{"id":2117,"date":"2022-03-12T16:24:01","date_gmt":"2022-03-12T16:24:01","guid":{"rendered":"https:\/\/blog.metu.edu.tr\/eresmech\/?page_id=2117"},"modified":"2022-12-06T00:59:48","modified_gmt":"2022-12-06T00:59:48","slug":"3-4","status":"publish","type":"page","link":"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-4\/","title":{"rendered":"3-4"},"content":{"rendered":"<div id=\"pl-gb2117-6a1b8f8b00fc2\"  class=\"panel-layout\" ><div id=\"pg-gb2117-6a1b8f8b00fc2-0\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-gb2117-6a1b8f8b00fc2-0-0\"  class=\"panel-grid-cell\" ><div id=\"panel-gb2117-6a1b8f8b00fc2-0-0-0\" class=\"so-panel widget widget_sow-editor panel-first-child panel-last-child widgetopts-SO\" data-index=\"0\" ><div\n\t\t\t\n\t\t\tclass=\"so-widget-sow-editor so-widget-sow-editor-base\"\n\t\t\t\n\t\t>\n<div class=\"siteorigin-widget-tinymce textwidget\">\n\t<h1><b>3.4<\/b> Vector Loops of a Mechanism<\/h1>\n<p>The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints form closed polygons that we shall call a\u00a0<b>loop<\/b>. The motion analysis of mechanisms is based on expressing these loops mathematically.<\/p>\n<p>In kinematic analysis we shall assume that all the necessary dimensions of each link is given and link length dimensions (i.e. the distance between the joints or the angles) can be determined from the given dimensions using the geometry of the link.<\/p>\n<p>In Section 2.1.2, we have seen that it is sufficient to represent the position of each link (rigid body) by describing the position of any two points on that link. One way of selecting these two points on a link is to use the permanently coincident points. It is obvious that in such a procedure, the origin of a vector will be defined by the previous vector and thus the number of parameters to define the link positions will be decreased.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2119\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/image7.gif\" alt=\"\" width=\"689\" height=\"282\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\"><div class=\"su-image-carousel  su-image-carousel-has-spacing su-image-carousel-has-lightbox su-image-carousel-has-outline su-image-carousel-adaptive su-image-carousel-slides-style-default su-image-carousel-controls-style-dark su-image-carousel-align-center\" style=\"max-width:550px\" data-flickity-options='{\"groupCells\":true,\"cellSelector\":\".su-image-carousel-item\",\"adaptiveHeight\":true,\"cellAlign\":\"left\",\"prevNextButtons\":true,\"pageDots\":false,\"autoPlay\":false,\"imagesLoaded\":true,\"contain\":false,\"selectedAttraction\":1,\"friction\":1}' id=\"su_image_carousel_6a1b8f8b03bf6\"><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_1.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_1.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_2.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_2.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_3.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_3.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_4.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_4.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_5.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_5.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_6.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_6.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_7.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_7.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_8.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_8.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_9.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_9.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_10.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_10.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_11.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"550\" height=\"400\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/fourbar1e_11.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><\/div><script id=\"su_image_carousel_6a1b8f8b03bf6_script\">if(window.SUImageCarousel){setTimeout(function() {window.SUImageCarousel.initGallery(document.getElementById(\"su_image_carousel_6a1b8f8b03bf6\"))}, 0);}var su_image_carousel_6a1b8f8b03bf6_script=document.getElementById(\"su_image_carousel_6a1b8f8b03bf6_script\");if(su_image_carousel_6a1b8f8b03bf6_script){su_image_carousel_6a1b8f8b03bf6_script.parentNode.removeChild(su_image_carousel_6a1b8f8b03bf6_script);}<\/script><\/p>\n<p>Let us consider a four-bar mechanism as shown above as a simple example. In this mechanism A<sub>0<\/sub>\u00a0,is a permanently coincident point between links 1 and 2, A is peranently coincident point between links 2 and 3, B between 3 and 4 and B<sub>0<\/sub>\u00a0between 1 and 4. Let us disconnect joint B. In such a case we will obtain two open kinematic chains A<sub>0<\/sub>AB (links 2, 3) with two degrees of freedom and A<sub>0<\/sub>B<sub>0<\/sub>B (links 1, 4) with one degree of freedom (see figure). To determine the positions of the links we must have a reference frame. One obvious choice is to select the fixed pivots A<sub>0<\/sub>, B<sub>0<\/sub>\u00a0as one of the co-ordinate axes and select A<sub>0<\/sub>\u00a0or B<sub>0<\/sub> as the origin. Next,in order to define the position of link 2, we must define angle \u03b8<sub>12<\/sub>, which is related with the degree of freedom of the joint between links 1 and 2. To determine the position of link 3, since the location of the permanently coincident point A between 2 and 3 can be determined when \u03b8<sub>12<\/sub> defined, we must now define \u03b8<sub>13<\/sub> , which is related to the freedom of the joint between links 2 and 3. Similarly \u03b8<sub>14<\/sub> must be defined to determine the position of link 4. Hence we need 3 parameters (\u03b8<sub>12<\/sub>, \u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>\u00a0) which are all related to the joint freedoms for the open kinematic chains obtained when we disconnect a joint to eliminate a loop.<\/p>\n<p>Each link can be defined by a vector fixed on that link, let us select the permanently coincident points between the links as the tips of these vectors and define vector\u00a0<b>A<sub>0<\/sub>A<\/b>\u00a0(for link 2),\u00a0<b>AB\u00a0<\/b>(for link 3),<b>\u00a0B<sub>0<\/sub>B\u00a0<\/b>(for link 4) and\u00a0<b>A<sub>0<\/sub>B<sub>0<\/sub>\u00a0<\/b>(for link 1). Except\u00a0<b>A<sub>0<\/sub>B<sub>0<\/sub>\u00a0<\/b>, the other three vectors will be a function of time (since the distances between the two points on the same link are fixed, the magnitudes will remain constant but the directions of these vectors will change in time). Since the mechanism contains revolute joints only, The magnitude of the vectors are constant link lengths (a<sub>2<\/sub> = |A<sub>0<\/sub>A|, a<sub>3<\/sub> = |AB|, a<sub>1<\/sub> = |A<sub>0<\/sub>B<sub>0<\/sub>| and a<sub>4<\/sub> = |B<sub>0<\/sub>B|). The angular orientation of these vectors will be rotation variables (\u03b8<sub>12<\/sub>, \u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>\u00a0). When the joint at B is disconnected, B<sub>3<\/sub>\u00a0and B<sub>4<\/sub>\u00a0may not be coincident. For the open kinematic chain, the position of point B may be defined in two different forms as:<\/p>\n<p style=\"padding-left: 40px;text-align: center\"><strong>A<sub>0<\/sub>A<\/strong> + <strong>AB <\/strong>= <strong>A<sub>0<\/sub>B<sub>3<\/sub><\/strong> (1, 2, 3 open loop)<\/p>\n<p style=\"padding-left: 40px;text-align: center\"><strong>A<sub>0<\/sub>B<sub>0<\/sub><\/strong> + <strong>B<sub>0<\/sub>B<\/strong> = <strong>A<sub>0<\/sub>B<sub>4<\/sub><\/strong> (1, 4 open loop)<\/p>\n<p>However, at every instant the revolute joint between links 3 and 4 must exist and point B must remain a permanently coincident point for different values of the position variables if the system we are considering is a mechanism. Therefore the vector\u00a0<b>A<sub>0<\/sub>B<sub>3<\/sub><\/b>\u00a0and\u00a0<b>A<sub>0<\/sub>B<sub>4<\/sub><\/b> obtained from the two equations using the two open kinematic chains must be equal and this results with the vector equation:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>0<\/sub>A<\/strong> + <strong>AB<\/strong> = <strong>A<sub>0<\/sub>B<sub>0<\/sub><\/strong> +\u00a0<strong>B<sub>0<\/sub>B<\/strong><\/p>\n<p>This vector equation must be valid for all positions due to the permanetly coincident points. If this vector equation can nott be satisfied for a given input angle, then that position cannot exist (mechanism cannot be assembled at that position).<\/p>\n<p>In a four-bar mechanism there is a single loop formed and the vector equation describes the closure of this loop mathematically.The equation(s) that describes the closure of the loop(s) formed in the mechanism are known as\u00a0<b>loop closure equation(s)<\/b>. The variables in the loop closure equations are always related by the joint freedoms and we can solve for two position variables from any loop equation. In plane the vector equation will correspond to two scalar equations. In the four-bar example there are three variables (\u03b8<sub>12<\/sub>, \u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>) which we shall call &#8220;<b>position variables<\/b>&#8220;. If one of the position variable (say \u03b8<sub>12<\/sub>), the other position variables (\u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>) can be solved from this vector loop equation. The number of independent parameters that are required will always be equal to the degree-of-freedom of the mechanism. The relation between the position variables is a nonlinear, trigonometric relation.<\/p>\n<p>One simple and concise form of writing the vector loop equations is to use complex numbers. for example, If the length of the vector\u00a0<b>A<sub>0<\/sub>A\u00a0<\/b>is a<sub>2<\/sub> and if the vector makes an angle \u03b8<sub>12<\/sub>:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>0<\/sub>A<\/strong> = a<sub>2<\/sub>cos\u03b8<sub>12<\/sub>\u00a0+ ia<sub>2<\/sub>sin\u03b8<sub>12<\/sub><\/p>\n<p>or, using Euler&#8217;s equation:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>0<\/sub>A<\/strong>\u00a0= a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup><\/p>\n<p>In a similar fashion if the link lengths are denoted as a<sub>i<\/sub> olarak (a<sub>1<\/sub>= |A<sub>0<\/sub>B<sub>0<\/sub>|, a<sub>2<\/sub>=|A<sub>0<\/sub>A|, etc.) the vector loop equation in complex numbers can be written as:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> + a<sub>3<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup>\u00a0= a<sub>1<\/sub>\u00a0+ a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup><\/p>\n<p>If required, the equation can be written in cartesian coordinates as:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>cos\u03b8<sub>12<\/sub><strong>i<\/strong> + a<sub>2<\/sub>sin\u03b8<sub>12<\/sub><strong>j<\/strong> + a<sub>3<\/sub>cos\u03b8<sub>13<\/sub><strong>i<\/strong> + a<sub>3<\/sub>sin\u03b8<sub>13<\/sub><strong>j<\/strong> = a<sub>1<\/sub><strong>i <\/strong>+ a<sub>4<\/sub>cos\u03b8<sub>14<\/sub><strong>i<\/strong> + a<sub>4<\/sub>sin\u03b8<sub>14<\/sub><strong>j<\/strong><\/p>\n<p>x and y components can be equated separetely two yield two scalar equations in the form:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>cos\u03b8<sub>12<\/sub>\u00a0+ a<sub>3<\/sub>cos\u03b8<sub>13<\/sub>\u00a0= a<sub>1<\/sub>\u00a0+ a<sub>4<\/sub>cos\u03b8<sub>14<\/sub><br \/>\na<sub>2<\/sub>sin\u03b8<sub>12<\/sub>\u00a0+ a<sub>3<\/sub>sin\u03b8<sub>13<\/sub>\u00a0= a<sub>4<\/sub>sin\u03b8<sub>14<\/sub><\/p>\n<p>In case of a four-bar, the vectors in the loop closure equation have fixed magnitudes. However, the angular inclinations of the three vectors representing the moving links will change. Hence, there are three position variables (\u03b8<sub>12<\/sub>, \u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>). If one of these variables is defined, the remaining two variables can be solved from the vector equation. If we refer to the definition of the degree-of-freedom of a mechanism, the variable that must be defined is the input variable and for a constrained motion the number of input variables must be equal to the degree-of-freedom of the joints involved. In case of a four-bar, since all the connections are revolute joints, the variables are all rotation variables. In case of a prismatic joint, the variable will be the magnitude of a vector or a vector component. Consider a slider-crank mechanism as shown in Fig. A. Let us disconnect the revolute joint at B (Fig. B). In order to determine the positions of links 2 and 3 we must define \u03b8<sub>12<\/sub>\u00a0and \u03b8<sub>13<\/sub>. To locate the position of link 4 its displacement along the slider axis must be known and the position variable s<sub>14<\/sub>\u00a0must be defined. The resulting loop closure equation is:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>o<\/sub>A<\/strong>\u00a0+\u00a0<strong>AB<\/strong>\u00a0=\u00a0<strong>A<sub>o<\/sub>B<\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2120\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/image8.gif\" alt=\"\" width=\"738\" height=\"250\" \/><\/p>\n<p>Again there are 3 variables (\u03b8<sub>12<\/sub>, \u03b8<sub>13<\/sub> and \u03b8<sub>14<\/sub>)\u00a0one of which must be specified as the input. In this case the vectors\u00a0<b>A<sub>o<\/sub>A<\/b>\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">and\u00a0<\/span><b>AB<\/b>\u00a0have fixed magnitudes and varying directions. The vector\u00a0<b>A<sub>o<\/sub>B<\/b>\u00a0has a fixed y component (length c) and a changing x component (s<sub>14<\/sub>). Depending on the applications either \u03b8<sub>12<\/sub>\u00a0(i.e. in pumps) or s<sub>14<\/sub> (i.e. internal combustion engines) is the input. In complex numbers the vector loop equation will be:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> + a<sub>3<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup> = s<sub>14<\/sub> + ic<\/p>\n<p>The vectors defined and the variables used in the loop closure equations are not unique. For example, for the slider crank mechanism, rather than disconnecting the revolute joint at B, one can as well disconnect the revolute joint at A between links 2 and 3 (Fig\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">C<\/span>). We must now define the angle \u03b8<sub>13<\/sub>\u2032 = \u2220xBA instead of the angle \u03b8<sub>13<\/sub>\u00a0o determine the position of link 3. Note that the angles. \u03b8<sub>13<\/sub>\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">and \u03b8<sub>13<\/sub>\u2032<\/span>\u00a0differ by a constant angle (In this case by 180<sup>o<\/sup>). The resulting loop equation is:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>o<\/sub>A<\/strong> = <strong>A<sub>o<\/sub>B<\/strong> +\u00a0<strong>BA<\/strong><\/p>\n<p>or in complex numbers:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> = s<sub>14<\/sub> + ic + a<sub>3<\/sub>e<sup>i\u03b8<sub>13<\/sub>\u2032<\/sup><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-831\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image9.gif\" alt=\"\" width=\"434\" height=\"231\" \/><\/p>\n<p><strong>Example:<\/strong> <span style=\"text-decoration: underline\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-4\/3-4_ornek\/\"><span style=\"color: #0000ff\"><b>The loop closure equations of a six link mechanism<\/b><\/span><\/a><\/span><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-832\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image10.gif\" alt=\"\" width=\"447\" height=\"293\" \/><\/p>\n<p>Referring to the four-bar mechanism above, one can write a vector equation in the form:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>o<\/sub>A<\/strong> +\u00a0<strong>AB\u00a0<\/strong>=\u00a0<strong>A<sub>o<\/sub>B<\/strong><\/p>\n<p>Considering the vector\u00a0<b>A<sub>o<\/sub>B<\/b>\u00a0the magnitude and direction of this vector are variables (or its x and y components) and these variables are not related with the joint freedoms.we can solve for the vector\u00a0<b>A<sub>o<\/sub>B<\/b>\u00a0provided that the magnitudes and the directions of the other two vectors are known. Such an equation will not help us for the solution of position variables. Although it is a valid vector equation,\u00a0<span style=\"color: #ff0000\"><em><strong>it is not a loop closure equation<\/strong><\/em><\/span>. One can identify such loops by noting the variables involved are not related with the joint freedoms and these equations are not obtained by disconnecting the joints.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-833\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image11.gif\" alt=\"\" width=\"323\" height=\"264\" \/><\/p>\n<p>A similar argument can also be made for the vector equations:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>o<\/sub>A<\/strong>\u00a0+ <strong>AC<\/strong> = <strong>A<sub>o<\/sub>C<\/strong>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(i)<\/p>\n<p style=\"text-align: center\"><strong>AB<\/strong>\u00a0+\u00a0<strong>BC<\/strong>\u00a0=\u00a0<strong>AC\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong>(ii)<\/p>\n<p style=\"text-align: left\">In the later case, all three vectors are on the same link and they have a fixed orientation with respect to each other. If the origin and the angular orientation of one of the vectors is known, due to rigidity the orientation of the other vectors will be known.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2121\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/image12.gif\" alt=\"\" width=\"558\" height=\"249\" \/><\/p>\n<p>In certain other cases we may have to use instantaneously coincident points or other points on the links as the tips of the vectors, as shown in the figure. The variable involved in such a case is the relative displacement of one link with respect to another link. The vector <b>B<sub>o<\/sub>A<\/b>\u00a0can be separated into two components:\u00a0<b>B<sub>o<\/sub>C<\/b>\u00a0and\u00a0<b>CA<\/b>\u00a0so that one component is of constant magnitude and the magnitude of the other vector is related to the displacement of the prismatic joint between links 3 and 4. The loop closure equation can than be written in the form:<\/p>\n<p style=\"text-align: center\"><strong>A<sub>o<\/sub>A<\/strong>\u00a0=\u00a0<strong>A<sub>o<\/sub>B<sub>o<\/sub><\/strong>\u00a0+\u00a0<strong>B<sub>o<\/sub>C<\/strong>\u00a0+\u00a0<strong>CA<\/strong><\/p>\n<p>or in complex numbers:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> = a<sub>1<\/sub> + a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup> + s<sub>43<\/sub>e<sup>i(\u03b8<sub>14 <\/sub>+<sub>\u00a0<\/sub>\u03b1<sub>4<\/sub>)<\/sup><\/p>\n<p>Points A<sub>2<\/sub>\u00a0ve A<sub>3<\/sub>\u00a0are permanently coincident points. Point A<sub>4<\/sub>\u00a0is instantaneously coincident with point A<sub>3<\/sub> when the mechanism is moved from this position the two points will be displaced by a distance \u0394s along the slider axis relative to each other. Points A<sub>2<\/sub>\u00a0and A<sub>3<\/sub> will be two other different points that will be coincident with (A\u2032) (Fig. C).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2122\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/image13.gif\" alt=\"\" width=\"453\" height=\"294\" \/><\/p>\n<p>The vector\u00a0<b>CA<\/b>\u00a0changes both its magnitude and direction. However its orientation with respect to the vector\u00a0<b>B<sub>0<\/sub>C<\/b>\u00a0will be fixed, and no new variable is needed. If the orientation of the vector\u00a0<b>B<sub>0<\/sub>C<\/b> is whown by the variable \u03b8<sub>14<\/sub>\u00a0measured from the positive x-axis of our reference, the orientation of the vector\u00a0<b>CA<\/b> with respect to positive x axis is \u03b8<sub>14<\/sub> + \u03b1<sub>4 <\/sub>and angle \u03b1<sub>4<\/sub>\u00a0is a constant angle measured on link 4 between two lines B<sub>o<\/sub>C and CP (point P is any point on the slider axis on link 4). The position variables in the loop equation will be \u03b8<sub>12<\/sub>, \u03b8<sub>14<\/sub>\u00a0and s<sub>43<\/sub>.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-836\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image14.gif\" alt=\"\" width=\"579\" height=\"303\" \/><\/p>\n<p>Note that the same vector loop equation can be derived for the swinging block mechanism shown above. Therefore, the inverted slider crank mechanism is the same as the swinging block mechanism although their construction is different. Although it is a different construction, if the link dimensions (a<sub>2<\/sub>, a<sub>4<\/sub>, \u03b1<sub>4<\/sub>) are the same, the motion of the two mechanisms will be the same.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2123\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/image15.gif\" alt=\"\" width=\"542\" height=\"294\" \/><\/p>\n<p>When writing the vector loop equations one must be sure that the equations are valid for every position of the mechanism. The mechanism may be at a special position such that one or more of the links are collinear as shown above\u00a0(Although links 2 and 1 are collinear, they will have different orientations at some other instant). In such a case you may redraw the mechanism slightly offset from the critical position or show the variable angle \u03b8<sub>12<\/sub>, as shown. In some cases if the constant link angles between two vectors are of a certain simple value (such as 90\u00b0), then it is advisable to simplifiy the equations accordingly. For example if the angle \u03b1<sub>4<\/sub>\u00a0of the inverted slider crank mechanism is a right angle, The loop closure equation must be written as:<\/p>\n<p style=\"text-align: center\">a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> = a<sub>1<\/sub> + a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup> + is<sub>43<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup><\/p>\n<p>The solution to the loop equations may not exist for every value of the independent parameter. This will mean that for that particular link lengths the mechanism cannot be assembled at the requested position.<\/p>\n<p><strong><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-4\/3-4_bds\/\">How will you determine the number of independent loops from the number of joints and links?<\/a><\/strong><\/p>\n<p>In planar mechanisms we can write vector loop equations for each loop of the mechanism. This corresponds to L (L= number of independent loops) vector equations or 2L scalar equations, if we equate the x and y components of vectors. The number of parameters involved in these equations will be 2L+ F, where F is the degree of freedom of the mechanism. If we now define F number of variables (independent variables or input parameters (variables)), then theoretically, we must be able to solve for the other variables (dependent position parameteres). We can change the input variable within a given range in certain increments and obtain the values for the dependent variables. For example if the input variable corresponds to the angle that defines the angular position of an input crank, we change this angle from 0 to 360<sup>o<\/sup>. If the input is the movement of a piston inside a cylinder, then we change this length of the piston starting from the closed position to the most extended position (the difference is the stroke of the piston).<\/p>\n<p>When we determine the values of all the position variables corresponding to a certain input variable, then we can determine the position of any point on any link of the mechanism.<\/p>\n<p><span style=\"color: #cc0000\"><b><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-19\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/important.gif\" alt=\"\" width=\"28\" height=\"27\" \/><\/b><strong>One need not draw the mechanism with dismantled joints to write the loop equations. After some practice one can conceptually disconect the joints, identify the loops and the write the necessary loop equations. Initially of course, as a visual aid, we have shown the joints are shown disconnected.<\/strong><\/span><\/p>\n<p>In the following examples the necessary loop equations for some mechanisms are written. The loop equations and the variables defined are not unique. You are to determine the joint that is disconnected when writing the given loop equation.<\/p>\n<p>In recent years different package programs are available for the analysis of mechanisms. When using these programs, you must input these loops by telling the program which link is connected to which link by what kind of a joint (i.e. what kind of freedom is permitted by that joint). If you are using mathematical packages such as matlab or mathcad, you must type these equations in one form or another. A slight mistake in the loop equations results with erroneous results. Please keep in mind that these equations define the mathematical model of an existing mechanism. This mathematical model can be solved in different ways as we shall see in the coming sections.<\/p>\n<p><span style=\"color: #cc0000\"><strong>Example I:<\/strong><\/span><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-838\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image16.gif\" alt=\"\" width=\"714\" height=\"478\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\">(a<sub>2<\/sub> = |A<sub>0<\/sub>A|, a<sub>3<\/sub> = |AB|, a<sub>4<\/sub> = |BC|)<\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>A<sub>0<\/sub>A<\/strong>\u00a0+ <strong>AB<\/strong> +\u00a0<strong>BC<\/strong>\u00a0= <strong>A<sub>0<\/sub>C<\/strong>\u00a0 \u21d2\u00a0 a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> + a<sub>3<\/sub>e<sup>i(\u03b8<sub>13<\/sub> \u2212 \u03b1<sub>3<\/sub>)<\/sup>\u00a0+ a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup> = s<sub>15<\/sub> + ia<sub>1<\/sub><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>A<sub>0<\/sub>A<\/strong>\u00a0+\u00a0<strong>AD <\/strong>= <strong>A<sub>0<\/sub>D<sub>0<\/sub><\/strong>\u00a0+ <strong>D<sub>0<\/sub>D<\/strong> \u21d2\u00a0 a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> + s<sub>36<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup> = c<sub>1<\/sub> + ib<sub>1<\/sub> \u2212 ia<sub>6<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup><\/p>\n<p><span style=\"color: #cc0000\"><strong>Example II:<\/strong><\/span><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1483\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17.png\" alt=\"\" width=\"306\" height=\"534\" srcset=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17.png 204w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17-172x300.png 172w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17-100x175.png 100w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17-150x262.png 150w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image17-200x349.png 200w\" sizes=\"auto, (max-width: 306px) 100vw, 306px\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\">(c<sub>1<\/sub> = |PB<sub>0<\/sub>|, a<sub>2<\/sub> = |A<sub>0<\/sub>A|, a<sub>4<\/sub> = |BC|, a<sub>5<\/sub> = |B<sub>0<\/sub>B|)<\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>A<sub>0<\/sub>A<\/strong> = <strong>A<sub>0<\/sub>B<sub>0<\/sub><\/strong>\u00a0+<strong>B<sub>0<\/sub>B <\/strong>+ <strong>BA<\/strong>\u00a0 \u21d2\u00a0 a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> = c<sub>1<\/sub> \u2212 ia<sub>1<\/sub> + a<sub>5<\/sub>e<sup>i\u03b8<sub>15<\/sub><\/sup> + s<sub>43<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>A<sub>0<\/sub>C <\/strong>=\u00a0<strong>A<sub>0<\/sub>B<sub>0<\/sub><\/strong> +\u00a0<strong>B<sub>0<\/sub>B<\/strong> +\u00a0<strong>BC<\/strong>\u00a0 \u21d2\u00a0 s<sub>16<\/sub> + ib<sub>1<\/sub>\u00a0= c<sub>1<\/sub> \u2212 ia<sub>1<\/sub> + a<sub>5<\/sub>e<sup>i\u03b8<sub>15<\/sub><\/sup> + a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup><\/p>\n<p style=\"text-align: left\" align=\"center\"><span style=\"color: #cc0000\"><strong>Example III:<\/strong><\/span><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-840\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/image18.gif\" alt=\"\" width=\"661\" height=\"526\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>B<sub>0<\/sub>B <\/strong>+ <strong>BC <\/strong>+ <strong>CD <\/strong>= <strong>B<sub>0<\/sub>A<sub>0<\/sub><\/strong> +\u00a0<strong>A<sub>0<\/sub>A <\/strong>+ <strong>AD<\/strong>\u00a0 \u21d2\u00a0 a<sub>3<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup> + a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup> + a<sub>6<\/sub>e<sup>i\u03b8<sub>16<\/sub><\/sup> = a<sub>1<\/sub> + ib<sub>1<\/sub> + a<sub>2<\/sub>e<sup>i\u03b8<sub>12<\/sub><\/sup> + a<sub>7<\/sub>e<sup>i\u03b8<sub>17<\/sub><\/sup><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>B<sub>0<\/sub>B <\/strong>+ <strong>BC <\/strong>= <strong>B<sub>0<\/sub>C<sub>0<\/sub><\/strong> +\u00a0<strong>C<sub>0<\/sub>C<\/strong>\u00a0 \u21d2\u00a0 a<sub>3<\/sub>e<sup>i\u03b8<sub>13<\/sub><\/sup> + a<sub>4<\/sub>e<sup>i\u03b8<sub>14<\/sub><\/sup> = \u2212d<sub>1<\/sub> + ic<sub>1<\/sub> + a<sub>5<\/sub>e<sup>i\u03b8<sub>15<\/sub><\/sup><\/p>\n<p>Due to the gear pair\u00a0: r<sub>3<\/sub>\u03b8<sub>13<\/sub> = \u2212r<sub>2<\/sub>(\u03b8<sub>12<\/sub> \u2212 \u03b1<sub>2<\/sub>) (\u03b8<sub>12<\/sub>\u00a0= \u03b1<sub>2<\/sub>\u00a0when \u03b8<sub>13<\/sub> = 0)<\/p>\n<\/div>\n<\/div><\/div><\/div><\/div><\/div>\n\n\n<p>  <a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-3\/\" data-type=\"page\"><img loading=\"lazy\" decoding=\"async\" width=\"38\" height=\"38\" class=\"wp-image-16\" style=\"width: 38px\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/back_button.gif\" alt=\"\"><\/a><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/\" data-type=\"page\" data-id=\"52\"><img loading=\"lazy\" decoding=\"async\" width=\"38\" height=\"38\" class=\"wp-image-17\" style=\"width: 38px\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/contents_button.gif\" alt=\"\"><\/a><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/\" data-type=\"page\" data-id=\"47\"><img loading=\"lazy\" decoding=\"async\" width=\"38\" height=\"38\" class=\"wp-image-18\" style=\"width: 38px\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/home_button.gif\" alt=\"\"><\/a><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-5\/\" data-type=\"page\" data-id=\"92\"><img loading=\"lazy\" decoding=\"async\" width=\"38\" height=\"38\" class=\"wp-image-20\" style=\"width: 38px\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/next_button.gif\" alt=\"\"><\/a><img loading=\"lazy\" decoding=\"async\" width=\"119\" height=\"40\" class=\"wp-image-15\" style=\"width: 119px\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/04\/ceres.gif\" alt=\"\">       <\/p>\n","protected":false},"excerpt":{"rendered":"<p>3.4 Vector Loops of a Mechanism The main difference between freely moving bodies and the moving links in a mechanism is that they have a constrained motion due to the joints in between the links. The links connected by joints &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch3\/3-4\/\"> <span class=\"screen-reader-text\">3-4<\/span> Devam\u0131n\u0131 Oku &raquo;<\/a><\/p>\n","protected":false},"author":7747,"featured_media":0,"parent":1950,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"full-width-page.php","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-2117","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/2117","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/users\/7747"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/comments?post=2117"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/2117\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/1950"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/media?parent=2117"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}