OVERVIEW
The purpose of modeling the motion is to get an understanding of what we should be looking for before doing the tests and calculations of our data. We also use the model to compare the results we get to the model. Because the data we are using for our calculations is the accelerations in the x and y axes, our goal is to calculate those points. We focused on modeling a time signature of 4/4 and a tempo of 100 bpm.
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In order to model this complex motion, we broke the motion down into 4 arcs to simplify calculations. To the right, you can see a figure displaying each of the 4 arcs. We made sure to account for the different starting angles of each arc. We also set a certain radius for each of the arcs, as that will be necessary in calculating acceleration.​
Technical Information-
MOdeling the motion
METHODS
We have to calculate a few things before being able to calculate the acceleration.
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1. We calculated the angular velocity by using the following equation. We knew the beats per minute was 100, so each arc was completed in .6 seconds. The two middle arcs have a different angular velocity than the two outer arcs because although the time is constant, the theta they are traveling is different.
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ω=dθ/dt
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2. We found the period, T, by using the following equation.
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T=2*pi/ω
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3. We set up vectors for theta and time such that the theta would range from the starting angle to the ending angle and time to correspond with each of these theta values. The time would start at 0 or after the last arc's time and end at T/4 or T/8 depending on whether it was a quarter circle or an eighth circle. This would allow us to have an acceleration as a function of time rather than an acceleration as a function of theta.
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4. We calculated the acceleration in the x-direction using this equation:
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ax=-r*ω^2*cos(theta)
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5. We calculated the acceleration in the z-direction using this equation:
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az=-r*ω^2*sin(theta)
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6. Finally, we concatenated the time and acceleration vectors in order to create a full measure.
RESULTS
Arc 1: Acceleration in the X and Z components
Arc 2: Acceleration in the X and Z components
Arc 3: Acceleration in the X and Z components
Arc 4: Acceleration in the X and Z components
Concatenated Acceleration in the X and Z components
We combined two full measures to get a proper sense of the pattern and to be able to easily compare it with the data.
DISCUSSION
When comparing the model to the data, we saw some differences as shown in the figure below. There are several reasons for these differences. It is nearly impossible to perfectly model a conductor's motion because there will be differences in every motion due to human error. The most significant error is likely due to not accounting for the accelerations due to change in motion. We assumed each arc to have a constant angular acceleration, which is inconsistent with the motion. Additionally, the motion of a conductor is not in perfect arcs - an arc is simply a good approximation of the motion.
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After testing this model against our data and realizing that they did not line up well, we changed a few things about our model. For one, we changed the radii of the arcs to be different. Additionally, we plotted two measures to get a better sense of how the plot looks as it goes from one measure to the next. This helped make the patterns a little bit more similar but still didn't get us to the point where the two plots lined up.
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This model helped us realize how significant accelerations due to changes in directions are in a conductor's motion. There is more than just slowing down - there is a little bit of punctuation during each beat for emphasis on the tempo. If we had more time, we would look into modeling these changes in direction, punctuations at the beat and the different angular accelerations. We would also work on the sampling rate such that all of the time vectors had the same step rate. This would allow us to get better results for the Fourier Transform. It could be possible that the frequency peaks line up but the acceleration peaks don't.
