Overview

Reaction Wheels (Torque Rods) are ways of controlling a vehicle in a microgravity environment by storing angular momentum in spinning disks and discharging it as necessary to apply control forces.

Theory of Operation

The relevant starting equations are:

$//<![CDATA[ \begin{array}{l}\tau = I\alpha\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}H = I \Omega\end{array} //]]>$ 

Where  $//<![CDATA[ \begin{array}{l}\tau\end{array} //]]>$is the torque on a vehicle,  $//<![CDATA[ \begin{array}{l}H\end{array} //]]>$ is the angular momentum of the vehicle, and  $//<![CDATA[ \begin{array}{l}I\end{array} //]]>$is the inertia tensor of the vehicle.

From conservation of angular momentum we know that  $//<![CDATA[ \begin{array}{l}H\end{array} //]]>$ must be constant. Supposing we define the system to include the vehicle, and a reaction wheel, then we could find that we can change the satellite's angular velocity  $//<![CDATA[ \begin{array}{l}\omega\end{array} //]]>$arbitrarily as long as  $//<![CDATA[ \begin{array}{l}I_{RW} \times \omega_{RW} = I_{vehicle} \times \omega_{vehicle}\end{array} //]]>$. Practically, since motors only operate around one axis, this equations can become scalar equations for a single principal axis of the vehicle.

If a perturbation acts on the vehicle, imparting angular velocity  $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$, you can impart  $//<![CDATA[ \begin{array}{l}H_{disturbance}\end{array} //]]>$ to the reaction wheels and nullify the disturbance velocity.

Useful Literature

https://ocw-mit-edu.ezproxyberklee.flo.org/courses/aeronautics-and-astronautics/16-83x-space-systems-engineering-spring-2002-spring-2003/projects/design_final_e.pdf

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19940008667.pdf

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