(27)
(The obliquity which is found to answer best in practice is about 141⁄2°; its cosine is about 31/22, and its sine about 1⁄4. These values though not absolutely exact, are near enough to the truth for practical purposes.)
| Fig. 102. |
Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P1IP2. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P1IP2, that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D1D1′, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D2D2′; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45.
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 P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if distances equal to the pitch be marked off either way from I towards P1 and P2 respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-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 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 141⁄2°, the addendum is about 3.1 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 connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.
§ 47. Teeth for a given Path of Contact: Sang’s 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 § 45, 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 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.