R=Q&@tan..

Let a represent the length of the arm AP, and a, the length of the perpendicular let fall from A upon the direction of a tangent to that point in the circumference of the wheel where the end EC of the band leaves it.

Then, neglecting the friction of the axis A, we have (Art. 5.)

P. a1 =Q. a2;

Pa, tan.
R= ε

.....

(207).

If P, represent any pressure applied to the circumference of the break wheel, and P, a pressure applied to the working point of the machine, whatever it may be, to which the break is applied. and if P1=aP,+b (Art. 152.) represent the relation between P, and P, in the inferior state bordering upon motion by the preponderance of P,; then, when P, is taken in this expression to represent the pressure W, whose action upon the working point of the machine the break is intended to control, P will represent that value R of the friction upon the break which must be produced by the intervention of the lever to control the action of the pressure W upon the machine; so that taking R to represent the same quantity as in equation (207), we have

R=aW+b.

Eliminating R between this equation and equation (207), and solving in respect to P,

THE BAND.

100. When the circular motion of any shaft in a machine, and the pressure which accompanies that motion, constituting together with it the work of the shaft, are to be communicated to any other distant shaft, this communication is

usually established by means of a band of leather, which passes round drums fixed upon the two shafts, and has its extremities drawn together with a certain pressure and united, so as to produce a tension, which should be just that necessary to prevent the band from slipping upon the drums, subject to the pressure under which the work is transferred. The facility with which this communication of rotatory motion may be established or broken at any distance and under almost every variety of circumstance, has brought the band so extensively into use in machinery, that it may be considered as a principal channel through which work is made to flow in its distribution to the successive stages of every process of mechanism, carried on in the same workshop or manufactory.

191. The sum of the tensions upon the two parts of a band is the same, whatever be the pressure under which the band is driven, or the resistance overcome, the tension of the driving part of the band being always increased by just so much as that of the driven part is diminished.

This principle was first given by M. Poncelet; it has since been amply confirmed by the experiments of M. Morin.* It may be proved as follows:-In the very commencement of the motion of that drum to which the driving pressure is applied, no motion is communicated by it to the other drum. Before any such motion can be communicated to the latter, a difference must be produced between the tensions of the two parts of the band sufficient to overcome the resistance, whatever it may be, which is opposed to the revolution of the driven drum. Now, an increase of the tension on the driving side of the band must be followed by an elongation of that side of the band (since the band is elastic), and by the revolution of the circumference of the driving drum through a space precisely equal to this elongation. Supposing, then, the other, or driven side of the band, to remain extended, as before, in a straight line between its two points of contact with the drums, this portion of the band

*Nouvelles Expériences sur le Frottement, &c. Metz.

No demonstration appears to have been given of it by M. Poncelet.

must evidently have contracted by precisely the length through which the circumference of the driving drum has revolved, or the driving side of the band elongated. Thus, the elongation of the driving side of the band is precisely equal to the contraction of the driven side. Now, the band being supposed perfectly elastic, the increase or diminution of its tension is exactly proportional to the increase or diminution of its length. The increase of tension on the one side, produced by a given elongation, is therefore precisely equal to the diminution of tension produced by a contraction equal to that elongation on the other side. Thus, if T represent the tension upon each side of the band before the driving pressure, whatever it may be, was applied, and if T, and T, represent the tensions upon the driving and the driven sides of the band after that pressure is applied; then, since T,-T represents the increase of tension on the one side, and T-T, the diminution of tension on the other, T,-T=T-T,;

It is a great principle of the economy of power in the use of the band to adjust this initial tension T, so that it may just be sufficient to prevent the band from slipping upon the drum under any pressure which it is required to transmit. The means of making this adjustment will be explained hereafter.

THE MODULUS OF THE BAnd.

192. For simplifying the consideration of this important element in machinery, we shall first consider a particular case of its application. Let the two drums, whose axis are C, and C, be supposed equal to one another, so that the two parts of the band which pass round them may be parallel.

Let, moreover, the centres of the two drums be in the same vertical straight line, so that the two parts of the band may be vertical.

Let P, and P, be pressures applied, in vertical directions, to turn the drums, and at perpendicular distances from their centres, represented by C,P, and C,P,; of which pressures P, is the working or driven pressure,

or that which is upon the point of yielding by the prepon derance of the other P. In fig. 1. P, is seen applied on the same side of the centre of the drums as P,, and in fig. 2. on the opposite side. Let T, and T, represent the tensions upon the two parts of the band, T, being that on the driving, and T, that on the driven side.

a1 =C,P,, a,=C,P2,
r-radius of each drum,
Wweight of each drum,

pradius of axis of each drum,

R, and R, resistances of axes of drums,
limiting angle of resistance.

Now, the parallel pressures P,, W, T,, T., R,, applied to the
lower drum, are in equilibrium; therefore (Art. 16.),
R1 = ±(T,+T,-P1-W);

or substituting for T,+T, its value 2T (equation 209), R1 = ±(2T-P,—W) . . . . . (210).

The sign being taken according as 2T is greater or less than P,+W, or according as the axis of the lower drum presses upon the upper surface of its bearings, as shown in fig. 1., or upon the lower surface, as shown in fig. 2. In like manner, the pressures P, W, T, T, R,, applied to the upper drum, being in equilibrium,

R=T,+T,P2+W,

or (equation 209) R,=2T‡P2+W . . . . (211),

where the sign is to be taken according as P, is applied on the same side of the axis as P,, or on the opposite side.

Since, moreover, R, and R, act, in the state bordering upon motion, at perpendicular distances from the centre of the axis, which are each represented by p sin. (Art. 153.), we have, by the principle of the equality of moments,

2

· P1a,+T2r=T12r+R1p sin. o \
Pa+T+Rp sin. Tr

(212),

observing that the resultant of all the pressures applied to each drum (excepting only the resistance of its axis) must be such as would alone communicate motion to it in the direc tion in which it actually moves, and therefore that the re sistance of the axis, which is opposite to this resultant, must tend to communicate motion to the drum in a direction oppo site to that in which it actually moves.

Subtracting the above equations, and transposing,
P1,a,-P,α=(R, +R2) p sin. 9.

Substituting the values of R, and R, from equations (210) and (211), we obtain, in the case in which the negative sign of R, is to be taken, or in which 2T is less than P, +W, the axis C, resting upon the lower surface of its collar as shown in fig. 2.,

P1a,-P2a,=(P,‡P,+2W) p sin. ❤ ;

and in the case in which the positive sign of R, is to be taken, 2T being greater than P+W, and the axis C, pressing against the upper surface of its collar, as shown in fig. 1., P1a,-P,a=(4T-P,FP) p sin. . Transposing and reducing, we obtain for the relation between the driving and driven pressures in these two cases respectively,

In all which equations the sign is to be taken according as P, is applied on the same side of the line C,C,, joining the axis as P, or on the opposite side.

193. To determine the initial tension T upon the band, so that it may not slip upon the surface of the drum when subjected to the given resistance opposed to its motion by the work.