What are the effects of the tilted lift vector in a banked plane?
In a banked turn under cetripetal force, what causes the change in the heading of the plane such that its longitudinal axis remains facing the relative wind?


The simplest case of a flat wing will create lift orthogonal to its direction of motion and the direction of wingspan, and some drag parallel to its direction of motion. Both lift and drag will act in the plane of symmetry. Let's assume that drag is compensated by a constant thrust, so the level wing is in equilibrium. A tilt will contribute a sideways lift component that is not compensated by weight and will accelerate the wing sideways.

This will move the wing into the direction of lift and reduce its angle of attack. Lift will drop which reduces sideways acceleration. Unfortunately, this also means that gravity will not be fully compensated. The wing will be accelerated downwards which increases its angle of attack. Both effects will settle at a point where the wing moves side- and downwards, the sink speed depending on the drag increase since more lift in total must be generated.

But the picture is incomplete without the contribution of the tail surfaces. The sideways motion will cause a side force on the vertical tail which will weathervane the aircraft into the direction of sideslip. Only that lateral stability contribution will start the turning motion. Now you get a continuous process in which the wing accelerates the whole airplane sideways and the tail will yaw it into the wind, resulting in a not well coordinated turn. The yawing motion will add a centripetal acceleration which will eventually balance the sideways acceleration from the tilted lift force.

Since we decided to leave thrust constant, the added lift requirement will cause more drag and the aircraft will sink in order to compensate for the energy loss from turning. If we now add the freedom to change the elevator deflection, we can trim the aircraft for the higher lift and a lower speed so that (depending on the position of the initial state on the power curve) the aircraft could fly a level turn at a lower flight speed.

I also neglected pitch stability so far. At higher bank angles it becomes important and contributes to the turn in proportion to the sine of the roll angle. With a constant elevator deflection, pitch stability will try to keep speed constant and pitch damping will reduce the turn rate. Once we allow to adjust the elevator, pulling will reduce pitch damping and trim a higher angle of attack, so the turning rate will pick up.