that is, the line of action; and let C1 P1, C, P2, be the common perpendiculars of the line of action and of the two axes respectively.

Then at the given instant, the components

along the line P, P2 of the velocities of the points P1, P2, are equal. Let 1, 2, be the angles which that line makes with the directions of the axes respectively. Let a1, a2, be the respective angular velocities of the moving pieces; then

which is the principle stated above.

When the line of action is perpendicular in direction to both axes, then sin i sin i2 = 1; and Equation 1 becomes

When the axes are parallel, i1 = i2 Let I be the point where the line of action cuts the plane of the two axes; then the triangles P1 C1 I, P2 C2 I, are similar; so that Equation 1 A is equivalent to the following:

1

CASE 3. A rotating piece and a shifting piece, in sliding contact, have their comparative motion regulated by the following principle: Let C P denote the perpendicular distance from the axis of the rotating piece to the line of action; i the angle which the direction of the line of action makes with that axis; a the angular velocity of the rotating piece; v the linear velocity of the sliding piece; j the angle which its direction of motion makes with the line of action; then

.(2.) When the line of action is perpendicular in direction to the axis of the rotating piece, sin i = 1; and

v=a • CP · sec j = α

∙1C;

(2A.)

where I C denotes the distance from the axis of the rotating piece to the point where the line of action cuts a perpendicular from that axis on the direction of motion of the shifting piece.

PITCH AND NUMBER OF TEETH.

81

142. Teeth of Wheels.-The most usual method of communicating motion between a pair of wheels, or a wheel and a rack, and the only method which, by preventing the possibility of the rotation of one wheel unless accompanied by the other, insures the preservation of a given velocity-ratio exactly, is by means of the projections called teeth.

The pitch surface of a wheel is an ideal smooth surface, intermediate between the crests of the teeth and the bottoms of the spaces between them, which, by rolling contact with the pitch surface of another wheel, would communicate the same velocity-ratio that the teeth communicate by their sliding contact. In designing wheels, the forms of the ideal pitch surfaces are first determined, and from them are deduced the forms of the teeth.

Wheels with cylindrical pitch surfaces are called spur wheels; those with conical pitch surfaces, bevel wheels; and those with hyperboloidal pitch surfaces, skew-bevel wheels.

The pitch line of a wheel, or, in circular wheels, the pitch circle, is a transverse section of the pitch surface made by a surface perpendicular to it and to the axis; that is, in spur wheels, by a plane perpendicular to the axis; in bevel wheels, by a sphere described about the apex of the conical pitch surface; and in skew-bevel wheels, by any oblate spheroid generated by the rotation of an ellipse whose foci are the same with those of the hyperbola that generates the pitch surface.

The pitch point of a pair of wheels is the point of contact of their pitch lines; that is, the transverse section of the line of contact of the pitch surfaces.

Similar terms are applied to racks.

That part of the acting surface of a tooth which projects beyond the pitch surface is called the face; that which lies within the pitch surface, the flank.

The radius of the pitch circle of a circular wheel is called the geometrical radius; that of a circle touching the crests of the teeth is called the real radius; and the difference between those radii, the addendum.

143. Pitch and Number of Teeth.-The distance, measured along the pitch line, from the face of one tooth to the face of the next, is called the PITCH.

The pitch, and the number of teeth in circular 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.

G

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.

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); secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b); and 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

a = m N = M n = Mmd; b = M; c=m......... .(2.)

Case 3. If N and n be prime to each other,

a=Nn; b=N; c=n...................

.(3.)

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 to make the numbers of teeth in each pair of wheels which work together such as to be either prime to each other, or to have their greatest common divisor as small as is possible consistently with the purposes of the machine.

VI. The smallest number of teeth which it is practicable to give to a pinion (that is, a small wheel), is regulated by the principle, that in order that the communication of motion from one wheel to another may be continuous, at least one pair of teeth should always be in action; and that in order to provide for the contingency of a tooth breaking, a second pair, at least, should be in action also. For reasons which will appear when the forms of teeth are considered, this principle gives the following as the least numbers of

A TRAIN OF WHEELWORK.

83

teeth which can be usually employed in pinions having teeth of the three classes of figures named below, whose properties will be explained in the sequel:

I. Involute teeth,.....

II. Epicycloidal teeth,......

III. Cylindrical teeth, or staves,.

.25.

12.

6.

144. Hunting Cog.-When the ratio of the angular velocties of two wheels, being reduced to its least terms, is expressed by small numbers, less than those which can be given to wheels in practice, and it becomes necessary to employ multiples of those numbers by a common multiplier, which becomes a common divisor of the numbers of teeth in the wheels, millwrights and engine-makers avoid the evil of frequent contact between the same pairs of teeth, by giving one additional tooth, called a hunting cog, to the larger of the two wheels. This expedient causes the velocity-ratio to be not exactly but only approximately equal to that which was at first contemplated; and therefore it cannot be used where the exactness of certain velocity-ratios amongst the wheels is of importance as in clockwork.

145. A Train of Wheelwork consists of a series of axes, each having upon it two wheels, one of which is driven by a wheel on the preceding axis, while the other drives a wheel on the following axis. If the wheels are all in outside gearing, the direction of rotation of each axis is contrary to that of the adjoining axes. In some cases, a single wheel upon one axis answers the purpose both of receiving motion from a wheel on the preceding axis and giving motion to a wheel on the following axis. Such a wheel is called an idle wheel: it affects the direction of rotation only, and not the velocity-ratio.

Let the series of axes be distinguished by numbers 1, 2, 3, &c. m; let the numbers of teeth in the driving wheels be denoted by N's, each with the number of its axis affixed; thus, N1, N2, &c. Nm-1; and let the numbers of teeth in the driven or following wheels be denoted by n's, each with the number of its axis affixed; thus, n2, no, &c. nm. Then the ratio of the angular velocity a of the mth axis to the angular velocity a, of the first axis is the product of the m− 1 velocity-ratios of the successive elementary combinations, viz. :—

that is to say, the velocity-ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers; and it is obvious that so long as the same drivers and followers constitute the train,

the order in which they succeed each other does not affect the resultant velocity-ratio.

Supposing all the wheels to be in outside gearing, then as each elementary combination reverses the direction of rotation, and as the number of elementary combinations, m - 1, is one less than the number of axes, m, it is evident that if m is odd, the direction of rotation is preserved, and if even, reversed.

It is often a question of importance to determine the numbers of teeth in a train of wheels best suited for giving a determinate velocity-ratio to two axes. It was shewn by Young, that to do this with the least total number of teeth, the velocity-ratio of each elementary combination should approximate as nearly as possible 3.59. This would in many cases give too many axes; and as a useful practical rule it may be laid down, that from 3 to 6 ought to be the limit of the velocity-ratio of an elementary combination in wheelwork.

B

C

Let be the velocity-ratio required, reduced to its least terms, and let B be greater than C.

B

If is not greater than 6, and C lies between the prescribed C minimum number of teeth (which may be called t), and its double 2 t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are if possible to be resolved into factors, and those factors, or if they are too small multiples of them, used for the numbers of teeth. Should B or C, or both, be at once inconveniently large, and prime, then instead of the exact ratio

B

C'

some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions. See MATHEMATICAL INTRODUCTION, page 2, Article 4.

B Should be greater than 6, the best number of elementary C combinations is found by dividing by 6 again and again till a quotient is obtained less than unity, when the number of divisions will be the required number of combinations, m – 1.

Then, if possible, B and C themselves are to be resolved each into m-1 factors, which factors, or multiples of them, shall be not less than t, nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity-ratio, found

B

by the method of continued fractions, is to be substituted for as C' before. When the prime factors of either B or C are fewer in