Page:The Kinematics of Machinery.djvu/170

 and determining by their means the reciprocally restraining figures. This problem is the one which occurs by far the most often in con- nection with machine design, and has frequently to be solved both for simple and for complex motions.

We shall here simply indicate the general methods of procedure in this case. These are very numerous, but admit of being classed under the seven headings examined in the following paragraphs, in which we shall consider cylindric rolling only in the first place.*

§31. First Method.—Determination of the Profile of one Element, that of the other being arbitrarily assumed.

If the profile of one element of a pair of which the centroids are known be arbitrarily assumed, the centroid of, the unknown element may be brought to rest, and that of the assumed one rolled


 * The following note may make clearer to some readers the nature of the problems

treated by Prof. Reuleaux in 31-37. Let A and B be any two centroids, and act,' and bb' the profiles of bodies whose relative motions the centroids represent. It is required so to form these profiles that during the rolling of the centroids they shall remain continuously in contact. The necessary condition for this may be thus shown. Let be the point ot contact of the centroids, and let the two pro- files be touching at a; draw their common tan- gent it', join a and draw T T perpendicular to it. Then suppose B fixed, and A free to roll upon it. The instantaneous motion of the point a can only take place about the instantaneous centre 0, that is in the direction T T' perpendicular to a. But if a move towards T it leaves the profile b, while it cannot move towards T because the point b restrains motion in that direction. Hence the assumed profiles a a' and b b', in no way fulfil the requirements of the problem. They show very clearly, however, the condition necessary for this fulfilment, for it is obvious that the point a, moving about 0, can remain in contact with b only if the tangent t f to that profile coincide with the line T T'. a is normal to T T' by construction, we may therefore express the condition generally by saying: in order that two elements may remain in contact during the rolling of their centroids the normal to the common tangent of their profiles must always pass through the in- stantaneous centre, or point of contact of the centroids. It must be remembered that (except in one special case), the profiles themselves do not roll upon one another, but slip or grind to a greater or less extent.