{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T20:43:48Z","timestamp":1775767428568,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,31]],"date-time":"2024-08-31T00:00:00Z","timestamp":1725062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["KFU241312"],"award-info":[{"award-number":["KFU241312"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper examines the stability behavior of the nonlinear dynamical motion of a vibrating cart with two degrees of freedom (DOFs). Lagrange\u2019s equations are employed to establish the mechanical regulating system of the examined motion. The proposed approximate solutions (ASs) of this system are estimated through the use of the multiple-scales method (MSM). These solutions are considered novel as the MSM is being applied to a new dynamical model. Secular terms have been eliminated to meet the solvability criteria, and every instance of resonance that arises is categorized, where two of them are examined concurrently. Therefore, the modulation equations are developed based on the representations of the unknown complex function in polar form. The solutions for the steady state are calculated using the corresponding fixed points. The achieved solutions are displayed graphically to illustrate the impact of manipulating the system\u2019s parameters and are compared to the numerical solutions (NSs) of the system\u2019s original equations. This comparison shows a great deal of consistency with the numerical solution, which indicates the accuracy of the applied method. The nonlinear stability criteria of Routh\u2013Hurwitz are employed to assess the stability and instability zones. The value of the proposed model is exhibited by its wide range of applications involving ship motion, swaying architecture, transportation infrastructure, and rotor dynamics.<\/jats:p>","DOI":"10.3390\/axioms13090596","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T12:54:42Z","timestamp":1725281682000},"page":"596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analyzing the Stability of a Connected Moving Cart on an Inclined Surface with a Damped Nonlinear Spring"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9076-9945","authenticated-orcid":false,"given":"Muneerah","family":"AL Nuwairan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6293-711X","authenticated-orcid":false,"given":"T. S.","family":"Amer","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7428-7409","authenticated-orcid":false,"given":"W. S.","family":"Amer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebeen El-Kom 32511, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1002\/(SICI)1521-4001(199901)79:1<65::AID-ZAMM65>3.0.CO;2-X","article-title":"On the motion of the pendulum on an ellipse","volume":"79","author":"Ismail","year":"1999","journal-title":"ZAMM"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/0167-2789(85)90015-6","article-title":"Experiments on periodic and chaotic motions of a parametrically forced pendulum","volume":"16","author":"Leven","year":"1985","journal-title":"Phys. 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