{"id":1165,"date":"2024-07-09T15:27:48","date_gmt":"2024-07-09T09:57:48","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=1165"},"modified":"2025-07-16T15:02:48","modified_gmt":"2025-07-16T09:32:48","slug":"klein-construction","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/me\/theory-of-machines\/klein-construction","title":{"rendered":"Kleins construction"},"content":{"rendered":"<p style=\"text-align: justify;\">For a slider-crank mechanism, a simpler method to construct constant velocity and acceleration polygon on the configuration diagram itself has been suggested by Klein. According to him, the line representing crank in the configuration diagram also represents velocity and acceleration of the moving end in the velocity and acceleration diagrams respectively.<\/p>\n<p style=\"text-align: justify;\">Consider a configuration diagram of a slider-crank mechanism as shown in Fig. Let <em>r <\/em>be the length of crank\u00a0<em>OC<\/em>.<\/p>\n<p style=\"text-align: justify;\"><strong>Velocity Polygon<\/strong> <strong>:<\/strong> Let <em>r <\/em>represents velocity V<sub>c<i>o<\/i><\/sub>. Extend the line <em>PC <\/em>to intersect line drawn perpendicular to <em>OP <\/em>through <em>O <\/em>at point <em>M<\/em>. The triangle <em>OCM <\/em>represents velocity polygon.<\/p>\n<p style=\"text-align: justify;\">In Fig.: <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1169 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/klein-construction.png\" alt=\"Klein\u2019s Construction\" width=\"272\" height=\"246\" \/><\/p>\n<ul style=\"text-align: justify;\">\n<li><em>OC <\/em>represents magnitude of velocity of point <em>C <\/em>relative to point O. (<i>V<sub>co<\/sub><\/i>)<\/li>\n<li><em>CM <\/em>represents magnitude of velocity of point <em>C <\/em>relative to point P. (V<sub>cp<\/sub>)<\/li>\n<li><em>OM <\/em>represents magnitude of slider velocity (V<sub>po<\/sub>)<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><strong>Acceleration Polygon : <\/strong>For acceleration polygon, let <em>r\u00a0<\/em>represents acceleration of point <em>C<\/em> with respect to point <em>O<\/em>, <em>a<sub>co<\/sub><\/em><em> (= \u03c9<sup>2<\/sup>r)<\/em>.\u00a0This assumption gives the scale for acceleration polygon. For construction of acceleration polygon, the following procedure may be used:<\/p>\n<ol style=\"text-align: justify;\">\n<li>Draw a circle of radius <em>CM <\/em>with point <em>C <\/em>as centre of the circle.<\/li>\n<li>Draw another circle with link <em>CP <\/em>as diameter and midpoint of the line <em>CP <\/em>as the centre of the circle.<\/li>\n<li>Join the points of intersection of these two circles by a straight line <em>KL<\/em>, which is chord common to both circle. Let this common chord cuts the links <em>CP <\/em>and <em>OP <\/em>at point <em>Q <\/em>and <em>N <\/em>respectively.<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Here, the quadrilateral <em>OCQN <\/em>is the required acceleration polygon,<\/p>\n<ul>\n<li style=\"text-align: justify;\"><em>ON<\/em> \u2192 Linear acceleration of slider <em>P<\/em><\/li>\n<li style=\"text-align: justify;\"><em>OC \u2192 <\/em>Centripetal acceleration of crank <em>OC <\/em><\/li>\n<li style=\"text-align: justify;\"><em>CQ \u2192 <\/em>Centripetal acceleration of link <em>CP <\/em><\/li>\n<li style=\"text-align: justify;\"><em>QN<\/em> \u2192 Tangential acceleration of link <em>CP<\/em><\/li>\n<li style=\"text-align: justify;\"><em>CN \u2192 <\/em>Total acceleration of link <em>CP<\/em><\/li>\n<\/ul>\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/me\/theory-of-machines\/the-slider-crank-chain\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/me\/theory-of-machines\/pantograph\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a><br \/>\n<strong> Must Read: <\/strong> <a href=\"https:\/\/study.madeeasy.in\/subjects\/what-is-the-theory-of-machines\/\" target=\"_blank\" rel=\"noopener\"><strong>What is the Theory of Machines?<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For a slider-crank mechanism, a simpler method to construct constant velocity and acceleration polygon on the configuration diagram itself has<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[287,10],"tags":[364,365],"class_list":["post-1165","post","type-post","status-publish","format-standard","hentry","category-theory-of-machines","category-me","tag-acceleration-polygon","tag-velocity-polygon"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/1165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=1165"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/1165\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=1165"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=1165"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=1165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}