Rudder Yaw Rate Coeff⚠ unverified

Understanding the Parameters

• What the rudder does — the vertical tail is a small wing; deflecting the rudder cambers it, generating a side force. Because the tail is well behind the CG, that side force makes a yawing moment that points the nose left or right. The rudder is the primary directional control: coordinating turns, countering adverse yaw, holding heading in a crosswind, and — critically — controlling yaw after an engine failure on a multi-engine aircraft.

• $C_{n\delta_r}$ — the effectiveness — the yawing moment per radian of rudder, set by the tail area, its moment arm, and the rudder's chord fraction. It is conventionally negative (positive/right rudder yaws the nose right, which is a negative $C_n$ in the standard sign convention). Its magnitude is what "rudder power" means.

• Coefficient, not rate — the key caveat — the output here is $\Delta C_n$, a moment coefficient. To get an actual yaw rate you would have to solve the yaw equation of motion: the moment accelerates the aircraft in yaw, directional stability $C_{n\beta}$ and yaw damping $C_{nr}$ resist it, and a steady rate emerges only in a sustained manoeuvre — all requiring inertia and airspeed this calculator does not take. The name overpromises; the number is authority, not rate.

• Dimensionalising it — like any moment coefficient, the physical yawing moment is $N = \Delta C_n\,q\,S\,b$ (dynamic pressure × wing area × span) — the same $qS\times$length structure as Control Power Moment but with span $b$ for the yaw axis. That would give the newton-metres driving the yaw response.

• Engine-out and $V_{MC}$ — the most safety-critical use of rudder authority: after losing an engine, the rudder must generate enough yawing moment to counter the asymmetric thrust. The minimum control speed $V_{MC}$ is set by where the available rudder $\Delta C_n\,qSb$ can just balance the engine-out yawing moment — a direct application of this quantity.

Derivation (Approaching a Proof)

The yawing-moment contribution of the rudder is, to first order, linear in its deflection through the control derivative $C_{n\delta_r}$:

$$\Delta C_n = \frac{\partial C_n}{\partial \delta_r}\,\delta_r = C_{n\delta_r}\,\delta_r,$$

which is exactly what the calculator returns. The derivative itself comes from the tail: deflecting the rudder by $\delta_r$ changes the tail's side-force coefficient by roughly $C_{L\alpha,v}\,\tau\,\delta_r$ (with $\tau$ the rudder effectiveness factor), and that side force acting at the tail arm $\ell_v$ gives

$$C_{n\delta_r} = -\,\eta\,\frac{S_v\,\ell_v}{S\,b}\,C_{L\alpha,v}\,\tau,$$

the vertical-tail volume ratio times the tail lift-curve slope times the rudder effectiveness — the same structure as the fin's contribution to directional stability.

Why this is not a rate. The actual yaw response obeys $I_{zz}\ddot\psi = N = \Delta C_n\,qSb$ plus damping ($C_{nr}$) and stiffness ($C_{n\beta}$) terms. Solving that for a rate requires the inertia, airspeed, and the other derivatives; the registry stops at $\Delta C_n$ and labels it a rate. So the honest reading is: this is the rudder yaw authority (a moment coefficient), the input to the yaw dynamics, not their output. $\qquad\blacksquare$

Dimensional check. $C_{n\delta_r}$ is per radian and $\delta_r$ radians, so $\Delta C_n = C_{n\delta_r}\delta_r$ is dimensionless — a moment coefficient, confirming it is not rad/s.

History and Development

• The vertical tail and rudder. Directional control by a movable rear surface is one of the oldest ideas in aeronautics; the rudder's role in coordinating turns and countering adverse yaw was understood by the Wright brothers, who linked rudder to wing-warping for coordinated flight.

• Engine-out control. With multi-engine aircraft, rudder authority became life-critical: the certification speed $V_{MC}$ (minimum control speed with the critical engine inoperative) is defined by the rudder's ability to counter asymmetric-thrust yaw, and drives vertical-tail sizing on every twin. Accidents from insufficient rudder authority at low speed shaped the regulations.

• Yaw dampers and fly-by-wire. Because the dynamic yaw response (the dutch roll) is often poorly damped, rudder is also driven automatically by yaw dampers; modern fly-by-wire blends pilot and automatic rudder commands. All of it starts from the rudder authority $\Delta C_n$ this calculator represents.

Related Concepts: Sideslip Due To Rudder, Weathercock Stability, Dutch Roll Damping, Dutch Roll Wn, Aileron Roll Rate, Control Power Moment

Notes: Returns the yawing-moment coefficient $\Delta C_n = C_{n\delta_r}\delta_r$ (rudder yaw authority), not a yaw rate — despite the name (registry latex honestly uses $\propto$; output labelled dimensionless). A real rate needs the yaw dynamics (inertia, $C_{n\beta}$, $C_{nr}$, airspeed). Dimensionalise via $N=\Delta C_n\,qSb$. $C_{n\delta_r}$ from the vertical-tail volume ratio; sets engine-out $V_{MC}$. Per radian; defaults give $\Delta C_n = -0.01$.