Tail moment-arm, airflow curvature, and spiral stability

August 2007 edition

Steve Seibel

steve at aeroexperiments.org

www.aeroexperiments.org

Consider an aircraft with a "conventional" configuration including a wing, fuselage, and vertical tail. We'll assume the pilot is not using the rudder.

Bear in mind that in turning flight, the relative wind is curved, following the arc of the turn.

When we say that some part of the aircraft is feeling a "slipping" airflow, we mean that the relative wind is impacting the surfaces that lie toward the inside of the turn. When we say that some part of the aircraft is feeling a "skidding" airflow, we mean that the relative wind is impacting the surfaces that lie toward the outside of the turn. If the flight path is curving, and the relative wind is tangent to the fuselage or keel at some point on the aircraft, there will be a slipping airflow ahead of the tangent point, a skidding airflow behind the tangent point, and no slip or skid right at the tangent point.

The fuselage and vertical tail will have a combined "center of yaw torque" somewhere near the vertical tail, well aft of the CG. When the curving relative wind is tangent to the fuselage at this point, so that the parts of the aircraft forward of this point feel a slipping airflow and the parts of the aircraft aft of this point feel a skidding airflow, the fuselage and fin will contribute no net yaw torque.

The outboard wingtip experiences more airspeed, and creates more drag, than the inboard wingtip. This creates a yawing-out torque. The fuselage and vertical tail together must create a yawing-in torque to counteract this. A yawing-in torque will arise if the curving relative wind is tangent to the fuselage at a point aft of the "center of yaw torque", so that the "center of yaw torque" feels a slipping airflow. In actual practice the tangent point will probably be well aft of the aft end of the fuselage, so that every part of the fuselage and fin feel a slipping airflow. The airflow will strike the inside of the fin and create a significant yawing-in torque.

Since the relative wind is curved, the slip angle at the CG will be larger than the slip angle at the tail. The curving relative wind does not push on the outside of the tail and force the nose to yaw outboard. However the curving relative wind allows there to be a large slip angle at the CG, even though there is only a small slip angle at the vertical tail. This allows the CG and nose to experience a large slip angle.

For a given slip angle at the vertical tail, the longer the tail moment arm, the larger the slip angle at the CG, because the relative wind has more distance to curve between the tail and the CG. However for a given vertical tail size, the longer the tail moment arm, the smaller the slip angle at the tail is likely to be, because the tail is more effective at generating yaw torque. For any given vertical tail size, there will be one tail moment-arm length that will create the least slip at the CG. If the tail moment-arm is longer than this, the slip angle at the CG will increase because the airflow has more distance to curve, for a given slip angle at the vertical tail. If the tail moment-arm is shorter than this, the slip angle at the CG will increase because the vertical fin will be less effective, so there will be a larger slip angle at the vertical tail. The larger the vertical tail, the shorter the tail moment-arm that is associated with the least slip at the CG. A very large vertical fin positioned immediately behind the CG would streamline itself perfectly with the airflow and ensure that there is no slip at the CG. Progressively smaller fins need to be progressively further back to maximize their yaw torque and minimize the slip angle at the fin, but if they are too far back they will allow extra slip at the CG because the airflow has more distance to curve. The theoretical minimum amount of slip at the CG occurs when the vertical fin is perfectly streamlined with the airflow and this theoretical minimum amount of slip is larger if the tail moment-arm is long than if it is short, because the airflow has more distance to curve when the tail moment-arm as long. However there are undoubtedly many cases where the vertical fin flies at a fairly substantial slip angle, and increasing the vertical fin's moment arm makes the fin more effective and decreases the fin's slip angle enough to decrease the slip angle at the CG.

Therefore for a given vertical fin size, increasing the vertical fin's moment-arm may either increase or decrease the slip angle at the CG.

For a given vertical fin moment-arm, increasing the vertical fin size will always decrease the slip angle at the vertical fin, which will always decrease the slip angle at the CG. (Strictly speaking this may be only absolutely true in the case of a very slender "boom" fuselage so the fuselage itself does not act like a vertical fin. If the fuselage itself acts like a vertical fin, if we change the size of the actual vertical fin we are also changing the moment-arm of the "total" effective vertical fin that represents the combined effects of the fuselage and the vertical fin. But for practical purposes the above rule probably applies nearly universally: for a given vertical fin moment-arm, increasing the vertical fin size will decrease the slip angle at the vertical fin, which will decrease the slip angle at the CG.)

If the aircraft has effective dihedral, the larger the slip angle at the CG, the more the dihedral will create a rolling-out torque, and the less spiral instability (or the more spiral stability) the aircraft will have. If the aircraft has effective anhedral, the larger the slip angle at the CG, the more anhedral will create a rolling-in torque, and the more spiral instability the aircraft will have.

Therefore for a given vertical fin size, increasing the vertical fin's moment-arm may either increase or decrease the aircraft's spiral stability or spiral instability.

A hang glider, which gets its yaw stability primarily from sweep, can be modeled as having a triangular fixed vertical fin that starts at the apex of the nose and extends back behind the CG just as far as the tip areas of the wing do. Adding a vertical fin at the end of the keel will shift the center of effective fin area aft, as well as increase the effective fin area. The fin's moment arm is undoubtedly so short that there is no risk that the added fin area, and the associated aft movement of the center of effective fin area, will increase the slip angle at the CG. Instead the increased effectiveness of the fin will decrease the slip angle at the CG. If the hang glider has effective dihedral, the fin should decrease the hang glider's spiral stability or increase the hang glider's spiral instability. If the hang glider has effective anhedral, the fin should decrease the hang glider's spiral instability. However the fin will also move the keel in relation to the rest of the airframe in a way that will tend create a rolling-out torque, which will tend to increase the glider's spiral stability or decrease the glider's spiral instability, regardless of whether the glider has effective dihedral or effective anhedral.