graduated into a scale of quarter-inches and tenths. If this bevil be laid upon the radius AT, so that its point T coincides with the pitch circle, the center point P will be found at once, by reading off the radius of the wheel in inches upon the reduced scale. Thus the radius AT in the figure, is two inches long, and the point P is found at 2 upon the scale. To describe the other teeth, draw with center A and radius AP, a circle within the pitch circle, dotted in the figure, this will be the locus of the centers of the teeth; then having previously divided the pitch circle, take the constant radius PT in the compasses, and keeping one point in the dotted circle, step from tooth to tooth and describe the arcs, first to the right and then to the left, as for example, mn is described from q and p0 from P. If Op were an arc of an involute having the circle Ppq for its base, PT would be its radius of curvature at T. These teeth, therefore, approximate to involute teeth, and they possess in common with them the oblique action, the power of acting with wheels of any number of teeth, and the adjustment of back-lash; but, as the sides of the teeth consist each of a single arc, there is but one position of action in which the angular velocity ratio is strictly constant, namely, when the point of contact is on the line of centers. 139. By making the side of each tooth consist of two arcs joined at the pitch circle, and struck in such wise that the exact point of action of the one shall lie a little before the line of centers, say at the distance of half the pitch, and the exact point of the other at the same distance beyond that line, an abundant degree of exactitude will be obtained for all practical purposes. To describe the teeth of such a pair of wheels, let A (fig.6) be the center of motion of a proposed wheel, B the center of motion of the wheel with which it is to work, T the point of contingence of the pitch circles. Draw QTq making an angle of 75° with the line of centers. (This angle is in fact arbitrary, but by various trials I find 75° to give the best form to the teeth.) Draw k TK perpendicular to QTq, and set off TK and Tk equal to each other, and less than either AT or TB. Join AK and BK, producing the latter to Q, then P and Q are a pair of tooth-centers. Take a Take a point m on the pitch circle a Te, at the distance of half the pitch from T, and on the opposite side to the tooth-centers. A convex arc struck from P through m on the outside of this pitch circle will work correctly with a concave arc struck from Q through the same point, and within the other pitch circle. To describe the faces of the teeth of the lower wheel we may proceed as in the last example, thus: draw with center A a circle through P, which will be the locus of the centers of the small arcs; and having previously divided the pitch circle for the reception of the teeth, take the constant radius Pm in the compasses, and keeping one point in the circle Pf, describe the faces of the teeth to the right and left outside the pitch circle, as shewn in the figure at t and s. A similar proceeding will give the flanks of the teeth of the wheel. upper To obtain the flanks of the lower wheel and faces of the upper wheel, join Bk and Ak, producing the latter to q, then will p and q be another pair of centers, from which let arcs be struck through a point n, at the distance of half the pitch beyond T, but within the pitch circle of A and without that of B. The action of these arcs will be exact at the distance of half the pitch from T. To complete the teeth of the lower wheel already begun, describe from A with radius Aq, a circle for the locus of the centers of the flanks of these teeth, and with the constant radius equal to qn step from tooth to tooth, describing the flanks in the manner shewn in the figure, as at r and q. 140. From the construction it appears that these teeth of the lower wheel would work correctly with a wheel of any radius, provided the points K and k remain constant; for a change in the position of B, on the line of centers, only affects the points Q, p, which belong to its own teeth, but does not disturb the points P, q, from which the teeth of the lower wheel have been described. In short, if any number of wheels be in the above manner described, in which the lines Qq, Kk, preserve the same angular position with respect to the line of centers and the same distances KT, kT, then any two of these wheels will work together. The distance KT may be determined for a set of wheels by considering that if A approach T, Aq becomes parallel to Tq, and q is at that moment at an infinite distance; the flank of the tooth becoming a right line perpendicular to Tq. If A approach still nearer, q appears on the opposite side of T, and the flank becomes convex, giving a very awkward form to the tooth. The greatest value therefore that can be given to KT, must be one which when employed with the smallest radius of the set, will make Aq parallel to Tq; therefore, if R, be this smallest radius, we have RKT = R, x sin QTA, or C = R, × sin 0; which substituted in the formula (Art. 136), gives _kT. R 141. By assuming constant values for R, and 0 in a set of wheels, the values of D and d which correspond to different numbers and pitches, may be calculated and arranged in tables for use, so as to supersede the necessity of making the construction in every case. Thus the tables which follow in fig. 62 were obtained by assuming twelve teeth as the least number to be given to a wheel, and 0 = 75o. The unit of length in which the values of D and d are expressed is one twentieth of an inch, that being sufficiently small to avoid errors of a practical magnitude. |