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## Sprocket and Chain Physics

Sprockets are one common way to transmit power and change the output torque or speed of a mechanical system. Understanding these basic concepts is required to make optimized design decisions. This section will briefly cover the definition of these concepts and then explain them in relationship to basic sprocket and chain designs.
Speed
Torque
Power
Speed is the measure of how fast an object is moving. The speed of an object is how far it will travel in a given amount of time. The SI unit for speed is meters per second but speed is also commonly expressed in feet per second.
Torque is roughly the measure of the turning force on an object like a sprocket or a wheel. Mathematically, torque is defined as the rate of change of the angular momentum of an object. A common example of torque is a wrench attached to a bolt produces a torque to tighten or loosen it. Torque is commonly expressed in N⋅m or in⋅lbs.
When torque is turning an object, like a sprocket, the sprocket will create a straight line (linear) force at the point where the teeth contact the chain. The magnitude of the torque created is the product of the rotational force applied and the length of the lever arm, which in the case of a sprocket, is half of the pitch diameter (the radius).
Power (P) is the rate of work over time. The concept of power includes both a physical change and a time period which the change occurs. This is distinct from the concept of work which only measures a physical change. It takes the same amount of work to carry a brick up a mountain whether you walk or run, but running takes more power because the work is done in a shorter amount of time. The SI unit for power is the watt(W) which is the same as one joule per second (J/s).
Often in competition robotics the total power is fixed by the motors and the batteries available. The maximum speed at which an arm can lift a certain load is dictated by the maximum system power.

## Chain Drive

Selecting sprockets with different sizes relative to the input sprocket varies the output speed and the output torque. However, total power is not effected through these changes.
Sprocket and chain is a very efficient way to transmit torque over long distances. Modest reductions can be accomplished using sprockets and chain, but gears typically provide a more space-efficient solution for higher ratio reductions.

### Sprocket Ratio

When a larger sprocket drives a smaller one, for every rotation of the larger sprocket, the smaller sprocket must complete more revolutions, so the output will be faster than the input. If the situation is reversed, and a smaller sprocket drives a larger output sprocket, then for one rotation of the input, the output will complete less than one revolution- resulting in a speed decrease from the input. The ratio of the sizes of the two sprockets is proportional to the speed and torque changes between them.
The ratio in size from the input (driving) sprocket to the output (driven) sprocket determines if the output is faster (less torque) or has more torque (slower). To calculate exactly how the sprocket size ratio effects the relationship from input to output, use the ratio of the number of teeth between the two sprockets.
In the image below, the ratio of the number of teeth from the input sprocket to the output sprocket is 20T:15T, which means the input needs to turn 1.3 rotations for the output to complete one rotation
$20T/15T =1.3$

### Compound Reduction

Some designs may require more reduction than is practical in a single stage. The ratio from the smallest sprocket available to the largest is 64:16, so if a greater reduction then 4x is required, multiple reduction stages can be used in the same mechanism which is called a compound gear reduction. There are multiple gear or sprocket pairs in a compound reduction with each pair linked by a shared axle. When using sprockets and chain in a multi stage reduction, it’s very common to use gears for the first stage and then use sprockets and chain for the last stage. The figure below is an example of a two-stage reduction using all gears, but one of the pairs could be replaced with sprockets and chain. The driving gear (input) of each pair is highlighted in orange.
Reduction is calculated the same for gears and sprockets based on the ratio of the number of teeth. To calculate the total reduction of a compound reduction, identify the reduction of each stage and then multiply each reduction together.
$\text{CR}=\text{R}_1×\text{R}_2 ×\text{…} ×\text{R}_n$
Where:
• CR is the total Compound Reduction
• Rn is the total reduction of each stage
Using the image above as an example, the compound reduction is 12:1.
$\text{CR}=\text{R}_1× \text{R}_2 =\frac{60}{30}× \frac{90}{15}=2 ×6=12$
For any gear system, there are a limited number of gear and sprocket sizes available, so in addition to being able to create greater reductions using compound reductions, it is also possible to create a wider range of reduction values or the same reduction of a single stage, but with smaller diameter motion components.
Each additional compound stage will result in a decrease in efficiency of the system.

## Spacing and Center to Center Distances

Chain Loops can be used with ION Sprockets and structure featuring the MAX Pattern. Any 1:1 ratio will have the correct center-to-center distance for a properly tensioned chain, without the need for tensioning bushings. To calculate how many links you will need, multiply the center-to-center distance by eight, and add the number of teeth on one sprocket.
• Links of #25 chain = (Center-to-center Distance x 8) + Teeth in one sprocket
If a ratio other than 1:1 is needed when using the REV ION Build System, use our Ratio Plates to accommodate for the change in center-to-center distance. An ION Ratio Plate provides an offset from the standard MAX Pattern pitch that creates the center-to-center distance.

### Spacing

In order for sprockets to work effectively, it’s important that the center-to-center distance is correctly adjusted. The sprocket and chain example with the red "X" in the image below may work under very light loads, but they will certainly not work and will skip under any significant loading. The sprockets in this example are too close together, so the chain is loose enough that it can skip on the sprocket teeth. The sprockets with the green check mark are correctly spaced, which will provide smooth and reliable operation.