{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T00:52:12Z","timestamp":1784076732173,"version":"3.55.0"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,6]],"date-time":"2020-08-06T00:00:00Z","timestamp":1596672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Gansu Provincial Natural Science Foundation of China","award":["18JR3RA140"],"award-info":[{"award-number":["18JR3RA140"]}]},{"name":"The Open Fund of Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering at Wuhan University of Science and Technology","award":["MTMEOF2019B02"],"award-info":[{"award-number":["MTMEOF2019B02"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51375320"],"award-info":[{"award-number":["51375320"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In order to theoretically research the tooth surface maximum contact stress of a Cylindrical Gear with Variable Hyperbolic Circular-Arc-Tooth-Trace (VH-CATT), the computing formula of maximum contact stress of VH-CATT cylindrical gear is investigated according to Hertz formula in this paper. Insufficient contact fatigue strength will lead to pitting corrosion, plastic deformation of tooth surface and other damages. Therefore, the maximum contact stress of tooth surface must be carried out. The contact stress calculation formula is particularly considering the effect of normal force, total carrying length, synthetical curvature radius, and position angle. The present work establishes analytical solutions to research the effect of different parameters for the contact stress of VH-CATT cylindrical gear incorporating elastic deformation on the tooth surface, and which have shown that the different module, transmission ratio, pressure angle, tooth width, and the cutter head radius have a crucial effect on the contact stress and contact ellipse of VH-CATT cylindrical gear along the tooth width direction. Moreover, a finite element analysis is carried out to verify the correctness of the theoretical computing formula of contact stress of VH-CATT cylindrical gear. By contrast with the theoretical calculated value and the stress value of finite element analysis, its error is very small. It is indicated that the derived formula of contact fatigue strength of VH-CATT cylindrical gear has high accuracy and can accurately reflect the real contact stress value of tooth surface, which is beneficial for research on tooth break reduction, pitting, wear resistance and fatigue life improvement of the VH-CATT cylindrical gear. The study results also have a certain reference value for the design and check calculation of the VH-CATT cylindrical gear.<\/jats:p>","DOI":"10.3390\/sym12081318","type":"journal-article","created":{"date-parts":[[2020,8,10]],"date-time":"2020-08-10T07:25:03Z","timestamp":1597044303000},"page":"1318","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Analytical Calculation of the Tooth Surface Contact Stress of Cylindrical Gear with Variable Hyperbolic Circular-Arc-Tooth-Trace"],"prefix":"10.3390","volume":"12","author":[{"given":"Yongqiao","family":"Wei","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rui","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongping","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changbin","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dawei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anran","family":"Wan","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gang","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan 430081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,6]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Analytical Investigation on Load Sharing Characteristics of Herringbone Planetary Gear Train with Flexible Support and Floating Sun Gear","volume":"144","author":"Mo","year":"2020","journal-title":"Mech. Mach. Theory"},{"key":"ref_2","first-page":"1","article-title":"Improvement on Meshing Stiffness Algorithms of Gear with Peeling","volume":"11","author":"Lingli","year":"2019","journal-title":"Symmetry"},{"key":"ref_3","first-page":"1","article-title":"Review of research on international gear transmissions","volume":"8","author":"Qin","year":"2014","journal-title":"J. Chongqing Univ."},{"key":"ref_4","first-page":"171","article-title":"Mathematical Modeling of Rotary Cutter Arc Tooth Line of Cylindrical Gear Shaped by Origin Face of Rotary Cutter","volume":"45","author":"Xiao","year":"2013","journal-title":"J. Sichuan Univ. Eng. Sci. Ed."},{"key":"ref_5","first-page":"210","article-title":"Modeling and strength analysis of cylindrical gears with curvilinear shape teeth for manufacture","volume":"2","author":"Wang","year":"2012","journal-title":"J. Sichuan Univ. Eng. Sci. Ed."},{"key":"ref_6","first-page":"1","article-title":"Modelling and strength analysis for manufacture-oriented cylindrical gear with arcuate tooth trace","volume":"26","author":"Tang","year":"2016","journal-title":"Aust. J. Mech. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11","DOI":"10.21278\/TOF.40402","article-title":"Method of processing and analysis of meshing and contact of circular-arc-tooth-trace cylindrical gears","volume":"40","author":"Qi","year":"2016","journal-title":"Trans. FAMENA"},{"key":"ref_8","first-page":"196","article-title":"Study on the tooth surface and curvature characteristics of cylindrical gear with variable hyperbolic arc-tooth-trace","volume":"49","author":"Wei","year":"2017","journal-title":"Adv. Eng. Sci."},{"key":"ref_9","first-page":"17","article-title":"Analysis on geometric characteristics of cylindrical gear with variable hyperbolic circular-arc-tooth-trace","volume":"50","author":"Luo","year":"2018","journal-title":"Adv. Eng. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"61","DOI":"10.21278\/TOF.42106","article-title":"Numerical Analysis for Isothermal Elastohydrodynamic Lubrication of Cylindrical Gears with Variable Hyperbolic Circular-Arc-Tooth-Trace","volume":"42","author":"Wei","year":"2018","journal-title":"Trans. FAMENA"},{"key":"ref_11","first-page":"451","article-title":"Research on Thermal Elastohydrodynamic Lubrication of Cylindrical Gears with Curvilinear Shaped Teeth","volume":"39","author":"Wei","year":"2018","journal-title":"J. Chin. Soc. Mech. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1016\/S0094-114X(01)00049-0","article-title":"Mathematical model and undercutting of cylindrical gears with curvilinear shaped teeth","volume":"36","author":"Tseng","year":"2001","journal-title":"Mech. Mach. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3901\/JME.2012.05.165","article-title":"CNC machining method of whole modified surface of cylindrical gears with arcuate tooth trace","volume":"48","author":"Ma","year":"2012","journal-title":"J. Mech. Eng."},{"key":"ref_14","first-page":"119","article-title":"Research on the forming theory analysis and digital model of circular arc gear shaped by rotary cutter","volume":"48","author":"Zhao","year":"2016","journal-title":"J. Sichuan Univ. Eng. Sci. Ed."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1115\/1.3260763","article-title":"Helical gears with circular arc teeth: Simulation of conditions of meshing and bearing contact","volume":"107","author":"Litvin","year":"1985","journal-title":"ASME J. Mech. Transm. Autom. Des."},{"key":"ref_16","first-page":"670","article-title":"Analysis on Contact Stress after Gear Meshing and Finite Element of Stiffness","volume":"3752","author":"Ru","year":"2015","journal-title":"Appl. Mech. Mater."},{"key":"ref_17","first-page":"21","article-title":"Stress Analysis of an Asymmetric Spur Gear Tooth with Series of Circular Hole using Finite Element Method","volume":"1","author":"Morrish","year":"2018","journal-title":"J. Innov. Mech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.mechmachtheory.2005.03.001","article-title":"Design, manufacture, stress analysis, and experimental tests of low-noise high endurance spiral bevel gear","volume":"41","author":"Litvin","year":"2006","journal-title":"Mech. Mach. Theory"},{"key":"ref_19","first-page":"30","article-title":"Non-standard cylindrical gears","volume":"21","author":"Parsons","year":"2004","journal-title":"Gear Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.cma.2013.12.017","article-title":"Computerized design, simulation of meshing, and finite element analysis of two types of geometry of curvilinear cylindrical gears","volume":"272","author":"Fuentes","year":"2014","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1177\/0954406219876954","article-title":"Analytical investigation on load-sharing characteristics for multi-power face gear split flow system","volume":"234","author":"Shuai","year":"2020","journal-title":"J. Mech. Eng. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1007\/s40430-014-0272-6","article-title":"Contact strength analysis of ciucular-arc-tooth-trace cylindrical gear","volume":"38","author":"Sun","year":"2016","journal-title":"J. Braz. Soc. Mech. Sci. 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