A Novel Analytical Homotopy Solution for Nonlinear Vibration of Porous GPL Reinforced Plate on Nonlinear Elastic Foundation

A Novel Analytical Homotopy Solution for Nonlinear Vibration of Porous GPL Reinforced Plate on Nonlinear Elastic Foundation | 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 A Novel Analytical Homotopy Solution for Nonlinear Vibration of Porous GPL Reinforced Plate on Nonlinear Elastic Foundation Nooshin Asadifar, Ali Reza Saidi, Mohammad Ali Sabahi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8015281/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 study presents a comprehensive analysis of the nonlinear free vibration behavior of porous plates reinforced with graphene platelets resting on a Winkler-Pasternak nonlinear elastic foundation, considering the in-plane and out of plane displacement component. The governing equations of motion of the thin plate are derived based on Von Kármán strain relations and Hamilton's principle. The Galerkin method is employed to transform the partial differential equations into ordinary ones. The system of nonlinear differential equations has been solved using the Homotopy Analysis Method (HAM) for the first time. The time response of the plate vibrations has been found analytically and compared with that predicted by the numerical method for the porous GPL-reinforced plate. It is shown that the results have very good agreement with the numerical solutions that obtained from the fourth-order Runge-Kutta method. The resulting nonlinear frequencies and time histories provide critical insights into the dynamic response of the reinforced plates. The findings reveal that neglecting in-plane displacements leads to significant errors, especially as the initial amplitude increases. Additionally, a symmetrical distribution pattern of graphene platelets substantially enhances the plate's stiffness and nonlinear frequency. The analysis also indicates that an increase in geometric parameters, such as the plate aspect ratio, yields a decrease in nonlinear frequency. These results underscore the importance of considering geometric parameters, boundary conditions, foundation parameters and the distribution patterns of graphene platelets for accurately assessing the nonlinear behavior of graphene-reinforced porous plates. Nonlinear vibration Porous plates Graphene platelets Galerkin method Homotopy analysis method System of nonlinear differential equations 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-8015281","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":545666441,"identity":"3a21642a-b069-4e13-94cf-dc2103a8a3a5","order_by":0,"name":"Nooshin Asadifar","email":"","orcid":"","institution":"Shahid Bahonar University of Kerman","correspondingAuthor":false,"prefix":"","firstName":"Nooshin","middleName":"","lastName":"Asadifar","suffix":""},{"id":545666442,"identity":"7d6f6a1c-9265-4b9f-9eaa-f7b969d7a024","order_by":1,"name":"Ali Reza 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Foundation","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":"Nonlinear vibration, Porous plates, Graphene platelets, Galerkin method, Homotopy analysis method, System of nonlinear differential equations","lastPublishedDoi":"10.21203/rs.3.rs-8015281/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8015281/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study presents a comprehensive analysis of the nonlinear free vibration behavior of porous plates reinforced with graphene platelets resting on a Winkler-Pasternak nonlinear elastic foundation, considering the in-plane and out of plane displacement component. The governing equations of motion of the thin plate are derived based on Von K\u0026aacute;rm\u0026aacute;n strain relations and Hamilton's principle. The Galerkin method is employed to transform the partial differential equations into ordinary ones. The system of nonlinear differential equations has been solved using the Homotopy Analysis Method (HAM) for the first time. The time response of the plate vibrations has been found analytically and compared with that predicted by the numerical method for the porous GPL-reinforced plate. It is shown that the results have very good agreement with the numerical solutions that obtained from the fourth-order Runge-Kutta method. The resulting nonlinear frequencies and time histories provide critical insights into the dynamic response of the reinforced plates. The findings reveal that neglecting in-plane displacements leads to significant errors, especially as the initial amplitude increases. Additionally, a symmetrical distribution pattern of graphene platelets substantially enhances the plate's stiffness and nonlinear frequency. The analysis also indicates that an increase in geometric parameters, such as the plate aspect ratio, yields a decrease in nonlinear frequency. These results underscore the importance of considering geometric parameters, boundary conditions, foundation parameters and the distribution patterns of graphene platelets for accurately assessing the nonlinear behavior of graphene-reinforced porous plates.\u003c/p\u003e","manuscriptTitle":"A Novel Analytical Homotopy Solution for Nonlinear Vibration of Porous GPL Reinforced Plate on Nonlinear Elastic Foundation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-26 18:51:08","doi":"10.21203/rs.3.rs-8015281/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"67c1190a-9509-4a78-9cfd-c46ffdb66e18","owner":[],"postedDate":"November 26th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-11-28T13:53:08+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-26 18:51:08","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8015281","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8015281","identity":"rs-8015281","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}