A bicycle wheel consists of a central hub and a round rim, joined by a number of spokes.
Spokes radiate from the hub to the rim, where they are anchored in a screw-thread attachment called a nipple. The nipple has flat faces which may be turned using a tool called a spoke key (available for a small cost from any good bicycle shop, in extremis a pair of pliers or small adjustable wrench will suffice); thus the tension in an individual spoke can be increased or reduced.
Spokes may be arranged in a variety of patterns, of which three-cross, four-cross and radial are the most common. The pattern affects the strength, weight and characteristics of the wheel but is not relevant to the process of truing.
The rim will be pulled to the side on which spokes are tightened. To true a wheel which is deflected to the right, tighten the left-side spokes near the deflection and loosen the right-side spokes.
It is important to maintain the balance of the wheel's tension -- if you tighten a spoke on one side, you should loosen one on the opposite side. All the spokes on one side of a wheel should be at approximately the same tension; a quick test is to "ping" the spokes with a fingernail: they should all sound the same. Watch out for pulling the wheel into an eccentric shape. Check roundness by observing vertical movement when mounted in the frame. An out-of-round condition can be corrected by slightly loosening spokes on either side of a high spot (going around the rim) and tightening those in the high spot to pull it down. Reverse the process for a low spot.
To avoid putting undue torsion (twisting force) on a spoke, it is good practice to slightly over tighten and then partially loosen. For example, to add a half-turn to a spoke, tighten to three-quarters of a turn and then loosen a quarter turn.
Bicycles with asymmetric rear wheels have shorter spokes on the drive side - the side with the cluster of sprockets - under more tension than the spokes on the non-drive side. Whatever the look of the wheel, it is correctly true when the rim is exactly on a centreline drawn between the wheelnuts (the exception to this is bicycles with offset rear triangles, an exotic touring variation; you'll know if you have one). A simple tool for this purpose is the carpenter's "folding rule". Unfold the ruler and put an angle at its middle. Place it simultaneously against the inner axle nut and opposite sides of the rim so it touches all three. Without changing the resulting angle, place the ruler on the opposite side of the wheel. Adjust the spokes' tension until the ruler touches the same on both sides.
If a rim is badly bent it will be difficult to straighten by spoke tensioning. It is best to replace the rim or entire wheel. However, one can do an acceptable job by completely loosening the spokes in the damaged area (note how they are crossed), and straightening the rim with pliers or even a hammer and anvil before respoking and tensioning.
Patience is a virtue and will greatly decrease the work necessary (and frustration). Make adjustments in small increments. One adjustment almost always affects a different part of the wheel, so check between each one.
A full, comprehensive discussion of bicycle wheel building and truing is found in Jobst Brand's book The Bicycle Wheel.
Note: This discussion of truing applies to conventionally spoked wheels, and may not be relevant for the new road-racing style of wheel with very few spokes.
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