A "holistic" view of how dihedral contributes to roll stability and anhedral contributes to roll instability
This page was last modified on September 14, 2006
Here is a "holistic" view of how dihedral contributes to roll stability and anhedral contributes to roll instability.
Dihedral or anhedral create a roll torque whenever there is a sideways component in the airflow over an aircraft, as we described in "Oblique and side views of aircraft with dihedral" and "Oblique and side views of aircraft with anhedral". When the nose of an aircraft points toward the outside or high side of a turn in relation to the actual direction of the flight path at any given moment--i.e. when the aircraft is sideslipping--the roll torque created by dihedral will tend to roll the aircraft toward a shallower bank angle, as we saw in "Roll torque created by dihedral during slips and skids". To understand how dihedral contributes to roll stability, we need to understand why an aircraft will sideslip after being tipped into a bank
When an aircraft tips from wings-level into a bank, this causes the flight path to start to curve. To keep the aircraft's nose aligned with the actual direction of the flight path and relative wind at any given moment, the aircraft's yaw rotation rate must increase. Until this happens, the aircraft's nose will point toward the outside or high side of the turn in relation to the actual direction of the flight path and relative wind at any given moment. In other words, in the absence of other competing factors, the aircraft's yaw rotational inertia will tend to cause the aircraft's heading to remain constant as the flight path starts to curve. This is a form of sideslip. The resulting sideways component in the relative wind will interact with the aircraft's dihedral geometry to create a difference in angle-of-attack between the two wings, which will create a roll torque, as described in "Oblique views and side views of aircraft with dihedral". The roll torque will tend to return the aircraft toward wings-level.
Another effect that can contribute to sideslip as an the aircraft rolls from wings-level into a bank is adverse yaw, as we described in "The main cause of adverse yaw during rolling motions: the 'twist' in the relative wind". Even in the simpler case of an intentional pilot roll input, the relationship between slip due to adverse yaw and slip due to yaw rotational inertia is complex, as we hinted at in "2 kinds of yaw". If adverse yaw is pronounced enough, the aircraft's yaw rotational inertia will actually tend to minimize the total amount of sideslip that occurs rather than enhance the total amount of sideslip that occurs. In the more complex case of a rolling motion caused by a gust rather than a roll input from the pilot, it's not clear to me how to best analyze the likely effects of adverse yaw. It seems that while the aircraft is being accelerated in the roll axis by the gust, the airflow may actually be twisted in the opposite direction as is described in the above link--after all, the airflow is now forcing the rolling motion, not resisting the rolling motion. At some point where the gust has ended and the aircraft still retains some rotational momentum about the roll axis, the relative wind would be twisted as described in the above link, which would create some adverse yaw. These complexities suggest that perhaps yaw rotational inertia is more important than adverse yaw in creating sideslip when an aircraft is thrown into a bank by a gust, but this is conjecture on my part.
The two effects listed above only operate when the aircraft's bank angle is changing or immediately after the aircraft's bank angle has changed. To a first approximation they can be thought of as factors that tend to inhibit an aircraft from entering a spiral dive, rather than factors that act to bring an aircraft back to a particular "preferred" bank angle after some sort of disturbance or pilot input has increased the bank angle. Yet some aircraft can be configured to seek a "preferred" bank angle in this manner; free-flight model airplanes are a great example. In such an aircraft, at lower bank angles the pilot will need to maintain a rolling-out control input to prevent the bank angle increasing, and at higher bank angles the pilot will need to maintain a rolling-in control input to prevent the bank angle from decreasing. Some aircraft with a great deal of dihedral even tend to return all the way to wings-level after a disturbance. It's clear that we haven't yet told the whole story--or even the most important part of the story--about why a sideslip arises when an aircraft banks.
In "Roll and yaw torques due to the difference in the airspeed of the the left and right wingtips", we saw that even when an aircraft is in a constant-banked turn, if the pilot isn't applying a touch of "inside" rudder to keep the nose of the aircraft (or more precisely the wing of the aircraft) aligned with the actual direction of the flight path and the relative wind, the faster-moving outside wingtip will tend to experience more drag than the slower-moving inside wingtip, which will tend to yaw the nose to point slightly toward the outside or high side of the actual direction of the flight path and relative wind at any given moment. In other words, the aircraft will tend to slip as it turns. This effect will be most pronounced in slow-flying, long-spanned aircraft. The resulting sideways component in the relative wind will interact with the aircraft's dihedral geometry to create a difference in angle-of-attack between the two wings, which will create a roll torque, as described in "Oblique views and side views of aircraft with dihedral". The roll torque will tend to return the aircraft toward wings-level. This effect is very important in understanding spiral stability in general.
The following statement is conjecture on my part: for long-spanned, slow-flying light aircraft, in cases where the pilot is not actually making a roll command to force the bank angle to change, the sideslip arising from the difference in drag between the inboard and outboard wingtips seems likely to be larger than the sideslip resulting from the other effects that we've discussed up to this point, except in certain unusual situations such as during or immediately after a large, rapid increase in bank angle.
Here's one more effect that will contribute to sideslip during a turn, but only when the turn is descending with respect to the airmass. In "A constant-banked climbing or descending turn involves a continual rolling motion", we noted that a constant-banked turn that is descending with respect to the airmass actually involves a continual rotation around the aircraft's roll (longitudinal) axis, with the extreme case being a vertical rolling dive. Like any other rolling motion, this creates a "twist" in the relative wind, which creates adverse yaw, so the aircraft's nose will tend to point slightly toward the outside or high side of the turn in relation to the actual direction of the flight path at any given moment. This effect will tend to be most pronounced at when the sink rate is high with respect to the airmass, which for a glider will correspond to high airspeeds and steep bank angles. These are exactly the situations where the difference in airspeed between the two wingtips is least important: the difference in airspeed between the two wingtips will be greatest when the airspeed is low and the bank angle is moderate. The sideslip due to adverse yaw due to the rolling motion that we're describing here will interact with dihedral or sweep--if present--to contribute a stabilizing roll torque.
This article has explored several different reasons why an aircraft may sideslip while banked. Regardless of which of these effects are most important in any given situation, if the aircraft has anhedral rather than dihedral, the sideslipping airflow will interact with the anhedral to create a rolling-in torque, which will tend to create a spiral dive with increasing bank angle. In other words, anhedral creates a destabilizing effect in roll just as dihedral and sweep create a stabilizing effect in roll.
However, dihedral is not the only factor that can act to limit the development of a diving spiral. We've noted that a constant-banked turn that is descending with respect to the airmass actually involves a continual rotation around the aircraft's roll (longitudinal) axis, with the extreme case being a vertical rolling dive. In "The main cause of adverse yaw during rolling motions: the 'twist' in the relative wind", we also noted that any rolling motion creates an "aerodynamic damping" effect, which creates a roll torque that acts against the direction of the rolling motion. When a descending turn or spiral dive builds to a certain bank angle and descent rate, this roll damping effect will become strong enough to prevent any further increase in the roll rate, which will prevent any further increase in the bank angle, which will prevent any further steepening of the diving spiral--assuming that the aircraft is still in one piece! Flex-wing hang glider pilots are very familiar with this effect: even though flex-wing hang gliders have anhedral, the bank angle generally tends to decrease during a high-speed, low-angle-of-attack spiral dive with the control bar well "pulled-in", unless the pilot maintains a strong rolling-in control input.
As a logical extension of the ideas in "Sweep creates a dihedral-like effect", we can see that sweep interacts with a sideways (sideslipping) airflow in a way that creates a stabilizing, rolling-out roll torque, just as dihedral does. Most of the comments that we've made up to this point in this article in relation to dihedral also apply to sweep. However, all other things being equal--i.e. for a given sideslip angle--the dihedral-like effects of sweep will be most pronounced when the wing's angle-of-attack is high, as we saw in "The dihedral-like effect of sweep depends strongly upon angle-of-attack".
Therefore if the aircraft has both sweep and anhedral, they will have competing effects on the aircraft's spiral stability characteristics, and these effects will be angle-of-attack dependent, with the stabilizing effects of sweep being strongest at high angles-of-attack and the destabilizing effects of anhedral being strongest at low angles-of-attack. This is analogous to the way that when an aircraft has both sweep and anhedral, the aircraft is will have the strongest positive (or the weakest negative) "coupling between yaw (slip) and roll" at high angles-of-attack, and will have the weakest positive (or the strongest negative) "coupling between yaw (slip) and roll" at low angles-of-attack. See "Competing effects of sweep and anhedral" for more.
This doesn't mean that a swept-wing aircraft will always show the greatest spiral instability at low angles-of-attack (high airspeeds) and the least spiral instability at high angles-of-attack (low airspeeds), because other important factors are also present. We noted above that flex-wing hang gliders actually show more spiral instability at low airspeeds (high angles-of-attack) than at high airspeeds (low angles-of-attack), due to the important stabilizing role played by "aerodynamic damping in the roll axis" when the airspeed--and therefore the sink rate with respect to the airmass--is high. This relationship will likely be true of many other aircraft as well, as long as the thrust or power setting is kept constant and the wing's sweep angle is not too extreme.
In general, increasing the size of an aircraft's fixed vertical fin will minimize the sideslip that occurs during a commanded or uncommanded roll from wings-level into a bank, or during a constant-banked descending turn or tightening descending spiral, regardless of the exact source of that sideslip. If the aircraft has dihedral, or if the aircraft has sweep and no anhedral, this means that increasing the size of the aircraft's fixed vertical fin will tend to decrease the aircraft's roll stability or increase the aircraft's spiral instability. If the aircraft has anhedral and no sweep, increasing the size of the aircraft's fixed vertical fin will tend to decrease the aircraft's spiral instability. If the aircraft has both anhedral and sweep, a fixed vertical fin might increase the aircraft's spiral instability at high angles-of-attack where the dihedral-like effects of sweep are most pronounced, and might decrease the aircraft's spiral instability at low angles-of-attack where the dihedral-like effects of sweep are least pronounced.
Here are some other sources that make note of the relationship between vertical fin size and spiral instability in the basic case of an aircraft with dihedral, as noted above:
Figure 148 "Spiral and directional divergence" from NASA's SP-367 "An introduction to the aerodynamics of flight". From RC Soaring Digest, January-March 2000.
"Swept Wings and Effective Dihedral" by Bill and Bunny Kuhlman. From RC Soaring Digest, January-March 2000.
"Roll Stability and Control" by Don Stackhouse, from the on-line feature "Ask J and D"-- see the paragraph that begins "Now let's talk about "Spiral stability" and "Dutch roll".
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