Bifurcation analysis of rotating machinery in non-stationary operations: an angular harmonic balance approach

Bifurcation analysis of rotating machinery in non-stationary operations: an angular harmonic balance approach | 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 Bifurcation analysis of rotating machinery in non-stationary operations: an angular harmonic balance approach Roberto Alcorta, Victor Clerc, Mojtaba Ahani, Didier Remond This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8251488/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract In this paper, we introduce an approach that extends the applicability of numerical continuation methods to the study of rotating machinery where the hypothesis of constant instantaneous angular speed is relaxed. Besides having a profound impact on the system’s dynamics, in particular close to bifurcations, this choice has the practical consequence of forcing the state of the system to include angular coordinates which increase without bounds over time, thus preventing the straightforward use of typical numerical continuation methods. We describe a series of transformations, inspired by the so-called angular approach used in the field of condition monitoring, which systematically recast the system in an equivalent form, in such a way as to render periodic solutions admissible and within the grasp of the harmonic balance method. Furthermore, we detail a practical implementation of this method which reduces the number of back-and-forth transformations between angle and angle-frequency domains to a minimum, an approach which we call the Angular Harmonic Balance Method (AHBM). For the selected test case, we report numerous numerical results in order to showcase the versatility, robustness, and performance of AHBM-based continuation on the simplest possible models which remain representative of the phenomenology expected from more complex rotating machines. This includes the prediction of bifurcations during a ramp response, bifurcation tracking to determine the validity of the simplified model, and the use of continuation as an aid for the study of defects in condition monitoring. AHBM rotating machines Non stationary continuation cyclic excitation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 08 Dec, 2025 Editor assigned by journal 05 Dec, 2025 Submission checks completed at journal 05 Dec, 2025 First submitted to journal 01 Dec, 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-8251488","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":557000278,"identity":"1fed0d30-8f6b-4626-b7e6-5a7ea261ee94","order_by":0,"name":"Roberto Alcorta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIie3RwUrDMBjA8W8Esktg14yN9BVSAq1i8VkyCs1lA4+KoIHCdprnPYaP4Mihl7KzR0tfYN4meDBp50VadTeR/KHJR+kPEgrg8/3JyHEfthvDzXbllsFLP5H2QdAMoiXcLYj/isz0TyRerbf1zSGBUY4qO1yqh8mq2gM3LAaE9x1kWu7SsJQZUIOFHdLFcloKaok41whtOgil82ispQEwBNsBLZZ0DpN3bmaPTyPTdTBL4jdHgpbcK0xVfYCGINRDooEjvCVGYioj+i0hu3Sss4yE7i46K0J7sOgMuBLc9JDhevuqk4SxIq8qndwGwUbVz3B9wXiRd5LPyNcX+fFPndDdid/7fD7fP+4DGcVVK7Ajxh4AAAAASUVORK5CYII=","orcid":"","institution":"Institut National des Sciences Appliquées de Lyon","correspondingAuthor":true,"prefix":"","firstName":"Roberto","middleName":"","lastName":"Alcorta","suffix":""},{"id":557000279,"identity":"c1a7d353-3da1-431c-bd04-8c11684664e2","order_by":1,"name":"Victor Clerc","email":"","orcid":"","institution":"Institut National des Sciences Appliquées de Lyon","correspondingAuthor":false,"prefix":"","firstName":"Victor","middleName":"","lastName":"Clerc","suffix":""},{"id":557000281,"identity":"805bb8d1-fe45-446b-8161-985052ea94dd","order_by":2,"name":"Mojtaba Ahani","email":"","orcid":"","institution":"Institut National des Sciences Appliquées de Lyon","correspondingAuthor":false,"prefix":"","firstName":"Mojtaba","middleName":"","lastName":"Ahani","suffix":""},{"id":557000282,"identity":"5eb409c3-b677-4c09-8493-72cd4870b24b","order_by":3,"name":"Didier Remond","email":"","orcid":"","institution":"Institut National des Sciences Appliquées de Lyon","correspondingAuthor":false,"prefix":"","firstName":"Didier","middleName":"","lastName":"Remond","suffix":""}],"badges":[],"createdAt":"2025-12-01 14:38:08","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8251488/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8251488/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100796132,"identity":"8ec81476-4140-4811-a7c4-7936931e35a9","added_by":"auto","created_at":"2026-01-21 13:40:45","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":11048452,"visible":true,"origin":"","legend":"","description":"","filename":"RAVCMADRNODYAHBM.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8251488/v1_covered_7c30846b-4d54-464e-bcae-fd8249eeeeab.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Bifurcation analysis of rotating machinery in non-stationary operations: an angular harmonic balance approach","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":"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},"keywords":"AHBM, rotating machines, Non stationary, continuation, cyclic excitation","lastPublishedDoi":"10.21203/rs.3.rs-8251488/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8251488/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, we introduce an approach that extends the applicability of numerical continuation methods to the study of rotating machinery where the hypothesis of constant instantaneous angular speed is relaxed. Besides having a profound impact on the system’s dynamics, in particular close to bifurcations, this choice has the practical consequence of forcing the state of the system to include angular coordinates which increase without bounds over time, thus preventing the straightforward use of typical numerical continuation methods. We describe a series of transformations, inspired by the so-called angular approach used in the field of condition monitoring, which systematically recast the system in an equivalent form, in such a way as to render periodic solutions admissible and within the grasp of the harmonic balance method. Furthermore, we detail a practical implementation of this method which reduces the number of back-and-forth transformations between angle and angle-frequency domains to a minimum, an approach which we call the Angular Harmonic Balance Method (AHBM). 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