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Conclusion/Improvements
To do more sophisticated analysis of this system, there are several aspects of the procedure that would be changed. First, we would design the apparatus so that completely vertical oscillations would take place with reasonable consistency. The slightest deviation from a strictly vertical movement can lead to erroneous data. A more automated approach to the release would need to be devised instead of the completely manual release that was used for our current data set. Along the same lines, a weight that is more of a point mass should be employed. Also such a mass would be less likely to engage in non-vertical oscillations during trial runs. Data that needs to be obtained to do a better modeling of the system is how a rubber band would oscillate by itself. This information would allow for better equations to solve modeling the overall systems behavior.
Using both methods of reconstructing the Case 4 solution, the first being the brute force MATLAB/Maple approach where the oscillations were examined piecewise to construct the graph, as well as the second method which involved using the actual data to generate a modulating amplitude, these results matched closely with the overall shape of the graph we'd expect. There are many parameters, including damping, mass, spring, and rubber band constants that when changed, could vary the results quite significantly. Future research into this experiment would utilize a wide array of variables, including the ones I just mentioned, in order to fully understand the behavior of this system in every possible case. Due to time and resource constraints, our analysis focused mostly on the simpler cases in order to understand the general behaviors with as few variables as possible.
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