Mechanical advantage of gear trains

Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system.


Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force.


The power into and out of the lever must be the same, so forces applied to points farther from the pivot must be less than when applied to points closer in.


If a and b are distances from the fulcrum to points A and B and if force FA applied to A is the input force and FB exerted at B is the output, the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by This is the law of the lever, which was proven by Archimedes using geometric reasoning.
It shows that if the distance a from the fulcrum to where the input force is applied (point A) is greater than the distance b from fulcrum to where the output force is applied (point B), then the lever amplifies the input force.


If the distance from the fulcrum to the input force is less than from the fulcrum to the output force, then the lever reduces the input force.
The mechanical advantage of a pair of a chain drive or timing belt drive with an input sprocket with NA teeth and the output sprocket has NB teeth is given by The mechanical advantage for friction belt drives is given by Chains and belts dissipate power through friction, stretch and wear, which means the power output is actually less than the power input, which means the mechanical advantage of the real system will be less than that calculated for an ideal mechanism.


If the sprockets at the crank and at the rear drive wheel are the same size, then the ratio of the output force on the tire to the input force on the pedal can be calculated from the law of the lever to be Now, consider the small and large front sprockets which have 28 and 52 teeth respectively, and consider the small and large rear sprockets which have 16 and 32 teeth each.
For an ideal block and tackle system there is no friction in the pulleys and no deflection or wear in the rope, which means the power input by the applied force FV must equal the power out acting on the load FV, that is The ratio of the output force to the input force is the mechanical advantage of an ideal gun tackle system, This analysis generalizes to an ideal block and tackle with a moving block supported by n rope sections, This shows that the force exerted by an ideal block and tackle is n times the input force, where n is the number of sections of rope that support the moving block.