Vibrating Beam Modeling Results

1
Vibrating Beam Modeling Results

• Prime (Group 7)
• Abby, Jacob, TJ, Leo

2
The model and data

• We are attempting to use the ordinary
  differential equation model

• This is assuming that the beam behaves
• like an under damped harmonic oscillator

3
C 0.68439333 standard error (C)
0.00897737 95 confidence interval (
0.66643859, 0.70234806 )
K 1525.693399 standard error (K)
0.350108 95 confidence interval (
1524.993183, 1526.393614 )

• s2 1.0559e-010

4

• notice the model only appears to estimate one
  frequency of the three that the data appears to
  contain.

5
The Optimization Algorithm

• The data vector is massaged by truncating at the
  max and adding the average back into it 5000
  times
• A loop using fminsearch adds random vectors to C
  and K. The values that produce the lowest
  least-squares cost are kept.

6
C1 -0.64450000 standard error (C1)
0.00908483 95 confidence interval (
-0.66266967, -0.62633033 )
K1 -1530.600000 standard error (K1)
0.354825 95 confidence interval
(-1531.309651, -1529.890349 )
s12 4.056e-011 compared to s2
1.0559e-010 cost11.9064e-007 cost
5.1366e-007
7
Residual Plots
New residual plot
Old residual plot

• Ideally the plot should be random about zero
• The occurrence of the diagonal pattern implies
  that there may be a better fit model

8
QQ Distribution
First QQ plot
Second QQ plot
The QQ or normal probability plot shows that the
function doesnt have a normal distribution.
9
Diagnostic Plots
First C, K values
Second C, K values

• Exhibits non-constant variance

10
The ODE Model

• The model appears to generally mimic the behavior
  of the system
• It seems to only capture one out of three
  frequencies displayed
• It shows residuals that are not evenly
  distributed or random about zero
• An improved model should be attempted

11
PDE Model

• Next, we attempted to fit the PDE model to the
  data

12
The PDE Model Initial parameter guesses
13
PDE ModelInitial parameter guess

• The model catches the first two, but misses the
  third frequency

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PDE ModelSecond Parameter Guess
15
PDE ModelSecond Parameter Guess
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Further Improvements

• Optimize the parameter selection
• Use statistical analysis for the PDE model of our
  data
• Attempt alternate statistical analysis techniques
• Collect multiple data sets and improve laboratory
  settings
• Research current literature for more accurate
  models