Hypocycloids for the flanks of the teeth maybe traced in a similar manner. Thus in Figure 239, 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. 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 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. The six divisions on O, from a to f, give at f a point in the curve. Five divisions on N, four on M, and so on, give, respectively, points in the curve.

Fig. 239.

Fig. 239.

There is this, however, to be noted concerning the construction of the last two figures. Since the circle described by the centre of the generating circle is of a different arc or curve to that of the pitch circle, the length of an arc having an equal radius 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 a, b, 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 f is formed in five steps, it will contain this error multiplied five times. Point d would contain it multiplied three times, because it has three 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. 240.

Fig. 240.

For showing the dimensions through the arms and hub, a sectional view of a section of the wheel may be given, as in Figure 240, which represents a section of a wheel, and a pinion, and on these two views all the necessary dimensions may be marked.

Fig. 240 a. (Page 203.)

Fig. 240 a. (Page 203.)

If it is desired to draw an edge view of a wheel (which the student will find excellent practice), the lines for the teeth may be projected from the teeth in the side view, as in Figure 240 a. Thus tooth E is projected by drawing lines from the corners A, B, C, in the side view across the face in the edge view, as at A, B, C in the latter view, and similar lines may be obtained in the same way for all the teeth.

When the teeth of wheels are to be cut to form in a gear-cutting machine, the thickness of the teeth is nearly equal to the thickness of the spaces, there being just sufficient difference to prevent the teeth of one wheel from becoming locked in the spaces of the other; but when the teeth are to be cast upon the wheel, the tooth thickness is made less than the width of the space to an amount that is usually a certain proportion of the pitch, and is termed the side clearance. In all wheels, whether with cut or cast teeth, there is given a certain amount of top and bottom clearance; that is to say, the points of the teeth of one wheel do not reach to the bottom of the spaces in the other. Thus in the Pratt and Whitney system the top and bottom clearance is one-eighth of the pitch, while in the Brown and Sharpe system for involute teeth the clearance is equal to one-tenth the thickness of the tooth.

In drawing bevil gear wheels, the pitch line of each tooth on each wheel, and the surfaces of the points, as well as those at the bottom of the spaces, must all point to a centre, as E in Figure 241, which centre is where the axes of the shafts would meet. It is unnecessary to mark in the correct curves for the teeth, for reasons already stated, with reference to the curves for a spur wheel. But if it is required to do so, the construction to find the curves is as shown in Figure 242, in which let A A represent the axis of one shaft, and B that of the other of the pair of bevil wheels that are to work together, their axes meeting at W; draw the line E at a right angle to A A, and representing the pitch circle diameter of one wheel, and draw F at a right angle to B, and representing the pitch circle of the other wheel; draw the line G G, passing through the point W and the point T, where the pitch circles or lines E F meet, and G G will be the line of contact of the tooth of one wheel upon the tooth of the other wheel; or in other words, the pitch line of the tooth.

Fig. 241.

Fig. 241.

Fig. 242.

Fig. 242.

Draw lines, as H and I, representing the tooth breadth. From W, as a centre, draw on each side of G G dotted lines, as P, representing the height of the tooth above and below the pitch line G G. At a right angle to G G draw the line J K; and from where this line meets B, as at Q, mark the arc a, which will represent the pitch circle for the large diameter of the pinion D. [The smallest wheel of a pair of gears is termed the pinion.] Draw the arc b for the height, and circle c for the depth of the teeth, thus defining the height of the tooth at that end. Similarly from P, as a centre mark (for the large diameter of wheel C,) arcs g, h, and i, arc g representing the pitch circle, i the height, and h the depth of the tooth. On these arcs draw the proper tooth curves in the same manner as for spur wheels; that is, obtain the curves by the construction shown in Figures 237, or by those in Figures 238 and 239.