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I understand that most turboshaft engines tend to run at (or close to) their most efficient rpm all of the time, and it is the load that is changed e.g. the pitch of the blade pitch on helicopters.

If everything else is kept the same and the load is increased, the rpm would obviously drop. Therefore, I assume that to maintain the constant rpm, the fuel flow must be increased as load is increased.

How exactly does the engine set the correct fuel flow to keep the engine at its most efficient rpm? I guess that on modern engines, this is probably done by a computer system, but is there also a mechanical way to do this, as would have been used in older engines?

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    $\begingroup$ Called a governor... $\endgroup$
    – Solar Mike
    Apr 28 '19 at 11:45
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In old steam machines a centrifugal governor was used: https://en.wikipedia.org/wiki/Centrifugal_governor

Later they used other governors with cables and mechanical devices like that.

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Setting aside newer turboshaft engines that because of advances in technology do use variable rpm, they control the rotational speed is governed by a Full Authority Digital Engine Control (FADEC).

the quote below is from Gianluigi Alberto Misté and Ernesto Benini

fuel control system which, by adjusting the amount of fuel injected, ensures that the rotor speed is as constant as possible for each type of flight maneuver. The main reason for choosing a constant rotational speed is linked to the supposed decrease in engine efficiency at part load (typical variations in speed do not exceed 15% 1), as well as because variable speed drives trains of resonant frequencies into the airframe

This in conjunction with the pilot control on adjusting the rotors angle of attack and speed ensure optimal operation..

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