{"id":8737,"date":"2020-07-30T09:58:36","date_gmt":"2020-07-30T13:58:36","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=8737"},"modified":"2022-07-10T10:31:38","modified_gmt":"2022-07-10T14:31:38","slug":"static-deformation-and-eoms","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/static-deformation-and-eoms\/","title":{"rendered":"Static deformation and EOMs"},"content":{"rendered":"

A homogeneous disk of mass m and outer radius R rolls without slipping on an inclined surface, with the center of the disk supported by two springs, as shown below.<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Say we let x<\/em> represent the displacement of the center O from where the springs are unstretched. The dynamical equation of motion (EOM) in terms of the coordinate x<\/em> is shown below. As we can see, the resulting EOM is inhomogeneous.<\/p>\n

\"\"<\/p>\n

From this EOM, we also see that the static equilibrium position of O is given by: x<\/em>static<\/sub> = mg\u00a0<\/em>sin(theta<\/em>)<\/span>\/3k<\/em>. That is, if the disk were placed at a position of x<\/em> =\u00a0x<\/em>static<\/sub>, the disk remains there in equilibrium.<\/p>\n

Suppose that we choose a new coordinate of z<\/em> = x<\/em> – x<\/em>static<\/sub>\u00a0, which represents the motion of O relative to the static equilibrium position. The EOM in terms of his new coordinate is shown above to be homogenous.<\/p>\n

 <\/p>\n

Physical interpretation<\/strong><\/em><\/p>\n

Shown \u00a0below is an animation from a simulation of the free response of the system with initial conditions of x<\/em>(0) = x<\/em>_dot(0) = 0. The red curve shown below is the response of the system, x<\/em>(t). Note that the response is harmonic with a non-zero mean value, as expected. From the above analysis, we know that the frequency of response is given by omega<\/em>n<\/sub> = sqrt(2k\/m<\/em>). Relevant to our discussion above is that the oscillations occur about the static equilibrium position of the disk (shown in grey). The motion about that position is z<\/em>(t<\/em>), which has a zero mean value.<\/p>\n

 <\/p>\n

<\/p>\n

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

A homogeneous disk of mass m and outer radius R rolls without slipping on an inclined surface, with the center of the disk supported by two springs, as shown below.   Say we let x represent the displacement of the center O from where the springs are unstretched. The dynamical equation of motion (EOM) in … Continue reading Static deformation and EOMs<\/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-8737","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8737","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=8737"}],"version-history":[{"count":5,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8737\/revisions"}],"predecessor-version":[{"id":16251,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/8737\/revisions\/16251"}],"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=8737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}