To SET OUT THE TEETH OF WHEELS.

204. All the teeth of the same wheel are constructed of the same form and of equal dimensions: it would, indeed, evidently be impossible to construct two wheels with different numbers of teeth, which should work truly with one another, if all the teeth on each wheel were not thus alike.

motion. AB represents a ring of metal, fixed in position, and having teeth cut upon its concave circumference. C is the centre of a wheel, having teeth cut in its circumference to work with those upon the circumference of the ring, and having the diameter of its pitch circle equal to half that of the pitch circle of the teeth of the ring. This being the case, it is evident, that if the pitch circle of the wheel C were made to roll upon that of the ring, any point in its circumference would describe a straight line passing through the centre D of the ring; but the circle C would roll upon the ring by the mutual action of their teeth as it would by the contact of their pitch circles; if the circle C then be made to roll upon the ring by the intervention of teeth cut upon both, any point in the circumference of C will describe a straight line passing through D. Now, conceive C to be thus made to roll round the ring by means of a double or forked link CD, between the two branches of which the wheel is received, being perforated at their extremities by circular apertures, which serve as bearings to the solid axis of the wheel. At its other extremity D, this forked link is rigidly connected with an axis passing through the centre of the ring, to which axis is communicated the circular motion to be converted by the instrument into an alternating rectilinear motion. This circular motion will thus be made to carry the centre C of the wheel round the point D, and, at the same time, cause it to roll upon the circumference of the ring. Now, conceive the axis C of the wheel, which forms part of the wheel itself, to be prolonged beyond the collar in which it turns, and to have rigidly fixed upon its extremity a bar CP. It is evident that a point P in this bar, whose distance from the axis C of the wheel equals the radius of its pitch circle, will move precisely as a point in the pitch circle of the wheel moves, and therefore that it will describe continually a straight line passing through the centre D of the ring. This point P receives, therefore, the alternating rectilinear motion which it was required to communicate.

All the teeth of a wheel are therefore set out by the workman from the same pattern or model, and it is in determining the form and dimensions of this single pattern or model of one or more teeth in reference to the mechanical effects which the wheel is to produce, when all its teeth are cut out upon it and it receives its proper place in the mechanical combination of which it is to form a part, that consists the art of the description of the teeth of wheels.

The mechanical function usually assigned to toothed wheels is the transmission of work under an increased or diminished velocity. If CD, DE, &c. represent arcs of the pitch circle

of a wheel intercepted between similar points of consecutive teeth (the chords of which arcs are called the pitches of the teeth), it is evident that all these arcs must be equal, since the teeth are all equal and similarly placed; so that each tooth of either wheel, as it passes through its contact with a corresponding tooth of the other, carries its pitch line through the same space CD, over the point of contact C of the pitch lines. Since, therefore, the pitch line of the one wheel is carried over a space equal to CD, and that of the other over a space equal to cd by the contact of any two of their teeth, and since the wheels revolve by the contact of their teeth as they would by the contact of their pitch circles at C, it follows that the arcs CD and cd are equal. Now let r Τι and r, represent the radii of the pitch circles of the two wheels,

then will 2r, and 2r, represent the circumferences of their pitch circles; and if n, and no represent the numbers of

2πτ, N1

teeth cut on them respectively, then CD= and cd=

(227);

Therefore the radii of the pitch circles of the two wheels must be to one another as the numbers of teeth to be cut upon them respectively.

Again, let m, represent the number of revolutions made by the first wheel, whilst ma revolutions are made by the second; then will 2arm, represent the space described by the circumference of the pitch circle of the first wheel while these revolutions are made, and 27rm, that described by the circumference of the pitch circle of the second; but the wheels revolve as though their pitch circles were in contact, therefore the circumferences of these circles revolve through equal spaces, therefore 2ar,m1 = 2πr2m2;

The radii of the pitch circles of the wheels are therefore inversely as the numbers of revolutions made in the same time by them.

Equating the second members of equations (227) and (228),

The numbers of revolutions made by the wheels in the same time are therefore to one another inversely as the numbers of teeth.

205. In a train of wheels, to determine how many revolutions the last wheel makes whilst the first is making any given number of revolutions.

When a wheel, driven

9

by another, carries its axis round with it, on which axis a third wheel is fixed, engaging with and giving motion to a fourth, which, in like manner, is fixed upon its axis, and carries round with it a fifth wheel fixed upon the same

axis, which fifth wheel engages with a sixth upon another axis,

and so on as shown in

the above figure, the combination Let n1, N2, N3,

forms a train of wheels. nap represent the numbers of teeth in the successive wheels forming such a train of p pairs of wheels; and whilst the first wheel is making m revolutions, let the second and third (which revolve together, being fixed on the same axis) make m, revolutions; the fourth and fifth (which, in like manner, revolve together) m, revolutions, the sixth and seventh m2, and so on; and let the last or 2p" wheel thus be made to revolve m, times whilst the first revolves m times. Then, since the first wheel which has n, teeth gives motion to the second which has no teeth, and that whilst the former makes m revolutions the latter makes m1 revolutions, therefore (equation 229), = and since, while the third wheel (which revolves with the second) makes m, revolutions, the fourth makes m, revolutions;

m

which has ng teeth, makes m, revolutions (revolving with the fourth), the sixth, which has ng teeth, makes m, revolu

nator of the first member of the equation which results from their multiplication, we obtain

The factors in the numerator of this fraction represent the numbers of teeth in all the driving wheels of this train, and those in the denominator the numbers of teeth in the driven wheels, or followers as they are more commonly called.

If the numbers of teeth in the former be all equal and represented by n,, and the numbers of teeth in the latter also equal and represented by n,, then

Having determined what should be the number of teeth in each of the wheels which enter into any mechanical combination, with a reference to that particular modification of the velocity of the revolving parts of the machine which is to be produced by that wheel*, it remains next to consider, what must be the dimensions of each tooth of the wheel, so that it may be of sufficient strength to transmit the work which is destined to pass through it, under that velocity, or to bear the pressure which accompanies the transmission of that work at that particular velocity; and it remains further to determine, what must be the dimensions of the wheel itself consequent upon these dimensions of each tooth, and this given number of its teeth.

206. To determine the pitch of the teeth of a wheel, knowing the work to be transmitted by the wheel.

Let U represent the number of units of work to be transmitted by the wheel per minute, m the number of revolutions

* The reader is referred for a more complete discussion of this subject (which belongs more particularly to descriptive mechanics) to Professor Willis's Principles of Mechanism, chap. vii., or to Camus on the Teeth of Wheels, by Hawkins, p. 90.