{"id":878,"date":"2024-07-06T14:45:49","date_gmt":"2024-07-06T09:15:49","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=878"},"modified":"2025-07-16T15:00:21","modified_gmt":"2025-07-16T09:30:21","slug":"degrees-of-freedom","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/me\/theory-of-machines\/degrees-of-freedom","title":{"rendered":"Degrees of freedom"},"content":{"rendered":"<p>The connection of a link with another imposes certain constraints on their relative motion thus,<\/p>\r\n<p>Degrees of freedom = 6 \u2013 number of restraints<\/p>\r\n<p><strong>Degree of Freedom of Space Mechanism (3-D)<\/strong><\/p>\r\n<p style=\"text-align: center;\"><em>F <\/em>= 6(<em>L <\/em>\u2013 1) \u2013 5<em>P<\/em><sub>1<\/sub> \u2013 4<em>P<\/em><sub>2<\/sub> \u2013 3<em>P<\/em><sub>3<\/sub> \u2013 2<em>P<\/em><sub>4<\/sub> \u2013 <em>P<\/em><sub>5<\/sub><\/p>\r\n<p style=\"text-align: center;\">Here,\u00a0\u00a0\u00a0\u00a0\u00a0 <em>F<\/em> = Degree of freedom (D.O.F.)<br \/><em>L <\/em>= Total number of links in mechanism<br \/><em>P<\/em><sub>1<\/sub> = Number of pairs having one D.O.F.<br \/><em>P<\/em><sub>2<\/sub> = Number of pairs having two D.O.F.<br \/><em>P<\/em><sub>3<\/sub> = Number of pairs having three D.O.F.<br \/><em>P<\/em><sub>4<\/sub> = Number of pairs having four D.O.F.<br \/><em>P<\/em><sub>5<\/sub> = Number of pairs having five D.O.F.<\/p>\r\n<ul>\r\n<li><strong>Degree of freedom of plane (2D) mechanism (Gruebler\u2019s Criterion)<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><em>F <\/em>= 3(<em>L<\/em>&#8211; 1) &#8211; 2<em>P<\/em><sub>1<\/sub> &#8211; <em>P<\/em><sub>2<\/sub><br \/>Here, <em>L <\/em>= Number of links in a mechanism<br \/><em>P<\/em><sub>1<\/sub> = Number of pairs having one degree of freedom<br \/><em>P<\/em><sub>2<\/sub> = Number of pairs having two degree of freedom<\/p>\r\n<ul>\r\n<li><strong>Kutzback\u2019s equation<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><em>F <\/em>= 3(<em>L\u00a0 <\/em>&#8211; 1) &#8211; 2<em>j <\/em>&#8211; <em>h<br \/><\/em>Here,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <em>L <\/em>= Number of link, <em>j <\/em>= Number of <strong>lower joints,<\/strong><br \/><em>h <\/em>= Number of <strong>higher joints<\/strong><\/p>\r\n<ul>\r\n<li><strong>Grubler\u2019s Equation:<\/strong> For those mechanisms which have <strong>single<\/strong> degree of freedom and <strong>zero<\/strong> higher pair.<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">3<em>l <\/em>&#8211; 2j &#8211; 4 = 0<br \/>Here, <em>l <\/em>= Number of links, <em>j <\/em>= Number of binary joints<\/p>\r\n<ul>\r\n<li><strong>Degree of Freedom:<\/strong> <em>F <\/em>= 0 (Frame), <em>F <\/em>&lt; 0 (Redundant frame)<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\"><em>F<\/em> &gt; 0 (constrained\/unconstrained frame)<br \/>All mechanisms have <strong>minimum 4<\/strong> number of links.<\/p>\r\n\r\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/me\/theory-of-machines\/types-of-constrained-motion\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/me\/theory-of-machines\/four-bar-mechanism\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a> <br \/><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>\r\n<p>&nbsp;<\/p>\r\n","protected":false},"excerpt":{"rendered":"<p>The connection of a link with another imposes certain constraints on their relative motion thus, Degrees of freedom = 6<\/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":[292,295,293,294],"class_list":["post-878","post","type-post","status-publish","format-standard","hentry","category-theory-of-machines","category-me","tag-degree-of-freedom","tag-degree-of-freedom-of-space-mechanism","tag-grublers-equation","tag-kutzbacks-equation"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/878","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=878"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/878\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=878"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=878"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=878"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}