Graphical Position Analysis of Linkages

 For any one DOF linkages, such as further, only one parameter is needed to completely define the positions of all the links. The parameters usually chosen is the angle of the input link. This is shown as θ₂ in figure 4-4. We want to find θ₃ and θ₄. The link lenghts are known. Note that we Will consistently number the ground link as 1 and the driver link as 2 in these examples.

The graphical análisis of this problemas trivial and can be done using only high school geometry. If we draw the linkage carefully to scale with rule, compass and protractor in particular position (given θ₂) the it is only necessary to measure angles of links 3 and 4 with the protractor. Note that all the link angles are measured from a positive X axis. In figure 4-4, a local XY axis system, parallel to the global XY system, has been created at point A to measure θ₃. The accuracy of this graphical solution will be limited by our care and drafting ability and by the crudity of the protractor used. Nevertheless, a very rapid approximate solution can be found for any one position.

Figure 4-5 shows the construction of the graphical position solution. The four link lengths a, b, c, d and the angle θ₂ of the input link are given. First, the ground link (1) and the input link (2) are drawn to a convenient scale such that they intersect at the original θ₂ of the global XY coordinate system with link 2 placed at the input angle θ₂. Link 1 is drawn along the X axis of for convenience. The compass is set to the scale length of link 3, and an arc of that radius swung about the end of link 2 (point A). Then the compass is set to the scaled length of link 4, and a second are swung about the end of link 1 (point O₄).

These two arcs will have two intersections at B and B´ that define the two solutions to the position problem for a fourbar linkage which can be assembled in two configurations, called circuits, labeled open and crossed in figure 4-5. Circuits in linkages will be discussed in later section,

The angles of links 3 and 4 can be measured with a protractor. One circuit has angles θ₃ and θ₄. A graphical solution is only valid for the particular value of the input angle used. For each additional position analysis, we must completely redraw the linkage. This can become burdensome if we need a complete analysis at every 1- or 2-degree increment of θ₂. In that case we will be better off to derive an analytical solution for θ₃ and θ₄ which can be solved by computer.