If the influence of the work accumulated in the arms of the wheel be given in, for an increase of the equalising power beyond the prescribed limits, that accumulated in the heavy rim or ring which forms its periphery being alone taken into the account; then (Art. 86.) Mk2=I=2bсR (R2++c2), where b represents the thickness, c the depth, and R the mean radius of the rim. But by Guldinus's first property (Art. 38.), 2 bсR=M; therefore k2=(R2+c2). Substituting in equation (343)

If the depth c of the rim be (as it usually is) small as compared with the mean radius of the wheel, c2 may be neglected as compared with R2, the above equation then

by which equation the weight W in tons of a fly-wheel of a given mean radius R is determined, so that being applied to an engine of a given horse power H, making a given number of revolutions per minute N, it shall cause the angular velo

city of that wheel not to vary by more than th from its mean

value. It is to be observed that the weight of the wheel varies inversely as the cube of the number of strokes made

*If the section of each arm be supposed uniform and represented by K, and the arms be pin number, it is easily shown from Arts, 79.81., that the momentum of inertia of each arm about its extremity is very nearly represented by (R—c), where c represents the depth of the rim; so that the P whole momentum of inertia of the arms is represented by *(R—¿c)3,

which expression must be added to the momentum of the rim to determine the whole momentum I of the wheel. It appears however expedient to give the inertia of the arms to the equalising power of the wheel.

by the engine per minute, so that an engine making twice as many strokes as another of equal horse power, would receive an equal steadiness of motion from a fly-wheel of one eighth the weight; the mean radii of the wheels being the

same.

If in equation (342) we substitute for I its value 2abc R3, or 2.KR3 (representing by K the section be of the rim), and if we suppose the wheel to be formed of cast iron of mean quality, the weight of each cubic foot of which may be assumed to be 450 lb., we shall obtain by reduction

by which equation is determined the mean radius R of a flywheel of cast iron of a given section K, which, being applied to an engine of given horse power H, making a given. number of revolutions N per minute, shall cause its angular velocity not to deviate more than 4th from the mean; or

n

conversely, the mean radius being given, the section K may be determined according to these conditions.

269. In the above equations, m is taken to represent the number of effective strokes made by the piston of the engine whilst the fly-wheel makes one revolution, and to represent that angle whose sine is

m

Let now the axis of the fly-wheel be supposed to be a continuation of the axis of the crank, so that both turn with the same angular velocity, as is usually the case; and let its application to the single-acting engine, the double-acting engine, and to the double crank engine, be considered separately.

1. In the single-acting engine, but one effective stroke of the piston is made whilst the fly-wheel makes each revolu1 tion. In this case, therefore, m=1, and sin. =-=0·3183098

с с 3

π

=sin. 18° 33'; therefore, cos. 9480460, also

=

=

18° 33'

180°

(347);

by which equations are determined, according to the pro-
posed conditions, the weight W in tons of a fly-wheel for a
single-acting engine, its mean radius in feet R being given,
and its material being any whatever; and also its mean radius
R in feet, its section (in square feet) K being given, and its
material being cast iron of mean quality; and lastly, the sec-
tion K of its rim in square feet, its mean radius R being
given, and its material being, as before, cast iron.

2. In the double-acting engine, two effective strokes are
made per minute by the piston, whilst the fly-wheel makes
2

one revolution. In this case, therefore m=2 and sin. »=

n

=

= 0.636619 sin. 39° 32'; therefore, cos. 7712549
39° 32'

=21963; therefore (1–27) =56074;

Substituting in equations (345) and (346),

by which equations the weight of the fly-wheel in tons, the
mean radius in feet, and the section of the rim in square feet,
are determined for the double-acting engine.

3. In the engine working with two cylinders and a double
crank, it has been shown (Art. 263.) that the conditions of
the working of the two arms of a double crank are precisely

the same as though the aggregate pressure 2P, upon their extremities, were applied to the axis of the crank by the intervention of a single arm and a single connecting rod; the instead of a,

length of this arm being represented by

a

√2

and its direction equally dividing the inclination of the arms of the double crank to one another.

Now, equations (345) and (346) show the proper dimensions of the fly-wheel to be wholly independent of the length of the crank-arm; whence it follows that the dimension of a fly-wheel applicable to the double as well as a single crank, are determined by those equations as applied to the case of a double-acting engine, the pressure upon whose piston rod is represented by 2P,. But in assuming Nm. 2P1a=33000H, we have assumed the pressure upon the piston rod to be represented by P1; to correct this error, and to adapt equations (345) and (346) to the case of a double crank engine, we must therefore substitute H for H in those equations. We shall thus obtain

by which equations the dimensions of a fly-wheel necessary to give the required steadiness of motion to a double crank engine are determined under the proposed conditions.

THE FRICTION OF THE FLY-wheel.

270. W representing the weight of the wheel and the limiting angle of resistance between the surface of its axis and that of its bearings, tan. will represent its coefficient of friction (Art. 138.), and W tan. p, the resistance opposed to its revolution by friction at the surface of its axis. Now, whilst the wheel makes one revolution, this resistance is overcome through a space equal to the circumference of the axis, and represented by 2p, if p be taken to represent the radius of the axis. The work expended upon the friction of the axis, during each complete revolution of the wheel, is therefore represented by

2 W tan. ; and if N represent the number of strokes N

made by the engine per minute, and therefore the number of revolutions made by the fly-wheel per minute, then will the number of units of work expended per minute, upon the friction of the axis be represented by NapW tan. ; and the number of horses' power, or the fractional part of a horse's power thus expended, by

If in this equation we substitute for W the weight in lbs. of the fly-wheel necessary to establish a given degree of steadiness in the engine, as determined by equations (347), (348), and (349), we shall obtain for the horse power lost by friction of the fly-wheel, in the single-acting engine, the double-acting engine, and the double crank engine, respectively, the formulæ

THE MODULUS of the CRANK AND FLY-wheel.

271. If S, represent the space traversed by the piston of the engine in any given time, and a the radius of the crank, W the weight of the fly-wheel in lbs., and p the radius of its axis,

then will 2a represent the length of each stroke,

S,

the num2a

ber of strokes made in that time, and 2pW tan. .

WS tan. 4 the work expended upon the friction of the flywheel during that time, which expression being added to the equation (329) representing the work necessary to cause the crank to yield a given amount of work U, to the machine driven by it (independently of the work expended on the friction of the fly-wheel), will give the whole amount of work which must be done upon the combination of the crank and fly-wheel, to cause this given amount of work to be