Comparing the change in length of gear and brake housing.
by Dr. Leslie BrownKnowing the surface area & stiffness constants for materials, the change in length of different housing can be compared to each other.
As cited elsewhere, sintered aluminium oxide possesses a Young’s modulus in the range of 300-400Gpa, while steel has a Young’s modulus of 210GPa. Even taking the conservative value 310Gpa for 96% Al2O3, these numbers indicate that all else being equal, vertebrae ceramic housing compresses 540% less than standard shimano gear housing (it has 360% more surface area and the material is 48% stiffer than steel to begin with). Put another way, that’s over 5 times more drivetrain accuracy.
But we all know that quantitative numbers are much better than qualitative ones. Given a set of brake lines on a bike, how much less compression are we actually talking about? The following equation can be used to estimate this.
The change in length of a material under load is given by the formula:
e = FL/EA
where
e = extension or change in length. Negative values represent compression while positive values represent tension.
F = externally applied force measured in Newtons (see below).
L = initial length ~ 0.75 metres for a rear brake line (assuming a frame with cable stops & external cable routing).
E = Young’s modulus (a stiffness constant).
A = cross-sectional area.
Externally Applied Force:
The external force is applied by our hands and fingers. This force is multiplied considerably by the brake lever. Obviously the amount we squeeze the brake lever is arbitrary, but each person has a physiological limit. Assuming we pull the lever with 5kg of ‘force’, the actual load transferred to the cable depends on the leverage ratio of the particular brake levers that are installed.
Brake Leverage Ratios:
This leverage ratio can be calculated by measuring 2 critical dimensions of the levers themselves.
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The distance between the pivot point and the approximate place where the force is normally exerted by the fingers… this measurement is naturally a little subjective.
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The distance between the pivot point and the cable attachment point.
Here are some cable-leverage ratios obtained from some common brake levers:
Campagnolo Centaur 2008 ~ 80mm/20mm = 4.00
Campagnolo Veloce 2009 ~ 100mm/23 = 4.35
Shimano Ultegra SL ~ 85mm/18mm = 4.72
Avid V-brake levers ~ 75mm/32.5mm = 2.31
5kg equals 49 Newtons of Force. Assuming we’re using the 2008 campagnolo levers, this force increases by a factor of four (196N) along the brake cable itself.
Plugging all the numbers in, we can determine that standard brake housing theoretically compresses by around 0.10mm (100 microns). Vertebrae ceramic housing would only compress 0.037mm (37µm) under the same conditions.
Nice as that is to know, in the real world, a physical test needs to be performed to confirm these numbers. Why you ask? Because the cross-sectional shape of each type of housing is not consistent and would likely affect the results.
Standard gear housing for instance is composed of separate steel strands which would tend to splay out upon compression (the individual wires would much rather bend along their length than compress). Hence the compressive strength of the multiple steel cores is actually limited by the circumferential strength of the plastic retaining shroud. Peel off that shroud and the compressive strength of those wires would drop considerably. Not so with vertebrae!
