i always thought about that watching the launches, with the giant candles slowly crawling up off the pad, being pushed by those flames BELOW their center of mass. I think from the days of Tsiolkovsky that was on all the rocket bilders' minds. And I think the solution was from the earliest days to automate the recentering around the plumb-line with the help of a gyroscopic or even quicksilver "autopilot". The Soviet system, if you think about it, was a bit more stable inherently, by giving a wider footprint to the propelling force.
From what I've heard of them the cluster was barely able to provide reliable running at all, let alone precise levels of force from specific engines. It would be interesting to find out whether the flow control hardware for balancing was responsible for the other problems.
The K2 had a system where if an engine failed, the opposite engine would be shut down, thus eliminating the uneven thrust vectoring problem. If you look at the Dragon 9 rocket by SpaceX today, you'll see a similar approach.
They were not static, no. The center of mass moved upwards as the fuel was used up, not downward, because the upper stages were still full while the bottom was emptied. But the center was always in the centerline of the vehicle. So while the vehicle became more and more sensitive to thrust vectoring as it ascended, it didn't need lots of correction to begin with (unlike the shuttle) since all the mass was inline with the engines. And the Saturn V was a fly by wire system, using gyroscopes and accelerometers to determine speed and orientation. This in the day before anyone even dreamed of a pocket calculator.
As for your question about mass, the lion's share of all rocket launches is the fuel. Easily 85% of the weight of any rocket is the fuel within it.
None of the liquid rockets had positive stability at all times. In addition to issues of changing center of mass during fuel burn, all liquid rockets have to function with the possibility of asymmetrical engine thrust and or shutdowns, so they all have active stability control systems. That said designers frequently chose to minimize the amount of work such a system would have to perform, thus the Saturn rockets had fins that made the design less unstable, this was I understand so that if one performance the analysis of everything fails, how quickly does the abort system have to react? The worse case scenario could be handled and the capsule ejected safety.
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Thinking about the Soviet rockets, sure, the force footprint was wider, but was it in any way self-correcting?
On the other hand, maybe having the ass end of the rocket wide like that dragged the center of pressure back some.
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As for your question about mass, the lion's share of all rocket launches is the fuel. Easily 85% of the weight of any rocket is the fuel within it.
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