{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:21:22Z","timestamp":1787239282196,"version":"build-2736575974"},"reference-count":50,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T00:00:00Z","timestamp":1756425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Hunan Provincial Natural Science Foundation of China","award":["2024JJ7113"],"award-info":[{"award-number":["2024JJ7113"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["23B0832"],"award-info":[{"award-number":["23B0832"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["23C0400"],"award-info":[{"award-number":["23C0400"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["202323016881"],"award-info":[{"award-number":["202323016881"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["202222015981"],"award-info":[{"award-number":["202222015981"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["HQ22011"],"award-info":[{"award-number":["HQ22011"]}]},{"name":"Hunan Provincial Natural Science Foundation of China","award":["KFA22014"],"award-info":[{"award-number":["KFA22014"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["2024JJ7113"],"award-info":[{"award-number":["2024JJ7113"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["23B0832"],"award-info":[{"award-number":["23B0832"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["23C0400"],"award-info":[{"award-number":["23C0400"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["202323016881"],"award-info":[{"award-number":["202323016881"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["202222015981"],"award-info":[{"award-number":["202222015981"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["HQ22011"],"award-info":[{"award-number":["HQ22011"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department of China","award":["KFA22014"],"award-info":[{"award-number":["KFA22014"]}]},{"name":"Guiding plan project of Hengyang city","award":["2024JJ7113"],"award-info":[{"award-number":["2024JJ7113"]}]},{"name":"Guiding plan project of Hengyang city","award":["23B0832"],"award-info":[{"award-number":["23B0832"]}]},{"name":"Guiding plan project of Hengyang city","award":["23C0400"],"award-info":[{"award-number":["23C0400"]}]},{"name":"Guiding plan project of Hengyang city","award":["202323016881"],"award-info":[{"award-number":["202323016881"]}]},{"name":"Guiding plan project of Hengyang city","award":["202222015981"],"award-info":[{"award-number":["202222015981"]}]},{"name":"Guiding plan project of Hengyang city","award":["HQ22011"],"award-info":[{"award-number":["HQ22011"]}]},{"name":"Guiding plan project of Hengyang city","award":["KFA22014"],"award-info":[{"award-number":["KFA22014"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["2024JJ7113"],"award-info":[{"award-number":["2024JJ7113"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["23B0832"],"award-info":[{"award-number":["23B0832"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["23C0400"],"award-info":[{"award-number":["23C0400"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["202323016881"],"award-info":[{"award-number":["202323016881"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["202222015981"],"award-info":[{"award-number":["202222015981"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["HQ22011"],"award-info":[{"award-number":["HQ22011"]}]},{"name":"Scientific Research Project of Hunan Institute of Technology","award":["KFA22014"],"award-info":[{"award-number":["KFA22014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>Regenerative chatter is an unfavorable phenomenon that severely affects machining efficiency and surface finish in milling operations. The prediction of chatter stability is an important way to obtain the stable cutting zone. Based on implicit multistep schemes, this paper presents the third-order and fourth-order implicit exponentially fitted methods (3rd IEM and 4th IEM) for milling stability prediction. To begin with, the delay differential equations (DDEs) with time-periodic coefficients are employed to describe the milling dynamics models, and the principal period of the coefficient matrix is firstly decomposed into two different subintervals according to the cutting state. Subsequently, the fourth-step and fifth-step implicit exponential fitting schemes are applied to more accurately estimate the state term. Two benchmark milling models are utilized to illustrate the effectiveness and advantages of the high-order implicit exponentially fitted methods by making comparisons with the three typical existing methods. Under different radial immersion conditions, the numerical results demonstrate that the 3rd IEM and the 4th IEM exhibit both faster convergence rates and higher prediction accuracy than the other three existing prediction methods, without much loss of computational efficiency. Finally, in order to verify the feasibility of the 3rd IEM and the 4th IEM, a series of experimental verifications are conducted using a computer numerical control machining center. It is clearly visible that the stability boundaries predicted by the 3rd IEM and the 4th IEM are mostly consistent with the cutting test results, which indicates that the proposed high-order exponentially fitted methods achieve significantly better prediction performance for actual milling processes.<\/jats:p>","DOI":"10.3390\/mi16090997","type":"journal-article","created":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T09:18:57Z","timestamp":1756977537000},"page":"997","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["High-Order Exponentially Fitted Methods for Accurate Prediction of Milling Stability"],"prefix":"10.3390","volume":"16","author":[{"given":"Yi","family":"Wu","sequence":"first","affiliation":[{"name":"School of Intelligent Manufacturing and Mechanical Engineering, Hunan Institute of Technology, Hengyang 421002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bin","family":"Deng","sequence":"additional","affiliation":[{"name":"School of Intelligent Manufacturing and Mechanical Engineering, Hunan Institute of Technology, Hengyang 421002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qinghua","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hunan Meibeida Technology Co., Ltd., Leiyang 421800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tuo","family":"Ye","sequence":"additional","affiliation":[{"name":"School of Intelligent Manufacturing and Mechanical Engineering, Hunan Institute of Technology, Hengyang 421002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anmin","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Intelligent Manufacturing and Mechanical Engineering, Hunan Institute of Technology, Hengyang 421002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenbo","family":"Jiang","sequence":"additional","affiliation":[{"name":"Hunan Meibeida Technology Co., Ltd., Leiyang 421800, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/S0007-8506(07)60032-8","article-title":"Chatter stability of metal cutting and grinding","volume":"53","author":"Altintas","year":"2004","journal-title":"CIRP Ann."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s00170-017-0186-4","article-title":"Stability analysis for milling operations using an Adams-Simpson-based method","volume":"92","author":"Qin","year":"2017","journal-title":"Int. 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