Efficiency in Rigid Body Systems
Any devices with work/power inputs and outputs will have some loss of work or power between that input and output due to inefficiencies such as friction. While energy is always conserved according to the principle of work and energy, some energies such as the heat generated via friction may not be considered useful. A measure of the useful work or power that makes it from the input of a device to the output is the efficiency. Specifically efficiency (denoted by the lowercase Greek letter eta) is defined as the useful work out of a device divided by the work into the device. With power being the work over time, efficiency can also be described as useful power out divided by the power in to a device (the time term would cancel out leaving us with our original definition).
| \[\eta=\frac{W_{out}}{W_{in}}=\frac{P_{out}}{P_{in}}\] |
Efficiency is a unitless measure that is always between zero and one, though it may also be denoted as a percentage. The higher the efficiency, the greater the percentage of work or power that makes it from the input to the output.
Because kinematics will usually not allow for a drop in velocities or angular velocities in mechanical systems (for example the teeth in gear systems can't pass through one another), the efficiency will play out as a loss of force or moments rather than a loss of velocities.
| Translational: | \[F_{out}=\eta\frac{F_{in}\omega_{in}}{\omega_{out}}\] |
| Rotational: | \[M_{out}=\eta\frac{M_{in}v_{in}}{v_{out}}\] |
It is impossible to have efficiencies greater than one (or 100%) because that would be a violation of the conservation of energy, however for most devices we wish to get the efficiencies as close to one as possible. This is not only because it wastes less work or power, but also because any work or power that is "lost" in the device will be turned into heat that may build up. If it is large enough, this heat build up may also require the addition of a cooling system to draw that heat away.
