for its rolling surface, as W2 in fig. 98, in which case it is a disk wheel. The rolling surfaces of actual wheels consist of frusta or zones of the complete cones or disks, as shown by W1, W2 in figs. 97 and 98.
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| Fig. 98. |
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| Fig. 99. |
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| Fig. 100. |
§ 42. Sliding Contact (lateral): Skew-Bevel Wheels.—An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids E, F, equal or unequal, be placed in the closest possible contact, as in fig. 99, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes AG, BH in opposite directions. The axes will not be parallel, nor will they intersect each other.
The motion of two such hyperboloids, turning in contact with each other, has hitherto been classed amongst cases of rolling contact; but that classification is not strictly correct, for, although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are parallel neither to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of contact which are unequal, and their difference constitutes a lateral sliding.
The directions and positions of the axes being given, and the required angular velocity ratio, the following construction serves to determine the line of contact, by whose rotation round the two axes respectively the hyperboloids are generated:—
In fig. 100, let B1C1, B2C2 be the two axes; B1B2 their common perpendicular. Through any point O in this common perpendicular draw OA1 parallel to B1C1 and OA2 parallel to B2C2; make those lines proportional to the angular velocities about the axes to which they are respectively parallel; complete the parallelogram OA1EA2, and draw the diagonal OE; divide B1B2 in D into two parts, inversely proportional to the angular velocities about the axes which they respectively adjoin; through D parallel to OE draw DT. This will be the line of contact.
A pair of thin frusta of a pair of hyperboloids are used in practice to communicate motion between a pair of axes neither parallel nor intersecting, and are called skew-bevel wheels.
In skew-bevel wheels the properties of a line of connexion are not possessed by every line traversing the line of contact, but only by every line traversing the line of contact at right angles.
If the velocity ratio to be communicated were variable, the point D would alter its position, and the line DT its direction, at different periods of the motion, and the wheels would be hyperboloids of an eccentric or irregular cross-section; but forms of this kind are not used in practice.
§ 43. Sliding Contact (circular): Grooved Wheels.—As the adhesion or friction between a pair of smooth wheels is seldom sufficient to prevent their slipping on each other, contrivances are used to increase their mutual hold. One of those consists in forming the rim of each wheel into a series of alternate ridges and grooves parallel to the plane of rotation; it is applicable to cylindrical and bevel wheels, but not to skew-bevel wheels. The comparative motion of a pair of wheels so ridged and grooved is the same as that of a pair of smooth wheels in rolling contact, whose cylindrical or conical surfaces lie midway between the tops of the ridges and bottoms of the grooves, and those ideal smooth surfaces are called the pitch surfaces of the wheels.
The relative motion of the faces of contact of the ridges and grooves is a rotatory sliding or grinding motion, about the line of contact of the pitch-surfaces as an instantaneous axis.
Grooved wheels have hitherto been but little used.
§ 44. Sliding Contact (direct): Teeth of Wheels, their Number and Pitch.—The ordinary method of connecting a pair of wheels, or a wheel and a rack, and the only method which ensures the exact maintenance of a given numerical velocity ratio, is by means of a series of alternate ridges and hollows parallel or nearly parallel to the successive lines of contact of the ideal smooth wheels whose velocity ratio would be the same with that of the toothed wheels. The ridges are called teeth; the hollows, spaces. The teeth of the driver push those of the follower before them, and in so doing sliding takes place between them in a direction across their lines of contact.
The pitch-surfaces of a pair of toothed wheels are the ideal smooth surfaces which would have the same comparative motion by rolling contact that the actual wheels have by the sliding contact of their teeth. The pitch-circles of a pair of circular toothed wheels are sections of their pitch-surfaces, made for spur-wheels (that is, for wheels whose axes are parallel) by a plane at right angles to the axes, and for bevel wheels by a sphere described about the common apex. For a pair of skew-bevel wheels the pitch-circles are a pair of contiguous rectangular sections of the pitch-surfaces. The pitch-point is the point of contact of the pitch-circles.
The pitch-surface of a wheel lies intermediate between the points of the teeth and the bottoms of the hollows between them. That part of the acting surface of a tooth which projects beyond the pitch-surface is called the face; that part which lies within the pitch-surface, the flank.
Teeth, when not otherwise specified, are understood to be made in one piece with the wheel, the material being generally cast-iron, brass or bronze. Separate teeth, fixed into mortises in the rim of the wheel, are called cogs. A pinion is a small toothed wheel; a trundle is a pinion with cylindrical staves for teeth.
The radius of the pitch-circle of a wheel is called the geometrical radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.
The distance, measured along the pitch-circle, from the face of one tooth to the face of the next, is called the pitch. The pitch and the number of teeth in wheels are regulated by the following principles:—
I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference.
In wheels which reciprocate without performing a complete revolution this condition is not necessary. Such wheels are called sectors.
II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each.
III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.
IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.
From this principle arise problems of a kind which will be referred to in treating of Trains of Mechanism.
V. Let n, N be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch-surfaces before t and T work together again (let this number be called a); and, secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b); thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c).
Case 1. If n is a divisor of N,
Case 2. If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = Md; then
Case 3. If N and n be prime to each other,
It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They therefore study that the numbers of teeth in each pair of wheels which work together shall either be prime to each other, or shall have their greatest common divisor as small as is consistent with a velocity ratio suited for the purposes of the machine.
§ 45. Sliding Contact: Forms of the Teeth of Spur-wheels and Racks.—A line of connexion of two pieces in sliding contact is a line perpendicular to their surfaces at a point where they touch. Bearing this in mind, the principle of the comparative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by applying the principles stated generally in §§ 36 and 37 to the case of parallel axes for a pair of spur-wheels, and to the case of an axis perpendicular to the direction of shifting for a wheel and a rack.
In fig. 101, let C1, C2 be the centres of a pair of spur-wheels; B1IB1′, B2IB2′ portions of their pitch-circles, touching at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower.
