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• Sajiv Shah

# Spacecraft Reaction Wheel Physics

Updated: Sep 12, 2020

A perfect example of how mechanisms utilize the conservation of angular momentum is a reaction wheel system. SOC-i, a nano-satellite team I'm interning for, recently assembled their reaction wheel system, which is shown above.

If you have 0 clue what a reaction wheel is, then you've come to the right place to learn the basics of how fast spinning wheels are used on tons of spacecraft for controlling orientation.

Many spacecraft have the requirement that they must be able to change the direction that they are facing. Any satellite that needs to orient cameras or sensors in certain directions have to evaluate what the best way to do that may be. For large spacecraft, like crew modules that hold astronauts or rockets used for launch, the best option is usually some sort of thruster that uses a fuel or gas to create some torque or rotation. However, for micro and nano satellites, holding enough onboard fuel to control maneuvers for periods longer than 5-6 months is not practical. The weight and amount of components involved in a gas thruster for spacecraft simply does not justify its use on a small satellite.

As a result, reaction wheels have become very popular. Famous spacecraft, such as the Hubble Space Telescope, utilize reaction wheels for control.

The physical build of a reaction wheel is usually some sort of motor attached to a metal disk that has a high moment of inertia. Here are some of the equations for moments of inertia of circular objects:

The assembly of the motor and disk is then placed inside a housing so the fast spinning disk is enclosed and harmless. That housing is attached directly to the satellite.

To understand how reaction wheels cause an object to rotate, it is very important to grasp a basic understanding of the requirements of reaction wheels. The end goal is to be able to rotate the reaction wheel so it's heading can be in any direction. In order to do that, the minimum number of axis that we need to be able to rotate the spacecraft around is 3. Those would simply be the traditional coordinate x, y, and z axis that are commonly used.

It's also to lay out some important physics concepts: "L" is used to represent angular momentum, which is the product of the angular velocity (omega) and rotational moment of inertia (I). The law of conservation of momentum states that whatever momentum is in a system will be conserved throughout. So if a rocket and the fuel inside it have 0 momentum to begin, the momentum of their system will also be 0 as the rocket propels upward and the fuel exits downward. This law can be derived from the fact that every action has an equal and opposite reaction, which is Newton's third law of motion. The math below explains how using Newton's third law can give the equation for the conservation of momentum.

This equation gives a simple explanation of how a reaction wheel spinning in one direction will cause a spacecraft to rotate in the other direction a certain amount. That said, having one of these on each axis allows for control and rotation about each axis. However, because of a lack of redundancy, many systems use four reaction wheels. An example of this is in the image above, where the SOC-i nano satellite is using four reaction wheels pointed in different axis to control their satellite.

However, there are some issues with reaction wheels. One of them is saturation, an issue where the reaction wheel reaches its max speed but is not able to rotate the satellite to a desired velocity. I'll be exploring its issues and possible solutions in the future.

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