axis greater than the length of crank arm, and so much

greater, that P, (1-2) may exceed W. These conditions

commonly obtain in the practical application of the crank.

261. Should it, however, be required to determine the modulus in the case in which P, is not, in every position of the arm, greater than P,+W, let it be observed, that this condition does not affect the determination of the modulus (equation 327) in respect to the descending, but only the ascending stroke; there being a certain position of the arm as it ascends in which the resultant pressure upon the axis represented by the formula {P,-(P,+W), passing through zero, is afterwards represented by (P,+W)-P; and when the arm has still further ascended so as to be again inclined to the vertical at the same angle, passes again through zero, and is again represented by the same formula as before. The value of this angle may be determined by substituting P, for P1+W in equation (324), and solving that equation in respect to . Let it be represented by ; let equation (325) be integrated in respect to the ascending stroke from 8=0 to =0,, the work of P, through this angle being represented by u,; let the signs of all the terms involving p, sin. , then be changed, which is equivalent to changing the formula representing the pressure upon the axis from {P,-(P,+W)} to {(P2+W)—P1}; and let the equation then be integrated

T

from=8, to=2, the work of P, through this angle being re

2'

presented by u,; 2(u,+u,) will then represent the whole work U, done by P, in the ascending arc. To determine this sum, divide the first integral by the co-efficient of u,, and the second by that of u,, add the resulting equations, and multiply their sum by 2; the modulus in respect to the ascending are will then be determined; and if it be added to the modulus in respect to the descending arc, the modu lus in respect to an entire revolution will be known.

THE DEAD POINTS IN THE CRANK.

362. If equation (324) be solved in respect to P, it be

comes

In that position of the arm, therefore, in which

2

the driving pressure P, necessary to overcome any given resistance P opposed to the revolution of the crank, assumes an infinite value. This position from which no finite pressure acting in the direction of the length of the connecting rod is sufficient to move the arm, when it is at rest in that position, is called its dead point.

Since there are four values of 4, which satisfy equation (330) two in the descending and two in the ascending semirevolution of the arm, there are, on the whole, four dead points of the crank.* The value of P, being, however, in all cases exceedingly great between the two highest and the two lowest of these positions, every position between the two former and the two latter, and for some distance on either side of these limits, is practically a dead point.

THE DOUBLE CRANK.

263. To this crank, when applied to the steam engine, are affixed upon the same solid shaft, two arms at right angles to one another, each of which sustains the pressure of the steam in a separate cylinder of the engine, which pressure is transmitted to it, from the piston rod, by the intervention of a beam and connecting rod as in the marine engine, or a guide and connecting rod as in the locomotive engine.

* It has been customary to reckon theoretically only two dead points of the crank, one in its highest and the other in its lowest position. Every practical man is acquainted with the fallacy of this conclusion.

Fig. 1.

In either case, the connecting rods may be supposed to remain constantly parallel to themselves, and the pressures applied to them in different planes to act in the same plane, without materially affecting the results about to be deduced.+

Let the two arms of the crank be supposed to be of the same length a; let the same driving pressure P, be supposed to be applied to each; and let the same notation be adopted in other respects as was used in the case of the crank with a single arm; and, first, let us consider the case represented in fig. 1, in which both arms of the crank are upon the same side of the centre C.

Let the angle W,CB=8; therefore W,CE=+: whence

2

it follows by precisely the same reasoning as in Art. 260., that the perpendicular upon the direction of the driving pressure applied by the connecting rod AB is represented (see equation 323) by a sin. -P, sin. 9, and the perpendicular upon the pressure applied by the rod ED by

a sin.(+)-P, sin. 9,, or a cos. §—p, sin. o̟„.

Let now a

be taken to represent the perpendicular distance from the axis C, at which a single pressure, equal to 2P,, must be applied, so as to produce the same effect to turn the crank as is produced by the two pressures actually applied to it by the two connecting rods; then, by the principle of the equality of moments,

2P,a,P,(a sin. —p, sin. 9,)+P(a cos. —p, sin. );

.. a,a (sin.+cos. 4)-p, sin..,;

This principle will be more fully discussed by a reference to the theory of statical couples. (See Pritchard on Statical Couples.)

The relative dimensions of the crank arm and connecting rod are here supposed to be those usually given to these parts of the engine; the supposition does not obtain in the case of a short connecting rod.

which expression becomes identical with the value of. a,, determined by equation (323), if in the latter equation a be

conditions of the equilibrium of the double crank in the state bordering upon motion, and therefore the form of the modulus, are, whilst both arms are on the same side of the centre. precisely the same as those of the single crank, the direction of whose arm bisects the right angle BCE, and the length of whose arm equals the length of either arm of the double crank divided by 4/2.

Now, if, be taken to represent the inclination W,CF of this imaginary arm to W,C, both arms will be found on the same side of the centre, from that position in which &

=

to that in which it equals (-). If, therefore, we substi

tute for e, in equations (326), and for a,

4

equations together, the symbol 2 U, in the resulting equation will represent the whole work yielded by the working pressure, whilst both arms remain on the same side of the centre, in the ascending and the descending arcs. We thus obtain, representing the sum of the driving pressures upon the two arms by P1,

2P, {a —— (p, sin.q, +p, sin.q,) } =2U,. . . . . (331).*

Throughout the remaining two quadrants of the revolution of the crank, the directions of the two equal and parallel pressures applied to it through the connecting rods being opposite, the resultant pressure upon the axis is represented by (P,+W), instead of {P,±(P+W)}; whilst, in other respects, the conditions of the equilibrium of the state bor

*Whewell's Mechanics, p. 25.

dering upon motion remain the same as before; that is, the

Fig. 2.

H

same as though the pressure P, were applied to an imaginary arm, whose

a

length is and whose position co

1/2

incides with CF. Now, referring to equation (324), it is apparent that this condition will be satisfied if, in that equation, the ambiguous sign of (P,+W) be suppressed, and the value of P, in the second member, which is multiplied by p, sin. „, be assumed 0; by which assumption the term -P2 sin. 91 will be made to disappear from the left-hand member of equation (325), and the ambiguous signs which affect the first and second terms of the right-hand member will become positive. Now, these substitutions being made, and the equation being then integrated, first, between the limits 0 and, and then be

3π

4'

tween the limits and, the symbol U, in it will evidently

4

represent the work done during each of those portions of a semi-revolution of the imaginary arm in which the two real arms of the crank are not on the same side of the centre. Moreover, the integral of that equation between the limits and

4 is evidently the same with its integral between the

3T

4

limits and. Taking, therefore, twice the former inte gral, we have

Dividing this equation by (a2+p, sin. ❤,), or by a,

P

(1+ A sin. 4, ),

sin. e, ), and neglecting terms above the first dimen

sion in sin. 9, and sin. ❤,,