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I’m trying to figure out the angle of a bank caused by a shift in the centre of gravity of an aircraft. How should I do this?

What avenues should I go down to calculate this? I know that the plane will bank until c.g. is below centre of lift.

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  • $\begingroup$ this also depends if the aircraft has dihedral stability as well. I would read up on what makes an aircraft stable in flight to better understand how to "destabilize" it when you wish to change direction. $\endgroup$
    – Robert DiGiovanni
    Commented Mar 4, 2019 at 20:06
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    $\begingroup$ you may wish to study hang gliders. Sorry for the tangle here. Shifting weight to the side will cause a rolling force. How far it progresses depends on roll stability of aircraft, and, of course, control inputs. This will be a unique property of each type of aircraft you study. Your initial thought is correct. You can learn more by building free flight models and adjusting the weight not only fore and aft, but also side to side (I use pennies). Good luck with this. $\endgroup$
    – Robert DiGiovanni
    Commented Mar 5, 2019 at 8:49
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    $\begingroup$ A sideways shift in the center of gravity will give you a rolling moment but not a stabilized bank angle. The rolling moment won't magically stop once a certain bank angle has been reached. What could stabilize the plane is a combination of speed, bank and sideslip, but not all designs will be stable at such an equilibrium. $\endgroup$
    – Peter Kämpf
    Commented Mar 6, 2019 at 16:06
  • $\begingroup$ @ RobertDiGiovanni and original poster-- Riffing on Robert's comment-- in most modern hang gliders, at LOW angle-of-attack (pilot's CG well forward, airspeed initially high), shifting pilot's weight say 12 inches to one side and keeping it there will result in a rather modest bank angle-- a wild guess would be maybe 15-20 degrees-- while at HIGH angle-of-attack (pilot's CG well forward, airspeed initially low), the same input will cause the glider to "wind up" into a rather steep spiral, eventually reaching a bank angle of over 60 degrees, in which case the airspeed will no longer be "low". $\endgroup$
    – quiet flyer
    Commented Apr 5, 2021 at 18:48
  • $\begingroup$ So-- "it depends". In conventionally-shaped model airplanes with lots of dihedral, I've been surprised how little visible change in the "trimmed" bank angle is obtained by a rather significant weight on one wingtip. In the hang glider case, we could probably go on for several pages about all the different factors playing a role here, including the way that the dihedral-like effect created by wing sweep is strongly dependent on angle-of-attack, and the way that the way that washout creates a roll torque toward wings-level is also strongly dependent on aoa, but in the opposite direction. $\endgroup$
    – quiet flyer
    Commented Apr 5, 2021 at 18:49

2 Answers 2

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You can't. Aircraft are not stable in roll!

Once there is an offset between the centre of gravity and the lift action line, there will be a rolling moment until the aircraft is banked at 90°, at which points the forces become perpendicular, so they can't generate any moment.

The moment will cause roll acceleration, but as the wing going down will have higher angle of attack, its lift will increase, causing opposing moment that will stabilize the aircraft at some roll rate. Which will slowly decrease as the moment between lift and gravity decreases with the increasing bank. But it will not vanish until 90° bank, so the aircraft keep rolling and end in a graveyard spiral.

If there is some roll-yaw and yaw-roll coupling, and the imbalance is small enough, the side-slip created by the incipient turn will generate enough opposing roll moment to return the plane to a dutch roll instead. But that is an oscillating motion, so you still won't have stable bank angle.

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    $\begingroup$ Some aircraft are not stable in roll, but most are. $\endgroup$
    – Michael Hall
    Commented Mar 4, 2019 at 21:33
  • $\begingroup$ @MichaelHall, it is not possible to have stability in pure roll. Aircraft do have roll-yaw and yaw-roll coupling that makes small excitations in roll produce dutch roll instead of runway bank, but it usually only works at low bank angles, because otherwise the dutch roll would be rather uncomfortable. $\endgroup$
    – Jan Hudec
    Commented Mar 4, 2019 at 21:38
  • $\begingroup$ I'm not saying that there isn't some small amount of dutch roll as the aircraft recovers from an upset, but to imply negative stability in roll as the norm is misleading. $\endgroup$
    – Michael Hall
    Commented Mar 4, 2019 at 21:42
  • $\begingroup$ @MichaelHall, the dutch roll is the only mechanism that provides any kind of positive stability. Above certain bank angle, the yaw-roll coupling provides a negative stability instead, but that is also yaw-roll coupling. In coordinated flight, roll stability is always neutral. $\endgroup$
    – Jan Hudec
    Commented Mar 4, 2019 at 22:03
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    $\begingroup$ @MichaelHall the Boeing 747-400. I don't think I was very clear about the maximum bank angle question. Sorry. I meant to ask how can you calculate the threshold bank angle for the aircraft rolling back to it's position. $\endgroup$
    – jim
    Commented Mar 5, 2019 at 2:16
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While it is true that a lateral displacement in the QG will produce some small rolling moment about the longitudinal axis, the effects are minimal and the built in roll stability and trim capability of most aircraft will adequately compensate.

While it is certainly possible to calculate this moment, it is generally a non-issue. Due to the longer moment arm, wing fuel imbalances have a greater effect than any weight shift within the fuselage. No weight and balance calculations are performed to determine where the lateral CG lies because in conventional aircraft it will never be out of limits in the way that a fore and aft CG might be.

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  • $\begingroup$ If shifted the centre of mass was caused by the fuel imbalance would there be a sufficient enough moment arm to calculate the bank? $\endgroup$
    – jim
    Commented Mar 5, 2019 at 0:34
  • $\begingroup$ @Jim, I'm still not sure what you are really trying to accomplish with this question. Yes, at some point an imbalance could easily exceed the ability to trim out the force, and in an extreme case could exceed the pilot's ability to counter with opposite aileron. However, it will never just roll to some calculable angle of bank and stop there. If it rolls on its own it is by definition unstable, and will continue to roll unless countered. Research positive, neutral, and negative stability and see if that clarifies things. $\endgroup$
    – Michael Hall
    Commented Mar 5, 2019 at 0:45
  • $\begingroup$ The weight imbalance is controlled aerodynamicly, so you would figure the (vertical) lift difference of left and right wing based on dihedral angle. Notice even if it were enough, the plane tilted would start to turn. If the plane had no dihedral and/or CG below the wing, it would simply roll into a spiral if no corrections were made. As you overcome the dihedral stability, and then ailerons and rudder with greater imbalance, the plane will crash. This is also true with fore and aft balance, and why it is so important to have CG in limits. Aerodynamic control diminishes at lower speed. $\endgroup$
    – Robert DiGiovanni
    Commented Mar 5, 2019 at 1:36
  • $\begingroup$ @MichaelHall The stability did clarify things. I had the misconception of the fact that the aircraft would roll until the centre of lift was above the centre of mass and stay at that bank. $\endgroup$
    – jim
    Commented Mar 5, 2019 at 2:19

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