{"id":11407,"date":"2021-04-26T10:20:27","date_gmt":"2021-04-26T14:20:27","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?p=11407"},"modified":"2024-10-05T18:09:18","modified_gmt":"2024-10-05T22:09:18","slug":"homework-6-l","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/chapter-6-discussions\/homework-6-l\/","title":{"rendered":"Homework H6.C.09a"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-11408 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/6L-300x101.png\" alt=\"\" width=\"443\" height=\"149\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/6L-300x101.png 300w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/6L-1024x346.png 1024w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/6L-768x259.png 768w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/6L.png 1055w\" sizes=\"auto, (max-width: 443px) 100vw, 443px\" \/><\/p>\n<p>Any questions?<\/p>\n<hr \/>\n<p>&nbsp;<\/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\/04\/H6C_09.gif\" width=\"476\" height=\"410\" \/><\/p>\n<p><em><strong>Discussion and hints<\/strong><\/em>:<\/p>\n<p>Shown above are animations of the response of this system from simulations performed with the exciting frequency <em>omega<\/em> being <em>0.5*omega_n<\/em> and <em>1.5*omega_n<\/em>. Note that for <em>0.5*omega_n<\/em>\u00a0the response is in-phase with the base motion <em>x_B(t)<\/em>, whereas for <em>1.5*omega_n\u00a0<\/em>\u00a0the response is 180\u00b0 out-of-phase with the base motion. This phase difference should be apparent from both the visualization of the motion, as well as from the plots provided for x(t) and x_B(t). Can you observe this difference?<\/p>\n<p><em><strong>Derivation of the EOM: the four-step plan<\/strong><\/em><\/p>\n<ol>\n<li><em><strong>FBD<\/strong><\/em>: Define a coordinate <em>x<\/em> that represents the displacement of the block as measured from its position when the springs are unstretched. \u00a0Draw a free body diagram (FBD) of the block. In doing so, take care to get the directions of the spring forces correct, and that the force in the spring on the left depends on the relative motion between B and the block. Also, draw the FBDs of the disks. Define some rotation coordinates, one for each disk.<\/li>\n<li><em><strong>Newton\/Euler<\/strong><\/em>: It is recommended that you use Euler&#8217;s equations for the disks, choosing their centers as the reference points. Use Newton&#8217;s second law for the block.<\/li>\n<li><em><strong>Kinematics<\/strong><\/em>: You will need to relate <em>x_ddot<\/em> to the angular accelerations of the disk<em>s<\/em>. Be sure to get the signs correct on these as you look back over your definitions of rotation coordinates in Step 1.<\/li>\n<li><em><strong>EOM<\/strong><\/em>: Combine the kinetics equation from Step 2 with the kinematics from Step 3 to arrive at the single differential equation of motion for the system in terms of the <em>x<\/em> coordinate.<\/li>\n<\/ol>\n<hr \/>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Any questions? &nbsp; Discussion and hints: Shown above are animations of the response of this system from simulations performed with the exciting frequency omega being 0.5*omega_n and 1.5*omega_n. Note that for 0.5*omega_n\u00a0the response is in-phase with the base motion x_B(t), whereas for 1.5*omega_n\u00a0\u00a0the response is 180\u00b0 out-of-phase with the base motion. This phase difference should &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/chapter-6-discussions\/homework-6-l\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Homework H6.C.09a<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":3969,"featured_media":0,"parent":15017,"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-11407","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11407","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\/3969"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=11407"}],"version-history":[{"count":5,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11407\/revisions"}],"predecessor-version":[{"id":18457,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11407\/revisions\/18457"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/15017"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=11407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}