{"id":2165,"date":"2022-03-13T13:05:20","date_gmt":"2022-03-13T13:05:20","guid":{"rendered":"https:\/\/blog.metu.edu.tr\/eresmech\/?page_id=2165"},"modified":"2022-09-02T16:17:36","modified_gmt":"2022-09-02T16:17:36","slug":"4-1-2","status":"publish","type":"page","link":"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch4\/4-1-2\/","title":{"rendered":"4-1-2"},"content":{"rendered":"<div id=\"pl-gb2165-69d60bed9afdc\"  class=\"panel-layout\" ><div id=\"pg-gb2165-69d60bed9afdc-0\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-gb2165-69d60bed9afdc-0-0\"  class=\"panel-grid-cell\" ><div id=\"panel-gb2165-69d60bed9afdc-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>4.1<\/b> VELOCITY AND ACCELERATION ANALYSIS &#8211; 2<\/h1>\n<p><strong><span style=\"color: #cc0000\"><u>Ge<span style=\"font-family: Arial, Helvetica, sans-serif\">neral Plane Motion<\/span>:<\/u><\/span><\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2168 size-full aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/img222-1.gif\" alt=\"\" width=\"314\" height=\"141\" \/><\/p>\n<p align=\"left\">General plane motion is the plane motion which is neither\u00a0 translation or rotation about a fixed axis but it can be analysed by the superposition of these motions using the\u00a0<strong><em><span style=\"color: #ff0000\">relative motion<\/span><\/em><\/strong>\u00a0concept.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2169 size-full aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/img222-2.gif\" alt=\"\" width=\"350\" height=\"156\" \/><\/p>\n<p>Consider a plane represented by two points AB. Finite displacement of the rigid body from position 1 to position 2 at a time interval t can be regarded as the sum of the translation from AB to A\u2019B\u201d and a fixed axis of rotation about an axis perpendicular to the plane and passing through A\u2019. One can also achieve the same final motion by translating from AB to A\u201dB\u2019 and then rotating about an axis passing through B\u2019, or one can first perform a rotation about an axis perpendicular to the plane and passing through any point on the rigid body and then perform the translation. The actual paths of points will of course not correspond to the paths described by these different combinations of rotations and translations, but the final position can be reached and when the two positions are infinitesimally close these motions will coincide with the instantaneous paths of the points on the rigid body.<\/p>\n<p>When performing translation, every point on the rigid body moves with the motion of the point selected and the velocity and acceleration of every point is the same for this motion. In case of rotation the rigid body is rotated about an axis passing through the particular point selected for translation. Now the rigid body is in a rotation. Since the point selected is not fixed, it is a <strong><em><span style=\"color: #ff0000\">relative motion<\/span><\/em><\/strong><em>.<\/em><\/p>\n<p style=\"text-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_69d60bed9d090\"><div class=\"su-image-carousel-item\"><div class=\"su-image-carousel-item-content\"><a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/genmotione_1.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"570\" height=\"398\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/genmotione_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\/genmoitone_2.gif\" target=\"_blank\" rel=\"noopener noreferrer\" data-caption=\"\"><img loading=\"lazy\" decoding=\"async\" width=\"570\" height=\"398\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/09\/genmoitone_2.gif\" class=\"\" alt=\"\" \/><\/a><\/div><\/div><\/div><script id=\"su_image_carousel_69d60bed9d090_script\">if(window.SUImageCarousel){setTimeout(function() {window.SUImageCarousel.initGallery(document.getElementById(\"su_image_carousel_69d60bed9d090\"))}, 0);}var su_image_carousel_69d60bed9d090_script=document.getElementById(\"su_image_carousel_69d60bed9d090_script\");if(su_image_carousel_69d60bed9d090_script){su_image_carousel_69d60bed9d090_script.parentNode.removeChild(su_image_carousel_69d60bed9d090_script);}<\/script><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>General Plane motion seperated to translation and rotation<\/strong><\/p>\n<p style=\"text-align: left\" align=\"center\">Let us analyse this relative motion by attaching a moving reference frame x-y at point A,\u00a0 that translates relative to the fixed frame X-Y .<\/p>\n<p>The motion is to be considered of two parts. First, the body translates from AB to A\u2019B\u201d with displacement\u00a0<strong><span style=\"font-family: Symbol\">D<\/span>r<\/strong><sub> A\u00a0<\/sub>which is the displacement of point A. This is similar to the rigid body translation discussed in part (a). Next the body rotates by an angle\u00a0<span style=\"font-family: Symbol\">Df<\/span>\u00a0about A\u2019. Since the distance of any other point, such as B, remains constant, the motion of B from B\u201d to B\u2019 will be on an arc with centre A\u2019 and will be displaced by\u00a0<strong><span style=\"font-family: Symbol\">D<\/span>r<\/strong><sub>\u00a0B\/A<\/sub>. This displacement is the motion of point B relative to point A.<\/p>\n<p style=\"text-align: center\">\u0394<strong>r<\/strong><sub>B<\/sub> = \u0394<strong>r<\/strong><sub>A<\/sub> +\u0394<strong>r<\/strong><sub>B\/A<\/sub><\/p>\n<p>Since the motion from B\u201d to B\u2019 is a rotation,\u00a0<strong><span style=\"font-family: Symbol\">D<\/span>r<\/strong><sub> B\/A\u00a0<\/sub>= |BA|<span style=\"font-family: Symbol\">Df<\/span>. Note that this relative motion is a function of absolute instantaneous angular motion\u00a0<span style=\"font-family: Symbol\">Df<\/span> of the rigid body. Therefore selection of point A will not effect this angular rotation, whereas the translation will depend on the point A that is selected. If one changes the order of rotation and translation the same equation will result. Dividing the above equation by the corresponding time interval,\u00a0<span style=\"font-family: Symbol\">D<\/span>t and taking the limit as\u00a0<span style=\"font-family: Symbol\">D<\/span>t tends to zero we obtain:<\/p>\n<p style=\"text-align: center\"><strong>V<sub>B<\/sub>\u00a0=\u00a0V<sub>A<\/sub>+\u00a0V<sub>B\/A<\/sub><\/strong><\/p>\n<p>where:<\/p>\n<p style=\"text-align: center\"><strong>V<sub>B\/A<\/sub> = \u03c9 \u00d7 r<sub>B\/A<\/sub><\/strong><\/p>\n<p>and<\/p>\n<p style=\"text-align: center\"><strong>\u00a0r<sub>\u00a0B\/A\u00a0<\/sub>= AB = r<sub>\u00a0B\u00a0<\/sub>&#8211; r<sub>\u00a0A<\/sub><\/strong><\/p>\n<p style=\"text-align: left\"><strong>V<sub>\u00a0B\/A<\/sub><\/strong>is a\u00a0<strong>relative velocity<\/strong>\u00a0between two points B and A belonging to the\u00a0<strong><em><u><span style=\"color: #cc0000\">same rigid body<\/span><\/u><\/em><\/strong><u>.<\/u>\u00a0<strong>V<sub>\u00a0B<\/sub><\/strong>\u00a0and\u00a0V<sub>\u00a0A<\/sub>are absolute <strong>velocities <\/strong>of points A and B on a rigid body in a general plane motion.<u><\/u><\/p>\n<p style=\"text-align: left\">The same result can be obtained using complex numbers by writing the position of point B relative to A in complex numbers:<\/p>\n<p style=\"text-align: center\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<strong>r<sub>\u00a0B<\/sub><\/strong>=\u00a0<strong>r<sub>\u00a0A<\/sub><\/strong>+be<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span><\/sup><\/p>\n<p style=\"text-align: left\">where b=\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">|<\/span>AB| and\u00a0\u00a0<span style=\"font-family: Symbol\">f<\/span>\u00a0is the angle between the line AB and the positive real axis. b<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span>\u00a0<\/sup>term shows the position of point B with respect to A and b length is constant only if both A and B are on the same rigid body. Since we are to be concerned with the relative motion, let us assume that the velocity and acceleration of point A is known. (<strong>V<sub>\u00a0A<\/sub><\/strong>\u00a0 and\u00a0<strong>a<sub>\u00a0A<\/sub><\/strong>are known). Differentiating the above equation with respect to time:<\/p>\n<p style=\"text-align: center\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<strong>V<sub>\u00a0B\u00a0<\/sub><\/strong>=\u00a0<strong>V<sub>\u00a0A\u00a0<\/sub><\/strong>+ib<span style=\"font-family: Symbol\">w<\/span>e<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span><\/sup><\/p>\n<p style=\"text-align: left\">where\u00a0<span style=\"font-family: Symbol\">w<\/span>=d<span style=\"font-family: Symbol\">f<\/span>\/dt. The second term has the magnitude b<span style=\"font-family: Symbol\">w<\/span>\u00a0and is in the direction ie<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span>\u00a0<\/sup>\u00a0 , which is a unit vector perpendicular to AB rotated in the sense of\u00a0<span style=\"font-family: Symbol\">w<\/span>. This is the relative velocity of point B with respect to A (Fig.<span style=\"font-family: Arial, Helvetica, sans-serif\">4.60<\/span>). Hence:<\/p>\n<p style=\"text-align: center\"><strong>V<sub>\u00a0B\/A\u00a0<\/sub><\/strong>= ib<span style=\"font-family: Symbol\">w<\/span>e<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span><\/sup><\/p>\n<p style=\"text-align: left\">and<\/p>\n<p style=\"text-align: center\"><strong>V<sub>\u00a0B\u00a0<\/sub><\/strong>=\u00a0<strong>V<sub>\u00a0A\u00a0<\/sub><\/strong>+<strong>\u00a0V<sub>\u00a0B\/A<\/sub><\/strong><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1098 aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/img222-6.gif\" alt=\"\" width=\"725\" height=\"318\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong><span style=\"font-family: Arial, Helvetica, sans-serif\">Velocity of point B in terms of velocity of point A<\/span><\/strong><\/p>\n<p align=\"center\">Differentiation of the velocity equation with respect to time yields:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub>\u00a0=\u00a0<strong>a<\/strong><sub>A<\/sub> + ib\u03b1e<sup>i\u03d5<\/sup> \u2212 b\u03c9<sup>2<\/sup>e<sup>i\u03d5<\/sup><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1099 aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/img222-7.gif\" alt=\"\" width=\"677\" height=\"226\" \/><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong><span style=\"color: #000000\">Acceleration of point B in terms of acceleration of point A<\/span><\/strong><\/p>\n<p>The first relative acceleration component has the magnitude b<span style=\"font-family: Symbol\">a<\/span>\u00a0and is in the direction ie<sup>\u00a0i<span style=\"font-family: Symbol\">f<\/span>\u00a0<\/sup>(Fig.<span style=\"font-family: Arial, Helvetica, sans-serif\">4.61<\/span>), which is perpendicular to the line AB and rotated in the sense of the angular acceleration\u00a0<span style=\"font-family: Symbol\">a<\/span>. This is relative acceleration<span style=\"font-family: Arial, Helvetica, sans-serif\">\u00a0<\/span>component is\u00a0<strong><em><span style=\"color: #cc0000\">tangential relative acceleration component<\/span><\/em><\/strong>\u00a0and is denoted by\u00a0<strong>a<sup>\u00a0t<\/sup><sub>\u00a0B\/A<\/sub><\/strong>\u00a0. The second relative acceleration component has the magnitude b<span style=\"font-family: Symbol\">w<\/span><sup>2<\/sup>\u00a0and is in the direction -e<sup>i<span style=\"font-family: symbol\">f<\/span><\/sup>\u00a0which is along the line AB towards the centre of rotation A. This direction is normal to the relative path of B with respect to A. Therefore it is called\u00a0<strong><em><span style=\"color: #cc0000\">normal relative acceleration component<\/span><\/em><\/strong>\u00a0and is denoted by\u00a0<strong>a<sup>\u00a0n<\/sup><sub>\u00a0B\/A<\/sub><\/strong>. Hence the acceleration of point B can be expressed as:<\/p>\n<p style=\"text-align: center\"><b><\/b><strong>a<\/strong><sub>B<\/sub>\u00a0=\u00a0<strong>a<\/strong><sub>A<\/sub>\u00a0+\u00a0<strong>a<sup>t<\/sup><\/strong><sub>B\/A<\/sub>\u00a0+\u00a0<strong>a<sup>n<\/sup><\/strong><sub>B\/A<\/sub><\/p>\n<p>and the relative acceleration is:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B\/A<\/sub>=\u00a0<strong>a<sup>t<\/sup><\/strong><sub>B\/A<\/sub>\u00a0+\u00a0<strong>a<sup>n<\/sup><\/strong><sub>B\/A<\/sub><\/p>\n<p>Therefore, for the relative motion between two points on a rigid body in a general plane motion the relative velocity and acceleration components are as if the body is in a fixed axis of rotation about one of the points on the rigid body.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2170 size-full aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002.png\" alt=\"\" width=\"374\" height=\"187\" srcset=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002.png 374w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002-300x150.png 300w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002-100x50.png 100w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002-150x75.png 150w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002-200x100.png 200w\" sizes=\"auto, (max-width: 374px) 100vw, 374px\" \/><\/p>\n<p>Another different type of relative motion occurs between two coincident points on two different rigid bodies but for which we know the path of one point relative to the other rigid body<span style=\"font-family: Arial, Helvetica, sans-serif\">.<\/span><\/p>\n<p align=\"left\">Consider plane 2 moved from position #1 to another position #2. During this time a point A<sub>3<\/sub>\u00a0 that is coincident with A<sub>\u00a02<\/sub> at position #1 but on a different plane, moves to position A\u2032<sub>3<\/sub> while A<sub>2<\/sub> moves to A\u2032<sub>2<\/sub>. Points that were instantaneously coincident in position #1 are not coincident at position #2. However, the path of on link 2 is known. The motion of point A<sub>3<\/sub> can be regarded as the sum of translation with A<sub>3<\/sub> to A\u2033<sub>3<\/sub> \u00a0 that is coincident with A\u2032<sub>2<\/sub>\u00a0 and a relative motion from A\u2033<sub>3<\/sub> to A\u2032<sub>3<\/sub>\u00a0relative to the plane 2. Hence, as shown\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">below<\/span>, point A<sub>3<\/sub>\u00a0first moves by a distance\u00a0<span style=\"font-family: Symbol\">D<\/span>r<sub>A2<\/sub> with A<sub>2<\/sub>\u00a0and then moves by a distance\u00a0\u00a0<span style=\"font-family: Symbol\">D<\/span>r<sub>A3\/2<\/sub> relative to A<sub>2<\/sub>\u00a0. The order of the two superimposed motions is not important.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2171 size-full aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000.png\" alt=\"\" width=\"408\" height=\"282\" srcset=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000.png 408w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000-300x207.png 300w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000-100x69.png 100w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000-150x104.png 150w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0000-200x138.png 200w\" sizes=\"auto, (max-width: 408px) 100vw, 408px\" \/><\/p>\n<p>The total displacement of point A<sub>3\u00a0<\/sub>\u00a0is:<\/p>\n<p style=\"text-align: center\">\u0394<strong>r<\/strong><sub>A<\/sub><sub>3<\/sub>\u00a0= \u0394<strong>r<\/strong><sub>A<\/sub><sub>2<\/sub> + \u0394<strong>r<\/strong><sub>A<\/sub><sub>3\/2<\/sub><\/p>\n<p>Dividing the above expression by the corresponding time interval,\u00a0<span style=\"font-family: Symbol\">D<\/span>t , and taking the limit as\u00a0\u00a0<span style=\"font-family: Symbol\">D<\/span>t tends to zero, we obtain:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>A<\/sub><sub>3<\/sub> = <strong>v<\/strong><sub>A<\/sub><sub>2<\/sub> + <strong>v<\/strong><sub>A<\/sub><sub>3\/2<\/sub><\/p>\n<p>where\u00a0<strong>V<sub>A3\/2<\/sub><\/strong> is the relative velocity which is always tangent to the relative path of point A<sub>3<\/sub>\u00a0with respect to the plane 2.<\/p>\n<p>In mechanisms such relative motion occurs when we have prismatic, cylinder-in-slot or cam pairs between two moving links. In case of cam pairs we utilise equivalent linkage concepts with which we can replace a cam pair instantaneously by an equivalent mechanism containing lower kinematic pairs (see section 2.3). For prismatic or cylinder-in-slot joints, the relative path is a straight line.<\/p>\n<p>In order to explain the relative motion between the two links, consider link 3 connected to link 2 by a prismatic joint . For simplicity, link 2 is connected to the fixed link by a revolute joint. Point B corresponds to two coincident points B<sub>2<\/sub> and B<sub>3<\/sub> on links 2 and 3. The position vector for point B (B<sub>2<\/sub> or B<sub>3<\/sub>) in complex numbers is:<\/p>\n<p style=\"text-align: center\"><b><\/b><strong>R<\/strong><sub>B<\/sub> = re<sup>i\u03b8<\/sup><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1102 aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/img222-10.gif\" alt=\"\" width=\"207\" height=\"265\" \/><\/p>\n<p>Note that\u00a0<strong>R<\/strong><sub>B<\/sub> \u00a0is the position vector for both B<sub>2<\/sub> and B<sub>3<\/sub>. However for B<sub>2<\/sub> \u00a0r is a constant but for B<sub>3<\/sub> r is a variable. Therefore, when differentiating this vector, we must specify which rate of change of position is sought. For example if we are to determine the rate of change of the position of point B<sub>2<\/sub>, then:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B2<\/sub>\u00a0= ir\u03c9e<sup>i\u03b8<\/sup><\/p>\n<p>where\u00a0<span style=\"font-family: Symbol\">w <\/span>= d<span style=\"font-family: Symbol\">q<\/span>\/dt . If we want to determine the rate of change of the position B<sub>3<\/sub>, then:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B2<\/sub>\u00a0= ir\u03c9e<sup>i\u03b8<\/sup> + (dr\/dt)e<sup>i\u03b8<\/sup><\/p>\n<p>The first term is the velocity of point B<sub>2<\/sub><strong>, V<\/strong><sub>B2<\/sub>. The second term is the relative velocity of point B<sub>3<\/sub>\u00a0with respect to link 2. It is tangent to the relative path. Hence:<\/p>\n<p style=\"text-align: center\"><b><\/b><strong>v<\/strong><sub>B<\/sub><sub>3<\/sub>\u00a0=\u00a0<strong>v<\/strong><sub>B<\/sub><sub>2<\/sub> +\u00a0<strong>v<\/strong><sub>B<\/sub><sub>3\/2<\/sub><\/p>\n<p>B<sub>3<\/sub>, which is fixed on link 3 will be coincident with another point B\u2019<sub>2<\/sub>\u00a0on link 2 after a certain time interval. The second derivative of the position vector with respect to time yields:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub><sub>3<\/sub> = d<strong>v<\/strong><sub>B<\/sub><sub>3<\/sub>\/dt = [ir\u03b1e<sup>i\u03b8<\/sup> \u2212 r\u03c9<sup>2<\/sup>e<sup>i\u03b8<\/sup> + i(dr\/dt)\u03c9e<sup>i\u03b8<\/sup>] + [(dr<sup>2<\/sup>\/dt<sup>2<\/sup>)e<sup>i\u03b8<\/sup> + i(dr\/dt)\u03c9e<sup>i\u03b8<\/sup>]\n<p align=\"left\">Where the terms in the first bracket are those obtained from the differentiation of<span style=\"font-family: Arial, Helvetica, sans-serif\">\u00a0<\/span><strong>V<\/strong><sub>B2<\/sub>\u00a0, and those in the second bracket are obtained when the relative velocity\u00a0<strong>V<\/strong><sub>B3\/2<\/sub>\u00a0 is differentiated. The first two terms are the tangential and normal accelerations components of point B<sub>\u00a02<\/sub> and they add up to the angular acceleration of point B<sub>2<\/sub>\u00a0(<strong>a<\/strong><sub>B2<\/sub>). The third term arises due to the fact that B<sub>2<\/sub> observed as the coincident point with B<sub>3<\/sub>, will be a different point in a differential displacement. At a certain time interval\u00a0<span style=\"font-family: Symbol\">\u00a0D<\/span>t, B<sub>3<\/sub>\u00a0will be coincident with B\u2019<sub>\u00a02<\/sub>, that is at a distance\u00a0\u00a0<span style=\"font-family: Symbol\">D<\/span>r from B<sub>2<\/sub> (direction being along the axis of the slider) . The velocity of B<sub>2<\/sub>\u00a0is ir\u03c9e<sup>i\u03b8<\/sup> and the velocity of B\u2032<sub>2<\/sub> is\u00a0 i(r + \u0394r)\u03c9e<sup>i\u03b8<\/sup>. The change in the velocity during the time period\u00a0\u00a0<span style=\"font-family: Symbol\">D<\/span>t is i\u0394r\u03c9e<sup>i\u03b8<\/sup>. Therefore there is an acceleration perpendicular to the slider axis by an amount ir\u03c9e<sup>i\u03b8<\/sup>.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1103\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/img222-11.gif\" alt=\"\" width=\"502\" height=\"409\" \/><\/p>\n<p>Consider the terms obtained when the relative velocity is differentiated. The first term is due to the change in magnitude of the relative velocity. This term is tangent to the relative path. The second term is due to the change in the direction of the relative velocity vector. In order to explain this term consider a constant magnitude for relative velocity. At a certain time interval\u00a0<span style=\"font-family: Symbol\">D<\/span>t, link 2 will rotate by an angle\u00a0<span style=\"font-family: Symbol\">Dq<\/span>. The direction of the relative velocity is now e<sup>i(\u03b8<sub>\u00a0<\/sub>+<sub>\u00a0<\/sub>\u0394\u03b8)<\/sup> instead of e<sup>i\u03b8<\/sup>\u00a0initially.\u00a0<span style=\"font-family: Arial, Helvetica, sans-serif\">T<\/span>he vectorial change of the relative velocity vector is ir\u0394\u03b8e<sup>i\u03b8<\/sup>, and its time rate of change is ir\u0394\u03b8e<sup>i\u03b8<\/sup>.\u00a0\u00a0 The last two terms in both of the brackets are the same in magnitude and direction. They can be combined into one term 2ir\u03c9e<sup>i\u03b8<\/sup>\u00a0This term is called\u00a0<strong><em>Coriolis acceleration component<\/em><\/strong> and is represented by a<sup>c<\/sup><sub>B3\/2<\/sub> . It is a relative acceleration component. Its direction is normal to the relative path described by B<sub>3<\/sub>\u00a0on link 2. Its direction is obtained by rotating the relative velocity vector\u00a0<strong>V<\/strong><sub>B3\/2<\/sub>, 90\u00b0 in the sense of the angular velocity of link 2. Hence the acceleration of point B<sub>3<\/sub>\u00a0can be written as:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B3<\/sub> = ir\u03b1e<sup>i\u03b8<\/sup> \u2212 r\u03c9<sup>2<\/sup>e<sup>i\u03b8<\/sup> + (dr<sup>2<\/sup>\/dt<sup>2<\/sup>)e<sup>i\u03b8<\/sup> + 2i(dr\/dt)\u03c9e<sup>i\u03b8<\/sup><\/p>\n<p style=\"text-align: center\" align=\"center\"><strong>a<\/strong><sub>B3<\/sub> = <strong><b>a<sup>t<\/sup><\/b><\/strong><sub>B<\/sub><sub>2<\/sub> + <strong><b>a<sup>n<\/sup><\/b><\/strong><sub>B<\/sub><sub>2<\/sub>\u00a0+ <strong> <b>a<sup>t<\/sup><\/b><\/strong><sub>B<\/sub><sub>3\/2<\/sub> + <strong>a<sup>c<\/sup><\/strong><sub>B<\/sub><sub>3\/2<\/sub><\/p>\n<p style=\"text-align: left\" align=\"center\">In the previous case, since the prismatic joint axis is along the radial line B<sub>0<\/sub>B, the normal acceleration component of point B<sub>2<\/sub>\u00a0(a<sup>n<\/sup><sub>2<\/sub>) and the tangential relative acceleration component a<sup>t<\/sup><sub>B3\/2<\/sub> are along the same direction (so is a<sup>c<\/sup><sub>B3\/2<\/sub>\u00a0and\u00a0 a<sup>t<\/sup><sub>B2<\/sub>). These directions need not be along the same direction in general. Consider a prismatic joint as shown in figure below<span style=\"font-family: Arial, Helvetica, sans-serif\">\u00a0<\/span>between links 2 and 3. In such a case:<\/p>\n<p style=\"text-align: center\" align=\"center\">\u00a0<strong>r<\/strong><sub>B<\/sub><sub>3<\/sub> = ae<sup>i\u03b8<\/sup> + se<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>where b and a are fixed parameters of link 2. Differentiating, we obtain:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B<\/sub><sub>3<\/sub> = ia\u03c9e<sup>i\u03b8<\/sup> + is\u03c9e<sup>i(\u03b8+\u03b2)<\/sup> + (ds\/dt)e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>which can be written in the form:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B<\/sub><sub>3<\/sub> = i\u03c9e<sup>i\u03b8<\/sup>(a + se<sup>i\u03b2<\/sup>) + (ds\/dt)e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>Note that:<\/p>\n<p style=\"text-align: center\">a + se<sup>i\u03b2<\/sup> = be<sup>i\u03b3<\/sup><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1104 aligncenter\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2021\/09\/img222-12.gif\" alt=\"\" width=\"361\" height=\"324\" \/><\/p>\n<p align=\"left\">Where b is the variable distance A<sub>0<\/sub>B and \u03b3\u00a0is the variable angle between A<sub>\u00a00<\/sub>A and A<sub>\u00a00<\/sub>B on link 2. This vector is the position of point B on a coordinate frame fixed on link 2. Hence:<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B<\/sub><sub>3<\/sub> = i\u03c9be<sup>i(\u03b8+\u03b3)<\/sup>\u00a0+ (ds\/dt)e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>or<\/p>\n<p style=\"text-align: center\"><strong>v<\/strong><sub>B<\/sub><sub>3<\/sub>\u00a0=\u00a0<strong>v<\/strong><sub>B<\/sub><sub>2<\/sub> +\u00a0<strong>v<\/strong><sub>B<\/sub><sub>3\/2<\/sub><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2173\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001.png\" alt=\"\" width=\"563\" height=\"314\" srcset=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001.png 450w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001-300x167.png 300w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001-100x56.png 100w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001-150x84.png 150w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0001-200x112.png 200w\" sizes=\"auto, (max-width: 563px) 100vw, 563px\" \/><\/p>\n<p align=\"left\">The first term has the magnitude \u03c9|A<sub>0<\/sub>B|= \u03c9b and its direction is perpendicular to the line A<sub>0<\/sub>B since the unit vector is ie<sup>i(<span style=\"font-family: Symbol\">q+g<\/span>)\u00a0<\/sup>. The second term has the magnitude (ds\/dt)\u00a0 which is the relative velocity of point B<sub>3<\/sub> with respect to link 2 and its direction is along the slider axis AB given by the unit vector e<sup>i(<span style=\"font-family: Symbol\">q+b<\/span>\u00a0)<\/sup>.<\/p>\n<p>Differentiating the velocity equation once more, we have the acceleration of point B<sub>3<\/sub>\u00a0as<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub><sub>3<\/sub> = ia\u03b1e<sup>i\u03b8<\/sup> \u2212 a\u03c9<sup>2<\/sup>e<sup>i\u03b8<\/sup> + is\u03b1e<sup>i(\u03b8+\u03b2)<\/sup> + i(ds\/dt)\u03c9e<sup>i(\u03b8+\u03b2)<\/sup> \u2212 s\u03c9<sup>2<\/sup>e<sup>i(\u03b8+\u03b2)<\/sup> + (d<sup>2<\/sup>s\/dt<sup>2<\/sup>)e<sup>i(\u03b8+\u03b2)<\/sup> + i(ds\/dt)\u03c9e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>the terms can be grouped as:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub><sub>3<\/sub> = i\u03b1(a + se<sup>i\u03b2<\/sup>)e<sup>i\u03b8<\/sup> \u2212 \u03c9<sup>2<\/sup>(a + se<sup>i\u03b2<\/sup>)e<sup>i\u03b8<\/sup> + 2i(ds\/dt)\u03c9e<sup>i(\u03b8+\u03b2)<\/sup>+ (d<sup>2<\/sup>s\/dt<sup>2<\/sup>)e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p>or substituting a + se<sup>i\u03b2<\/sup> = be<sup>i\u03b3<\/sup> :<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub><sub>3<\/sub> = i\u03b1be<sup>i(\u03b8+\u03b3)<\/sup> \u2212 \u03c9<sup>2<\/sup>be<sup>i(\u03b8+\u03b3)<\/sup> + 2i(ds\/dt)\u03c9e<sup>i(\u03b8+\u03b2)<\/sup>+ (d<sup>2<\/sup>s\/dt<sup>2<\/sup>)e<sup>i(\u03b8+\u03b2)<\/sup><\/p>\n<p align=\"left\">The first two terms are the tangential and normal acceleration components of point B<sub>2<\/sub>\u00a0whose direction are normal and parallel (directed towards A<sub>0<\/sub>) to the line A<sub>0<\/sub>B respectively . The third term is the relative Coriolis acceleration which is perpendicular to the slider axis AB (given by the unit vector ie<sup>\u00a0i<span style=\"font-family: Symbol\">(q+b)<\/span>\u00a0<\/sup>). The last term is the tangential relative acceleration with magnitude d<sup>2<\/sup>s\/dt<sup>2<\/sup> and direction e<sup>i<span style=\"font-family: Symbol\">(q+b)<\/span><\/sup>\u00a0which is along the slider axis AB. Therefore in vector form the acceleration of B<sub>3<\/sub>\u00a0is:<\/p>\n<p style=\"text-align: center\"><strong>a<\/strong><sub>B<\/sub><sub>3<\/sub> = <strong><b>a<sup>t<\/sup><\/b><\/strong><sub>B<\/sub><sub>2<\/sub> + <strong><b>a<sup>n<\/sup><\/b><\/strong><sub>B<\/sub><sub>2<\/sub>\u00a0+ <strong> <b>a<sup>c<\/sup><\/b><\/strong><sub>B<\/sub><sub>3\/2<\/sub> + <strong>a<sup>t<\/sup><\/strong><sub>B<\/sub><sub>3\/2<\/sub><\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2172\" src=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002.png\" alt=\"\" width=\"563\" height=\"249\" srcset=\"https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002.png 457w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002-300x133.png 300w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002-100x44.png 100w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002-150x66.png 150w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002-200x88.png 200w, https:\/\/blog.metu.edu.tr\/eresmech\/files\/2022\/03\/4-1-2_clip_image002_0002-450x199.png 450w\" sizes=\"auto, (max-width: 563px) 100vw, 563px\" \/><\/p>\n<\/div>\n<\/div><\/div><\/div><\/div><\/div>\n\n\n<p> <a href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch4\/4-1-1\/\" 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\/ch4\/\" 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\/ch4\/4-2-1\/\" 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>4.1 VELOCITY AND ACCELERATION ANALYSIS &#8211; 2 General Plane Motion: General plane motion is the plane motion which is neither\u00a0 translation or rotation about a fixed axis but it can be analysed by the superposition of these motions using the\u00a0relative &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/blog.metu.edu.tr\/eresmech\/mechanisms\/ch4\/4-1-2\/\"> <span class=\"screen-reader-text\">4-1-2<\/span> Devam\u0131n\u0131 Oku &raquo;<\/a><\/p>\n","protected":false},"author":7747,"featured_media":0,"parent":1964,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"full-width-page.php","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-2165","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/2165","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=2165"}],"version-history":[{"count":0,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/2165\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/pages\/1964"}],"wp:attachment":[{"href":"https:\/\/blog.metu.edu.tr\/eresmech\/wp-json\/wp\/v2\/media?parent=2165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}