Rolling dynamics – understanding an airplane’s reaction to a roll command
Updated: Jan 4, 2022
No matter if we look at an aerobatic trainer, a jet fighter or a transport category airliner, they all execute rolling maneuvers on a daily basis. Obviously to a different extent. It turns out, the reaction of an airplane to a roll command is much more complex than it might seem at a first glance. Here are some questions to get started:
Following an aileron deflection, what roll rate and roll acceleration will the airplane develop? How does that change with altitude?
Along which axis does the airplane roll?
The basics – aileron deflection
First, we look at single-degree-of-freedom, aileron rolls. This somewhat mathematical term aims to describe the reaction of an airplane to a given aileron input, for the moment ignoring the coupling effects (which we deal with later). When the ailerons are deflected, the lift distribution is altered in such a way, that there is a net rolling moment causing the airplane to build up a roll rate (see green part in Figure 1).
As soon as the airplane rolls, there is an opposing rolling moment, caused by the change in angle of attack on the wing section (see blue part in Figure 1). In a very simplified way, it can be said that after a given aileron deflection, the roll rate builds up to the point, where these two moments are balanced .
A key concept: The roll helix angle
The helical path that the wing tip follows (red annotation in Figure 1) is often referred to as the roll helix angle (pb/2v) . The fascinating part is that this remains essentially constant (steady state) for a given aileron deflection .
Figure 2 above shows the aileron step response (reaction to a sudden change) for different aileron deflections (25%, 50%, 75% and 100%). Two points are important here :
For a given airplane and configuration, a given aileron deflection yields (to a very good approximation) a certain roll helix angle in the steady-state.
The time constant tau (where ~63%) of the steady-state helix angle are reached is the same for different aileron deflections (all other parameters being equal).
Figure 3 below shows the aileron step response for the roll rate. The green line represents the reference aircraft at the reference altitude.
An interesting observation can be made, when the same airplane is exposed to the same control input, but at a higher altitude: The amber line shows the reaction, when TAS is maintained: Lower acceleration (due to the lower EAS) but same steady-state roll rate. The blue line depicts the situation at the same EAS: Similar acceleration, but higher steady-state roll rate (due to the higher TAS).
Here is the crux: If EAS is maintained, the steady-state roll rate increases with altitude as TAS increases. This results in the roll helix angle remaining constant! 
A special case: Maximum pilot effort (reversible, unboosted controls)
For airplanes with reversible, unboosted controls where the pilot uses muscular force to deflect the ailerons, a situation may arise where the maximum aileron deflection can no longer be maintained with increasing airspeed. This is depicted in Figure 4.
As depicted above, the aileron deflection remains constant until the maximum control force is reached. Then, the aileron deflection and roll rate begin to decrease. Also note, how the roll helix angle is very closely related to the aileron deflection.
Reality: More than one degree of freedom
In reality, the airplane is not limited to one degree of freedom and depending on the layout of the fuselage and flight controls, different side-effects occur . The list below provides an excerpt of the most relevant stability and control derivatives that are at play here. The magnitude of each one will depend on the airframe.
Cnb Yawing moment due to sideslip
Cnda Yawing moment due to aileron deflection (adverse/proverse yaw)
Cnp Yawing moment due to roll rate
Clb Rolling moment due to sideslip
Theoretical development of the roll rate after a pure aileron input (no turn coordination) is shown for different cases in Figure 5. The text in color below Figure 5 describes each situation.
High Cnb , low Clb , low Cnda , low Cnp
Strong directional stability, little adverse or proverse yaw, and weak dihedral effect :
Only minimal sideslip excursions are observed. As the dihedral effect is small, there is very little effect on roll rate. This type of airplane may be maneuvered without rudder coordination and will yield a similar result as the single-degree-of-freedom case.
Low Cnb, high Clb, high Cnda , high Cnp
Low directional stability, high yawing moments generated by lateral control deflection and roll rate, low dihedral effect :
Significant sideslip excursions, handling qualities degraded, lag observed.
Low Cnb, very high Clb, very high Cnda, and Cnp
Weak directional stability, strong dihedral effect, significant yawing moments generated by lateral control deflection and roll rate :
Serious handling problems. Possible Dutch roll oscillation. In extreme cases, the roll rate may be reversed!
In theory, all these airframes could be flown in a coordinated manner by the appropriate rudder inputs. The problem is that the amount of pilot effort to do so increases significantly for poor designs .
Advanced Roll coupling effects
Coming back to the second question of the introduction:
If the airplane is rolled by ailerons, along which axis does it roll? The answer is not as straight forward as one might think…
Many pilot training handbooks are silent about this topic, but it certainly deserves attention. Here is the reason why: The axis of rotation may vary for different situations. In fact, this depends on the mass distribution and stability of the airplane and how quickly it is rolled. History has shown that an insufficient understanding of these effects can cause significant discomfort or even loss of control…
We consider the following coupling effects:
An airplane with negligible stability and significant inertia, will roll almost exclusively along its body axes . This leads to an exchange of angle of attack (alpha) and sideslip angle (beta), as described by Pinsker . If the airplane had infinite stability and negligible inertia, it would roll along the wind axes, with constant alpha and beta. Figure 6 depicts the two extreme scenarios. A real aircraft will lie in between these two scenarios (usually closer to the “green” scenario).
In order to have a more modern way of depicting this, I reached out to a flight control expert from industry and got some great support from Rodney Rodríguez Robles. He kindly created the following animations, showing the rolling with infinite inertia and negligible stability and vice versa.
With infinite inertia and negligible stability, the airplane rolls along the body axis, this causes a change in alpha and beta (see Animation 1). This is purely a geometrical effect. The fuselage turns red when max. beta is exceeded.
With negligible inertia and infinite stability, the airplane rolls along the wind axes . It is important to realize that the aerodynamic stability of the airplane causes the angles alpha and beta to remain constant in Animation 2. A realistic combination of the two scenarios may be found at the end of this article. One more thing to note: When the roll rate is low compared to the airplane’s natural frequencies, the aerodynamic moments are sufficient to prevent significant alpha and beta excursions. But for higher rates of roll and low stability airplanes, this can become problematic .
This form of coupling occurs, if an airplane is subject to angular rates in two axes simultaneously i.e. rolled and yawed or rolled and pitched at the same time . The resultant axis of rotation is depicted in Figure 7. Depending on the mass distribution of the airplane, the moments of inertia will cause potentially significant coupling effects . The airplane mass is represented by dumbbell-shaped black points. When rotating around the blue vectors, this causes the moments (M) as shown in Figure 7 below:
It can be shown mathematically that inertial coupling effects can be reduced to zero, if the moments of inertia Ixx, Iyy and Izz are equal . The effects are more pronounced for airplanes with high pitch (Iyy) and yaw (Izz) inertia and low roll (Ixx) inertia . It should now become apparent, why a F-104 “Starfighter” is more susceptible to this than a C182…
Today’s fighters usually have digital flight control systems that take care of these effects and ensure safe handling characteristics.
By the nature of most airplanes, the mass distribution in the xz-plane is not symmetrical. Thus, the principal inertial axis is slightly offset from the x axis. This causes a “flyweight” effect when the airplane is rolled rapidly along the x axis . This effect is also countered by the aerodynamic stability derivatives and is usually quite weak .
A seemingly simple maneuver turns out to be quite complex when examined more closely. Hopefully, this article could shed some light onto the dynamic behavior of airplanes during rolling. Even without diving into mathematics, there are some key aspects that should be remembered, such as the roll helix angle and the change in roll behavior with altitude.
To wrap up, I will let you enjoy a slow flight, rapid rolling animation with more realistic inertia and stability properties, also kindly created by Rodney Rodríguez Robles.
 Kimberlin, Flight testing of fixed-wing aircraft, AIAA, 2003
 USNTPS-FTM-No. 103, U.S. Naval test pilot school flight test manual, fixed wing stability and control, naval air warfare center aircraft division, Patuxent River Maryland, 1997
 Etkin, Dynamics of atmospheric flight, Dover publications, 2000
 Phillips, Effect of steady rolling on longitudinal and directional stability, NACA technical note 1627, Langley Memorial Aeronautical Laboratory, 1948
 Pinsker, Critical flight conditions and loads resulting from inertia cross-coupling and aerodynamic stability deficiencies, Aeronautical Research Council CP 404, Her Majesty’s stationary office London, 1958