And once again, please show us how your evidence supports your claim that steering dynamics of the LM in solo flight were significantly altered by the addition of plume defelctors, or at least altered such that instability was a real concern.
Again, here is the equation you brought to the table:
M+X = 89D1 - 59D2
The instability that leads to a positive feedback loop is only present when M+X is negative, as stated in the memo. You agree with that, yes?
I don't think he does. Which is to say I don't get the impression he thinks one can compute stability, or that it makes sense to do so. I don't think he considers free-body dynamics to be a computable regime. This is something I run into all the time with hoax claimants trying to bluff their way through specialized knowledge. They usually don't know what's possible. Their extrapolation from intuition usually goes down the wrong path. Here it seems Jr Knowing has only a pidgin, cargo-cult understanding of stability. "Stability" is the binary condition that arises from pairs of jets placed exactly in opposition, acting exactly through the center of gravity (which can never vary). "Instability" is the binary condition that happens when any of these "constraints" is violated.
And in his version, one can never know whether one has achieved stability except by demonstration flight. The problem is solvable practically, but not prescriptively. This is why we keep pushing the math under his nose and he just keeps trying to figure out how it applies. It hasn't sunk in that there is a formalized model for this, and that math solves the problem in the abstract, not just individual cases. He hasn't figured out that engineers can
know there will be no stability problem (or, as in the memo, that a certain curious condition will arise in remote circumstances) by working it out on paper. He doesn't see how math solves what he thinks is a purely practical problem. The plume deflectors weren't flight-tested, so in his limited pseudo-engineering world they were untried.
We generalize the problem of free-body dynamics for most practical purposes using what's called a linearized state-space model. It's "linearized" in the sense that all the familiar Newtonian elements of the problem are represented as entities in linear algebra -- matrices and vectors. More accurately, many elements of the problem are matrix- or vector-valued functions of some other variable such as time. It's a "state space" in that it's a vector space of all possible inputs, outputs, and states (and their derivatives) that a system can be in, as represented in linear algebra terms. Ironically, the state-space class of mathematical solutions is also used in econometrics.
The beauty of such a system is that all possible effects are correctly modeled using a homogeneous (and small) vocabulary. You can abstract concepts like body axes and control axes -- and in the LM's case the control axes don't even have to be orthogonal (at right angles). Everything boils down to multiplying vectors and matrices. That's what linear algebra is for. A layman is probably not going to stumble onto this by himself. He was either taught it and thereby understands its power and simplicity, or else his concept of the quantitative nature of the problem is likely to be a bewildering melange of special-case formulations that would quickly become intractable for such a problem as controlling a spacecraft. In this system, the center of mass not being at the center of the control axes isn't a problem, because it's never assumed to be. Transforming between body axes and control axes is straightforward and never omitted. There are no special cases to consider. And the transformation can even be a time-parameterized function (or a function of some other variable such as fuel-on-board) with no loss of elegance.
With these techniques, the additional effect of plume impingement on the deflector simply becomes another vector in the problem, no different than the direct effect of the jet itself. It has discoverable, deducible physical properties, and these properties can be modeled easily in the language of linearized state spaces. The equation above is merely a matrix multiplication rendered out in its scalar decomposition. The fact that it also works out to be the definition of toque (a quantity of force acting a distance from the center of mass) is intentional. Torque is not some contrived concept. It's a mathematical expression of how we observe the universe to work. The algebraic equivalence between the basic expression of the concept and the model we use validates the model.
Apparently unaware of this, Jr not-Knowing figures that the engineers who came up with the plume deflector had no way to determine its effect on the control problem before flight. And in his world of perfectly-balanced jets and perfectly-located centers of mass and idealized structure, any disruption is disastrous. And if we can't see this, then we're just not at his level of understanding. (Well, that's true. But not in the way he wants.) The central theorem of state-space dynamics is not that a system rests at equilibrium or returns to it unaided, but rather than a system can be
driven to a desired state deterministically. The whole science of control theory would be obviated entirely if everything worked the way Jr Knowing imagined it does.
And that same misconception is behind the bravado with which he insinuates that we can't know that we're right and that he's wrong. Yes! Yes, we can! The same math by which the engineers originally determined the effect of the plume deflectors and predicted its feedback loop in extraordinary circumstances works just as well for us in determining that no possible location of the LM's center of mass in solo flight reverses the relevant moment. No, we aren't just gullible or brainwashed. I know what I know. I know
why I know it. I know that it works because I see it work. It's not just a thing I read, or a thing someone told me. And I'm not alone. These are common techniques, widely known and broadly applicable. Jr's ignorance of them doesn't make them invalid, doesn't make them go away, and doesn't make him the insightful genius he hopes to play.
