Stability Derivative Cm Q⚠ unverified

Understanding the Parameters

• A rate ("rotary") derivative — the aircraft's aerodynamic behaviour is captured by a set of stability derivatives, split into those that respond to angular displacement (stiffness: $C_{m\alpha}$, the static margin) and those that respond to angular rate (damping: $C_{mq}$, $C_{lp}$, $C_{nr}$). $C_{mq}$ is the pitch damping member — how much moment the aircraft makes because it is pitching, not because of where it is pointed.

• The non-dimensional pitch rate $\hat q = q_{\text{pitch}}\,c/2V$ — pitch rate is non-dimensionalised by the chord and airspeed so the derivative is a clean geometric/aerodynamic property independent of flight condition. This is why $C_{mq}$ is quoted "per radian" of $\hat q$ and why converting it back to a dimensional damping moment reintroduces the airspeed (the $1/V$ that Pitch Rate Damping must carry).

• Sign: negative = damping — a physically damping aircraft has $C_{mq} < 0$: a positive (nose-up) pitch rate makes a negative (nose-down) moment that opposes the motion. A positive $C_{mq}$ would be anti-damping — a divergent, unflyable pitch response. Typical values run from a few to tens (negative), dominated by the tail's contribution $\propto (\ell_t/c)^2$ (the square of the tail arm).

• Where it comes from — mostly the horizontal tail: pitching motion changes the tail's angle of attack (it moves up or down through the air), and the resulting moment opposes the pitch. The long tail arm makes this the dominant damping source; the wing and body add smaller contributions.

• How it is measured — being a dynamic derivative, $C_{mq}$ cannot be found from a static wind-tunnel balance; it needs forced-oscillation rigs, CFD with prescribed motion, or system identification from flight test. It is one of the trickier derivatives to pin down accurately.

Derivation (Approaching a Proof)

The calculator is an identity, so the content is the definition and physical origin of $C_{mq}$.

By definition it is the partial derivative of the pitching-moment coefficient with respect to the non-dimensional pitch rate:

$$C_{mq} \equiv \frac{\partial C_m}{\partial \hat q}, \qquad \hat q = \frac{q_{\text{pitch}}\,c}{2V}.$$

Tail contribution (the dominant term). When the aircraft pitches nose-up at rate $q_{\text{pitch}}$, the horizontal tail, at distance $\ell_t$ behind the CG, moves downward with speed $q_{\text{pitch}}\,\ell_t$, so its local angle of attack increases by

$$\Delta\alpha_t \approx \frac{q_{\text{pitch}}\,\ell_t}{V}.$$

This raises the tail's lift by $\Delta C_{L,t} = C_{L\alpha,t}\Delta\alpha_t$, producing a nose-down pitching moment $\propto \Delta C_{L,t}\,\ell_t$. Collecting the factors and non-dimensionalising by $\hat q$ gives the tail's damping derivative:

$$C_{mq,\text{tail}} = -\,2\,\eta\,\frac{S_t\,\ell_t}{S\,c}\,\frac{\ell_t}{c}\,C_{L\alpha,t} = -\,2\,\eta\,V_H\,\frac{\ell_t}{c}\,C_{L\alpha,t},$$

where $V_H = S_t\ell_t/(Sc)$ is the horizontal-tail volume ratio. The negative sign (damping) and the $(\ell_t/c)^2$-type dependence (long tail arm → strong damping) are the two key features. $\qquad\blacksquare$

The registry does none of this — it returns whatever $C_{mq}$ you give it. The formula above shows what a real estimate involves, and where the number comes from; the pass-through simply carries it to Pitch Rate Damping, which multiplies it by $q\,S\,c^2/(2VI_{yy})$ to get the dimensional damping.

Dimensional check. $C_m$ is dimensionless and $\hat q$ is dimensionless (a ratio of speeds), so $C_{mq} = \partial C_m/\partial\hat q$ is dimensionless — conventionally quoted "per radian" of pitch-rate angle ✓.

History and Development

• Bryan's equations (1911). G. H. Bryan cast aircraft dynamics as a set of linear differential equations whose coefficients are the stability derivatives, $C_{mq}$ among them — the framework that turned "does it fly nicely?" into an eigenvalue calculation.

• The pitch-damper era. Recognising that natural $C_{mq}$ damping falls with air density led to stability-augmentation systems that feed back measured pitch rate to the elevator, synthesising extra damping — effectively boosting the effective $C_{mq}$ electronically. This is standard on high-altitude and fly-by-wire aircraft.

• Hard to measure, central to models. Because it is a dynamic derivative, $C_{mq}$ has driven the development of forced-oscillation tunnels, rotary balances, and modern flight-test system identification — all to nail down this one number, which then drives the short-period damping every handling-qualities assessment depends on.

Related Concepts: Pitch Rate Damping, Static Margin, Roll Mode Time Constant, Dutch Roll Damping, Control Power Moment, Moment Coefficient

Notes: Pass-through — returns $C_{mq}$ unchanged; the content is its meaning. $C_{mq}=\partial C_m/\partial\hat q$, $\hat q=q_{\text{pitch}}c/2V$; a rate (damping) derivative, negative for a stable aircraft (nose-up rate → nose-down moment). Supplied mainly by the horizontal tail ($\propto V_H\,\ell_t/c$). Dynamic → hard to measure (forced-oscillation/CFD/flight ID). Used dimensionally by Pitch Rate Damping (which reintroduces the airspeed).