Nonlinear Stability and Vibration Analysis of Flexible Spacecraft Solar Arrays under Thermally Induced Deformations within the Thermal Flutter Regime during the Penumbra Phase

Nonlinear Stability and Vibration Analysis of Flexible Spacecraft Solar Arrays under Thermally Induced Deformations within the Thermal Flutter Regime during the Penumbra Phase | 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 Article Nonlinear Stability and Vibration Analysis of Flexible Spacecraft Solar Arrays under Thermally Induced Deformations within the Thermal Flutter Regime during the Penumbra Phase Omid Motaharifard, Kamran Daneshjou, Majid Bakhtiari This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7900165/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Feb, 2026 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract This study develops a nonlinear analytical model of a spacecraft with flexible solar arrays to investigate complex vibration phenomena occurring within the thermal flutter regime, particularly during the satellite’s transition through the penumbra phase. The model captures dynamic behaviors driven by transient thermal gradients and structural flexibility key contributors to self-excited oscillations in orbit. Two major nonlinear phenomena are examined: limit cycle oscillations (LCOs) self-sustained periodic motions arising under subcritical thermal conditions and internal resonance, which emerges within the flutter velocity range and strongly influences modal interactions. Special attention is given to the Hubble Space Telescope (HST), where nonlinearities stem from large elastic deformations and thermally induced loads and torques. A combined analytical–numerical approach is adopted, incorporating nonlinear dynamic modeling and time-domain simulations. The focus is on the penumbra phase, where the thermal wave propagation velocity is less than the speed of solar radiation. The system’s coupled bending–bending–twisting motion equations include quadratic and cubic nonlinearities. Through model reduction, a tractable set of nonlinear ordinary differential equations is derived. Using the method of multiple scales (MMS), internal resonance conditions are analytically identified to characterize energy exchanges among modes. To further assess the system’s dynamic behavior, phase portraits, Poincaré sections, and bifurcation diagrams are employed. These analyses elucidate the onset and evolution of LCOs, offering critical insights into the coupled thermomechanical instabilities of flexible space structures and contributing to the design of more robust spacecraft systems under dynamic thermal environments. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics flexible solar arrays structural nonlinearities thermally induced nonlinear Limit Cycle Oscillations Thermal Flutter Penumbra Phase Poincaré maps bifurcation diagrams Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 19 Feb, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 14 Dec, 2025 Reviews received at journal 12 Dec, 2025 Reviewers agreed at journal 03 Dec, 2025 Reviews received at journal 22 Nov, 2025 Reviewers agreed at journal 19 Nov, 2025 Reviewers agreed at journal 14 Nov, 2025 Reviewers invited by journal 13 Nov, 2025 Editor assigned by journal 10 Nov, 2025 Editor invited by journal 10 Nov, 2025 Submission checks completed at journal 07 Nov, 2025 First submitted to journal 04 Nov, 2025 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-7900165","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":549404893,"identity":"c1ca6dc0-7dcb-457c-9bba-c2e91b1910ee","order_by":0,"name":"Omid Motaharifard","email":"","orcid":"","institution":"Iran University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Omid","middleName":"","lastName":"Motaharifard","suffix":""},{"id":549404894,"identity":"c92b8d61-0541-44ab-89d1-3c5d06082f17","order_by":1,"name":"Kamran Daneshjou","email":"","orcid":"","institution":"Kamran Daneshjou, Iran University of Science 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Phase","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"flexible solar arrays, structural nonlinearities, thermally induced nonlinear, Limit Cycle Oscillations, Thermal Flutter, Penumbra Phase, Poincaré maps, bifurcation diagrams","lastPublishedDoi":"10.21203/rs.3.rs-7900165/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7900165/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study develops a nonlinear analytical model of a spacecraft with flexible solar arrays to investigate complex vibration phenomena occurring within the thermal flutter regime, particularly during the satellite\u0026rsquo;s transition through the penumbra phase. The model captures dynamic behaviors driven by transient thermal gradients and structural flexibility key contributors to self-excited oscillations in orbit. Two major nonlinear phenomena are examined: limit cycle oscillations (LCOs) self-sustained periodic motions arising under subcritical thermal conditions and internal resonance, which emerges within the flutter velocity range and strongly influences modal interactions. Special attention is given to the Hubble Space Telescope (HST), where nonlinearities stem from large elastic deformations and thermally induced loads and torques. A combined analytical\u0026ndash;numerical approach is adopted, incorporating nonlinear dynamic modeling and time-domain simulations. The focus is on the penumbra phase, where the thermal wave propagation velocity is less than the speed of solar radiation. The system\u0026rsquo;s coupled bending\u0026ndash;bending\u0026ndash;twisting motion equations include quadratic and cubic nonlinearities. Through model reduction, a tractable set of nonlinear ordinary differential equations is derived. Using the method of multiple scales (MMS), internal resonance conditions are analytically identified to characterize energy exchanges among modes. To further assess the system\u0026rsquo;s dynamic behavior, phase portraits, Poincar\u0026eacute; sections, and bifurcation diagrams are employed. These analyses elucidate the onset and evolution of LCOs, offering critical insights into the coupled thermomechanical instabilities of flexible space structures and contributing to the design of more robust spacecraft systems under dynamic thermal environments.\u003c/p\u003e","manuscriptTitle":"Nonlinear Stability and Vibration Analysis of Flexible Spacecraft Solar Arrays under Thermally Induced Deformations within the Thermal Flutter Regime during the Penumbra Phase","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-25 16:52:50","doi":"10.21203/rs.3.rs-7900165/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-12-14T21:38:41+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-12-13T03:08:38+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"267340753469297083214632145481522888917","date":"2025-12-04T02:47:31+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-11-22T07:44:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"143416811041459086808295029269192047014","date":"2025-11-19T09:05:25+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"23259617310289370772786497954552831751","date":"2025-11-14T06:36:27+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-11-13T13:45:51+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-11-11T01:05:21+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-11-10T19:52:46+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-11-07T16:31:16+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-11-04T15:48:54+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b5f835dc-3c65-4efd-b2da-267ebf5d7167","owner":[],"postedDate":"November 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":58456921,"name":"Physical sciences/Engineering"},{"id":58456922,"name":"Physical sciences/Mathematics and computing"},{"id":58456923,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-02-26T13:29:53+00:00","versionOfRecord":{"articleIdentity":"rs-7900165","link":"https://doi.org/10.1038/s41598-026-38274-4","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-02-19 15:58:38","publishedOnDateReadable":"February 19th, 2026"},"versionCreatedAt":"2025-11-25 16:52:50","video":"","vorDoi":"10.1038/s41598-026-38274-4","vorDoiUrl":"https://doi.org/10.1038/s41598-026-38274-4","workflowStages":[]},"version":"v1","identity":"rs-7900165","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7900165","identity":"rs-7900165","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}