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Flight mechanics

solid pillar 1 · Physical modeling and domain theory

In one minute

  • Performance follows from the drag polar plus thrust: range, endurance, climb rate and turn are all algebra once you have \(C_D(C_L)\).
  • Static margin — distance from CG to neutral point, in chords — is the single number governing longitudinal stability. Positive is stable, and it moves as fuel burns.
  • Linearised about trim, the longitudinal dynamics split into a fast short period and a slow phugoid; the lateral into roll subsidence, Dutch roll and spiral.
  • Stability and controllability trade against each other. More static margin costs trim drag and control authority.
  • Breguet range assumes constant \(L/D\) and \(\mathrm{SFC}\) — a cruise-climb idealisation, not a flight plan.

Key results

Steady level flight and the polar:

\[ L = W = \tfrac{1}{2}\rho V^2 S C_L, \qquad D = \tfrac{1}{2}\rho V^2 S\left(C_{D_0} + kC_L^2\right), \qquad k = \frac{1}{\pi e \mathrm{AR}} \]

Breguet range, jet aircraft:

\[ R = \frac{V}{c_t}\,\frac{L}{D}\,\ln\frac{W_i}{W_f} \qquad\text{(propeller: } R = \frac{\eta_p}{c_p}\frac{L}{D}\ln\frac{W_i}{W_f}\text{)} \]

Maximum range for a jet occurs at \(\max(V\,L/D)\), i.e. \(\max(C_L^{1/2}/C_D)\) — faster than best \(L/D\). Maximum endurance is at \(\max(L/D)\).

Climb and energy:

\[ \text{RoC} = \frac{P_{\text{avail}} - P_{\text{req}}}{W}, \qquad h_e = h + \frac{V^2}{2g} \]

Specific excess power \(P_s = V(T-D)/W\) contours are the standard way to see what an aircraft can actually do across its envelope.

Turning:

\[ n = \frac{1}{\cos\phi}, \qquad R_{\text{turn}} = \frac{V^2}{g\sqrt{n^2-1}}, \qquad \dot{\psi} = \frac{g\sqrt{n^2-1}}{V} \]

Longitudinal static stability. With \(h\) the CG position and \(h_n\) the neutral point, both in chords from the leading edge:

\[ \frac{dC_m}{dC_L} = h - h_n, \qquad \text{static margin} = h_n - h > 0 \ \text{for stability} \]

Dynamic modes:

Mode Character Typical Driven by
Short period fast, well damped 1–5 s pitch stiffness \(M_\alpha\), damping \(M_q\)
Phugoid slow, lightly damped 30–100 s exchange of altitude and speed
Roll subsidence fast, non-oscillatory <1 s roll damping \(L_p\)
Dutch roll moderate, oscillatory 3–10 s yaw stiffness vs dihedral effect
Spiral very slow, often divergent 20–100+ s balance of \(L_\beta\) and \(N_\beta\)

A mildly divergent spiral is acceptable and common — it is slow enough for the pilot or autopilot to correct.

Mental model

Performance is a bookkeeping exercise on energy: thrust adds it, drag removes it, and the difference is available for climbing or accelerating. Once that framing is in place, the specific-excess-power chart replaces a dozen separate formulae.

Stability is about restoring moments. Displace the aircraft in pitch, and a stable configuration generates a moment that reduces the displacement. The neutral point is where that restoring moment vanishes; putting the CG behind it makes the aircraft divergent. Everything about static margin follows from that one picture.

Numerics / practice

  • Track CG travel across the whole loading envelope. Fuel burn, payload and stores all move it. The binding case is usually aft CG at low fuel.
  • Linearised models are valid near trim only. Large-amplitude manoeuvres, stall and spin need the full nonlinear equations.
  • Non-dimensionalise consistently. Sign and reference-length conventions for stability derivatives differ between textbooks and between tools; mixing them is a classic source of sign errors.
  • Check mode characteristics against handling-qualities criteria (e.g. Cooper–Harper, MIL-STD-1797 bands) rather than judging damping ratios in isolation.
Failure modes

Static margin sign and reference confusion. Different sources define it from the leading edge, the quarter chord, or the CG, and some flip the sign. A sign error here inverts the stability conclusion entirely. Always state the reference and sanity-check against a known-stable configuration.

CG checked at one loading only. An aircraft stable at takeoff can be unstable at aft-CG, low-fuel conditions. This is a loading-envelope question, not a single-point one.

Breguet applied to a real mission. It assumes constant \(L/D\), constant \(\mathrm{SFC}\) and cruise-climb. Real missions have step climbs, reserves, and off-design cruise. Symptom: range overpredicted by 10–20% against a segment-by-segment integration.

Best range and best endurance confused. For a jet, best endurance is at \(\max(L/D)\) and best range at \(\max(C_L^{1/2}/C_D)\) — a noticeably higher speed. Cruising at best \(L/D\) costs range.

Stability equated with good handling. A very stable aircraft is sluggish and needs large control surfaces and more trim drag. Modern fighters are deliberately relaxed-stability or unstable, closed-loop stabilised. Stability is a design variable, not a virtue.

Linearised dynamics used past their range. Small-perturbation derivatives evaluated at cruise say nothing about post-stall behaviour. Departure and spin are nonlinear phenomena.

Ignoring thrust-line offset. A thrust line above or below the CG produces a pitching moment that varies with power setting, coupling throttle to trim. Easy to omit and very visible in flight.

Worked example

Best-range and best-endurance speeds are not the same point:

import numpy as np

CD0, e, AR = 0.022, 0.80, 9.0
k  = 1.0 / (np.pi * e * AR)

CL_endurance = np.sqrt(CD0 / k)             # max L/D
CL_range     = np.sqrt(CD0 / (3 * k))       # max CL^(1/2)/CD  (jet)

for name, CL in (("endurance (max L/D)", CL_endurance), ("range (jet)", CL_range)):
    CD = CD0 + k * CL**2
    print(f"{name:22s} CL={CL:.3f}  L/D={CL/CD:5.2f}  V/V_ref={1/np.sqrt(CL):.3f}")

endurance (max L/D)    CL=0.705  L/D=16.03  V/V_ref=1.191
range (jet)            CL=0.407  L/D=13.88  V/V_ref=1.567

Best range flies about 32% faster than best endurance, and accepts a 13% lower \(L/D\) to do it — the extra speed more than pays for the lost efficiency.

Connections

Sources

  • Anderson, Introduction to Flight — performance fundamentals.
  • Etkin & Reid, Dynamics of Flight: Stability and Control.
  • Archive: Aerospace/AERSP420 and related; see the course archive.