Would pressure oscillations at the inlet of a liquid engine turbo-pump subside at the outlet, or would they get amplified?
What are the factors that decides the change in the amplitude of pressure oscillation at the outlet?
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They can get amplified. That is a contributor to the classic "pogo" resonance phenomenon.
Another rocket system instability, of very low frequency (order of a few hertz), is caused by propellant flow rate oscillations which result from pump amplification of the fluctuations of the pump inlet pressure (the pump inlet pressure variations are due to the g-loading of the liquid column extending back to the tank).
The important factors are listed here.
PUMPS: The proper treatment of pumps requires more than just the simulation of the head-flow relationships. The capacitance and inertance can be equally important, particularly in the high-speed pumps used in rocket engine systems.