APPLIED MECHANICS.] MECHANICS 759 a straight tangent to the pitch-circle at that point; R the internal and R the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID be the path of con tact, consisting of the arc of approach DI and the arc of recess ID . In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The obliquity of the action in passing the line of centres is no thing; the maximum obliquity is the angle EID = E ID; and the mean obliquity is one-half of that angle. It appears from experience that the mean obliquity should not exceed 15 ; therefore the maximum obliquity should be about 30; therefore the equal arcs DI and ID should each be one-sixth of a circumference ; therefore the circumference of the describing circle should be six times the pitch. It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth. 60. Nearly Epicycloidal Teeth Willis s Method. -To facili tate the drawing of epicycloidal teeth in practice, Willis has shown how to approximate to their figure by means of two circular arcs, one concave, for the flank, and the other convex, for the face, and each having for its radius the mean radius of curvature of the epicycloidal arc. Willis s formulae are founded on the fol lowing properties of epicycloids : Let R be the radius of the pitch-circle ; r that of the describing circle ; 6 the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I ; that is, the obliquity of the action at T. Then the radius of curvature of the epicycloid at T is For an internal epicycloid, p = 4rsin0 p - r- For an external epicycloid, p =4? sin0p -- iv ~T T Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2? sin0 ; and therefore R For the flank, p- For the face, p -c = 2rsinO R-2r R (29). X P 2 N - 12 p N 2 N + 12 (30). R + 2r For the proportions approved of by Willis, sin0 = nearly; r=p (the pitch) nearly ; c = p nearly ; and, if N be the number of r 6 teeth in the wheel, p = ^ nearly ; therefore, approximately, Hence the following construction (fig. 19). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac = p . Draw radii bd, ce; draw/6, eg, making angles of 75| with those radii. Make bf= p -c, cy=p- c. From /, with the radius fa, draw the circular arc ah ; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required. To facilitate the application of this rale, Willis published tables of p-c and p -c, and invented an instrument called the odontograph. " 61. Trundles and Pin-Wheels. If a wheel or trundle have cylin drical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epi cycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels. 62. Backs of Teeth and Spaces. Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch-circle, is made about 1th part wider than the thickness of the tooth on the pitch -circle ; that is to say, Thickness of tooth = T B r pitch ; Width of space = T T pitch . The difference of ^ of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about y^th of the pitch. 63. Stepped and Helical Teeth. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel- work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end ; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has com menced. Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and ex pensive workmanship than common teeth ; and helical teeth nre, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery. 64. Teeth of Bevel- Wheels, The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex. The operations of describing the exact figures of the teeth of bevel- wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bi vel-wliecls all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substituting poles for centres, and great circles for straight lines. In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used : Let (fig. 20) be the common apex of a pair of bevel-wheels; OBJ, OB 2 I their pitch cones ; OC^ OC 2 their axes ; 01 their line of contact. Perpendicular to 01 draw AJA,, cutting the axes in Aj, A 2 ; make the outer rims of the patterns and of the wheels portions of the cones A 1 B 1 I, A 2 B 2 I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about for practical pur poses. To find the figures of the teeth, draw on a flat surface circular arcs IDj, ID 2 , with the radii A X I, A 2 I ; those arcs will be the developments of arcs of the pitch-circles BJ, B. 2 I, when the conical surfaces AJ^I, A S B 2 I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels ; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces. 65. Teeth of Skew-Bevel Wheels. The, crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface ; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch-surface is a cone touching the hyperboloidal surface at the given circle. 66. Cams. A cam is a single tooth, oither rotating continuously or oscillating, and driving a sliding or turning piece either con stantly or at intervals. All the principles which have been stated in sect. 55 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into considera tion the form of the ideal pitch -surf ace, which would give the same comparative motion by rolling contact that the cam gives by sliding contact. 67. Screws. The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, windinc round it. Convex and concave screws are distinguished technically by the respective names of male and female ; a short concave screw is called a nut ; and when a screw is spoken of with out qualification a convex screw is usually understood. The relation between the advance and the rotation, which comfo: the motion of a screw working in contact with a fixed s helical guide, has already been demonstrated in sect. 41 ; and the same relation exists between the magnitudes of the rotation ot a screw about a fixed axis and the advance of a shifting nut in wluct it rotates. The advance of the nut takes place in the opposifc direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the mean-
in" assigned to it in that section, viz. - the distance, measured