Spectral Analysis of Large-Amplitude Nonlinear Free Vibrations of Beams: Influence of Boundary Conditions on Displacement Localization and Frequency–Amplitude Relationships

Spectral Analysis of Large-Amplitude Nonlinear Free Vibrations of Beams: Influence of Boundary Conditions on Displacement Localization and Frequency–Amplitude Relationships | 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 Spectral Analysis of Large-Amplitude Nonlinear Free Vibrations of Beams: Influence of Boundary Conditions on Displacement Localization and Frequency–Amplitude Relationships VOWOGBE KOSSI HUBERT KOSSIHUBERT, MERRIMI EL BEKKAYE This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8457869/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 This paper presents a comparative spectral analysis of large-amplitude geometrically nonlinear free vibrations of beams under different boundary conditions, namely clamped–clamped (CC) and clamped–simply supported (CSS) configurations. Based on Hamilton’s principle, a nonlinear formulation accounting for bending and axial strain energies induced by large transverse displacements is developed. The nonlinear eigenvalue problem is solved using an iterative numerical procedure, allowing the evaluation of frequency–amplitude relationships and associated transverse displacement fields. Particular attention is paid to the spatial localization of maximum displacement and its dependence on boundary conditions and vibration symmetry. The results demonstrate that boundary conditions significantly affect both the nonlinear frequency shift and the position of peak transverse displacement along the beam span. For CC beams, symmetric and antisymmetric modes exhibit distinct displacement localizations, whereas CSS beams show a pronounced shift of maximum displacement toward intermediate span locations. Quantitative comparisons reveal notable differences in frequency ratios and displacement amplitudes for identical vibration levels. These findings provide physical insight into the role of boundary conditions in nonlinear beam dynamics and offer useful guidance for the design and optimization of slender structural components operating under large vibration amplitudes. Mechanical Engineering Geometrically nonlinear vibrations Euler–Bernoulli beam Large-amplitude vibrations Boundary conditions Spectral method Nonlinear frequency–amplitude relationship Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted 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. 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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-8457869","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":566003935,"identity":"1368afd8-1473-4eb6-86f5-be86b586e7b3","order_by":0,"name":"VOWOGBE KOSSI HUBERT 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Relationships\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"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":"Geometrically nonlinear vibrations, Euler–Bernoulli beam, Large-amplitude vibrations, 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