758 MECHANICS [APPLIED MECHANICS. position, the path of contact coincides with it, and is straight ; in other cases the path of contact is curved. It is divided by the pitch-point I into two parts, the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres. During the approach, the flank D 1 B 1 of the driving tooth drives the face D.,B 2 of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face ~BA. of the driving tooth drives the flank B .,A 2 of the following tooth, and the teeth are sliding/rowi each other. The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess ter minates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact ; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact. V. The obliqiiity of the action of the teeth is the angle EIT = ICiPi-ICjPj. In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15*. nor the maximum value 30. It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation 26 the difference of the angular velocities should be substituted for their sum. 56. Involute Teeth. The simplest form of tooth which fulfils the conditions of sect. 55 is obtained in the following manner (see fig. 16). Let C x , C 2 be the centres of two wheels, BJB j, B 2 IB 2 their pitch-circles, I the pitch-point ; let the obliquity of action of the teeth be constant, so that the same straight line PjIPg shall represent at once the constant line of connexion of teeth and the path of contact. Draw C^P^ C 2 P 2 perpendicular to PjIPg, and with those lines as radii describe about the centres of the wheels the circles DjD j, D.jD 2 , called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that CjP^IC^ . cos obliquity C 2 P 2 = IC 2 . cos obliquity (The obliquity which is found to answer best in practice is about 14^ ; its cosine is about f-|, and its sine about . These values, though not absolutely exact, are near enough to the truth for practical purposes.) Suppose the base-circles to bi a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is PJPo. As the line of connexion of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity ratio. B 2 Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact PiIP. 2 , that point will trace on a plane rotating along with the wheel 1 part of the in volute of the base-circle DjD j, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D 2 D 2 ; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the con dition required by Principle I. of sect. 55. Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle ; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel. All involute teeth of the same pitch work smoothly together. To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along PjIP 2 , is less than the pitch iu the ratio of cos obliquity: 1 ; and consequently that, if distances equal to the pitch be marked off either way from I towards Pj and P., respectively, as the extremities of the path of contact, and if, according to Principle IV. of sect. 55, the adden dum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz., about 2pyth times the pitch ; and with this length of path, and the obliquity already mentioned of 14.^, the addendum is about f^ths of the pitch, The teeth of a rack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of con nexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action. 57. Teeth for a given Path of Contact Mr Sancfs Method. In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz., the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of sect. 55, and by equation 25. This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Mr Edward Sang. All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set. 58. Teeth traced by Rolling Curves. If any curve R (fig. 17) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from sect. 37, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that con sequently the line AT, traced by a tracing- point T, fixed to the rolling curve upon the plane of the wheel, will lie everywhere perpen dicular to the straight line TI ; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If the same rolling curve R, with the same tracing-point T, be rolled on the outside of any other pitch-circle, it will have the face of a tooth suitable to work with the flank AT. In like manner, if either the same or any other rolling curve R be rolled the opposite way, on the outside of the pitch-circle BB, so that the tracing point T shall start from A, it will trace the face AT of a tooth suitable to work with a flank traced by rolling the same curve R with the same tracing-point T inside any other pitch- circle. The figure of the path of contact is that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E l E , as the case may be) at a fixed point I (or I ). If the same rolling curve and tracing point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form a set. The teeth of a rack, of the same set, are traced by rolling the rolling curve on both sides of a straight line. The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces ; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-sur faces are in rolling contact. 59. Hpicycloidal Teeth. The most convenient rolling curve is- the circle. The path of contact which it traces is identical with itself ; and the flanks of the teeth are internal and their faces ex ternal epicycloids for wheels, and both flanks and faces are cycloids for a rack. For a pitch-circle of twice the radius of the rolling or describing circle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex and incurved or under-cut, which would be inconvenient ; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing Fig- 18. circle, so that the flanks may be either straight or concave. In fig. 18, let BB be part of the pitch-circle of a wheel with epi-
cycloidal teeth ; CIC the line of centres ; I the pitch-point ; EIE A