I’m starting a new project and its bike related so I thought I’d share some of the research on here.  For this first part I’m looking for the mechanical advantage of different styles of brake calipers for road bikes.  The numbers are not exact by any means.  The calculations COULD be erroneous so let me know if you find anything wrong.  Overall it gives good insight into the science of some of the different caliper designs you and I are using.
 
The first is a common side pull brake caliper.  These have been used up until around the early 90s when Shimano dug up the double pivot design.  The side pull caliper design is light, simple and operates well when manufactured and installed properly.  This design is basically pliers with slightly unequal arms.
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To find the caliper’s mechanical advantage I first isolated the two arms.  Then the distance between where the force of the cable is applied to the pivot point is measured.  Conversely the distance from the pivot point to where the normal force from the wheel pushes onto the caliper is measured.  Then moments are calculated and viola, a ratio of the output (Q) and input (F) forces can be found. 

For this case Arm A and Arm B are slightly different as their physical geometries are slightly different.  The average of the two ratios can be taken to find the ratio of the caliper as a whole.  In the case of this Dia-compe caliper with the pads mounted all the way down the mechanical advantage is 0.92 meaning for every pound of force fed through the cables you get only .92 pounds squeezing the rim.  This is why calipers work better with shorter reaches as that will dramatically increase the mechanical advantage of the system. 
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The second caliper is the Dura Ace 7800, touted as the performance standard to which all other calipers are measured by.  These calipers use a dual pivot design in which the two arms rotate on different pivots but are connected physically by a lever arm.  Again, the arms are first isolated, with forces F and Q labeled.  Additionally, there is a Force P that is the connecting force between the two calipers. 

For the system the mechanical advantage comes out to be 1.225.  This is a drastic increase from the side pivot design.  Keep in mind that this design is typically heavier, works better on narrower rims,  stays centered better but does not track an off true wheel.
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Lastly there is the Zero Gravity single pivot with cam lever design.  This design is touted as an elegant way of getting the mechanical advantage of the dual pivot setup with the low weight and simplicity of the side pull setup.  In this design the two arms pivot on the same center pivot just as the side pull design but they are also connected via a third “helper” arm that greatly increases the leverage of the system.

Getting to the point, the caliper has an average Output to Input ratio of 1.354!  that’s greater than the dual pivot design!  Don’t be fooled though.  The reason for this lies in the design of the cam “helper” arm.  At a static fully opened position the cam arm is at its highest leverage position.  As the caliper is squeezed the cam’s stroke decreases the leverage that it supplies to the main arms.  Basically what it comes down to is proper sizing of rim width with the brake setup to avoid running into low power areas within the caliper’s stroke.
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Keep in mind that these numbers are very rough.  Also, much of the mechanical advantage of the braking system is provided through the brake levers themselves.  Brake pad and rim material also plays a huge role in how a brake system operates.

Shoot me an email if you have any questions/concerns.