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Static deflection of the generic wing under its polar design load

The FSI subpackage is blade oriented, but at Omega = 0 its beam machinery reduces to a classic clamped cantilever: no centrifugal tension, no propeller moment. This example applies it to the project's generic wing, the synthetic NACA 0012 of pyflightstream.qa.geometry (chord 1 m, span 8 m, AR 8) that flies the steady polar example, and prescribes structural inputs sized for reasonable numbers:

  1. flight point and elliptical lift from the measured polar slope;
  2. structural prescription: EI from a 3 percent tip deflection target, GJ from a half degree twist target, plausible mass;
  3. static solve: bending and torsion along the span;
  4. modal sanity: first bending and torsion frequencies.

Everything is synthetic and self-contained: no solver run, no real aircraft data. The load is prescribed, not iterated: this is the static response to a frozen load, not a divergence analysis.

import math

from pyflightstream.fsi import beam
from pyflightstream.fsi.config import BladeProperties, FsiConfig
from pyflightstream.qa.geometry import WingSpec

1. The wing and its design point

Same wing and flight point as the steady polar example: 30 m/s at sea level. The lift slope 4.83/rad was measured on FlightStream 26.120 (HND-016); at 6 degrees the wing carries about 2230 N. Each clamped half wing takes half of that, distributed elliptically, which is both the classic ideal and consistent with an AR 8 rectangular wing.

wing = WingSpec(naca="0012", chord_m=1.0, span_m=8.0)
HALF_SPAN_M = wing.span_m / 2.0
RHO_KG_M3 = 1.225
VELOCITY_M_S = 30.0
ALPHA_DEG = 6.0
CL_SLOPE_PER_RAD = 4.83

q_pa = 0.5 * RHO_KG_M3 * VELOCITY_M_S**2
cl = CL_SLOPE_PER_RAD * math.radians(ALPHA_DEG)
half_lift_n = 0.5 * q_pa * wing.area_m2 * cl
peak_n_per_m = 4.0 * half_lift_n / (math.pi * HALF_SPAN_M)
print(f"design point: q {q_pa:.0f} Pa, CL {cl:.3f}, half-wing lift {half_lift_n:.0f} N")

2. Structural prescription

Sized backwards from the response we want, the numbers a real preliminary design would iterate on:

  • EI = 6.0e4 N m^2: an elliptical load of this magnitude on a 4 m uniform cantilever gives a tip deflection near 12 cm, 3 percent of the half span, typical of a semi-rigid light aircraft wing.
  • GJ = 3.0e4 N m^2: with the elastic axis at 40 percent chord and the aerodynamic center at 25 percent, the nose-up torque arm of 0.15 c twists the tip about half a degree.
  • mu = 8 kg/m: 64 kg of wing structure for a 227 kg design lift, putting the first bending mode near 3 Hz; the sectional inertias I1 + I2 = 0.5 kg m (radius of gyration about a quarter chord) put first torsion near 15 Hz, an order above bending, as it should be.

The CG offsets stay zero: they only enter the centrifugal terms, inert at Omega = 0. blade_count 2 stands for the two half wings.

N_STATIONS = 21
EI_N_M2 = 6.0e4
GJ_N_M2 = 3.0e4
MU_KG_PER_M = 8.0
I1_KG_M = 0.45
I2_KG_M = 0.05
EA_ARM_M = 0.15 * wing.chord_m

n = N_STATIONS
radii = [i * HALF_SPAN_M / (n - 1) for i in range(n)]
config = FsiConfig(
    blade_count=2,
    omega_rad_per_s=0.0,
    blade=BladeProperties(
        station_radii_m=radii,
        chord_m=[wing.chord_m] * n,
        mass_per_length_kg_per_m=[MU_KG_PER_M] * n,
        inertia_major_kg_m=[I1_KG_M] * n,
        inertia_minor_kg_m=[I2_KG_M] * n,
        bending_stiffness_n_m2=[EI_N_M2] * n,
        torsion_stiffness_n_m2=[GJ_N_M2] * n,
        elastic_axis_offset_chordwise_m=[0.0] * n,
        elastic_axis_offset_normal_m=[0.0] * n,
        cg_offset_chordwise_m=[0.0] * n,
        cg_offset_normal_m=[0.0] * n,
        geometric_pitch_deg=[0.0] * n,
    ),
)

3. Static solve

The elliptical lift bends the wing up; the same lift acting at the aerodynamic center, ahead of the elastic axis, twists it nose up (the wash-in direction that drives divergence on swept-forward wings; here it stays half a degree).

lift_n_per_m = [peak_n_per_m * math.sqrt(max(0.0, 1.0 - (r / HALF_SPAN_M) ** 2)) for r in radii]
torsion_n_m_per_m = [EA_ARM_M * value for value in lift_n_per_m]

model = beam.build_beam_model(config)
beam.apply_station_loads(
    model, config, flap_load_n_per_m=lift_n_per_m, torsion_moment_n_m_per_m=torsion_n_m_per_m
)
beam.solve_static(model)
solution = beam.extract_solution(model, config)

tip_w = solution.flap_deflection_m[-1]
tip_theta_deg = math.degrees(solution.elastic_twist_rad[-1])
print(f"tip deflection: {tip_w * 100:.1f} cm ({tip_w / HALF_SPAN_M:.1%} of the half span)")
print(f"tip twist: {tip_theta_deg:+.2f} deg (nose up)")
print(f"{'y [m]':>6} {'w [cm]':>8} {'theta [deg]':>12}")
for r, w, theta in zip(
    solution.station_radii_m,
    solution.flap_deflection_m,
    solution.elastic_twist_rad,
    strict=True,
):
    if r in (radii[0], radii[5], radii[10], radii[15], radii[20]):
        print(f"{r:6.1f} {w * 100:8.2f} {math.degrees(theta):12.3f}")

4. Modal sanity

The same model gives the natural frequencies. For this prescription the closed forms of the uniform cantilever put first bending at 3.0 Hz and first torsion at 15.3 Hz; the model reproduces them within a percent (that is the WP3 benchmark).

modal = beam.modal_frequencies(model, config, n_modes=4)
for f, kind in zip(modal.frequencies_rad_per_s, modal.kinds, strict=True):
    print(f"{kind:8s} {f / (2.0 * math.pi):6.2f} Hz")

5. Spanwise plot (optional, extra [plot])

try:
    import matplotlib.pyplot as plt
except ModuleNotFoundError:
    print("matplotlib not installed (extra [plot]); skipping the figure")
else:
    fig, (ax_w, ax_t) = plt.subplots(2, 1, sharex=True, figsize=(7.0, 6.0))
    y = solution.station_radii_m
    ax_w.plot(y, [w * 100 for w in solution.flap_deflection_m], "o-")
    ax_w.set_ylabel("deflection w [cm]")
    ax_w.set_title("Generic wing at the polar design point, prescribed elliptical load")
    ax_t.plot(y, [math.degrees(t) for t in solution.elastic_twist_rad], "s-", color="tab:red")
    ax_t.set_ylabel("elastic twist [deg]")
    ax_t.set_xlabel("span position y [m]")
    fig.tight_layout()
    out = "wing_static_deflection.png"
    fig.savefig(out, dpi=150)
    print(f"figure written to {out}")