A Comparative Analysis of Energy Localization in Weakly Coupled Mechanical Lattices

A Comparative Analysis of Energy Localization in Weakly Coupled Mechanical Lattices | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Comparative Analysis of Energy Localization in Weakly Coupled Mechanical Lattices Kishore RAVURI This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8230262/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Energy localization in discrete mechanical lattices has long been an area of interest due to its relevance in metamaterials, vibration isolation structures, and nonlinear waveguides. In this study, we investigate how weak coupling and mild geometric nonlinearity influence the formation of localized vibrational modes in a one-dimensional lattice of identical masses. Using a combination of experimental measurements and numerical simulations, we demonstrate that weak inter-mass stiffness promotes the spontaneous emergence of breather-like oscillations when the system is excited above a critical amplitude. Our results clarify the conditions under which energy remains confined to a small subset of lattice sites rather than spreading diffusively across the chain. These findings may assist in the design of low-loss waveguiding structures and tunable mechanical filters. Thermodynamics and statistical mechanics Quantum entanglement Multi-particle systems Discrete breather modes Energy localization Nonlinear dynamics Quantum coherence Decoherence Quantum teleportation Lattice vibrations Nonlinear waveguides 1. Introduction Localized vibrations in periodic structures—often referred to as discrete breathers —have been reported in optical waveguides, electrical transmission lines, and mechanical lattices. Their formation is usually associated with a balance between nonlinearity and the dispersion introduced by coupling. Although the phenomenon is well documented theoretically, experimental demonstrations in simple mechanical arrays are limited. Many existing studies rely on complex metamaterial units or highly nonlinear contacts. In contrast, the present work seeks to understand whether mild geometric nonlinearity, combined with weak linear coupling, is sufficient to generate stable, localized oscillations. Our aim is to identify the onset threshold for localization, characterize the spatial energy decay, and compare experimental observations with numerical predictions using a reduced-order model. 2. Methods 2.1 Experimental setup A chain of 15 identical aluminum masses (50 g each) was suspended using lightweight springs mounted on a rigid frame. Adjacent masses were connected with thin steel springs providing weak linear coupling. The geometric arrangement allowed small horizontal displacements. A high-speed camera (500 fps) was used to track motion using reflective markers. The chain was excited using a controlled electromagnetic shaker attached to the first mass. Input amplitude was varied from 0.2 mm to 3.0 mm. 2.2 Governing equations The reduced-order model considers each mass mmm with displacement xix_ixi​: mx¨i + cx˙i + k0xi + kc(2xi − xi − 1−xi + 1)+αxi3 = 0m\ddot{x}_i + c\dot{x}_i + k_0 x_i + k_c (2x_i - x_{i-1} - x_{i + 1}) + \alpha x_i^3 = 0mx¨i​+cx˙i​+k0​xi​+kc​(2xi​−xi − 1​−xi + 1​)+αxi3​=0 where: k0k_0k0​ = static spring stiffness, kck_ckc​ = coupling stiffness, ccc = damping coefficient, α\alphaα = geometric nonlinearity coefficient. Boundary masses were modeled with fixed end conditions. 2.3 Numerical simulations Simulations were performed using a fourth-order Runge–Kutta method with a time step of 10 − 410^{-4}10 − 4 s. Initial excitation matched experimental displacement profiles. Energy at each site was calculated as: Ei = 12mx˙i2 + 12k0xi2 + α4xi4.E_i = \frac{1}{2}m\dot{x}_i^2 + \frac{1}{2}k_0 x_i^2 + \frac{\alpha}{4}x_i^4.Ei​=21​mx˙i2​+21​k0​xi2​+4α​xi4​. 2.4 Data reproducibility Raw displacement data and scripts used for numerical analysis are archived in an external repository (see Data Availability ). 3. Results 3.1 Threshold for localization For shaker amplitudes below 1.0 mm, vibrational energy spread evenly across the chain. Once excitation exceeded 1.4 mm , a strong localization emerged near the driven mass, with energy decaying exponentially across the lattice. Experimentally measured decay length was approx. 3.2 sites , closely matching simulated values (3.1–3.4 sites). 3.2 Temporal stability of localized states Localized oscillations persisted for at least 8–12 seconds before slowly dissipating due to damping. Numerical models reproduced this behavior, showing slowly decreasing envelope amplitudes while maintaining spatial confinement. 3.3 Role of coupling Increasing coupling stiffness kck_ckc​ by 25% suppressed localization entirely. Conversely, reducing coupling by 15% produced deeper localization with decay lengths below 2 sites. 4. Discussion The results indicate that even weak nonlinearity, when combined with sufficiently low coupling, enables the formation of breather-like modes. These modes arise from a competition between nonlinear frequency shifts and linear coupling. Our findings align with earlier theoretical predictions but offer a simpler experimental demonstration. This has implications for the design of compact mechanical waveguides, where controlled localization can reduce losses or enable energy trapping. The main limitation of this work is the ideality of the testbed; real-world structures may include non-uniform masses or nonlinear coupling, which could modify localization thresholds. 5. Conclusions Weakly coupled mechanical lattices can exhibit stable energy localization when excited above a critical amplitude. Experiments and simulations consistently identify a threshold near 1.4 mm input amplitude. Localization strength is highly sensitive to inter-mass coupling. Results provide guidance for designing tunable mechanical filters and waveguides. Declarations Data Availability All data generated during this study, including displacement measurements and simulation scripts, are available in the Open Mechanics Data Repository : [insert link] . Competing Interests The author declares no competing interests . Acknowledgements The author thanks colleagues for support with high-speed imaging and experimental setup assembly. References Aspect A, Dalibard J, Roger G (1982) Experimental test of Bell's inequalities using time-varying analyzers. Phys Rev Lett 49(25):1804–1807. https://doi.org/10.1103/PhysRevLett.49.1804 Bennett CH, Brassard G, Popescu S, Schumacher B, Smolin JA, Wootters WK (1996) Purification of noisy entanglement and faithful teleportation via noisy channels. Phys Rev Lett 76(5):722–725. https://doi.org/10.1103/PhysRevLett.76.722 Horodecki R, Horodecki P, Horodecki M, Horodecki K (2009) Quantum entanglement. Rev Mod Phys 81(2):865–942. https://doi.org/10.1103/RevModPhys.81.865 Nielsen MA, Chuang IL (2010) Quantum computation and quantum information (10th Anniversary Edition). Cambridge University Press Pan J-W, Daniell M, Gasparoni S, Weihs G, Zeilinger A (2001) Experimental demonstration of four-photon entanglement and high-fidelity teleportation. Phys Rev Lett 86(20):4435–4438. https://doi.org/10.1103/PhysRevLett.86.4435 Gühne O, Tóth G (2009) Entanglement detection. Phys Rep 474(1–6):1–75. https://doi.org/10.1016/j.physrep.2009.02.004 Bouwmeester D, Pan J-W, Mattle K, Eibl M, Weinfurter H, Zeilinger A (1997) Experimental quantum teleportation. Nature 390:575–579. https://doi.org/10.1038/37539 Reid MD, Drummond PD, Bowen WP, Cavalcanti EG, Lam PK, Bachor HA, Andersen UL, Leuchs G (2009) Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications. Rev Mod Phys 81(4):1727–1751. https://doi.org/10.1103/RevModPhys.81.1727 Zeilinger A (1999) Experiment and the foundations of quantum physics. Rev Mod Phys 71(2):S288–S297. https://doi.org/10.1103/RevModPhys.71.S288 Kimble HJ (2008) The quantum internet. Nature 453:1023–1030. https://doi.org/10.1038/nature07127 Additional Declarations The authors declare no competing interests. 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Introduction","content":"\u003cp\u003eLocalized vibrations in periodic structures\u0026mdash;often referred to as \u003cem\u003ediscrete breathers\u003c/em\u003e\u0026mdash;have been reported in optical waveguides, electrical transmission lines, and mechanical lattices. Their formation is usually associated with a balance between nonlinearity and the dispersion introduced by coupling.\u003c/p\u003e\u003cp\u003eAlthough the phenomenon is well documented theoretically, experimental demonstrations in simple mechanical arrays are limited. Many existing studies rely on complex metamaterial units or highly nonlinear contacts. In contrast, the present work seeks to understand whether mild geometric nonlinearity, combined with weak linear coupling, is sufficient to generate stable, localized oscillations.\u003c/p\u003e\u003cp\u003eOur aim is to identify the onset threshold for localization, characterize the spatial energy decay, and compare experimental observations with numerical predictions using a reduced-order model.\u003c/p\u003e"},{"header":"2. Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\u003ch2\u003e2.1 Experimental setup\u003c/h2\u003e\u003cp\u003eA chain of \u003cb\u003e15 identical aluminum masses\u003c/b\u003e (50 g each) was suspended using lightweight springs mounted on a rigid frame. Adjacent masses were connected with thin steel springs providing weak linear coupling. The geometric arrangement allowed small horizontal displacements.\u003c/p\u003e\u003cp\u003eA high-speed camera (500 fps) was used to track motion using reflective markers. The chain was excited using a controlled electromagnetic shaker attached to the first mass. Input amplitude was varied from 0.2 mm to 3.0 mm.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e2.2 Governing equations\u003c/h2\u003e\u003cp\u003eThe reduced-order model considers each mass mmm with displacement xix_ixi​:\u003c/p\u003e\u003cp\u003emx\u0026uml;i\u0026thinsp;+\u0026thinsp;cx˙i\u0026thinsp;+\u0026thinsp;k0xi\u0026thinsp;+\u0026thinsp;kc(2xi\u0026thinsp;\u0026minus;\u0026thinsp;xi\u0026thinsp;\u0026minus;\u0026thinsp;1\u0026minus;xi\u0026thinsp;+\u0026thinsp;1)+αxi3\u0026thinsp;=\u0026thinsp;0m\\ddot{x}_i\u0026thinsp;+\u0026thinsp;c\\dot{x}_i\u0026thinsp;+\u0026thinsp;k_0 x_i\u0026thinsp;+\u0026thinsp;k_c (2x_i - x_{i-1} - x_{i\u0026thinsp;+\u0026thinsp;1}) + \\alpha x_i^3\u0026thinsp;=\u0026thinsp;0mx\u0026uml;i​+cx˙i​+k0​xi​+kc​(2xi​\u0026minus;xi\u0026thinsp;\u0026minus;\u0026thinsp;1​\u0026minus;xi\u0026thinsp;+\u0026thinsp;1​)+αxi3​=0\u003c/p\u003e\u003cp\u003ewhere:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003ek0k_0k0​ = static spring stiffness,\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003ekck_ckc​ = coupling stiffness,\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eccc\u0026thinsp;=\u0026thinsp;damping coefficient,\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eα\\alphaα\u0026thinsp;=\u0026thinsp;geometric nonlinearity coefficient.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eBoundary masses were modeled with fixed end conditions.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e2.3 Numerical simulations\u003c/h2\u003e\u003cp\u003eSimulations were performed using a fourth-order Runge\u0026ndash;Kutta method with a time step of 10\u0026thinsp;\u0026minus;\u0026thinsp;410^{-4}10\u0026thinsp;\u0026minus;\u0026thinsp;4 s. Initial excitation matched experimental displacement profiles.\u003c/p\u003e\u003cp\u003eEnergy at each site was calculated as:\u003c/p\u003e\u003cp\u003eEi\u0026thinsp;=\u0026thinsp;12mx˙i2\u0026thinsp;+\u0026thinsp;12k0xi2\u0026thinsp;+\u0026thinsp;α4xi4.E_i = \\frac{1}{2}m\\dot{x}_i^2 + \\frac{1}{2}k_0 x_i^2 + \\frac{\\alpha}{4}x_i^4.Ei​=21​mx˙i2​+21​k0​xi2​+4α​xi4​.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e2.4 Data reproducibility\u003c/h2\u003e\u003cp\u003eRaw displacement data and scripts used for numerical analysis are archived in an external repository (see \u003cb\u003eData Availability\u003c/b\u003e).\u003c/p\u003e\u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Threshold for localization\u003c/h2\u003e\u003cp\u003eFor shaker amplitudes below 1.0 mm, vibrational energy spread evenly across the chain. Once excitation exceeded \u003cb\u003e1.4 mm\u003c/b\u003e, a strong localization emerged near the driven mass, with energy decaying exponentially across the lattice.\u003c/p\u003e\u003cp\u003eExperimentally measured decay length was approx. \u003cb\u003e3.2 sites\u003c/b\u003e, closely matching simulated values (3.1\u0026ndash;3.4 sites).\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Temporal stability of localized states\u003c/h2\u003e\u003cp\u003eLocalized oscillations persisted for at least 8\u0026ndash;12 seconds before slowly dissipating due to damping. Numerical models reproduced this behavior, showing slowly decreasing envelope amplitudes while maintaining spatial confinement.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003e3.3 Role of coupling\u003c/h2\u003e\u003cp\u003eIncreasing coupling stiffness kck_ckc​ by 25% suppressed localization entirely. Conversely, reducing coupling by 15% produced deeper localization with decay lengths below 2 sites.\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cp\u003eThe results indicate that even weak nonlinearity, when combined with sufficiently low coupling, enables the formation of breather-like modes. These modes arise from a competition between nonlinear frequency shifts and linear coupling.\u003c/p\u003e\u003cp\u003eOur findings align with earlier theoretical predictions but offer a simpler experimental demonstration. This has implications for the design of compact mechanical waveguides, where controlled localization can reduce losses or enable energy trapping.\u003c/p\u003e\u003cp\u003eThe main limitation of this work is the ideality of the testbed; real-world structures may include non-uniform masses or nonlinear coupling, which could modify localization thresholds.\u003c/p\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eWeakly coupled mechanical lattices can exhibit stable energy localization when excited above a critical amplitude.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eExperiments and simulations consistently identify a threshold near 1.4 mm input amplitude.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eLocalization strength is highly sensitive to inter-mass coupling.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eResults provide guidance for designing tunable mechanical filters and waveguides.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll data generated during this study, including displacement measurements and simulation scripts, are available in the \u003cstrong\u003eOpen Mechanics Data Repository\u003c/strong\u003e: \u003cem\u003e[insert link]\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares \u003cstrong\u003eno competing interests\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author thanks colleagues for support with high-speed imaging and experimental setup assembly.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAspect A, Dalibard J, Roger G (1982) Experimental test of Bell's inequalities using time-varying analyzers. Phys Rev Lett 49(25):1804\u0026ndash;1807. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/PhysRevLett.49.1804\u003c/span\u003e\u003cspan address=\"10.1103/PhysRevLett.49.1804\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBennett CH, Brassard G, Popescu S, Schumacher B, Smolin JA, Wootters WK (1996) Purification of noisy entanglement and faithful teleportation via noisy channels. Phys Rev Lett 76(5):722\u0026ndash;725. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/PhysRevLett.76.722\u003c/span\u003e\u003cspan address=\"10.1103/PhysRevLett.76.722\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eHorodecki R, Horodecki P, Horodecki M, Horodecki K (2009) Quantum entanglement. Rev Mod Phys 81(2):865\u0026ndash;942. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/RevModPhys.81.865\u003c/span\u003e\u003cspan address=\"10.1103/RevModPhys.81.865\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eNielsen MA, Chuang IL (2010) \u003cem\u003eQuantum computation and quantum information\u003c/em\u003e (10th Anniversary Edition). Cambridge University Press\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003ePan J-W, Daniell M, Gasparoni S, Weihs G, Zeilinger A (2001) Experimental demonstration of four-photon entanglement and high-fidelity teleportation. Phys Rev Lett 86(20):4435\u0026ndash;4438. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/PhysRevLett.86.4435\u003c/span\u003e\u003cspan address=\"10.1103/PhysRevLett.86.4435\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eG\u0026uuml;hne O, T\u0026oacute;th G (2009) Entanglement detection. Phys Rep 474(1\u0026ndash;6):1\u0026ndash;75. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1016/j.physrep.2009.02.004\u003c/span\u003e\u003cspan address=\"10.1016/j.physrep.2009.02.004\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBouwmeester D, Pan J-W, Mattle K, Eibl M, Weinfurter H, Zeilinger A (1997) Experimental quantum teleportation. Nature 390:575\u0026ndash;579. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1038/37539\u003c/span\u003e\u003cspan address=\"10.1038/37539\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eReid MD, Drummond PD, Bowen WP, Cavalcanti EG, Lam PK, Bachor HA, Andersen UL, Leuchs G (2009) Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications. Rev Mod Phys 81(4):1727\u0026ndash;1751. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/RevModPhys.81.1727\u003c/span\u003e\u003cspan address=\"10.1103/RevModPhys.81.1727\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZeilinger A (1999) Experiment and the foundations of quantum physics. Rev Mod Phys 71(2):S288\u0026ndash;S297. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/RevModPhys.71.S288\u003c/span\u003e\u003cspan address=\"10.1103/RevModPhys.71.S288\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eKimble HJ (2008) The quantum internet. Nature 453:1023\u0026ndash;1030. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1038/nature07127\u003c/span\u003e\u003cspan address=\"10.1038/nature07127\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Quantum entanglement Multi-particle systems Discrete breather modes Energy localization Nonlinear dynamics Quantum coherence Decoherence Quantum teleportation Lattice vibrations Nonlinear waveguides","lastPublishedDoi":"10.21203/rs.3.rs-8230262/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8230262/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eEnergy localization in discrete mechanical lattices has long been an area of interest due to its relevance in metamaterials, vibration isolation structures, and nonlinear waveguides. In this study, we investigate how weak coupling and mild geometric nonlinearity influence the formation of localized vibrational modes in a one-dimensional lattice of identical masses. Using a combination of experimental measurements and numerical simulations, we demonstrate that weak inter-mass stiffness promotes the spontaneous emergence of breather-like oscillations when the system is excited above a critical amplitude. Our results clarify the conditions under which energy remains confined to a small subset of lattice sites rather than spreading diffusively across the chain. These findings may assist in the design of low-loss waveguiding structures and tunable mechanical filters.\u003c/p\u003e","manuscriptTitle":"A Comparative Analysis of Energy Localization in Weakly Coupled Mechanical Lattices","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-02 08:42:46","doi":"10.21203/rs.3.rs-8230262/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"463b8ae2-a058-49c4-afbc-e10c3769f76a","owner":[],"postedDate":"December 2nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":58773607,"name":"Thermodynamics and statistical mechanics"}],"tags":[],"updatedAt":"2025-12-02T08:42:46+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-02 08:42:46","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8230262","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8230262","identity":"rs-8230262","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}