the obliquity as small as possible; and, on the other hand, there is a connection between the obliquity of action and the number of teeth which makes it impracticable to use pinions of fewer than a certain number of teeth with less than a certain maximum obliquity of action. Mr. Willis, from an examination of the results of ordinary practice, concludes that the best value on the whole for the mean obliquity of action in toothed gearing is between 14° and 15. Such an angle may be easily constructed by drawing a rightangled triangle whose three sides bear to each other the proportion of the numbers

65: 63: 16;

when the required angle will lie opposite to the shortest side of the triangle. The values of its chief trigonometrical functions are—

The corresponding angle is 14° 15'; being a little less than one-25th part of a revolution.

130. The Teeth of Spur-Wheels and Racks have acting surfaces of the class called cylindrical surfaces, in the comprehensive sense of that term; and their figures are designed by drawing the traces of their surfaces on a plane perpendicular to the axes of the wheels (or, in the case of a rack, to the axis of the wheel that is to gear with the rack); which plane contains the pitch-lines and the line of connection, and may be represented by the plane of the paper in fig. 82, page 115. The path of contact, also, is situated in the same plane; and the angle of obliquity of action is at each instant equal to the angle IC P, which the common perpendicular, C P, of the line of connection and one of the axes makes with the line of centres, CIC. Because of the comparative simplicity of the rules for drawing the figures of the teeth of spur-wheels, those rules are used, with the aid of certain devices to be afterwards described, for drawing the figures of the teeth of bevel wheels and skew-bevel wheels also.

131. Involute Teeth for Circular Wheels. (A. M., 457.) —The simplest of all forms for the teeth of circular wheels is that in which the path of contact is a straight line always coinciding with the line of connection, which makes a constant angle with the line of centres, and is inclined at a constant angle of obliquity to the common tangent of the pitch-lines.

In fig. 84, let C1, C2, be the centres of two circular wheels, whose

pitch-circles are marked B, B. Through the pitch-point I draw the intended line of connection, P, P2, making, with the line of centres, the angle CIP the complement

of the intended obliquity.

=

From C, and C, draw C, P, and C2 P perpendicular to P, P2, with which two perpendiculars as radii describe circles (called base-circles) marked D1, D2.

D

B2

P

C2

P

2.

Suppose the base-circles to be a pair of, circular pulleys, connected by means of a cord whose course from pulley to pulley is PIP As the line of connection of those pulleys is the same with that of the proposed teeth, they will rotate with the required P velocity-ratio. Now, suppose a tracing point, T, to be fixed to the cord, so as to be carried along the path of contact, P, I P. That point will trace, on a plane rotating along with the wheel 1, part of the involute of the base-circle D1, and on a plane rotating along with the wheel 2, part of the involute of the base-circle D2, and the two curves so traced will always cut the line of connection at right angles, and touch each other in the required point of contact T, and will therefore fulfil the condition required by Article 122, page 114. The teeth thus traced are called Involute Teeth.

Fig. 84.

All involute teeth of the same pitch work smoothly together. The following is the process by which the figures of involute teeth are to be drawn in practice :

In fig. 85, let C represent the centre of the wheel, I the pitchpoint, CI the geometrical radius, BIB the pitch-circle, and let the intended angle of obliquity of action be given, and also the pitch. (In the example represented by the figure, the obliquity is supposed to be 1410, as stated in Article 129, page 120; and the wheel has 30 teeth.) Then proceed by the following rules:I. To draw the base-circle and the line of connection. About C, with the radius C P CI x cosine of obliquity (that is to say, in

=

63

65

the present example, CI), draw a circle, D P D; this is the base-circle. Then about I, with a radius I PCI x sine of

16 65

obliquity (that is to say, in the present example, C I), draw a

short circular arc, cutting the base-circle in P. Draw the straight line PFI E; this will be the line of connection; and it will touch the base-circle at P.

TI. To find the normal pitch, the addendum, and the real

122

radius, and to draw the addendum-circle and the flank-circle. At the pitch-point, I, draw the straight line I A, touching the pitch

A

E

G'

circle, and lay off upon it the length I A equal to the pitch. From A let fall A E perpendicular to I E. Then I E will be what may be called the NORMAL PITCH, being the distance, as measured along the line of connection, from the front of one tooth to the front of the next tooth.

The normal pitch is also the pitch on the base-circle; that is, the distance, as measured on the base-circle, between the front of one tooth and the front of the next.

The ratio of the normal pitch of involute teeth to the circular pitch is equal to the ratio of the radius of the base-circle to that of the pitch-circle; that is to say,

I E C P

= cosine of obliquity (=

=

63

65

In order that two pairs of teeth at least may always be in action, the arc of contact is to consist of two halves, each equal to the pitch (see Article 128, page 119). Lay off on the line of connection, E P, the distance I F = I E. Then E F will be the path of contact (Article 127, page 119), consisting of two halves, each equal to the normal pitch.

Draw the straight line CE; this will be the real radius, and the circle E G G', drawn with that radius, will be the addendumcircle, which all the crests of the teeth are to touch. Then, with the radius C F, draw the circle F H (marked with dots in the figure); this may be called the FLANK-CIRCLE, for it marks the inner ends of the flanks of all the teeth.

III. To draw the ROOT CIRCLE; that is, the circle which the bottoms of all the hollows between the teeth (or CLEARING CURVES, as they are called) are to touch. First find, by drawing or by calculation, the greatest addendum of any wheel with which the given wheel may have to gear; that is, the addendum of the smallest practicable pinion of the same pitch and obliquity; that is, the addendum of a pinion in which the pitch subtends at the centre an angle approximately equal to the obliquity. With the obliquity already stated, such a pinion has 25 teeth. To find the addendum of such a pinion by drawing:-Through F, parallel to P C, draw F L, as cutting IC in L. Join LE; then LE LI will be the required greatest addendum. To find the greatest addendum by calculation, let denote the obliquity, and p the pitch; then

Ф

LE LI

=

-

cotan
P { √ (3 sin2 + 1) − 1 } .

With the angle of obliquity already stated, this gives

LE LI = 0.343 p, very nearly;

and this is the origin of the value 0.35 p, which is very commonly used for the addendum of teeth.

To the greatest addendum, thus found, add a suitable allowance for clearance (Article 125, page 116), and lay off the sum I K inwards from the pitch-circle along the radius. Then CK will be the radius of the required root-circle.

IV. To draw the traces of the teeth. Mark the pitch-points of the fronts of the teeth (I, I', &c.), according to the principles of Article 121, page 113, and those of their backs, by laying off a suitable thickness on the pitch-circle (see Article 125, page 116). Obtain a "templet," or thin flat disc of wood or metal, having its edge accurately shaped to the figure of the base-circle. Such a templet is represented in plan by C D D, fig. 86, and in elevation

by D' D'. A piece of watch-spring, marked P M in plan, and P' M' in elevation, is to have its edges filed so as to leave a pair of sharp

projecting tracing-points, marked T', t, in elevation, and T in plan. One end of that spring, P, P', is to have a round hole drilled in it, and to be fixed to the middle of the edge of the templet by means of a screw, about which the spring is to be free to turn; and the other end, M, M', is to be fitted with a knob to hold it by. Place the templet on the drawing (or pattern, as the case may be), so that C shall coincide with the centre of the wheel, and D D with the base-circle; and also so that the lower of the two tracing-points, when the spring is moved to and fro, shall pass through the pitchpoint of a tooth; then that tracing-point will draw the trace of the front of the tooth; and by turning the templet about C, and repeating the process, the traces of the fronts of any required number of teeth may be drawn.

To draw the traces of the backs of the teeth, the position of the spring relatively to the templet is to be reversed, by turning it about the screw at P, so as to use the tracing-point that was previously uppermost.

The distance, PT, from the screw to the tracing-points should not be less than twice the normal pitch.

V. The Clearing Curves are the traces of the hollows which lie inside the flank-circle, F H, fig. 85. Their side parts ought to be tangents to the inner ends of the flanks of the teeth (at F and H, for example), and their bottom parts ought to coincide with the root-circle through K. Those different parts may be joined to each other by means of small circular arcs. In connection with the figures of the side parts of those clearing curves, it may be observed, that FL is a tangent to the inner end of the flank I' F, and therefore to the clearing curve at that point; and that tangents to the inner ends of other flanks may be drawn by re