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Wheelbuilding is the art and science of assembling a wheel (generally a bicycle wheel, although such wheels are also used on motorcycles, wheelchairs and some cars) from its component rim, spokes, nipples and hub.
Goals
A good wheelwright will ensure that the wheel is "true" in three ways: "runout" (sideways wobble), "dish" (accurate centering of the rim between the hub locknuts) and "hop" (roundness). Each spoke must have equal tension, and this tension must be sufficiently high that spokes can never come loose under shock loads. Spokes should have no residual twist from tightening the nipples. Finally, the spokes must be "stress relieved", i.e. subjected to a greater tension during building than they are ever likely to encounter in use - usually by squeezing pairs of spokes together very hard. This yields the spokes into a permanent shape where they bend around the hub flanges and each other.
Lacing
Spoking patterns may be radial or semi-tangential. Only the latter is suitable for a wheel that has to transmit torque from the hub to the rim, as with rear wheels or hub brakes. Rear wheels may also incorporate radial spoking on the non-drive side and semi-tangential spoking on the drive side. The most common spoking pattern is "three-cross" where each spoke crosses three others on the same flange of the hub before meeting the rim The last cross is normally "interlaced" by wrapping the spoke around the one from the other side of the flange. Radial-spoked wheels, where the spokes do not cross each other, save the weight of two spokes compared to a three-cross wheel but run the risk of tearing hub flanges apart, unless the hub is specifically designed for this pattern.
Bicycle wheels
Most conventional bicycle wheels now use 32 or 36 spokes front and rear, although the asymmetry of the rear wheel (to allow for the cluster of sprockets), and the additional weight it carries, means it benefits from having more spokes than the front.
Wheels can be built by machine instead of by hand. However, machine-built wheels are rarely as satisfactory as handbuilt wheels, partly because it is uneconomic to allow the machine to spend long enough on each wheel for a perfect result but also because most machines leave spokes with some residual twist. More modern "factory built" wheels such as Mavic's Ksyrium series are of quite different construction from that of a conventional wheel, trading a deeper and stronger rim for fewer spokes. They are popular, being aerodynamic and quite light (in the more expensive models) but not as durable, readily repairable or maintainable as a conventional wheel.
The canonical text on building conventional bicycle wheels is "The Bicycle Wheel" by Jobst Brandt, published by Avocet.
Motorcycle wheels
Motorcycle wheels typically use 36 or 40 spokes, of much heavier gauge than those on a bicycle. They are never "interlaced" in the manner described above, nor are strait-laced "radial" builds recommended (except on wheels without brakes, which are not themselves a good idea). Adequate spoke tension is very important with motorcycle wheels, because of the greater torque applied to the wheel by an engine or disc brake. Loose spokes on a wheel fatigue rapidly and break, usually at the bend where they attach to the hub. When this happens, the wheel must be rebuilt using all new spokes, because even unbroken spokes in such a wheel are fatigued, and will usually break when tightened during the truing operation.
Truing
Truing is best done using one or two dial indicators. An indicator is set to ride on the side of the rim, and the spokes are tightened or loosened to pull the rim straight. An indicator is then placed on the tire seating surface of the rim to measure axial runout, and spokes are tightened or loosened until this measurement is acceptable. During this step, spokes from the left and right sides of the hub are tightened equally, to avoid throwing the straightness out. For a new rim, deviations of 0.010 to 0.015 inch (0.25 to 0.40 mm) are good. Used rims often cannot be trued to this standard. Deviations greater than 0.150 inch (3.8 mm) may require rim replacement. (Tires often exceed this amount of deviation, even when mounted on a perfectly true alloy wheel. It is unlikely, however, that wheel deviation will cancel out any of the tire deviation, and it's not usually a good idea to try to do that by truing the wheel to the tire.) When both indicator readings are acceptable, all the spokes on one side of the hub are tightened to center the rim on the hub (or to create the proper offset, on wheels that should have one). To finish truing, all the spokes are tightened equal amounts, until they are very tight. When this is complete, all the spokes should "ring" when struck, but do not expect them to make the same tone.
Spoke length
Selecting spokes of the correct length requires a calculation based on rim diameter, hub diameter, hub width, lacing pattern, number of spokes, etc.Sheldon Brown
Notes
References
Sheldon Brown Cycling
"Wheelbuilding"