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if I have a statically stable aircraft, it will always return to its neutral position. What happens now if I do not trim my aircraft? Will it move/osicallate around its neutral position and be statically stable but dynamically neutral?

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Trimming or not trimming your aircraft should not change its inherent stability. It will, however, change the airspeed, vertical speed, and attitude of the aircraft based on power settings. Control input would have to be applied in order to maintain these parameters. The more out of trim the aircraft is, the more force/pressure will need to be applied to the controls.

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An aircraft is always trimmed for some airspeed. Manual control input allows you to force a different airspeed from what the plane is trimmed for, but if you let go of the controls, the plane will return to its trimmed speed. Depending on power setting, that will result in either a climb (to slow down) or descent (to speed up). This is why you can climb or descend by adjusting power without ever touching the yoke/stick.

(I assume that by “trim” you’re talking about elevator trim; rudder or aileron trim would have corresponding effects in their own axes.)

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