Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption

Adam Peichl; Mat\v{e}j Ku\v{r}e; Tom\'a\v{s} Vyhl\'idal; Wim Michiels

REVIEW 1 major objections 5 minor 46 references

Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A delayed resonator that stops a non-collocated target leaves the chain's energy spread fixed by structure alone; redesigning masses and springs, not retuning control, cuts fatigue and actuator power tenfold.

desk verdict The main independence result is real and the paper deserves refereeing, but the reported optimizations rest on an unproven delay-branch rule and a swapped normalization pair. read the letter →

arxiv 2505.23608 v1 pith:HO44JYUA submitted 2025-05-29 eess.SY cs.SYmath.OC

Adam Peichl , Matěj Kuře , Wim Michiels , Tomáš Vyhlídal This is my paper

classification eess.SYcs.SYmath.OC MSC 93C2370Q05

keywords Non-collocatedvibrationabsorptionDelayedresonatorStructuralanalysisConstrainedoptimizationFatigueEnergyefficiencyTime-delayfeedbackSpectralabscissa

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 claims that in a chain of flexibly linked masses with a delayed resonator absorbing vibration at a non-collocated target, the control effort cannot redistribute the oscillation energy: once the target mass is ideally stopped, the maximal elastic potential energy in every link is determined only by the disturbance and by the masses, springs, and dampers. The absorber force needed for full absorption is unique for a given structure, and retuning the resonator changes nothing about where fatigue accumulates. The remedy is to treat structural parameters as design variables, which the authors do: they derive the complete gain-delay set of the resonator and the actuator power needed, then solve a constrained nonlinear optimization that minimizes the worst-link energy and actuator power while keeping a prescribed stability margin. Experimental (three-mass) and numerical (five-mass) case studies show both objectives can drop by roughly an order of magnitude simultaneously.

What carries the argument

The central object is the phasor-domain decomposition of the linear chain at the motionless target mass, which splits the system into a resonating subsystem (the masses between the absorber and the target), a vibrating subsystem (the masses beyond the target), and the absorber itself. The load-bearing identity is the closed-form force balance (19)-(20) fixing the absorber link force phasor $\vec{f}_a$ required for full absorption, together with the matching condition $g e^{-j\omega\tau} = Q(\omega)$ of Proposition 3 that converts this force into the delayed-resonator feedback pair $(g, \tau)$. Fatigue is quantified as the maximal elastic potential energy in each link, (35)-(36), and stability is measured by the spectral abscissa $\alpha$ of the delay-differential-algebraic model (72). The optimization (80) eliminates the control parameters via the matching condition, so the decision variables are purely structural, and the nonsmooth, nonconvex problem is solved by a quasi-Newton sequential quadratic programming method with gradients approximated by finite differences.

What would settle it

Run the three-mass experimental setup at fixed $\omega$ with two different control implementations that both bring $x_s$ to near zero, for example a delayed-resonator feedback and a directly injected link force, and compare the measured per-link maximal elastic potential energies: if the distributions differ beyond measurement noise, Theorem 1 is false. A second check is to re-tune the optimized five-mass design at $\omega \pm 5\%$ and measure worst-link energy and $P_{\max}$; if the stability margin collapses or the improvements vanish, the single-frequency and smallest-delay-branch assumptions are the cause.

Watch

Extended reading notes

Core claim

At the stage where the target mass $m_s$ is perfectly still, the system separates at that mass into a resonating subsystem driven by the absorber link force and a vibrating subsystem driven by the disturbance, and the required force phasor $\vec{f}_a$ is given in closed form by (19) or (20). Theorem 1 then shows that all maximal elastic potential energies (35)-(36) depend on the disturbance and on the structural parameters only. Consequently, as stated in Corollary 1, the elastic energy distribution cannot be altered by the absorber's control authority; the only lever is the structure itself. Building on this, the paper derives the full set of delayed-resonator gain and delay pairs (45)-(46) that generate this force, quantifies the actuator power (64), and formulates the integrated optimization (80) in which the gain and delay are eliminated in favour of the structural parameters, stability being imposed through the spectral abscissa of the delay-differential-algebraic model. In the five-mass case the optimized design reduces the normalized objective from 1.0 to 0.111, with the worst-link elastic energy falling from 0.01115 J to 0.00128 J and the actuator power from 0.06067 W to 0.00653 W while the spectral abscissa stays below $-0.2$ s$^{-1}$. This is what the authors set out to establish: structural redesign, not control retuning, is the way to reconcile vibration absorption with fatigue resistance and energy efficiency.

Load-bearing premise

The analysis assumes a single known excitation frequency, exact steady-state harmonic motion, an ideal linear chain, and an actuator that can deliver the required force without saturation, and it selects the smallest positive delay branch on the strength of the collocated-case rule of thumb; if the real disturbance differs in frequency, parameters drift, or that branch choice is poor in the non-collocated setting, the predicted fatigue and power improvements will not be realized.

Editorial extensions

If this is right

At the full-absorption stage the distribution of link energies is parameter-free with respect to control: any absorber force, not just a delayed-resonator force, leaves the maximal elastic potential energies governed by (35)-(36), so swapping the absorber type cannot cure fatigue.
The delayed-resonator gain and delay are not free design knobs once full absorption and stability are demanded; they are functions of the structural parameters, so closed-loop stability has to be bought by structure, which is exactly why the stability constraint enters the optimization.
Integrated structural and control design can reduce the worst-link elastic energy and the actuator power simultaneously, by roughly an order of magnitude in the five-mass example, while keeping the rightmost characteristic root below a preset negative bound.
For $p < s = d$, with the disturbance acting directly on the target mass, the required absorber force (20) does not involve the vibrating subsystem, which simplifies the design substantially.
The same force-balance reasoning extends, via the transfer-function form (39), to more general topologies than linear chains, so the no-redistribution theorem is not limited to the serial configuration.

Reading between the lines

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

A testable extension the authors leave implicit: because the analysis is a steady-state one at a single frequency, the no-redistribution statement should be re-derived for multi-frequency excitation; for a broadband disturbance the total fatigue measure is a sum of per-frequency contributions, and it is not immediate that control remains unable to shift it.
The branch-selection rule, that the smallest positive delay gives the best stability posture, is taken by analogy from the collocated case; a direct comparison of all delay branches inside the non-collocated optimization could either confirm the rule or reveal different fatigue-power trade-offs.
The optimization assumes the disturbance frequency is known exactly; a frequency-mismatch constraint, for example bounding the spectral abscissa over a neighbourhood of $\omega$, would be the natural industrial follow-up since excitation frequencies drift in practice.
The paper optimizes the steady-state envelope, but the simulations show the transient after the feedback is switched on still carries substantial energy; the design could be extended to penalize transient link energies as well as the equilibrium ones.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core result here holds up. Theorem 1 and Corollary 1 are the real contribution: once the target mass is stopped, the distribution of elastic potential energy across the structure is fixed by the structural parameters and the disturbance, and no amount of control authority can change it. The phasor derivations in Propositions 1–3 are algebraically consistent, and the closed-form expressions for the required force, gain/delay pair, and actuator power go beyond the simulation-only treatments in Olgac and Jenkins. The experimental three-mass study is genuine physical support, not just a simulation of the same model. That is worth credit.

The soft spots are real but mostly addressable. First, the normalization constants in Section 5 are inconsistent with Table 4: Wnom and Pnom are swapped relative to the nominal values, so the reported J = 0.111 is not the normalized objective you would get from the stated formulas. That does not overturn the qualitative conclusion—the optimized system is clearly better on both partial objectives—but the numbers need correcting. Second, the stress-test concern about the delay branch is legitimate. After Eq. (46) the paper says "as a rule of thumb, the pair with the lowest positive delay τ is selected," citing the collocated analysis in [23]. The optimization then eliminates g and τ using that rule, and the numerical study uses (46) with k = 0 without comparing the g+ family or other branches. No theorem establishes that the smallest-delay branch gives the best stability margin or lowest power in the non-collocated setting. The paper is honest that this is a rule of thumb, but because it is load-bearing for the optimization results, the reported reductions are conditional on that choice. A comparison of branches or a proof would settle it. Minor points: the experimental evidence is single runs without error bars, and no code or data are shipped, though the public TDS-CONTROL and GRANSO packages make the numerical portion easier to reproduce.

Who is this for? People working on non-collocated vibration absorption, especially the delayed-resonator community, and anyone interested in joint structural/control design for harmonic disturbances. It deserves a serious referee. The central theoretical claim is new, the experimental evidence is meaningful, and the flaws are fixable rather than fatal. I would accept it for review with a request to fix the normalization, discuss or justify the delay-branch choice, and ideally release the code/data.

Referee Report

1 major / 5 minor

Summary. The paper addresses non-collocated vibration absorption in a serial chain of flexibly linked masses, where a delayed resonator (DR) mounted at one mass is used to fully stop the vibration of a target mass under harmonic excitation. The authors derive phasor-domain expressions for the required total absorber-link force, the maximal elastic potential energies in the structural links, and the complete set of DR gain/delay parameter pairs. They prove that, at the ideal fully suppressed stage, the link energy maxima depend only on the disturbance and on structural parameters, and cannot be altered by the control law. Based on this, they formulate a constrained nonlinear optimization problem that jointly adjusts selected structural parameters and DR parameters to minimize a weighted sum of worst-link fatigue energy and actuator power, subject to a spectral-abscissa stability margin and an absorber-link energy limit. The method is illustrated by an experimental three-mass setup, where a two-parameter grid search is used, and by a five-mass numerical case study solved with GRANSO and TDS-CONTROL.

Significance. If the claims are accepted, the paper makes a useful contribution: Theorem 1 gives a crisp structural-design insight, namely that fatigue redistribution in the non-collocated absorption stage cannot be achieved by control and requires structural parameter changes. The phasor derivations in Propositions 1–3 are algebraically consistent, and the paper provides both experimental validation on a physical three-mass setup and a reproducible numerical optimization pipeline. The explicit inclusion of the spectral-abscissa constraint is a valuable practical safeguard. The main weakness is that the reduction of the optimization problem relies on an unproven branch-selection rule for the DR delay, which makes the reported optimality claims conditional; this is a load-bearing issue that needs to be fixed or explicitly scoped in a revision.

major comments (1)

[Section 2.3 / Section 3, Eqs. (45)–(46) and (79e)–(80)] The optimization problem eliminates the DR parameters g and tau by asserting that the pair with the smallest positive delay is uniquely determined by theta, but no theorem or numerical study establishes that this branch is optimal or even feasible in the non-collocated setting. The text after Eq. (46) explicitly calls this a 'rule of thumb' based on analogy with the collocated case [23], and the stability constraint (80c) and power objective Pmax in (64) depend on the selected branch. Since different branches of (45)–(46) give different characteristic equations (73) and different Pmax values, the feasible set of (80) is conditional on this unproven choice. Moreover, Section 5 states that the DR parameters were 'determined by (46) with k = 0,' which is not the globally smallest positive delay across both families in (45)–(46); it is a single element of one family. Consequently, the optimization may exclude admissible designs where another branch is stable or yields lower power, and the reductions reported in Table 4 are only demonstrated for this particular branch. The authors should either prove the branch-selection rule for the non-collocated case, optimize explicitly over the branch index, or clearly reformulate the claim as a heuristic design procedure rather than a global minimization.

minor comments (5)

[Section 5, text before Table 4] The normalization constants are inconsistent with Table 4: the text states Wnom(theta0) = 0.0607 J and Pnom(theta0) = 0.0111 W, while Table 4 lists the nominal maximal link energy as 0.01115 J and the nominal Pmax as 0.06067 W. The two values appear to be swapped; with the table values the nominal objective equals 1.0 as intended, whereas with the text values it would not. Please correct the text and verify all normalized objective values.
[Section 2.2, Proposition 2] The definition of e_v,j in Eq. (36) appears to contain an index error: for j > s, the expression e_v,j = [o^T_{s-j-1} 1 o^T_{d-j}]^T uses a negative index; it should presumably be e_v,j = [o^T_{j-s-1} 1 o^T_{d-j}]^T to encode the position of mass m_j inside the vibrating subsystem. This does not affect the subsequent formulas, but it should be corrected for clarity.
[Appendix A, Eq. (A.10)] The equation for W^p_a(t) writes the mean term as \bar W^p_{p,mean}, but the surrounding text and the definition just below refer to \bar W^p_{a,mean}; the notation should be made consistent.
[Section 4] There is a duplicated reference in the hardware description: 'more detailed description of the setup hardware can be found in [38], [38], and [25]' should presumably cite [38] only once.
[Section 2, Eq. (2) and general terminology] The word 'dumper' (in 'a spring ki and a dumper ci') should be 'damper' throughout the text.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the unique-force, fatigue-energy, and DR-tuning results follow by direct algebra from the stated model, so nothing reduces to its own input or to a fitted value; the only flagged item is the smallest-positive-delay branch rule adopted by analogy from the authors' collocated analysis [23], which the paper openly labels as a rule of thumb and which is independently…

full rationale

The derivation chain is self-contained. Proposition 1 (Eqs. (19)-(20)) is obtained by substituting x_s = 0 into the block-partitioned phasor equations (21)-(27) and eliminating x_R and x_V; the 'unique' force fa is a direct algebraic consequence, not an assumed design variable. Theorem 1 and Corollary 1 then follow by substituting (24)-(25) into (33)-(36): once x_s = 0, the vibrating subsystem is driven only by fd and the resonating subsystem only by fa, with fa itself fixed by (19)-(20), so the claim that the control cannot redistribute structural fatigue is derived rather than posited. Proposition 3 derives the complete (g, tau) families (45)-(46) by matching magnitude and phase of ge^{-jomega tau} = Q(omega) from (51)-(53); no parameter is fitted to the later objectives. The only passage approaching a circular move is the branch selection after Eq. (46): 'In [23], it was derived for the collocated deployment of the DR that the best stability posture and robustness against nominal and true frequency mismatch is achieved for the first delay branch. Analogous result can be expected for the non-collocated case under consideration. Therefore, as a rule of thumb, the pair with the lowest positive delay tau is selected.' This heuristic is carried over by analogy from [23] (a partly overlapping-author work), and it conditions the elimination step in problem (80), where 'by choosing the solution (45)-(46) with smallest positive delay, they are uniquely determined by theta.' However, it is not a circular reduction: the rule is explicitly presented as an expectation rather than a theorem, the selected pair must still pass the spectral-abscissa constraint (80c) evaluated on the full DDAE (72), and the reported improvements are genuine for the designs obtained, with only the global optimality (not the validity) being conditional on the rule. The five-mass numerical study simulates the same model used in the derivation, so it checks internal consistency only, but the three-mass hardware experiments in Section 4 (Fig. 4) provide independent physical support. No quantity in the paper is defined in terms of the result it is supposed to predict, and no fitted parameter is renamed as a prediction; the score of 1 reflects the single minor, essentially non-load-bearing self-citation in the branch-rule heuristic.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivations are self-contained given the linear chain model; the only hand-chosen quantities are user design weights and bounds (gamma, xi_alpha, xi_a). The delay-branch rule of thumb is an unproven analogy imported from collocated DR theory. No new physical entities are introduced.

free parameters (3)

gamma (objective weighting) = 0.5 (case study)
User-chosen weight balancing fatigue and power objectives in (79a)/(80a); it changes the trade-off but is not fitted to data.
xi_alpha (stability margin) = -0.2 s^-1 (numerical case study)
Hand-chosen spectral abscissa bound in (79c)/(80c); defines the required distance from instability.
xi_a (absorber link energy limit) = 0.01 J (numerical case study)
Hand-chosen safety limit for the absorber link energy, constraint (79d)/(80d).

assumptions (5)

domain assumption Linear time-invariant chain of lumped masses connected by springs and dampers, with harmonic excitation and steady-state motion.
The entire phasor analysis, Propositions 1-3 and the energy formulas, assume this linear model; nonlinearities or time-varying parameters are outside the claimed validity.
domain assumption Perfect vibration suppression of the target mass is achievable, i.e. x_s=0 in steady state.
All derivations are performed at the ideal final stage; transient behavior and imperfect suppression are only checked by simulation/experiment.
standard math The submatrices A_R(omega) and A_V(omega) are invertible, as assumed in Proposition 1.
The force formulas (19)-(20) require invertibility; this is typically satisfied for damped systems but is not proven for all parameter ranges.
domain assumption The DR feedback law has the exact form u(t)=g x_a(t-tau), and the actuator can realize any required force without saturation.
Proposition 3 and the power assessment assume this structure and an ideal actuator; actuator limits are not modeled.
ad hoc to paper The smallest-positive-delay branch in (45)-(46) gives the best stability posture, by analogy with the collocated case in [23].
Stated as a rule of thumb in Section 2.3; it is not proven for the non-collocated setting and is load-bearing for the numerical stability and optimization results.

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Cite this review

Pith. "Pith review of Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption." pith.science (2026). https://pith.science/paper/HO44JYUA

@misc{pith2026250523608,
  author       = {Pith},
  title        = {Pith review of: Integrated design of system structure and delayed resonator towards efficient non-collocated vibration absorption},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HO44JYUA}},
  note         = {Machine review of arXiv:2505.23608}
}

read the original abstract

The problem of non-collocated vibration absorption by a delayed resonator is addressed with emphasis on system fatigue resistance and energy efficiency of control actions. The analysis is performed for a system consisting of an arbitrary large series of flexibly linked single-degree-of-freedom masses. For the stage where the vibration of the target mass is fully absorbed by the non-collocated resonator, key forces, motion amplitudes and potential energies across the system structure are assessed. Next, a complete parameter set of the resonator gain and delay is derived, and the actuation force and power needed by the resonator for the full vibration absorption is determined. The derived quantities are utilized in forming an optimization problem to balance minimal risk of fatigue across the system structure and power needed by the resonator, under the closed loop stability and parameter constraints. Next to the gain and delay of the resonator, selected structural parameters of the system are used as variables in the constrained nonlinear optimization problem. Experimental and numerical case studies are included to demonstrate benefits of the proposed integrated structural and control design.

Figures

Figures reproduced from arXiv: 2505.23608 by the authors.

**Figure 2.** Experimental setup with three flexibly linked carts actuated by the voice coils 4. Structural analysis and optimization of experimental setup In order to explain the mechanisms by which the optimized parameters influence the optimization objectives, we analyse and optimize parameters of the experimental set-up depicted in [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** Left column — objectives, maxima of potential elastic energies in links Wi,max as a function of m3 (top), maxima of power as a function of m3 and ma. Right column — non linear inequalities as a function of m3 and ma, spectral abscissa (top) and maxima of potential elastic energy in absorber link (bottom). Red polygons define structural constraints (linear inequalities), small red dots represent considered grid with … view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Left — Measured transients of potential elastic energies in links i = 1, . . . , 4 Right — Measured transients (from top to bottom): position of target mass m2, control force u, potential elastic energy in absorber link Wa, absolute value of power |P. For all plots: No…

**Figure 5.** Figure 5: Spectrum distribution of the system with the DR, represented by DDAE (72) [PITH_FULL_IMAGE:figures/full_fig_p038_5.png]

**Figure 6.** Figure 6: Motion of the masses mi , i = 1, . . . , 5 for the whole system described by (72) with nominal parameters (blue) and optimized parameters (green). Left — Simulated transients from zero initial conditions: passive regime at t ∈ [0, 15) s; active regime t ∈ [15, 30] s. R…

**Figure 7.** Figure 7: Potential elastic energies in links i = 1, . . . , 6 for the whole system (72) with nominal parameters (blue) and optimized parameters (green). Left — Simulated transients from zero initial conditions: passive regime at t ∈ [0, 15) s; active regime t ∈ [15, 30] s. Righ…

**Figure 8.** Figure 8: Absorber’s characteristics: motion of mass [PITH_FULL_IMAGE:figures/full_fig_p041_8.png]

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

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