Passive Vibration Isolation Characteristics of Negative Extensibility Metamaterials

Hemant Sharma; Jinkyu Yang; Seokgyu Yang; Somya Ranjan Patro

REVIEW 3 major objections 6 minor 36 references

Passive Vibration Isolation Characteristics of Negative Extensibility Metamaterials

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A threshold-triggered stiffness switch turns a forced Duffing oscillator into a passive low-frequency vibration isolator, cutting resonant amplitude by about 87% and beating asymmetric bistable isolators by 80–90% at high excitation.

desk verdict A reasonable extension of countersnapping-based isolation ideas, but the headline amplitude reductions rest on a nondimensionalization mismatch in the strain-softening branch that needs correcting before the quantitative claims can be trusted. read the letter →

arxiv 2507.00396 v1 pith:XNCYHRXT submitted 2025-07-01 nlin.AO

Somya Ranjan Patro , Hemant Sharma , Seokgyu Yang , Jinkyu Yang This is my paper · ORCID

classification nlin.AO MSC 34C1570K4074H45

keywords negativeextensibilitypassivevibrationisolationcountersnappinginstabilityBraessparadoxstrainhardening/softeningDuffingoscillatorasymmetricbistablesystemsharmonicbalance

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 argues that a single spring-mass-damper whose stiffness switches from strain hardening to strain softening when the displacement crosses a fixed threshold can isolate low-frequency vibrations with no active control. The switch is the dynamic counterpart of the Braess spring paradox: at fixed displacement the restoring force jumps, so the oscillator countersnaps away from resonance rather than resonating. On the harmonic-balance frequency response, crossing the threshold $x_T=1$ drops the amplitude from 1 to 0.134 (86.59%), and the return switch drops it to 0.17 (83%). Compared with two types of asymmetric bistable isolators with matched linear stiffness, the proposed negative-extensibility system reduces transient and steady-state displacement by roughly 80–90% under high-amplitude excitation, and by 73–77% in the reverse switching direction.

What carries the argument

The load-bearing object is the countersnapping instability (CSI), the displacement-driven version of the Braess spring paradox: severing a supporting string reconfigures two springs from series to parallel, quadrupling stiffness so that, at fixed displacement, the restoring force jumps. In the oscillator, CSI is implemented as a threshold rule: the restoring force is $x+\tilde{\kappa}x^3$ with $\tilde{\kappa}=1$ below $x_T$ and $\tilde{\kappa}=-0.25$ above it, and both branches share the same equilibrium at the dual-stiffness point. The harmonic-balance frequency-amplitude relation supplies the two stable branches between which the system snaps, and the drop from the high-amplitude hardening branch to the low-amplitude softening branch is what produces isolation.

What would settle it

Drive a physical spring-string negative-extensibility cell near the dimensionless resonance frequency $\lambda\approx1.1$ with forcing amplitude large enough to cross $x_T=1$, and measure the steady-state peak displacement; the paper's claim predicts a drop to roughly 0.13–0.17 times the threshold amplitude, so a prototype that shows no such drop, or measurable energy loss during switching, would refute the quantitative isolation claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that counter-snapping instability—the displacement-controlled version of the Braess spring paradox—can be turned into a passive vibration-isolation mechanism. The model is the forced Duffing-type oscillator $x''+2\zeta x'+x+\tilde{\kappa}x^3=\Gamma\cos(\lambda\tau)$ with $\tilde{\kappa}=1$ (strain hardening) below the threshold and $\tilde{\kappa}=-0.25$ (strain softening) above it. Harmonic balance gives two response branches, and whenever the hardening branch would drive the amplitude past $x_T=1$, the system jumps to the softening branch and avoids resonance, with amplitude reductions of 86.59% and 83% for the two jump directions. The paper then reports that this switching outperforms asymmetric bistable systems by approximately 80–90% in transient and steady-state displacement amplitude under high-amplitude vibrations, and by 73–77% when the switch direction is from softening back to hardening.

Load-bearing premise

The result assumes the stiffness switch is instantaneous, lossless, and triggered purely by the displacement crossing the threshold; if real switching takes finite time, dissipates energy, or requires passing through an unstable configuration, the predicted amplitude reductions no longer carry over physically.

Editorial extensions

If this is right

Resonance avoidance becomes an automatic property of the oscillator itself: the vibration amplitude triggers the stiffness change, so no sensor, controller, or external energy input is needed to tune the isolator.
Near the natural frequency the response is capped near the threshold value instead of growing without bound, so high-amplitude forcing is converted into a jump to a much smaller steady-state amplitude.
The three-dimensional parametric space gives a design rule: the negative-extensibility effect activates only in low-damping, high-amplitude, near-resonance conditions, and the activation volume shrinks and moves to higher excitation frequencies as the baseline stiffness increases.
With matched linear stiffness, the negative-extensibility system's steady-state peak displacement is about 80–90% lower than that of asymmetric bistable systems under high excitation, and 73–77% lower for the reverse switching scenario.

Reading between the lines

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

A natural extension is to couple many negative-extensibility cells into a chain or lattice; if the threshold switch survives coupling, countersnapping fronts could convert large traveling waves into small-amplitude motion, giving broadband passive attenuation beyond the single-degree-of-freedom result.
The reported 80–90% advantage is for peak displacement at matched linear stiffness; transmitted force and vibration-energy transmissibility could give a different margin, and computing those quantities is a natural follow-up not present in the paper.
The model switches at the same threshold in both directions; inserting a deliberate gap between the forward and reverse thresholds would add hysteresis and create a tunable isolation band, which is a straightforward modification of the current rule.

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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

I think the reader's report is about right, and the stress-test note lands. The paper is a clear, well-organized theoretical study of a switching Duffing oscillator framed as a negative-extensibility metamaterial. What is genuinely new is the three-dimensional activation surface in amplitude-frequency-damping space and the head-to-head comparison with two asymmetric bistable isolators; the countersnapping mechanism itself comes from Ducarme et al., and the authors say so. The HBM and time-domain simulations are standard and the paper is honestly written, including an explicit statement in the conclusions that practical realization needs future development.

The soft spot is load-bearing. Equation (3) as used cannot represent the stated SS branch with delta = 4 beta in the same nondimensional time coordinate as the SH branch. Either the linear stiffness factor of 4 is missing, or the SS branch is normalized by its own natural frequency, which changes tau and lambda by a factor of two relative to SH. The paper does not specify which convention it is using, and the time-domain switching plots in Figs. 4 and 8 appear to combine both branches in a single coordinate without that rescaling. So the quoted 86.59% drop and the 80-90% advantage over bistable systems are not yet supported by a consistent model.

The switching rule itself is also idealized: instantaneous, lossless, reversible at |x| = x_T. No energy cost, finite switch time, or passage through an unstable configuration is discussed. The 4:1 stiffness ratio and the threshold x_T = 1 are chosen so that switching moves the resonance away, so part of the performance is baked into the design. That is a legitimate design idea, but the quantitative superiority claim needs a more physical switching model or at least a clear statement of the idealization and its consequences.

Who should read this: people working on mechanical metamaterials, passive vibration isolation, and switching nonlinear oscillators. It deserves a serious referee rather than a desk reject. My recommendation is major revision: redo the nondimensionalization with one global time scale, re-plot the frequency responses and time traces, and report what the amplitude reductions actually are. If they survive, this is a solid incremental contribution; if not, the qualitative mechanism still stands but the headline numbers will need to come down.

Referee Report

3 major / 6 minor

Summary. This paper proposes a single-degree-of-freedom oscillator with a piecewise nonlinear stiffness that switches from strain-hardening (SH) to strain-softening (SS) when the displacement amplitude exceeds a threshold, motivated by the Braess spring paradox and countersnapping instability. The authors use harmonic balance and time-domain simulations to characterize the frequency response, produce a three-dimensional activation diagram, and compare the proposed negative extensibility (NE) system with two types of asymmetric bistable systems. The central claim is that the NE system achieves passive, low-frequency vibration isolation with an 86.59% reduction in resonant amplitude and an 80–90% improvement over bistable isolators at high excitation amplitudes.

Significance. If the quantitative results were correct, the paper would offer an interesting passive strategy for resonance avoidance that exploits countersnapping instability. The paper usefully draws attention to the dynamic potential of negative extensibility metamaterials and provides a parameter study. However, the reliability of the paper's central quantitative claims depends on a correct and consistent nondimensionalization of the two stiffness branches, and the current treatment has an inconsistency that affects the reported amplitude reductions and the comparison with bistable systems.

major comments (3)

[Sec. 3, Eq. (3); Sec. 5.1, Fig. 4] The strain-softening branch equation is incorrect under the stated parameter choices. With δ=4β and a common time scale based on the initial SH natural frequency, the SS equation of motion should be x~''+2ζx~'+4x~−x~^3=Γcos(λτ), not x~''+2ζx~'+x~−0.25x~^3 as in Eq. (3). The derivation of Eq. (3) uses the branch-specific κ1 in the nondimensionalization, but the switching simulations use a single dimensionless time and a common threshold x_T=1. As a result, the SS resonance in Fig. 4 is placed at λ≈1 instead of λ≈2, and the reported 86.59% reduction at λ≈1.1 is a comparison between the SH branch and a softened branch with the same linear stiffness, not the stated δ=4β branch. This issue also shifts the parametric space in Fig. 5 and changes the comparison with bistable systems in Sec. 5.3.
[Sec. 5.3, Table 3] The comparison with bistable systems uses NE systems whose SS linear stiffness is set to δ~=2.6 and 15 to match the second stable states of the bistable systems. If the time-domain simulations for the NE system use Eq. (3) with κ~=−0.25, then the simulated SS branch has a linear stiffness of 1, not 2.6 or 15, so the comparison is not with the model described in the text. If instead the simulations use δ~=2.6 or 15, the SS equation must include that factor in the linear term, and the claimed 80–90% improvements must be recomputed. The authors need to specify the exact equations they actually simulated for the NE system in this section.
[Sec. 4, Fig. 3] The switching model is an instantaneous, reversible, lossless switch at |x~|=x_T. For the claimed 'passive' isolation mechanism, the physical spring-paradox realization involves severing a string, which is not reversible without external action, and real snap-through transitions generally involve energy dissipation and hysteresis. The manuscript does not provide an energy balance for the switch, nor does it discuss finite switching time or dissipation during switching. The reported amplitude reductions are therefore ideal-limit results; the paper should state this limitation explicitly and ideally quantify the sensitivity of the isolation performance to switching losses.

minor comments (6)

[Eq. (2)] The definition Γ=F0√(κ3/κ1^3) becomes imaginary for the SS branch because κ3=γ<0; the authors should use |κ3| or specify that the SH branch value is used throughout.
[Sec. 5.1, Fig. 4] The sentence about switching back to the SHFD regime, causing a drop from 1 to 0.17, does not specify the frequency ratio at which this occurs; please add the value.
[Fig. 4(b) caption] The caption says 'strain-softening (SHPP)' where it should say 'SSPP'; also verify that the black-to-red and red-to-black direction labels are consistent between the time-domain and phase-portrait descriptions.
[Sec. 5.3] The text says 'From all the twelve-time domain responses' but Fig. 8 has six panels; please clarify whether this counts the NE and bistable responses separately.
[Abstract/Conclusions] The abstract claims improvements in both transient and steady-state displacement amplitudes, whereas the Conclusions state only steady-state improvement; align the wording.
[Introduction] The novelty relative to Ref. [9] (Ducarme et al.) should be made explicit; a parameter sweep and comparison with bistable systems are useful additions, but the introduction should state clearly what new physical mechanism or result is being claimed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stiffness-switching model and its claimed vibration-isolation performance are computed forward from stated inputs, not reduced to those inputs by definition.

full rationale

The derivation chain is a forward model: Eqs. (1)-(3) define a Duffing oscillator with strain-hardening and strain-softening branches, the parameters (kappa~ = 1 or -0.25, delta = 4 beta, gamma = -alpha, threshold x_T = 1) are explicitly stated inputs chosen to mimic the spring-paradox stiffness ratio, and the countersnapping switching rule is an operational threshold condition. The reported quantitative outcomes (86.59% reduction, 83% reduction, 80-90% improvement over bistable systems) are computed outputs of the stated equation of motion via harmonic balance and time-domain integration; they are not fitted parameters or renamed inputs. The bistable comparison is external: the NE system's linear stiffnesses are matched to those of the Type I/II bistable wells and the NE thresholds are set from the bistable equilibrium geometry, so the comparison does not presuppose the claimed advantage. The only author self-citation, reference [18] (Zhang-Yang-Zhu), appears in the introduction as background on bistable isolators and is not load-bearing in the derivation. Concerns about the strain-softening branch nondimensionalization (the delta/beta = 4 factor) or the physical realizability of instantaneous lossless switching are correctness or modeling-fidelity issues, not circular reductions in which an output is equivalent to an input by construction.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The quantitative results (86.59%, 83%, 80-90%) are conditional on several hand-chosen parameters and idealized assumptions: the threshold x_T=1, the 4:1 stiffness ratio, cubic nonlinearity, and an instantaneous lossless switch. No parameter is fitted to experimental data, but none is derived from a physical prototype either.

free parameters (8)

displacement threshold x_T = 1
Set by hand in Table 1; the switch triggers at this amplitude, so the claimed reductions depend on it.
SS to SH linear stiffness ratio delta/beta = 4
Chosen in Sec. 3 to mimic the spring paradox; the resonance shift and amplitude drop follow directly from this ratio.
SH cubic coefficient alpha/beta = 1
Implied by kappa~=1 in Table 1; cubic nonlinearity shape is assumed for simplicity.
SS cubic coefficient gamma/alpha = -1
Chosen as gamma=-alpha in Sec. 3.
damping ratio zeta = 0.05
Table 1; damping strongly controls trigger region in parametric space.
forcing amplitude Gamma = 0.5 and 0.01-3 for comparisons
Selected to demonstrate different response regimes; the 80-90% figure depends on these choices.
bistable threshold displacements x_T^I, x_T^II = 0.25, 1
Table 3; chosen from distance to unstable equilibrium of each bistable system, which affects the comparison.
quintic polynomial coefficients eta_Ij, eta_IIj = Table 2
Fitted to the chosen bistable force-displacement curves; the comparison depends on these fits.

assumptions (4)

domain assumption Harmonic balance with a single harmonic (Eq. 4) adequately represents the response
Higher harmonics are neglected; for strongly nonlinear switching the single-harmonic approximation may miss chaos and superharmonic resonances.
domain assumption Cubic force-displacement law captures strain-hardening and strain-softening behavior
The actual NE metamaterial may have piecewise-linear or other constitutive behavior; cubic is an idealization.
domain assumption The spring paradox network gives an exactly fourfold stiffness ratio with massless, frictionless strings
Sec. 2; real strings and springs have mass, friction, and finite stiffness, so the 4:1 ratio is idealized.
ad hoc to paper Switching occurs instantly and reversibly at the displacement threshold
Sec. 4; no energy balance or transition dynamics is provided.

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Cite this review

Pith. "Pith review of Passive Vibration Isolation Characteristics of Negative Extensibility Metamaterials." pith.science (2026). https://pith.science/paper/XNCYHRXT

@misc{pith2026250700396,
  author       = {Pith},
  title        = {Pith review of: Passive Vibration Isolation Characteristics of Negative Extensibility Metamaterials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XNCYHRXT}},
  note         = {Machine review of arXiv:2507.00396}
}

read the original abstract

Negative extensibility refers to the category of mechanical metamaterials having an unusual phenomenon where the system contracts upon expansion. The dynamic analysis of such systems is crucial for exploring the vibration isolation characteristics, forming the prime focus of the present study. Inspired by the Braess paradox, the mechanical model incorporates coupled tunable nonlinear spring stiffness properties (strain hardening and softening), which alternate when a certain displacement threshold is exceeded. This stiffness switching mechanism facilitates low frequency passive vibration isolation using the phenomenon of countersnapping instability. The vibration isolation characteristics resulting from the stiffness switching mechanism are investigated using time and frequency domain plots. Furthermore, the relationship between the stiffness switching mechanism and various system parameters is visualized using a three dimensional parametric space. The efficacy of the proposed system is evaluated by comparing it with the existing bistable systems, revealing superior performance in isolating high-amplitude vibrations. The proposed mechanism enhances the understanding of dynamic behaviors in critical structural elements for multistable mechanical metamaterials, providing insights and opportunities for innovative adaptive designs.

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Works this paper leans on

36 extracted references · 32 canonical work pages

[1]

P. Jiao, J. Mueller, J.R. Raney, X. Zheng, A.H. Alavi, Mechanical metamaterials and beyond, Nature C ommunications 14 (2023)

work page 2023
[2]

Bertoldi, V

K. Bertoldi, V. Vitelli, J. Christensen, M. Van Hecke, Flexible mechanical metamaterials, Nature Reviews Materials 2 (2017) 1–11. https://www.nature.com/articles/natrevmats201766 (accessed May 22, 2025)

work page 2017
[3]

Frenzel, M

T. Frenzel, M. Kadic, M. Wegener, Three -dimensional mechanical metamaterials with a twist, Science 358 (2017) 1072 –1074. https://doi.org/10.1126/science.aao4640

doi:10.1126/science.aao4640 2017
[4]

Surjadi, L

J.U. Surjadi, L. Gao, H. Du, X. Li, X. Xiong, N.X. Fang, Y. Lu, Mechanical Metamaterials and Their E ngineering Applications, Adv Eng Mater 21 (2019) 1800864. https://doi.org/10.1002/adem.201800864

doi:10.1002/adem.201800864 2019
[5]

Barchiesi, M

E. Barchiesi, M. Spagnuolo, L. Placidi, Mechanical metamaterials: a state of the art, Mathematics and Mechanics of Solids 24 (2019) 212–234. https://doi.org/10.1177/1081286517735695

doi:10.1177/1081286517735695 2019
[6]

X. Yu, J. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review, Progress in Materials Science 94 (2018) 114 –173. https://www.sciencedirect.com/science/article/pii/S0079642517301445 (accessed May 26, 2025)

work page 2018
[7]

J.C. Ji, Q. Luo, K. Ye, Vibration control based metamaterials and origami structures: a state-of-the-art review, Mechanical Systems and Signal Processing 161 (2021) 107945. https://www.sciencedirect.com/science/article/pii/S088832702100340X (accessed May 26, 2025)

work page 2021
[8]

Zadpoor, Mechanical meta -materials, Materials Horizons 3 (2016) 371 –381

A.A. Zadpoor, Mechanical meta -materials, Materials Horizons 3 (2016) 371 –381. https://pubs.rsc.org/en/content/articlehtml/2016/mh/c6mh00065g (accessed May 29, 2025)

work page 2016

Show all 36 references

[9]

Ducarme, B

P. Ducarme, B. Weber, M. Van Hecke, J.T.B. Overvelde, Exotic mechanical properties enabled by counte rsnapping instabilities, Proc. Natl. Acad. Sci. U.S.A. 122 (2025) e2423301122. https://doi.org/10.1073/pnas.2423301122

2025 doi
[10]

Nicolaou, F

Z.G. Nicolaou, F. Jiang, A.E. Motter, Metamaterials with negative compressibility highlight evolving interpretations and opportunities, Nature Communications 15 (2024) 8573. https://www.nature.com/articles/s41467 -024-52853-x (accessed May 22, 2025)

2024
[11]

J. Zha, Z. Zhang, Reversible negative compressibility metamaterials inspired by Braess’s paradox, Smart Materials and Structures 33 (2024) 075036. https://iopscience.iop.org/article/10.1088/1361-665X/ad59e6/meta (accessed March 20, 2025)

2024 doi
[12]

Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung Operations Research 12 (1968) 258 –268

D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung Operations Research 12 (1968) 258 –268. https://doi.org/10.1007/BF01918335

1968 doi
[13]

Penchina, L.J

C.M. Penchina, L.J. Penchina, The Braess paradox in mechanical, traffic, and other networks, American Journal of Physics 71 (2003) 479–482. https://pubs.aip.org/aapt/ajp/article-abstract/71/5/479/1044666 (accessed May 22, 2025)

2003
[14]

Cohen, P

J.E. Cohen, P. Horowitz, Paradoxical behaviour of mechanical and electrical networks, Nature 352 (19 91) 699 –701. https://www.nature.com/articles/352699a0 (accessed March 20, 2025). 11

2025
[15]

Chen, E.G

M.L. Chen, E.G. Karpov, Bistability and thermal coupling in elastic metamaterials with negative compressibility, Phys. Rev. E 90 (2014) 033201. https://doi.org/10.1103/PhysRevE.90.033201

2014 doi
[16]

Caprini, F

D. Caprini, F. Battista, P. Zajdel, G. Di Muccio, C. Guardiani, B. Trump, M. Carter, A.A. Yakovenko, E. Amayuelas, L. Bartolomé, Bubbles enable volumetric negative compressibility in metastable elastocapillary systems, Nature Communications 15 (2024) 5076. https://www.nature.c...

2024
[17]

Karpov, L.A

E.G. Karpov, L.A. Danso, J.T. Klein, Negative extensibility metamaterials: Occurrence and design-space topology, Phys. Rev. E 96 (2017) 023002. https://doi.org/10.1103/PhysRevE.96.023002

2017 doi
[18]

Zhang, J

M. Zhang, J. Yang, R. Zhu, Origami -based bistable metastructures for low -frequency vibration control, Journal of Applied Mechanics 88 (2021) 051009. https://asmedigitalcollection.asme.org/appliedmechanics/article -abstract/88/5/051009/1096924 (accessed May 26, 2025)

2021
[19]

J. Zhao, G. Zhou, D. Zhang, I. Kovacic, R. Zhu, H. Hu, Integrated design of a lightweight metastruct ure for broadband vibration isolation, International Journal of Mechanical Sciences 244 (2023) 108069. https://www.sciencedirect.com/science/article/pii/S002074032200947X (acce...

2023
[20]

P. Ling, L. Miao, B. Ye, J. You, W. Zhang, B. Yan, Ultra-low frequency vibration isolation of a novel click-beetle-inspired structure with large quasi -zero stiffness region, Journal of Sound and Vibration 558 (2023) 117756. https://www.sciencedirect.com/science/article/pii/S0...

2023
[21]

Z. Wei, Y. Wu, H. Lai, J. Qian, Compression-Twist Coupling Mechanical Metamaterials with Programmed Bistability, Acta Mech. Solida Sin. (2025). https://doi.org/10.1007/s10338-025-00583-y

2025 doi
[22]

Kovacic, M.J

I. Kovacic, M.J. Brennan, The Duffing equation: nonlinear oscillators and their behaviour, John Wile y & Sons, 2011. https://books.google.com/books?hl=en&lr=&id=f6oZ0cwjTs8C&oi=fnd&pg=PT6&dq=Georg++%22Duffing+Equation%22&ots =8ViD6IoydI&sig=yeVJJTpf6CAgE7NaH2rc-Ai-WYs (accesse...

2011
[23]

Ramlan, M.J

R. Ramlan, M.J. Brennan, I. Kovacic, B.R. Mace, S.G. Burrow, Exploiting knowledge of jump -up and jump-down frequencies to determine the parameters of a Duffing oscillator, Communications in Nonlinear Science and Numerical Simulation 37 (2016) 282–

2016
[24]

Brennan, I

M.J. Brennan, I. Kovacic, A. Carrella, T.P. Waters, On the jump-up and jump-down frequencies of the Duffing oscillator, Journal of Sound and Vibration 318 (2008) 1250 –1261. https://www.sciencedirect.com/science/article/pii/S0022460X08003805 (accessed May 25, 2025)

2008
[25]

Harne, K.-W

R.L. Harne, K.-W. Wang, Harnessing bistable structural dynamics: for vibration control, energy harvesting and sensing, John Wiley & Sons, 2017. https://books.google.com/books?hl=en&lr=&id=gGXbDQAAQBAJ&oi=fnd&pg=PA1&dq=Harnessing+Bistable+Structural+Dyn amic&ots=lLPexrkgCO&sig=...

2017
[26]

Hasan, S

M.N. Hasan, S. Paul, T.E. Greenwood, R.G. Parker, Y.L. Kong, P. Wang, Harmonically induced shape morphing of bistable buckled beam with static bias, Extreme Mechanics Letters 76 (2025) 102299. https://www.sciencedirect.com/science/article/pii/S2352431625000112 (accessed March ...

2025
[28]

J. Liu, M. Wang, H. Pu, S. Zhou, Z. Li, Y. Sun, J. Ding, Y. Peng, S. Xie, J. Luo, Theoretical and experimental analysis of symmetric and asymmetric magnet -based bistable vibration isolators, Mechanical Systems and Signal Processing 224 (2025) 111956. https://www.sciencedirect...

2025
[29]

J. Dou, Z. Li, Y. Cao, H. Yao, R. Bai, Magnet based bi -stable nonlinear energy sink for torsional vibration suppression of rotor system, Mechanical Systems and Signal Processing 186 (2023) 109859. https://www.sciencedirect.com/science/article/pii/S088832702200927X (accessed M...

2023
[30]

Masana, S

R. Masana, S. Khazaaleh, H. Alhussein, R.S. Crespo, M.F. Daqaq, An origami-inspired dynamically actuated binary switch, Applied Physics Letters 117 (2020). https://pubs.aip.org/aip/apl/article/117/8/081901/39687 (accessed March 31, 2025)

2020
[31]

Agarwal, K.W

V. Agarwal, K.W. Wang, On the nonlinear dynamics of a Kresling -pattern origami under harmonic force excitation, Extreme Mechanics Letters 52 (2022) 101653. https://www.sciencedirect.com/science/article/pii/S2352431622000 293 (accessed May 31, 2025)

2022
[32]

L. Ren, W. Zhang, T. Dong, Y. Zhang, Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell: an experimental and numerical study, Appl. Math. Mech.-Engl. Ed. 45 (2024) 779–794. https://doi.org/10.1007/s10483-024- 3111-7

2024 doi
[33]

Kumar, P.M

A.P. Kumar, P.M. Anilkumar, A. Haldar, S. Scheffler, E.L. Jansen, B.N. Rao, R. Rolfes, Tailoring bis tability in unsymmetrical laminates using an additional composite strip, Thin -Walled Structures 168 (2021) 108212. https://www.sciencedirect.com/science/article/pii/S026382312...

2021
[34]

Bashir, P.M

D. Bashir, P.M. Anilkumar, S. Scheffler, A. Haldar, B.N. Rao, R. Rolfes, A Review of the Dynamic Behavior of Thermally Induced Bistable Configurations of Unsymmetrical Composite Laminates and their Applications, Arch Computat M ethods Eng 32 (2025) 1635–1677. https://doi.org/1...

2025 doi
[35]

Hasan, T.E

M.N. Hasan, T.E. Greenwood, R.G. Parker, Y.L. Kong, P. Wang, Fractal patterns in the parameter space of a bistable Duffing oscillator, Phys. Rev. E 108 (2023) L022201. https://doi.org/10.1103/PhysRevE.108.L022201

2023 doi
[291]

https://www.sciencedirect.com/science/article/pii/S1007570416300041 (accessed May 25, 2025)

2025
[6004]

https://www.nature.com/articles/s41467-023-41679-8 (accessed March 20, 2025)

2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

P. Jiao, J. Mueller, J.R. Raney, X. Zheng, A.H. Alavi, Mechanical metamaterials and beyond, Nature C ommunications 14 (2023)

work page 2023

[2] [2]

Bertoldi, V

K. Bertoldi, V. Vitelli, J. Christensen, M. Van Hecke, Flexible mechanical metamaterials, Nature Reviews Materials 2 (2017) 1–11. https://www.nature.com/articles/natrevmats201766 (accessed May 22, 2025)

work page 2017

[3] [3]

Frenzel, M

T. Frenzel, M. Kadic, M. Wegener, Three -dimensional mechanical metamaterials with a twist, Science 358 (2017) 1072 –1074. https://doi.org/10.1126/science.aao4640

doi:10.1126/science.aao4640 2017

[4] [4]

Surjadi, L

J.U. Surjadi, L. Gao, H. Du, X. Li, X. Xiong, N.X. Fang, Y. Lu, Mechanical Metamaterials and Their E ngineering Applications, Adv Eng Mater 21 (2019) 1800864. https://doi.org/10.1002/adem.201800864

doi:10.1002/adem.201800864 2019

[5] [5]

Barchiesi, M

E. Barchiesi, M. Spagnuolo, L. Placidi, Mechanical metamaterials: a state of the art, Mathematics and Mechanics of Solids 24 (2019) 212–234. https://doi.org/10.1177/1081286517735695

doi:10.1177/1081286517735695 2019

[6] [6]

X. Yu, J. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review, Progress in Materials Science 94 (2018) 114 –173. https://www.sciencedirect.com/science/article/pii/S0079642517301445 (accessed May 26, 2025)

work page 2018

[7] [7]

J.C. Ji, Q. Luo, K. Ye, Vibration control based metamaterials and origami structures: a state-of-the-art review, Mechanical Systems and Signal Processing 161 (2021) 107945. https://www.sciencedirect.com/science/article/pii/S088832702100340X (accessed May 26, 2025)

work page 2021

[8] [8]

Zadpoor, Mechanical meta -materials, Materials Horizons 3 (2016) 371 –381

A.A. Zadpoor, Mechanical meta -materials, Materials Horizons 3 (2016) 371 –381. https://pubs.rsc.org/en/content/articlehtml/2016/mh/c6mh00065g (accessed May 29, 2025)

work page 2016