{"id":2207,"date":"2012-04-25T09:01:05","date_gmt":"2012-04-25T13:01:05","guid":{"rendered":"http:\/\/blogs.vassar.edu\/magnes\/?p=2207"},"modified":"2013-07-11T10:28:28","modified_gmt":"2013-07-11T14:28:28","slug":"results-graph-of-forces-and-final-analysis","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2012\/04\/25\/results-graph-of-forces-and-final-analysis\/","title":{"rendered":"Results IV – Graph of Forces and Final Analysis"},"content":{"rendered":"

Using the equation for the lifting force on a magnetic dipole by a conducting plane, as well as the relationship between lifting force and drag force (which happens to apply for all<\/em>\u00a0coil configurations) given by:<\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

(stated by\u00a0Jayawant<\/em>, using the same variables as previously)<\/p>\n

I produced the following graph in Mathematica, a plot of the ratio of the force and the ideal image force with respect to speed, ignoring the constant term out front:<\/p>\n

\"\"<\/a><\/p>\n

where the blue line represents the lifting force \"F_{L}\", the red line represents the drag force\u00a0\"F_{D}\", and the yellow line represents the ratio of the lifting force and the drag force\u00a0\"\frac{F_{L}}{F_{D}}\".<\/p>\n

From this graph, it appears that our theory for the limiting case (as \"v>>\omega\") holds true, since the lifting force starts off proportional to\u00a0\"v^{2}\", but then quickly becomes asymptotic toward the ideal image force\u00a0\"F_{I}\" as speed continues to increase. Meanwhile, the drag force is proportional to\u00a0\"v\" at low velocities, and is actually greater than the lifting force within this low speed interval. However, at around \"2m/s\", the lifting force surpasses the drag force, as the lifting force continues to rise while the drag force reaches a peak. Then, as speed increases, the drag force appears to fall off as\u00a0\"\frac{1}{\sqrt{v}}\". Additionally, for this magnetic dipole approximation, the ratio of lifting force to drag force remains fairly linear and very steep.<\/p>\n

If I were able to compare these results with those for a rectangular coil of wire – a configuration more appropriate to maglev trains – I would expect the same patterns to appear, though the lifting force might take longer to approach the limiting value of the ideal image force, since the rectangular coil situation represents a real case rather than an ideal case. Also, the drag peak might vary, thus causing a change in the train speed at which the lifting force surpasses the drag force. Because the ratio of lifting force to drag force is one measure of the efficiency of a maglev system (i.e., because it measured some\u00a0desired<\/em>\u00a0outcome with respect to an undesired<\/em>\u00a0outcome, both dependent upon the speed of the train), I would expect the ratio for this perfect dipole case to always be greater than any nonideal case: thus, the slope of the ratio curve should be lower for real EDS maglev systems.<\/p>\n","protected":false},"excerpt":{"rendered":"

Using the equation for the lifting force on a magnetic dipole by a conducting plane, as well as the relationship between lifting force and drag force (which happens to apply for all\u00a0coil configurations) given by: (1)   (stated by\u00a0Jayawant, using the same variables as previously) I produced the following graph in Mathematica, a plot of […]<\/p>\n","protected":false},"author":582,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,26720,29905],"tags":[],"class_list":["post-2207","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-joe","category-spring-2012"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2207","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\/582"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=2207"}],"version-history":[{"count":8,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2207\/revisions"}],"predecessor-version":[{"id":2384,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2207\/revisions\/2384"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=2207"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=2207"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=2207"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}