Analytical study on periodic and quasi-periodic dynamics of vari-potential bi-stable nonlinear energy sink

Analytical study on periodic and quasi-periodic dynamics of vari-potential bi-stable nonlinear energy sink | 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 Analytical study on periodic and quasi-periodic dynamics of vari-potential bi-stable nonlinear energy sink Lin Chen, Xin Liao, Limeng Yin, Anping Ji, Juan Yan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8616903/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 12 You are reading this latest preprint version Abstract Vari-potential bi-stable nonlinear energy sink (VBNES) has broadband and high-efficiency vibration absorption performance. This paper presents the first comprehensive analytical investigation on the periodic and quasi-periodic dynamics of the system coupling a VBNES with a linear oscillator (LO). Firstly, the exact and approximate theoretical models of the LO-VBNES are established based on Lagrange equation and Taylor series expansion. The potential energy and restoring force surfaces are predicted, and vari-potential barrier effect (VPBE) is verified by numerical response trajectory tracking on these surfaces. Secondly, the complexification-averaging (CX-A) and multiple scales method (MSM) are used to derive the complex-averaged modulation equations of the LO-VBNES. The slow invariant manifold (SIM) of the LO-VBNES is predicted by detecting the fast average system, and the periodic and non-periodic orbits are validated with numerical results. The influences of system parameters on the SIM and the fold singularities of the SIM are discussed. Furthermore, the amplitude-frequency curves of periodic responses in the 1:1 resonance region are investigated via the slow average system, which has the single and multiple responses separated by saddle-node (SN) bifurcations. The analytical and numerical results indicate that the multi-responses may be low- and high-energy harmonics, chaotic, or strongly modulated responses (SMR). The influences of different parameters on the amplitude-frequency response and vibration suppression performance are analyzed. Finally, the analytical expressions of the excitation thresholds of the SMR are predicted, and the numerical verification shows that the analytical thresholds are conservative and effective. Bi-stable nonlinear energy sink Vari-potential barrier effect Slow invariant manifold Amplitude-frequency response Strongly modulated oscillation Inter-well orbit Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 09 Feb, 2026 Reviews received at journal 09 Feb, 2026 Reviews received at journal 08 Feb, 2026 Reviewers agreed at journal 21 Jan, 2026 Reviewers agreed at journal 19 Jan, 2026 Reviewers agreed at journal 19 Jan, 2026 Reviewers agreed at journal 19 Jan, 2026 Reviewers agreed at journal 18 Jan, 2026 Reviewers invited by journal 18 Jan, 2026 Editor assigned by journal 16 Jan, 2026 Submission checks completed at journal 16 Jan, 2026 First submitted to journal 16 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8616903","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":576668827,"identity":"175910d7-0b99-42a8-853e-517356de446d","order_by":0,"name":"Lin Chen","email":"","orcid":"","institution":"Chongqing Three Gorges University","correspondingAuthor":false,"prefix":"","firstName":"Lin","middleName":"","lastName":"Chen","suffix":""},{"id":576668839,"identity":"d307cb59-ede4-464b-8967-9b3370740d9a","order_by":1,"name":"Xin 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sink","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Bi-stable nonlinear energy sink, Vari-potential barrier effect, Slow invariant manifold, Amplitude-frequency response, Strongly modulated oscillation, Inter-well orbit","lastPublishedDoi":"10.21203/rs.3.rs-8616903/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8616903/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Vari-potential bi-stable nonlinear energy sink (VBNES) has broadband and high-efficiency vibration absorption performance. This paper presents the first comprehensive analytical investigation on the periodic and quasi-periodic dynamics of the system coupling a VBNES with a linear oscillator (LO). Firstly, the exact and approximate theoretical models of the LO-VBNES are established based on Lagrange equation and Taylor series expansion. The potential energy and restoring force surfaces are predicted, and vari-potential barrier effect (VPBE) is verified by numerical response trajectory tracking on these surfaces. Secondly, the complexification-averaging (CX-A) and multiple scales method (MSM) are used to derive the complex-averaged modulation equations of the LO-VBNES. The slow invariant manifold (SIM) of the LO-VBNES is predicted by detecting the fast average system, and the periodic and non-periodic orbits are validated with numerical results. The influences of system parameters on the SIM and the fold singularities of the SIM are discussed. Furthermore, the amplitude-frequency curves of periodic responses in the 1:1 resonance region are investigated via the slow average system, which has the single and multiple responses separated by saddle-node (SN) bifurcations. The analytical and numerical results indicate that the multi-responses may be low- and high-energy harmonics, chaotic, or strongly modulated responses (SMR). The influences of different parameters on the amplitude-frequency response and vibration suppression performance are analyzed. Finally, the analytical expressions of the excitation thresholds of the SMR are predicted, and the numerical verification shows that the analytical thresholds are conservative and effective.","manuscriptTitle":"Analytical study on periodic and quasi-periodic dynamics of vari-potential bi-stable nonlinear energy sink","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-21 09:28:12","doi":"10.21203/rs.3.rs-8616903/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-10T00:13:16+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-09T08:46:14+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-02-08T21:14:45+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"37690617561044302534158630286033786236","date":"2026-01-21T14:52:43+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"24458583072273859563366732429591727697","date":"2026-01-19T08:47:13+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"173283191470105134926935618148962587905","date":"2026-01-19T08:00:55+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"71158980298233025968315659455279799939","date":"2026-01-19T06:58:30+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"94890545843302764492404094080343032183","date":"2026-01-19T03:29:28+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-19T03:09:07+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-16T10:04:27+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-16T10:03:11+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2026-01-16T07:26:52+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"ddbd2290-e962-469f-a943-4eaa0ccce30f","owner":[],"postedDate":"January 21st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T20:23:15+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-21 09:28:12","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8616903","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8616903","identity":"rs-8616903","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}