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From what I understand, pogo oscillation was already a known problem for Saturn V rocket's first stage (and likely other stages, not sure) long before, say, Apollo 11, especially on its central out of 5 main F-1 engines due to, I guess, not sturdy enough support structure (cruciform) and the central engine's force moving it upwards, shortening the fuel line during high thrust and vice-versa when the engine was off.

It might have been a known problem even before the first Saturn V flight. They (von Braun et al.) were dealing with combustion instabilities that they were trying to solve (or at least establish where the problem is) by adding a small explosive charge to the engines assembly and detonating it during hot fire testing of the first stage engines. That didn't go too well, and this pogo oscillation problem persisted at least also for the first Saturn V flight of Apollo 4.

My question is, did they make structural adjustments to the Saturn V main stage cruciform during the Apollo program, or somehow otherwise dampen the pogo oscillation and sloshing problems, or perhaps make adjustments to its ascent (thrust) profile, or they just flew them as is (which would be rather risky, wouldn't it - the term rapid unplanned disassembly comes to mind)?

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Apparently the issue could not be addressed in time for the Apollo 13 mission as is evidenced by this note in the NASA archives available online.

During the Apollo 13 flight the pogo effect had reappeared, this time on the second stage. Severe oscillations had forced an early shutdown (two minutes ahead of schedule) of the inboard engine.

The NASA archives write to say about the Apollo 14 pogo oscillation fix

Marshall and North American engineers devised three changes to the second stage. They installed a helium gas accumulator in the LOX line of the center engine. This reservoir served to dampen fluid pressure oscillations, keeping them out of phase with the vibrations of the thrust structure and engines. North American added a cutoff device to shut down the center engine in case the accumulator failed to control the oscillations. Finally, simplified propellant valves were installed on all five J-2 engines. The valves controlled the propellant mixture to the engines, providing a rich mixture for high thrust during the early portion of the burn and a leaner mixture later.

More reading in this context are posted on this web-page. The author there writes to say

Second, there was no one single mission before which pogo was a serious problem, and after which it wasn't. Pogo occurred in the Saturn V all the way up to Apollo 17. But during the program engineers took a series of steps to make pogo less of a problem for each successive mission. But there was often a delay because engineers needed time to analyze the results from a Saturn V flight, and in most cases the next Saturn V was already "stacked" in the assembly building and couldn't be easily modified.

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