{"paper_id":"1167e818-489c-4290-bd3a-7f2e1fad4cea","body_text":"Principal and Combination Parametric Resonances of an Electromagnetically Suspended Vehicle subject to Base Excitation | 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 Principal and Combination Parametric Resonances of an Electromagnetically Suspended Vehicle subject to Base Excitation Jithu Paul, Karel N. van Dalen, Andrei B. Fărăgău, Rens J. van Leijden, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7943406/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 13 You are reading this latest preprint version Abstract This paper investigates the dynamic stability of an electromagnetically suspended vehicle, encountered in Hyperloop and Maglev systems, subject to periodic excitations caused by surface irregularities or vibration of the support induced by external noise. The narrow clearance between the vehicle and the support can make it highly sensitive to small oscillations, since the admissible amplitudes of the vehicle oscillations can be comparable to external excitation amplitude. The vehicle is modelled as a three-degree-of-freedom model where the vehicle is suspended via two identical electromagnetic actuators from a rigid support that oscillates. The governing equations are derived using force and torque balances, incorporating nonlinear electromagnetic forces, and Kirchhoff’s law for the electromagnets with PD control strategy on the airgap. The equations of motion are linearized around the steady state induced by the surface oscillation, yielding a system with time-periodic coefficients. We analytically explore both principal and combination parametric resonances using an extended Hill’s method, and Floquet theory is used for numerical validation. The stability boundaries are obtained as ellipses in control gain parameter space, and the influence of system parameters on these boundaries is characterized. For the principal parametric resonance, the ratio of the sizes of the two obtained ellipses is three to one, whereas for the combination parametric resonance, the ratio is fourteen to one. When all ellipses are simultaneously present, one of the ellipses associated with the combination parametric resonance is the largest. Moreover, we found that in all cases, the relative sizes of the ellipses are independent of the excitation frequency, when normalized by the local width of the stable domain. Additionally, the impact of using hybrid magnets in the supports—combining electromagnets with permanent magnets—on the parametric resonance is analysed, showing that they preserve the same stability boundaries while changing the steady-state response. Results reveal critical conditions under which each type of resonance dominates, offering key insights into the safe design and operation of magnetically suspended vehicles. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 10 Dec, 2025 Reviews received at journal 10 Dec, 2025 Reviews received at journal 07 Dec, 2025 Reviews received at journal 07 Dec, 2025 Reviews received at journal 04 Dec, 2025 Reviewers agreed at journal 09 Nov, 2025 Reviewers agreed at journal 07 Nov, 2025 Reviewers agreed at journal 07 Nov, 2025 Reviewers agreed at journal 07 Nov, 2025 Reviewers invited by journal 07 Nov, 2025 Editor assigned by journal 25 Oct, 2025 Submission checks completed at journal 25 Oct, 2025 First submitted to journal 24 Oct, 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. 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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-7943406\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":542246877,\"identity\":\"87ab9746-842d-4a57-ae50-1ea52f3592c3\",\"order_by\":0,\"name\":\"Jithu Paul\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1UlEQVRIiWNgGAWjYBACxmYGNjCDj72BgYGHwYIfJHgAvxZmiBY2nsMgLRKSPUAOXi0MDDAtEslEamFu5z/2mHeHXWKb5PtjEm8qJCTsGXgPEHIYuzHvmeTENulkNsk5ZyQkeBj4Egj6RZq3jRmsBciQqONh4DEgRks90GGHgYx/IFuI03I4sU0CxGggTouZ5Nwzx43beJKNLeccA2o5TECLYf/BZxJvd1TL9rMffHjjTY2NBHt7j+EDvFoaQFY1IAsx41MPBPIMGFpGwSgYBaNgFKABAFP8PDL3I0VeAAAAAElFTkSuQmCC\",\"orcid\":\"\",\"institution\":\"Delft University of Technology\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Jithu\",\"middleName\":\"\",\"lastName\":\"Paul\",\"suffix\":\"\"},{\"id\":542246878,\"identity\":\"69bb4a10-9cb0-4292-a3e4-0dacf9a70805\",\"order_by\":1,\"name\":\"Karel N. van Dalen\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Delft University of Technology\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Karel\",\"middleName\":\"N. van\",\"lastName\":\"Dalen\",\"suffix\":\"\"},{\"id\":542246879,\"identity\":\"577bd0f1-b749-4848-b008-a3421d89e7c9\",\"order_by\":2,\"name\":\"Andrei B. 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Suspended Vehicle subject to Base Excitation\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"nonlinear-dynamics\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"nody\",\"sideBox\":\"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)\",\"snPcode\":\"11071\",\"submissionUrl\":\"https://submission.nature.com/new-submission/11071/3\",\"title\":\"Nonlinear 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The narrow clearance between the vehicle and the support can make it highly sensitive to small oscillations, since the admissible amplitudes of the vehicle oscillations can be comparable to external excitation amplitude. The vehicle is modelled as a three-degree-of-freedom model where the vehicle is suspended via two identical electromagnetic actuators from a rigid support that oscillates. The governing equations are derived using force and torque balances, incorporating nonlinear electromagnetic forces, and Kirchhoff\\u0026rsquo;s law for the electromagnets with PD control strategy on the airgap. The equations of motion are linearized around the steady state induced by the surface oscillation, yielding a system with time-periodic coefficients. We analytically explore both principal and combination parametric resonances using an extended Hill\\u0026rsquo;s method, and Floquet theory is used for numerical validation. The stability boundaries are obtained as ellipses in control gain parameter space, and the influence of system parameters on these boundaries is characterized. For the principal parametric resonance, the ratio of the sizes of the two obtained ellipses is three to one, whereas for the combination parametric resonance, the ratio is fourteen to one. When all ellipses are simultaneously present, one of the ellipses associated with the combination parametric resonance is the largest. Moreover, we found that in all cases, the relative sizes of the ellipses are independent of the excitation frequency, when normalized by the local width of the stable domain. Additionally, the impact of using hybrid magnets in the supports\\u0026mdash;combining electromagnets with permanent magnets\\u0026mdash;on the parametric resonance is analysed, showing that they preserve the same stability boundaries while changing the steady-state response. Results reveal critical conditions under which each type of resonance dominates, offering key insights into the safe design and operation of magnetically suspended vehicles.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Principal and Combination Parametric Resonances of an Electromagnetically Suspended Vehicle subject to Base Excitation\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2025-11-18 06:32:38\",\"doi\":\"10.21203/rs.3.rs-7943406/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0},{\"type\":\"decision\",\"content\":\"Revision requested\",\"date\":\"2025-12-10T18:46:10+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"editorInvitedReview\",\"content\":\"\",\"date\":\"2025-12-10T09:40:41+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"editorInvitedReview\",\"content\":\"\",\"date\":\"2025-12-07T16:36:58+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"editorInvitedReview\",\"content\":\"\",\"date\":\"2025-12-07T06:53:48+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"editorInvitedReview\",\"content\":\"\",\"date\":\"2025-12-04T14:01:02+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewerAgreed\",\"content\":\"232409455513381560549634918951736831190\",\"date\":\"2025-11-09T09:06:41+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewerAgreed\",\"content\":\"77975021916930935737801498604850713407\",\"date\":\"2025-11-07T09:45:12+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewerAgreed\",\"content\":\"61055449248780765924172570266064315479\",\"date\":\"2025-11-07T09:12:25+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewerAgreed\",\"content\":\"198957955909876416564167035158206195333\",\"date\":\"2025-11-07T09:08:43+00:00\",\"index\":\"hide\",\"fulltext\":\"\"},{\"type\":\"reviewersInvited\",\"content\":\"\",\"date\":\"2025-11-07T09:00:26+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"editorAssigned\",\"content\":\"\",\"date\":\"2025-10-25T18:41:52+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"checksComplete\",\"content\":\"\",\"date\":\"2025-10-25T12:53:42+00:00\",\"index\":\"\",\"fulltext\":\"\"},{\"type\":\"submitted\",\"content\":\"Nonlinear Dynamics\",\"date\":\"2025-10-24T21:49:16+00:00\",\"index\":\"\",\"fulltext\":\"\"}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"nonlinear-dynamics\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"nody\",\"sideBox\":\"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)\",\"snPcode\":\"11071\",\"submissionUrl\":\"https://submission.nature.com/new-submission/11071/3\",\"title\":\"Nonlinear Dynamics\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"stoa\",\"reportingPortfolio\":\"Springer Hybrid\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false}}],\"origin\":\"\",\"ownerIdentity\":\"cca39b62-11f7-4b22-b5b4-20a9be18c38f\",\"owner\":[],\"postedDate\":\"November 18th, 2025\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"in-revision\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2025-12-10T18:53:34+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2025-11-18 06:32:38\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-7943406\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-7943406\",\"identity\":\"rs-7943406\",\"version\":[\"v1\"]},\"buildId\":\"8U1c8b4HqxoKbykW_rLl7\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}