{"id":8763,"date":"2020-07-31T08:57:42","date_gmt":"2020-07-31T12:57:42","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=8763"},"modified":"2025-04-26T19:07:44","modified_gmt":"2025-04-26T23:07:44","slug":"forced-response-harmonic-excitation","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/forced-response-harmonic-excitation\/","title":{"rendered":"Forced response – harmonic excitation"},"content":{"rendered":"

Consider an undamped single degree-of-freedom system experiencing harmonic excitation whose equation of motion (EOM) is given by:
\n\"\"
\nHere omegan<\/sub><\/em> is known as the natural frequency of response.<\/p>\n

The particular solution for this EOM is give by:
\n\"\"<\/p>\n

where the magnitude of \u00a0A<\/em> is known as the “amplitude of response”. A plot of A<\/em> vs. omega\u00a0<\/em>the frequency of excitation omega<\/em> is shown below.<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Observations on A as a function of excitation frequency (for undamped response)<\/strong><\/em><\/span><\/p>\n

For low frequencies of excitation (omega < omegan<\/sub>\u00a0<\/em>), the response amplitude A<\/em> is approximately F<\/em>0<\/sub>\/K<\/em>, or the static response for a constant load of F<\/em>0<\/sub>, as shown in the animation below. It is also seen that the response is “in phase” with the excitation (since A<\/em> > 0).<\/p>\n\n\n\n
Frequency of excitation near zero<\/span>
\n<\/b><\/span><\/em>The response is in phase with the excitation and the amplitude of response approaches the static response.\"\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

\n

For near-resonant \u00a0frequencies of excitation (omega <\/em>approximately equal to omegan<\/sub><\/em>), the magnitude of \u00a0A<\/em> is large, as shown in the animation below. It can be shown that the response is “90 degrees out of phase” with the excitation.<\/p>\n\n\n\n
Frequency of excitation near resonance<\/span>
\n<\/b><\/span><\/em>For excitation frequencies near the natural frequency, the amplitude of response is large. Shown below is the particular solution of the EOM for the excitation frequency \u03c9 slightly less than \u03c9n<\/sub>. Note for since \u03c9 < \u03c9n<\/sub>, the response is still in phase with the excitation.<\/p>\n\n\n\n
\"\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

\n

For high frequencies of excitation (omega > omegan<\/sub>\u00a0<\/em>), the magnitrude of the response amplitude A<\/em> is very small, as shown in the animation below. It is also seen that the response is “180 degrees out of phase” with the excitation (since A<\/em>\u00a0< 0).<\/p>\n\n\n\n
Large frequency of excitation<\/span>
\n<\/b><\/span><\/em>The response is 180 degrees out of phase with the excitation and the amplitude of response is small.\"\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Consider an undamped single degree-of-freedom system experiencing harmonic excitation whose equation of motion (EOM) is given by: Here omegan is known as the natural frequency of response. The particular solution for this EOM is give by: where the magnitude of \u00a0A is known as the “amplitude of response”. A plot of A vs. omega\u00a0the frequency … Continue reading Forced response – harmonic excitation<\/span> →<\/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-8763","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8763","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=8763"}],"version-history":[{"count":9,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8763\/revisions"}],"predecessor-version":[{"id":22764,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8763\/revisions\/22764"}],"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=8763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}