{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T01:10:44Z","timestamp":1781226644301,"version":"3.54.1"},"reference-count":37,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,26]],"date-time":"2021-11-26T00:00:00Z","timestamp":1637884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The frequency of a nonlinear vibration system is nonlinearly related to its amplitude, and this relationship is critical in the design of a packaging system and a microelectromechanical system (MEMS). This paper proposes a straightforward frequency prediction method for nonlinear oscillators with arbitrary initial conditions. The tangent oscillator, the hyperbolic tangent oscillator, a singular oscillator, and a MEMS oscillator are chosen to elucidate the simple solving process. The results, when compared with those obtained by the homotopy perturbation method, exhibit a good agreement. This paper introduces a very convenient procedure for attaining quick and accurate insight into the vibration property of a nonlinear vibration system.<\/jats:p>","DOI":"10.3390\/axioms10040320","type":"journal-article","created":{"date-parts":[[2021,11,28]],"date-time":"2021-11-28T22:19:16Z","timestamp":1638137956000},"page":"320","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":103,"title":["A Simple Frequency Formulation for the Tangent Oscillator"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1636-0559","authenticated-orcid":false,"given":"Ji-Huan","family":"He","sequence":"first","affiliation":[{"name":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"},{"name":"School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454150, China"},{"name":"National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou 215123, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qian","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chun-Hui","family":"He","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6386-6181","authenticated-orcid":false,"given":"Yasir","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin 31991, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1177\/1461348418822135","article-title":"A modification of homotopy perturbation method for a hyperbolic tangent oscillator arising in nonlinear packaging system","volume":"38","author":"Song","year":"2019","journal-title":"J. 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