{"id":11240,"date":"2021-04-14T10:20:16","date_gmt":"2021-04-14T14:20:16","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=11240"},"modified":"2024-10-05T18:09:20","modified_gmt":"2024-10-05T22:09:20","slug":"more-on-static-deformations-and-vibrations","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/more-on-static-deformations-and-vibrations\/","title":{"rendered":"More on static deformations and vibrations"},"content":{"rendered":"<p>Consider a particle of mass <em>m<\/em> that is suspended in a vertical plane by a spring of stiffness <em>k<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-11251 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/04\/figure20.jpg\" alt=\"\" width=\"189\" height=\"173\" \/><\/p>\n<ul>\n<li>When writing in terms of the <em>x<\/em>-coordinate measured from the unstretched position of the particle, the right-hand side of the equation of motion (EOM) includes the weight term, <em>mg: \u00a0m*x_ddot + k*x = mg,<\/em> where <em>x<\/em> is measure positively downward.<\/li>\n<li>The static deformation, <em>x_st<\/em>, is found by setting <em>x_ddot = 0<\/em>, giving: <em>x_st = mg\/k<\/em>.<\/li>\n<li>Motion about the static equilibrium state is described by the coordinate <em>z = x &#8211; x_st<\/em>. \u00a0As seen in the lecture book, this produces a <span style=\"text-decoration: underline\"><em>homogeneous<\/em><\/span> EOM in terms of z: \u00a0<em>m*z_ddot + k*z = 0<\/em>.<\/li>\n<li>Since the EOM in terms of <em>z<\/em> is homogeneous, the free oscillations are centered on the position of static equilibrium.<\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>Consider the animation below:<\/p>\n<ul>\n<li>On the left is the response of the system if we release it from rest at the static equilibrium point: this produces no motion, as expected, since that is the position where the system remains at rest.<\/li>\n<li>In the animation second from the left, the block is released from a position where the spring is unstretched. In this case, the block has oscillatory motion centered on the static equilibrium point.<\/li>\n<li>In the animations third and fourth from the left, the block is released from rest with a general initial displacement: one with the spring compressed, and the other with the spring stretched. In both cases, the oscillations are still centered on the static equilibrium point.<\/li>\n<li>In all cases for which oscillations occur, the motions are centered on the static equilibrium point, and they all are of the SAME frequency of <em>omega_n = sqrt(k\/m)<\/em>. The amplitude of the motion is dictated by the amount of initial displacement from the static equilibrium point, NOT by the displacement from the unstretched state.<\/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\/2021\/04\/static_deformation.gif\" width=\"621\" height=\"313\" \/><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a particle of mass m that is suspended in a vertical plane by a spring of stiffness k. When writing in terms of the x-coordinate measured from the unstretched position of the particle, the right-hand side of the equation of motion (EOM) includes the weight term, mg: \u00a0m*x_ddot + k*x = mg, where x &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/more-on-static-deformations-and-vibrations\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">More on static deformations and vibrations<\/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-11240","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11240","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=11240"}],"version-history":[{"count":7,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11240\/revisions"}],"predecessor-version":[{"id":11252,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/11240\/revisions\/11252"}],"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=11240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}