6.2 Longitudinal Stability & Centre of Gravity Limits

Longitudinal Stability & Centre of Gravity Limits

Longitudinal stability describes an aircraft's stability in pitch about its lateral (transverse) axis. Of the three rotational dimensions of flight, longitudinal stability is the most critical for pilot control and flight safety. It is directly controlled by aircraft design geometry and, crucially, by the location of the Centre of Gravity (CG) relative to the aircraft's aerodynamic center.

---

1. Forces and Moments in Pitch Equilibrium

In steady level flight, three main pitch moments act on the aircraft about its CG:

1. Wing Pitching Moment ($M_{ac}$): Most cambered aerofoils exhibit a natural nose-down pitching moment about their aerodynamic center ($AC_{wing}$), which remains relatively constant regardless of angle of attack.
2. Wing Lift Moment: The lift vector generated by the wing acts through $AC_{wing}$. If the CG is located behind $AC_{wing}$, wing lift creates a nose-up pitching moment; if the CG is ahead of $AC_{wing}$, wing lift creates a nose-down moment.
3. Tailplane Moment: The horizontal tailplane (stabilizer) operates at an arm distance ($l_t$) behind the CG. On conventional aircraft, the tailplane generates downward lift (downforce) to counter the wing's nose-down pitching moment.

$\sum M_{CG} = M_{ac} + L_{wing} \cdot (x_{CG} - x_{AC}) - L_{tail} \cdot l_t = 0$

                     L_wing (Lift)
                       ^ 
                       |
      [Nose] ----------+----------[Tail]
                       |            |
                       v            v
                      W (CG)     L_tail (Downforce)

The Tailplane Downwash Effect

As the main wing generates lift, it creates a downward deflection of airflow behind the trailing edge known as downwash ($\epsilon$). The horizontal tailplane operates within this downwash field. When the aircraft's angle of attack ($\alpha$) increases, downwash increases proportionally:

$\alpha_{tail} = \alpha_{wing} - i_{tail} - \epsilon$

Because downwash reduces the effective angle of attack seen by the tailplane, it reduces the restoring tail downforce increment. High-tail or T-tail configurations elevate the tailplane out of the main wing downwash field during normal flight, enhancing tailplane efficiency, but present risks during deep stall conditions.

---

2. The Neutral Point & Static Margin

The Neutral Point (NP) is defined as the aerodynamic center of the complete aircraft (wing, fuselage, and tailplane combined). It represents the position along the longitudinal axis where the net pitching moment coefficient is independent of the angle of attack ($dC_m/d\alpha = 0$).

The relationship between the CG location ($x_{CG}$) and the Neutral Point ($x_{NP}$) dictates the degree of static longitudinal stability, quantified as the Static Margin (SM):

$\text{Static Margin} = \frac{x_{NP} - x_{CG}}{\text{MAC}}$

where $\text{MAC}$ is the Mean Aerodynamic Chord.

[Forward CG Limit] <---> [Aft CG Limit] <---> [Neutral Point]
       |                       |                    |
  Statically              Minimum Safe           dCm/dalpha = 0
 High Stability          Static Margin         (Neutral Stability)

• Positive Static Margin ($x_{CG} < x_{NP}$): The CG is located forward of the Neutral Point. The aircraft is statically stable ($dC_m/d\alpha < 0$). Any increase in $\alpha$ produces a nose-down restoring moment.
• Zero Static Margin ($x_{CG} = x_{NP}$): The CG lies exactly on the Neutral Point. The aircraft is neutrally stable ($dC_m/d\alpha = 0$). Stick forces per g drop to zero.
• Negative Static Margin ($x_{CG} > x_{NP}$): The CG is loaded aft of the Neutral Point. The aircraft is statically unstable ($dC_m/d\alpha > 0$). Any pitch disturbance causes continuous, self-amplifying pitch divergence.

3. Forward vs. Aft CG Limits: Operational Trade-offs

Aircraft flight manuals enforce strict Forward and Aft CG boundaries. Operating outside these certified limits degrades safety and handling characteristics.

The EASA Exam Trap: CG and Stall Speed

A common Part-66 exam trap involves the effect of CG on stall speed:

Why does a forward CG increase stall speed? To balance the forward CG, the tailplane must push down harder. The main wing must produce enough lift to support both the aircraft's actual weight AND this downward tail load. Since the wing is carrying a heavier total aerodynamic load, it reaches its critical angle of attack ($\alpha_{crit}$) at a higher indicated airspeed ($V_s$ increases).

---

4. Dynamic Longitudinal Stability Modes

Longitudinal dynamic stability manifests in two distinct oscillatory modes that differ drastically in frequency and amplitude:

Short-Period Pitch Oscillation

• Frequency: High frequency, short period ($T \approx 1 \text{ to } 3 \text{ seconds}$).
• Characteristics: Characterized by rapid pitching motion about the CG with rapid changes in angle of attack ($\alpha$). Airspeed remains virtually constant during the brief motion.
• Damping: Heavily damped by tailplane aerodynamic forces in a correctly designed aircraft. If un-damped, it can cause severe Pilot-Induced Oscillations (PIO) during precision maneuvering or landing flare.

Phugoid (Long-Period) Oscillation

• Frequency: Very low frequency, long period ($T \approx 20 \text{ to } 100 \text{ seconds}$).
• Characteristics: A slow, gradual trade between potential energy (altitude) and kinetic energy (airspeed). The aircraft pitches up, climbs, loses airspeed, pitches down, descends, gains airspeed, and repeats.
• Angle of Attack: The angle of attack remains nearly constant throughout the entire phugoid wave.
• Damping: Lightly damped by skin friction and parasite drag. Because the oscillation is so slow, pilots can effortlessly control or damp out a phugoid manually.