Machines are all around us. Here is some of their very interesting lore.

- Introduction
- Real Machines
- The Lever Family
- The Inclined Plane Family
- The Pulley Family
- The Miscellaneous Family
- Mechanisms and Complex Machines
- Perpetual Motion
- Exercises
- References

In ancient Greece, μηχανη was a device, a contrivance, perhaps for lifting weights or making gods appear in air on the stage, and μηχανικοs meant ingenious, inventive, resourceful, or indeed the engineer himself. The word appeared in Latin as *machina*, from which it came into English. The principles of machines were known emprically from ancient times, and the theory was elaborated and put on a rational basis by Archimedes (3rd century BC). Simple machines have always been an elementary part of Physics instruction, and are of continuing importance in daily life.

It is good to start with clear definitions, so that we know what we are talking about. Let us say that a *machine* is a collection of resistant bodies arranged to change the magnitude, direction or point of application of moving forces. Motion is an essential part of a machine; without it, at least in principle, we have no machine, but a structure. The restriction to resistant bodies sets hydraulic and other fluid machines aside; these deserve special treatment. Some authors classify the hydraulic press as a machine. In a sense it is, of course, and depends on statics, but we will leave it aside. An *ideal machine* is one in which the parts are considered to be weightless, frictionless and rigid. Real machines are not ideal, but ideal machines aid thought and analysis, and in many cases are adequate approximations, so they are quite useful. A *simple machine* is a machine from which no part can be removed without destroying it as a machine. A *mechanism* is a machine considered solely from the point of view of its motions, *kinematically*, without consideration of loads. Some authors say a mechanism is a machine with "that does no useful work," but this is not a helpful distinction. A steam engine valve gear is essentially a mechanism to obtain a particular motion, but it also does useful work on the valve. A *structure* transmits force without motion.

It may be useful to define an *engine* as a machine in which the input is not in the form of mechanical energy, but which is converted into forces and torques by the machine. For example, the input could be electrical, or provided by a heat engine. It could also include machines worked by men or animals considered as part of the machine and not as users of it. A *prime mover* is an engine whose power is derived from some nonmechanical source, such as a heat engine. A prime mover is capable of motion, or being moved, without connection to any other system. However, windmills, water wheels and turbines are considered to be prime movers, as clearly are men and animals. Fundamentally, engine and machine are actually two words for the same thing, derived from Latin and Greek, respectively.

There are many ways of transmitting forces in machines, such as fluids pressing on a piston in a cylinder, or exerting their weight in buckets, flexible agents such as ropes, belts, cables and chains, springs, and weights themselves. These means are not part of the machine itself. Weight is the force of gravity on massive bodies, and form a very common load on a machine. A *link* is a member that transmits an axial force of compression or tension, and is connected by pins or sliders at its ends. A link is not a machine by itself (it does not transform its input), but is a typical part of a mechanism, and may transmit forces between simple machines. A *slotted link* with a sliding block may permit a variable amount of motion to be transmitted.

Every machine has an input and an output, and the output is a modification of the input, not a simple replication of it. A machine is a processor or transformer in some sense. The motion of the output is fully constrained by the motion of the input, by their kinematic connection. The force at the input is called the *effort*, and the force at the output, a *load*. The *mechanical advantage*, which we shall call simply the *advantage*, is the ratio of the load to the effort. The *velocity ratio* is the ratio of the movement of the load to the movement of the effort, in linear displacement or rotation. In an ideal machine the product of the advantage and the velocity ratio is unity, as we shall see. There is a trade-off between force and speed. In a real machine the product is less than unity. As a consequence, an ideal machine in equilibrium (when the effort and the load balance) can be moved by the least impetus, as well in one direction as in the other, so the machine is *reversible*. A real machine, however, requires a certain effort to move it in either direction; it is *irreversible*, and there is an unavoidable loss of energy whenever it moves.

It is reasonable to exclude from our definition those devices that depend essentially on inertial forces. The pendulum is one such device, as is the whole family of fluid turbines, and perhaps sails and airfoils as well. Simple machines can, however, form a part of such devices. These devices all deserve special consideration, and involve matters not essential to the machines that will be discussed here. Therefore, our machines depend only on the principles of statics and kinematics, not dynamics. Dynamics may have to be considered in connection with the design of machine elements, however.

The inputs and outputs of a machine may be either forces or torques, and a machine may convert one into the other. A torque or moment tends to cause rotation, while a force causes linear motion. The work done is either torque times angle of rotation, or force times distance. The dimensions of torque are force times distance, and this should be carefully distinguished from work, which has the same dimensions. Sometimes, torque is stated in, for example, pound-foot while work is in foot-pound to make this clear. A fundamental property of machines is that the input and output work are the same, except for frictional losses that make the output work smaller. This *principle of the conservation of energy* is a very important generalization, and will be considered in more detail later.

To understand the magnitudes of the forces in a machine, the methods of *statics* are used. If you already know statics, then the application to machines will be easy. If you do not, machines are an excellent and graphic way to learn about statics, and will help you to understand it. Briefly, we note that forces add according to the parallelogram rule, and can be resolved trigonometrically into components in many ways, the most useful being the rectangular components. The *moment* of a force about an axis is the product of the force and the shortest distance between its line of action and the axis. A body is in equilibrium if the sum of the forces acting upon it is zero, and the moment of these forces about any axis is zero. This gives up to six equations that may be used to find the magnitude and direction of unknown forces. In applying these principles, it is best to draw the body in question isolated from all others, and show the forces acting on it, and only those forces.

Since ancient times, simple machines have been classified as lever, wedge, wheel and axle, pulley and screw. Sometimes the wedge and screw are considered special cases of the inclined plane, so there are either four or six simple machines. This is no more than an arbitrary and incomplete taxonomy. Since classifications should be useful, you should try to make your own classification that reminds you of the principal similarities and differences. I prefer to divide simple machines into three families, those of the lever, the inclined plane and the pulley, and will treat machines in that order in this paper. Each family has various tribes, and some tribes are descendants of two families. There is also a miscellaneous family in which mechanisms are put that fit nowhere else. In complex machines, the families are mixed and connected in glorious variety.

There are ingeneous devices that, while not machines in themselves, are very important parts of machines. These include bearings, couplings, clutches, cams, springs and gears, which are conveniently studied in connection with machines.

Friction is usually the most important reason real machines are different from ideal ones, and in some machines friction plays an essential role. For example, belt drives would not work without friction. Friction cannot be described accurately in a few words, since it is a very complex and variable phenomenon. However, Coulomb's assertions of the general nature of frictional forces between solid surfaces are often adequate, at least qualitatively. The minimum tangential force F that will cause movement between two solid surfaces pressed together is proportional to the normal force N, F = μN, where μ is the *coefficient of friction*, which depends on the nature and treatment of the surfaces. The area over which N is spread is of no significance. Once motion starts, the minimum force required to maintain motion is less than F, and may depend on the speed, usually decreasing with increasing speed. This is *sliding* friction. *Rolling* friction is much less, practically vanishing if the surfaces in contact are hard and smooth, and there is no deflection.

*Journal bearings*are good examples of minimizing sliding friction. The metals in contact must be different, or the bearing will seize, especially at high bearing pressures and with poor lubrication. One metal should be hard, the other soft. Many bearings use a soft metal (babbitt) in recesses in a harder metal to confine it, working against polished steel. Steel and brass are a common pair for journal bearings. Lubrication can be supplied by greases that form an adherent thin film between the surfaces, by graphite flakes, or other means. An excellent bearing uses oil forced between the mating surfaces, dragged in by the relative movement. Now the contact is metal-oil, and the coefficient of friction drops sharply. Such a bearing is called *hydrodynamic*. Ball and roller bearings, called *anti-friction* bearings, take advantage of rolling contact between the inner and outer races, and require only minimal lubrication. Their friction is comparable to that of a hydrodynamic bearing when the latter is moving, but they require far less maintenance, besides having low friction when starting from rest. Leonardo da Vinci sketched a supposed anti-friction bearing, shown in the Figure, that seems to have been the basis for several later attempts to avoid journal bearings. It does not work, of course, merely subdividing the friction. Rolling contact is essential.

A second limiting factor in machines is the elasticity of their parts. In a rod or bar stressed along its length, the *stress*, or force F divided by cross-sectional area A, is proportional to the *strain*, or change of length ΔL divided by the length L. The constant of proportionality Y is Young's modulus (30 x 10^{6} psi for steel). Hence, ΔL = FL/YA. If a machine is scaled up proportionately, the loads F increase as L^{3}, the areas only as L^{2}, so the deflection increases as the square of the size. Elephant legs are proportionally much thicker than gazelles' legs. Bending due to transverse forces leads to much larger deflections than simple tension or compression, and these deflections increase more rapidly with size. The effect is even greater in bending. Scaling a machine up in size must be done with care.

Materials also crush under bearing pressure. Wood is particularly subject to crushing perpendicularly to the grain. One sees modern enthusiasts on television tackling ancient engineering problems by using wooden rollers, thinking of them as round. Until the load is applied, they are, but then become rather square, or sink into the even softer mud beneath them. These 'experts' are blissfully unaware of the limitations of their materials and the propagation of stress. Lever fulcrums are particularly subject to distortion. Ropes are very extensible when strained, but this is of no concern when they are used with pulleys, which allow the ropes to stretch arbitrarily.

In summary, machines are far more likely to be limited by friction and elasticity than by the strength of their parts. It is usually a simple matter to make sure that the parts of a machine are strong enough to support their loads without failure.

The lever is the most familiar machine, and its family of related machines is widespread and varied. Famously, Archimedes sang the virtues of the lever, and explained the relation betweeen the advantage and the velocity ratio. As shown in the Figure, the lever consists of the lever proper and a fulcrum, to which are applied the load W and the effort F. For an ideal lever 1, when the effort F moves a distance x vertically, the load W moves a distance y = -(b/a)x vertically as well. The negative sign only means that the movements are in opposite directions. For the lever to remain in equilibrium, the moments of the forces acting upon it, with respect to any axis, must be zero. Taking the axis through the fulcrum (which eliminates the reaction force R = F + W) this means that Fa - Wb = 0, or W = (a/b)F. The product of the advantage a/b and the velocity ration b/a is indeed unity (in absolute value). If we multiply the two equation we have obtained by considering the movement and the forces separately, we find Wy = -(b/a)(a/b)Fx, or Wy + Fx = 0. If we define the product of a force and a displacement in the direction of the force to be the *work* done by the force, we can conclude that the total work done by all the *external* forces in a (small) displacement of a machine is zero.

The three kinds of levers shown in the Figure are called *classes* of levers. This is not a very important distinction, but seems always to be presented when levers are taught. In Class I the mechanical advantage can be less or greater than unity; in Class II it is always greater than unity, while in Class III it is always less than unity. It is easy to find examples of all three classes in daily life. For example, a pry bar is Class I, a wheelbarrow is Class II, while a forearm is Class III.

The powerful tool presented in the paragraph preceding the last is called the *principle of virtual work*, very often the easiest way to analyze a machine. Even if the product of force and distance were significant only in this respect, work would still be a valuable concept. However, it also appears in other circumstances. Newton's equation of motion, md^{2}x/dt^{2} = F, when multiplied by v = dx/dt, gives Mvdv/dt = Fv, or mvdv = Fdx (after multiplying through by dt). this integrates to Mv^{2}/2 = Fx, if F is constant while accelerating M from rest through a distance x. If F is not constant, or the initial velocity is not zero, then the integral gives the desired result. Now, Mv^{2}/2 is the *kinetic energy*, and it looks as if work is the flow of energy into the body. The conservation of energy is now a commonplace, though it was a very late invention. The connection of work with energy is what gives the concept of work most of its importance.

James Watt introduced the *horsepower*, hp, to express the capability of his engines, and it has remained a very popular unit. It is conventionally 33,000 ft-lb/min or 550 ft-lb/s. A *metric horsepower* is 75 kgf-m/s, but these days the SI watt is more commonly used. 1 hp may be taken as 746 W. The *efficiency* of a machine is the ratio of the output work to the input work, often expressed as a percentage.

Work in rotational motion is torque times angle of rotation. There are rotational analogues to all the expressions for linear motion, in which moment of inertia of mass (the product of mass and the square of its distance from the axis of rotation) takes the place of mass, and the angle of rotation in place of linear displacement. The angle is expressed in radians so that the product of radius and angle is a distance, with radius and distance in the same units.

After this digression into energy, let us return to the lever. The Figure shows two other arrangements, labelled 2 and 3, in which both load and effort are on the same side of the fulcrum. The only change from case 1 is that the displacements are in the same direction, instead of opposite, which is no essential difference. The advantage of a lever is the ratio of the output, such as the raising of a weight W, to the input, which is the effort F. The ideal lever can be arranged to give any advantage from positive infinity to negative infinity, though the real lever has a much more limited range. The traditional classification of levers into three classes has no deep sigificance, and is useful only as a qualitative description. The lever itself can be bent into any form, and the fulcrum placed at any point. For example, a lever that is bent at a right angle, called a *bell crank*, can divert a force through 90°. Obviously, any angle is also possible, and the input and output forces can be in parallel planes.

As an example of the use of levers, consider the problem of operating a device at some distance by a steel rod or pipe. The length of a 300 yard run will change by about 3-1/2 inches for a temperature change of 50 °F. This surely will do mischief unless it is *compensated*. The best way to do this is to have half the run in compression, and the other half in tension. If the two halves expand by the same amount, there will be no change in the total length. Therefore, we need a mechanism that will change tension into compression. Several methods of doing this with levers are shown in the Figure. The simple straight lever is obvious, but the rods are displaced. If the rods have to be in the same line, the *lazy jack* compensator shown below will do. Two 90° cranks can be used if the rods are to be displaced an arbitrary distance.

The wheel and axle, or crank and axle, operates precisely like a lever, except that the members are adapted to continuous rotation, instead of the limited motion of the lever. The load and input can be on the same or opposite sides of the axle, which takes the place of the fulcrum. The wheel and axle normally requires some sort of journal bearing, in place of the rocking bearing of the lever. The output is commonly taken by a rope, so that any length of rope can be wound up on the axle, while rotating a crank repeatedly. The force on the input wheel can be exerted by water, either by its weight while descending in buckets on the rim, or by its dynamic impact on peripheral buckets. Reversed, the device becomes a pump. The capstan is another form of the crank and axle, with a vertical axle. Belt drives may also be included in this family, since they involve driven wheels of different diameters. They are not in the pulley family, since the very essence of a belt drive is different tensions on different parts of the belt.

Two wheels can be made to rotate in step by providing them with interpenetrating teeth on the rims, and the wheels are then called *gears*. For the teeth to mesh properly, they must have the same spacing, or *linear pitch*, on driving and driven gears. Instead of specifying the pitch directly, the *diametral pitch*, or the number of teeth divided by the diameter of the gear (rather than the more logical reciprocal of this!) is used. Gears of the same diametral pitch will mesh with one another. The pitch diameter of a gear is the diameter of the equivalent cylinder that would give the same average velocity ratio when in frictional contact with the pitch cylinder of a mating gear. The velocity ratio, output to input, is the ratio of the numbers of teeth, N_{in}/N_{out}. The form of the gear teeth must be carefully designed to give a constant velocity ratio, if vibration and wear are to be avoided. This is a profound statement, and the design of practical gear teeth to transmit large forces at reasonable speeds is not elementary. Gears can be arranged for input and output shafts at different angles or even normal to different planes. Very commonly, *mitre* gears connect shafts at 90° or some other angle. A *rack* is part of a gear of infinite radius, which produces linear motion. Gears are not traditionally included in the ancient five machines, but are indeed very important machines.

The second great family of machines is the inclined plane family. The fundamental concept here is that of coupled, constrained motions in a plane. The reaction to a weight W on an inclined plane that rises a distance b in a horizontal distance a can be resolved into a reaction normal to the plane, N = W cos theta;, and a force along the plane, F = W sin θ. In the absence of friction, W can be held in place or moved by an applied force F. This gives an advantage W/F = csc θ over raising the weight vertically. The real inclined plane is modified by friction. If F < μN, or θ < arctan μ, then W will not slide down the plane by itself, but can be pushed down the plane with a relatively small force. The largest angle for which a weight on an inclined plane will not break free and slide down is called the *angle of repose*. To push W up the plane requires a force of F plus the frictional force, or F = W(sin θ + μ cos θ). Hence, the real advantage is usually considerably less than the ideal. If μ is made small (perhaps by using rolling friction), the inclined plane is much closer to ideal.

The inclined plane was the ancient method for raising heavy loads, as for the pyramids of Egypt. It remained in nearly exclusive use for loading and unloading ships until early modern times, and is still often seen for similar uses. Barrels can be rolled up an inclined plane very conveniently, with an additional mechanical advantage. The crane was not used until Roman times, and then mainly in construction, where the space was not usually available for inclined planes. In the form of the screw, the inclined plane is probably still the most-used machine. Incidentally, heavy loads were usually moved on the horizontal in ancient times by sliding them on wet, slippery clay surfaces. Rollers and such require hard paved roads and are easily crushed, as we have already noted.

The wedge is an inclined plane used in a different way. Here F is applied horizontally to resist a vertical force W by an inclined interface. In the ideal case, with θ = 45°, W = F, and the machine only changes the direction of the force. With a smaller wedge angle, friction will maintain the force W when the input force F is removed. The limiting angle is θ = 2 arctan μ, since friction now acts on two surfaces. There are, correspondingly, two varieties of wedge, one for producing motion at right angles, and one for making a firm, but removable, connection. Wedges are very ancient machines, for fixing axe heads to hafts, and for splitting logs. They were used in split bearings to allow adjustment of the clearance for wear. The cone clutch is also related to the wedge.

The most common and ingenious use of the inclined plane is in the screw, which is simply an inclined plane wrapped around a cylinder. The virtues of this arrangement were demonstrated by Archimedes, and the Roman world was the only culture ever to use the screw until modern times. The same two types of applications that we mentioned for the wedge also exist for the screw. By rotating the screw, a nut with internal threads can be made to move back and forth. The nut can be the input, moving the screw back and forth, as in a printing press. As a fastener joining two parts, the nut can be tightened by rotating it, creating a large normal force so that friction will successfully prevent the nut from loosening. If more than one inclined plane is wrapped around the cylinder (imagine another beween the windings of the one in the Figure), the advance in one turn is a multiple of the distance between threads. This gives more threads in the nut, for added strength, or a greater advance per turn, for more rapid motion with the same frictional holding power. Plastic bottle caps often have multiple screws. They open quickly, but hold securely. Screws are also used to provide a controlled translational motion, as in a lathe for cutting screw threads, as well as in accurate instruments.

It is instructive to analyze the screw in detail, using the principles of statics. We'll assume the forces as concentrated, although they are actually distributed over the thread surfaces of the screw. They are assumed to act at a radius r, the pitch radius of the screw. The supported load is denoted by W. The force F acting at radius r rotates the screw, so the applied torque is T = Fr. T may be applied by a force acting on a handle, or by any other method. One rotation of the screw through 2π radians raises the load by a distance L, the *lead* of the screw. The helix angle of the screw is λ, where tan λ = L/2πr. For a single thread, 1/L is the number of threads per unit distance, and L is also the *pitch* of the screw.

The vector diagram at the left in the figure shows a transverse section of the thread, and defines the angle φ. It is drawn for a 60° V-thread, for which φ = 30°. An acme thread has φ = 14.5°, and a square thread φ = 0. The wedging caused by the load W increases the normal force on the thread by a factor 1/cos φ, which at most is 1.155, for the 60° thread. This is equivalent to replacing the coefficient of friction f by f/cos φ.

The vector diagram at the right is the usual one for the inclined plane, as unwrapped from the pitch surface. The load is represented by the point P, acted upon by gravity W, the normal force N on the plane, the applied force F, and the frictional force F' shown directed down the plane, as for impending upward motion of P. That is, F will be the force required to raise the load W. The forces are resolved in the vertical and horizontal directions. N is eliminated, and the result expressed as F/W, or what is equivalent, T/Wr. Note the appearance of f/cos φ as the coefficient of Coulombic friction, which we have assumed independent of contact area and only depending on the normal force. The two terms of the numerator correspond to frictional and ideal forces. If there is a thrust collar of radius r' and coefficient of friction f', add the term f'r'/r to the expression for F/W. The threads and the thrust collar both handle the same load W.

If we reverse the sign of the thread friction term, we have a result for impending downward motion. Since the frictional force is usually greater than the ideal, F will now be negative. This is the force required to start the load down the plane. If λ is increased, the frictional and ideal forces may become equal. Now F = 0, and the load is just held from descending. Further increase in λ makes F positive again; this is the force required to prevent the load from moving. All this is valid for an ordinary inclined plane, as well as for a screw, of course.

The efficiency e is easily found using the expression for T/Wr, as shown in the diagram. Normally, it may be around 0.15. This means that 0.15 of the input work raises the load, while 0.85 opposes friction. The friction is quite desirable in a lifting screw, since it holds the load steady when the applied force is removed; that is, it prevents *overhauling*. Roughly speaking, overhauling will not occur when the efficiency is less than 0.50.

The worm gear combines the inclined plane wrapped around a cylinder with the lever in the form of a gear. The linear pitch of the worm gear, the distance between teeth, must match the linear pitch of the spur gear. The teeth on the spur gear must be inclined at the pitch angle of the worm. Each turn of the worm advances the gear by one tooth, so the velocity ratio is N:1. Worm gears can be made with multiple threads, so that in one revolution of the worm the gear moves a space of n teeth, giving a velocity ratio of N:n. Worm gears are a good way to get large velocity ratios, and also prevent the mechanism from driving in the reverse direction, something desired in steering mechanisms, for example.

The pulley family features a grooved wheel or *sheave*, supported on its axis in a *block*, over which a flexible tension element--a rope, cord, cable, chain or belt--passes. The wheel is used only as an equal-armed lever, since the basic function of the pulley is only to change the direction of a force, and to add the forces on the rope together. In an ideal pulley system, the tension in the rope (the rope is often called the *fall*) is the same at all points. Advantage and velocity ratio are obtained by supporting the load by multiple ropes. The effort is applied to either end of the rope, or to a block holding the pulleys. An end of the rope can be fastened to a fixed support. A block fastened to a fixed support is called a *standing block*, while one that moves up and down is a *running block*. A pulley system is, in principle, reversible. The simplest pulley system has a single sheave in a standing block supported overhead, and serves to divert an applied downward effort into an equal upward pull, which is usually advantageous. Indeed, the term *pulley* often refers to just this arrangement. As another example, a lift may consist of two cages on opposite ends of a rope that passes over a pulley, so that the weights of the lift cages balance, and the only extra force that is necessary is equal to the load. In fact, a weight such as a full tank of water can pull up useful cargo. Self-acting inclined planes worked on this principle.

If one end of the rope is fastened overhead, and the load suspended from a block through which the rope is run to an effort exerted upwards, the effort required will be only half the load, since it is supported by two ropes. This arrangement is called a *runner*, and may be useful for raising a load to an upper level. An additional pulley can be used to return the rope to the ground, as in the top Figure. The result may be called a *gun tackle*, from a common use for it. By adding more sheaves to the standing and running blocks, additional advantage can be gained. An arrangement for an advantage of 3 is shown in the lower Figure. Such arrangements are called *block and tackle*, a nautical term for this kind of *purchase*. Curiously, the traditional nautical pronunciation of tackle is tayckle. With good bearings, sheaves can give good efficiency, making this a very practical way to lift heavy loads, and one that is often seen today. The velocity ratio is in the inverse ratio, the motion of the effort being multiplied by the number N of ropes supporting the load. The differential hoist, which can have a large advantage, is presented in the Exercises.

The purchase with mechanical advantage 3 shown in Block and Tackle uses one double pulley and one single pulley. It is commonly called a *luff*, from its use with sails. Suppose you must raise the same load, but have only two single pulleys. It isn't difficult to make a purchase with mechanical advantage 2, but can you devise one with mechanical advantage 3 that does not require a double pulley? Answer in the Exercises.

Where a pulley is used simply to give a downward pull, the tackle is said to be *rove to disadvantage*, while in the other case, as in a simple runner, it is said to be *rove to advantage*. Two examples of purchases with a larger mechanical advantage are shown in the Figure. Both are rove to disadvantage, but this has little effect. Note that only single and double blocks are used, instead of the more obvious purchases with triple and larger blocks. Both use some way to get a higher tension in the rope before the final pulleys. In all the diagrams, double pulleys are shown with the sheaves side by side for clarity. Of course, the sheaves are parallel and rotate on the same axis.

An arrangement that can be very convenient is to have two gun tackles (say) side by side. The load is raised by using one tackle, and then it can be moved sideways by taking the load with the second tackle, as on to an elevated floor. This is called a *yard and stay* tackle.

Another common mechanism is the wheel and rail. Of course, this means any wheel running on a hard road, which is the only circumstance in which wheels are superior to legs for transport. If the wheel and road are ideally hard, and the road is level, the effort is zero for any load W. In practice, the road must be inclined, so this machine is combined with the inclined plane, and could be brought into that family, but as a machine it shares nothing with that family, and is best considered by itself. The ball or roller bearing can also be considered as part of this clan, where the road has been rolled up into a circle that is travelled repeatedly. The load does not move in its direction, so the velocity ratio is zero while the advantage is infinite, the limit of the product of these being unity.

Practically, the contact cannot be an ideal line or point, but is spread out into a small region by elastic deformation, provided the pressures are kept within bounds. The creation and annihilation of this deformation of the wheel tread and the rail head causes small frictional losses, which produce a *rolling resistance*. For rubber treads on concrete roads, the rolling resistance is considerable, though much less than friction, of course. This resistance should not depend on speed, as long as the speed is less than the speed of sound in the solid materials. Another source of resistance arises if the rail deflects under the load. Then, the wheel is effectively climbing an incline at all times. However, it is also descending an incline at the same time, so the net effect depends on the elasticity and rebound of the rail. If the rebound is slow or frictional, the climbing dominates. The resistance in this case is found to be roughly proportional to the forward speed.

The oar and the paddle-wheel can probably be included with our machines. Both use the water only as a reaction medium. The oar is a lever, with fulcrum at the blade in the water, and giving an advantage a bit greater than unity. The paddle-wheel is a wheel and axle used in the reverse direction to its ordinary application. The output is the reaction on the bearings of the paddle shaft. The sails of ships and mills, and the airfoils of kites, are outside our compass, except as sources of force on a machine. Such devices have no parts that move relatively, and are similar to structures in many respects.

The cam is usually considered as an element of a mechanism, since it is used principally to create a certain motion, but it is very much like a machine, and I think it should be considered as one. The essence of a cam is a rotating member, the *cam*, in sliding contact with a second member, the *follower*. It has some similarities with the wedge, but is not a wedge. A cam is very useful if a variable velocity ratio is desired. The uniform rotation of a shaft can be transformed into an arbitrary displacement. The *eccentric*, used to obtain reciprocating motion from rotating motion where a crank in the shaft is not desirable, or where the crank angle must be adjustable, is a kind of cam, because of the sliding motion, though usually considered as equivalent to a crank.

The simple machines can be combined in limitless variety to produce complex machines to answer any requirements. From this huge field, I will adduce only a few examples to make the process clear. Hand tools offer excellent examples of machines. Pliers, scissors and similar tools are two levers joined at their common fulcrum, with the effort applied at one end by the closing of a hand. A knife is a wedge. A hand drill has a crank and axle, where the crank is part of a large gear wheel, driving a small gear wheel with mitred teeth so that the output rotation is perpendicular to the rotation of the handle. The chuck is a nut on a screw, that drives the jaws forward as wedges to grip the shank of the twist drill.

Roberval's Enigma, shown in the Figure, looks something like an equal-armed balance with equal weights on each side. The enigma, or paradox, seems to be that the weights can be moved up and down without destroying the balance. The parallelogram linkage ensures that the weights remain at equal distances from the centre. Also, when one weight is raised, the other descends an equal distance, so no net work is done. An equal-armed lever has the same properties, it seems to me.

The balance is a machine, a simple lever. An equal-armed balance detects equality of the load and the effort in the form of weights placed on opposite balance pans. There are numerous examples from all cultures and times. The more convenient *steelyard* uses a single weight that is moved along the balance arm to balance the load in the balance pan, so that a set of weights is not necessary. Another kind of balance uses a weight that is rotated at the end of an arm (or otherwise moved by a mechanism) so that the torque it exerts varies with the depression of the weighing pan. This is called a self-indicating balance, since the balance can be calibrated directly to show the weight in the pan. An example is Leonardo da Vinci's self-indicating balance, shown in the Figure, which was one of several designs of his for such balances.

The clock is a fascinating sort of machine, full of simple machines according to our definitions. The input is the effort exerted by a weight or spring, and the output is simply movement proportional to time, or the triggering of subsidiary actions such as ringing bells. The essential part, the *escapement*, involves inertial forces, but simple machines can be identified in it. The earliest mechanical escapement was the foliot and verge escapement, shown in the Figure. It appeared in the 14th century, and its inventor is unknown. The crown wheel is part of a wheel and axle, driven by a falling weight. The foliot, or "balance," is not actually a balance. It supports two weights at a distance from the axis formed by the verge. These weights are driven back and forth through a limited arc by pressure against the pallets on the verge. The pallets are cammed by the steep edge of a tooth, at first forcing the wheel backwards slightly as the weights are brought to a rest, recovering their energy, then forwards as the weights are accelerated. The recoil of the crown wheel is a feature of all early escapements. When a pallet moves free, the opposite pallet then comes into play as the wheel snaps forward a little. This pallet then decelerates the weights, and the cycle continues. The action is like that of a *relaxation oscillator* in electronics. It has no natural frequency, and its speed depends on the force with which it is operated. Its very great advantage is the positive nature of its motion, very useful with friction and rude construction. A train of gears driven by the crown wheel, with final ratios of 1:12:60, drives hour, minute and second hands. The clock is regulated by moving the weights out or in. The application of the pendulum, which has a natural frequency, to control the escapement led to much more accurate clocks. At first, the pendulum worked a verge, but the anchor escapement allowed a much smaller pendulum swing, which meant greater accuracy. Now, a vibrating quartz crystal is usually the controlling agent, and the 'gearing' is electronic.

The Watt parallel motion, shown in the Figure, is a famous mechanism. Watt had to guide the piston rod accurately vertically for both pushes and pulls in a simple, practical way. It was not yet possible to make accurate slides, so this mechanism was his answer. The point X moves very closely vertically over a range of motion. It is shown here near the top of the range. The basic mechanism is the three-bar linkage ABCD, with points A and D fixed on the engine frame. Adding members EF and FB to make a parallelogram provides a point F that moves just like point X, but is more convenient for attaching the piston rod. Previous engines, such as the Newcomen atmospheric engine, used an arch bar on the walking beam with a chain link connection to the piston. This worked only for a pull. A toothed rack and arch bar would transmit both pushes and pulls, but would be heavy and cumbersome. The Watt motion was an elegant solution.

A second famous mechanism is the crank and slider, which converts reciprocating motion into rotary motion. It was patented by James Pickard in 1780, prompting Watt to devise the sun and planet gear as an alternative. Watt's first rotary engines used the parallel motion and the sun and planet gear. In the Figure, x is the displacement of the slider from the position of Front Dead Centre (FDC). The angle the crank has rotated from FDC is θ, and the inclination of the connecting rod is φ. The relation between the angles is given by a sin φ = r sin θ, and x = a(1 - cos φ) + r(1 - cos θ). This mechanism is a combination of the slider and link, typical components of mechanism, and the crank and axle, a simple machine.

The sun-and-planet mechanism is shown in the Figure. Gear A, the sun, is connected to the shaft of the engine and rotates with it. Gear B, the planet, is connected rigidly to the connecting rod, and constrained to rotate about A by the link. This is the simplest example of an *epicyclic* gear train, whose analysis is made interesting by the rotation of the axes of some of the gears. If the gears are of the same diameter, the shaft rotates twice for each reciprocation of the slider. You can see two of Watt's engines using the sun and planet gearing, as well as the parallel motion, in the Science Museum, London. It is stated there that the 1797 engine for John Maud of London, 16" bore by 48" stroke, developing 8 hp, operated at 25 double strokes of the piston per minute, and gave 50 rpm to the flywheel. It is possible to learn a lot from such examples, which always show practical, successful designs.

Note the flyball governors on these engines. This mechanism is shown at the left. The vertical rotating shaft is rotated at a speed proportional to the speed of the engine shaft. When it is not rotating, the spring pushes the weights close to the axis, and the throttle is wide open. As the rotation speed increases, the heavy balls fly outward, opposed by the spring, whose force is proportional to its compression, and the movement closes the throttle. The speed control is independent of the load on the engine. Should the load increase and the engine slow down, the throttle is opened enough to restore the previous speed. In this way, the rotating engine maintains a constant speed. There is some change in the engine speed with load, enough to produce a sufficient change in throttle opening, but the change in speed is much less than it would be if the throttle setting were constant. This is an example of feedback control. The engine speed is converted to a control action that is fed back to the input. To change the engine speed, the resting spring length is changed. The greater the pressure exerted by the spring, the faster the engine runs.

Inventors have tried to devise machines that work by themselves, and move without applied effort. The basic idea is to move a weight horizontally away from an axis, which requires much less effort than lifting the weight. The weight is then allowed to fall. At its lowest point, it is moved back closer to the axis, where an equal weight farther from the axis can lift it back up without external effort. Many means of doing this were proposed, and models built, but they all refused stubbornly to work. Some of us can see clearly that the falling of the weight can do a certain amount of work, but an equal amount of work must be spent to raise the weight to its original level, so the best we can do is break even. Inventors are led astray by the introduction of lever arms and torques, which do not change the situation. The principle of conservation of energy is so deeply engrained these days that it provides an irresistible refutation of any device that produces work with no other change in the universe. For most people, even scientists, it is accepted uncritically, like something said loudly three times before breakfast. However, the common observation that mechanical energy does seem to disappear without trace nourished the hope that perhaps the opposite also occurred. We can get energy from temperature differences, which is as close to perpetual motion as the universe comes. It is, in fact, the inverse to the disappearance of mechanical energy by friction. Oddly, it depends on the fact that matter is molecular, but that is another story!

Now that mechanics is so well known, perpetual motion is looked for in less well understood effects, involving electricity, magnetism or gravity, or fuels from water, and such like. The Dean Drive of a few decades ago was a vibrating device that was supposed to jump up more than it jumped down, so that it could propel a rocket without any external effect. The United States Patent Office will issue you a patent for perpetual motion. An early decision to throw out all such claims *prima facie* was reversed when a disappointed inventor shot a patent officer. Perpetual movers tend to be unbalanced, and it isn't worth human life.

The solution to the question posed in Pulleys is shown at the left.

1. The differential hoist is often found. It gives a large advantage, and is little affected by friction. The two sheaves in the top block, of different radii R and r, rotate as one, and are toothed so that the chain cannot slip. In the hoist shown, r = 0.9R. If the hoist is considered ideal, find the effort E necessary to raise the weight W. Answer: E = W/20. [Hint: use a free-body diagram for the upper sheave. Also, solve by virtual work.]

2. In the machine shown, the weight W is raised by the effort E applied horizontally. Show that the advantage is 2 cotθ, and the tension in the cord is (W/2)secθ, using a free-body diagram. Now solve the problem using virtual work. Note that E is not constant, but varies with θ, so an integration is necessary. If you assume that all the work due to horizontal motion is due to E, you will not get the correct result.

3. An epicyclic train of gears is shown in the Figure. Gears A and C are connected to shafts, as is the yoke holding the gears B. Any two of the shafts can be inputs, and the remaining shaft an output. Movement of the gears is subject to the constraint of rolling where they touch, which gives a linear relation between the angular velocities of the shafts. Let us suppose that A has 30 teeth, B 10 teeth, and C 50 teeth (all gears must have the same diametral pitch, of course). If gear C is held fast, what is the velocity ratio of gear A and the rotating yoke? Such a problem is famously confusing to engineering students. One method of solution is to imagine infinitesimal rotations, and to apply the constraint of rolling. An easier way is to appreciate that if we have any two possible motions, then the superposition of the two motions is also possible. We choose the two motions to be specially simple. For example, let one be a motion in which the yoke does not rotate, and the other one in which the yoke rotates, but gears A, B and C do not move relatively. Make a table showing the rotations of the other parts when gear A receives one complete revolution in the first case, and when the yoke is rotated once in the second case. Now take a linear combination, where the constants are chosen to make the rotations of C zero, and this will give the desired answer. The velocity ratio of A with respect to the yoke is 8:3. [The table is 0,1,-3,-3/5 and 1,1,0,1 for yoke, A, B and C, respectively.]

4. Design a compensator (see the section on Levers) using racks. Why is compensation rendered difficult if the actuation is by wires in tension, instead of by rods or pipes?

*Encyclopedia Britannica*, 15th ed. (1995), v. 7, article Machine.

*McGraw-Hill Encyclopedia of Science and Technology*, 7th ed. (1992) v. 10, article Machine, which sends you to v. 16, article Simple Machine. This is a short introduction, directing you to further articles on the individual types of machines, based on the canonical five.

P. Schwamb, A. L. Merrill, W. H. James and V. L. Doughtie, *Elements of Mechanism*, 6th ed. (New York: John Wiley & Sons, 1947). Traditional engineering texts of this kind mentioned much about simple machines, in addition to comprehensive treatments of linkages, gears, wheel trains and cams.

S. Strandh, *A Hisory of the Machine* (New York: A&W Publishers, 1979). Profusely illustrated.

Bureau of Naval Personnel, *Basic Machines and How They Work* (New York: Dover, 1971).

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Composed by J. B. Calvert

Created 12 June 2000

Last revised 21 March 2007