A blade's structural properties from its sections and a material¶

The FSI configuration carries, per radial station, the blade's running mass, its sectional mass moments of inertia, its bending and torsional stiffness and the offsets of its elastic axis. This example generates all of them from the blade's GEOMETRY and a MATERIAL instead of typing them, and keeps the record of where every number came from:

a synthetic blade: NACA 24xx sections thinning from 15 to 9 percent, linear chord and linear twist, written to a section file;
the section properties of one station, and the torsion constant converging as its grid is refined;
the whole blade from the file and titanium Ti-6Al-4V, with its provenance;
the configuration round trip, and the natural frequencies of the generated blade (needs the [fsi] extra, as every FSI example).

The sections are SOLID and homogeneous, and the elastic axis is taken at each section's centroid; both are stated hypotheses the generated configuration records. Everything is synthetic: no real blade enters the repository.

import csv
import math
import tempfile
from pathlib import Path

from pyflightstream.fsi import beam
from pyflightstream.fsi.config import FsiConfig, config_sha256, dump_config, load_config
from pyflightstream.fsi.materials import material
from pyflightstream.fsi.sections import (
    airfoil_section_contour,
    blade_properties_from_sections,
    solid_section_properties,
)
from pyflightstream.qa.geometry import naca4_contour

1. The synthetic blade and its section file¶

Seven stations from 0.15 m to 0.75 m. The chord tapers linearly from 80 mm to 40 mm, the geometric pitch from 35 to 11 degrees, and the thickness from 15 to 9 percent of chord (NACA 2415 at the root to NACA 2409 at the tip). airfoil_section_contour places each unit airfoil in the section frame of the FSI configuration: chordwise toward the LEADING edge, normal toward the suction side, origin on the pitch axis at the quarter chord.

The contours are written to a file and read back, the way a real geometry arrives; the file's sha256 goes into the provenance.

N_STATIONS = 7
ROOT_M, TIP_M = 0.15, 0.75
radii = [ROOT_M + i * (TIP_M - ROOT_M) / (N_STATIONS - 1) for i in range(N_STATIONS)]
chords = [0.080 + (0.040 - 0.080) * i / (N_STATIONS - 1) for i in range(N_STATIONS)]
pitches = [35.0 + (11.0 - 35.0) * i / (N_STATIONS - 1) for i in range(N_STATIONS)]
designations = [f"24{15 - i:02d}" for i in range(N_STATIONS)]

scratch = tempfile.TemporaryDirectory(prefix="pyfs_solid_blade_")
workdir = Path(scratch.name)
section_file = workdir / "blade_sections.csv"
with section_file.open("w", newline="", encoding="utf-8") as handle:
    writer = csv.writer(handle)
    writer.writerow(["station", "chordwise_m", "normal_m"])
    for index, (designation, chord) in enumerate(zip(designations, chords, strict=True)):
        contour = airfoil_section_contour(naca4_contour(designation, 120), chord, 0.25)
        for chordwise, normal in contour:
            writer.writerow([index, f"{chordwise:.9e}", f"{normal:.9e}"])

sections: list[list[tuple[float, float]]] = [[] for _ in range(N_STATIONS)]
with section_file.open(newline="", encoding="utf-8") as handle:
    for row in csv.DictReader(handle):
        sections[int(row["station"])].append((float(row["chordwise_m"]), float(row["normal_m"])))
print(f"{section_file.name}: {N_STATIONS} sections, {len(sections[0])} points each")

The area, the centroid and the second moments of a polygon are exact closed forms. The torsion constant J is not: it is solved from the Prandtl stress function on a grid over the section, and the grid is part of the answer. Doubling the cells across the thickness divides the error by about four, so the coarser grids show how far the default one (64 cells across) is from converged: a second-order scheme leaves at 64 cells about a third of the change from 32 to 64. The thin-section formula (1/3) integral t^3 ds is printed beside it as a cross-check only.

root = solid_section_properties(sections[0])
moments = root.moments
print(f"root section NACA {designations[0]}, chord {chords[0] * 1000:.0f} mm")
print(f"  area            {moments.area_m2 * 1e6:10.3f} mm^2")
print(
    f"  centroid        {moments.centroid_chordwise_m * 1000:+10.3f} mm chordwise, "
    f"{moments.centroid_normal_m * 1000:+.3f} mm normal (from the pitch axis)"
)
print(f"  I flap          {moments.second_moment_flap_m4 * 1e12:10.1f} mm^4")
print(f"  I chord         {moments.second_moment_chord_m4 * 1e12:10.1f} mm^4")
print(f"  principal angle {math.degrees(moments.principal_angle_rad):+10.3f} deg")
for cells in (16, 32, 64):
    torsion = solid_section_properties(sections[0], grid_cells=cells).torsion
    j_mm4 = torsion.torsion_constant_m4 * 1e12
    print(f"  J on {torsion.grid_cells[0]:4d} x {cells:3d} cells: {j_mm4:9.2f} mm^4")
print(f"  thin-section estimate:    {root.thin_section_torsion_m4 * 1e12:9.2f} mm^4 (cross-check)")

3. The whole blade, from the file and a material¶

blade_properties_from_sections multiplies every section by the material: EI = E I about the chordwise axis, GJ = G J, running mass rho A, and the mass moments of inertia rho times the principal second moments. The material is an entry of the versioned database in pyflightstream.fsi.materials, each carrying the data set its four numbers were read from.

titanium = material("ti-6al-4v-grade5-annealed")
print(
    f"{titanium.name}: rho {titanium.density_kg_per_m3:.0f} kg/m^3, "
    f"E {titanium.youngs_modulus_pa / 1e9:.1f} GPa, G {titanium.shear_modulus_pa / 1e9:.1f} GPa, "
    f"nu {titanium.poisson_ratio}"
)
print(f"  source: {titanium.source.document}")

blade = blade_properties_from_sections(
    radii,
    sections,
    chords,
    pitches,
    titanium,
    geometry_source="synthetic NACA 24xx blade of examples/fsi_solid_blade_properties.py",
    geometry_file=section_file,
)
print(
    f"{'r [m]':>6} {'mu [kg/m]':>10} {'EI [N m^2]':>11} {'GJ [N m^2]':>11} "
    f"{'I1 [kg m]':>10} {'I2 [kg m]':>10} {'e_c [mm]':>9} {'thin/J':>7}"
)
for i in range(N_STATIONS):
    print(
        f"{blade.station_radii_m[i]:6.2f} {blade.mass_per_length_kg_per_m[i]:10.4f} "
        f"{blade.bending_stiffness_n_m2[i]:11.2f} {blade.torsion_stiffness_n_m2[i]:11.2f} "
        f"{blade.inertia_major_kg_m[i]:10.3e} {blade.inertia_minor_kg_m[i]:10.3e} "
        f"{blade.elastic_axis_offset_chordwise_m[i] * 1000:9.2f} "
        f"{blade.provenance.thin_section_torsion_ratio[i]:7.3f}"
    )

The provenance travels inside the configuration: the material and its source, the geometry file's name and sha256, the solid-section and centroid hypotheses, and the torsion method with its grid per station.

provenance = blade.provenance
entry = provenance.material
print(f"material: {entry.name} (database version {entry.database_version})")
print(f"geometry: {provenance.geometry_file_name}, sha256 {provenance.geometry_sha256[:16]}...")
print(f"elastic axis: {provenance.elastic_axis[:80]}...")
print(f"torsion grids: {provenance.torsion_grid_cells}")

4. Into the configuration, and what the blade does¶

The generated distributions go where typed ones go. The configuration round trips through config.json with its provenance, and its sha256 covers it. A configuration written by hand, with no provenance, keeps the digest it always had.

config = FsiConfig(blade_count=2, omega_rad_per_s=0.0, blade=blade)
path = workdir / "config.json"
dump_config(config, path)
reloaded = load_config(path)
assert reloaded == config
print(f"config.json round trip: identical, sha256 {config_sha256(reloaded)[:16]}...")

The beam of pyflightstream.fsi.beam takes the stiffnesses as they are, through its unit-moduli material, so the modulus is applied once. At rest, the first flap and torsion frequencies of the titanium blade:

model = beam.build_beam_model(config)
beam.solve_static(model)
modal = beam.modal_frequencies(model, config, n_modes=2 * N_STATIONS)
for family in ("flap", "torsion"):
    first = next(
        frequency
        for frequency, kind in zip(modal.frequencies_rad_per_s, modal.kinds, strict=True)
        if kind == family
    )
    print(f"first {family:8s} {first / (2.0 * math.pi):8.1f} Hz")

scratch.cleanup()

A blade's structural properties from its sections and a material¶

1. The synthetic blade and its section file¶

2. One section, and its torsion constant under refinement¶

3. The whole blade, from the file and a material¶

4. Into the configuration, and what the blade does¶