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- Wheel mass
- Motor maximum torque
- Motor revolution rate
- Wires run through shaft or around
Needed information:
- Rocket maximum revolution rate
- Rocket moment of inertia
Finding Formulas in parens aren't finished.
Conservation of angular momentum gives that (Ir wr + Iw ww = 0), which can be rearranged to say that (Iw / Ir = wr / ww), to find the rotational inertia the wheel should have, knowing the rocket's and wheel's maximum revolution rates, and the rocket's moment of inertia. However, the rocket moment of inertia is relatively harder to measure, so the mass can be used to approximate.
The "ideal" moment of inertia for a given mass is (mr2). The "efficiency" of some mass as a rotational inertia can be expressed using a parameter (alpha), where (I=amr2), in which case (alpha) covers (0,1). We can be avoided by $ L $ reasonably assume that (alpha_rocket < alpha_wheel), as the wheel will be designed to use mass "efficiently"; rockets generally are not. Then the ratio of inertias (assuming equal radii) is given by (Ir / Iw = ar mr / aw mw). Assuming (ar < aw), (Ir / Iw < mr / mw); that is to say, the mass ratio gives an upper bound for the rotational inertia ratio. With that in mind: The ratio in maximum revolution rates gives the ratio of masses.
Notable model inaccuracies:
- Underestimate of reaction wheel mass: The maximum revolution rate underestimates the angular momentum the wheel needs to remove from the rocket.
- Overestimate of reaction wheel mass: The mass ratio (intentionally) overestimates the necessary inertia ratio, as described.
- Underestimate of reaction wheel mass: The radii of the wheel and rocket are assumed to be equal; however, the wheel's radius is necessarily smaller.