Modern Machine-Shop Practice, Volumes I and II

by Joshua Rose

Fig. 142.

Each of these parts is marked with the number of teeth the wheel is to contain, and with the pitch of the teeth as shown in [Fig. 140], which represents part c full size. Now suppose it is required to find the thickness at the root, for a tooth of a wheel having 60 teeth of one inch pitch, the circles from the point a, pitch line b and root c being drawn, and a radial line representing the middle of the tooth being marked, as is shown in [Fig. 142], the compass points are set to the distance f b, [Fig. 140]—f being at the junction of line 1 with line 60; the compasses are then rested at g, and the points h i are marked. Then, from the portion b, [Fig. 139] of the diagram, which is shown full-size in [Fig. 141], the compasses may be set to half the thickness at the pitch circle, as in this case (for ordinary teeth) from e to e, and the points j k, [Fig. 142], are marked. By a reference to the portion d of the diagram, half the thickness of the tooth at the point is obtained, and marked as at l m in [Fig. 142]. It now remains to set compasses to the radius for the face and that for the flank curves, both of which may be obtained from the part a of the diagram. The locations of the centres, wherefrom to strike these curves, are obtained as in [Fig. 142]. The compasses set for the face curve are rested at h, and the arc n is struck; they are then rested at j and the arc o struck; and from the intersection of n o, as a centre, the face curve h j is marked. By a similar process, reference to the portion d of the diagram, half the thickness of the tooth at the point is obtained, and marked as at l m in [Fig. 142]. It now remains to set the compasses to the radius to strike the respective face and flank curves, and for this purpose the operator turns to the portion a, [Fig. 139], of the diagram or scale, and sets the compasses from the marks on that portion to the required radii.

It now remains to find the proper location from which to strike the curves.

The face curve on the other side of the tooth is struck. The compasses set to the flank radius is then rested at m, and the arc p is marked and rested at k to mark the arc q; and from the intersection of p q, as a centre, the flank curve k m is marked: that on the other side of the tooth being marked in a similar manner.

Additional scales or diagrams, not shown in [Fig. 139], give similar distances to set the compasses for the teeth of internal wheels and racks.

It now remains to explain the method whereby the author of the scale has obtained the various radii, which is as follows: A wheel of 200 teeth was given the form of tooth curve that would be obtained by rolling it upon another wheel, containing 200 teeth of the same pitch. It was next given the form of tooth that would be obtained by rolling upon it a wheel having 10 teeth of the same pitch, and a line intermediate between the two curves was taken as representing the proper curve for the large wheel. The wheel having 10 teeth was then given the form of tooth that would be obtained by rolling upon it another wheel of the same diameter of pitch circle and pitch of teeth. It was next given the form of tooth that would be given by rolling upon it a wheel having 200 teeth, and a curve intermediate between the two curves thus obtained was taken as representing the proper curve for the pinion of 10 teeth. By this means the inventor does not claim to produce wheels having an exactly equal velocity ratio, but he claims that he obtains a curve that is the nearest approximation to the proper epicycloidal curve. The radii for the curves for all other numbers of teeth (between 10 and 200) are obtained in precisely the same manner, the pinion for each pitch being supposed to contain 10 teeth. Thus the scale is intended for interchangeable cast gears.

The nature of the scale renders it necessary to assume a constant height of tooth for all wheels of the same pitch, and this Mr. Walker has assumed as .40 of the pitch, from the pitch line to the base, and .35 from the pitch line to the point.

The curves for the faces obtained by this method have rather more curvature than would be due to the true epicycloid, which causes the points to begin and leave contact more easily than would otherwise be the case.