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April 6, 2010 1

Vicki SpringmannEdward Montiel

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Advisor: Matt Pennybacker

Dr. Ildar Gabitov

Larry Hoffman, Department of Physics

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IntroductionTwo ModelsApplications

Simple Model TheoryLagrangianStability

Experimental DataParameters

StroboscopeComputer ModelingRunge-KuttaSimple Model Results

Error AnalysisApril 6, 2010 3

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Moving Cart Oscillating Base

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Segway (moving cart)Early Seismometers (moving cart)Neck and Spine in biomechanics (oscillating)

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Oscillating BaseTreated as a point massat its center of mass.

The Lagrangiansummarizes the

dynamics of the system.Must determine the kineticand potential energy

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The x and y coordinates of the mass are

Differentiating with respect to time,

Kinetic Energy is

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Substituting in the velocity,

But , so

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However,

So

+ (complete derivative) + (function of only t)

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On Earth, the potential gravitational energy is

So the potential energy of the pendulum is

Or,+ (function of only t)

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The equation of motion is obtained by

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Can be separated into two oscillations:

large small (amplitude)

slow fast (frequency)

The interaction between these oscillationscauses the vertical position to become stable.

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may be written as

where

therefore

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is small, but is large.is large, but it is slow.

Balancing, we have that

And because can be treated as a constant,

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We also have

Averaging this, we get

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So we have an equation for the effectivepotential

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The derivative of with respect to is

From this, we see that andare stable.The stability condition for is

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The potential has two maxima and two minimafor if the stability condition is met

Minimum at and

Maxima are between 0 and , and andTheir precise location depends on the parameters.Find by taking the derivative and setting it equal to 0.

The potential vs. angle graph is shown after theparameters have been determined.

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g = -9.8 m/s 2

a = 0.009 ± 0.0005 m

Pendulum is a trapezoidal prismlength = 28cmtop: 0.9cm x 0.9cmbottom: 0.9cm x 0.4cm

l = 0.12m (from center of mass)

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An instrument used to make a cyclicallymoving object appear slow-moving orstationary.

The lamp emits brief and rapid flashes of light.The frequency of the flash is adjusted until it is

equal to the object’s frequency.This makes it appear stationary.

Example with cpu fan:http://www.youtube.com/watch?v=_eoDVpC67Rc&feature=related

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Used to measure the frequency of fastoscillations.

When the pendulum did not appear to move

vertically, frequency can be read off device.Measurements were in units of min -1

Minimum stable frequency measured as1970 ± 49 min -1

206 ± 5.1 rad/s

Stability range measurements at 2097 min -1

220.0 rad/s

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Necessary for inputting equation of motioninto MatLab.

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Family of iterative methods to solve first orderODEs

Fourth order Runge-Kutta is the standardmethod

Error per step: O(h5); Total Error: O(h

4)

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Given an initial value problem

The slope is the weighted average of the slopes:At the beginning of the interval

At the midpoint of the interval

At the end of the interval

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Ω = 220.0 [rad/s]

φ

t

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Theoretically, the pendulum is stable when

Experimentally, minimum frequency ofstability is 206 ± 5.1 rad/s

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rad/s

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Ω = 220.0 [rad/s]

φ

u

Stable Region

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φ

d u

/ d φ

Ω = 220.0 [rad/s]

Stable Region

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StroboscopeDifficult to fine tuneResonance problem

Pendulum set upInterference from guidePendulum loosely attached to base (wobbling)Tape used to hold constant frequency loosens fromthe vibrations.

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The pendulum was treated as though it were apoint mass at its center of mass.

This is more appropriate for a sphere at the endof a thin, rigid rod.

The distributed mass of our pendulum needs tobe taken into account.This will be the focus of our future work.

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Landau L. D. & Lifshitz E. M., Mechanics ,(Pergamon, NY, 1960) pp 93-95. Motion in arapidly oscillating field

Smith H.J.T. & Blackburn J.A., Am J Phys 60 , 909(1992). Experimental study of an invertedpendulum