Symmetrical solution to Continuous transmission pulley problem

Symmetrical solution to Continuous transmission pulley problem | 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 Symmetrical solution to Continuous transmission pulley problem Aleksandar Hatzivelkos This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8454064/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 We study a geometric closure problem arising in belt–pulley mechanisms, where a prescribed belt length induces an implicit relation between the radii of two pulleys. For a given discrete radius profile r(t), the corresponding profile R(t) is defined pointwise on a uniform grid by solving an implicit belt-length equation, which yields an explicit mapping R = g(r) via a one-dimensional root-finding step. To enforce symmetry, we propose a simple discrete symmetrization iteration that couples only the symmetric grid pairs (t i , 1−t i ) and preserves the closure relation at every iteration. For each fixed pair, we prove geometric decay of the symmetry error under a Lipschitz condition on g in the open-belt regime. Based on the observed regularity of the computed profiles, we additionally derive a low-order quadratic approximation for the symmetric discrete radius samples and provide an a posteriori bound on the maximum deviation using standard interpolation error estimates. Numerical experiments illustrate the convergence of the iteration and the accuracy of the quadratic approximation for representative parameter values. Applied Mathematics belt–pulley mechanism geometric closure implicit equation root finding discrete symmetrization fixed-point iteration geometric convergence quadratic approximation a posteriori error bound Full Text Additional Declarations The authors declare no competing interests. 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-8454064","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":565778150,"identity":"fad72110-0117-4915-923e-7be6c2778fd9","order_by":0,"name":"Aleksandar Hatzivelkos","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVRIiWNgGAWjYDACCRBhwCAHopiJ1cLYANRiTKoWBobEBqK18M9ufv7wR8Hh9A030i8wF+4hxpI7xwwbJAwO5264kVPAPOMZMdbcSDBsMDBIy53ZcyaBmecAETrkb6R/bEgwSEuXJFqLwY0cw4YDBjYJ/OztB4jTYngjp3Bmg4GNYT97D8NhorTI3Ujf8PHHHwl5Nmb2h4+J0oIEeAxI1MDAwP6AVB2jYBSMglEwQgAAbnA3AuxI4T0AAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0003-4759-7177","institution":"University of Applied Sciences Velika Gorica","correspondingAuthor":true,"prefix":"","firstName":"Aleksandar","middleName":"","lastName":"Hatzivelkos","suffix":""}],"badges":[],"createdAt":"2025-12-26 09:27:47","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8454064/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8454064/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":99152238,"identity":"b01fc307-fc42-4ce5-b1ca-ea6ca24e3600","added_by":"auto","created_at":"2025-12-29 10:40:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":526988,"visible":true,"origin":"","legend":"","description":"","filename":"CMMcontinuouspulleyproblem.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8454064/v1_covered_b9689a83-204d-4a71-8553-78d1076b3e49.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eSymmetrical solution to Continuous transmission pulley problem\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Veleučilište Velika Gorica","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":"belt–pulley mechanism, geometric closure, implicit equation, root finding, discrete symmetrization, fixed-point iteration, geometric convergence, quadratic approximation, a posteriori error bound","lastPublishedDoi":"10.21203/rs.3.rs-8454064/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8454064/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study a geometric closure problem arising in belt–pulley mechanisms, where a prescribed belt length induces an implicit relation between the radii of two pulleys. For a given discrete radius profile r(t), the corresponding profile R(t) is defined pointwise on a uniform grid by solving an implicit belt-length equation, which yields an explicit mapping R = g(r) via a one-dimensional root-finding step. To enforce symmetry, we propose a simple discrete symmetrization iteration that couples only the symmetric grid pairs (t\u003csub\u003ei\u003c/sub\u003e, 1−t\u003csub\u003ei\u003c/sub\u003e) and preserves the closure relation at every iteration. For each fixed pair, we prove geometric decay of the symmetry error under a Lipschitz condition on g in the open-belt regime. Based on the observed regularity of the computed profiles, we additionally derive a low-order quadratic approximation for the symmetric discrete radius samples and provide an a posteriori bound on the maximum deviation using standard interpolation error estimates. 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