Page:The Kinematics of Machinery.djvu/173

 (allgemeine Verzahnung). In this case the centroids are commonly circles, which renders some simplification possible. The following is the way in which I have used* this method for the construction of wheel-teeth upon circular centroids.

In Fig. 105 a c O e g is the given profile, A the corresponding centroid drawn ahout the centre A l; B is the centroid, drawn about B v of the element whose profile is to be found. After drawing the normal ab to any point a in the profile we must next find the position in which a will be when it itself becomes the point of contact with the as yet unknown profile, for which purpose we may suppose both centroids to be turning about their centres A l and B^ assumed to be fixed points in the plane of the paper, The point of contact 1 lies necessarily at the intersection of a circle described about A 1 with a radius A l a, with a second circle described about the instantaneous centre with the central distance b a, for at the moment of contact the normal a b must pass through 0. B has meanwhile turned through an arc 0\ Ob. The new profile point ! which corresponds to a must therefore be at the intersection of a circle drawn from B l with radius B l 1 with another circle drawn from Jj_ with radius 01. In the same way the points of contact 2, 3, 4. . and the corresponding points c v e v g 1 ... in the profile, can be found. The series of points 1, 2, 3, 4 ... give us the line of contact, or locus of all the successive points of contact of the two profiles. The line joining the point of contact with the instantaneous centre is for each instant both the direction of restraint, and the direction of the pressure between the two profiles.

The method of which we have here. given three applications furnishes an immense variety of profile forms, among them many which are of little or no practical use. Those curves especially which contain cusps or loops, or foim contracted spirals etc., (see Fig. 106) are commonly unsuitable; they are not useful although they are geometrically c o r r e c t, v fulfilling the required conditions \f y*\

as to continuous restraint in mo- f (J

tions determined by the given F IG< 106<

pair of centroids. If the applica- tion of this method furnish us in any case with such impracticable profile-curves it becomes necessary to choose some more suitable


 * Published first in Der Constructor, 2nd Ed. 1865.