Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions

Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions | 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 Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions János Lelkes, Bendegúz Dezső Bak, Tamás Kalmár-Nagy This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-2692520/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 A novel approach for solving the equations describing the longitudinalvibration of functionally graded rods with viscous and elastic boundaryconditions is proposed. The characteristic equation of the system is derivedand a homotopy method is applied to compute the eigenvalues and mode shapesfor any type of boundary condition, including fixed, free, elastic and/orviscous. Examples are provided for different graded rods with elastic andviscous boundary conditions. The optimal damping of the system is computedbased on the spectral abscissa criterion. It is shown that the qualitativebehavior depends on whether the damping of the system exceeds the optimaldamping. The energy density distribution of the different graded rods is alsodiscussed. An energy measure, the mean scaled energy density distribution isintroduced to characterize the energy distribution along the rod in theasymptotic time limit. It is shown that the energy distribution alsoqualitatively depends on the relation between the actual damping and theoptimal damping of the system. Full Text 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. 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-2692520","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":185934201,"identity":"ee3ccfb0-3a83-4031-bd0b-d7aa8add54f6","order_by":0,"name":"János Lelkes","email":"","orcid":"","institution":"Budapest University of Technology and Economics Faculty of Mechanical Engineering: Budapesti Muszaki es Gazdasagtudomanyi Egyetem Gepeszmernoki Kar","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"János","middleName":"","lastName":"Lelkes","suffix":""},{"id":185934202,"identity":"02bc0fd4-9f12-4917-a6b5-e222423c5242","order_by":1,"name":"Bendegúz Dezső Bak","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArUlEQVRIiWNgGAWjYBACAwY2xgcQBgMDM7FamA1I1sImQZoWc/ZjaRU/99gxmIsdPsBc2EaEFsuetGM3e54lM1jOTktgnkmMFoMD6W03eA4wMxjczjFg5iVKy/nnbYV/DtQDteR/IFLLjbRjzDwHDoNsYSBOi+WMZ8nSMgeO8wD9YnCY5xwRWsz50ww/vjlQLWcunfzwMU8ZEVpggAdEHCBBwygYBaNgFIwCfAAAJjQy2Vpr/30AAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0003-1378-875X","institution":"Budapest University of Technology and Economics Faculty of Mechanical Engineering: Budapesti Muszaki es Gazdasagtudomanyi Egyetem Gepeszmernoki Kar","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Bendegúz","middleName":"Dezső","lastName":"Bak","suffix":""},{"id":185934203,"identity":"ab1222d1-db36-493f-aadb-ab79cc90fb38","order_by":2,"name":"Tamás Kalmár-Nagy","email":"","orcid":"","institution":"Budapest University of Technology and Economics Faculty of Mechanical Engineering: Budapesti Muszaki es Gazdasagtudomanyi Egyetem Gepeszmernoki Kar","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Tamás","middleName":"","lastName":"Kalmár-Nagy","suffix":""}],"badges":[],"createdAt":"2023-03-14 15:16:38","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-2692520/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-2692520/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":34875606,"identity":"05f8db31-0e43-43fc-bc0b-add1cd5da0c9","added_by":"auto","created_at":"2023-03-27 16:31:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4176756,"visible":true,"origin":"","legend":"","description":"","filename":"graded20230307.pdf","url":"https://assets-eu.researchsquare.com/files/rs-2692520/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"","lastPublishedDoi":"10.21203/rs.3.rs-2692520/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-2692520/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"A novel approach for solving the equations describing the longitudinalvibration of functionally graded rods with viscous and elastic boundaryconditions is proposed. 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