Continuous Turbulence Gust Loads

Continuous-Turbulence Gust Loads (structures/loads/continuous-turbulence-gust-loads)

Use when the continuous-turbulence design condition of the transport gust and turbulence loads rules must be evaluated: the von Karman or Dryden power spectral density of the vertical gust velocity, the rigid-aircraft heave gust-response transfer function, the response PSD, the rms-response ratio A, the design limit incremental load factor and the equivalent discrete-gust velocity handed to the discrete 1-cosine envelope method. This is the spectral, continuous- turbulence counterpart of the discrete gust and maneuver case; pair it with structures/loads/gust-maneuver-loads for the discrete-gust and V-n envelope side of the same certification loads set.

Domain quick reference

• Von Karman one-sided vertical-gust velocity PSD (spatial frequency, Hoblit convention): Phi_vK(Omega) = sigma_w^2 (L/pi) (1 + (8/3)(1.339 L Omega)^2) / (1 + (1.339 L Omega)^2)^(11/6).
• Dryden one-sided vertical-gust velocity PSD, same convention: Phi_D(Omega) = sigma_w^2 (L/pi) (1 + 3 (L Omega)^2) / (1 + (L Omega)^2)^2. Both share the DC ordinate Phi(0) = sigma_w^2 L/pi and integrate (Parseval) to sigma_w^2; von Karman decays as Omega^(-5/3) rather than Dryden's Omega^(-2), so it carries more high-frequency energy.
• Rigid-aircraft heave gust-response transfer, quasi-steady aerodynamics: |H(omega)|^2 = h_inf^2 omega^2 / (omega^2 + K^2), with h_inf = rho V S a_lift / (2 W) the frozen-gust gain and K = rho V S a_lift / (2 m) the heave-follow rate (vehicle inertia and lift-curve slope, not an oscillator transmissibility).
• Rms-response ratio A: A^2 = (1/pi) integral_0^inf |H(V Omega)|^2 Phi_hat(Omega) dOmega, evaluated by a fixed 16384-panel Simpson quadrature in ln(Omega) with analytic tails (Dryden also has an exact closed form, A_D^2 = h_inf^2 (3 y_c + 2) / (2 (1 + y_c)^2)).
• Design rule (linear model, continuous-turbulence design condition, named by name only): delta_n_limit = U_sigma A, n_limit = 1 + delta_n_limit, with U_sigma the limit turbulence intensity (true airspeed); the limit increment equals 2.5 times the rms response to the 0.4 U_sigma field.
• Equivalent discrete-gust velocity (hand-off): the algebraic inversion of the discrete 1-cosine formula, U_de_eq = 2 (W/S) delta_n / (rho V_e a_lift K_g), evaluated only to report the value the discrete-gust envelope method takes as input.

Workflow

1. Build the turbulence PSD ordinates with von_karman_psd or dryden_psd (or spectrum_psd_ordinates for a grid of spatial frequencies) from the scale of turbulence L and sigma_w.
2. Form the rigid-aircraft gust-response parameters with rigid_aircraft_response_params (h_inf, K, mass) and, if needed, the transfer magnitude squared with gust_response_transfer_squared.
3. Integrate the response PSD to the rms-response ratio A with rms_response_ratio (spectrum choice "von-karman" or "dryden"), and inspect the response PSD ordinates with response_psd_ordinates.
4. Scale to the rms incremental load factor of a given rms field with rms_load_factor_response(sigma_w, a_ratio).
5. Scale to the design limit incremental load factor with design_incremental_load_factor(a_ratio, u_sigma) and read off n_limit = 1 + delta_n_limit, the continuous-turbulence design criterion result.
6. Report the hand-off value with equivalent_discrete_gust_velocity (using gust_alleviation_factor internally) for the discrete-gust envelope analysis to consume.
7. Run the one-shot continuous_turbulence_report for the full set of ordinates, ratios and load factors in a single call.

Worked example

Typical transport at the VB-class sea-level condition (rho = 1.225 kg/m^3, V = 154.33 m/s, W/S = 4800 Pa, S = 120 m^2, cbar = 3.81 m, a_lift = 5.7/rad, L = 762 m, U_sigma = 27.432 m/s TAS, no alleviation):

• Rigid-aircraft response: h_inf = 0.11225096093750002 per (m/s), K = 1.1008058860777343 /s, reduced filter corner y_c = K L / V = 5.435197856484374.
• Von Karman: A_vK = 0.06000490538363879 per (m/s), response factor A/h_inf = 0.5345602824464799, delta_n_limit = 1.6460545644839792, n_limit = 2.646054564483979, U_de_eq = 19.063820591364568 m/s EAS (about 62.5 fps), with mu_g = 36.79718848898816 and k_g = 0.7692087531874008 evaluated inside the inversion.
• Dryden: A_D = 0.0527721846394613 per (m/s), n_limit = 2.4476465690297022, U_de_eq = 16.765953612441937 m/s EAS (about 55.0 fps). The von Karman ratio sits 13.705554912291994 percent above the Dryden ratio because its spectrum decays more slowly at high reduced frequency; both stay below the frozen bound h_inf.
• The continuous-turbulence limit condition (n = 2.65 von Karman) sits in the same design band as the discrete VB gust of the sibling method (2.4-2.7 class); the equivalent discrete-gust velocities (about 62.5 and 55.0 fps EAS) sit just below the 66 fps VB design gust, so the discrete condition marginally drives at this point and the continuous condition is the second envelope input.

Verification

Deterministic checks (see the contract test): the Parseval closure of the normalized von Karman spectrum against its analytic Beta-function value, the exact Dryden closure and DC-ordinate identity, the Dryden closed-form response integral against the quadrature, the response factor small-c and large-c limits, the von-Karman-above-Dryden ordering with both bounded by h_inf, linear scaling of the rms and limit load factors, grid convergence of the fixed Simpson quadrature under panel doubling, determinism of repeated report calls, and ValueError rejection of non-physical inputs (non-positive or boolean rho, V, W, S, a_lift, g, L, cbar, U_sigma, sigma_w, delta_n; negative Omega; a negative or invalid A; an unknown spectrum name; an empty Omega grid).

Pitfalls

• Treating the response transfer as an SDOF oscillator transmissibility: it is the rigid-AIRCRAFT heave response built from vehicle inertia and lift-curve slope, not a test-input acceleration PSD response (that machinery belongs to structures/loads/random-vibration-analysis, confined to SDOF base-excitation equipment screening).
• Claiming the discrete 1-cosine load factor, the gust alleviation factor, or the V-n envelope as a product of this leaf: they appear here only inside the algebraic inversion that reports the equivalent discrete-gust velocity; the discrete method and its envelope belong to gust-maneuver-loads.
• Using the published rounded von Karman constant 1.339 and expecting the Parseval closure to equal 1.0 exactly: the closure ratio is 0.9999890060233615 for the rounded constant, exactly 1.0 only for the unrounded VK_A_EXACT; assert within 1e-3 of unity, never exact equality.
• Mixing the spatial frequency Omega (rad/m, spectrum argument) with the encounter circular frequency omega = V Omega (rad/s, transfer function argument): rms_response_ratio and response_psd_ordinates handle this conversion internally, but a direct call to gust_response_transfer_squared needs omega in rad/s.
• Forgetting that A is independent of the turbulence intensity: A depends only on the aircraft and spectrum parameters; sigma_w or U_sigma scale the response and limit load factor linearly after A is computed.
• Feeding an unknown spectrum name or an empty Omega grid: only "von-karman" and "dryden" are accepted, and the ordinate functions reject an empty grid rather than returning an empty list silently.

Behavior contract (gate 3)

The behavior contract is scripts/test_continuous_turbulence_gust_loads.py against scripts/continuous_turbulence_gust_loads_logic.py (stdlib unittest, offline, deterministic). Run:

python3 scripts/test_continuous_turbulence_gust_loads.py

It asserts: the worked-example rms-response ratios, h_inf, K, y_c and design limit load factors for both spectra within 1e-6 relative; the equivalent discrete-gust velocity hand-off and the one-shot report key set; the von Karman turbulence PSD and response PSD ordinates at the default Omega grid within 1e-6 relative; the shared DC ordinate of both spectra; the Parseval closures (von Karman within 1e-3 of unity for the rounded constant, Dryden within 1e-6 of exact unity) checked against their analytic values, never by exact-float equality; the Dryden closed-form identity and its response-factor limits; the von-Karman-above-Dryden ordering and the frozen bound; linear scaling of the rms and limit load factors including the 2.5x quotient at the 0.4 U_sigma field; grid convergence under panel doubling and determinism of the one-shot report; and ValueError rejection of every non-physical input with the real message prefixes.

Related leaves

• structures/loads/gust-maneuver-loads: owns the discrete 1-cosine gust load factor, the gust alleviation factor and the V-n envelope; this leaf's equivalent discrete-gust velocity is the reported hand-off value that envelope analysis takes as input.
• structures/loads/random-vibration-analysis: the SDOF base-excitation equipment-qualification response (transmissibility, Miles equation, g-rms screening), disjoint from this leaf's rigid-aircraft continuous-turbulence gust response.
• aerodynamics/aeroelasticity/aeroelastic-gust-response: the flexible typical-section time-domain response to a discrete gust (Wagner/Kussner indicial states), distinct from this leaf's rigid closed-form spectral method.
• aerodynamics/aeroelasticity/sears-function-gust-lift: the single-frequency Sears function gust gain and phase of a rigid airfoil section; this leaf is the spectral, continuous-turbulence implementer that leaf's Related leaves section names as the PSD machinery home.

Compliance

• Standards: far-25 and cs-25, both reference-only (STANDARDS-REF), gated: false. The continuous-turbulence design condition of 14 CFR 25.341(b) and CS 25.341(b) is referenced by name only, never reproduced; FAA AC 25.341-1 is named as method context only.
• The von Karman and Dryden spectrum relations, the heave transfer function and the design rule are standard published methodology (Hoblit, Gust Loads on Aircraft); no verbatim regulation text, no proprietary tables, no test data.