![\"\"](\"http:\/\/pages.vassar.edu\/magnes\/files\/2012\/04\/sine-wave.jpg\")
<\/a><\/p>\nNow I will move on to the work that I have done with my triangle wave. \u00a0The 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. \u00a0This 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. \u00a0Increasing 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.<\/p>\n It appears that the key is finding the balance between number and size of spikes (higher or lower relative rotation frequencies) that yields\u00a0the highest overall total $B$, and thus the highest induced emf. \u00a0To 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 $d$. \u00a0Unfortunately, Mathematica is unable to process this computation. \u00a0I 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. \u00a0Now that I know what exactly I’m looking for (in this vein at least), getting to it should not be too difficult. \u00a0For 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.<\/p>\n <\/p>\n Mathematica code for the all images: \u00a0continuous plots<\/a>, data and discrete plot<\/a>, sine and triangle wave<\/a>.<\/p>\n 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. \u00a0Note 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 […]<\/p>\n","protected":false},"author":1599,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,29899,29905],"tags":[],"class_list":["post-1608","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-jacob","category-spring-2012"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1608","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/users\/1599"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=1608"}],"version-history":[{"count":13,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1608\/revisions"}],"predecessor-version":[{"id":2374,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/1608\/revisions\/2374"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=1608"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=1608"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=1608"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/a><\/p>\nThe link to the mathematica code for both of these can be found at the bottom of the page.<\/h6>\n
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