Page:The Kinematics of Machinery.djvu/171

 upon it, any number of consecutive positions of the latter may be drawn, and a curve enveloping these, if rigidly connected to the stationary centroid, gives a figure with which the known element remains in continuous contact during its motion. Such a figure will serve as the profile of the stationary element, if it can be provided with a sufficient number of points of restraint. This may readily happen, indeed the figure may have even more such points than are necessary. In these cases only such portions of the.%ure need be constructed as suffice to make the restraint continuous. We have seen this in the case of the curve-triangle and square, and also with the duangle and triangle, where the enveloping curves were not drawn in the corners of the square and triangle, their omission not affecting the restraint. The method is therefore available; we must examine the manner in which it can be practically carried out.

If A and B (Fig. 103) be the two given centroids, and a I the assumed profile of the element corresponding to A, then if any point, as b, of the assumed profile be also a point in the centroid, the corresponding point of contact of the centroid B gives at once one point in the profile to be found. In order to deter- mine a second point in it, that for instance which shall correspond to a, let a normal a c be drawn to the given profile at a. It cuts the corresponding centroid A in c. If we roll the centroid B upon A until c becomes the point of contact, i.e. the instantaneous centre, then a must be the point of contact of the profiles, and a c, which we may call the central distance of a, must be the common distance of the two touching profile points from the con- tact points of their respective centroids. The distance at which the point of the cen- troid B originally at must then be from a can at once be /

seen when J9is in the position. 103.

\ c (in which contact occurs

at c); it is simply a b v the centroidal arc\ c being = I c. If now we have b ^ = ^ c = b c, and describe from c x with