Nonlinear Dynamics Laboratory Equipment

Andrew Dick Joe Houtz

Jeremy Redlecki Ashley Rice

James Streeter

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

Presentation Outline

• Nonlinear Dynamics and Chaos• Project Goals• Design Process• Preliminary Design• Expenses and Budget• Spring Quarter Schedule• Conclusion

Nonlinear Dynamics and Chaos

• Nonlinear Dynamics• Physical systems will nonlinear electrical/mechanical

components• State Equations contain nonlinear terms• Closed form solutions are unlikely• Variety of dynamic behavior

• Chaos• Deterministic systems, not stochastic• Long-term apparent erratic behavior• Dynamics governed by ‘strange’ attractor• Sensitive Dependence on initial conditions

Project Goals

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

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

Description

• Dynamic systems governed by nonlinear equations

• Ability to demonstrate nonlinear and chaotic behavior

• Equipped with data acquisition equipment to collect data for analysis

• Compatible with three tier system of experimentation1. Demonstration

2. Experimentation

3. Design oriented experimentation

• Extensive testing to validate equipment and gather data required for the completion of the project

Equipment Set

• Chua’s Circuit• Nonlinear Electric Circuit

• Multi-well Oscillator• Magneto-Elastic System

• Chaotic Waterwheel• Mechanical-Fluid System

Design Facets

• Recognize & quantify the need

• Concept development

• Feasibility assessment

• Establishing design objectives & criteria

• Analysis of problems & synthesis into the design

• Preliminary design

Facet One: Needs Assessment

• Laboratory equipment usage• One, two and three tier experiments

• Design constraints

• Financial parameters• $2000 budget

• Performance parameters

Facet Two: Concept Development

• Brainstorming

• Research existing systems

• Group consensus of preliminary concepts

• Develop drawings & descriptions of each preliminary concept• Group exercises

• Concept drawing

• Empathy method

• Preliminary BOM

Facet Three: Feasibility Assessment

• Technical aspects

• Economical aspects

• Market

• Schedule

• Performance

Facet Four: Design Objectives & Performance Specifications

• Safety codes and standards

• List of design objectives • Yes/no question format

• Evaluation checklist

• Performance specification table

Chua’s Circuit• Kirchoff’s laws

)( 1121

1 VgR

VV

dt

dVC

LIR

VV

dt

dVC

212

2

2Vdt

dIL L

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3 Chua's Circuit I-V Characteristic

V (volts)

I (a

mps

)

xydt

dx

xz y rxdt

dy

bzxydt

dz

Lorenz Equations:

Chua’s Circuit:Simulation and Analysis

• PSpice simulation

• Matlab• Frequency Spectrum

Time

0s 5ms 10ms 15ms 20ms 25ms 30ms 35ms 40ms 45ms 50msV(V1)

-3.0V

-2.0V

-1.0V

-0.0V

1.0V

2.0V

3.0V

V(V1)

-3 -2 -1 0 1 2 3-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Chua's Circuit V1 vs. V2

V1 (volts)

V2

(vol

ts)

Chua’s Circuit Apparatus Design

• Op-amps represent Chua’s Diode.• Output voltages can be connected to an

oscilloscope, LabView, or speakers.• Setup:

• A base is to be fabricated • Transparent cast acrylic cover• Two speakers

Multi-Well Oscillator

• Research• Magneto-elastic system

• Theory

)cos(32

2

tFxxdt

dx

dt

xd

Multi-well Oscillator:Simulation and Analysis

• Control parameters:• , Damping Factor

• F, Force Amplitude

Multi-well Oscillator Apparatus Design

• Analyses• Stress

• Fatigue

• Frequency

• Magnetic

Chaotic Waterwheel

• Willem Malkus and Lou Howard developed the first chaotic waterwheel at MIT in the 1970’s.

Chaotic Waterwheel:Simulation and Analysis

111 Kaba

1111 qKbab

I

grav 1

Waterwheel Equations:

Chaotic Waterwheel Apparatus Design

• General operation

• Analyses

• Flow analysis

• Moment of inertia analysis

• Stress analysis

• Brake

• Encoder Bearing

Encoder bearing

Brake

In flow

Overflow

Needle valve

Out flow

Expenses and Budget

• Bill of material created for each device• Raw material, purchased components, electrical components, sensors,

nuts and bolts.

• Cost per device and complete set• Chua’s Circuit Apparatus 141 parts $267.71

• Multi-well Oscillator Apparatus 153 parts $597.01

• Chaotic Waterwheel Apparatus 447 parts $1135.81

• Complete set of three Apparatuses 741 parts $1895.29

• Available Balance$104.71

Spring Quarter Schedule

• Phase One: Construction 4 weeks• Quality/performance test components• Fabricate components• Assemble equipment

• Phase Two: Testing 4 weeks• Confirm/optimize design• Determine critical parameters• Develop experimentation procedures

• Phase Three: Documentation 2 weeks• Complete user manual• Final Design Report• Critical Design Presentation

Conclusion

• Successfully completed design facets 1 through 6

• Conducted many different analyses on each of the devices

• Completion of a preliminary design

• Expenses and Budget

• Spring Quarter Schedule