Page:The Kinematics of Machinery.djvu/89

 the point M l in the second centroid which corresponds to O v and which must be at their intersection.

Example 1. The nature of the kinematic chain (Fig 20) discussed in 3 allows none hut con-plane motions; it may therefore serve us here for an illustration. Let it he required to find its centroids. As each of its links can have a motion relatively to all the three others, there are in all six pairs of centroids "belonging to the chain, four for the motions of adjacent and two for those of opposite links. The four first are very simple, each curve heing a point only; the two others are not so readily found. We will here examine the pair of centroids belonging to the

links a h and d e. For this purpose we first bring the link

a h to rest (we may suppose it connected with a fixed pedestal, as

shown in Fig. 21); then a d rotates about a, while e h swings

to and fro in circular arcs about h. The centres of the elements c and /

. 20.

describe therefore paths to which the normals are always radii passing through the centres of a and h. By producing these radii until they intersect we can consequently obtain any number of points in the centroid of the fixed link a - - h. The curve found in this way is .shown in Fig. 22. or M is the pefe for the original position

a - - d - - e h obtained by producing a d and h - e

until they cut each other. The whole figure 1. . . . 4 is not simple in form. It contains four infinitely distant points, corresponding to the

two parallel positions of a d and h e. The second centroid,

that of the link d - - e, drawn in the way above described, is shown in MM l. . . . J/ 4; it also contains, necessarily, four infinitely distant points. The two centroids which in the engraving touch at M, roll upon one another as the mechanism moves (0 O l . . . . 4 remaining stationary), and supply completely the means of examining the whole

complicated motion of the link d e. As regards easy comprehension

this geometric representation still leaves something to be wished, the infinitely distant points impair its clearness not a little. But the