic WheelAcrylic Wheel Pillow Block BearingPillow Block Bearing Encoder BearingEncoder Bearing BrakeBrake Steel Base and FrameSteel Base and Frame Submersible Water Submersible Water

PumpPump Flow MeterFlow Meter


RedesignRedesign

Rare Earth MagnetsRare Earth Magnets Multi-well OscillatorMulti-well Oscillator

Printed Circuit BoardPrinted Circuit Board Chua’s CircuitChua’s Circuit

Ground Fault Circuit Ground Fault Circuit InterruptInterrupt

Chaotic WaterwheelChaotic Waterwheel Printed Circuit Board Design


RedesignRedesign

Optical EncoderOptical Encoder Chaotic WaterwheelChaotic Waterwheel

Air BearingAir Bearing Chaotic WaterwheelChaotic Waterwheel

Air Bearing

Brake Assembly

Optical Encoder

Coupling


FabricationFabrication


Equipment Testing – Chua’s CircuitEquipment Testing – Chua’s Circuit

Testing ProcedureTesting Procedure Modeled circuitModeled circuit

PSpice softwarePSpice software Simulated circuit while Simulated circuit while

varying resistorvarying resistorProduced output voltage Produced output voltage time seriestime series

Built circuitBuilt circuitTroubleshootingTroubleshooting


Equipment Testing – Chua’s CircuitEquipment Testing – Chua’s Circuit

0 0.2 0.4 0.6 0.8 1

x 10-6

0

0.5

1

1.5

2

2.5

3x 10

5

Frequency, Hz

Am

pliti

ude

Frequency Spectrum

0 0.2 0.4 0.6 0.8 1

x 10-6

0

0.5

1

1.5

2

2.5

3x 10

5 Frequency Spectrum

Am

pliti

ude

Frequency, Hz

0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Two-Dimensional Projection of Attractor

Voltage V1, volts

Vol

tage

V2,

vol

ts

0 0.5 1 1.5 2 2.5 3 3.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4


Voltage V1, voltsV

olta

ge V

2, v

olts

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4


Voltage V1, volts

Vol

tage

V2,

vol

ts

Single Period Behavior High Period Behavior Chaotic Behavior

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-6

0

0.5

1

1.5

2

2.5

3x 10

5

Frequency, Hz

Am

pliti

ude

Frequency Spectrum


Equipment Testing – Multi-well OscillatorEquipment Testing – Multi-well Oscillator

Testing ProcedureTesting Procedure Built electromagnet test Built electromagnet test

bracketbracket Varied unbalanced mass Varied unbalanced mass

connected to motorconnected to motor Collected strain gauge Collected strain gauge

output though LabVIEWoutput though LabVIEW Analyzed steel beam using Analyzed steel beam using

Laser VibrometerLaser Vibrometer


Equipment Testing – Multi-well OscillatorEquipment Testing – Multi-well Oscillator

56 56.5 57 57.5 58 58.5 59 59.5 60-80

-60

-40

-20

0

20

40

60

80Time Series collected from strain gauge with LabVIEW

Time, sec

Am

plitu

de o

f st

rain

gau

ge v

alue

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-80

-60

-40

-20

0

20

40

60


Time, sec

Am

plitu

de o

f st

rain

gau

ge v

alue

50 51 52 53 54 55 56 57 58 59 60-80

-60

-40

-20

0

20

40

60


Time, sec

Am

plitu

de o

f st

rain

gau

ge v

alue

Periodic above one magnet Chaotic above both magnets

Periodic above two magnets


Chaotic OscillationChaotic Oscillation


Equipment Testing – Chaotic WaterwheelEquipment Testing – Chaotic Waterwheel


Equipment Testing – Chaotic WaterwheelEquipment Testing – Chaotic Waterwheel

Testing ProcedureTesting Procedure Varied system parametersVaried system parameters Observe a range of Observe a range of

nonlinear behaviorsnonlinear behaviors

Water Inflow Rate IncreaseDecrease

System Stability DecreaseIncrease

Control ParametersControl Parameters Primary: Water Flow Primary: Water Flow

RateRate Secondary: Wheel Secondary: Wheel

damping (brake)damping (brake)


Experiments and DemonstrationsExperiments and Demonstrations

DemonstrationsDemonstrations Show nonlinear dynamics and chaotic behavior to Show nonlinear dynamics and chaotic behavior to

studentsstudents

ExperimentsExperiments ObjectivesObjectives BackgroundBackground PrelabPrelab ExperimentsExperiments AnalysisAnalysis QuestionsQuestions

Design Oriented ExperimentsDesign Oriented Experiments

R·I·TMechanical Engineering



-4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4Rossler System a=b=0.2, c=2.5

X value of system

Y v

alue

of

syst

em

Chua's CircuitChua's Circuit Period-Doubling Period-Doubling

BifurcationBifurcation Orbit DiagramOrbit Diagram

-6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

Rossler System a=b=0.2, c=3.5

X value of system

Y v

alu

e o

f sys

tem

Control Parameter

Orbit DiagramOrbit Diagram


Experiments and DemonstrationsExperiments and DemonstrationsChaotic WaterwheelChaotic Waterwheel Nonlinear and Chaotic Nonlinear and Chaotic

BehaviorBehavior Attractor Attractor

ReconstructionReconstructionX(t)=[B(t),B(t+X(t)=[B(t),B(t+)])]

2-D Attractor Reconstructed Attractor

Chaotic Behavior

Nonlinear Behavior



-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

2Poincare Map of Forced Double Well Oscilator Attractor, Reconstructed

Position

Del

ayed

Pos

ition

Multi-well OscillatorMulti-well Oscillator Dynamics of Forced OscillatorDynamics of Forced Oscillator Attractor ReconstructionAttractor Reconstruction Poincaré MapPoincaré Map

PP((XX)=)=XX(k*n) for forcing period k(k*n) for forcing period k

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Strange Attractor of Forced Double Well Oscilator, Reconstructed

Position

Del

ayed

Pos

ition

Chaotic Behavior

Reconstructed Attractor Poincaré Map

0 50 100 150 200 250-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Position Time Series for Forced Double-well Oscillator

Time, sec

Pos

ition


ConclusionConclusion

Created 3 unique devices to display nonlinear Created 3 unique devices to display nonlinear dynamicsdynamics Chua’s CircuitChua’s Circuit Multi-well OscillatorMulti-well Oscillator Chaotic WaterwheelChaotic Waterwheel

On schedule and within budgetOn schedule and within budget

Potential for future innovationsPotential for future innovations

May 16th, 2003 © 2002-2003, Team 02021 Nonlinear Dynamics Laboratory Equipment Andrew DickJoe Houtz...

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