Aileron Roll Rate⚠ unverified

Understanding the Parameters

• Ailerons make a rolling moment, damping sets the steady rate — deflect the ailerons and the aircraft accelerates in roll; as it rolls, the down-going wing gains and the up-going wing loses angle of attack, creating an opposing roll-damping moment ($C_{lp}<0$). The roll rate rises until the damping moment exactly cancels the aileron moment — that equilibrium is the steady roll rate. So roll rate is fundamentally an aileron-power-versus-roll-damping balance, not just aileron power alone.

• What the registry keeps and drops — it keeps the aileron moment ($C_{l\delta_a}\delta_a$) but drops the roll damping and airspeed that actually set the steady rate. That is why it needs an externally-supplied $p_{\max}$ cap to stop the "rate" being unbounded — a stand-in for the physics it omits.

• Roll rate scales with airspeed — in the real relation $p_{ss} \propto V$: faster flight gives a higher steady roll rate for the same deflection (more dynamic pressure driving the ailerons, up to structural and reversal limits). The registry model, lacking $V$, cannot show this.

• The $p_{\max}$ ceiling — real roll rate is limited by aileron travel, by aileron reversal at high speed (wing twist can cancel or reverse the aileron's effect), and by structural/actuator limits. Military roll- performance specs (e.g. time to bank 90°) drive fighter design; the $p_{\max}$ here is a lumped stand-in for all these ceilings.

• Adverse yaw companion — deflecting ailerons also yaws the aircraft against the roll (the down-aileron wing has more drag), which is why coordinated rolls use rudder too. This calculator models only the roll rate, not the coupled yaw.

Derivation (Approaching a Proof)

The honest derivation is the roll-damping balance the registry approximates. Consider the roll degree of freedom about the body x-axis (Torque, Roll Mode Time Constant):

$$I_{xx}\,\dot p = L = q\,S\,b\,\big(C_{l\delta_a}\,\delta_a + C_{lp}\,\hat p\big), \qquad \hat p = \frac{p\,b}{2V},$$

where the rolling moment $L$ has an aileron part ($C_{l\delta_a}\delta_a$, driving the roll) and a damping part ($C_{lp}\hat p$, opposing it), $\hat p$ being the non-dimensional roll rate. In the steady roll ($\dot p = 0$) the two moments balance:

$$C_{l\delta_a}\,\delta_a + C_{lp}\,\frac{p_{ss}\,b}{2V} = 0 \quad\Longrightarrow\quad p_{ss} = -\frac{C_{l\delta_a}}{C_{lp}}\,\delta_a\,\frac{2V}{b}.$$

This is the real steady roll rate: proportional to aileron deflection and effectiveness, inversely proportional to roll damping, and — crucially — proportional to airspeed $V$ and inversely to span $b$. Since $C_{lp}<0$, $p_{ss}$ has the expected sign.

The registry formula discards $C_{lp}$, $V$, and $b$ (none are inputs), taking the aileron moment coefficient itself as the "rate" and imposing a hard ceiling:

$$p = \min\!\big(C_{l\delta_a}\,\delta_a,\; p_{\max}\big).$$

It reproduces the correct proportionality to $C_{l\delta_a}\delta_a$ and the existence of a maximum, but not the damping/airspeed dependence — hence a rough model, not a dimensionally-honest roll rate. This is the flag.

Dimensional check. The real $p_{ss}$: $\dfrac{C_{l\delta_a}}{C_{lp}}$ is dimensionless, $\delta_a$ radians, $\dfrac{2V}{b} = \dfrac{\text{m/s}}{\text{m}} = 1/\text{s}$, giving rad/s ✓. The registry product $C_{l\delta_a}\delta_a$ is dimensionless (a moment coefficient) and is simply labelled rad/s — the dimensional inconsistency that the missing $2V/b$ would have supplied.

History and Development

• Roll performance as a design driver. How fast an aircraft rolls has been a combat discriminator since WWI; the WWII Fw 190's superb roll rate, and the Spitfire's later fabric-to-metal aileron change to fix poor high-speed roll, are famous examples. Modern military specs (MIL-STD-1797) set explicit time-to-bank requirements that this quantity must meet.

• Aileron reversal. At high dynamic pressure a deflected aileron twists the wing enough to reduce or reverse its rolling moment — aileron reversal, a structural-aeroelastic limit discovered painfully in the 1930s–40s. It caps the usable roll rate and motivated stiffer wings, inboard ailerons, and spoilers for high-speed roll — physics behind the $p_{\max}$ ceiling.

• From the roll-damping model to CFD. The steady-roll balance $p_{ss} = -(C_{l\delta_a}/C_{lp})\delta_a(2V/b)$ is the classical flight-dynamics result (Etkin, Nelson); modern design refines $C_{l\delta_a}$ and $C_{lp}$ with CFD and flight test, but the airspeed-and-damping structure — which this simplified calculator omits — remains the correct picture.

Related Concepts: Roll Mode Time Constant, Rudder Yaw Rate Coeff, Sideslip Due To Rudder, Dihedral Effect, Dynamic Pressure, Control Power Moment, Torque

Notes: Crude model — the real steady roll rate is $p_{ss}=-\tfrac{C_{l\delta_a}}{C_{lp}}\delta_a\tfrac{2V}{b}$ (aileron power vs roll damping $C_{lp}$, $\propto$ airspeed $V$, $\div$ span $b$); the registry drops $C_{lp},V,b$ (not inputs), takes the moment coefficient $C_{l\delta_a}\delta_a$ as the "rate" and caps it at $p_{\max}$. So the output is dimensionally a coefficient labelled rad/s. Ceiling $p_{\max}$ stands in for travel/ reversal/structural limits. Ignores adverse yaw. Per radian; defaults $1.0$.