Divergence Speed

Static Aeroelastic Divergence (aerodynamics/aeroelasticity/divergence-speed)

Use when the task is the static aeroelastic divergence condition of a lifting surface: the divergence dynamic pressure from the torsional stiffness and the aerodynamic center to shear center offset, the divergence speed, and the divergence margin against the design dive speed.

Domain quick reference

• Static divergence: the lift acts at the aerodynamic center, ahead of the shear center (elastic axis). Its torsional moment about the shear center twists the surface nose-up, which raises the local angle of attack and the lift. The destabilizing moment grows with the dynamic pressure; at the divergence dynamic pressure it overcomes the torsional stiffness and the twist grows without bound.
• Divergence dynamic pressure: q_div = k_theta / (S * c * C_Lalpha * e), where k_theta is the torsional stiffness about the shear center axis (N m per rad; for a beam model the product G * K_theta of the shear modulus and the torsion constant), S the reference area (m^2), c the chord (m), C_Lalpha the lift curve slope per radian, and e the offset ratio, the aerodynamic-center-to-shear-center distance divided by the chord, positive when the aerodynamic center lies ahead of the shear center. Worked example, k_theta = 40000 N m per rad, S = 16 m^2 (2 m chord over an 8 m strip), c = 2 m, C_Lalpha = 5.0 per radian, e = 0.2 (aerodynamic center at 25 percent chord, shear center at 45 percent chord): q_div = 1250 Pa.
• Divergence speed: V_div = sqrt(2 * q_div / rho) with the flight density; the ISA sea level value rho = 1.225 kg/m^3 is the default. For the example: V_div = 45.18 m/s. Thinner air at altitude raises the speed for the same q_div.
• Divergence margin: m = V_div / V_design. Common design practice keeps the margin at or above 1.15, a rule of thumb (not a regulatory limit; the airworthiness aeroelastic requirements belong to the certification standards and are referenced, not reproduced, in the standards map). For the example with V_design = 40 m/s: m = 1.129, below the threshold, so the surface is flagged at divergence risk and needs more torsional stiffness.
• Stiffness for a target margin: k_theta_req = q_target * S * c * C_Lalpha * e with q_target = 0.5 * rho * (margin * V_design)^2. For V_design = 40 m/s and margin 1.15 (V_div target 46 m/s): q_target = 1296.05 Pa, k_theta_req = 41473.6 N m per rad, which restores the exact 1.15 margin.
• Offset sign: an aerodynamic center at or aft of the shear center (e <= 0) gives a restoring torsion; such a configuration has no divergence and the formulas are out of domain.

Workflow

1. Collect the inputs: torsional stiffness k_theta (N m per rad), reference area S, chord c, lift curve slope C_Lalpha per radian, and the offset ratio e = (x_AC - x_SC) / c with x_AC the aerodynamic center station and x_SC the shear center station.
2. Confirm e > 0: a zero or negative offset means no divergence mechanism (aerodynamic center at or aft of the shear center).
3. Compute the divergence dynamic pressure with divergence_dynamic_pressure(k_theta, area, chord, cl_alpha, offset_ratio).
4. Convert to the divergence speed with divergence_speed(q_div) at the sea level density, or pass the flight density for altitude cases.
5. Assess the margin with assess_divergence_margin(v_div, v_design), which returns (margin, acceptable) against the 1.15 threshold; a margin below 1.15 flags divergence risk.
6. When the surface is flagged, size the required torsional stiffness with stiffness_for_margin(v_design, area, chord, cl_alpha, offset_ratio) and re-run steps 3 to 5 to confirm the margin closes.
7. Record the divergence dynamic pressure, the divergence speed, and the margin verdict in the aeroelastic clearance assessment.

Pitfalls

• Routing control reversal there: aileron control reversal, where the aileron effectiveness vanishes at the reversal dynamic pressure, belongs to flight-mechanics/stability-control/aileron-reversal; this leaf is the pure torsion divergence condition, not the control effectiveness problem.
• Routing dynamic flutter there: coupled bending-torsion oscillation and flutter clearance testing belong to the flight-test-operations/flutter leaves; divergence is a static aeroelastic divergence, not an oscillation.
• Sign of the offset: with the aerodynamic center aft of the shear center the torsion is restoring and divergence does not occur; forcing the formula gives a negative q_div, which is meaningless.
• Wrong density: V_div depends on the flight density; report the value used (sea level 1.225 kg/m^3 is the default, thinner air gives a higher speed).
• Treating 1.15 as a regulation: the 1.15 margin is a common design practice rule of thumb; the airworthiness aeroelastic requirements live in the certification standards (FAR/CS part 25), which this leaf references but never reproduces.
• Mixing speeds and pressures: the margin compares speeds at the same density, which is equivalent to comparing the dynamic pressures only when the density is fixed; convert consistently.
• Units on the stiffness: k_theta must be N m per rad; a stiffness expressed per unit span combined with an area that already includes the span double-counts the geometry.
• Forgetting the strip convention: in a per-unit-span typical-section model set the area S equal to the chord c, so q_div = k_theta / (c^2 * C_Lalpha * e).

Behavior contract (gate 3)

The divergence logic is exercised by the gate 3 contract test: scripts/test_divergence_speed.py against scripts/divergence_speed_logic.py (stdlib unittest, offline). Run: python3 scripts/test_divergence_speed.py

Compliance

• The typical section divergence analysis is public-domain textbook methodology (Bisplinghoff, Ashley and Halfman, Aeroelasticity; Hodges and Pierce, Introduction to Structural Dynamics and Aeroelasticity); NACA TR 824 is referenced as the pack's public-domain anchor for the section lift data, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.