Is the turning motion of a banked airplane caused by true centripetal force?

Question

Virtually every explanation I have read of the aerodynamics of a turning airplane ascribe the turning motion simply to centripetal force, without further comment. I'm hoping someone can check my understandings and conclusions whether this is true or not.

As I understand it, centripetal force is defined as the net of the forces on an object that keep it on a circular path. Put another way, it's the resultant of the force vectors on the object. Further, centripetal force always acts in a direction orthogonal to the object's direction of motion.

Now, with a banked airplane, the horizontal component of lift, in and of itself, will simply create a lateral force that introduces sideslip. The heading of the airplane must be continuously changed in order to change the direction of that lateral force, and thereby change the force vector from a lateral one to a radial (centripetal) one. This is done independently with the addition of yaw, either coordinated via rudder input, or uncoordinated via weathervaning.

Assuming all this is correct (again please check me), is it mathematically correct to state that true centripetal force is created by the resultant of the force vectors here (the lateral forces adding sideslip and the rotational forces adding yaw)?

Thank you very much for your help.

(EDIT: Through the answers below, I understand now that my third paragraph above is incorrect. The horizontal lift vector remains perpendicular to the direction of movement (relative wind), not the airplane's heading. Therefore, while yaw is necessary to maintain coordinated flight, the rotational force vectors behind it are not addends in the centripetal force calculation.)

Answer 1

Centripetal force is defined as the component of the total force acting on the object that is causing it to follow a circular path. It is the force exactly perpendicular to the velocity vector.

To create a circular path, the force needs to turn with the velocity. If you have a force that keeps its spatial orientation, less of it will be centripetal and more of it will just accelerate the object as it moves and the resulting path will be parabolic (e.g. the ballistic curve under gravity).

The horizontal component of lift always contributes wholly to the centripetal force, but this is again quirk of the definitions: lift is the component of the aerodynamic force on the wing that is perpendicular to the relative wind and drag is the component that is parallel to it. So analysing the situation in the reference frame of the air mass, relative wind is the velocity vector.

Now if the plane didn't turn with its velocity vector, it would just gain side-slip, the lift would decrease—banking into side-slip reduces the angle of attack—and the aircraft would stop turning—it's flight path does turn initially, even if the body does not—and settle in sideways flight. So we need to make the aircraft rotate with the turn.

Now just like linear motion, rotating motion stays constant¹ unless acted upon by a moment of force (a.k.a torque). So when initiating the turn, a yawing moment of force needs to be created to also initiate the rotation.

A moment of force may be created by a pair of force of opposite direction and different action lines, so it does not have to cause any linear acceleration, but in case of aircraft usually an initially unbalanced force is created on the tail that starts the rotation, but also accelerates the aircraft sideways until balanced by drag on the fuselage. Note that this force is out of the turn (and up), so it reduces the centripetal force a bit.

But in a steady turn the torque is zero. Or oscillating around zero as the correct angular velocity is maintained via feedback loop. So the horizontal component of lift is the only centripetal force in a coordinated turn. In not coordinated turn, some side force on the fuselage is added or subtracted depending on whether it is a slip or skid.

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¹ Angular momentum is conserved but the angular velocity does not have to be. Depending on the axis of rotation and mass distribution (moment of inertia) you may get strange phenomena like the Dzhanibekov effect.

Answer 2

If the airplane banks without a change in pitch, then it will accelerate sideways and build up a sideslip angle while at the same time losing altitude because the cosine of the lift vector will be too small to counterbalance all weight. This downwards motion will increase the angle of attack a bit so the airplane settles at a higher angle of attack, a slight descent speed and a growing sideways motion.

In order to start a turn, the elevator position has to change, too, so the tail lift is reduced (or its downforce increased) and the airplane starts a pitching motion. Also, the rudder should be a bit off-center to start and maintain a yawing motion, however, with sufficient lateral stability this yawing also happens with a bit of sideslip angle at the tail.

It is maybe easiest to view a turn at the extreme bank angle of 90°, forgetting the need to keep some of the lift vector in the vertical direction. Now the circle is actually an inside loop in the horizontal plane. Every turn is a combination of a horizontal loop and a yawing motion.

Now the difference between banked flight and a turn should become clearer: The lift vector has to act at a more rearward location for a turn to happen! It's backward shift must compensate for the pitch damping resulting from the pitching motion which is part of every turn. If that compensation does not happen, pitch damping will stop the rotation and instead result in a sideways acceleration.

With the proper backward shift, the horizontal component of lift is indeed the centripetal force wich causes the airplane to turn.

Answer 3

A turn in level flight can be modeled as an earth referenced circle. Therefor any forces that move the plane in curved flight around that circle can be broken into forward and inward vectors. The inward vector is centripetal force.

It is important to stay with the earth referenced result of forces created by the aircraft, because there are many ways (some better than others) to turn an aircraft in a level circle.

The best way is the "coordinated" turn, where lateral (bank/slip with ailerons) and rotational (yaw/skid with rudder) are balanced, keeping the G forces (ball) towards floor of aircraft.

However, flying the same circle with worst technique in history will produce the same (earth referenced) centripetal force.

So, that leaves one with mass, radius, and angular velocity to determine forces mathematically. Poor technique simply creates more drag, using more fuel.

Two formulas for centripetal acceleration:

Ac = v$^2$/ r and Ac = r x (angular velocity)$^2$

Answer 4

There are some subtle issues at play here.

I'm not sure that your question is really recognizing the difference between an established turn and a developing turn, i.e. a turn entry.

The fundamental characteristic of a turn is a curvature in the flight path, and the fundamental cause of a curvature in the flight path is a net centripetal force. The primary effect of banking the wing is to generate a centripetal force that curves the flight path, just as the pull of the sun's gravity causes a planet to follow a curving path through space. It's misleading to think of the wing's lift vector as primarily acting to drive a sideslip, as one of your statements seems to suggest.

Yet you are right that turning flight often involves some sideslip, or some forces generated to eliminate sideslip, and this does have some effect on the net balance of forces acting on an aircraft in turning flight.

I'd suggest that a good way to approach this question is by looking at three different "models" or paradigms, each more refined than the last.

Model 1 -- in the absence of pilot rudder inputs to the contrary, an aircraft naturally tends to stay "coordinated", i.e. naturally tends to point directly into the relative wind, so that there is zero sideslip.

Model 2 -- when entering a turn, yaw rotational inertia tends to hold the nose on its original heading even as the flight path starts to curve, which causes some sideslip. Adverse yaw associated with rolling also contributes to these effects. Appropriate rudder inputs can cancel these effects and keep the nose of the aircraft pointing directly into the relative wind, so that there is no sideways airflow over the fuselage and vertical fin. These effects are absent once the turn is established, so the rudder input need not be continued after this point. Note that the net yaw torque must be zero in a steady-state established turn-- in an established turn, the yaw rotation rate is constant, so there is no longer any need to apply a net yaw torque to overcome the aircraft's yaw rotational inertia.

Model 3-- even in an established turn, subtle aerodynamic effects often tend to cause some slip or skid, unless the pilot holds a corrective rudder input. Even though the net yaw torque must be zero in an established turn, the deflected rudder or the sideways airflow over the fuselage and vertical fin may still generate a horizontal force component, that adds or subtracts to the horizontal force generated by the banked wing.

We can see from model 3 that even in an established turn, it will often be the case that either the rudder is slightly deflected, or the vertical fin is meeting the airflow at a slight angle. This does generate some sideways force, which does add or subtract to the net centripetal force generated by the aircraft. Therefore the net centripetal force generated by the aircraft is not exactly equal to the wing's lift vector times cosine (bank angle). This may have been what you were getting at with your statement "the true centripetal force is created by the resultant of the force vectors here (the lateral forces adding sideslip and the rotational forces adding yaw)".

But there's no conflict between the idea that the net centripetal force generated by the aircraft is not exactly equal to the horizontal force component generated by the banked wing, and the idea that a turn is driven purely by the net centripetal force generated by the aircraft. Despite the complexities we've been exploring here, the answer to the question posed by your actual title, "Is the turning motion of a banked airplane caused by true centripetal force?", is a definitive yes.

In practice, in a coordinated turn, the force from the banked wing does dominate the overall force balance, dwarfing the force from the deflected rudder or the vertical fin. Thus the total centripetal force is almost exactly equal to the wing's lift vector times cosine (bank angle). That's why for most practical purposes, we assume that when the slip-skid ball is centered, the aircraft is pointing directly into the relative wind, so that the fuselage is generating no aerodynamic sideforce.

One instance where this is strikingly not true, is when we are dealing with a conventional twin-engine prop plane with one failed engine. Here the rudder must be deflected so strongly to cancel the yaw torque from the one good engine, that it makes a significant contribution to the net horizontal force acting perpendicular to the flight path. This can drive a turn toward the failed engine. The most efficient way to counteract this force and bring the net horizontal force to zero is to bank the aircraft (typically about 5 degrees) in the opposite direction. When the optimal bank angle is selected, the aircraft will fly in a straight line, even with no sideslip as measured by a yaw string at the nose. If any other bank angle is selected (or if the wings are level), if the aircraft is flying in a straight line, then sideslip (as measured by a yaw string at the nose) cannot be zero, so drag is not being minimized. In this situation it may be tempting to view the banked wing as trying to drive sideslip in one direction and the rudder as trying to drive sideslip in the other direction, resulting in zero sideslip when the two effects are balanced, but I'd argue that that's not really what is going on. Rather, the bank is trying to drive a turn in one direction, and the sideforce from the deflected rudder is trying to drive a turn in the other direction-- even though the rudder is directed in the same direction as the aircraft is banked. When you've wrapped your mind around this, you are starting to understand some the more obscure issues involved in the dynamics of turning flight.

When a sailplane is circling at low airspeed, the rudder often must be deflected toward the center of the turn to keep the yaw string centered. To accommodate the horizontal force generated by the deflected rudder in this situation, one author has recommended leaving the slip-skid ball deflected about half a diameter toward the inside or low side of the turn. The argument is made that this minimizes sideslip over the overall length of the fuselage, just as is the case in the twin-engine aircraft with one failed engine. For more, search the web to find the article in "Soaring" magazine entitled "Circling the Holighaus Way", by Richard Johnson.¹

In actual practice, unless we are dealing with a failed engine on a twin- or multi-engine aircraft, or are trying to thermal at a low airspeed in a long-spanned aircraft, in coordinated flight the fin or rudder makes a negligible contribution to horizontal forces acting perpendicular to the flight path, meaning that the net centripetal force can be assumed to be equal to the wing's lift vector times cosine (bank), or to weight times (tangent) bank, and the turn can be assumed to be "coordinated" (in the sense that nose of the aircraft is pointing directly into the airflow) whenever the slip-skid ball is centered.

Footnotes--

1. This article is currently difficult to find on-line. Here's a related article by Richard Johnson from the magazine "Kiwi"-- "Do you really want to keep the yaw string centered?"