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arxiv: 2604.22889 · v2 · submitted 2026-04-24 · 📡 eess.IV · cond-mat.mtrl-sci

Recognition: unknown

Fixed-phase Resonance Tracking for Fast Nonlinear Resonant Ultrasound Spectroscopy

Jan Kober , Radovan Zeman , Marco Scalerandi

Authors on Pith no claims yet

Pith reviewed 2026-05-08 09:14 UTC · model grok-4.3

classification 📡 eess.IV cond-mat.mtrl-sci
keywords resonance trackingnonlinear resonant ultrasound spectroscopyNRUSphase-based controldiscrete-time methodsandstone conditioningnonlinear acoustics
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The pith

A discrete-time method tracks instantaneous resonance in nonlinear ultrasound tests by updating frequency from phase alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a resonance tracking procedure for Nonlinear Resonant Ultrasound Spectroscopy that keeps the excitation at the current resonance condition without repeating full frequency sweeps. This matters because material properties in samples such as sandstone change during the test due to fast and slow dynamic effects, so slow conventional sweeps can produce inaccurate or inconsistent nonlinear indicators. Resonance is defined by a fixed phase difference between drive and response; a linearized model then predicts the small frequency adjustment needed at each step. Optional feedforward terms suppress transient buildup when an external control parameter changes. Tests on sandstone bars show that the faster protocol yields different resonance frequency and damping estimates than traditional methods.

Core claim

The central claim is that a model-assisted discrete-time controller, driven by a linearized frequency-phase relation, can maintain a resonant system at its instantaneous resonance condition and thereby extract resonance frequency and damping estimates without acquiring complete frequency sweeps at each measurement point.

What carries the argument

The linearized frequency-phase model that converts observed phase error into an excitation-frequency update to enforce a prescribed phase relation between drive and response.

If this is right

  • Measurement duration drops because full sweeps are replaced by single-frequency steps.
  • Transient wave buildup can be actively suppressed by optional feedforward correction tied to an external control parameter.
  • Inferred nonlinear indicators become sensitive to the chosen measurement speed and mode stability.
  • The same tracking logic applies to any resonant system whose parameters evolve slowly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could support continuous monitoring of damage or conditioning in materials under repeated loading.
  • It may extend naturally to other resonant measurement domains such as mechanical vibration testing or acoustic levitation.
  • Combining the tracker with adaptive model updates could handle faster parameter drifts than the current fixed linearization assumes.

Load-bearing premise

The linearized frequency-phase model must accurately predict the required frequency update and material parameters must change slowly enough that the discrete steps keep the system near resonance without accumulating lag.

What would settle it

Running the tracker in parallel with occasional full frequency sweeps on the same sandstone sample and finding that the tracked frequency consistently misses the actual peak location by more than the expected discretization error would falsify the method.

Figures

Figures reproduced from arXiv: 2604.22889 by Jan Kober, Marco Scalerandi, Radovan Zeman.

Figure 1
Figure 1. Figure 1: Experimental data from NRUS, selection of excitation amplitudes from loading view at source ↗
Figure 2
Figure 2. Figure 2: Resonance curve (experimental data) fitted using MoDaNE (calibration in view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the initial steps of the procedure. a) Phase vs. view at source ↗
Figure 4
Figure 4. Figure 4: Experimental setup. 2.3.4. Summary of the procedure To summarize the resonance tracking procedure, the measurement con￾sists of repeated evaluation of the following algorithm: 1. Acquire the signals, calculate ai and ∆ϕi and estimate fres,i and αi using MoDaNE inversion (Eqs. 4 and 5). 2. Update the phase slope ki using Eq. 12. 3. Update the amplitude coefficient ℓi using Eq. 13. 4. Update the excitation f… view at source ↗
Figure 5
Figure 5. Figure 5: Measurement protocol and resonance tracking. Subplots in second and third view at source ↗
Figure 6
Figure 6. Figure 6: Phase difference (a) and distribution of excitation frequency deviations from the view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of NRUS results measured using resonance tracking, monochromatic view at source ↗
Figure 8
Figure 8. Figure 8: Results obtained using resonance tracking in measurements with various du view at source ↗
Figure 9
Figure 9. Figure 9: Results obtained using resonance tracking in measurements with various du view at source ↗
Figure 10
Figure 10. Figure 10: Conditioning and relaxation experiment. a) Excitation (blue) and measured view at source ↗
Figure 11
Figure 11. Figure 11: Relative resonance frequency variations during a) conditioning, b) relaxation. view at source ↗
read the original abstract

Nonlinear Resonant Ultrasound Spectroscopy (NRUS) experiments that rely on repeated sampling of resonance curves are inherently sensitive to measurement protocol due to evolution of material parameters caused by fast and slow dynamic effects. We introduce a model-assisted discrete-time resonance tracking method that maintains a system at its instantaneous resonance condition without the need to acquire full frequency sweeps. Resonance is defined through a prescribed phase relation between excitation and response, and the excitation frequency is iteratively updated using a linearized frequency--phase model. The procedure allows controlled suppression of transient wave buildup using optional feedforward correction with respect to an external control parameter. The method is demonstrated on NRUS and on conditioning--relaxation protocol conducted on a sandstone bar, providing estimates of resonance frequency and damping. Comparison with conventional approaches shows that measurement speed and mode stability significantly influence the inferred nonlinear indicators. The proposed framework is not limited to nonlinear acoustics and can be applied to arbitrary resonant systems with slowly evolving parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a model-assisted discrete-time resonance tracking method for nonlinear resonant ultrasound spectroscopy (NRUS) that maintains a system at its instantaneous resonance condition by defining resonance via a fixed phase relation between excitation and response and iteratively updating the drive frequency using a linearized frequency-phase model. The approach optionally incorporates feedforward correction to suppress transients with respect to an external control parameter. It is demonstrated on NRUS and conditioning-relaxation protocols on a sandstone bar, providing estimates of resonance frequency and damping, with comparisons to conventional full-sweep methods indicating that measurement speed and mode stability affect inferred nonlinear indicators. The framework is presented as applicable to arbitrary resonant systems with slowly evolving parameters.

Significance. If the tracking accuracy and assumptions hold under validation, the method could enable substantially faster NRUS measurements with reduced sensitivity to material evolution during acquisition, improving the consistency of nonlinear parameter estimates. The discrete-time fixed-phase approach and optional feedforward control represent a practical advance over repeated full sweeps, with potential extension to other resonant systems in acoustics or related fields.

major comments (3)
  1. [Methods] The central claim that the method maintains the instantaneous resonance condition relies on the linearized frequency-phase model for discrete updates and the assumption of sufficiently slow parameter evolution. However, no quantitative bounds on tracking error, lag, or overshoot are provided, nor is there sensitivity analysis to step size or evolution rate (see skeptic note on validity of the linearization).
  2. [Results/Demonstrations] The demonstrations on sandstone NRUS and conditioning-relaxation protocols compare to conventional sweeps and note influences on nonlinear indicators, but lack reported error bars, explicit tracking accuracy metrics, or tests of when the linear approximation breaks, preventing assessment of whether results support the claims of improved stability and speed.
  3. [Method description] The abstract and description state that the procedure allows controlled suppression of transient wave buildup, but no equations, implementation details, or validation of the optional feedforward correction are given to show how it interacts with the discrete updates without introducing additional error.
minor comments (2)
  1. [Abstract] The abstract claims 'estimates of resonance frequency and damping' but provides no specific values, figures, or tables in the summary, making it hard to gauge the quantitative output of the method.
  2. [Methods] Notation for the prescribed phase relation and the linearized model coefficients could be introduced with explicit equations early in the methods to improve clarity for readers implementing the approach.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments highlight important aspects of rigor and clarity that we will address in the revision. Below we respond point-by-point to the major comments.

read point-by-point responses

  1. Referee: [Methods] The central claim that the method maintains the instantaneous resonance condition relies on the linearized frequency-phase model for discrete updates and the assumption of sufficiently slow parameter evolution. However, no quantitative bounds on tracking error, lag, or overshoot are provided, nor is there sensitivity analysis to step size or evolution rate (see skeptic note on validity of the linearization).

    Authors: We agree that explicit quantitative bounds would strengthen the central claim. The linearization follows directly from the first-order Taylor expansion of the phase-frequency relation at resonance and is valid when the frequency step is small compared with the resonance bandwidth. In the revised manuscript we will add a dedicated subsection containing (i) an analytic expression for the steady-state tracking lag under constant parameter drift and (ii) numerical sensitivity results obtained by simulating the discrete-time controller with controlled rates of resonance-frequency evolution. These additions will also address the validity of the linear approximation. revision: yes

  2. Referee: [Results/Demonstrations] The demonstrations on sandstone NRUS and conditioning-relaxation protocols compare to conventional sweeps and note influences on nonlinear indicators, but lack reported error bars, explicit tracking accuracy metrics, or tests of when the linear approximation breaks, preventing assessment of whether results support the claims of improved stability and speed.

    Authors: The experimental section emphasizes comparative differences in inferred nonlinear parameters rather than absolute accuracy. We will augment the results with (a) standard-error bars derived from repeated tracking runs on the same specimen, (b) a quantitative metric of tracking fidelity (rms phase deviation from the target value), and (c) a short discussion, supported by the new simulation analysis mentioned above, of the regime in which the linear update remains accurate. These additions will allow readers to evaluate the claimed improvements in stability and speed. revision: yes

  3. Referee: [Method description] The abstract and description state that the procedure allows controlled suppression of transient wave buildup, but no equations, implementation details, or validation of the optional feedforward correction are given to show how it interacts with the discrete updates without introducing additional error.

    Authors: The feedforward term is presented only conceptually in the current text. In the revised methods section we will supply the explicit discrete-time feedforward equation, describe its coupling to the phase-based frequency update, and include both simulated and experimental validation demonstrating that the combined controller does not degrade steady-state phase tracking when the external parameter changes at the rates typical of NRUS conditioning protocols. revision: yes

Circularity Check

0 steps flagged

No significant circularity in resonance tracking method derivation

full rationale

The paper presents a model-assisted discrete-time resonance tracking method defining resonance via a prescribed phase relation between excitation and response, with frequency updated iteratively using a linearized frequency-phase model. No equations, derivations, or self-citations are shown in the abstract or description that reduce by construction to fitted inputs renamed as predictions, self-definitional loops, or load-bearing self-citations. Demonstrations on sandstone NRUS and conditioning-relaxation protocols include comparisons to conventional full-sweep approaches, indicating the central claim retains independent content and is self-contained against external benchmarks rather than forced by internal definitions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only abstract available; method relies on a linearized frequency-phase model and slow parameter evolution assumptions whose details and parameters are unspecified.

free parameters (1)
  • linearized frequency-phase model coefficients
    Iterative updates depend on a linearized model whose specific coefficients or fitting process are not detailed in abstract.
axioms (2)
  • domain assumption Resonance condition is defined by a prescribed phase relation between excitation and response.
    Basis for the tracking procedure as stated in abstract.
  • domain assumption Material parameters evolve slowly compared to discrete tracking updates.
    Required for maintaining instantaneous resonance without full sweeps.

pith-pipeline@v0.9.0 · 5462 in / 1354 out tokens · 53771 ms · 2026-05-08T09:14:39.714703+00:00 · methodology

discussion (0)

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