APPLIED MECHANICS.] MECHANICS 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 with th-it 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. 54. 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 insures 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, Kind 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 con tiguous 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, ie 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 stares 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 adden dum 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 Uiat, 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 con tinuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be ex pressible 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 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 J) ; 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, N > = ; c = l (20). CASE 2. If the greatest common divisor of N and n be d, a number less than n, so that n = md, N = M<Z ; then CASE 3 If X and n be prime to each other, a = N; 6=N; c = n (22). It is considered desirable by millwrights, with a view to the pre servation 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. 55. Sliding Contact Forms of the Teeth of Spur-u-hcels and Racks. A line of connexion of two pieces in sliding contact is a line perpendicular to their sur faces at a point where they touch. Bearing this in mind, the principle of the com parative motion of a pair of teeth belonging to a pair of spur-wheels, or to a spur-wheel and a rack, is found by apply ing the principles stated gene- rally in sects. 46 and 47 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. 15, let C,, C 2 be the centres of a pair of spur- wheels ; BjIB j, B 2 IB 2 portions of their pitch-circles, touch ing at I, the pitch-point. Let the wheel 1 be the driver, and the wheel 2 the follower. Let D^BjAj, D 2 TB 2 A 2 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 PjP 2 , perpendicular to their surfaces at T. Let CjPj, C 2 P 2 be perpendiculars let fall from the centres of the wheels on the line of contact. Then, by sect. 46, the angular velocity- ratio is Fi ^=^ (23). dj L^ 2 i 2 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 PjP 2 cut the line of centres at I ; then, by similar triangles, which is also the angular velocity ratio due to the rolling contact of the circles BjIB ,, B 2 1B 2 . 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 C 1 1C 8 , 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 C P = Oa C P (25) III. The relative velocity perpendicular to PjP 2 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 CjC 2 to rotate backwards round C t with the angular velocity a 1} and the wheel 2 to rotate round C 2 as before, with the angular velocity a., relatively to the line of centres CjC,,, 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 tctal motion given by the principles of sects. 37 and 38, that is, a rotation about the instantaneous axis I, with the angular velocity oj + a. 2 . Hence the velocity of sliding is that due to this rotation about I, with the radius IT ; that is to say, its value is (aj + a,). 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