Modeling Vibrating Beam

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Modeling Vibrating Beam

• -using the harmonic Oscillator equation verses
  collected data

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Initial Data Collected (unmodified)
-Our initial data collected started with a
zeroing out area
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First Attempt to Model Data
What went wrong with this attempt?
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First Attempt Cont.

• Used fminsearch with initial conditions on C01
  and K02 resulted in C-.1163 and K-4.4527
• How can we improve on this model?
• Take out the zeroing out area.
• Shift graph to time0.
• Rerun fminsearch.

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Taking Out Zeroing Area
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Rerunning fminsearch

• CO1 K02 cost6.3941x10-7
• C -.67926355551960
• K -1.5227892111787423x103
• But are these initial conditions the best ones to
  estimate C and K?

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Finding better estimates of C and K

• Rewrite code with double for loops to check
  initial conditions between -5 and 5 for both C0
  and K0
• Stored CO and KO conditions with cost values into
  a matrix.
• This program ran overnight and did not finish in
  time.
• Checked the resulting (and incomplete) data for
  the smallest cost.

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The Better C and K

• We were able to look at 633 additional different
  initial C0 and K0 conditions.
• Found C0 -5 and K0-2.41 to have a lower cost
  of 6.3875x10-7.
• This cost is lower than for C01 and K02 so we
  used the resulting C and K of
• C -0.68407341326426
• K -1.522633494063630x103

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Determine Standard Error and Confidence Intervals
for C and K
Using the least squares method, we were able to
determine the confidence interval of both C and K.

• Model fitted
• d2 y / dt2 - C dy/dt - K y
• n 5019
• C -0.68407341
• s.e. 0.00979855
• 95 perc. ci ( -0.70367051, -0.66447632 )
• K -1522.633494
• s.e.(K hat) 0.383006
• 95 perc. ci ( -1523.399506, -1521.867482 )
• sigma2 1.2666e-010

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What do our estimates tell us?
We can see that a harmonic oscillator model fits
this data somewhat.
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A closer look
By taking a closer look, we can see that there
are many discrepancies that our model does not
account for.
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Looking at the differences

• The residuals gives us the difference between our
  data and the values estimated by the model.
• From the stats.m program that we were given, we
  have a graph of the residuals vs. time, and
  residuals vs. fitted data.

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Residuals vs. Fitted Value
Ideally we expect this plot to be bounded in a
region of residual values and have a cloud-like
spread. Were looking for a constant variance of
the error terms.
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Residual vs. Time
This plot of residual vs. time is used to check
independence and non-independence. Here is we see
that the error terms are non-independent.
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Sample Data vs. Standard Normal
If this were a normal distribution, then our
sample data would agree with the standard normal
line. We see that it only follows the normal line
for awhile and the tails do not.
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Deviation From Normal Distribution
Since there is a deviation from the normal
distribution, we see the tails at the ends of the
above red line.
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So what could we do better?

• A final technique to better model our data was
  introduced to our group by Professor Smith as the
  beam model.
• To look at this, we examined his data and his
  beam model which provided a best fit.

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Better Model Fit of Data Example
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A Closer Look
Notice here with this other data set that
multiple nodes are captured by the model.
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Summary

• This analysis took multiple steps in which
  careful examination is needed to better improve
  our models.
• By taking into consideration other variables, we
  should be able to better model our own data using
  the beam model program.