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In addition, the mutual influence of gear stress states and the cumulative effect of fatigue damage can lead to the correlation between failure modes on the gear, thereby affecting the confidence level of reliability assessments. To address this, the performance degradation of gears over their lifecycle is considered, and time\u2010dependent limit state functions for contact and bending fatigue failures are established. The Kendall rank correlation coefficient is also employed to characterize the time\u2010dependent correlation between failure modes. A time\u2010dependent mixed Copula function is developed to analyze the reliability of gears, elucidating the evolution of reliability throughout the gear's operational lifecycle.<\/jats:p>","DOI":"10.1002\/qre.70061","type":"journal-article","created":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T18:10:47Z","timestamp":1756750247000},"page":"3415-3429","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reliability Analysis of Gear Under Multiple Failure Modes Based on Time\u2010Dependent Mixed Copula Function"],"prefix":"10.1002","volume":"41","author":[{"given":"Zhe","family":"Deng","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center For System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tudi","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center For System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hua\u2010Ming","family":"Qian","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center For System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4162-6387","authenticated-orcid":false,"given":"Ying","family":"Zeng","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center For System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4478-8349","authenticated-orcid":false,"given":"Hong\u2010Zhong","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering University of Electronic Science and Technology of China  Chengdu Sichuan China"},{"name":"Center For System Reliability and Safety University of Electronic Science and Technology of China  Chengdu Sichuan China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,9]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ymssp.2022.109896"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfailanal.2021.105802"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.3431"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00158-015-1239-7"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1061\/JSDEAG.0001577"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1080\/03601217908905329"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.oceaneng.2016.11.048"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109529"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109485"},{"key":"e_1_2_9_11_1","first-page":"229","article-title":"Fonctions De Repartition a n Dimensions Et Leurs Marges","volume":"8","author":"Sklar M.","year":"1959","journal-title":"Publications De L'Institut De Statistique De L'Universite De Paris"},{"key":"e_1_2_9_12_1","first-page":"1","volume-title":"An Introduction to Copulas","author":"Nelsen R. 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