# Net Angular Momentum of Satellite with Reaction Wheel

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1. Aug 9, 2015

### swagbowl

I am modelling the attitude dynamics of a satellite. The satellite has a reaction wheel in 1 plane to help control the attitude. There is significant debate about the equation for the net angular momentum of the satellite and what inertia tensors should be used regarding parallel axis theorems and relative velocities.

For simplicity consider the satellite in only the plane that contains the reaction wheel (e.g. the plane in which the wheel rotates). The system is considered to be the combination of the satellite (excluding the reaction wheel) and the reaction wheel.

The net angular momentum of the system is H.

The inertia tensor of the satellite (excluding the reaction wheel) about its principle axis is JSat.
The mass of the satellite is mSat.
The displacement of the satellite's COM to the system's COM is dSat.
The angular velocity of the satellite in inertial space is ωSat.

The inertia tensor of the reaction wheel about its principle axis is JRW.
The mass of the reaction wheel is mRW.
The displacement of the reaction wheel's COM to the system's COM is dRW.
The angular velocity of the wheel with respect to the satellite is ωRW//Sat.

Some believe that the net angular momentum of the system is:
H = [(JSat + mSatdSat2Sat] + [(JRW + mRWdRW2)(ωSatRW/Sat)]

While others believe that the net angular momentum of the system is:
H = [(JSat+mSatdSat2Sat] + [JRWωRW/Sat+mRWdRW2ωSat]

Can anyone shed some light on the correct answer? A first principles derivation or supporting source would be beneficial to my case of proving the correct answer. As my head will be on the chopping block if its wrong I would like some piece of mind...

2. Aug 9, 2015

### swagbowl

Sorry, reaction wheel should really be replace by momentum wheel as it will have an angular velocity during flight