RT, T P, and C T, lie in one straight line, when equations 2, 4, and 5 will still hold.

In order to diminish the lateral pressure Q, and the friction arising from it, to the least possible amount, the mechanism should be so arranged as to make P and R act parallel to each other at the same side of the axle.

In most actual cases, sin =ƒ: √1 +ƒa differs from tan in a proportion too small to be of any practical importance.

f

The bearings of axles should be made of materials which, though hard enough to resist the rubbing without abrasion, are not so hard as the axle. Hence for wrought iron axles, bronze bearings are commonly used. Bearings of cast iron, millboard, and hardwood, such as elm, with the grain set radially, have also been used with advantage.

675. Friction of a Pivot.—A pivot is the termination of an axle, which presses endways against a bearing called a step, or footstep. Pivots require great hardness, and are usually made of steel.

A flat pivot is a short cylinder of steel, having a plane circular end for a rubbing surface. If the pressure Q be equally distributed over that surface whose radius is r, the moment of friction is easily found by integration to be

In flat pivots, the intensity of the pressure, which is given by the equation

Q p =

is usually limited to 2,240 lbs. per square inch.

.(2.)

In the cup and ball pivot, the end of the shaft, and the step, present two recesses facing each other, into which are fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere, or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable, from the extreme smallness of the radius of the circles of contact of the ball and cups; but as they wear, that radius and the moment of friction increase.

676. Friction of a Collar.-When it is impracticable or inconvenient to sustain the pressure which acts along a shaft by means of a pivot at its end, that pressure is

collars, or rings projecting from the by means of one or more

shaft, and pressing against corresponding ring-shaped bearings, for which, in the case of shafts of screw propellers, hardwood set with the grain endways has been

FRICTION OF COLLARS OF TEETH-OF BANDS.

found a good material amongst others.

617

Let r be the external, and

'the internal radius of a collar; its moment of friction for the pressure Q is given by the formula

677. Friction of Teeth.—When a pair of wheels work together, let P be the pressure exerted between each pair of their teeth which comes into action, s the distance through which each pair of teeth slide over each other, as found in Articles 453, 455, 458, and 462 A, and n the number of pairs of teeth which pass the line of centres in a given interval of time. Then in that interval, the work lost by the friction of the teeth is

678. Friction of a Band.- A flexible band, such as a cord, rope, belt, or strap, may be used either to exert an effort or a resistance upon a drum or pulley round which it wraps. In either case, the tangential force, whether effort or resistance, exerted between the band and the pulley, is their mutual friction, caused by and proportional to the normal pressure between them.

In fig. 264, let C be the axis of a pulley A B, round an arc of which there is wrapped a band, T, A B T.; let the outer arrow represent the direction in which the band slides, or tends to slide, relatively to the pulley, and the inner arrow the direction in which the pulley slides, or tends to slide, relatively to

the band.

Let T, be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; the ratio of the length of that arc to the radius of the pulley; d the ratio of an indefinitely small element of that arc to the radius; RT, - T2, the total friction between the band and the pulley; d R the elementary portion of that friction due to the elementary arc d;f the co-efficient of friction between the materials of the band and pulley.

Fig. 264.

Then according to a principle proved in Articles 179 and 271, it is known that the normal pressure at the elementary arc d e is

Tde;

T being the mean tension of the band at that elementary arc; consequently, the friction on that arc is

dR=fTdo.

Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc; or

dT=dR=fTd0;

which equation being integrated throughout the entire arc of contact, gives the following formulæ :

When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest; and when the pulleys are set in motion, so that one of them drives the other by means of the belt; it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula:—

which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.

If the arc of contact between the band and pulley, expressed in turns and fractions of a turn, be denoted by n,

When the band is used to resist the motion of the pulley, it constitutes a kind of brake called a friction strap. In this case the rubbing surfaces of the band and pulley may either be both of iron, or may be protected by a covering made of pieces of wood, which is renewed from time to time as it wears out.

679. In Frictional Gearing, described in Article 445, it appears that when the angle of the grooves is 40°, and when their surfaces are smooth, clean, and dry, the tangential force transmitted between the wheels is once and a-half the force with which their axes are pressed together. This proportion is much greater than that due to ordinary friction, and must arise partly from adhesion.

680. Friction Couplings are used to communicate rotation be tween pieces having the same axis, where sudden changes of force or of velocity take place; being so adjusted as to limit the force transmitted within the bounds of safety. Contrivances of this kind

STIFFNESS OF ROPES-ROLLING FRICTION-CARRIAGES.

619

are very numerous; one of the most common and most useful is that called a pair of friction cones. The angle made by the sides of the cones with the axis should not be less than the angle of repose.

681. Stiffness of Ropes.-Ropes offer a resistance to being bent, and when bent to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.

The work lost in pulling a given length of rope over a pulley, is found by multiplying the length of the rope in feet, by its stiffness in pounds; that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.

The following empirical formulæ for the stiffness of hempen ropes have been deduced by General Morin from the experiments of Coulomb :

Let R be the stiffness in pounds avoirdupois ;

d, the diameter of the rope, in inches;

n = 48 d for white ropes, 35 d2 for tarred ropes; r, the effective radius of the pulley, in inches;

T, the tension, in pounds; then,

For white

=

n

ropes, R (0·0012 + 0·001026 n + 0·0012 T);

For tarred ropes, Ꭱ

(1.)

682. Rolling Resistance of Smooth Surfaces.-By the rolling of two surfaces over each other without sliding, a resistance is caused, which is called rolling friction. It is of the nature of a couple resisting rotation; its moment is found by multiplying the normal pressure between the rolling surfaces by an arm whose length depends on the nature of the rolling surfaces; and the work lost in an unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot :

Oak upon oak,.....
Lignum-vitæ on oak,..

Cast iron on cast iron,....

.....0·006 (Coulomb).

...0'004

.......0002 (Tredgold).

683. The Resistance of Carriages on Roads consists of a constant part, and a part increasing with the velocity. According to General Morin, it is given approximately by the following formula :—

R = { a + b (v - 3·28)} ;......

..(1.)

where Q is the gross load, r the radius of the wheels in inches, v the velocity in feet per second, and a and b two constants, whose

On gravel roads the resistance is about double, and on sandy and gravelly soft ground, five times the resistance on good broken stone roads.

684. Resistance of Railway Trains. In the following formula, which are all empirical

E denotes the weight of the engine;

T

V

R

f

с

the gross load drawn by it;

the velocity, in miles an hour;

the radius of curvature of the line, in miles;
the resistance in pounds;

a co-efficient of friction;

a co-efficient for resistance due to curvature.

Then for single carriages with cylindrical wheels, at velocities up to 12 miles an hour, according to the experiments of Lieutenant David Rankine and the Author,

R =ƒ

where f= 0.002; and c = 0.3.

(+)T;.

.(1.)

For an engine and train, the following is an empirical formula deduced from the experiments of various authors:

:

where franges from 0027 to 004, according to the state of the line and carriages, and c from 0.3 to 0.1. (See Rankine's Manual of Civil Engineering; see also Appendix.)

685. Heat of Friction. The work lost in friction produces heat in the proportion of one British thermal unit, being so much heat as raises the temperature of a pound of water one degree of Fahrenheit, for every 772 foot pounds of lost work.

Excessive heating is prevented by a constant and copious supply of a good unguent.