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I am jist trying to understand the thresholds involved in pogo oscillation. I'm not sure where to start. But I will give a guess.

As rocket size increases the tension exterted on rocket walls increases due to pressure diameter relation. If we assume 1000 tons of fuel pressing against a 1m2 tank control valve that's 1000 ton per sq m. If the rocket doubles in size, then its fuel doubles and tank surface area increases as the 2/3rds power due to the pi d relation between pi and diameter. So we get an increase to the 1/3 power in area per force. Furthermore the rocket is accelerating at roughly g so the pressure is doubled.

So we have 20 mpa of force on this valve. Assume that there is a 1% error in the fuel flow and the resulting shockwave moves at 25 meters per second up the 25m tank. This causes a 1% amplification every second. This is a double exponential function due to the flow error affecting thrust.

After 100 seconds the total pressure has increased 4x growing at 2% per second. After 200 seconds the pressure has increased 32x growing at 4% for seconds. This is now a 640 mpa force on steel with a 200mpa strength.

So it looks like something the size of apollo is barely tractable in engineering terms.

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    $\begingroup$ "If we assume 1000 tons of fuel pressing against a 1m2 tank control valve that's 1000 ton per sq m." - only if the valve has the same size as the tank. Usually, the valve is much smaller than the tank diameter. Pressure is only the weight of the fuel column right on top of the valve. The rest of the fuel presses against the aft tank dome, not the valve. $\endgroup$ – Hobbes Sep 30 '18 at 11:08

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