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arxiv: 2604.13920 · v1 · submitted 2026-04-15 · ❄️ cond-mat.mes-hall

Experimental Quantification of Nonlinear Mode Coupling in Nanomechanical Resonators using Multi-tone Excitation

Chris F. D. Wattjes , Zichao Li , Minxing Xu , Richard A. Norte , Peter G. Steeneken , Farbod Alijani This is my paper

Pith reviewed 2026-05-10 12:25 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords nonlinear mode couplingnanomechanical resonatorsmulti-tone excitationsideband responsesreduced-order modelsmodal interactionsexperimental quantificationtensioned nanostrings
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The pith

Multi-tone spectroscopy quantifies ten pairwise nonlinear couplings across five modes in nanomechanical resonators.

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

The paper introduces an experimental technique that uses multiple driving tones to generate observable sideband signals whose amplitudes encode the strengths of specific nonlinear interactions between vibration modes. Dual-tone excitation near chosen resonances, paired with probing tones at other modes, produces these signatures, which an inverse reconstruction procedure then converts into quantitative coupling coefficients. The method was applied to highly tensioned nanostrings, yielding ten distinct pairwise parameters that allow construction of device-specific nonlinear reduced-order models. These experimental models match closely with independent numerical predictions from finite-element analysis. Accurate knowledge of such couplings matters because they govern energy exchange, frequency shifts, and stability limits in resonators used for sensing, filtering, and signal processing.

Core claim

Dual-tone excitation near selected resonances combined with probing tones at higher-order modes generates sideband responses associated with specific modal couplings. An inverse reconstruction procedure applied to these spectral signatures quantitatively determines the corresponding nonlinear coupling strengths. Applied to five modes of highly tensioned nanostrings, the approach yields ten pairwise nonlinear coupling parameters, enabling the reconstruction of fully experimental, device-specific nonlinear reduced-order models that show excellent agreement with values obtained from finite-element-based numerical models.

What carries the argument

Multi-tone spectroscopy that generates and inverts sideband responses to extract pairwise nonlinear coupling coefficients in the frequency domain.

If this is right

  • Fully experimental nonlinear reduced-order models can replace or supplement numerical ones for device-specific predictions.
  • The technique provides a generic route to characterize diverse modal and intermodal couplings in mechanical and hybrid resonant systems.
  • Quantitative coupling parameters enable direct prediction of nonlinear phenomena such as modal energy transfer and amplitude-dependent frequency shifts.
  • Ten measurable couplings across five modes demonstrate that the method scales to multimode systems without requiring exhaustive numerical fitting.

Where Pith is reading between the lines

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

  • The same sideband signatures could be tracked over time to monitor how couplings evolve with temperature, stress, or fabrication variations.
  • Adding more simultaneous tones might allow extraction of selected three-mode or higher-order couplings once the pairwise terms are known.
  • The close match between experiment and simulation suggests the method can serve as a calibration tool to improve the accuracy of numerical models for complex geometries.

Load-bearing premise

Observed sideband responses arise solely from the targeted pairwise nonlinear couplings, and the inverse reconstruction can isolate their strengths without significant interference from higher-order nonlinearities or other unaccounted dynamics.

What would settle it

Extracted coupling values that deviate substantially from independent finite-element calculations or from measurements obtained by an alternative experimental protocol on the same device.

Figures

Figures reproduced from arXiv: 2604.13920 by Chris F. D. Wattjes, Farbod Alijani, Minxing Xu, Peter G. Steeneken, Richard A. Norte, Zichao Li.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of measurement setup, consisting of a MSA-500 Polytec laser Doppler vibrometer (LDV), Intermodulation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic flow diagram of the nonlinear parameter estimation methodology. Different combinations of two-tone [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Nonlinear parameter estimation with multi-tone excitation. (a) The amplitude of the multi-tone excitation is increased [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Nonlinear parameter estimation of two coupled degrees of freedom of the nanostring. (a) Single tone frequency sweep [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Nonlinear parameter estimation for the first five modes of a string with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: Moreover, the ability to obtain ROMs directly [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Nonlinear modal interactions in resonant systems govern a wide range of phenomena, with broad relevance across modern physics and engineering. Yet, experimentally determining the strength of nonlinear coupling in multimode resonators remains highly challenging. Here, we introduce a multi-tone spectroscopy method for identifying nonlinear coupling coefficients directly from experimental data. Our approach employs dual-tone excitation near selected resonances which, in combination with additional probing tones at higher-order modes, generates sideband responses associated with specific modal couplings. These spectral signatures are analyzed using an inverse reconstruction procedure to quantitatively determine the corresponding nonlinear coupling strengths in the frequency domain. Using this method, we determine ten pairwise nonlinear coupling parameters across five modes of highly tensioned nanostrings, enabling the reconstruction of fully experimental, device-specific nonlinear reduced-order models. Our experimentally derived models show excellent agreement with values obtained numerically using finite element based nonlinear reduced-order models. Our method is generic and can be used for the characterization of diverse modal and intermodal couplings in mechanical and hybrid resonant systems.

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

2 major / 2 minor

Summary. The paper introduces a multi-tone spectroscopy technique for experimentally extracting nonlinear coupling coefficients in multimode nanomechanical resonators. Dual-tone excitation near selected resonances combined with probing tones at higher modes generates sideband responses that are inverted via a reduced-order model containing quadratic and cubic intermodal terms to determine ten pairwise coupling parameters across five modes in highly tensioned nanostrings. These experimental coefficients are used to construct device-specific nonlinear models that are reported to agree closely with independent finite-element simulations.

Significance. If the inverse procedure reliably isolates the targeted couplings, the work provides a valuable experimental route to parameterizing nonlinear reduced-order models for complex resonators, reducing reliance on purely numerical approaches. The direct comparison between measured and FEM-derived coefficients is a strength, offering cross-validation that supports broader applicability to MEMS/NEMS characterization and nonlinear dynamics studies.

major comments (2)
  1. [inverse reconstruction / multi-tone method] The section describing the inverse reconstruction procedure: the central claim that sideband amplitudes can be uniquely inverted to the ten pairwise coefficients assumes dominance by the included quadratic/cubic terms. The manuscript should include explicit tests (e.g., synthetic spectra with added quartic self- or cross-terms) showing that unaccounted higher-order nonlinearities or frequency-dependent damping do not produce degenerate solutions with comparable sideband fits.
  2. [results / parameter extraction] Results on extracted parameters and FEM comparison: while agreement is reported as excellent, both the experimental fit and the FEM use the same modal truncation and basis; this does not test robustness of the experimental inverse when the physical system contains additional nonlinear channels omitted from the model. A sensitivity analysis quantifying bias from such omissions is needed to support the quantitative accuracy claim.
minor comments (2)
  1. [figures] Figure presentation: sideband spectra and fit residuals should include explicit labels for all probed modes, drive frequencies, and amplitude scales to allow readers to assess potential spectral overlaps or confounding features.
  2. [methods / equations] Notation and equations: all symbols in the reduced-order model (e.g., coupling coefficients, modal amplitudes) should be defined in a single table or early in the methods to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the significance of our multi-tone spectroscopy method. We address each major comment below and will revise the manuscript to strengthen the validation of the inverse procedure.

read point-by-point responses

  1. Referee: The section describing the inverse reconstruction procedure: the central claim that sideband amplitudes can be uniquely inverted to the ten pairwise coefficients assumes dominance by the included quadratic/cubic terms. The manuscript should include explicit tests (e.g., synthetic spectra with added quartic self- or cross-terms) showing that unaccounted higher-order nonlinearities or frequency-dependent damping do not produce degenerate solutions with comparable sideband fits.

    Authors: We agree that explicit validation of the inverse procedure against unmodeled higher-order terms is valuable. Our reduced-order model is based on the perturbative expansion for tension-dominated strings, where quadratic and cubic intermodal couplings dominate the observed sidebands at the drive amplitudes employed. Higher-order terms produce sidebands at distinct frequency combinations not present in the measured spectra. In the revised manuscript we will add a dedicated subsection (with supporting synthetic data in the supplement) that generates multi-tone spectra from an extended model including quartic self- and cross-terms whose magnitudes are taken from the same FEM framework. Inverting these synthetic spectra with the original quadratic/cubic model recovers the input coefficients to within experimental uncertainty and yields no degenerate solutions that fit the sideband amplitudes equally well. Frequency-dependent damping is ruled out by the linear response data and does not affect the nonlinear sideband analysis at the reported precision. revision: yes

  2. Referee: Results on extracted parameters and FEM comparison: while agreement is reported as excellent, both the experimental fit and the FEM use the same modal truncation and basis; this does not test robustness of the experimental inverse when the physical system contains additional nonlinear channels omitted from the model. A sensitivity analysis quantifying bias from such omissions is needed to support the quantitative accuracy claim.

    Authors: We acknowledge that the direct comparison uses the same modal truncation. To quantify possible bias from omitted nonlinear channels, the revised manuscript will include a sensitivity study in which small quartic and higher-order couplings (consistent with the full FEM) are added to the underlying equations before generating synthetic spectra. Re-inversion with the truncated model then provides explicit bounds on the systematic error in the extracted quadratic/cubic coefficients. The close agreement already obtained with the independent FEM-derived coefficients (computed from the complete nonlinear FEM before reduction) supplies additional cross-validation that significant omitted channels would have produced visible discrepancies. We will report the outcome of this analysis to support the quantitative accuracy of the reported values. revision: yes

Circularity Check

0 steps flagged

No significant circularity; experimental inverse reconstruction is data-driven and cross-validated against independent FEM.

full rationale

The derivation proceeds from measured sideband spectra under multi-tone drive, through an inverse procedure that solves for the ten pairwise coefficients in a truncated quadratic/cubic reduced-order model, to comparison against separate finite-element nonlinear ROMs computed from device geometry and material parameters. No equation or step reduces to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on self-citation chains. The method is presented as newly introduced and the FEM benchmark is numerically independent of the experimental fit. Potential non-uniqueness from omitted higher-order terms is a modeling-assumption issue, not a circularity issue.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the inverse reconstruction procedure, which depends on assumptions about the form of nonlinear interactions and the uniqueness of sideband signatures.

axioms (1)
  • domain assumption Nonlinear modal interactions produce distinct, identifiable sideband responses under the described multi-tone excitation scheme.
    This assumption enables the mapping from measured spectra back to specific coupling coefficients.

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

  1. [1]

    Bachtold, J

    A. Bachtold, J. Moser, and M. Dykman, Mesoscopic physics of nanomechanical systems, Reviews of Modern Physics94, 045005 (2022)

    work page 2022

  2. [2]

    Houri, M

    S. Houri, M. Asano, H. Okamoto, and H. Yamaguchi, A kuramoto network in a single nonlinear microelectrome- chanical device, arXiv preprint arXiv:2201.01913 (2022)

    work page arXiv 2022

  3. [3]

    Shoshani and S

    O. Shoshani and S. W. Shaw, Resonant modal inter- actions in micro/nano-mechanical structures, Nonlinear Dynamics104, 1801 (2021)

    work page 2021

  4. [4]

    M. Wang, D. J. Perez-Morelo, D. Lopez, and V. A. Aksyuk, Persistent nonlinear phase-locking and non- monotonic energy dissipation in micromechanical res- onators, Physical Review X12, 041025 (2022)

    work page 2022

  5. [5]

    Y. Yan, X. Dong, L. Huang, K. Moskovtsev, and H. B. Chan, Energy transfer into period-tripled states in coupled electromechanical modes at internal resonance, Phys. Rev. X12, 031003 (2022)

    work page 2022

  6. [6]

    Ke¸ skekler, O

    A. Ke¸ skekler, O. Shoshani, M. Lee, H. S. van der Zant, P. G. Steeneken, and F. Alijani, Tuning nonlin- ear damping in graphene nanoresonators by parametric– 9 direct internal resonance, Nature communications12, 1099 (2021)

    work page 2021

  7. [7]

    M. H. Matheny, J. Emenheiser, W. Fon, A. Chapman, A. Salova, M. Rohden, J. Li, M. Hudoba de Badyn, M. P´ osfai, L. Duenas-Osorio,et al., Exotic states in a simple network of nanoelectromechanical oscillators, Sci- ence363, eaav7932 (2019)

    work page 2019

  8. [8]

    Shoshani, S

    O. Shoshani, S. Strachan, D. Czaplewski, D. Lopez, and S. W. Shaw, Extraordinary frequency stabilization by resonant nonlinear mode coupling, Physical Review Ap- plied22, 054055 (2024)

    work page 2024

  9. [9]

    Ke¸ skekler, H

    A. Ke¸ skekler, H. Arjmandi-Tash, P. G. Steeneken, and F. Alijani, Symmetry-breaking-induced frequency combs in graphene resonators, Nano Letters22, 6048 (2022)

    work page 2022

  10. [10]

    D. A. Czaplewski, C. Chen, D. Lopez, O. Shoshani, A. M. Eriksson, S. Strachan, and S. W. Shaw, Bifurcation gen- erated mechanical frequency comb, Physical review let- ters121, 244302 (2018)

    work page 2018

  11. [11]

    M. H. de Jong, J. Li, C. G¨ artner, R. A. Norte, and S. Gr¨ oblacher, Coherent mechanical noise cancellation and cooperativity competition in optomechanical arrays, Optica9, 170 (2022)

    work page 2022

  12. [12]

    X. Jin, C. G. Baker, E. Romero, N. Arora, N. P. Mauranyapin, T. M. Hirsch, G. I. Harris, and W. P. Bowen, Nanomechanical error correction, arXiv preprint arXiv:2509.11560 (2025)

    work page arXiv 2025

  13. [13]

    Grollier, D

    J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor- Sitte, S. Fukami, and M. D. Stiles, Neuromorphic spin- tronics, Nature electronics3, 360 (2020)

    work page 2020

  14. [14]

    M. I. Younis,MEMS linear and nonlinear statics and dynamics, Vol. 20 (Springer Science & Business Media, 2011)

    work page 2011

  15. [15]

    Asadi, J

    K. Asadi, J. Yu, and H. Cho, Nonlinear couplings and en- ergy transfers in micro-and nano-mechanical resonators: intermodal coupling, internal resonance and synchroniza- tion, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences376, 20170141 (2018)

    work page 2018

  16. [16]

    S. K. Das, N. Arora, H. A. S, A. Naik, and C. Samanta, Nonlinear mode coupling in silicon nitride membrane res- onators (2026), arXiv:2603.08689 [cond-mat.mes-hall]

    work page arXiv 2026

  17. [17]

    Ke¸ skekler, V

    A. Ke¸ skekler, V. Bos, A. M. Arag´ on, P. G. Steeneken, and F. Alijani, Multimode nonlinear dynamics of graphene resonators, Physical Review Applied20, 064020 (2023)

    work page 2023

  18. [18]

    Y. Yang, E. Ng, P. Polunin, Y. Chen, S. Strachan, V. Hong, C. H. Ahn, O. Shoshani, S. Shaw, M. Dykman, et al., Experimental investigation on mode coupling of bulk mode silicon mems resonators, in2015 28th IEEE International Conference on Micro Electro Mechanical Systems (MEMS)(IEEE, 2015) pp. 1008–1011

    work page 2015

  19. [19]

    Z. Li, M. Xu, R. A. Norte, A. M. Arag´ on, P. G. Steeneken, and F. Alijani, Cascade of modal interactions in nanomechanical resonators with soft clamping, Phys. Rev. Lett.136, 097202 (2026)

    work page 2026

  20. [20]

    C. Chen, D. H. Zanette, D. A. Czaplewski, S. Shaw, and D. L´ opez, Direct observation of coherent energy transfer in nonlinear micromechanical oscillators, Nature commu- nications8, 15523 (2017)

    work page 2017

  21. [21]

    M. H. de Jong, A. Cupertino, D. Shin, S. Gr¨ oblacher, F. Alijani, P. G. Steeneken, and R. A. Norte, Beat- ing ringdowns of near-degenerate mechanical resonances, Physical Review Applied20, 024053 (2023)

    work page 2023

  22. [22]

    A. A. Barakat, A. Chowdhury, A. T. Le, and E. M. Weig, Fundamental and second-subharmonic autler- townes splitting in classical systems, Phys. Rev. A , (2026)

    work page 2026

  23. [23]

    A. T. Le, A. Chowdhury, H. Ribeiro, and E. M. Weig, Precise estimation of the coupling strength between two nanomechanical modes from four ramsey fringes, arXiv preprint arXiv:2601.13415 (2026)

    work page arXiv 2026

  24. [25]

    Platz, E

    D. Platz, E. A. Thol´ en, D. Pesen, and D. B. Haviland, In- termodulation atomic force microscopy, Applied Physics Letters92, 153106 (2008)

    work page 2008

  25. [26]

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discover- ing governing equations from data by sparse identifica- tion of nonlinear dynamical systems, Proceedings of the National Academy of Sciences of the United States of America113, 3932 (2016)

    work page 2016

  26. [27]

    Cenedese, J

    M. Cenedese, J. Ax˚ as, B. B¨ auerlein, K. Avila, and G. Haller, Data-driven modeling and prediction of non- linearizable dynamics via spectral submanifolds, Nature communications13, 872 (2022)

    work page 2022

  27. [28]

    A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data

    M. Bosso, G. Franzese, K. Swamy, M. Theulings, A. M. Arag´ on, and F. Alijani, A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data (2026), arXiv:2604.06081 [cs.LG]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  28. [29]

    Jain and G

    S. Jain and G. Haller, How to compute invariant man- ifolds and their reduced dynamics in high-dimensional finite element models, Nonlinear dynamics107, 1417 (2022)

    work page 2022

  29. [30]

    Z. Li, F. Alijani, A. Sarafraz, M. Xu, R. A. Norte, A. M. Arag´ on, and P. G. Steeneken, Finite element-based nonlinear dynamic optimization of nanomechanical res- onators, Microsystems & Nanoengineering11, 16 (2025)

    work page 2025

  30. [31]

    Pozzi, J

    M. Pozzi, J. Marconi, S. Jain, M. Li, and F. Braghin, Topology optimization of nonlinear structural dynamics with invariant manifold-based reduced order models: M. pozzi et al., Structural and Multidisciplinary Optimiza- tion68, 72 (2025)

    work page 2025

  31. [32]

    Houri, D

    S. Houri, D. Hatanaka, M. Asano, and H. Yamaguchi, Demonstration of multiple internal resonances in a micro- electromechanical self-sustained oscillator, Physical Re- view Applied13, 014049 (2020)

    work page 2020

  32. [33]

    X. Yao, M. H. J. de Jong, J. Li, and S. Gr¨ oblacher, Long- range optomechanical interactions in sin membrane ar- rays, Phys. Rev. X15, 011014 (2025). Supplementary Information: Experimental Quantification of Nonlinear Mode Coupling in Nanomechanical Resonators using Multi-tone Excitation Chris F. D. Wattjes*,1, Zichao Li1, Minxing Xu1,2, Richard A. Norte...

    work page 2025

  33. [34]

    Next, to estimate the coupling parameters, three driving tones are applied following the procedure described in the main text

    Repeating this for other modes gives all linear and Duffing parameters for higher-order modes. Next, to estimate the coupling parameters, three driving tones are applied following the procedure described in the main text. To estimate cubic couplings between modes 1 and 2, the Eq. 4 in matrix form is then constructed: 5   jΩ 1ˆq(1) 1 ˆq(1) 1 ˆg(1) 11...

    work page

  34. [35]

    Therefore, instead of estimating b(1) 122 andb(2) 112 separately, we obtain ˜γdirectly

    Since terms in the equation of motion follow from the derivative of Unl with respect to generalized coordinates q1 andq2, it can be found that b(1) 122 =b(2) 112 = ˜γ. Therefore, instead of estimating b(1) 122 andb(2) 112 separately, we obtain ˜γdirectly. This is done by taking Eq. S14 for each mode, and combining them in an equation of the form: [ H1,unc...

    work page

  35. [36]

    Catalini, J

    L. Catalini, J. del Pino, S. S. Kumar, V. Dumont, G. Margiani, O. Zilberberg, and A. Eichler, Slow and fast topological dynamical phase transitions in a duffing resonator driven by two detuned tones, Phys. Rev. Res. 7, 033058 (2025)

    work page 2025

  36. [37]

    Z. Li, M. Xu, R. A. Norte, A. M. Arag´ on, P. G. Steeneken, and F. Alijani, Strain engineering of nonlinear nanoresonators from hardening to softening, Communications Physics 7, 10.1038/s42005-024-01543-7 (2024)

    work page doi:10.1038/s42005-024-01543-7 2024

  37. [38]

    Z. Li, M. Xu, R. A. Norte, A. M. Arag´ on, P. G. Steeneken, and F. Alijani, Cascade of modal interactions in nanomechanical resonators with soft clamping, Phys. Rev. Lett. 136, 097202 (2026)

    work page 2026

  38. [39]

    D. Shin, A. Cupertino, M. H. de Jong, P. G. Steeneken, M. A. Bessa, and R. A. Norte, Spiderweb nanomechani- cal resonators via bayesian optimization: Inspired by nature and guided by machine learning, Advanced Materials 34, 10.1002/adma.202106248 (2022). 9 TABLE S3. Experimentally obtained parameters for the first two modes of devices with different suppo...

    work page doi:10.1002/adma.202106248 2022