faced with wooden blocks on the side next the drum. When tension is applied to the ends of the chain, the blocks clasp the drum and produce friction; when that tension is removed, the blocks are drawn back from the drum by springs to which they are attached, and the friction ceases.

The following formulæ are exact for perfectly flexible continuous bands, and approximate for elastic straps and for chains of blocks. For their demonstration, the reader is referred to treatises on mechanics.

In fig. 18, let A B be the drum, and C its axis, and let the direction of rotation of the drum be indicated by the arrow. Let T1 and T2 represent the tensions at the two ends of the strap, which embraces the rim of the drum throughout the arc A B. The tension T1 exceeds the tension To by an amount equal to the friction between the strap and drum, R; that is,

Let c denote the ratio which the arc of contact AB bears to the circumference of the drum ; f the co-efficient of friction between the strap and drum; then the ratio T1. T, is the number whose common logarithm is 2·7288 fc, or

å

which number having been found, is to be used in the following formulæ for finding the tensions T1, T2, required in order to produce a given resistance R:

The following cases occur in practice:—

I. When it is desired to produce a great resistance compared with the force applied to the brake, the backward end of the brake, where the tension is T1, is to be fixed to the framework of the machinery, and the forward end moved by means of a lever or other suitable mechanism; when the force to be applied by means of that mechanism will be T, which, by making N sufficiently great, may be made small as compared with R

II. When it is desired that the resistance shall always be less than a certain given force, the forward end of the brake is to be fixed, and the backward end pulled with a force not exceeding the given force. This will be T1; and, as the equation 2 shows, how great soever N may be, R will always be less than T,. This is the principle of the brake applied by Sir William Thomson, to apparatus for paying out submarine telegraph cables, with a view to limiting the resistance within the amount which the cable can safely bear.

In any case in which it is desired to give a great value to the ratio N, the flexible brake may be coiled spirally round the drum, so as to make the arc of contact greater than one circumference.

50. Pump Brakes.-The resistance of a fluid, forced by a pump through a narrow orifice, may be used to dispose of superfluous energy.

The energy which is expended in forcing a given weight of fluid through an orifice is found by multiplying that weight into the height due to the greatest velocity which its particles acquire in that process, and into a factor greater than unity, which for each kind of orifice is determined experimentally, and whose excess above unity expresses the proportion which the energy expended in overcoming the friction between the fluid and the orifice bears to the energy expended in giving velocity to the fluid.

The following are some of the values of that factor, which will be denoted by 1 + F:

For an orifice in a thin plate, i + F

For a straight uniform pipe of the length 7, and whose hydraulic mean depth, that is, the area divided by the circumference of its cross-section, is m,

For cylindrical pipes, m is one-fourth of the diameter.

The factor f in the last formula is called the co-efficient of friction of the fluid. For water in iron pipes, the diameter d being expressed in feet, its value, according to Darcy, is

The greatest velocity of the fluid particles is found by dividing the volume of fluid discharged in a second by the area of the outlet

at its most contracted part. When the outlet is a cylindrical pipe, the sectional area of that pipe may be employed in this calculation; but when it is an orifice in a thin plate, there is a contracted vein of the issuing stream after passing the orifice, whose area is on an average about 0.62 of the area of the orifice itself; and that contracted area is to be employed in computing the velocity. Its ratio to the area of the orifice in the plate is called the co-efficient of contraction.

The computation of the energy expended in forcing a given quantity of a given fluid in a given time through a given outlet, is expressed symbolically as follows:

:

Let V be the volume of fluid forced through, in cubic feet per second.

D, the density, or weight of a cubic foot, in pounds.

A, the area of the orifice, in square feet.

c, the co-efficient of contraction.

v, the velocity of outflow, in feet per second.

R, the resistance overcome by the piston of the pump

the water, in pounds.

u, the velocity of that piston, in feet per second.

in driving

.(5.)

2.2

Ru = DV (1 + F)

(6.)

2 g

the factor 1+ F being computed by means of the formulæ 1, 2, 3, 4. To find the intensity of the pressure (p) within the pump, it is to be observed, as in Article 6, that if A' denotes the area of the piston,

that is, the intensity of the pressure is that due to the weight of a vertical column of the fluid, whose height is greater than that due to the velocity of outflow in the ratio 1 + F : 1.

To allow for the friction of the piston, about one-tenth may in general be added to the result given by equation 6.

The piston and pump have been spoken of as single; and such may be the case when the velocity of the piston is uniform. When & piston, however, is driven by a crank on a shaft rotating at an

uniform speed, its velocity varies; and when a pump brake is to be applied to such a shaft, it is necessary, in order to give a sufficiently near approximation to an uniform velocity of outflow, that there should be at least, either three single acting pumps, driven by three cranks making with each other angles of 120°, or a pair of double acting pumps, driven by a pair of cranks at right angles to each other; and the result will be better if the pumps force the fluid into one common air vessel before it arrives at the resisting orifice.

That orifice may be provided with a valve, by means of which its area can be adjusted, so as to cause any required resistance.

A pump brake of a simple kind is exemplified in the apparatus called the "cataract," for regulating the opening of the steam valve in single acting steam engines. It will be more fully described under the head of those engines.*

51. Fan Brakes.-A fan, or wheel with vanes, revolving in water, oil, or air, may be used to dispose of surplus energy; and the resistance which it causes may be rendered to a certain extent adjustable at will, by making the vanes so as to be capable of being set at different angles with their direction of motion, or at different distances from their axis.

Fan brakes are applied to various machines, and are usually adjusted so as to produce the requisite resistance by trial. It is, indeed, by trial only that a final and exact adjustment can be effected; but trouble and expense may be saved by making, in the first place, an approximate adaptation of the fan to its purpose by calculation.

The following formulæ are the results of the experiments of Duchemin, and are approved of by Poncelet in his Mécanique Industrielle.

For a thin flat vane, whose plane traverses its axis of rotation, let A denote the area of the vane;

, the distance of its centre of gravity from the axis of rotatio 1; s, the distance from the centre of gravity of the entire vane, to the centre of gravity of that half of it which lies nearest the axis of rotation;

v, the velocity of the centre of gravity of the vane (= a l,

the angular velocity of rotation);

D, the density of the fluid in which it moves;

R, the moment of resistance;

k, a co-efficient, whose value is given by the formula

if a is

* Continuous brakes are now applied in railway work. (See Addendum, page 557.)

When the vane is oblique to its direction of motion, let i denote the acute angle which its surface makes with that direction; then the result of equation 2 is to be multiplied by

It appears that the resistance of a fan with several vanes increases nearly in proportion to the number of vanes, so long as their distances apart are not less at any point than their lengths. Beyond that limit the law is uncertain.

SECTION 6.—Of Fly Wheels.

52. Periodical Fluctuations of Speed in a machine (A. M., 689) are caused by the alternate excess and deficiency of the energy exerted above the work performed in overcoming resisting forces, which produce an alternate increase and diminution of actual energy, according to the law explained in Article 30.

D'

H

K

E

b

To determine the greatest fluctuations of speed in a machino moving periodically, take ABC, in fig. 19, to represent the motion of the driving point during one period; let the effort P of the prime mover at each instant be represented by the ordinate of the curve D G EIF; and let the sum of the resistances, reduced to the driving point as in Article 9, at each instant, be denoted by R, and represented by the ordinate of the curve DHEK F, which cuts the former curve at the ordinates AD, BE, CF. Then the integrai

j (P - R) d 8,

B

Fig. 19.

being taken for any part of the motion, gives the excess or deficiency of energy, according as it is positive or negative. For the entire period ABC this integral is nothing. For A B, it denotes an excess of energy received, represented by the area DGEH; and for BC, an equal excess of work performed, represented by the equal area EK FI. Let those equal quantities be each represented by