(313.) The classes of simple machines denominated mechanic powers, have relation chiefly to the peculiar principle which determines the action of the power on the weight or resistance. In explaining this arrangement various other reflections have been incidentally mixed up with our investigations; yet still much remains to be unfolded before the student can form a just notion of those means by which the complex machinery used in the arts and manufactures so effectually attains the ends, to the accomplishment of which it is directed.

By a power of a given energy to oppose a resistance of a different energy, or by a moving principle having a given velocity to generate another velocity of a different amount, is only one of the many objects to be effected by a machine. In the arts and manufactures the kind of motion produced is generally of greater importance than its rate. The latter may affect the quantity of work done in a given time, but the former is essential to the performance of the work in any quantity whatever. In the practical application of machines, the object to be attained is generally to communicate to the working point some peculiar sort of motion suitable to the uses for which the machine is intended; but it rarely happens that the moving power has this sort of motion. Hence, the machine must be so contrived that, while that part on which this power acts is capable of moving in obedience to it, its connection with the other parts shall be such that the working point may receive that motion which is necessary for the purposes to which the machine is applied.

To give a perfect solution of this problem it would be necessary to explain, first, all the varieties of moving powers which are at our disposal; secondly, all the variety of motions which it may be necessary to produce; and, thirdly, to show all the methods by which each variety of prime mover may be made to produce the several species of motion in the working point. It is obvious that such an enumeration would be impracticable, and even an approximation to it would be unsuitable to the present treatise. Nevertheless, so much ingenuity has been displayed in many of the contrivances for modifying motion, and an acquaintance with some of them is so essential to a clear comprehension of the nature and operation of complex machines, that it would be improper to omit some account of those at least which most frequently occur in machinery, or which are most conspicuous for elegance and simplicity.

The varieties of motion which most commonly present themselves in the practical application of mechanics may be divided into rectilinear and rotatory. In rectilinear motion the several parts of the moving body proceed in parallel straight lines with the same speed. In rotatory motion the several points revolve round an axis, each performing a complete circle, or similar parts of a circle, in the same time.

Each of these may again be resolved into continued and reciprocating. In a continued motion, whether rectilinear or rotatory, the parts move constantly in the same direction, whether that be in parallel straight lines, or in rotation on an axis. In reciprocating motion the several parts move alternately in opposite directions, tracing the same spaces from end to end continually. Thus, there are four principal species of motion which more frequently than any others act upon, or are required to be transmitted by, machines:—

1. Continued rectilinear motion.
2. Reciprocating rectilinear motion.
3. Continued circular motion.
4. Reciprocating circular motion.

These will be more clearly understood by examples of each kind.

Continued rectilinear motion is observed in the flowing of a river, in a fall of water, in the blowing of the wind, in the motion of an animal upon a straight road, in the perpendicular fall of a heavy body, in the motion of a body down an inclined plane.

Reciprocating rectilinear motion is seen in the piston of a common syringe, in the rod of a common pump, in the hammer of a pavier, the piston of a steam-engine, the stampers of a fulling mill.

Continued circular motion is exhibited in all kinds of wheel-work, and is so common, that to particularise it is needless.

Reciprocating circular motion is seen in the pendulum of a clock, and in the balance-wheel of a watch.

We shall now explain some of the contrivances by which a power having one of these motions may be made to communicate either the same species of motion changed in its velocity or direction, or any of the other three kinds of motion.

(314.) By a continued rectilinear motion another continued rectilinear motion in a different direction may be produced, by one or more fixed pulleys. A cord passed over these, one end of it being moved by the power, will transmit the same motion unchanged to the other end. If the directions of the two motions cross each other, one fixed pulley will be sufficient; see fig. 113., where the hand takes the direction of the one motion, and the weight that of the other. In this case the pulley must be placed in the angle at which the directions of the two motions cross each other. If this angle be distant from the places at which the objects in motion are situate, an inconvenient length of rope may be necessary. In this case the same may be effected by the use of two pulleys, as in fig. 158.

If the directions of the two motions be parallel, two pulleys must be used as in fig. 158., where P′ A′ is one motion, and B W the other. In these cases the axles of the two wheels are parallel.

It may so happen that the directions of the two motions neither cross each other nor are parallel. This would happen, for example, if the direction of one were upon the paper in the line P A, while the other were perpendicular to the paper from the point O. In this case two pulleys should be used, the axle of one O′ being perpendicular to the paper, while the axle of the other O should be on the paper. This will be evident by a little reflection.

In general, the axle of each pulley must be perpendicular to the two directions in which the rope passes from its groove; and by due attention to this condition it will be perceived, that a continued rectilinear motion may be transferred from any one direction to any other direction, by means of a cord and two pulleys, without changing its velocity.

If it be necessary to change the velocity, any of the systems of pulleys described in chap. XV. may be used in addition to the fixed pulleys.

By the wheel and axle any one continued rectilinear motion may be made to produce another in any other direction, and with any other velocity. It has been already explained (250.) that the proportion of the velocity of the power to that of the weight is as the diameter of the wheel to the diameter of the axle. The thickness of the axle being therefore regulated in relation to the size of the wheel, so that their diameters shall have that proportion which subsists between the proposed velocities, one condition of the problem will be fulfilled. The rope coiled upon the axle may be carried, by means of one or more fixed pulleys, into the direction of one of the proposed motions, while that which surrounds the wheel is carried into the direction of the other by similar means.

(315.) By the wheel and axle a continued rectilinear motion may be made to produce a continued rotatory motion, or vice versâ. If the power be applied by a rope coiled upon the wheel, the continued motion of the power in a straight line will cause the machine to have a rotatory motion. Again, if the weight be applied by a rope coiled upon the axle, a power having a rotatory motion applied to the wheel will cause the continued ascent of the weight in a straight line.

Continued rectilinear and rotatory motions may be made to produce each other, by causing a toothed wheel to work in a straight bar, called a rack, carrying teeth upon its edge. Such an apparatus is represented in fig. 159.

In some cases the teeth of the wheel work in the links of a chain. The wheel is then called a rag-wheel, fig. 160.

Straps, bands, or ropes, may communicate rotation to a wheel, by their friction in a groove upon its edge.

A continued rectilinear motion is produced by a continued circular motion in the case of a screw. The lever which turns the screw has a continued circular motion, while the screw itself advances with a continued rectilinear motion.

The continued rectilinear motion of a stream of water acting upon a wheel produces continued circular motion in the wheel, fig. 93, 94, 95. In like manner the continued rectilinear motion of the wind produces a continued circular motion in the arms of a windmill.

Cranes for raising and lowering heavy weights convert a circular motion of the power into a continued rectilinear motion of the weight.

(316.) Continued circular motion may produce reciprocating rectilinear motion, by a great variety of ingenious contrivances.

Reciprocating rectilinear motion is used when heavy stampers are to be raised to a certain height, and allowed to fall upon some object placed beneath them. This may be accomplished by a wheel bearing on its edge curved teeth, called wipers. The stamper is furnished with a projecting arm or peg, beneath which the wipers are successively brought by the revolution of the wheel. As the wheel revolves the wiper raises the stamper, until its extremity passes the extremity of the projecting arm of the stamper, when the latter immediately falls by its own weight. It is then taken up by the next wiper, and so the process is continued.

A similar effect is produced if the wheel be partially furnished with teeth, and the stamper carry a rack in which these teeth work. Such an apparatus is represented in fig. 161.

It is sometimes necessary that the reciprocating rectilinear motion shall be performed at a certain varying rate in both directions. This may be accomplished by the machine represented in fig. 162. A wheel upon the axle C turns uniformly in the direction A B D E. A rod mn moves in guides, which only permit it to ascend and descend perpendicularly. Its extremity m rests upon a path or groove raised from the face of the wheel, and shaped into such a curve that as the wheel revolves the rod mn shall be moved alternately in opposite directions through the guides, with the required velocity. The manner in which the velocity varies will depend on the form given to the groove or channel raised upon the face of the wheel, and this may be shaped so as to give any variation to the motion of the rod mn which may be required for the purpose to which it is to be applied.

The rose-engine in the turning-lathe is constructed on this principle. It is also used in spinning machinery.

It is often necessary that the rod to which reciprocating motion is communicated shall be urged by the same force in both directions. A wheel partially furnished with teeth, acting on two racks placed on different sides of it, and both connected with the bar or rod to which the reciprocating motion is to be communicated, will accomplish this. Such an apparatus is represented in fig. 163., and needs no further explanation.

Another contrivance for the same purpose is shown in fig. 164., where A is a wheel turned by a winch H, and connected with a rod or beam moving in guides by the joint ab. As the wheel A is turned by the winch H the beam is moved between the guides alternately in opposite directions, the extent of its range being governed by the length of the diameter of the wheel. Such an apparatus is used for grinding and polishing plane surfaces, and also occurs in silk machinery.

An apparatus applied by M. Zureda in a machine for pricking holes in leather is represented in fig. 165. The wheel A B has its circumference formed into teeth, the shape of which may be varied according to the circumstances under which it is to be applied. One extremity of the rod ab rests upon the teeth of the wheel upon which it is pressed by a spring at the other extremity. When the wheel revolves, it communicates to this rod a reciprocating rectilinear motion.

Leupold has applied this mechanism to move the pistons of pumps.5 Upon the vertical axis of a horizontal hydraulic wheel is fixed another horizontal wheel, which is furnished with seven teeth in the manner of a crown wheel (263.). These teeth are shaped like inclined planes, the intervals between them being equal to the length of the planes. Projecting arms attached to the piston rods rest upon the crown of this wheel; and, as it revolves, the inclined surfaces of the teeth, being forced under the arm, raise the rod upon the principle of the wedge. To diminish the obstruction arising from friction, the projecting arms of the piston rods are provided with rollers, which run upon the teeth of the wheel. In one revolution of the wheel each piston makes as many ascents and descents as there are teeth.

(317.) Wheel-work furnishes numerous examples of continued circular motion round one axis, producing continued circular motion round another. If the axles be in parallel directions, and not too distant, rotation may be transmitted from one to the other by two spur-wheels (263.); and the relative velocities may be determined by giving a corresponding proportion to the diameter of the wheels.

If a rotary motion is to be communicated from one axis to another parallel to it, and at any considerable distance, it cannot in practice be accomplished by wheels alone, for their diameters would be too large. In this case a strap or chain is carried round the circumferences of both wheels. If they are intended to turn in the same direction, the strap is arranged as in fig. 100.; but if in contrary directions it is crossed, as in fig. 101. In this case, as with toothed wheels, the relative velocities are determined by the proportion of the diameters of the wheels.

If the axles be distant and not parallel, the cord, by which the motion is transmitted, must be passed over grooved wheels, or fixed pulleys, properly placed between the two axles.

It may happen that the strain upon the wheel, to which the motion is to be transmitted, is too great to allow of a strap or cord being used. In this case, a shaft extending from the one axis to another, and carrying two bevelled wheels (263.), will accomplish the object. One of these bevelled wheels is placed upon the shaft near to, and in connection with, the wheel from which the motion is to be taken, and the other at a part of it near to, and in connection with, that wheel to which the motion is to be conveyed, fig. 166.

The methods of transmitting rotation from one axis to another perpendicular to it, by crown and by bevelled wheels, have been explained in (263.).

The endless screw (299.) is a machine by which a rotatory motion round one axis may communicate a rotatory motion round another perpendicular to it. The power revolves round an axis coinciding with the length of the screw, and the axis of the wheel driven by the screw is at right angles to this.

The axis to which rotation is to be given, or from which it is to be taken, is sometimes variable in its position. In such cases, an ingenious contrivance, called a universal joint, invented by the celebrated Dr. Hooke, may be used. The two shafts or axles A B, fig. 167., between which the motion is to be communicated, terminate in semicircles, the diameters of which, C D and E F, are fixed in the form of a cross, their extremities moving freely in bushes placed in the extremities of the semicircles. Thus, while the central cross remains unmoved, the shaft A and its semicircular end may revolve round C D as an axis; and the shaft B and its semicircular end may revolve round E F as an axis. If the shaft A be made to revolve without changing its direction, the points C D will move in a circle whose centre is at the middle of the cross. The motion thus given to the cross will cause the points E F to move in another circle round the same centre, and hence the shaft B will be made to revolve.

This instrument will not transmit the motion if the angle under the directions of the shafts be less than 140°. In this case a double joint, as represented in fig. 168., will answer the purpose. This consists of four semicircles united by two crosses, and its principle and operation is the same as in the last case.

Universal joints are of great use in adjusting the position of large telescopes, where, while the observer continues to look through the tube, it is necessary to turn endless screws or wheels whose axes are not in an accessible position.

The cross is not indispensably necessary in the universal joint. A hoop, with four pins projecting from it at four points equally distant from each other, or dividing the circle of the hoop into four equal arcs, will answer the purpose. These pins play in the bushes of the semicircles in the same manner as those of the cross.

The universal joint is much used in cotton-mills, where shafts are carried to a considerable distance from the prime mover, and great advantage is gained by dividing them into convenient lengths, connected by a joint of this kind.

(318.) In the practical application of machinery, it is often necessary to connect a part having a continued circular motion with another which has a reciprocating or alternate motion, so that either may move the other. There are many contrivances by which this may be effected.

One of the most remarkable examples of it is presented in the scapements of watches and clocks. In this case, however, it can scarcely be said with strict propriety that it is the rotation of the scapement-wheel (266.) which communicates the vibration to the balance-wheel or pendulum. That vibration is produced in the one case by the peculiar nature of the spiral spring fixed upon the axis of the balance-wheel, and in the other case by the gravity of the pendulum. The force of the scapement-wheel only maintains the vibration, and prevents its decay by friction and atmospheric resistance. Nevertheless, between the two parts thus moving there exists a mechanical connection, which is generally brought within the class of contrivances now under consideration.

A beam vibrating on an axis, and driven by the piston of a steam-engine, or any other power, may communicate rotary motion to an axis by a connector and a crank. This apparatus has been already described in (311.). Every steam-engine which works by a beam affords an example of this. The working beam is generally placed over the engine, the piston rod being attached to one end of it, while the connecting rod unites the other end with the crank. In boat-engines, however, this position would be inconvenient, requiring more room than could easily be spared. The piston rod, in these cases, is, therefore, connected with the end of the beam by long rods, and the beam is placed beside and below the engine. The use of a fly-wheel here would also be objectionable. The effect of the dead points explained in (311.) is avoided without the aid of a fly, by placing two cranks upon the revolving axle, and working them by two pistons. The cranks are so placed that when either is at its dead point, the other is in its most favourable position.

A wheel A, fig. 169., armed with wipers, acting upon a sledge-hammer B, fixed upon a centre or axle C, will, by a continued rotatory motion, give the hammer the reciprocating motion necessary for the purposes to which it is applied. The manner in which this acts must be evident on inspecting the figure.

The treddle of the lathe furnishes an obvious example of a vibrating circular motion producing a continued circular one. The treddle acts upon a crank, which gives motion to the principal wheel, in the same manner as already described in reference to the working beam and crank in the steam-engine.

By the following ingenious mechanism an alternate or vibrating force may be made to communicate a circular motion continually in the same direction. Let A B, fig. 170., be an axis receiving an alternate motion from some force applied to it, such as a swinging weight. Two ratchet wheels (253.) m and n are fixed on this axle, their teeth being inclined in opposite directions. Two toothed wheels C and D are likewise placed upon it, but so arranged that they turn upon the axle with a little friction. These wheels carry two catches p, q, which fall into the teeth of the ratchet wheels m, n, but fall on opposite sides conformably to the inclination of the teeth already mentioned. The effect of these catches is, that if the axis be made to revolve in one direction, one of the two toothed wheels is always compelled (by the catch against which the motion is directed) to revolve with it, while the other is permitted to remain stationary in obedience to any force sufficiently great to overcome its friction with the axle on which it is placed. The wheels C and D are both engaged by bevelled teeth (263.) with the wheel E.

According to this arrangement, in whichever direction the axis A B is made to revolve, the wheel E will continually turn in the same direction, and, therefore, if the axle A B be made to turn alternately in the one direction and the other, the wheel E will not change the direction of its motion. Let us suppose that the axle A B is turned against the catch p. The wheel C will then be made to turn with the axle. This will drive the wheel E in the same direction. The teeth on the opposite side of the wheel E being engaged with those of the wheel D, the latter will be turned upon the axle, the friction, which alone resists its motion in that direction, being overcome. Let the motion of the axle A B be now reversed. Since the teeth of the ratchet wheel n are moved against the catch q, the wheel D will be compelled to revolve with the axle. The wheel E will be driven in the same direction as before, and the wheel C will be moved on the axle A B, and in a contrary direction to the motion of the axle, the friction being overcome by the force of the wheel E. Thus, while the axle A B is turned alternately in the one direction and the other, the wheel E is constantly moved in the same direction.

It is evident that the direction in which the wheel E moves may be reversed by changing the position of the ratchet wheels and catches.

(319.) It is often necessary to communicate an alternate circular motion, like that of a pendulum, by means of an alternate motion in a straight line. A remarkable instance of this occurs in the steam engine. The moving force in this machine is the pressure of steam, which impels a piston from end to end alternately in a cylinder. The force of this piston is communicated to the working beam by a strong rod, which passes through a collar in one end of the piston. Since it is necessary that the steam included in the cylinder should not escape between the piston rod and the collar through which it moves, and yet, that it should move as freely and be subject to as little resistance as possible, the rod is turned so as to be truly cylindrical, and is well polished. It is evident that, under these circumstances, it must not be subject to any lateral or cross strain, which would bend it towards one side or the other of the cylinder. But the end of the beam to which it communicates motion, if connected immediately with the rod by a joint, would draw it alternately to the one side and the other, since it moves in the arc of a circle, the centre of which is at the centre of the beam. It is necessary, therefore, to contrive some method of connecting the rod and the end of the beam, so that while the one shall ascend and descend in a straight line, the other may move in the circular arc.

The method which first suggests itself to accomplish this is, to construct an arch-head upon the end of the beam, as in fig. 171. Let C be the centre on which the beam works, and let B D be an arch attached to the end of the beam, being a part of a circle having C for its centre. To the highest point B of the arch a chain is attached, which is carried upon the face of the arch B A, and the other end of which is attached to the piston rod. Under these circumstances it is evident, that when the force of the steam impels the piston downwards, the chain P A B will draw the end of the beam down, and will, therefore, elevate the other end.

When the steam-engine is used for certain purposes, such as pumping, this arrangement is sufficient. The piston in that case is not forced upwards by the pressure of steam. During its ascent it is not subject to the action of any force of steam, and the other end of the beam falls by the weight of the pump-rods drawing the piston, at the opposite end A, to the top of the cylinder. Thus the machine is in fact passive during the ascent of the piston, and exerts its power only during the descent.

If the machine, however, be applied to purposes in which a constant action of the moving force is necessary, as is always the case in manufactures, the force of the piston must drive the beam in its ascent as well as in its descent. The arrangement just described cannot effect this; for although a chain is capable of transmitting any force, by which its extremities are drawn in opposite directions, yet it is, from its flexibility, incapable of communicating a force which drives one extremity of it towards the other. In the one case the piston first pulls down the beam, and then the beam pulls up the piston. The chain, because it is inextensible, is perfectly capable of both these actions; and being flexible, it applies itself to the arch-head of the beam, so as to maintain the direction of its force upon the piston continually in the same straight line. But when the piston acts upon the beam in both ways, in pulling it down and pushing it up, the chain becomes inefficient, being from its flexibility incapable of the latter action.

The problem might be solved by extending the length of the piston rod, so that its extremity shall be above the beam, and using two chains; one connecting the highest point of the rod with the lowest point of the arch-head, and the other connecting the highest point of the arch-head with a point on the rod below the point which meets the arch-head when the piston is at the top of the cylinder, fig. 172.

The connection required may also be made by arming the arch-head with teeth, fig. 173., and causing the piston rod to terminate in a rack. In cases where, as in the steam-engine, smoothness of motion is essential, this method is objectionable; and under any circumstances such an apparatus is liable to rapid wear.

The method contrived by Watt, for connecting the motion of the piston with that of the beam, is one of the most ingenious and elegant solutions ever proposed for a mechanical problem. He conceived the motion of two straight rods A B, C D, fig. 174., moving on centres or pivots A and C, so that the extremities B and D would move in the arcs of circles having their centres at A and C. The extremities B and D of these rods he conceived to be connected with a third rod B D united with them by pivots on which it could turn freely. To the system of rods thus connected let an alternate motion on the centres A and C be communicated: the points B and D will move upwards and downwards in the arcs expressed by the dotted lines, but the middle point P of the connecting rod B D will move upwards and downwards without any sensible deviation from a straight line.

To prove this demonstratively would require some abstruse mathematical investigation. It may, however, be rendered in some degree apparent by reasoning of a looser and more popular nature. As the point B is raised to E, it is also drawn aside towards the right. At the same time the other extremity D of the rod B D is raised to E′, and is drawn aside towards the left. The ends of the rod B D being thus at the same time drawn equally towards opposite sides, its middle point P will suffer no lateral derangement, and will move directly upwards. On the other hand, if B be moved downwards to F, it will be drawn laterally to the right, while D being moved to F′ will be drawn to the left. Hence, as before, the middle point P sustains no lateral derangement, but merely descends. Thus, as the extremities B and D move upwards and downwards in circles, the middle point P moves upwards and downwards in a straight line.6

The application of this geometrical principle in the steam-engine evinces much ingenuity. The same arm of the beam usually works two pistons, that of the cylinder and that of the air-pump. The apparatus is represented on the arm of the beam in fig. 175. The beam moves alternately upwards and downwards on its axis A. Every point of it, therefore, describes a part of a circle of which A is the centre. Let B be the point which divides the arm A G into two equal parts A B and B G; and let C D be a straight rod, equal in length to G B, and fixed on a centre or pivot C on which it is at liberty to play. The end D of this rod is connected by a straight bar with the point B, by pivots on which the rod B D turns freely. If the beam be now supposed to rise and fall alternately, the points B and D will move upwards and downwards in circular arcs, and, as already explained with respect to the points B D, fig. 174., the middle point P of the connecting rod B D will move upwards and downwards without lateral deflection. To this point one of the piston rods which are to be worked is attached.

To comprehend the method of working the other piston, conceive a rod G P′, equal in length to B D, to be attached to the end G of the beam by a pivot on which it moves freely; and let its extremity P′ be connected with D by another rod P′ D, equal in length to G B, and playing on points at P′ and D. The piston rod of the cylinder is attached to the point P′, and this point has a motion precisely similar to that of P, without any lateral derangement, but with a range in the perpendicular direction twice as great. This will be apparent by conceiving a straight line drawn from the centre A of the beam to P′, which will also pass through P. Since G P′ is always parallel to B P, it is evident that the triangle P′ G A is always similar to P B A, and has its sides and angles similarly placed, but those sides are each twice the magnitude of the corresponding sides of the other triangle. Hence the point P′ must be subject to the same changes of position as the point P, with this difference only, that in the same time it moves over a space of twice the magnitude. In fact, the line traced by P′ is the same as that traced by P, but on a scale twice as large. This contrivance is usually called the parallel motion, but the same name is generally applied to all contrivances by which a circular motion is made to produce a rectilinear one.