1:2 internal resonance between higher modes triggered by fundamental mode in a cantilevered pipe conveying fluid with tip mass

1:2 internal resonance between higher modes triggered by fundamental mode in a cantilevered pipe conveying fluid with tip mass | 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 1:2 internal resonance between higher modes triggered by fundamental mode in a cantilevered pipe conveying fluid with tip mass Tong Shen, Kiyotaka Yamashita, Eisuke Higuchi, Hiroshi Yabuno This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9426816/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 Fluid-conveying cantilevered pipes are slender and highly flexible structures that can undergo large overall deformations and exhibit complex dynamic behavior. As non-conservative and non-self-adjoint systems, they show rich post-critical phenomena, and therefore have attracted considerable attention.In this study, we consider a cantilevered flexible pipe conveying fluid with a concentrated mass attached at the free end. Linear analysis shows that higher-order modes may approach a near 1:2 frequency relation as the flow velocity varies. Because the nonlinearities are dominated by cubic terms, the resonance condition cannot generally be formed only by the higher modes. Instead, an additional lower-frequency mode is required to satisfy a combination relation among three modes. In the present system, the fundamental mode exhibits a very low natural frequency or an overdamped characteristic and therefore can participate in the internal resonance among the higher modes.A nonlinear analysis considering cubic nonlinear terms in both the equation of motion and the boundary conditions is carried out based on a projection method employing a generalized orthogonality relation among the mass matrix, eigenfunctions and adjoint eigenfunctions.The derived amplitude equations reveal that the linearly stable fundamental mode can be excited through nonlinear coupling with the destabilized higher modes.Steady-state analysis further shows that, due to the occurrence of internal resonance, the fundamental mode and the higher modes can attain stable non-trivial amplitudes.Experiments using a flexible silicone rubber pipe with a concentrated tip mass confirm these theoretical predictions. Pipe conveying fluid Nonlinear dynamics Internal resonance Self-excited oscillation Full Text Additional Declarations No competing interests reported. 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-9426816","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":629116570,"identity":"a336d9ee-de8e-4424-824d-43a3eafd8b34","order_by":0,"name":"Tong Shen","email":"","orcid":"","institution":"University of Tsukuba","correspondingAuthor":false,"prefix":"","firstName":"Tong","middleName":"","lastName":"Shen","suffix":""},{"id":629116571,"identity":"26e89981-62a0-41c7-a1a9-46f7911e9b79","order_by":1,"name":"Kiyotaka Yamashita","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIiWNgGAWjYLACxgYJBgb2xgeMDQwSMDE2glokGHgOG0C0sBGnBahUIhmkhbBiBoPjpxM//txhUcc/8zGb5Mw2CwaD+w2MH34w8OXh1HImd7M07xkJCYnbyWySG9skGAyOMTBL9jCwFePUciB3gzRjG9Avt/OPST5sk6jfcIyBQRrovMQGXFrOv9388ydQi/zNw2wgLWBbfuPVciN3mwQvUIvBDWa4w9jw2iJ54+02a6AWyY1nkpktZ5yTYJA8lthm2WOA2y9853M33/zZVscvd/ww482esjoGvsOHD9/4UXEMZ4gpHMAUA0WPwbEEXFrkcbm4BqeWUTAKRsEoGHEAABhsVQxjzUnDAAAAAElFTkSuQmCC","orcid":"","institution":"Fukui University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Kiyotaka","middleName":"","lastName":"Yamashita","suffix":""},{"id":629116572,"identity":"e88523e2-c48b-4011-b583-48a68749a5d3","order_by":2,"name":"Eisuke Higuchi","email":"","orcid":"","institution":"University of Tsukuba","correspondingAuthor":false,"prefix":"","firstName":"Eisuke","middleName":"","lastName":"Higuchi","suffix":""},{"id":629116573,"identity":"1c4265a0-3dc3-4c42-a46e-01881c676df1","order_by":3,"name":"Hiroshi Yabuno","email":"","orcid":"","institution":"University of Tsukuba","correspondingAuthor":false,"prefix":"","firstName":"Hiroshi","middleName":"","lastName":"Yabuno","suffix":""}],"badges":[],"createdAt":"2026-04-15 12:10:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9426816/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9426816/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108491850,"identity":"f9f3831f-33c7-40ea-b8df-b635ab38146b","added_by":"auto","created_at":"2026-05-05 09:55:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1028701,"visible":true,"origin":"","legend":"","description":"","filename":"ActaMechanica.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9426816/v1_covered_0fed92f7-7fcf-43ac-95ee-bd128ad2a2ba.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"1:2 internal resonance between higher modes triggered by fundamental mode in a cantilevered pipe conveying fluid with tip mass","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":"Pipe conveying fluid, Nonlinear dynamics, Internal resonance, Self-excited oscillation","lastPublishedDoi":"10.21203/rs.3.rs-9426816/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9426816/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFluid-conveying cantilevered pipes are slender and highly flexible structures that can undergo large overall deformations and exhibit complex dynamic behavior. As non-conservative and non-self-adjoint systems, they show rich post-critical phenomena, and therefore have attracted considerable attention.In this study, we consider a cantilevered flexible pipe conveying fluid with a concentrated mass attached at the free end. Linear analysis shows that higher-order modes may approach a near 1:2 frequency relation as the flow velocity varies. Because the nonlinearities are dominated by cubic terms, the resonance condition cannot generally be formed only by the higher modes. Instead, an additional lower-frequency mode is required to satisfy a combination relation among three modes. 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