Vibration fatigue design of an engine-mounted L-bracket (CalculiX FEA)

Static, modal, random-vibration and fatigue analysis of a parametric steel L-bracket that carries a 2 kg component, followed by a 400-design study to find the lightest bracket that meets stiffness and fatigue requirements. Everything is scripted in Python and runs with open-source tools: gmsh (OpenCASCADE geometry and quadratic tet meshing) and CalculiX 2.23 (an Abaqus-style solver; the input decks use *STATIC, *FREQUENCY, *STEADY STATE DYNAMICS, *RIGID BODY, *MASS).

The selected design is then released as a parametric CAD model with an assembly, ASME Y14.5 drawings, a CAD -> FEA round trip, a tolerance study and bolted-joint checks (FreeCAD, scripted in Python; see CAD design package).

The dataset produced by the design study is also used to train a graph neural network surrogate in the companion repository gnn-fea-surrogate.

Problem

Geometry	L-bracket, 100 x 70 mm base, 100 mm upright, two M8 bolt holes, M10 mounting hole
Design variables	base thickness t_b 4-12 mm, upright thickness t_w 4-10 mm, inner fillet r_f 2-10 mm, two optional gussets (leg L_g 20-45 mm, thickness t_g 3-8 mm), optional lightening hole d_h 8-20 mm
Material	AISI 1020 hot-rolled steel, E = 207 GPa, S_y = 210 MPa, S_ut = 380 MPa
Component	2 kg point mass, 30 mm in front of the upright, rigidly tied to the mounting hole
Supports	bolt hole surfaces fixed
Static load	5 g fore-aft + 5 g lateral + 10 g vertical on bracket and component
Random vibration	base excitation, flat 0.02 g²/Hz from 20 to 1000 Hz (4.4 g RMS), each axis separately, 2 % modal damping
Requirements	peak von Mises <= S_y / 1.5 = 140 MPa; f1 >= 100 Hz; fatigue life >= 1000 h (worst axis)

These loads and requirements are assumptions for a representative engine-mounted bracket, not a real product specification. Fatigue life scales with the input PSD level to the power -m/2 = -3.8, so the absolute hours matter less than the comparison between designs.

Method

1. Geometry and mesh (bracketfea/geometry.py): parametric solid built with OpenCASCADE booleans; the gusset attachment edges get 2 mm fillets so there are no re-entrant sharp edges. Quadratic tetrahedra (C3D10), refined at fillets and holes, about two elements through the thinnest plate.
2. CalculiX deck (bracketfea/ccx.py): static step, then 10 modes (*FREQUENCY). The component is a *MASS element tied to the mounting hole with *RIGID BODY.
3. Random vibration (bracketfea/vibration.py): base-excitation response by modal superposition with the participation factors and modal stresses from CalculiX. The stress PSD at every surface node is the equivalent von Mises PSD of Preumont and Piéfort (1994).
4. Fatigue (bracketfea/fatigue.py): S-N curve from the Shigley method (surface and size factors, extended below the endurance limit for random loading), Dirlik's rainflow-range distribution and Miner's rule. Narrow-band and Steinberg three-band estimates are reported for comparison.
5. Stress evaluation region: stresses within 2 bolt diameters of the bolt axes and 1.5 hole diameters of the mounting hole are not used. The fixed supports and the rigid coupling are idealisations; the peak stress there grows without bound as the mesh is refined (see below). The bolted joint itself would be checked separately.

Verification

Check	Reference	Result
Cantilever tip deflection (V1)	Timoshenko beam	-0.6 %
Cantilever mid-span bending stress (V1)	beam theory	+0.01 %
Cantilever bending frequencies f1, f2, lateral f1 (V1)	Euler-Bernoulli	+0.2 %, -0.9 %, -0.5 %
Cantilever + tip mass on a rigid end face (V1)	exact frequency equation	+0.2 %
Plate with hole, d/W = 0.2 (V2)	Heywood formula, Kt = 2.519	Kt = 2.530 (+0.4 %)
Modal superposition code vs CalculiX *STEADY STATE DYNAMICS (V3)	CalculiX	< 0.001 % of peak
10-mode quasi-static sum vs direct static 1 g (V3)	CalculiX static	0.2-0.4 % at hot spot
Dirlik vs rainflow counting of synthesized signals (V3)	time domain	-3 % (narrow band), +22 % (bimodal), -5 % (bracket hot spot)
Base plate bending stress of the bracket (hand calc)	beam theory at x = 40, 50 mm	-0.8 %

Mesh convergence (studies/mesh_convergence.py). On the design-study mesh the static peak stress and f1 are within 0.5 % and 0.1 % of the finest mesh. The RMS stress at the hot spot is 3-4 % low, which makes the fatigue life 20-35 % too long because life ~ stress^-7.65. The design study therefore uses a life margin of 1.5, and the selected design is re-checked on a finer mesh. The peak stress at the fixed bolt hole does not converge (95 -> 139 MPa), which is the expected support singularity.

Results

Design study (400 designs, Latin hypercube)

About 8.5 CPU-hours of CalculiX runs (25k-183k nodes per design). Every design passed the static check (largest peak 122 MPa vs 140 MPa allowed), so the design is driven by vibration:

Requirement	Designs that pass
static, von Mises <= 140 MPa	400 / 400
f1 >= 100 Hz	145 / 400
fatigue life >= 1000 h	26 / 400
all three	17 / 400 (16 with the 1.5 life margin)

• Base plate thickness controls both stiffness and fatigue. Spearman correlation with f1 is 0.93 and with log life 0.66; the upright thickness follows (0.36 and 0.32).
• Gussets raised f1 but lowered fatigue life. None of the 272 designs with gussets met the fatigue requirement (correlation of gusset size with log life: -0.43). In 154 of them the peak static stress sits at the gusset toe on the base plate, which is a stress raiser on a plate that bends.

Selected design

	baseline	thinnest plain bracket that passes	lightest feasible DOE design
t_b / t_w / r_f (mm)	6 / 6 / 5	9 / 9 / 5	9.9 / 7.1 / 8.6, 9.7 mm lightening hole
mass (kg)	0.632	0.932	0.884
f1 (Hz)	57	104	108
fatigue life, worst axis (h)	163	1,659	3,262 (2,894 on a finer mesh with 2.5x the nodes)
feasible	no	yes	yes

The selected design is 5 % lighter than simply thickening both plates. Its fatigue hot spot is at the ends of the inner fillet under lateral (y) excitation:

Two follow-up checks

Gusset detail. With the rest of the geometry fixed, increasing the fillet radius on the gusset edges from 2 to 6 mm raised the fatigue life 14x (21.7 -> 312 h) and 20x (2.9 -> 59.5 h) for two gusseted designs, for +2 % mass. Both stay below 1000 h, because the toe still ends on the bending base plate. A gusset that is meant to help fatigue would need to run closer to the bolts.

Size of the excluded bolt zone. Changing the excluded radius from 1.5 to 3 bolt diameters changes the static peak of the plain t = 9 bracket from 21.2 to 16.5 MPa, but does not change the fatigue life of either the selected or the plain design: the fatigue hot spot is at the fillet, away from the bolts.

CAD design package (FreeCAD)

The lightest feasible design is released as part BRK-001 with an assembly and drawings. Everything in cad/ is generated by Python scripts that drive FreeCAD 1.1 (PartDesign, Assembly, TechDraw) and the FE code above.

Step	What it does	Result
Release dimensions (cad/released_design.json)	round the optimizer values to dimensions that are easy to make: t_b 9.902 -> 10, t_w 7.071 -> 7, R 8.598 -> 8 (16 mm end mill), lightening hole 9.738 -> 10 (standard drill)	fine-mesh life 2,894 -> 2,312 h (-20 %), f1 107.6 -> 107.8 Hz; still meets all requirements
Parametric part (cad/build_part.py)	PartDesign feature tree: fully constrained sketch -> Pad -> ISO 273 clearance Hole features -> Pocket; every dimension driven by a spreadsheet	cad/out/bracket.FCStd, .step; mass 0.884 kg
Same solid as the FE model (cad/check_geometry.py)	build 4 designs in FreeCAD and in gmsh, compare volume and bounding box, re-import the STEP	volume difference 2e-15 (relative), i.e. identical
CAD -> FEA round trip (cad/fea_from_step.py)	mesh the FreeCAD STEP file, find holes and fillet by surface type, run static/modal/random/fatigue	vs. native geometry: f1 +0.001 %, peak static stress -0.4 %, worst-axis life +2.6 %
Assembly (cad/build_assembly.py)	mounting plate, bracket, 2 kg component envelope, ISO 4017 screws and ISO 7089 washers in separate part files linked into one assembly; pairwise interference check; BOM	no interference except the expected thread engagement; 0.5 mm radial clearance in all clearance holes
Drawings (cad/make_drawings.py)	TechDraw sheets (ANSI C): part drawing with datums A/B/C, position tolerances at MMC, flatness, perpendicularity, basic dimensions, notes; assembly drawing with balloons and BOM	BRK-001_part_drawing.pdf, ASM-001_assembly_drawing.pdf
Tolerance study (cad/tolerance_study.py)	FEA central differences for t_b, t_w, r_f, d_h; worst case and Monte Carlo (tolerance = 3 sigma); FEA check of the worst-case corners	worst-case f1 103.9 Hz (linear model 103.9); worst-case life 1,583 h (linear 1,617)
Bolted joints (cad/joint_check.py)	preload from torque (T = K F d), tipping about the contact edges, slip, thread engagement in the aluminium housing	M10: safety factor 40 on separation, 12 on slip; 2 x M8: 12 and 5.1

Fine-mesh check of the CAD geometry (75,799 nodes): f1 = 107.8 Hz (>= 100), peak static von Mises 16.9 MPa (<= 140), fatigue life 2,312 h in the worst axis (>= 1000).

Tolerances. The upright thickness and the fillet radius control the fatigue life; the base and upright thickness control f1. The fillet radius gets its own tolerance on the drawing (R8 +/-0.5) because the general radius tolerance (+/-1) would cost up to 25 % of the life. With every dimension at its worst limit the bracket still meets the requirements, and the linear model is within 0.03 Hz and about 2 % of life of the FEA at those corners.

Mesh noise in fatigue life. Two meshes of the same solid (native vs. STEP) give the same f1 to 0.001 % but a 13 % different life for the non-governing x axis, because life ~ stress^-7.65 turns a 2 % stress difference into 13 %. For t_w and r_f the finite-difference steps change the life by 24-61 %, well above this noise. For t_b and d_h the change is under 5 %, inside the noise, so those two life sensitivities are only known to be small.

Limits of the CAD package: threads are cosmetic, parts are placed with placements rather than assembly constraints, the GD&T frames are drawn as SVG symbols (TechDraw has no feature control frame object), and the joints are hand calculations, not FE contact models. The fatigue S-N curve uses the hot-rolled surface factor (k_a = 0.81); the machined part would have k_a = 0.93, so this is conservative.

Reproduce

python -m venv .venv && .venv/bin/pip install numpy scipy matplotlib gmsh pyvista pandas rainflow pytest
# CalculiX: FreeCAD ships ccx (brew install --cask freecad), or set CCX_PATH
PYTHONPATH=. .venv/bin/python -m pytest tests
PYTHONPATH=. .venv/bin/python verification/v1_cantilever.py
PYTHONPATH=. .venv/bin/python verification/v2_plate_hole.py
PYTHONPATH=. .venv/bin/python verification/v3_vibration_fatigue.py
PYTHONPATH=. .venv/bin/python studies/mesh_convergence.py
PYTHONPATH=.:studies .venv/bin/python studies/doe.py --workers 3
PYTHONPATH=. .venv/bin/python studies/hand_calcs.py
PYTHONPATH=. .venv/bin/python studies/plot_verification.py
PYTHONPATH=. .venv/bin/python studies/optimize.py              # DOE analysis, plain reference, fine-mesh check
PYTHONPATH=. .venv/bin/python studies/gusset_detail.py
PYTHONPATH=. .venv/bin/python studies/exclusion_sensitivity.py
PYTHONPATH=. .venv/bin/python studies/design_report.py --name baseline
# CAD package (FreeCAD 1.1: cad/fc.sh runs its headless Python, cad/fc_gui.sh the GUI for TechDraw)
cad/fc.sh cad/build_part.py
.venv/bin/python cad/check_geometry.py
.venv/bin/python cad/fea_from_step.py --threads 4
.venv/bin/python cad/tolerance_study.py --threads 4
cad/fc.sh cad/build_assembly.py
.venv/bin/python cad/joint_check.py
cad/fc_gui.sh cad/make_drawings.py
.venv/bin/python cad/render.py
.venv/bin/python cad/plot_tolerance.py

The design study (plus 40 OOD designs) took under 3 hours with 4 parallel jobs on a 16 GB M3 MacBook; more parallel jobs ran out of memory.

Limitations

• Linear elastic, small deflections; bolted joint and contact are not modelled in the FE model (the joints are checked by hand in cad/joint_check.py).
• The component is a point mass (no rotary inertia) on a rigid hole.
• Fatigue: no mean-stress correction (the static 1 g stress is small), no weld or surface-finish details beyond the Shigley factors, damping assumed (2 %).
• The design study uses a mesh that over-predicts life by 20-35 %; see the margin above.

License

MIT. See LICENSE.