{"id":22765,"date":"2025-04-26T19:13:03","date_gmt":"2025-04-26T23:13:03","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=22765"},"modified":"2025-04-26T20:14:43","modified_gmt":"2025-04-27T00:14:43","slug":"forced-response-of-an-base-excited-system","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/forced-response-of-an-base-excited-system\/","title":{"rendered":"Forced response of a base-excited system"},"content":{"rendered":"<p>Consider the undamped single-degree-of-freedom system shown below that is excited by harmonic base motion <em>x<sub>B<\/sub>(t) = b sin\u03c9t<\/em>, where \u03c9 is the frequency of the base motion:<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-2025-04-26-at-7.57.19\u202fPM.jpg\" width=\"349\" height=\"299\" \/><\/p>\n<p>Here <em>kbsin\u03c9t<\/em> plays the role of <em>f(t)<\/em> for the &#8220;excitation&#8221; of this undamped system.<\/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><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-2025-04-26-at-8.13.35\u202fPM.jpg\" width=\"349\" height=\"299\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the undamped single-degree-of-freedom system shown below that is excited by harmonic base motion xB(t) = b sin\u03c9t, where \u03c9 is the frequency of the base motion: Here kbsin\u03c9t plays the role of f(t) for the &#8220;excitation&#8221; of this undamped system. Shown below is an animation showing the amplitude\/frequency plot, |X(\u03c9)|, for the particular solution &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/forced-response-of-an-base-excited-system\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Forced response of a base-excited system<\/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-22765","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22765","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=22765"}],"version-history":[{"count":7,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22765\/revisions"}],"predecessor-version":[{"id":22778,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/22765\/revisions\/22778"}],"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=22765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}