{"id":22708,"date":"2025-04-25T10:32:55","date_gmt":"2025-04-25T14:32:55","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=22708"},"modified":"2025-04-25T11:32:45","modified_gmt":"2025-04-25T15:32:45","slug":"amplitude-and-phase-in-undamped-forced-response","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/amplitude-and-phase-in-undamped-forced-response\/","title":{"rendered":"Amplitude and phase in undamped forced response"},"content":{"rendered":"<p>Consider the undamped single-degree-of-freedom system shown below that is excited by harmonic forcing, where \u03c9 is the frequency of excitation:<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2025\/04\/Screenshot-2025-04-25-at-11.03.00\u202fAM-1.jpg\" width=\"349\" height=\"299\" \/><\/p>\n<p>Shown below is an animation showing the amplitude\/frequency plot, |<em>X(\u03c9)<\/em>|, for the particular solution response and the corresponding time history for <em>x<sub>P<\/sub>(t) <\/em>as we sweep through the frequency of excitation <em>\u03c9<\/em>:<\/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\/2025\/04\/undamped_forced_response.gif\" width=\"689\" height=\"520\" \/><\/p>\n<p>From this animation, you can see the following characteristics of the particular solution discussed in lecture:<\/p>\n<ul>\n<li>For low frequencies of excitation (\u03c9 &lt;&lt; \u03c9<sub>n<\/sub>), the response is <em><strong>in-phase<\/strong><\/em> with the forcing, and the response amplitude approaches that equilibrium displacement due to a constant forcing <em>f<sub>0<\/sub><\/em> as \u03c9 goes to zero.<\/li>\n<li>For high frequencies of excitation (\u03c9 &gt;&gt; \u03c9<sub>n<\/sub>), the response is <em><strong>180\u00b0 out-of-phase<\/strong><\/em> with the forcing, and the response amplitude tends to zero as \u03c9 becomes large.<\/li>\n<li>For near the &#8220;resonant&#8221; frequency (\u03c9 = \u03c9<sub>n<\/sub>), the response amplitude becomes large (in theory, becomes infinite) and the response changes from &#8220;in-phase&#8221; to &#8220;out-of-phase&#8221; with respect to the forcing.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the undamped single-degree-of-freedom system shown below that is excited by harmonic forcing, where \u03c9 is the frequency of excitation: Shown below is an animation showing the amplitude\/frequency plot, |X(\u03c9)|, for the particular solution response and the corresponding time history for xP(t) as we sweep through the frequency of excitation \u03c9: From this animation, you &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/amplitude-and-phase-in-undamped-forced-response\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Amplitude and phase in undamped forced response<\/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-22708","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22708","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=22708"}],"version-history":[{"count":9,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22708\/revisions"}],"predecessor-version":[{"id":22724,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22708\/revisions\/22724"}],"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=22708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}