Lateral Directional Stability

Lateral-Directional Stability (flight-mechanics/stability-control/lateral-directional-stability)

Use when the task is lateral-directional stability analysis: the directional stability derivative from the vertical tail, the dihedral contribution to roll stability, and the roll, Dutch roll, and spiral mode characteristics.

Domain quick reference

Documented convention (stability axes): x forward, y out the right wing, z down. The sideslip angle beta is positive when the relative wind comes from the left, so the velocity vector points to the right of the plane of symmetry. The yawing moment coefficient C_n is positive when the nose turns toward the relative wind; the rolling moment coefficient C_l is positive when the right wing rolls down.

• Directional (yaw) stability: a positive sideslip must produce a restoring yawing moment, so the total derivative C_n_beta > 0. The vertical tail contributes C_n_beta_vt = eta_vt * V_v * a_vt * (1 + k_s), where V_v = (S_vt * l_vt) / (S * b) is the vertical tail volume coefficient (dimensionless), eta_vt is the fin efficiency (sidewash and dynamic pressure ratio, dimensionless, 0 < eta_vt <= 1), a_vt is the fin lift slope (1/rad), and k_s is the sidewash gradient (dimensionless, >= 0). The fuselage adds a usually negative term C_n_beta_fuselage. Texts that define beta with the opposite sign write the fin term with a leading minus; the magnitude is unchanged.
• Lateral (roll) stability: a positive sideslip must produce a restoring rolling moment, so the total derivative C_l_beta < 0. The first-order dihedral contribution is C_l_beta_gamma = -C_L * gamma, with C_L the wing lift coefficient and gamma the dihedral angle in radians; the windward panel gains lift and the leeward panel loses it for positive dihedral. Anhedral (negative gamma) reverses the effect and destabilizes the roll response.
• Roll mode: the simplified roll equation p_dot = L_p * p gives the roll subsidence time constant tau = -1 / L_p, with the roll damping derivative L_p = (q_bar * S * b^2 * C_lp) / (2 * V * I_xx) built from the roll damping coefficient C_lp (negative), the dynamic pressure q_bar, the wing area S, the span b, the speed V, and the roll inertia I_xx. Roll damping always stabilizes the roll mode.
• Dutch roll: the simplified yaw-sideslip model (states beta and yaw rate r) gives the natural frequency omega_n = sqrt(N_beta + (N_r * Y_beta - N_beta * Y_r) / V) and the damping ratio zeta = -(Y_beta / V + N_r) / (2 * omega_n), where N_beta is the yaw stiffness (1/s^2), N_r and Y_r are the rate derivatives (1/s), Y_beta is the side force derivative (m/s^2), and V is the speed (m/s). Dutch roll is an oscillatory yaw-sideslip motion with a lightly damped character on many aircraft.
• Spiral mode: the slow lateral root approximates lambda_s = (g / V) * (L_beta * N_r - L_r * N_beta) / (N_beta * L_p); with N_beta > 0 and L_p < 0 the spiral mode is convergent (stable) when L_beta * N_r - L_r * N_beta > 0, and divergent otherwise.
• FAR-25 and CS-25 require positive directional and lateral static stability and adequate lateral-directional oscillation damping for transport aeroplanes; the derivative and mode computations above are the standard methodology used to check those characteristics.

Workflow

1. Collect the vertical tail area and arm (S_vt, l_vt) and the wing reference area and span (S, b); compute the volume coefficient with vertical_tail_volume.
2. Compute the fin contribution with cn_beta_vertical_tail, add any fuselage term with cn_beta_total, and check the verdict with directionally_stable.
3. Take the wing lift coefficient and dihedral angle, compute the dihedral contribution with cl_beta_dihedral, and check the verdict with laterally_stable.
4. Compute the roll damping derivative with roll_damping_derivative and the roll mode time constant with roll_mode_time_constant.
5. Assemble the yaw and side force derivatives and compute the Dutch roll frequency and damping ratio with dutch_roll_frequency and dutch_roll_damping_ratio.
6. Classify the spiral mode with spiral_mode_stable and report the approximate root with spiral_eigenvalue.
7. Gate the lateral-directional stability assessment on the three verdicts (directional, lateral, spiral) and the Dutch roll damping ratio.

Pitfalls

• Reversing the derivative signs: directional stability needs C_n_beta > 0 and lateral (roll) stability needs C_l_beta < 0 in this convention; flipping either sign flips the verdict.
• Mixing sign conventions: some textbooks define beta with the opposite sign and write the fin term with a leading minus; state the convention before comparing numbers.
• Converting the dihedral angle twice: cl_beta_dihedral takes degrees and converts internally; passing radians already converted misstates the derivative by a factor of 57.3.
• Confusing the volume coefficient with the derivative: V_v is a geometric ratio; C_n_beta_vt also needs the fin efficiency and lift slope.
• Accepting a non-negative roll damping derivative: L_p must be negative; roll_mode_time_constant rejects non-negative values because the roll mode would not subside.
• Reading the spiral criterion backwards: with L_p < 0 the spiral mode converges when L_beta * N_r - L_r * N_beta > 0; the opposite sign diverges.

Behavior contract (gate 3)

The lateral-directional stability logic is exercised by the gate 3 contract test: scripts/test_lateral_directional.py against scripts/lateral_directional_logic.py (stdlib unittest, offline). Run:

python3 scripts/test_lateral_directional.py

Compliance

• Standards referenced, not reproduced: FAR-25 and CS-25 require positive directional and lateral stability and adequate lateral-directional oscillation damping for transport aeroplanes; the derivative and mode computations are common flight mechanics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.