Tutorial 2 — Designing an OWChirp waveform - hermes-rheo

An optimally windowed chirp (OWChirp) is a constant-amplitude sinusoid whose instantaneous frequency sweeps exponentially from $\omega_0$ to $\omega_1$ over a duration $T$ ,

\omega(t) = \omega_0 \left(\frac{\omega_1}{\omega_0}\right)^{t/T},

(1)

multiplied by a cosine-tapered (Tukey) window that suppresses spectral leakage and nonlinear artefacts at the signal edges (Geri et al., 2018). The window tapers a fraction $r$ of the signal: $r = 0$ disables tapering and $0.05 \le r \le 0.15$ works well in practice.

The time–bandwidth product

TB = \frac{T(\omega_1 - \omega_0)}{2\pi}

(2)

controls the spectral resolution of the chirp and should satisfy $TB \gtrsim 100$ to keep leakage low. OWChirpGeneration computes the waveform, $TB$ and the piecewise zone equations needed to program the signal in a TA Instruments TRIOS Arbitrary Wave method.

Setup¶

Generated files (parameter summary, waveform samples and figure) are written to the gitignored results/ directory next to this notebook.

from pathlib import Path

from hermes_rheo import OWChirpGeneration
from hermes_rheo.visualization import use_science_style

use_science_style()
RESULTS = Path("results")
RESULTS.mkdir(parents=True, exist_ok=True)

Define the signal¶

OWChirpGeneration is a helper class, not a piblin transform: configure it once, then inspect, export and plot the signal.

waiting_time — quiet interval before the chirp (s). Data acquired here quantify systematic offsets (motor bias, encoder drift) that are removed before the Fourier transform.
signal_duration — chirp duration $T$ (s).
signal_amplitude — imposed strain (dimensionless, 0.01 = 1 %) or stress (Pa), depending on the controlled variable.
initial_frequency, final_frequency — $\omega_0$ and $\omega_1$ (rad/s).
tapering_parameter — Tukey window fraction $r$ .
initial_phase_shift — starting phase (degrees).

print_parameters reports the derived quantities: $TB$ , the piecewise zone equations and the shortest chirp duration compatible with $\omega_0$ .

signal = OWChirpGeneration(
    output_file_name="owchirp_parameters.txt",
    waiting_time=1.0,
    signal_duration=14.0,
    signal_amplitude=8.5e-5,
    initial_frequency=0.3,
    final_frequency=60.0,
    tapering_parameter=0.1,
    initial_phase_shift=0.0,
)
signal.print_parameters()

Shorter chirp signal: 13.9626 (s), TB = 133.0
Total length of chirp signal: 14.0 (s)
Initial Frequency: 0.3 (rad/s)
Final Frequency: 60.0 (rad/s)
Signal amplitude: 8.5e-05 
Waiting time: 1.0 (s)
Tapering parameter: 0.1 
Time Bandwidth: 133.0
Initial Phase Shift: 0.0 (°)
-------------------------------------------------------------------------
For t = waiting_time = 1.0s write: 0
For the first segment duration = 0.7s write: 8.5e-05*cos(2.244*t -3.8148)^2*sin(0.5428*exp(0.3785*t)-0.7926+0.0)
For the second segment duration = 12.6s write: 8.5e-05*sin(0.5428*exp(0.3785*t)-0.7926+0.0)
For the third segment duration = 0.7s write: 8.5e-05*cos(2.244*t -32.0891)^2*sin(0.5428*exp(0.3785*t)-0.7926+0.0)
--------------------------------------------------------------------------

Export and inspect¶

to_txt writes every parameter and the piecewise zone equations to enter in the TRIOS Arbitrary Wave dialog. visualize plots the tapered chirp next to the untapered ( $r = 0$ ) reference and saves the waveform samples and a .png snapshot. The dashed line marks the end of the waiting window. Use y_axis="stress" for a stress-controlled chirp.

signal.to_txt(RESULTS)
signal.visualize(display_r0=True, filepath=RESULTS, y_axis="strain")

References¶

Geri, M., Keshavarz, B., Divoux, T., Clasen, C., Curtis, D. J. and McKinley, G. H., Time-Resolved Mechanical Spectroscopy of Soft Materials via Optimally Windowed Chirps, Physical Review X 8, 041042 (2018). DOI: 10.1103/PhysRevX.8.041042.
Ghiringhelli, E., Roux, D., Bleses, D., Galliard, H. and Caton, F., Optimal Fourier rheometry, Rheologica Acta 51, 413–420 (2012). DOI: Ghiringhelli et al. (2012).
Perego, A., Vadillo, D. C., Mills, M. J. L., Das, M. and McKinley, G. H., Evaluation of Optimally Windowed Chirp Signals in Industrial Rheological Measurements: Method Development and Data Processing, Rheologica Acta 64, 391–406 (2025). DOI: Perego et al. (2025).

References¶

Ghiringhelli, E., Roux, D., Bleses, D., Galliard, H., & Caton, F. (2012). Optimal fourier rheometry: Application to the gelation of an alginate. Rheologica Acta, 51(5), 413–420. 10.1007/s00397-012-0616-z
Perego, A., Vadillo, D. C., Mills, M. J. L., Das, M., & McKinley FRS, G. H. (2025). Evaluation of optimally windowed chirp signals in industrial rheological measurements: method development and data processing. Rheologica Acta, 64(8), 391–406. 10.1007/s00397-025-01511-0