Pitch Rate Damping⚠ unverified

Understanding the Parameters

• Damping opposes the motion, not the displacement — stability derivatives come in two flavours: stiffness terms that resist angular displacement (like $C_{m\alpha}$, the static margin) and damping terms that resist angular rate. $C_{mq}$ is a damping term: it produces a moment proportional to, and opposing, the pitch rate. It is what makes a disturbed pitch oscillation die out rather than ring forever.

• The tail is the damper — when the aircraft pitches nose-up at rate $q_{\text{pitch}}$, the tail (well behind the CG) moves downward through the air, so it meets the flow at an increased angle of attack, makes more download, and produces a nose-down moment opposing the pitch-up. The long tail arm makes this effect strong, which is why $C_{mq}$ is large and negative — the horizontal tail is the aircraft's pitch shock absorber.

• Why the $1/V$ belongs there — the damping derivative is defined against the non-dimensional pitch rate $\hat q = q_{\text{pitch}}\,c/2V$. Converting back to a dimensional rate divides by $V$: faster flight means the same pitch rate corresponds to a smaller change in tail angle of attack, so the per-rate damping moment scales as $1/V$. Dropping $V$ (as the registry does) loses this and breaks the units.

• Sets short-period damping — this term is a principal contributor to the damping ratio of the short-period mode (the fast pitch oscillation). Together with the stiffness ($C_{m\alpha}$) it determines whether pitch response is crisp and well-damped (good handling) or sluggish/oscillatory. Handling-qualities specs put tight bounds on short-period damping precisely because of terms like this.

• Degrades at altitude — because the dimensional damping moment scales with $q = \tfrac12\rho V^2$, thin air at high altitude weakens aerodynamic pitch damping, which is one reason high-flying aircraft often need pitch dampers (stability augmentation) to restore acceptable short-period damping.

Derivation (Approaching a Proof)

The pitch-damping moment arises because pitching motion changes the angle of attack seen by the tail. When the aircraft pitches at rate $q_{\text{pitch}}$, a surface a distance $\ell$ behind the CG moves with vertical speed $q_{\text{pitch}}\,\ell$, changing its local angle of attack by $\approx q_{\text{pitch}}\ell/V$. This is captured non-dimensionally by referencing the pitch rate to $\hat q = q_{\text{pitch}}\,c/2V$.

The damping moment coefficient is $\Delta C_m = C_{mq}\,\hat q$, so the dimensional damping moment is

$$M_q = C_{mq}\,\hat q\,q\,S\,c = C_{mq}\,\frac{q_{\text{pitch}}\,c}{2V}\,q\,S\,c = \frac{C_{mq}\,q\,S\,c^2}{2V}\,q_{\text{pitch}}.$$

This is a moment proportional to the pitch rate $q_{\text{pitch}}$; the coefficient in front is the damping per unit rate. Dividing by the pitch inertia $I_{yy}$ gives its contribution to the pitch angular acceleration per unit rate — the damping eigenvalue-like term that appears in the short-period equation:

$$\frac{M_q}{I_{yy}\,q_{\text{pitch}}} = \frac{C_{mq}\,q\,S\,c^2}{2\,V\,I_{yy}}.$$

This is the correct pitch-damping term, with units of $1/\text{s}$. The registry drops the airspeed $V$ (not an input):

$$\frac{C_{mq}\,q\,S\,c^2}{2\,I_{yy}},$$

so its result equals the true term times $V$ — reproducing the right dependence on $C_{mq}$, $q$, $S$, $c^2$, and $I_{yy}$, but not the airspeed scaling, and not the clean units. This is the flag.

Dimensional check. Correct form: $\dfrac{(-)\,\text{Pa}\cdot\text{m}^2\cdot\text{m}^2}{(\text{m/s})\,\text{kg}\cdot\text{m}^2} = \dfrac{\text{N}\cdot\text{m}^2}{\text{m/s}\cdot\text{kg}\cdot\text{m}^2} = \dfrac{\text{kg}\cdot\text{m}/\text{s}^2}{\text{kg}/\text{s}} = \dfrac{1}{\text{s}}$ ✓. Registry form (no $V$) loses the $/(\text{m/s})$, leaving $\text{m/s}^2$ — dimensionally not the $1/\text{s}$ (or rad/s²) a damping term should be, which is exactly the missing-airspeed symptom.

History and Development

• Rotary derivatives. The rate ("rotary") derivatives $C_{mq}$, $C_{lp}$, $C_{nr}$ entered flight dynamics with the linearised equations of motion (G. H. Bryan, 1911), which first cast aircraft stability as an eigenvalue problem where damping terms decide whether the modes converge. $C_{mq}$ is the pitch member of this set.

• Short-period handling. As aircraft got faster and flew higher, weak natural pitch damping made the short-period mode unpleasant or dangerous, and pitch dampers / stability augmentation became standard — feeding back pitch rate to the elevator to synthesise the damping that thin air no longer provided. This term is exactly what the augmentation supplements.

• From derivative estimates to CFD/flight test. $C_{mq}$ has always been one of the harder derivatives to predict (it needs dynamic, not static, testing); forced-oscillation wind-tunnel rigs, and now CFD and system-identification from flight data, refine it. The moment-over-inertia structure here is the classical textbook form (Etkin, Nelson).

Related Concepts: Stability Derivative Cm Q, Roll Mode Time Constant, Dutch Roll Damping, Static Margin, Dynamic Pressure, Control Power Moment, Torque

Notes: Aerodynamic pitch damping — the tail opposes pitch rate (real $C_{mq}<0$; the tail's shock absorber), setting short-period damping. Registry omits the airspeed $1/V$: the correct term is $\dfrac{C_{mq}qSc^2}{2VI_{yy}}$ (units $1/$s); the registry $\dfrac{C_{mq}qSc^2}{2I_{yy}}$ is that × $V$ (dimensionally m/s², not rad/s²) — same defect as Roll Mode Time Constant. Weakens with altitude ($\propto q$) → needs pitch dampers. $\hat q = q_{\text{pitch}}c/2V$. Per radian; defaults $1.0$.