Experimental Vibration System Identification

A steel ruler, a $2 optical sensor, an Arduino — and a Young modulus that came out 7.6× too small. This is the story of finding the missing factor.

I built the bench, adapted the sensor, acquired the data and wrote the full Python pipeline — a home replication of a professional lab experiment, to see how close a low-cost rig can get to laboratory precision. The pipeline turns a raw optical signal into natural frequencies, damping, dynamic models and, ultimately, a material's identity.

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Act 1 — The rig

A TCRT5000 reflective optical sensor (modified for analog readout) stares at a clamped steel ruler; an Arduino Uno streams the raw signal at ~1 kHz over serial (tempo_us,leitura_bruta). All calibration and modeling happen later in Python. The sketch is in hardware/arduino/AMM.ino.

Experimental rig	Sensor circuit	Sensor alignment

Act 2 — The measurement

Pluck the ruler and it rings for 20+ seconds. Wavelet denoising, sub-bin FFT interpolation and a 39-run ensemble reduce the fundamental to one very solid number: 4.982 Hz, stable to ±0.001 Hz across runs.

Act 3 — The verdict that couldn't be right

Feed that frequency into the Euler–Bernoulli cantilever model with the ruler's documented thickness (1.5 mm) and out comes E ≈ 27 GPa — a factor of 7.6 below steel, stranded in a region of the chart where no engineering material lives. A photo of the ruler's edge suggested 1.0 mm instead, which moves the estimate to E ≈ 61 GPa — the more dangerous number, because it sits just below aluminum (68–72 GPa) and looks plausible. The frequency was beyond suspicion. So either the model's boundary condition was wrong (a soft clamp reads as a softer material), or the geometry was.

Act 4 — Interrogating the boundary

Instead of fudging the number, I designed a forced-vibration experiment: drive the same ruler at its first four resonances. Cantilever modes are not integer harmonics — they follow the βₙ² ladder (1 : 6.27 : 17.5 : 34.4), and that ladder is a fingerprint of the boundary condition, independent of the material. A soft clamp or a tip mass would push the higher ratios up, off the ideal line.

The measured ratios sit on the ideal ladder to within ~1%. The clamp walks free — which leaves the geometry as the only suspect.

Act 5 — The missing factor of 7.6

A micrometer settled it: the blade measures 0.55 mm, not the 1.5 mm on record or the 1.0 mm the photo suggested. Since E ∝ 1/h², correcting the thickness multiplies the estimate by 7.6 — the whole missing factor. With the true thickness:

E = 205.3 ± 5.3 GPa — squarely stainless/carbon steel. The uncertainty budget makes the lesson explicit: thickness contributes ±1.8%, length ±1.3%, density ±1.3% — and the frequency only ±0.2%. The $2 sensor was never the limitation; the ruler's metrology was. The reusable diagnostic lives in beam_modes.py; the full investigation is in docs/ORIGINAL_CODE_AUDIT.md.

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Beyond the headline

The same records feed a full identification toolbox:

Sensor nonlinearity	HAVOK reconstruction	PINN inverse problem

Signal processing & spectral analysis — windowed FFT with sub-bin parabolic interpolation; Welch PSD; wavelets (CWT scalograms, DWT energy, denoising); Hilbert-envelope damping with the physical quality factor Q = 2πf₀/γ and a half-power estimator.

Data-driven system identification — SINDy (linear and free-Duffing, with a dependency-free STLSQ); HAVOK (Hankel + SVD + regression); a two-stage physics-informed neural network (PyTorch) with trainable physical parameters; a 3-stage global Duffing fit; and a sensor output map / residual analysis that separates mechanical dynamics from the sensor's static nonlinearity.

Where the nonlinearity lives — reading the instantaneous frequency and amplitude off the ring-down settles what the model fits only hint at: the backbone curve is flat (the frequency moves less than 1% across a threefold amplitude decay, so the cubic stiffness term is a sensor-map artifact, not beam physics), while the envelope prefers amplitude-dependent damping over a single exponential (R² ≈ 0.95 → 0.998, Q ≈ 120). Linear stiffness, nonlinear dissipation — reproducible with run_backbone_damping.py.

Digital twin — the identified pieces compose into one end-to-end forward model: physical parameters → Euler–Bernoulli modal frequencies → nonlinear-damped oscillator → cubic sensor map → the signal you'd measure. Seeded with the identified parameters it reproduces the real ruler — the synthetic ring-down, run through the same pipeline, recovers f₀ = 4.982 Hz to four digits, the quality factor, the βₙ² modal ladder and the steel verdict — then runs virtual experiments: pluck, forced resonance sweep, geometry/material what-if, and tip-mass sensing.

There's a no-build interactive version: drag E, L, h, tip mass, Q and the drive frequency and watch the signal, spectrum, resonance sweep and material verdict update live. Reproduce the figure with run_digital_twin.py.

Physics-based modeling — Euler–Bernoulli cantilever (forward + inverse) with a Rayleigh tip-mass correction and the modal-ladder clamp diagnostic.

Engineering — typed modular package · 40 regression tests · ruff lint + format · GitHub Actions CI (3.10 / 3.12 + a job with the pysindy/torch extras).

Results

Context	Frequency	Geometry	Young modulus	Verdict
Plastic ruler	7.060 Hz	L=0.300 m, h=2.33 mm	2.99 GPa	acrylic / PVC / polystyrene range
Inox ruler (raw)	4.982 Hz	L=0.300 m, h=0.55 mm	205.3 ± 5.3 GPa	stainless / carbon steel (190–210 GPa)
Inox ruler (synchronized)	4.982 Hz	L=0.300 m, h=0.55 mm	205.3 GPa	reproduces the raw inox result
18 Hz baseline	18.737 Hz	not documented	n/a	signal-analysis validation only

Quick start

python -m venv .venv
source .venv/bin/activate      # Windows: .venv\Scripts\activate
pip install -e .
python scripts/run_basic_analysis.py

pip install -e . puts vibration_id on the path; the scripts also add src/ to sys.path, so they run from a clean checkout without installing. Generated figures land in figures/generated/.

# tests
pip install -r requirements-dev.txt && pytest -q

# advanced methods (SINDy / HAVOK / PINN; the PINN demo trains on the inox sample)
pip install -r requirements-advanced.txt
python scripts/run_advanced_analysis.py --run-pinn

# reproducible Duffing + sensor identification and the global fit
python scripts/run_sensor_identification.py
python scripts/run_global_fit.py
python scripts/run_sensor_residual.py

# where the nonlinearity lives: flat backbone (linear stiffness) + nonlinear damping
python scripts/run_backbone_damping.py

# digital twin: validated forward model + virtual experiments
# (interactive version: open interactive/digital_twin.html in a browser)
python scripts/run_digital_twin.py

# material study from trial metadata
python scripts/run_material_study.py

# the story figures and the cover GIF above
python scripts/make_story_figures.py
python scripts/make_cover_gif.py

The CSV loader accepts the project's format variants (tempo_us,posicao_mm · tempo_s,posicao_mm_cent · tempo_corrigido,volt · Second,Volt) and normalizes everything to time_s,signal.

Documentation

• docs/EXPERIMENTAL_NARRATIVE.md — hardware, sensor calibration and the identification workflow
• docs/ORIGINAL_CODE_AUDIT.md — audit of the original exploratory code and how it was ported / corrected here
• docs/FIGURE_PROVENANCE.md — figure-by-figure origin and reproducibility
• docs/PROJECT_STRUCTURE.md — module map

License

Released under the MIT License. See LICENSE.