What are the effects of the tilted lift vector in a banked plane?

Question

I want to clarify the effects of the change in direction of the lift vector caused by banking in a plane.

Banking rotates the lift vector creating a horizontal component. I have read that some effects of this horizontal component are as follows: 1) It acts as a centripetal forcing inducing circular motion in the plane. 2) It generates sideslip such that in dihedral planes there is an increased angle of attack in the lowered wing, and a decreased angle of attack in the raised wing. This generates a restoring moment facilitating roll stability.

So my questions are as follows: How does the horizontal component of the lift have both a centripetal effect and a slide slip effect? (A more general question) 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?

Perhaps I have somewhat answered the first question with the second: initially under the centripetal force the plane does not yaw to have its longitudinal axis aligned with the new air velocity vector and therefore sideslip occurs? I am not sure, please let me know.

Thanks

Answer 1

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.

Answer 2

In straight flight, the lift is not causing any rotation of the aircraft. When you start a bank (using ailerons), the centre of lift does not move, only the direction changes, so the plane does not yet start turning, it starts to side-slip (effect 2).

Now the side-slip has two effects:

• It changes lift differently on the windward and leeward wing. The effect is rather complex depending on wing position and dihedral, but generally aircraft are designed so the result is slightly restoring.
• It creates a side force on the vertical stabiliser that will turn the aircraft towards the relative wind (weathervaning).

And only now when the aircraft started turning, the lateral component of lift will provide the centripetal force to sustain the turn (effect 1).

Of course in practice it does not occur in distinct steps. As you start rolling, the side-slip will start developing, but it will be immediately causing yaw, so not much side-slip develops.

Or the pilot (or yaw damper) will do their job properly and work the rudder to initiate the turn together with banking, avoiding the side-slip altogether.