{"id":83,"date":"2014-01-01T19:57:15","date_gmt":"2014-01-01T19:57:15","guid":{"rendered":"http:\/\/www.eversholt.org.uk\/blog\/?p=83"},"modified":"2014-01-01T19:57:15","modified_gmt":"2014-01-01T19:57:15","slug":"rising-gate-geometry-2-wonkish","status":"publish","type":"post","link":"https:\/\/www.eversholt.org.uk\/blog\/rising-gate-geometry-2-wonkish\/","title":{"rendered":"Rising Gate Geometry 2 (Wonkish)"},"content":{"rendered":"

The previous post<\/a> demonstrated that gate hinges should lie in a vertical plane normal to the bisector of the open and closed gate positions. Here’s the plan view of the hinges in the open position: (click for a bigger version)<\/p>\n

\"gate5-1\"<\/a>

Plan view of hinges, gate fully open<\/p><\/div>\n

Triangles BFG<\/span> and BFK<\/span>, lying in the horizontal plane, are mirror images. So, length FK<\/span> must be ONS<\/sub><\/span><\/p>\n

\"gate7-1\"<\/a><\/p>\n

 <\/p>\n

The points labelled here are exactly the same as the ones in the previous diagram. Length FK<\/span> in the horizontal BFK<\/span> plane is ONS<\/sub><\/span>, because triangle BFK<\/span> and BFG<\/span> are mirror images. Length KM<\/span> is also ONS<\/sub><\/span>, because that’s how we made the spacer for the gate. Triangles AFK<\/span> and AMK<\/span> are therefore mirror images, because they are both right triangles, they share a hypotenuse and have side KM<\/span> equal to side KF<\/span>. So, angle KAF<\/span> is equal to angle KAM<\/span>. Let’s call this angle\u00a0\u03b8<\/span>. And, call distance AF<\/span>, the vertical distance between the hinges, OUD<\/sub><\/span>.<\/p>\n

tan\u00a0\u03b8<\/span> = KF<\/span> \/ AF<\/span> = ONS<\/sub><\/span> \/ OUD<\/sub><\/span><\/p>\n

\u03b8<\/span> = tan-1<\/sup> (ONS<\/sub><\/span> \/ OUD<\/sub><\/span>)<\/p>\n

Here’s how the gate looks when fully open.<\/p>\n

\"gate8\"<\/a><\/p>\n

 <\/p>\n

The gate rises up at an angle 2\u03b8<\/span> from the horizontal. The effective length of the gate, JK<\/span>, is OGATE<\/sub><\/span> + ONS<\/sub><\/span>.<\/p>\n

JN<\/span> \/ JK<\/span> = sin (2\u03b8<\/span>)<\/p>\n

Rise<\/span> = JN<\/span> = (OGATE<\/sub><\/span> + ONS<\/sub><\/span>) * sin (2\u03b8<\/span>)<\/p>\n

Rise<\/span> = (OGATE<\/sub><\/span> + ONS<\/sub><\/span>) * sin(2 *\u00a0tan-1\u00a0(ONS<\/sub><\/span>\u00a0\/ OUD<\/sub><\/span>)) \u00a0 \u2666 Equation 2
\nOddly, this doesn’t depend on\u00a0\u03b2. Is that right?<\/small><\/em><\/span><\/p>\n

So now I have some equations for the variables ONS<\/sub><\/span> and OWE<\/sub><\/span>. Here are the parameters fixed by the requirements of the gate design:<\/p>\n\n\n\n\n\n\n\n
Symbol<\/td>\nMeaning<\/td>\n<\/tr>\n
OUD<\/sub><\/span><\/td>\nVertical pitch between gate hinges<\/td>\n<\/tr>\n
OGATE<\/sub><\/span><\/td>\nWidth of gate when closed, measured from centre of top hinge pin to outside edge of gate<\/td>\n<\/tr>\n
Rise<\/span><\/td>\nRequired Rise of gate when fully open<\/td>\n<\/tr>\n
\u03b2<\/span><\/td>\nAngle between gate open and gate closed<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Here are the parameters I’m trying to determine:<\/p>\n\n\n\n\n\n
Symbol<\/td>\nMeaning<\/td>\n<\/tr>\n
ONS<\/sub><\/span><\/td>\nOffset of bottom hinge parallel to the gate<\/td>\n<\/tr>\n
OWE<\/sub><\/span><\/td>\nOutset of bottom hinge at right angles to the gate<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

And here are my equations:<\/p>\n

OWE<\/sub><\/span>\u00a0=\u00a0ONS<\/sub><\/span>\u00a0* tan (90\u00b0 –\u00a0\u03b2<\/span>\/2) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u2666 Equation 1, from previous post<\/a><\/p>\n

Rise<\/span> = (OGATE<\/sub><\/span> + ONS<\/sub><\/span>) * sin(2 *\u00a0tan-1\u00a0(ONS<\/sub><\/span>\u00a0\/ OUD<\/sub><\/span>)) \u00a0 \u00a0\u00a0\u2666 Equation 2<\/p>\n

I don’t actually know of a symbolic solution of (2) for ONS<\/sub><\/span>. Fortunately, that doesn’t stop me solving it numerically, which will come in a post real soon now.<\/p>\n

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

The previous post demonstrated that gate hinges should lie in a vertical plane normal to the bisector of the open and closed gate positions. Here’s the plan view of the hinges in the open position: (click for a bigger version) Triangles BFG and BFK, lying in the horizontal plane, are mirror images. So, length FK […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4,5],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","hentry","category-rising-gates","category-wonkish"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/posts\/83","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/comments?post=83"}],"version-history":[{"count":1,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/posts\/83\/revisions"}],"predecessor-version":[{"id":88,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/posts\/83\/revisions\/88"}],"wp:attachment":[{"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/media?parent=83"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/categories?post=83"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.eversholt.org.uk\/blog\/wp-json\/wp\/v2\/tags?post=83"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}