196. The modulus of the band, when the two parts of it, which intervene between the drums, are made to cross one another.

འ་

If the directions of the two parts of the band be made to cross, as shown in the accompanying figure, the moving pressure T, upon the second drum is applied to it on the side opposite to that on which it is applied when the bands do not cross; so that in this case, in order that the greatest economy of power may be attained (Art. 168.), the working pressure or resistance P, should be applied to it on the side opposite to that in which it was applied in the other case, and therefore on the side of the line C,C,, opposite to that on which the moving pressure P, upon the first drum is applied. This disposition. of the moving and working pressures being supposed, and this case being investigated by the same steps as the preceding, we shall arrive at precisely the same expressions (equations 223 and 224) for the relation of the moving and the working pressures, and for the modulus.

In estimating the value of the initial tension T (equation 225) it will, however, be found, that the angle , subtended at the centre C, of the second drum by the arc KML, which is embraced by the band, is no longer in this case represented by -, but by +a,. This will be evident if we consider that the four angles of the quadrilateral figure

C,KIL being equal to four right angles, and its angles at K and L being right angles, the remaining angles KIL and KCL are equal to two right angles, so that KC,L=”—α1; but the angle subtended by KML equals 2π-KC,L; it

equals therefore +a,. If this value be substituted for π-a, in equation (225) it becomes

Now the fraction in the denominator of this expression being essentially greater in value than that in the denominator of the preceding (equation 225), it follows that the initial tension T, which must be given to the band in order that it may transmit the work from the one drum to the other under a given resistance P, is less when the two parts of the band cross than when they do not, and, therefore, that the modulus (equation 224) is less; so that the band is worked with the greatest economy of power (other things being the same) when the two parts of it which intervene between the drums are made to cross one another. Indeed it is evident, that since in this case the arc embraced by the band on each drum subtends a greater angle than in the other case, a less tension of the band in this case than in the other is required (Art. 185.) to prevent it from slipping under a given resistance, so that the friction upon the axis of the drums which results from the tension of the band is less in this case than the other, and therefore the work expended on that friction less in the same proportion.

THE TEETH OF WHEELS.

197. Let A, B represent two circles in contact at D, and moveable about fixed centres at C, and C2. It is evident, that if by reason of the friction of these two circles upon one another at D any motion of rotation given to A be communicated to B, the angles PC,D and QC,D described in the same time by these two circles will be such as will make

the ares PD and QD which they subtend at the circumfer

ences of the circles equal to one another. Let the angle PC,D* be represented by 0,, and the angle QC,D by 2; also let the radii C,D and C,D of the circles be represented by r, and r. Now, arc PD=r,,, arc QD=r202; and since PD=QD, therefore r11=r02;

The angles described, in the same time, by two circles. which revolve in contact are therefore inversely proportional to the radii of the circles, so that their angular velocities (Art. 74.) bear a constant proportion to one another; and if one revolves uniformly, then the other revolves uniformly; if the angular revolution of the one varies in any proportion, then that of the other varies in like proportion.

When the resistance opposed to the rotation of the driven circle or wheel B is considerable, it is no longer possible to give motion to that circle by the friction on its circumference of the driving circle. It becomes therefore necessary in the great majority of cases to cause the rotation of the driven wheel by some other means than the friction of the circumference of the driving wheel.

One expedient is the band already described, by means of which the wheels may be made to drive one another at any distances of their centres, and under a far greater resistance than they could by their mutual contact. When, however, the pressure is considerable, and the wheels may be brought, into actual contact, the common and the more certain method

is to transfer the motion from one to the other by means of projections on the one wheel called TEETH, which engage in similar projections on the other.

In the construction of these teeth the problem to be solved is, to give such shapes to their surfaces of

* Or rather the arc which this angle subtends to radius unity.

(211), we obtain, in the case in which the negative sign of R1 is to be taken, or in which 2T is less than P1+W, the axis C1 resting upon the lower surface of its collar as shown in fig. 2.,

P11-P2α= (P1FP2+2W)p sin. & ;

and in the case in which the positive sign of R1 is to be taken, 2T being greater than P1 + W, and the axis C, pressing against the upper surface of its collar, as shown in fig. 1.,

=

P1a-P2a, (4T-P, FP2)p sin. p.

Transposing and reducing, we obtain for the relation between the driving and driven pressures in these two cases

respectively,

P1 =P2 a2 p sin.

Pap sin.+2Wp sin.

P1=P,

ap sin.

a1-p sin. p

a1+p sin. o

(214),

(a+p sin.) + 4Tp sin.

and therefore (by equation 121), for the moduli in the two

In all which equations the sign is to be taken according

as P, is applied on the same side of the line C,C2, joining the

2

axis as P1, 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.

Suppose the maximum resistance which may, during the

action of the machine, be opposed to the motion of the drum to be represented by a pressure P applied at a given distance a from its centre C. Suppose, moreover, that the band has received such an initial tension T as shall just cause it to be on the point of slipping when the motion of the drum is subjected to this. maximum resistance; and let t, and to be the tensions upon the two parts of the band when it is thus just in the act of slipping and of overcoming the resistance P. Now, the two parts of the band being parallel, it embraces one half of the circumference of each drum; the relation between t, and t, is therefore expressed (equation 205) by the equation.

Also, the relation between the resistance P, opposed to the motion of the upper drum, and the tensions t, and t, upon the two parts of the band, when this resistance is on the point of being overcome, is expressed (equation 212) by the equation

Pa+tar+Rap sin. =t1r;

or substituting the value of R, (equation 211), and transposing,

Pa+(2T+P+W)p sin. =(t1—tą)r ;

whence, substituting the value of t,-t,. determined above, and transposing, we have