matthew moelter - cal polymmoelter/nonlinear/gordonmoelterppt.pdf• “lecture” • 3 x 1 hr •...

Lab-based course innonlinear dynamics

Matthew MoelterAssoc. Professor, Department of Physics

California Polytechnic State Univ

San Luis Obispo, CA

June 16, 2004 Phys Educ&Rsch: Classical Mech/Nonlinear Dynamics 2

Colleagues/Support

• Physics: N. Sungar, J. Sharpe, N. Fleishon

• Math: K. Morrison, J. McDill

• Chemistry: R. Schoonover

• NSF DUE 9594909

• Am. J. Phys. 69, 591-597, May 2001


Today’s Plan

• Introduction

• Objectives

• Course development

• Laboratory: experiments, projects

• How it went

• Future


Cal Poly - “Learn by doing”

• Calif. State Univ. system (24 campuses)– Agriculture, Architecture, Business, Engineering,

Liberal Arts, Science and Math

• 17,000 students

• Highly selective

• ≈100 years old

• California “central coast” (SF-SLO-LA)


SF

LA

SLO

nonlinear


Objectives

• Interdisciplinary, upper-division

• Intuition: “typical” nonlinear behavior(s)

• Geometrical approach– Phase space, fixed points, bifurcations,

limit cycles, attractors

• Different physical systems, similar math


Objectives (cont)

• "in-time" examples– simulation and experimentation– math symbols↔knobs

• Data display/analysis techniques– Power spectra, Poincare sect, return maps


Developing course

• Studio classroom (in place)

• NSF grant– Faculty summer $alary

• Content/Laboratory experiences

– release time

– equipment

• Strogatz visit (text author)


Course: Physics 417

• Interdisciplinary– crosslisted math/phys

– approved elective (science/engineering)

• Prerequisites– 1 yr calculus (analytical)

– Junior-level course in major or diff eqn (*)

• Quarter (10 weeks)


Course (cont.)

• Student population– Physics: 16

– Engineering (various): 12

– Mathematics: 2

– Archit., Undec, Phys Sci: 4

• 3 times: 1997, 99, 2002


Text

• Strogatz, Nonlinear Dynamicsand Chaos– Non-physics students

– Qualitative/numerical solutions

– Phase space

– Bifurcations

– Chaos late


Nonlinear in the studio

• “Lecture”• 3 x 1 hr

• Activities: groups, problem-solving, simulations

• Lab• 1 x 3 hr

• 8-12 students (= 4x2)

• “Real time” experiments(symbol knob)


Physics Studio

20 computers, carpet, A/C

A/V: broadcast instructor/student screen


Real time data

• Computers– Macintosh IIci's,

PPC, G4

• Interface– ULI or LabPro

• Probes• Motion

• Rotation

• Temperature

• Sound

• Pressure

• Voltage/Current

• pH

probe interface computer

experiment


Software

• Software– Acquisition

• LoggerPro, MacMotion, ULITimer

– Analysis: Matlab, Excel

– Simulation• Differential Systems

• MacMath


Overall schedule (weeks)• 1D (non)linear (2)

– phase space, fixed pts, terminology

• Bifurcation (1)– parameter space, types of bifurcations

• 2D linear (1)– Eigenvalue/vector

• MultiD nonlinear (3)– Linearize at fixed pt, topology, limit cycles


Schedule (cont)

• 1D maps (1)– discrete time steps, return maps

• Bifurcation in 2D (1)– Hopf bifurcation, Poincare maps

• Lorenz (1)– Strange attractors, chaos


Experiments (weeks)

• Num. Methods (0.5)

• 1D systems (1.5)

• Bifurcation (0.5)

• 2D systems (1.5)

• MultiD systems (0.5)

• Written weeklyreports

• Stability (1)

• Nonlinear (1)

• Discrete systems (1)

• Project (3)

• symbol↔knob


E: Numerical methods

• Numerical integration by hand

• Differential Systems– slope fields

– trajectories

• Matlab– Plotting/analysis

– ODE solver


E: 1D Systems

• dx/dt = f(x)

• Dissimilar systems, same math

• Fixed points from dx/dt vs x

• Student data


E: 1D Systems-linear

• Newton’s lawof cooling

• RC circuit

˙ q

q


E: 1D Systems- nonlinear

• Titration • Terminal velocity

pH

dpH

/dt

velocityacce

lera

tion


E: Bifurcation-Bead on a hoop

Video the motion: time consuming

frequency

angl

e


E: 2D systems

• dx/dt = f(x,y); dy/dt=g(x,y)

• Dissimilar systems, same math

• Phase plane/relation to motion

• Study of oscillations/orbits/spirals


E: 2D systems-linear

• Mass-spring SHO

• Vary mass, x0,damping

• RLC circuit


E: MultiD systems

• Coupled oscillators– Linearization, normal modes, eigenvalues,

eigenvectors, etc.

– Fourier spectral analysis

x1 x2 x1 x2


E: MultiD systems

x1, x2 vs. time

η1, η2 vs. time

antisymmetric mode


E: Ship Stability (computer)

• Forced,damped,nonlinear

Forcing frequency

Forc

ing

ampl

itude


E: Nonlinear system

• Magnet responds to field: B=B0cos(ωt)

• Fourier, phase space, Poincare

signal generator amplifier

current probe

Helmholtz coils

rotation probe

shaft

permanent magnet

data to computer

data to computer

i

Spinning Magnet Schematic


E: Nonlinear system

drive frequency

periodic doubled chaotic

pow

er s

pect

rum

frequency


E: Discrete system-drippy faucet

• Display/analysistechniques– periodic

– period 2

– chaotic photogate

pump

drain tank

water tank

∆t


E: Drippy faucet - ∆t vs. t∆t

Midpoint time

periodic

periodic w/fluct

period 2

chaotic


E: Return maps ∆tn+1 vs. ∆tn

periodic

periodic w/fluct

period 2

chaotic

∆tn+

1

∆tn


Projects (3 weeks)

• Individuals (a few pairs)

• From list, make new, extend existing– Proposal

– Construction/assembly/program

– Data/simulation and analysis

– Formal report


P: Chaotic waterwheel

Difficult: construction details, reproducibilityspontaneously use the tools/techniques


P: Chua Circuit

Circuits can be problematic.-diode = op amps-no intuition, why surprised?


P: Chua Circuit (cont)


P: Thrust Nozzle Exhaust PlumeRick BurnesAero. Engin.

∆t vs treturn map


P: Inverted pendulum (Duffing)


Inverted pendulumphase space power spectrum


Other Student Projects

• Buckling beams

• Curie-point pendulum

• Double pendulum

• Patterns in heated fluids

• Terminal velocity w/ diff shapes/fluids

• Computer control of chaos

• Mercury beating heart


Outcomes-Students

Level of difficulty: about as expected.

Workload: appropriate.

Experiments

8. Experiments helpful in understanding concepts? strongly agree.

9. The experiments taught material not covered in lectures: agree.

Project

10. The project was a good learning experience: strongly agree.

11. I enjoyed working on the project: agree to strongly agree.


Outcomes-Students15. Computer was helpful in assisting the learning of material: agree-

strongly agree.

19. Amount learned in this course compared to other courses: more to alot more.

20. Experiments liked most: drippy faucet, coupled oscillators

⇒22. Best thing about course: labwork

23. Overall feeling about course: positive (4 out of 5, 5 = very positive).


Issues for students

• New approach– Qualitative/numerical vs. analytical

• Application to real physical systems– symbols↔knobs

– “messy” and “noisy”

• Project time goes too quickly


Future• New robust experiments

– Chemical oscillation

– Pendulum (Laws)

– Circuits (Sprott)

– Another mechanical bifurcation

• Variety of experiments– parallel tracks for different students


Future (cont)

• Curriculum in physics– Some choice in upper division

• More/different faculty→experiments– biology, engineering,...

• Time management on “project”


Conclusions

• Lab-based nonlinear dynamics course

• Uses readily available equipment

• “Typical” behaviors/characteristics

• Enthusiastic students and faculty

• Try it, you’ll like it


Experiments not done

• Chemical oscillation– Actually messy, difficult quantitative

– “oooh, cool…”

• Inverted pendulum (Duffing)– Special care needed

• Diode circuit


Outcomes-Students (I)

1. Level of difficulty of lecture: about as expected.

2. Level of difficulty of lab experiments: about as expected.

3. Course workload (time to complete assignments): appropriate.

4. Lab workload (time to complete lab reports): appropriate to longer thanappropriate.

5. Time required to complete the lab project: appropriate.

6. The pace of lecture instruction: slow to ok.

7. Clarity and ease of use of lab handouts: no consensus.

8. Experiments helpful in understanding concepts?: strongly agree.

9. The experiments taught material not covered in lectures : agree.

10. The project was a good learning experience: strongly agree.

11. I enjoyed working on the project: agree to strongly agree.


Outcomes-Students (II)12. Work accurately evaluated by lecture instructor: agree.

13. Work accurately evaluated by lab instructor: agree.

14. I have been strongly motivated to learn course content: agree.

15. Computer was helpful in assisting the learning of material: agree-stronglyagree.

16. Text presented ideas clearly: neutral-strongly agree.

17. Class discussions of homework were useful: agree to strongly agree

18. Course has changed the way you think about and approach problems in yourfield compared to any other academic experience: a bit to a lot more.

19. Amount learned in this course compared to other courses: more to a lot more.

20. Experiments liked most: drippy faucet, coupled oscillators

21. Experiments liked least: ship stability

22. Best thing about course: labwork

23. Overall feeling about course: positive (4 out of 5, 5 = very positive).

matthew moelter - cal polymmoelter/nonlinear/gordonmoelterppt.pdf• “lecture” • 3 x 1 hr •...

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