{"id":4853,"date":"2019-01-14T17:44:33","date_gmt":"2019-01-14T22:44:33","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/?page_id=4853"},"modified":"2019-01-22T16:13:10","modified_gmt":"2019-01-22T21:13:10","slug":"convolution-integral-interpreting-resonance","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/chapter-iv-animations\/convolution-integral-interpreting-resonance\/","title":{"rendered":"Convolution integral: interpreting resonance"},"content":{"rendered":"<h5 style=\"text-align: center;\"><!--This file created 11\/4\/02 12:11 AM by Claris Home Page version 3.0-->\u00a0<em>The Convolution Integral<\/em><\/h5>\n<p>Recall \u00a0that the convolution integral process is broken down into four steps:<\/p>\n<ul>\n<li><span style=\"text-decoration: underline;\"><em>folding<\/em><\/span> the impulse response function: <em>h(tau)<\/em> folds to <em>h(-tau)<\/em><\/li>\n<li><em><span style=\"text-decoration: underline;\">shifting<\/span><\/em> the impulse response function: <em>h(-tau)<\/em> shifts to <em>h(t-tau)<\/em><\/li>\n<li><em><span style=\"text-decoration: underline;\">multiplying<\/span><\/em> the folded\/shifted impulse response function with the excitation:\u00a0<em>h(t-tau) f(tau)<\/em><\/li>\n<li><em><span style=\"text-decoration: underline;\">integrating<\/span><\/em> to find the area under the <em>h(t-tau) f(tau)\u00a0<\/em>curve<\/li>\n<\/ul>\n<p>In the following animation, we see this four-step process to aid us in interpreting the resonance response of a single-DOF oscillator to harmonic excitation. From this, we see that the response amplitude is linearly increased as we move along in time. This is due to the lining-up of the shifted\/folded impulse response function with the excitation.<\/p>\n<p><center><\/center><center><\/center><center><br \/>\n<span style=\"color: #0000af;\"><b>(ANIMATION AUTOMATICALLY REPLAYS)<\/b><\/span><\/center><\/p>\n<table style=\"border-collapse: collapse; width: 125.79281183932346%; height: 439px;\" border=\"1\">\n<tbody>\n<tr>\n<td style=\"width: 100%;\"><span style=\"color: #af0000;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4855\" src=\"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-content\/uploads\/sites\/18\/2019\/01\/cmk.gif\" alt=\"\" width=\"576\" height=\"432\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><center>\u00a0<\/center><\/p>\n<hr \/>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0The Convolution Integral Recall \u00a0that the convolution integral process is broken down into four steps: folding the impulse response function: h(tau) folds to h(-tau) shifting the impulse response function: h(-tau) shifts to h(t-tau) multiplying the folded\/shifted impulse response function with the excitation:\u00a0h(t-tau) f(tau) integrating to find the area under the h(t-tau) f(tau)\u00a0curve In the following &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/ervibrations\/chapter-iv-animations\/convolution-integral-interpreting-resonance\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Convolution integral: interpreting resonance<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":4784,"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-4853","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/pages\/4853","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/comments?post=4853"}],"version-history":[{"count":5,"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/pages\/4853\/revisions"}],"predecessor-version":[{"id":5325,"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/pages\/4853\/revisions\/5325"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/pages\/4784"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/ervibrations\/wp-json\/wp\/v2\/media?parent=4853"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}