Let D1TB1A1, D2TB2A2 be the positions, at a given instant, of
the acting surfaces of a pair of teeth in the driver and follower
respectively, touching each other
at T; the line of connexion of
those teeth is P1P2, perpendicular
to their surfaces at T. Let
C1P1, C2P2 be perpendiculars let
Fig. 101.fall from the centres of the
wheels on the line of contact.
Then, by § 36, the angular
velocity-ratio is
| a2/a1 = C1P1/C2P2. | (23) |
The following principles regulate the forms of the teeth and their relative motions:—
I. The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connexion of the teeth cuts the line of centres at the pitch-point.
For, let P1P2 cut the line of centres at I; then, by similar triangles,
| α1 : α2 : : C2P2 : C1P1 : : IC2 : : IC1; | (24) |
which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′, B2IB2′.
This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle.
II. The component of the velocity of the point of contact of the teeth T along the line of connexion is
| α1 · C1P1 = α2 · C2P2. | (25) |
III. The relative velocity perpendicular to P1P2 of the teeth at their point of contact—that is, their velocity of sliding on each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C1C2 to rotate backwards round C1 with the angular velocity α1, and the wheel 2 to rotate round C2 as before, with the angular velocity α2 relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circle rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α1 + α2. Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is
| (α1 + α2) · IT; | (26) |
so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.
IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same 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 D1B1 of the driving tooth drives the face D2B2 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 B1′A1′ of the driving tooth drives the flank B2′A2′ of the following tooth, and the teeth are sliding from each other.
The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates 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 obliquity of the action of the teeth is the angle EIT = IC1, P1 = IC2P2.
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.
§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C1, C2 be the centres of two wheels, B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P1IP2 shall represent at once the constant line of connexion of teeth and the path of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′, 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
C2P2 = IC2 · cos obliquity.
 |
| Fig. 102. |
(The obliquity which is found to answer best in practice is about 1412°; its cosine is about 3132, and its sine about 14. These values though not absolutely exact, are near enough to the truth for practical purposes.)
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 1412°, 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.
 |
| Fig. 103. |
§ 48. Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, 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 consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular 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
