The Primitive Roots Kelly Rae Murray Stephen Small Eamonn Tweedy Anastasia Wong

Our Goal: my’’+ cy’ + ky=0 Fit data into a model to estimate parameters at a certain confidence level Fit data into a model to estimate parameters at a certain confidence level Find problem areas in order to discover the best possible model Find problem areas in order to discover the best possible model Create a better model with our data Create a better model with our data

The Experiment A PZT patch was used to vibrate a metal beam. A PZT patch was used to vibrate a metal beam. The proximity sensor measured the vibrations of the beam. The proximity sensor measured the vibrations of the beam. The vibrations were recorded by computer software, creating a file giving us a graph of our data. The vibrations were recorded by computer software, creating a file giving us a graph of our data.

Our Data

Before We can Model… Truncating Truncating –We found the maximum value of our data. –This became the initial point for our graph. –Set the time to begin at this point and end at 6 seconds. Oscillation Around Zero Oscillation Around Zero –We found the mean of our graph, which initially was below zero. –Then we shifted our data up to oscillate around zero.

Our Modified Data

Fitting the Model Guess and Check… Guess and Check… –Start graphing with the same amplitude as our model –To oscillate with our model –Damp at the same rate as our model

Our Best Fit For the Spring- Mass-Dashpot model… For the Spring- Mass-Dashpot model… –C = c/m = -.7 –K = k/m = –These values were found by finding the q through the fminsearch code.

Our Confidence Intervals Non-Linear Least Square Results C=c/m C=c/m –95% confidence ( , ) –Standard Error = –Sigma^2 = e-010 K = k/m K = k/m –95% confidence at ( , ) –Standard Error =.333

Residual Graphs

Testing Our Residuals Residuals Vs. Time Residuals Vs. Time –Shows too much of a pattern. –The periodicity tells us that there are modes that are not being represented. –Seems to have a higher frequency at the beginning and end. Residuals Vs. Fitted Value Residuals Vs. Fitted Value –Appears to be too much variation in the data. –Should be random around zero. –The higher spread at the beginning and end of the graph shows over-prediction and under-prediction

QQ Plot Comparing Error with Normal Distribution Comparing Error with Normal Distribution –Once again falls off at beginning and end. –This gives us enough proof that the Spring- Mass-Dashpot Model is not the best model for our experiment.

The Beam Model Since this model is a PDE it accounts for higher modes not covered by the simpler model. Since this model is a PDE it accounts for higher modes not covered by the simpler model. Hopefully, this model will give us a more accurate model for the motion of the beam. Hopefully, this model will give us a more accurate model for the motion of the beam.

Beam Model - Optimization  We first ran the beam simulation code without parameter optimization (fminsearch), using the ‘input’ data. The resulting fit didn’t quite match the latter portion of the data:

Beam Model - Optimization  Using some starting parameter guesses (conveniently close to those given to us), we ran an fminsearch that gave us a much closer fit to the actual data, even on the tail end:

Beam Model - Conclusion The beam model provided a much closer fit to the data set. The beam model provided a much closer fit to the data set. The PDE model clearly gives us a much more realistic approximation of the beam’s vibration. The PDE model clearly gives us a much more realistic approximation of the beam’s vibration.

Thanks!