The broomstick analogy is a fallacy because your hand always applies "thrust" upwards, not often along the broomstick axis. This means the as the broomstick tips and its center of mass departs the footprint of your hand, the rotational component of the force you are applying increases; the solution diverges rapidly and you lose control. This is the classical balancing problem. However, rockets do not operate this way. The thrust is always along the axis of the rocket. Any departure from the ideal situation in which the thrust vector points through the center of mass is preserved through the rotation. The controlling moments do not diverge, and this makes all the difference in the world. This is because the engine (and its thrust vector) rotate with the craft, which is not what happens in the balancing problem. While it remains true that the error moment increases in gravity as the tilt increases, this is not solved by moving the center of thrust to a point above the center of mass. Why? because the thrust vector is still fixed, the corrective moments are still fixed, and the dynamics remain literally unchanged.
Putting the center of drag behind the center of mass in a rocket flying through air achieves passive aerodynamic stability, but this is not meant to correct somehow for an inherently unstable thrust dynamic.
I was aware of this, and realize that the rotating direction of thrust to always point through the COM is BETTER than the traditional broomstick model (with force always upwards). If you draw out the static diagram of forces acting upon this rigid body, gravity pulls down "uniformly" on all mass, so can be modeled as pulling down on the COM, correct? While the thrust, which is "behind the COM" (or below when upright), is NOT providing a counter balance to gravity when LM is pitched at an angle. Therefore the RCS + Engine thrust must work together to maintain balance (attitude) along with maintaining a high-fidelity targeted trajectory to landing.
During Landing with LM pitch over, Gravity is ACTIVELY pulling on the LM. In this case, it MATTERS VERY MUCH the relative location of the Center-of-Thrust. If Above the LM COM, then this thrust would REDUCE the amount of RCS force required to maintain attitude/pitch. If "Exactly AT the COM", then RCS itself could be mostly passive.. But if BELOW the COM, then the Gravity becomes a DESTABILIZING FORCE, against which the RCS must constantly counter-act!
In the case of the LLTV, the Jet Engine was much closer to the COM... such that this instability from Pitching did NOT occur, making it easier to fly.
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This would have been beyond-bleeding-edge-tech for the 1960's... not coming to fruition until decades later.
But Apollo skipped this POC entirely. Real-world performance isn't the same as "theory on paper"... and requires a lot of rework and adjustments -- especially for bleeding-edge tech.... never done before, and not to be done again for decades.
