New Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting

New Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting | 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 Original Article New Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting Hongshen Zhang, Jianing He This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-145370/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 The calculation equations for the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are inaccurately described and researched in many studies. These errors are knowledge blind spots in mechanical engineering that have been introduced into the design and manufacturing of standard involute gears. Furthermore, these errors are continually imparted to tens of thousands of students in colleges every yearly. Racks have been incorrectly regarded as the rack-shaped cutters in the derivation of the calculation equations. The theoretical research, numerical simulation and experiments of the undercutting phenomenon of standard involute gears manufactured using rack-shaped cutters are presented. Firstly, theoretical analysis and calculation of the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are performed. Subsequently, the correct minimum teeth number and the minimum modification coefficient without undercutting are deduced. The computer simulations of the theoretical calculations are then examined considering the necessary cases. Experiments of standard involute spur gears with z = 10, 12, 17 and 22 are conducted, thereby validating the research. This study addresses the long-standing errors regarding the traditional undercutting phenomenon and provides an effective theoretical guide for gear design and manufacturing. Mechanical Engineering standard involute gear design standard involute gear manufacturing undercutting phenomenon minimum teeth minimum modification coefficient Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Full Text 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. 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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-145370","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Original Article","associatedPublications":[],"authors":[{"id":7996860,"identity":"f32874e5-8ae8-447a-8fd2-290d287bb95c","order_by":0,"name":"Hongshen Zhang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYBACAxDB2MDAwM8DEQCxidQi2QNkHCBJi8EZYrWYs/cefvlzh02e8Zkzxp8/MNjIbjjA/OwBPi2WPefSrHnPpBWbne0xkzjAkGa84QCbuQFeh93IMTNmbDucuO08jxnQYYcTNxzgYZMgpMXwZ9v/xM39PMYfDjD8J0qL8QPetgOJG3h7DIAOO0CEljNnzJh525ITZ5w5ViZxxiDZeOZhNjP8Wo73GH/82WaX2N+TvPlDRYWdbN/x5md4tQABsjNAQcVMQD1IyQfCakbBKBgFo2BEAwBaA0+n5rQa9AAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-3363-0319","institution":"昆明理工大学","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Hongshen","middleName":"","lastName":"Zhang","suffix":""},{"id":7996861,"identity":"efbace2a-5171-44ba-8a53-89a08631f3fa","order_by":1,"name":"Jianing He","email":"","orcid":"","institution":"Kunming University of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jianing","middleName":"","lastName":"He","suffix":""}],"badges":[],"createdAt":"2021-01-11 21:35:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-145370/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-145370/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":5004700,"identity":"9fc7f50e-a7c5-41ea-b505-5d47155b69f3","added_by":"auto","created_at":"2021-01-15 19:09:48","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":67212,"visible":true,"origin":"","legend":"Undercut and modification of standard involute gear","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/ba25f7c7fb7babb1fa6894a5.jpg"},{"id":5004701,"identity":"20ffebb8-0bdb-4077-a849-4391ca194d91","added_by":"auto","created_at":"2021-01-15 19:09:48","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":29509,"visible":true,"origin":"","legend":"Diagram of traditional involute gear manufacturing with standard rack","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/c4cd17504280d93ba1e8a72a.jpg"},{"id":5004574,"identity":"e6a5b7a2-b6f0-46f6-a03b-8f822aeb8539","added_by":"auto","created_at":"2021-01-15 19:06:48","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":31936,"visible":true,"origin":"","legend":"Standard involute gear manufacturing by a standard rack-shaped cutter","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/2aca54af226cdb24d2776d55.jpg"},{"id":5004704,"identity":"c033e15d-a6a2-4780-a5a0-3da13928e8e5","added_by":"auto","created_at":"2021-01-15 19:09:48","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":67119,"visible":true,"origin":"","legend":"Undercut and modification of standard involute gear in practice","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/fd0d6ef1de8db7092aadeb30.jpg"},{"id":5004566,"identity":"3ef17a20-0bd5-4a9e-a2d9-76cba2b52991","added_by":"auto","created_at":"2021-01-15 19:06:48","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":356333,"visible":true,"origin":"","legend":"Representative simulated standard involute spur gear with z=10. A) Standard involute spur gear of z=10 without modification. B) Enlarged tooth profile of z=10 without modification. C) xmin=0.41. D) Enlarged tooth profile corrected with xmin=0.41. E) Standard involute spur gears of z=10 with xmin=0.68. F) Enlarged tooth profile of z=10 with xmin=0.68.","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/52eab33a1fffc81566f8b399.jpg"},{"id":5004577,"identity":"66d8a8ae-9c83-4a8c-b706-9a5915ee44c4","added_by":"auto","created_at":"2021-01-15 19:06:49","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":135592,"visible":true,"origin":"","legend":"Representative simulated manufacturing drawing of spur gear with z=12. A) xmin=0. B) Enlarged tooth profile without modification. C) xmin=0.29. D) Enlarged tooth profile corrected with xmin=0.29. E) xmin=0.57. F) Enlarged tooth profile corrected with xmin=0.57.","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/bc3272458f4104699c7958bd.jpg"},{"id":5004960,"identity":"3cc519ad-5433-49ef-b64c-f12bca5d44bc","added_by":"auto","created_at":"2021-01-15 19:12:48","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":159439,"visible":true,"origin":"","legend":"Representative simulated manufacturing drawing of z=17. A) Diagram of z=17 in conventional theory. B) Original tooth profile. C) Diagram of z=17 with xmin=0.28 (considering the effect of bottom clearance). D) Enlarged corrected tooth profile.","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/e7cce69c86b35849bd363e6a.jpg"},{"id":5004705,"identity":"b1580234-6cf5-48ce-a8e5-63a1e38ed84b","added_by":"auto","created_at":"2021-01-15 19:09:48","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":113355,"visible":true,"origin":"","legend":"Representative simulated manufacturing drawing of z=22. A) Diagram of z=22 in conventional theory; B) Enlarged tooth profile.","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/cc80df304002ffe978c6a707.jpg"},{"id":5004575,"identity":"46bd3038-3889-41a4-9909-6485133cf092","added_by":"auto","created_at":"2021-01-15 19:06:48","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":70582,"visible":true,"origin":"","legend":"Representative manufactured standard involute spur gear with z=10. A) Standard involute spur gear of z=10 without modification. B) Meshing diagram of two standard involute spur gears of z=10 without modification. C) Standard involute spur gears of z=10 with xmin=0.68. D) Enlarged tooth profile without modification. E) Enlarged tooth profile with modification.","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/3ca3e5276a2e69970f9dfd94.jpg"},{"id":5004572,"identity":"79988ae9-4441-45b0-b773-5c7003f4ce3c","added_by":"auto","created_at":"2021-01-15 19:06:48","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":72603,"visible":true,"origin":"","legend":"Representative manufactured standard involute spur gear with z=12. A) Standard involute spur gear of z=12 without modification. B) Meshing diagram of two standard involute spur gears of z=12 without modification. C) Standard involute spur gears of z=12 with xmin=0.57. D) Enlarged tooth profile without modification. E) Enlarged tooth profile with modification.","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/a9b712b6b08cc12f9b38d15d.jpg"},{"id":5004576,"identity":"b290094d-3e2b-4609-95b7-540260ce1114","added_by":"auto","created_at":"2021-01-15 19:06:49","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":118455,"visible":true,"origin":"","legend":"Diagrams of manufactured standard involute spur gear with z=17. A) Standard involute spur gear of z=17 without modification. B) Meshing diagram of two standard involute spur gears of z=14 without modification. C) Standard involute spur gears of z=12 with xmin=0.28. D) Enlarged tooth profile without modification. E) Enlarged tooth profile with modification.","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/c7929b4d1e7fbda92b4ec445.jpg"},{"id":5004702,"identity":"e6662614-e944-44c2-87a0-7249be5237ec","added_by":"auto","created_at":"2021-01-15 19:09:48","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":43970,"visible":true,"origin":"","legend":"Representative manufactured standard involute spur gear with z=22. A) Standard involute spur gear of z=22 without modification. B) Meshing figure of two standard involute spur gears of z=22 without modification. C) Enlarged tooth profile without modification.","description":"","filename":"12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1/3c3a2ec6aab9ca6172d7ff49.jpg"},{"id":13576015,"identity":"82796858-85ab-4d88-9c3c-3fd7e62c8017","added_by":"auto","created_at":"2021-09-17 04:05:21","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3081632,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptNEW.pdf","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1_covered.pdf"},{"id":5005094,"identity":"80020421-06a0-440c-a9e4-dd918792b99d","added_by":"auto","created_at":"2021-01-15 19:15:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5042666,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptNEW.pdf","url":"https://assets-eu.researchsquare.com/files/rs-145370/v1_stamped.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eNew Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting\u003c/p\u003e","fulltext":[{"header":"Full Text","content":"\u003cp\u003eThis preprint is available for \u003ca href='/article/rs-145370/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e.\u003c/p\u003e"}],"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":false,"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":"standard involute gear design, standard involute gear manufacturing, undercutting phenomenon, minimum teeth, minimum modification coefficient","lastPublishedDoi":"10.21203/rs.3.rs-145370/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-145370/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe calculation equations for the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are inaccurately described and researched in many studies. These errors are knowledge blind spots in mechanical engineering that have been introduced into the design and manufacturing of standard involute gears. Furthermore, these errors are continually imparted to tens of thousands of students in colleges every yearly. Racks have been incorrectly regarded as the rack-shaped cutters in the derivation of the calculation equations. The theoretical research, numerical simulation and experiments of the undercutting phenomenon of standard involute gears manufactured using rack-shaped cutters are presented. Firstly, theoretical analysis and calculation of the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are performed. Subsequently, the correct minimum teeth number and the minimum modification coefficient without undercutting are deduced. The computer simulations of the theoretical calculations are then examined considering the necessary cases. Experiments of standard involute spur gears with \u003cem\u003ez\u003c/em\u003e\u0026thinsp;=\u0026thinsp;10, 12, 17 and 22 are conducted, thereby validating the research. This study addresses the long-standing errors regarding the traditional undercutting phenomenon and provides an effective theoretical guide for gear design and manufacturing.\u003c/p\u003e","manuscriptTitle":"New Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2021-01-15 19:06:45","doi":"10.21203/rs.3.rs-145370/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":"14ebd57a-e763-4913-91dd-adeaac6c46f9","owner":[],"postedDate":"January 15th, 2021","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":1887470,"name":"Mechanical Engineering"}],"tags":[],"updatedAt":"2021-01-15T19:06:47+00:00","versionOfRecord":[],"versionCreatedAt":"2021-01-15 19:06:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-145370","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-145370","identity":"rs-145370","version":["v1"]},"buildId":"WrCJVZZCHTDjtuVLN7oU0","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}