First, to set up my initial findings, I will show a comparison of a sine wave vs the Fourier Transform that I used to approximate a triangle wave.  Note how the transform is much closer to a linear function than the sine wave.

The link to the mathematica code for both of these can be found at the bottom of the page.

Now I will move on to the work that I have done with my triangle wave.  The first thing that I examined, as it seemed the most interesting, was the relative velocity $\omega_s – \omega_r$ which I simply called $d$ in my simulation.  This seemed the most interesting off the bat as it’s one of the only variables that appears both inside the sine term, and as a factor of the total expression’s amplitude.  Increasing the $d$ value on my plot of $B$ over a single cycle shrank the period dramatically but also increased the amplitude, yielding a much larger number of much skinnier spikes in $B$, as can be seen below.

It appears that the key is finding the balance between number and size of spikes (higher or lower relative rotation frequencies) that yields the highest overall total $B$, and thus the highest induced emf.  To this end, I tried to plot the integral of the absolute value of $B$ (the absolute value being introduced to account for all negative and positive values of $B$ as both contribute to the total emf) against an increasing $d$.  Unfortunately, Mathematica is unable to process this computation.  I have attempted to plug in points in between $d = 1$ and $d = 10$ to pinpoint a local max, or find that it continues to increase, but mathematica is being uncooperative with this computation as well.  Now that I know what exactly I’m looking for (in this vein at least), getting to it should not be too difficult.  For my next post, I will attempt to solve this problem using a 3-dimensional graph to include both $d$ and $t$ (the variable that cycles the sine waves) as variables.

Mathematica code for the all images:  continuous plots, data and discrete plot, sine and triangle wave.