Wheel, a solid piece or frame of wood or metal, usually circular, fixed to or movable upon a solid axis, about the centre line of which in either case it is intended to turn. The solid axis, when the wheel moves freely upon it, is commonly called an axle; when the wheel is fixed to and turns with it, an arbor or shaft. The true or mathematical axis is alwavs the fixed line about which the revolution of the wheel occurs. This line, or a point in it, is also called the centre of the wheel. When, as is ordinarily the case, this centre of motion coincides with the centre of form, we have a centred wheel; in case the centre of motion is to one side of the centre of form, an eccentric wheel. Both these sorts of wheels are circular; but for peculiar purposes wheels which are elliptical, or of a variety of curved outlines, are employed. Wagon wheels are usually made of wood, with a band of iron. Water wheels are made of wood, supported by iron. Wheels for heavy machinery, whether to be connected with cogs or by bands, are usually made of iron, but sometimes of iron and wood, and sometimes of brass. Wheels for clocks and watches are usually made of brass, as they also are in philosophical apparatus.

The various kinds of wheels are designated according to their uses; as fly wheels, for equalizing the motion of machinery by employing the reserved force of the momentum of a rapidly rotating wheel; balance wheels, such as are used in clocks and watches as parts of the escapement; crown wheels, having the cogs or teeth upon the face; spur wheels, having them upon the periphery; band or belt wheels, which communicate motion to each other by means of bands; ratchet wheels, which are held from moving in one direction by means of a ratchet, etc. When two wheels are placed upon a common axle, and one is much larger than the other, the former is called a wheel and the latter a pinion; and the cogs of the pinion are technically called leaves. By a train of wheels, or of wheelwork, is usually meant more than two wheels through which motion is successively transmitted. Evidently, the teeth upon two wheels or a wheel and pinion intended to engage, must be of like size and of corresponding form. The cutting and forming of the teeth, so as to secure continued rolling and action, with the least practicable jar, needless friction, and wear, is a consideration of much importance, and to which much study has been given.

Two general forms have been found best to satisfy these conditions: 1, that in which the general outline of the teeth is that of epicycloids or hypocycloids; 2, that in which they have the form of involutes of a circle. For the manner of determining these curves in practice for teeth of wheels having various sizes, and the use of the odontograph, by aid of which the curves are described, as well as for specific information respecting wheel work and the variety of other connections in machinery which cannot here be detailed, the reader is referred to Willis's "Principles of Mechanism," Buchanan's "Practical Essays on Mill Work and other Machinery," Mosely's " Mechanical Principles of Engineering," etc, Eankine's "Applied Mechanics," and other similar works. - Carriage wheels, in order to bear without fracture the concussions to which they are subject, require to be exceedingly strong, and somewhat elastic. Ordinary carriage wheels consist of a cylindrical block at the middle, the nave, turning on an axle, and having spokes in the direction of radii, which unite the nave with the wooden circular segments or fellies constituting the rim, which are enclosed and held together by a wrought-iron tire.

The tire, being made slightly small for the rim, is expanded by heating to redness, and in this condition is driven upon the rim and bolted to it; the contraction of the metal in cooling binds the fellies very firmly to one another and upon the spokes. Carriage or wagon wheels may be made flat; but they are most commonly " dishing " from the nave outward to the rim, for the double purpose of securing width of base to the vehicle, so as to lessen the danger of overturning, and of enabling the wheel better to resist lateral shocks. M. A. Morin found, as the result of many experiments, the ratio of the resistance to be overcome by the team to the whole load, with various styles of heavy-loaded wagons, to vary from 1/21 on a wet road with ruts, to 1/51 on a solid, dry road of hard gravel; the friction at the axles being, of course, reduced by the best lubricants. Generally, it may be stated that the ratio of draught to load on well macadamized roads in good order will be 1/35 to 1/40; on fresh gravelled roads, often as great as 1/16; on gravelled roads beaten hard, 1/35; and on the best paved or hard earth roads, about 1/70. Of the most important of Morin's results the following is a brief summary: 1, the resistance to wagons on solid metalled roads or pavements, taken with reference to the axle, and in a direction parallel to the ground, is sensibly proportional to the pressure or total weight of vehicle and load, and inversely proportional to the diameter of the wheels; 2, on such roads the resistance is very nearly independent of the width of the tires; 3, upon compressible bottoms, such as earths, sands, gravel, etc, the resistance decreases with increased width of tire; upon soft earths, such as loam or sand, the resistance is independent of the velocity; 5, upon metalled roads and upon pavements the resistance increases with the velocity, but the resistance is less as the wagon is better hung (with good springs) and the road more smooth; 6, the inclination of the line of draught (the direction in which the pull of the team takes effect) should approach the horizontal for all roads, and for common wagons so far as the construction will admit.

These results or laws are to be regarded as approximations, and as, practically, varied with the conditions. It is usual not to bring the line of draught nearer than by about 15° to the horizontal. (See the chapter on "Draught of Vehicles" in Morin's "Fundamental Ideas of Mechanics, and Experimental Data," New York, 1860.) - Oar wheels were at first made like ordinary spoke wheels, and were guided by flanges on the rails, as on the Sheffield colliery railroad in 1767, where the rails were of cast-iron. In 1789 car wheels were made with flanges to run on an edge rail, that is, a rail which rested upon its edge instead of lying flat. These rails were first made of cast iron and used at Loughborough, England. Car wheels with cast-iron hubs and rims and wrought-iron spokes were patented by Stevenson and Losh in 1816. A wrought-iron tire was shrunk on to the rim, and secured in its seat by a dovetailed depression. At the present time there are in use in the United States the following kinds of car wheels, which may be mentioned as examples of different modes of construction. The Washburne wheel, patented by Sax and Kear, is composed of a cast-iron centre surrounded by a steel tire.

It is made by first raising the tire to a white heat and almost to the fusing point, and casting the centre into it, which causes the union of the two parts. These wheels are heavy and durable, and capable of running 200,000 miles. As is the case with all car wheels, the centre may be composed of a hub and spokes, or of a hub and disk. In Moore's patent there is a packing of wood for diminishing the jar between the cast-iron centre and the steel rim, the latter being put on with hydraulic pressure. There are many wheels now made with a cast-iron centre and a steel tire shrunk on. Atwood's patent wheel, recently introduced, consists of a cast-iron centre and steel tire, between which there is a highly compressed oakum packing for the purpose of diminishing the jar. The kind in most general use is the chilled iron wheel, made of different patterns by different car wheel companies, but in all cases cast into a cold cast-iron mould, which chills the surface of the rim. This wheel has no separate tire, and cannot in consequence be "turned up " in a lathe when the rim becomes worn, as all steel-tired wheels can. The usual mileage of chilled iron wheels is 60,000 miles.

The Hamilton steel wheel company are introducing a wheel made of a mixture of cast steel and cast iron, melted and cast together, whereby the strength is supposed to be increased. It is intended as a substitute for the chilled iron wheel. Oar wheels are also made entirely of cast steel, which can, like steel-tired wheels, be " turned up " when worn. The driving wheels of locomotives are made with cast-iron centres and steel tires shrunk on. Formerly car wheels were keyed on their axles, but now the hole in the hub is turned slightly smaller than the perfectly cylindrical axle, which is then thrust in with hydraulic pressure. Wheels so treated have the advantage of never getting loose, while the keyed wheel always will in time with constant use. The bearings of locomotive wheels, both small and drive wheels, are on the inside, but truck wheels have the bearings on the outside. The bearing consists of a brass or other composition box in which the wrought-iron axle turns. - Water wheels are intended to impart to connected machinery the moving force due to the weight or momentum of water, or to both these combined. They are divided into two general sorts, according as they have horizontal or vertical axes.

The latter, most of which are also reaction wheels, are considered under Turbine. The former class, or those with horizontal axes, include the earliest known forms of water wheel; and they are generally the simpler in construction. If the natural current of a stream be employed, it readily appears that, supposing the conditions of depth and friction along the bed to correspond, the moving force in any case will be as the product of the volume of water that would in a given time strike and act on the float boards of a wheel into the amount of fall, or the descent of the stream within a given distance. But owing to the irregularities in the volume and velocity of. streams at different seasons, and the loss of momentum by friction against their beds, it becomes desirable to accumulate and retain a certain supply or head of water, and to develop the impelling force of this, due to volume and gravity, at a fixed position. These results are secured by constructing a weir or dam across the stream, and allowing the water collecting in a pond above this to fall upon the wheel at one of three points, which points of application of the force give to the wheel the name of overshot, undershot, or breast wheel. The tide wheel is a variety of undershot.

The undershot wheel is set directly in a running stream, or it is placed close to a fall or a dam; in the last case, the water is admitted to its lower side by a gate from the bottom of the pond. It is simply a large and strong wheel having stays projecting from its rim, upon which stout planks, called floats, and also palettes, extend along its length. The current or issuing water strikes these floats, and imparts movement to the wheel. Owing to friction, irregular flow of the issuing body of water, and the impossibility of consuming on the floats its entire moving force, the performance of common undershot wheels never exceeds from 25 to 33 per cent, of the power in the acting body of water. In Poncelet's wheel, however, with curved floats, into which the issuing body of water rises with more uniform movement, until its velocity is nearly or quite consumed, and from which it also escapes with less of back pressure and irregularity, the floats being in number about double those of the common form, a larger utilization of the water, rising to 50 or 60 per cent., is secured. Of overshot wheels, the summit is placed at or a little below the upper lever of the water; and the flow upon the rim of the wheel can be regulated by a flood gate.

The water is received into cavities formed by stout planks extending between the two ends of the wheel, and placed at an angle or curved toward the stream; these are buckets proper,. and the wheels are sometimes called bucket wheels. Borda showed that the maximum effect with overshot wheels is secured when the diameter equals the height of fall; but, other things being equal, the useful effect is greater the slower the revolution allowed; since, in such case, the water can enter the buckets more regularly, it is not flung from them by centrifugal force, and its velocity and impulsion are almost wholly consumed upon the wheel before it is released. Under the best conditions, this wheel utilizes 75 per cent, of the moving power of the water. The breast wheel resembles in form and construction the undershot, but has its floats closer together, and usually inclined toward the stream. It is set so that about one quadrant of it is turning close to a curved channel corresponding to its form, down which the supply of water (regulated by a gate) descends; and it is therefore impelled. both by the weight and momentum of the water. As it is thus intermediate in its action between the undershot and overshot wheels, so it is also in its value.

Being less loaded with the weight of water than the overshot, it moves with less strain and friction on its bearings, and under the best circumstances affords about 65 per cent, of the moving power. At the Burden nail works, Troy, N. Y., the overshot wheel furnishing the power required is 60 ft. in diameter and 22 ft. in breadth. The largest water wheel in the world is probably one employed in working a lead and silver mine in the Isle of Man. This is an overshot wheel, 72 ft. 6 in. in diameter, 6 ft. in breadth, with a crank stroke of 10 ft.; it is estimated to give 200 horse power, and pumps 250 gallons of water a minute 400 yards high.