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A State of the Art on Recent Progress and Emerging Challenges on Energy Transfer Between Vibrating Modes Under an External Mechanical Force With Time-Varying Frequency From 2020 to 2025

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that when a vibrating system is driven by a non-ideal source whose angular velocity oscillates sinusoidally in time, the Jacobi-Anger expansion rewrites the excitation as an infinite sum of fixed-frequency harmonic…

desk verdict Useful review of 2020-2025 RNIS work, but the central Jacobi-Anger derivation is internally inconsistent and needs major revision. read the letter →

arxiv 2506.01469 v1 pith:PTPFS5L5 submitted 2025-06-02 nlin.CD cs.SYeess.SY

Jose Manoel Balthazar , Jorge Luis Palacios Felix , Mauricio A. Ribeiro , Angelo Marcelo Tusset , Jeferson Jose de Lima , Vinicius Piccirillo , Julijana Simonovic , Nikola D. Nevsic
show 4 more authors
Marcos Varanis Clivaldo de Oliveira Raphaela C. Machado Gabriella O M Oliveira
This is my paper
classification nlin.CDcs.SYeess.SY MSC 34C1570K3070K40
keywords non-idealenergysourcestransferbetweenvibratingmodesJacobi-Angerexpansiontime-varyingfrequencyexcitationSommerfeldeffectsaturationphenomenonfractionaldampingnonlinearvibrations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper surveys non-ideal vibrating systems—machines whose driving motor has limited power and whose frequency changes with the motion of the structure—covering work from 2020 to 2025 on the Sommerfeld effect, saturation, fractional damping, and energy harvesting. Its central constructive claim is that a time-varying-frequency excitation of the form $\cos[\Omega_0 t + a_0 \sin(b_0 \Omega_0 t + c_0)]$ can be decomposed by the Jacobi-Anger expansion into a sum of ordinary harmonic terms with Bessel-function amplitudes. That turns the non-ideal forcing into an equivalent superposition of ideal fixed-frequency excitations, one per integer $k$, at frequencies $\Omega_k=\Omega_0(1+k b_0)$. The paper presents this as a generalization of earlier results and as a route for studying energy transfer between vibration modes, including a two-degree-of-freedom portal frame where quadratic nonlinearities couple the modes.

What carries the argument

The carrying object is the Jacobi-Anger expansion, $\cos(z\sin\theta)=\sum_{k=-\infty}^{\infty} J_k(z)\cos(k\theta)$, with $J_k$ the $k$-th Bessel function of the first kind. Applied to the phase-modulated cosine of Eq. (8), it converts one chirp-like forcing term into an infinite Fourier-type series indexed by $k$, with sideband frequencies $\Omega_k=\Omega_0(1+k b_0)$ and amplitudes $f_0 J_k(a_0)$. The assumption that feeds the identity is the averaged angular-velocity law $\dot{\theta}=\Omega_0+a_0\cos(b_0\Omega_0 t+c_0)$, whose parameters $a_0$, $b_0$, $c_0$ are treated as control parameters set by the interaction between the oscillator and the source.

What would settle it

Integrate the full two-equation electromechanical model, Eqs. (2)–(3), with a realistic motor torque law through resonance, then compute the Fourier spectrum of the support displacement: the representation Eq. (15) is falsified if spectral lines appear at frequencies not of the form $\Omega_0(1+k b_0)$ with amplitudes near $f_0 |J_k(a_0)|$, or if the numerically extracted instantaneous angular velocity is not a single sinusoid. Equivalently, in a laboratory shaker experiment, drive the base with the phase-modulated signal and check whether the response sidebands follow the Bessel weights $J_k(a_0)$.

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Extended reading notes

Core claim

The paper's central discovery is the reduction expressed in Eq. (15): a damped nonlinear oscillator with fractional-type damping, $\ddot{x}+2\zeta\,\operatorname{sgn}(\dot{x})|\dot{x}|^n+\omega_1^2 x = f_0 \cos[\Omega_0 t + a_0 \sin(b_0\Omega_0 t+c_0)]$, is equivalent, through the Jacobi-Anger identity $\cos(z\sin\theta)=\sum_k J_k(z)\cos(k\theta)$, to $\ddot{x}+2\zeta\,\operatorname{sgn}(\dot{x})|\dot{x}|^n+\omega_1^2 x = f_0 \sum_{k=-\infty}^{\infty} J_k(a_0)\cos(\Omega_k t + k c_0)$, with sideband frequencies $\Omega_k=\Omega_0(1+k b_0)$. Thus a non-ideal source whose instantaneous frequency wobbles sinusoidally acts as infinitely many simultaneous harmonic excitations, each weighted by a Bessel function $J_k(a_0)$; the same construction is applied to the two coupled modes of a portal frame driven by an eccentric rotating mass, where quadratic terms $\alpha_1 x y$ and $\alpha_2 x^2$ couple the modes. The paper states that this is a generalization of earlier results and identifies it as a future direction for studying energy transfer between vibrating modes under non-ideal excitation.

Load-bearing premise

The load-bearing premise is that the motor's instantaneous angular velocity takes the single-sinusoid form $\dot{\theta}=\Omega_0+a_0\cos(b_0\Omega_0 t+c_0)$, with $a_0$, $b_0$, $c_0$ treated as given control parameters; if the actual motor-structure dynamics produce a different phase modulation, the Jacobi-Anger sideband series does not represent the real forcing.

Editorial extensions

If this is right

  • Resonance of a non-ideal system can occur at any sideband frequency $\Omega_k=\Omega_0(1+k b_0)$, not only at the nominal motor frequency, because each Bessel-weighted harmonic is an independent forcing term.
  • Standard perturbation, averaging, and harmonic-balance methods designed for ideal harmonic excitation can be applied to non-ideal systems by treating the response as the superposition of responses to the individual sidebands.
  • For the two-mode portal frame, energy transfer governed by 2:1 internal resonance and the saturation phenomenon can be analyzed mode by mode, with the infinite harmonic series acting on each coupled equation.
  • The formulation extends directly to electrodynamic-shaker-driven supports governed by coupled mechanical and electrical equations, since the same phase-modulated forcing appears in that model.
  • Within the 2020–2025 literature, studies of the Sommerfeld effect, fractional damping, and energy harvesting can be re-expressed in this sideband picture rather than through direct numerical integration of the full motor-structure equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because $a_0$, $b_0$, and $c_0$ are not derived from the motor equations, an immediate testable extension is to extract them from a numerical or experimental time series of the motor speed and compare the predicted sideband amplitudes $f_0|J_k(a_0)|$ with measured response spectra.
  • The infinite series must be truncated in practice; the paper does not provide an error bound, so a natural extension is to determine how many sidebands are needed for a given modulation depth $a_0$ and damping.
  • If the physical motor-structure coupling produces phase modulation with more than one sinusoidal component, the same expansion can be iterated or generalized to a multi-tone phase modulation, but the single-sinusoid form is the version the paper develops.
  • The sideband picture suggests a control strategy: instead of suppressing the motor's fundamental resonance only, one could tune the modulation parameters to shift Bessel weights away from resonant sidebands—an implication the paper does not pursue.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript presents a state-of-the-art review of non-ideal vibrating systems (RNIS) for 2020–2025, covering the Sommerfeld effect, saturation phenomenon, fractional damping, energy harvesting, and frequency-varying excitations. In Section 4 it proposes a 'nowadays formulation' in which a frequency-modulated excitation is expanded via the Jacobi-Anger identity into a superposition of harmonic forcing terms, and it sketches applications to single-DOF, two-DOF portal frame, and electrodynamic shaker models. The paper's central claim is that each harmonic component resulting from the expansion can be treated as an independent harmonic excitation for energy-transfer studies.

Significance. If the formulation in Section 4 were correct, the Jacobi-Anger representation would be a convenient way to analyze non-ideal excitations with time-varying frequency by reducing them to a set of harmonic terms. The paper also compiles a useful list of recent references on RNIS. However, the technical presentation contains internal inconsistencies between the angular-velocity model and the phase-modulation model, and the key displayed equations contain summation errors. The review component is largely descriptive rather than critical. Strengths include the breadth of covered topics and the use of the Jacobi-Anger identity as a mathematical tool, but the manuscript requires a major revision before it can be accepted.

major comments (4)
  1. [Section 4, Eqs. (7)-(8)] Equations (7) and (8) are mutually inconsistent. Equation (7) defines the time-varying angular velocity as dθ/dt = Ω0 + a0 cos(b0 Ω0 t + c0), whose integral is θ(t) = Ω0 t + [a0/(b0 Ω0)] sin(b0 Ω0 t + c0). The forcing term in Eq. (8) uses the phase Ω0 t + a0 sin(b0 Ω0 t + c0), which requires the phase-deviation amplitude a0 to be dimensionless and to appear directly, not scaled by 1/(b0 Ω0). Consequently, the Bessel argument in the expansion leading to Eq. (15) is wrong unless Eq. (7) is amended or a0 is redefined. Because Eq. (15) is the central result of Section 4, this inconsistency is load-bearing.
  2. [Section 4, Eq. (15) and Eq. (16b)] The final equalities in Eq. (15) and Eq. (16b) drop the summation over k and are therefore false as written. Equation (15) states f0 Σ_k J_k(a0) cos(Ω_k t + k c0) = f0 J_k(a0) cos(Ω_k t + k c0), which only holds for a single selected k, not for the general expansion. The same problem appears twice in Eq. (16b). The text explains the intended meaning ('like a system with harmonic excitation for each k'), but the displayed equations must carry an explicit summation over k or use a subscripted notation such as f0 J_k(a0) cos(...) with a 'for each k' qualifier.
  3. [Section 4, paragraph after Eq. (7)] The parameters a0, b0, and c0 are asserted to be 'defined by the active interaction between the oscillating system and the excitation source,' but no derivation from the electromechanical model (e.g., Eqs. (2)-(3) or Eq. (19)) is provided. The sinusoidal phase-modulation form in Eq. (8) is therefore an assumption rather than a consequence of the non-ideal source dynamics. The authors should either derive this form from the motor-structure equations or explicitly identify it as a phenomenological approximation and discuss the conditions under which a single-sinusoid phase modulation is justified. As it stands, the claimed representation of a non-ideal excitation as f0 Σ_k J_k(a0) cos(Ω_k t + k c0) is not established.
  4. [Section 2, Eq. (5)] Equation (5) is garbled and contains typographical errors: the damping terms are written with misplaced parentheses and unclear notation, and the equation does not match the linear damping form of Eq. (1) when n = 1. The dimensionless parameters in Eq. (4) are also difficult to read. Because these equations are the basis for the numerical results in Figure 3, the authors must provide clean, correct versions of the governing equations and the parameter definitions.
minor comments (6)
  1. [Section 4, first paragraph] 'Felix et al, 1976' appears to be a typo for 'Felix et al., 2016' based on reference [10].
  2. [Section 4, subsection numbering] The subsection numbering is duplicated: both '4.1 Extension of Governing equations...' and '4.1 A Brief State of the art...' appear; renumber the latter as 4.2.
  3. [Section 4, Eq. (15)] Equation (15) contains a duplicated damping coefficient '2ζ1 2ζ1'; the factor '2ζ1' appears twice.
  4. [Section 2, Eq. (4)] The dimensionless parameter definitions in Eq. (4) are not legible due to OCR artifacts; please retypeset them.
  5. [Section 2, Figure 3 and Table 1] Figure 3 is said to use 'the parameter given by Table 1', but the table contents are not correctly reproduced in the text.
  6. [References] Reference [23] has an incomplete DOI; all bibliographic entries should be checked for completeness and consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Section 4 applies the standard Jacobi-Anger identity to an assumed phase-modulated excitation; self-citations are present but not load-bearing.

full rationale

The paper's central formal step in Section 4 is the replacement of the harmonic forcing cos[Ω0 t + a0 sin(b0 Ω0 t + c0)] by Σ J_k(a0) cos(Ω_k t + k c0), with Ω_k = Ω0 + k b0 Ω0. This is the classical Jacobi-Anger expansion stated in Eq. (9); it is an identity applied to the assumed phase, not a fitted quantity and not a prediction derived from the electromechanical model. The parameters a0, b0, c0 are declared control parameters fixed by the interaction (Section 4, after Eq. (7)) and are not estimated from response data, so no fitted-input-called-prediction occurs. The citations to Felix et al. 2016 and other works by the same group are self-citations, but the expansion is standard mathematics and is derived in the text before being attributed to Felix et al. 2016; the central claim therefore does not rest on an unverified self-citation. A separate internal-model issue exists but is not circularity: Eq. (7) gives dθ/dt = Ω0 + a0 cos(b0 Ω0 t + c0), whose integral is θ = Ω0 t + [a0/(b0 Ω0)] sin(b0 Ω0 t + c0), while Eqs. (8) and (7a) use phase modulation a0 sin(...) without the 1/(b0 Ω0) factor. This is a dimensional/consistency error in the modeling assumption, not a reduction of a prediction to an input. For the same reason, no renaming of a known result as a new prediction is present: the paper explicitly states that it is using the Jacobi-Anger expansion. Overall, the derivation chain is self-contained and no circularity is found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on standard math (Jacobi-Anger) and domain assumptions about the form of frequency modulation and the two-mode structural model. The free parameters a_0, b_0, c_0 are the main ad hoc inputs; they are not fitted to data or derived from first principles.

free parameters (3)
  • a_0
    Amplitude of the frequency modulation in Eq. (7). Treated as a control parameter 'defined by the active interaction' but never computed from the motor-structure model.
  • b_0
    Frequency ratio of the modulation in Eq. (7). Introduced ad hoc as a control parameter; no physical derivation or fitted value.
  • c_0
    Phase offset of the modulation in Eq. (7). Chosen by hand; no independent determination from dynamics or data.
assumptions (4)
  • standard math Jacobi-Anger expansion: cos(z sin theta) = sum_k J_k(z) cos(k theta)
    Invoked in Eq. (9) as the core expansion used to transform the frequency-modulated forcing into a harmonic sum.
  • domain assumption Angular velocity approximation of Kononenko: d theta/dt = Omega_0 + a_0 cos(b_0 Omega_0 t + c_0)
    Eq. (7) assumes the motor speed varies as a single sinusoid around a constant average. This is asserted without derivation from the electromechanical coupling equations.
  • domain assumption Averaging ansatz: x = a(t) cos(theta(t) + beta(t))
    Eq. (6) adopts a slowly varying amplitude-phase representation standard in nonlinear vibration analysis; used implicitly to justify the form of Eq. (7).
  • domain assumption Two-mode portal frame model with quadratic coupling and 2:1 internal resonance
    Section 4.1 sets up Eqs. (16a) with x-y quadratic coupling and natural frequencies omega_1, omega_2; this is a specific modeling choice for the energy-transfer discussion, not a general result.

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Cite this review

Pith. "Pith review of A State of the Art on Recent Progress and Emerging Challenges on Energy Transfer Between Vibrating Modes Under an External Mechanical Force With Time-Varying Frequency From 2020 to 2025." pith.science (2026). https://pith.science/paper/PTPFS5L5

@misc{pith2026250601469,
  author       = {Pith},
  title        = {Pith review of: A State of the Art on Recent Progress and Emerging Challenges on Energy Transfer Between Vibrating Modes Under an External Mechanical Force With Time-Varying Frequency From 2020 to 2025},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTPFS5L5}},
  note         = {Machine review of arXiv:2506.01469}
}
read the original abstract

In this paper, we discuss an example of current importance with a future perspective in engineering, in which excitation sources always have limited power, limited inertia, and their frequencies vary according to the instantaneous state of the vibrating system. Practical examples of non-ideal systems are considered. The most common phenomenon for this kind of system is discussed. The period considered is from 2020 to 2025. The specific properties of various models are also discussed. Directions for future investigations are provided. In this paper, the authors revisited some publications based on the assumption that the external excitations are produced by non-ideal sources (RNIS), that is, with limited power supply. Among these applications, nonlinear phenomena such as the Sommerfeld effect and saturation phenomenon were observed, considering fractional damping. Energy harvesters and the Jacobi-Anger expansion were used in the governing equations of motion. We also used the Jacobi-Anger expansion in the case of energy transfer between vibrating modes under an external force with time-varying frequency, which represents one of the future directions of research on non-ideal vibrating systems (RNIS).

Figures

Figures reproduced from arXiv: 2506.01469 by the authors.

Figure 1
Figure 1. (RNIS) motor with unbalanced rotor and its elas [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Torque of the motor we will obtain [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. First four Bessel Functions So, using the trigonometric funcƟon can de wriƩen as: 𝑐𝑜𝑠[ 𝛺଴𝑡 + 𝑎଴ 𝑠𝑖𝑛( 𝑏଴𝛺଴𝑡 + 𝑐଴)] = ෍ 𝐽௞(𝑎଴) 𝑐𝑜𝑠( 𝛺௞𝑡 + 𝑘𝑐଴) ஶ ௞ୀିஶ With 𝛺௞ = 𝛺଴ + 𝑘𝑏଴𝛺଴, and we will obtain (Felix et al, 2016): 0 1 2 3 4 5 6 7 8 9 10 -0.5 0 0.5 1 x Jk(x) J0 J1 J2 J3 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

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Works this paper leans on

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    IntroducƟon The features commonly found in vibraƟng engineering are nonlineariƟes (geometric or physical characterisƟcs); DissipaƟon of energy (internal or external); Gyroscopic Systems; ImperfecƟons of material; StaƟonary and non-staƟonary modes; Unlimited power sources (or Ideal sources) or limited (non-ideal sources (RNIS)) and non-conservaƟve dynamic ...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.