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Derailleur Allowances

Derailleur allowances are the amount of extra chain required to go around a derailleur rear gearchange mechanism.

Screenshot of derailleur allowance calculator

The Machinehead Chain Length Calculator has this handy pop up derailleur allowance calculator to calculate extra length of chain in a way that takes the most important dimensions of the rear mech into account:

The Industry Standard Approximation

A widely accepted approximation for the amount of extra chain required to go around a derailleur rear mech is "Add an inch/link". And its not a bad approximation for most of the rear mechs commonly in use i.e. 10 teeth on each cog/pulley/jockey wheel and about 2 1/4 to 3 3/4 inch between jockey wheel centres. But the size of the jockey wheels can vary. I have seen some with 13T inner and 15T outer wheels so an inch must surely be an under estimate for these.

The Triangular Approximation

Shortest length derailleur approximation diagram In the diagram on the right the two small circles represent the derailleur jockey wheels.
d1 is the tangential distance between the biggest front and rear gear cogs (chain ring and cassette or freewheel). I.e. The straight blue line from A to D represents a tightly stretched chain without a derailleur. This value, displayed as 'Tangential Length =' is taken from the last calculation in the main chain length calculator window.
d2 is the distance between the centres of the two jockey wheels. Typically about 2.25" for a short arm racing mech up to about 3.75" for a long arm touring mech. (Easy to measure with a ruler).
d3 is the gap between the biggest rear gear cog and the inner derailleur jockey wheel. Tricky to measure but always a pretty small gap. I'm currently defaulting to 0.1" as a suggested minimum gap. (See Also: Determining The Inner Jockey Wheel Position).

The extra amount of chain required is approximately a bit more than the difference between the length of the red zigzag line from A to B to C to D and the blue straight length from A to D. So a simple solution can be found by resolving three triangles. It will always under estimate very slightly because the reality is the chain follows a curved path around the jockey wheels. But it will suggest bigger allowances for rear mechs with bigger jockey wheels. Even calculating the straight bits and curved bits separately this would still be effectively as tight as possible which is still too tight. So another rule of thumb is to add a small allowance to give some slack - 1/4 to 1/2 a link ought to be about right.

The Rigorous Approximation

Works in basically the same way as the triangular approximation described above but now resolves some extra triangles and angles to calculate the length of the red wavy line with precision. This is the version I recommend using. Not so quick and easy to explain all the math, or check on your pocket calculator, slide rule or abacus but gives accurate answers for any rear mech... Then still add about 1/4" to let the rear mech relax a bit. Because the answer is usually only a few thou more than with the triangular approximation either method will do for practical purposes. The extra few thou pale into insignificance in comparison with the amount that is added to give some slack. However, if you want a warm fuzzy feeling from calculating chain length properly use the rigorous approximation ;-)
(See Also: Determining The Inner Jockey Wheel Position).

The Circular Approximation

Diagram of circular approximation This method seems to work rather well. It gets within a few thou of the rigorous method and requires the simplest interface. It calculates the length of chain between the two jockey wheel centres accurately (i.e. as two straight and two curved sections) then adds on the amount of chain required to wrap an extra 75.5 degrees segment of each jockey wheel. r1 and r2 are the radius of the two jockey wheels to the centre of a chain pin and can have different values.
NOTE:- The rigorous approximation also calculates the length of chain to wrap the jockey wheels from the highest and lowest points in this diagram in exactly the same way. But without the extra 75.5 degrees obviously.

Determining The Inner Jockey Wheel Position

Freewheel gear tangent to jockey wheel centre distance

To accurately fix the position of the inner derailleur jockey wheel and the tangential point on the biggest freewheel/cassette cog ChainLengthCalc needs to know about the minimum clearance (C) when using the biggest rear cog that the rear mech was designed to handle (Rear Mech Capacity). Theoretically C can be zero because r1 and r2 are distances to chain pin centres. But in practice some clearance is desirable i.e. a bit more than zero. By default Chainlengthcalc assumes that C = 0.1" This value has been arbitrarily plucked out of the ether because is seems about right. It is a bit more than the difference between the radius of a 32T cog and a 31T cog. When better information becomes availiable the value of this constant can be refined by opening the file ChainLengthCalc.ini in notepad and adding an entry JW2RC Clearance=?? in the [frmRearMech] section. The following changes 'C' to 0.01":
JW2RC Clearance=0.01

If the capacity of your derailleur is bigger than the number of teeth on the biggest rear cog, the distance C increases by the difference between the radius of the actual cog used and the radius that the biggest cog could be. Thus any small error due to inaccuracy in the C value do not become exaggerated.

Both the triangular and rigorous approximations determine the jockey wheel position and the length 'd' in this way.

The Scale Drawing And Bit Of String Approximation

If you haven't got a computer, abacus, slide rule, calculator etc but you have got a pair of compasses and a set square. An adequate estimate of the extra chain can be made by producing a scale drawing and wrapping a piece of string around the chain path. Click here for a table of radius values in inches for different cog sizes for 1/2 inch pitch bicycle chain. When you have finished with the string you can always use it to suspend a weather prediction cone.

See Also
Chain Length Calculator
Derailleur Limits
Bicycle Frame Dropouts
Fractions To Decimal

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