Fig. 134.
Hypocycloids for the flanks of the teeth may be traced in a similar manner. Thus in [Fig. 134] p p is the pitch circle, and b c the line of motion of the centre of the generating circle to be rolled within p p, and r a radial line. From 1 to 6 are points of equal division on the pitch circle, and d to i are arc locations for the centre of the generating circle. Starting from a, which represents the supposed location for the centre of the generating circle, the point of contact between the generating and base circles will be at b. Then from 1 to 6 are points of equal division on the pitch circle, and from d to i are the corresponding locations for the centres of the generating circle. From these centres the arcs j, k, l, m, n, o, are struck. From 6 mark the six points of division from a to f, and f is a point in the curve. Five divisions on n, four on m, and so on, give respectively points in the curve which is marked in the figure from a to f.
There is this, however, to be noted concerning the constructions of the last two figures. Since the circle described by the centre of the generating circle is of different arc or curve to that of the pitch circle, the chord of an arc having an equal length on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter. Suppose, for example, that the difference between the arc n 5 ([Fig. 133]) and its chord is .1, and that the difference between the arc 4 5, and its chord is .01, then the error in one step is .09, and, as the point v is formed in 5 steps, it will contain this error multiplied five times. Point d would contain it multiplied four times, because it has 4 steps, and so on.
The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels (so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive), it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.
Fig. 135.
An instrument much employed in the best practice to find the radius which will strike an arc of a circle approximating the true epicycloidal curve, and for finding at the same time the location of the centre wherefrom that curve should be struck, is found in the Willis’ odontograph. This is, in reality, a scale of centres or radii for different and various diameters of wheels and generating circles. It consists of a scale, shown in [Fig. 135], and is formed of a piece of sheet metal, one edge of which is marked or graduated in divisions of one-twentieth of an inch. The edge meeting the graduated edge at o is at angle of 75° to the graduated edge.
On one side of the odontograph is a table (as shown in the cut), for the flanks of the teeth, while on the other is the following table for the faces of the teeth: