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 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 ) σ 2 = 1.0559e-010 σ 2 = 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 ) σ 2 = 1.0559e-010 σ1 2 = 4.056e-011 compared to σ 2 = 1.0559e-010 cost1=1.9064e-007 cost = 5.1366e-007

7. 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 Residual Plots

8. QQ Distribution First QQ plotSecond QQ plot The QQ or normal probability plot shows that the function doesn’t have a normal distribution.

9. Diagnostic Plots First C, K valuesSecond 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 Model Initial parameter guess The model catches the first two, but misses the third frequency

14. PDE Model Second Parameter Guess

16. 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