{"id":12315,"date":"2021-09-09T09:56:38","date_gmt":"2021-09-09T13:56:38","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=12315"},"modified":"2024-10-05T18:09:19","modified_gmt":"2024-10-05T22:09:19","slug":"angular-acceleration-in-3d","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/angular-acceleration-in-3d\/","title":{"rendered":"Angular acceleration in 3D"},"content":{"rendered":"<p>Consider the rotor of a gyroscope. The rotor is attached to a frame that is rotating about a fixed vertical axes with a constant rate of <em>\u03c9<\/em><sub>1<\/sub>, and with the rotor rotating relative to that frame at a constant rate of <em>\u03c9<\/em><sub>2<\/sub>. The goal of the problem is to calculate the angular acceleration of the rotor.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7787 aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2024\/02\/Screenshot-2024-02-06-at-1.22.47-PM.jpg\" alt=\"\" width=\"315\" height=\"311\" \/><\/p>\n<p>The analysis for this is provided below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7787 aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2024\/02\/Screenshot-2024-02-06-at-2.31.30-PM.jpg\" alt=\"\" width=\"594\" height=\"498\" \/><\/p>\n<p>This result shows that, although both rotation components for the rotor are constant, the rotor is experiencing a non-zero angular acceleration. How is that possible? Consider the explanation that follows.<\/p>\n<p><strong><em>Visualizing the motion<\/em><\/strong><\/p>\n<ul>\n<li>Shown in left video below is the single rotation (<em><strong><span style=\"color: #ff0000\">RED<\/span><\/strong><\/em>) <em>\u03c9<\/em><sub>1<\/sub> about the fixed <em>Z<\/em>-axis.<\/li>\n<li>Shown in middle video is the single rotation (<em><strong><span style=\"color: #0000ff\">BLUE<\/span><\/strong><\/em>) <em>\u03c9<\/em><sub>2<\/sub> about the moving <em>x<\/em>-axis.<\/li>\n<li>Shown in right video is the TOTAL angular velocity vector (<em><strong><span style=\"color: #ff00ff\">MAGENTA<\/span><\/strong><\/em>) of the rotor, which is the vector sum of the red and blue angular velocity components.<\/li>\n<\/ul>\n<p>Note that the <em><span style=\"text-decoration: underline\">magnitude<\/span><\/em> of the total angular velocity is a constant; however, since the angular velocity vector changes in <em><span style=\"text-decoration: underline\">direction<\/span><\/em>, the angular acceleration is NOT zero. Here, with the magnitude of <em><strong>\u03c9<\/strong><\/em>\u00a0being constant, then <em><strong>\u03b1<\/strong><\/em> = d<em><strong>\u03c9<\/strong><\/em>\/dt is perpendicular to <em><strong>\u03c9<\/strong><\/em>. In this case, <em><strong>\u03b1<\/strong><\/em>\u00a0points in the positive y-direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/09\/gyro_comb.gif\" width=\"621\" height=\"313\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the rotor of a gyroscope. The rotor is attached to a frame that is rotating about a fixed vertical axes with a constant rate of \u03c91, and with the rotor rotating relative to that frame at a constant rate of \u03c92. The goal of the problem is to calculate the angular acceleration of the &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/angular-acceleration-in-3d\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Angular acceleration in 3D<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":14,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-12315","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/12315","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=12315"}],"version-history":[{"count":10,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/12315\/revisions"}],"predecessor-version":[{"id":20726,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/12315\/revisions\/20726"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/14"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=12315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}