Page:The Kinematics of Machinery.djvu/174

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form for the assumed profile, proceeding thus by trial. Absolute freedom in the choice of the form of the first profile is to this extent limited, and a further limitation arises from the fact that such curves as have normals which cut the centroid at the point of con- tact at too great an angle (as e.g. the normal 01 in Fig. 105) are not suitable for the profile, for with these most injurious friction will occur, if not complete "jamming." Those parts of profiles, lastly, whose normals do not pass through the centroid, and therefore cannot be normals of restraint, are entirely unusable. Thus in the employment of this method in Applied Kinematics a number of unsuitable and unusable forms must be withdrawn from those which can be used for the (otherwise) arbitrarily chosen form of the assumed profile.

§32.
Second Method.—Auxiliary Centroids.

The method just described gives the profile of a single element only; by that which we have now to examine the two profile forms possessing the required property of continuous restraint are determined at the same time.

Let A and B Fig. 107 be again the pair of centroids in contact at 0. If any third curve C touch A and B at the same point, and roll with them as they roll, the three curves will always have a common point of contact, and that point will always be the instan- taneous centre. Then any point D fixed to G describes a roulette relatively to each of the centroids A and B. The two curves thus obtained, a D . . . and ID.... have in any position a common point, as D, and also a common normal, as D 0. They may there- fore serve as profiles for elements,- for their common normal always passes through the point of contact of the centroids. Their practical usefulness depends on the same conditions as those mentioned in the last section.

The centroid C, by the help of which we have obtained the profiles, we may call an auxiliary centroid. If the describing point D be taken on this curve or within it, the roulettes obtained remain always upon one side of the primary centroid to which they belong; there is then always space upon the opposite side of the same curve to construct similarly another pair of roulette profiles. The