Rolling Dynamics
The dynamics of a rolling airplane are remarkably complex for a seemingly simple maneuver. The reason is lateral stability is highly coupled between roll, yaw, and sideslip. In real life rolling induces yaw and sideslip while yaw and sideslip also induces roll. Hence many variables are involved in determining roll performance. The following diagram illustrates how roll rate changes when you consider yaw and sideslip and stability to sideslip (dihedral effects) compared to a coordinated roll with rudder input to remove center yaw and remove sideslip.
Understanding what limits roll performance is therefore a rather complicated task if we operate in 6 degrees of freedom as in real life. In designing aircraft some simplifying assumptions are often made the most basic of these are to assume no yaw or sideslip when rolling. Essentially the simplification is to constrain roll to just one degree of freedom to reduce the complexity in assessing roll rate. When we do so we are able to reduce roll rate to the following general equation:
In plane English:
Roll rate = (rolling moment due to ailerons / roll damping moment ) * ( 1 – e ^- [time/roll transient time constant]) * aileron_deflection
The equation is the general one degree of freedom form for roll rate which includes both roll acceleration and steady state roll. A typical time history of a roll looks like the following:
Assuming one degree of freedom when an aircraft rolls it accelerates into the roll with aileron deflection. Roll rate increases from zero until it reaches a constant / steady state roll. Roll acceleration is when roll rate accelerates and increases from zero. Steady roll is when roll acceleration stops and roll rate becomes constant.
Understanding limits on roll performance means looking at factors for both these phases of roll acceleration and steady roll.
Roll Acceleration Phase:In our general equation roll acceleration damping is determined by the (1- e^[-t/T]) term . Roll time constant T determines how fast a roll reaches a steady state after being disturbed. Delving into the details we can substitute 1/T with the following equation:
where:
Q = dynamic pressure (.5*air_density*velocity^2)
S = wing surface area
b = wing span
Clp = coefficient of rolling moment due to rolling
Ixx = rolling moment of inertia
V = velocity
Actually the V on the bottom cancels out by V^2 in dynamic pressure so there is only Velocity left in the numerator but for simplicity sake we will leave it so that we can leave Q as is in the equation.
These are the variables that influence how fast a roll accelerates. Roll acceleration varies directly with speed, wing area, square of wing span, and rolling coefficient Clp while it varies inversely with rolling moment of inertia. What this means is the greater the speed, wing area, wing span, and Clp the faster the roll acceleration while the greater the rolling moment of inertia the slower the roll acceleration. Clp and Ixx are the variables that are hard to come by. More on that later. This is how Cthulhu’s statement about inertia comes into play on roll acceleration.
The above diagram shows you the impact of greater or lesser moment of inertia has. The greater the inertia the slower a roll accelerates.
Steady Roll Phase:Roll acceleration transient quickly goes to zero and roll rate becomes steady. When this occurs rolling moment due to aileron deflection equals the opposite rolling moment due to wing damping from angle of attack differential in a roll mentioned by Cthulhu. In this case the equation then simplifies to:
where
V = velocity
b = wing span
Clalpha_a = rolling moment coefficient due to ailerons
Clp = rolling moment coefficient due to rolling (also known as roll damping moment)
delta_a = aileron deflection angle
These then are the key variables that influence steady roll rate of an airplane: directly with airspeed, Clalpha_a, aileron deflection angle and inversely with Clp and wingspan. Clalpha_a depends on aileron and aileron to wing geometry while Clp depends on wing geometry. Neither of these values are easy to come by in the real world and are derived from flight testing for a real aircraft. In design mode various of ways of estimating these values exist for preliminary estimates of what they might be. Actual values need to come from wind tunnel and flight testing.
Cthulhu is right in saying that roll damping (Clp term) is a factor limiting maximum steady rolling rate. However the ratio of Clalpha_a/Clp is not the only factor in determining steady roll rate. Velocity, wing span, and aileron deflection angle also are important. If we look at some steady roll performance charts like that from NACA 868 Widewing refers to we can see how these variables interact, some obvious, some not so obvious.
As per the equation if airspeed increases then roll rate increases. Why then does roll rates begin to reduce after we exceed a certain airspeed? The answer is that roll rate is also a function of aileron deflection angle. Aileron deflection is affected by dynamic pressure which is a function of air density and air speed. The faster we go the greater the dynamic pressure which means the greater the force needed to deflect the aileron. Aileron deflection then depends on the amount of force that a pilot and associated control mechanisms can exert to deflect the aileron. The NACA graph above is done with a 50-lb stick force and we can see for different aircraft that the control forces are not enough to deflect the ailerons at their maximum any longer as airspeed increases.
So none of this takes into account any of the other degrees of freedom that exist! What Stoney and Bozon mention about dihedral effect etc. aren’t even factored in because we’ve assumed no yaw or sideslip involved. It get’s even more complicated when you factor those and other variables in!
As for looking for real life Clalpha_a, Clp, and Ixx data for the WW2 aircraft there definitely isn’t a single source I know where this is collected. In fact most of it would have to be derived from flight test data for these airplanes.
I hope that sheds some light on the matter!
Tango, XO
412th FS Braunco Mustangs
