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 Family
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, Nin /Nout. 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 Inclined Plane Family
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 Return to Tech Index
Composed by J. B. Calvert
Created 12 June 2000
Last revised 21 March 2007