(320.) With a view to the simplification of the elementary theory of machines, the consideration of several mechanical effects of great practical importance has been postponed, and the attention of the student has been directed exclusively to the way in which the moving power is modified in being transmitted to the resistance independently of such effects. A machine has been regarded as an instrument by which a moving principle, inapplicable in its existing state to the purpose for which it is required, may be changed either in its velocity or direction, or in some other character, so as to be adapted to that purpose. But in accomplishing this, the several parts of the machine have been considered as possessing in a perfect degree qualities which they enjoy only in an imperfect degree; and accordingly the conclusions to which by such reasoning we are conducted are infected with errors, the amount of which will depend on the degree in which the machinery falls short of perfection in those qualities which theoretically are imputed to it.

Of the several parts of a machine, some are designed to move, while others are fixed; and of those which move, some have motions differing in quantity and direction from those of others. The several parts, whether fixed or movable, are subject to various strains and pressures, which they are intended to resist. These forces not only vary according to the load which the machine has to overcome, but also according to the peculiar form and structure of the machine itself. During the operation the surfaces of the movable parts move in immediate contact with the surfaces either of fixed parts or of parts having other motions. If these surfaces were endued with perfect smoothness or polish, and the several parts subject to strains possessed perfect inflexibility and infinite strength, then the effects of machinery might be practically investigated by the principles already explained. But the materials of which every machine is formed are endued with limited strength, and therefore the load which is placed upon it must be restricted accordingly, else it will be liable to be distorted by the flexure, or even to be destroyed by the fracture of those parts which are submitted to an undue strain. The surfaces of the movable parts, and those surfaces with which they move in contact, cannot in practice be rendered so smooth but that such roughness and inequality will remain as sensibly to impede the motion. To overcome such an impediment requires no inconsiderable part of the moving power. This part is, therefore, intercepted before its arrival at the working point, and the resistance to be finally overcome is deprived of it. The property thus depending on the imperfect smoothness of surfaces, and impeding the motion of bodies whose surfaces are in immediate contact, is called friction. Before we can form a just estimate of the effects of machinery, it is necessary to determine the force lost by this impediment, and the laws which under different circumstances regulate that loss.

When cordage is engaged in the formation of any part of a machine, it has hitherto been considered as possessing perfect flexibility. This is not the case in practice; and the want of perfect flexibility, which is called rigidity, renders a certain quantity of force necessary to bend a cord or rope over the surface of an axle or the groove of a wheel. During the motion of the rope a different part of it must thus be continually bent, and the force which is expended in producing the necessary flexure must be derived from the moving power, and is thus intercepted on its way to the working point. In calculating the effects of cordage, due regard must be had to this waste of power; and therefore it is necessary to enquire into the laws which govern the flexure of imperfectly flexible ropes, and the way in which these affect the machines in which ropes are commonly used.

To complete, therefore, the elementary theory of machinery, we propose in the present and following chapter to explain the principal laws which determine the effects of friction, the rigidity of cordage, and the strength of materials.

(321.) If a horizontal plane surface were perfectly smooth, and free from the smallest inequalities, and a body having a flat surface also perfectly smooth were placed upon it, any force applied to the latter would put it in motion, and that motion would continue undiminished as long as the body would remain upon the smooth horizontal surface. But if this surface, instead of being every where perfectly even, had in particular places small projecting eminences, a certain quantity of force would be necessary to carry the moving body over these, and a proportional diminution in its rate of motion would ensue. Thus, if such eminences were of frequent occurrence, each would deprive the body of a part of its speed, so that between that and the next it would move with a less velocity than it had between the same and the preceding one. This decrease being continued by a sufficient number of such eminences encountering the body in succession, the velocity would at last be so much diminished that the body would not have sufficient force to carry it over the next eminence, and its motion would thus altogether cease.

Now, instead of the eminences being at a considerable distance asunder, suppose them to be contiguous, and to be spread in every direction over the horizontal plane, and also suppose corresponding eminences to be upon the surface of the moving body; these projections incessantly encountering one another will continually obstruct the motion of the body, and will gradually diminish its velocity, until it be reduced to a state of rest.

Such is the cause of friction. The amount of this resisting force increases with the magnitude of these asperities, or with the roughness of the surfaces; but it does not solely depend on this. The surfaces remaining the same, a little reflection on the method of illustration just adopted, will show that the amount of friction ought also to depend upon the force with which the surfaces moving one upon the other are pressed together. It is evident, that as the weight of the body supposed to move upon the horizontal plane is increased, a proportionally greater force will be necessary to carry it over the obstacles which it encounters, and therefore it will the more speedily be deprived of its velocity and reduced to a state of rest.

(322.) Thus we might predict with probability, that which accurate experimental enquiry proves to be true, that the resistance from friction depends conjointly on the roughness of the surfaces and the force of the pressure. When the surfaces are the same, a double pressure will produce a double amount of friction, a treble pressure a treble amount of friction, and so on.

Experiment also, however, gives a result which, at least at first view, might not have been anticipated from the mode of illustration we have adopted. It is found that the resistance arising from friction does not at all depend on the magnitude of the surface of contact; but provided the nature of the surfaces and the amount of pressure remain the same, this resistance will be equal, whether the surfaces which move one upon the other be great or small. Thus, if the moving body be a flat block of wood, the face of which is equal to a square foot in magnitude, and the edge of which does not exceed a square inch, it will be subject to the same amount of friction, whether it move upon its broad face or upon its narrow edge. If we consider the effect of the pressure in each case, we shall be able to perceive why this must be the case. Let us suppose the weight of the block to be 144 ounces. When it rests upon its face, a pressure to this amount acts upon a surface of 144 square inches, so that a pressure of one ounce acts upon each square inch. The total resistance arising from friction will, therefore, be 144 times that resistance which would be produced by a surface of one square inch under a pressure of one ounce. Now, suppose the block placed upon its edge, there is then a pressure of 144 ounces upon a surface equal to one square inch. But it has been already shown, that when the surface is the same, the friction must increase in proportion to the pressure. Hence we infer that the friction produced in the present case is 144 times the friction which would be produced by a pressure of one ounce acting on one square inch of surface, which is the same resistance as that which the body was proved to be subject to when resting on its face.

These two laws, that friction is independent of the magnitude of the surface, and is proportional to the pressure when the quality of the surfaces is the same, are useful in practice, and generally true. In very extreme cases they are, however, in error. When the pressure is very intense, in proportion to the surface, the friction is somewhat less than it would be by these laws; and when it is very small in proportion to the surface, it is somewhat greater.

(323.) There are two methods of establishing by experiment the laws of friction, which have been just explained.

First. The surfaces between which the friction is to be determined being rendered perfectly flat, let one be fixed in the horizontal position on a table T T′, fig. 176.; and let the other be attached to the bottom of a box B C, adapted to receive weights, so as to vary the pressure. Let a silken cord S P, attached to the box, be carried parallel to the table over a wheel at P, and let a dish D be suspended from it. If no friction existed between the surfaces, the smallest weight appended to the cord would draw the box towards P with a continually increasing speed. But the friction which always exists interrupts this effect, and a small weight may act upon the string without moving the box at all. Let weights be put in the dish D, until a sufficient force is obtained to overcome the friction without giving the box an accelerated motion. Such a weight is equivalent to the amount of the friction.

The amount of the weight of the box being previously ascertained, let this weight be now doubled by placing additional weights in the box. The pressure will thus be doubled, and it will be found that the weight of the dish D and its load, which before was able to overcome the friction, is now altogether inadequate to it. Let additional weights be placed in the dish until the friction be counteracted as before, and it will be observed, that the whole weight necessary to produce this effect is exactly twice the weight which produced it in the former case. Thus it appears that a double amount of pressure produces a double amount of friction; and in a similar way it may be proved, that any proposed increase or decrease of the pressure will be attended with a proportionate variation in the amount of the friction.

Second. Let one of the surfaces be attached to a flat plane A B, fig. 177., which can be placed at any inclination with an horizontal plane B C, the other surface being, as before, attached to the box adapted to receive weights. The box being placed upon the plane, let the latter be slightly elevated. The tendency of the box to descend upon A B, will bear the same proportion to its entire weight as the perpendicular A E bears to the length of the plane A B (286.). Thus if the length A B be 36 inches, and the height A E be three inches, that is a twelfth part of the length, then the tendency of the weight to move down the plane is equal to a twelfth part of its whole amount. If the weight were twelve ounces, and the surfaces perfectly smooth, a force of one ounce acting up the plane would be necessary to prevent the descent of the weight.

In this case also the pressure on the plane will be represented by the length of the base B E (286.), that is, it will bear the same proportion to the whole weight as B E bears to B A. The relative amounts of the weight, the tendency to descend, and the pressure, will always be exhibited by the relative lengths of A B, A E, and B E.

This being premised, let the elevation of the plane A B be gradually increased until the tendency of the weight to descend just overcomes the friction, but not so much as to allow the box to descend with accelerated speed. The proportion of the whole weight, which then acts down the plane, will be found by measuring the height A E, and the pressure will be determined by measuring the base B E. Now let the weight in the box be increased, and it will be found that the same elevation is necessary to overcome the friction; nor will this elevation suffer any change, however the pressure or the magnitude of the surfaces which move in contact may be varied.

Since, therefore, in all these cases, the height A E and the base B E remain the same, it follows that the proportion between the friction and pressure is undisturbed.

(324.) The law that friction is proportional to the pressure, has been questioned by the late professor Vince of Cambridge, who deduced from a series of experiments, that although the friction increases with the pressure, yet that it increases in a somewhat less ratio; and from this it would follow, that the variation of the surface of contact must produce some effect upon the amount of friction. The law, as we have explained it, however, is sufficiently near the truth for most practical purposes.

(325.) There are several circumstances regarding the quality of the surfaces which produce important effects on the quantity of friction, and which ought to be noticed here.

This resistance is different in the surfaces of different substances. When the surfaces are those of wood newly planed, it amounts to about half the pressure, but is different in different kinds of wood. The friction of metallic surfaces is about one fourth of the pressure.

In general the friction between the surfaces of bodies of different kinds is less than between those of the same kind. Thus, between wood and metal the friction is about one fifth of the pressure.

It is evident that the smoother the surfaces are the less will be the friction. On this account, the friction of surfaces, when first brought into contact, is often greater than after their attrition has been continued for a certain time, because that process has a tendency to remove and rub off those minute asperities and projections on which the friction depends. But this has a limit, and after a certain quantity of attrition the friction ceases to decrease. Newly planed surfaces of wood have at first a degree of friction which is equal to half the entire pressure, but after they are worn by attrition it is reduced to a third.

If the surfaces in contact be placed with their grains in the same direction, the friction will be greater than if the grains cross each other.

Smearing the surfaces with unctuous matter diminishes the friction, probably by filling the cavities between the minute projections which produce the friction.

When the surfaces are first placed in contact, the friction is less than when they are suffered to rest so for some time; this is proved by observing the force which in each case is necessary to move the one upon the other, that force being less if applied at the first moment of contact than when the contact has continued. This, however, has a limit. There is a certain time, different in different substances, within which this resistance attains its greatest amount. In surfaces of wood this takes place in about two minutes; in metals the time is imperceptibly short; and when a surface of wood is placed upon a surface of metal, it continues to increase for several days. The limit is larger when the surfaces are great, and belong to substances of different kinds.

The velocity with which the surfaces move upon one another produces but little effect upon the friction.

(326.) There are several ways in which bodies may move one upon the other, in which friction will produce different effects. The principal of these are, first, the case where one body slides over another; the second, where a body having a round form rolls upon another; and, thirdly, where an axis revolves within a hollow cylinder, or the hollow cylinder revolves upon the axis.

With the same amount of pressure and a like quality of surface, the quantity of friction is greatest in the first case and least in the second. The friction in the second case also depends on the diameter of the body which rolls, and is small in proportion as that diameter is great. Thus a carriage with large wheels is less impeded by the friction of the road than one with small wheels.

In the third case, the leverage of the wheel aids the power in overcoming the friction. Let fig. 178. represent a section of the wheel and axle; let C be the centre of the axle, and let B E be the hollow cylinder in the nave of the wheel in which the axle is inserted. If B be the part on which the axle presses, and the wheel turn in the direction N D M, the friction will act at B in the direction B F, and with the leverage B C. The power acts against this at D in the direction D A, and with the leverage D C. It is therefore evident, that as D C is greater than B C, in the same proportion does the power act with mechanical advantage on the friction.

(327.) Contrivances for diminishing the effects of friction depend on the properties just explained, the motion of rolling being as much as possible substituted for that of sliding; and where the motion of rolling cannot be applied, that of a wheel upon its axle is used. In some cases both these motions are combined.

If a heavy load be drawn upon a plane in the manner of a sledge, the motion will be that of sliding, the species which is attended with the greatest quantity of friction; but if the load be placed upon cylindrical rollers, the nature of the motion is changed, and becomes that in which there is the least quantity of friction. Thus large blocks of stone, or heavy beams of timber, which would require an enormous power to move them on a level road, are easily advanced when rollers are put under them.

When very heavy weights are to be moved through small spaces, this method is used with advantage; but when loads ore to be transported to considerable distances, the process is inconvenient and slow, owing to the necessity of continually replacing the rollers in front of the load as they are left behind by its progressive advancement.

The wheels of carriages may be regarded as rollers which are continually carried forward with the load. In addition to the friction of the rolling motion on the road, they have, it is true, the friction of the axle in the nave; but, on the other hand, they are free from the friction of the rollers with the under surface of the load, or the carriage in which the load is transported. The advantages of wheel carriages in diminishing the effects of friction is sometimes attributed to the slowness with which that axle moves within the box, compared with the rate at which the wheel moves over the road; but this is erroneous. The quantity of friction does not in any case vary considerably with the velocity of the motion, but least of all does it in that particular kind of motion here considered.

In certain cases, where it is of great importance to remove the effects of friction, a contrivance called friction-wheels, or friction-rollers, is used. The axle of a friction-wheel, instead of revolving within a hollow cylinder, which is fixed, rests upon the edges of wheels which revolve with it; the species of motion thus becomes that in which the friction is of least amount.

Let A B and D C, fig. 179., be two wheels revolving on pivots P Q with as little friction as possible, and so placed that the axle O of a third wheel E F may rest between their edges. As the wheel E F revolves, the axle O, instead of grinding its surface on the surface on which it presses, carries that surface with it, causing the wheels A B, C D, to revolve.

In wheel carriages, the roughness of the road is more easily overcome by large wheels than by small ones. The cause of this arises partly from the large wheels not being so liable to sink into holes as small ones, but more because, in surmounting obstacles, the load is elevated less abruptly. This will be easily understood by observing the curves in fig. 180., which represent the elevation of the axle in each case.

(328.) If a carriage were capable of moving on a road without friction, the most advantageous direction in which a force could be applied to draw it would be parallel to the road. When the motion is impeded by friction, it is better, however, that the line of draught should be inclined to the road, so that the drawing force may be expended partly in lessening the pressure on the road, and partly in advancing the load.

Let W, fig. 181., be a load which is to be moved upon the plane surface A B. If the drawing force be applied in the direction C D, parallel to the plane A B, it will have to overcome the friction produced by the pressure of the whole weight of the load upon the plane; but if it be inclined upwards in the direction C E, it will be equivalent to two forces expressed (74.) by C G and C F. The part C G has the effect of lightening the pressure of the carriage upon the road, and therefore of diminishing the friction in the same proportion. The part C F draws the load along the plane. Since C F is less than C E or C D the whole moving force, it is evident that a part of the force of draught is lost by this obliquity; but, on the other hand, a part of the opposing resistance is also removed. If the latter exceed the former, an advantage will be gained by the obliquity; but if the former exceed the latter, force will be lost.

By mathematical reasoning, founded on these considerations, it is proved that the best angle of draught is exactly that obliquity which should be given to the road in order to enable the carriage to move of itself. This obliquity is sometimes called the angle of repose, and is that angle which determines the proportion of the friction to the pressure in the second method, explained in (323.). The more rough the road is, the greater will this angle be; and therefore it follows, that on bad roads the obliquity of the traces to the road should be greater than on good ones. On a smooth Macadamised way, a very slight declivity would cause a carriage to roll by its own weight: hence, in this case, the traces should be nearly parallel to the road.

In rail roads, for like reasons, the line of draught should be parallel to the road, or nearly so.

(329.) When ropes or cords form a part of machinery, the effects of their imperfect flexibility are in a certain degree counteracted by bending them over the grooves of wheels. But although this so far diminishes these effects as to render ropes practically useful, yet still, in calculating the powers of machinery, it is necessary to take into account some consequences of the rigidity of cordage which even by these means are not removed.

To explain the way in which the stiffness of a rope modifies the operation of a machine, we shall suppose it bent over a wheel and stretched by weights A B, fig. 182., at its extremities. The weights A and B being equal, and acting at C and D in opposite ways, balance the wheel. If the weight A receive an addition, it will overcome the resistance of B, and turn the wheel in the direction D E C. Now, for the present, let us suppose that the rope is perfectly inflexible; the wheel and weights will be turned into the position represented in fig. 183. The leverage by which A acts will be diminished, and will become O F, having been before O C; and the leverage by which B acts will be increased to O G, having been before O D.

But the rope not being inflexible will yield partially to the effects of the weights A and B, and the parts A C and B D will be bent into the forms represented in fig. 184. The form of the curvature which the rope on each side of the wheel receives is still such that the descending weight A works with a diminished leverage F O, while the ascending weight resists it with an increased leverage G O. Thus so much of the moving power is lost, by the stiffness of the rope, as is necessary to compensate this disadvantageous change in the power of the machine.