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 model
for this is the law of the lever. Machine components designed to manage forces and movement
in this way are called mechanisms. An ideal mechanism transmits power without adding to
or subtracting from it. This means the ideal mechanism does not include a power source,
and is frictionless and constructed from rigid bodies that do not deflect or wear. The performance
of a real system relative to this ideal is expressed in terms of efficiency factors that
take into account friction, deformation and wear.
The law of the lever The lever is a movable bar that pivots on
a fulcrum attached to or positioned on or across a fixed point. The lever operates by
applying forces at different distances from the fulcrum, or pivot.
As the lever pivots on the fulcrum, points farther from this pivot move faster than points
closer to the pivot. The power into and out of the lever must be the same. Power is the
product of force and velocity, 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 is greater
than the distance b from fulcrum to where the output force is applied, 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. Recognizing
the profound implications and practicalities of the law of the lever, Archimedes has been
famously attributed with the quotation "Give me a place to stand and with a lever I will
move the whole world." The use of velocity in the static analysis
of a lever is an application of the principle of virtual work.
Speed ratio
The requirement for power input to an ideal mechanism to equal power output provides a
simple way to compute mechanical advantage from the input-output speed ratio of the system.
The power input to a gear train with a torque TA applied to the drive pulley which rotates
at an angular velocity of ωA is P=TAωA. Because the power flow is constant, the torque
TB and angular velocity ωB of the output gear must satisfy the relation
which yields
This shows that for an ideal mechanism the input-output speed ratio equals the mechanical
advantage of the system. This applies to all mechanical systems ranging from robots to
linkages. Gear trains
Gear teeth are designed so that the number of teeth on a gear is proportional to the
radius of its pitch circle, and so that the pitch circles of meshing gears roll on each
other without slipping. The speed ratio for a pair of meshing gears can be computed from
ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear,
its gear ratio.
The velocity v of the point of contact on the pitch circles is the same on both gears,
and is given by
where input gear A has radius rA and meshes with output gear B of radius rB, therefore,
where NA is the number of teeth on the input gear and NB is the number of teeth on the
output gear. The mechanical advantage of a pair of meshing
gears for which the input gear has NA teeth and the output gear has NB teeth is given
by
This shows that if the output gear GB has more teeth than the input gear GA, then the
gear train amplifies the input torque. And, if the output gear has fewer teeth than the
input gear, then the gear train reduces the input torque.
If the output gear of a gear train rotates more slowly than the input gear, then the
gear train is called a speed reducer. In this case, because the output gear must have more
teeth than the input gear, the speed reducer will amplify the input torque.
Chain and belt drives Mechanisms consisting of two sprockets connected
by a chain, or two pulleys connected by a belt are designed to provide a specific mechanical
advantage in power transmission systems. The velocity v of the chain or belt is the
same when in contact with the two sprockets or pulleys:
where the input sprocket or pulley A meshes with the chain or belt along the pitch radius
rA and the output sprocket or pulley B meshes with this chain or belt along the pitch radius
rB, therefore
where NA is the number of teeth on the input sprocket and NB is the number of teeth on
the output sprocket. For a toothed belt drive, the number of teeth on the sprocket can be
used. For friction belt drives the pitch radius of the input and output pulleys must be used.
The mechanical advantage of a pair of a chain drive or toothed 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. A chain or belt drive
can lose as much as 5% of the power through the system in friction heat, deformation and
wear, in which case the efficiency of the drive is 95%.
Example: bicycle chain drive
Consider the 18-speed bicycle with 7 in cranks and 26 in wheels. 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. Using these
numbers we can compute the following speed ratios between the front and rear sprockets
The ratio of the force driving the bicycle to the force on the pedal, which is the total
mechanical advantage of the bicycle, is the product of the speed ratio and the crank-wheel
lever ratio. Notice that in every case the force on the
pedals is greater than the force driving the bicycle forward. This low mechanical advantage
keeps the pedal crank speed low relative to the speed of the drive wheel, even in low
gears. Block and tackle
A block and tackle is an assembly of a rope and pulleys that is used to lift loads. A
number of pulleys are assembled together to form the blocks, one that is fixed and one
that moves with the load. The rope is threaded through the pulleys to provide mechanical
advantage that amplifies that force applied to the rope.
In order to determine the mechanical advantage of a block and tackle system consider the
simple case of a gun tackle, which has a single mounted, or fixed, pulley and a single movable
pulley. The rope is threaded around the fixed block and falls down to the moving block where
it is threaded around the pulley and brought back up to be knotted to the fixed block.
Let S be the distance from the axle of the fixed block to the end of the rope, which
is A where the input force is applied. Let R be the distance from the axle of the fixed
block to the axle of the moving block, which is B where the load is applied.
The total length of the rope L can be written as
where K is the constant length of rope that passes over the pulleys and does not change
as the block and tackle moves. The velocities VA and VB of the points A and
B are related by the constant length of the rope, that is
or
The negative sign shows that the velocity of the load is opposite to the velocity of
the applied force, which means as we pull down on the rope the load moves up.
Let VA be positive downwards and VB be positive upwards, so this relationship can be written
as the speed ratio
where 2 is the number of rope sections supporting the moving block.
Let FA be the input force applied at A the end of the rope, and let FB be the force at
B on the moving block. Like the velocities FA is directed downwards and FB is directed
upwards. 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 FAVA must equal the power out acting on the load FBVB, 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.
Efficiency Mechanical advantage that is computed using
the assumption that no power is lost through deflection, friction and wear of a machine
is the maximum performance that can be achieved. For this reason, it is often called the ideal
mechanical advantage. In operation, deflection, friction and wear will reduce the mechanical
advantage. The amount of this reduction from the ideal to the actual mechanical advantage
is defined by a factor called efficiency, a quantity which is determined by experimentation.
As an ideal example, using a block and tackle with six ropes and a 600 pound load, the operator
would be required to pull the rope six feet and exert 100 pounds of force to lift the
load one foot. Both the ratios Fout / Fin and Vin / Vout from below show that the IMA
is six. For the first ratio, 100 pounds of force in results in 600 pounds of force out;
in the real world, the force out would be less than 600 pounds. The second ratio also
yields a MA of 6 in the ideal case but fails in real world calculations; it does not properly
account for energy losses. Subtracting those losses from the IMA or using the first ratio
yields the AMA. The ratio of AMA to IMA is the mechanical efficiency of the system.
Ideal mechanical advantage The ideal mechanical advantage, or theoretical
mechanical advantage, is the mechanical advantage of a device with the assumption that its components
do not flex, there is no friction, and there is no wear. It is calculated using the physical
dimensions of the device and defines the maximum performance the device can achieve.
The assumptions of an ideal machine are equivalent to the requirement that the machine does not
store or dissipate energy; the power into the machine thus equals the power out. Therefore,
the power P is constant through the machine and force times velocity into the machine
equals the force times velocity out--that is,
The ideal mechanical advantage is the ratio of the force, or effort, out of the machine
relative to the force or effort into the machine, or
Applying the constant power relationship yields a formula for this ideal mechanical advantage
in terms of the speed ratio:
The speed ratio of a machine can be calculated from its physical dimensions. The assumption
of constant power thus allows use of the speed ratio to determine the maximum value for the
mechanical advantage. Actual mechanical advantage
The actual mechanical advantage is the mechanical advantage determined by physical measurement
of the input and output forces. Actual mechanical advantage takes into account energy loss due
to deflection, friction, and wear. The AMA of a machine is calculated as the
ratio of the measured force output to the measured force input,
where the input and output forces are determined experimentally.
The ratio of the experimentally determined mechanical advantage to the ideal mechanical
advantage is the efficiency η of the machine,
See also Outline of machines
Lever Compound lever
Simple machine Mechanical advantage device
Gear ratio Chain drive
Belt Roller chain
Bicycle chain Bicycle gearing
Transmission References
Fisher, Len, How to Dunk a Doughnut: The Science of Everyday Life, Arcade Publishing, ISBN 978-1-55970-680-3 .
United States Bureau of Naval Personnel, Basic machines and how they work, Courier Dover
Publications, ISBN 978-0-486-21709-3 . External links
Gears and pulleys Mechanical engineering — pulleys
Mechanical advantage — video