Study on local bifurcation of nonlinear energy sink with inerter and grounded stiffness

Study on local bifurcation of nonlinear energy sink with inerter and grounded stiffness | 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 Study on local bifurcation of nonlinear energy sink with inerter and grounded stiffness Peng Sui, Yong-Jun Shen, Xiaona Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1605075/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 Novel and efficient vibration control units such as inerter and grounded stiffness have contributed significantly to structural vibration reduction. However, as the performance of vibration suppression systems gradually improves, their structures become more complex. And the coupling effects among complex structures, as well as the effects on the system dynamics, are hazy. This aims to investigate the influence of combined structure of inerter and grounded stiffness on the bifurcation behaviors of nonlinear energy sink. The damping of the primary system, a parameter that has been neglected in the majority of studies, is also included in the model. The closed-form solutions of the system steady-state response are derived by the complexification-averaging method and verified numerically. Then, the control equations of stability judgment, saddle-node bifurcation, and Hopf bifurcation are calculated. Sensitivity analysis is performed for each parameter and is visualized in the form of two-dimensional and three-dimensional bifurcation diagrams. It is found that two kinds of bifurcation boundaries on the ( ξ 2 , f ) plane move slightly up and Hopf bifurcation boundaries become complicated both owing to the introduction of the damping in the primary system. The bifurcation zones on the ( ξ 2 , f ) and ( K 2 , f ) planes enlarge with the increase of inerter coefficient. Increasing grounded stiffness has the opposite influence on areas of the bifurcation boundary about ( ξ 2 , f ) and ( μ , f ), except for the special case that the area of saddle-node bifurcation on the ( μ , f ) plane increases accordingly. The amplitude of external force leading to bifurcations will increase when the inerter and grounded stiffness get greater jointly. Nonlinear energy sink Inerter Grounded stiffness Bifurcation Complexification-averaging method 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-1605075","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":104494695,"identity":"89449430-cb4a-4bb4-b067-13f220c89852","order_by":0,"name":"Peng Sui","email":"","orcid":"","institution":"Shijiazhuang Tiedao University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Peng","middleName":"","lastName":"Sui","suffix":""},{"id":104494696,"identity":"67ad7cb7-3e52-4b77-8ba1-c2cd95c1157b","order_by":1,"name":"Yong-Jun Shen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYDACCSjN2MDA+CChwoY0LcwGD86kkaAFCNgkH7YdIqxDfnbzscc8FXfsmmfkHqtIYDvAwN/enYBXi8GdY+nGPGeeJTfOyEu7kcBzh0HizNkN+LVI5JhJ87YdTmackWN2I0HiGVAkF78W+Rn53+BaChIMDhPWwnAjhw2kxQ6khSEhgQgtBjfSzCTnnDmcwNjzxlgi4UAaD0G/yM9IfibxpuKwvWF7juHHn/9s5Pjbewk4DAiYeBgYEjc2QDg8BJWDAOMPBgZ7eaKUjoJRMApGwYgEAJUFS6ZT9D9iAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-8768-1958","institution":"Shijiazhuang Tiedao University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yong-Jun","middleName":"","lastName":"Shen","suffix":""},{"id":104494697,"identity":"53f06c66-9358-41b3-a6f4-0f147ef1a615","order_by":2,"name":"Xiaona Wang","email":"","orcid":"","institution":"Shijiazhuang Tiedao University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xiaona","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2022-04-28 13:53:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1605075/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1605075/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":21395564,"identity":"527f9d2c-6b37-48a4-a730-8efb3dd79586","added_by":"auto","created_at":"2022-05-12 15:20:15","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3488100,"visible":true,"origin":"","legend":"","description":"","filename":"Originalmanuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1605075/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Study on local bifurcation of nonlinear energy sink with inerter and grounded stiffness","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1605075/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"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":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":"Nonlinear energy sink, Inerter, Grounded stiffness, Bifurcation, Complexification-averaging method","lastPublishedDoi":"10.21203/rs.3.rs-1605075/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1605075/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Novel and efficient vibration control units such as inerter and grounded stiffness have contributed significantly to structural vibration reduction. However, as the performance of vibration suppression systems gradually improves, their structures become more complex. And the coupling effects among complex structures, as well as the effects on the system dynamics, are hazy. This aims to investigate the influence of combined structure of inerter and grounded stiffness on the bifurcation behaviors of nonlinear energy sink. The damping of the primary system, a parameter that has been neglected in the majority of studies, is also included in the model. The closed-form solutions of the system steady-state response are derived by the complexification-averaging method and verified numerically. Then, the control equations of stability judgment, saddle-node bifurcation, and Hopf bifurcation are calculated. Sensitivity analysis is performed for each parameter and is visualized in the form of two-dimensional and three-dimensional bifurcation diagrams. It is found that two kinds of bifurcation boundaries on the ( ξ 2 , f ) plane move slightly up and Hopf bifurcation boundaries become complicated both owing to the introduction of the damping in the primary system. The bifurcation zones on the ( ξ 2 , f ) and ( K 2 , f ) planes enlarge with the increase of inerter coefficient. Increasing grounded stiffness has the opposite influence on areas of the bifurcation boundary about ( ξ 2 , f ) and ( μ , f ), except for the special case that the area of saddle-node bifurcation on the ( μ , f ) plane increases accordingly. 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