{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T05:00:44Z","timestamp":1775365244057,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T00:00:00Z","timestamp":1749081600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, the Proportional and Derivative Controller (PD) is presented as a modified control that combines features of the Proportional Controller (P-Controller) and the Derivative Controller (D-Controller) to suppress the vibrations of the Smooth and Discontinuous Oscillator (SD). The investigated model has been derived as a one-degree-of-freedom. The frequency response equation of the controlled system has been obtained using the perturbation technique up to a second approximation. The influence of the P-Controller, the D-Controller, and the PD-Controller on the SD-Oscillator amplitude has been studied by plotting the time histories. The numerical and approximate simulation established that the PD-Controller can inhibit the system vibration. Finally, there is high closeness between the numerical solutions (from time histories) and the approximate solutions (from perturbation analysis).<\/jats:p>","DOI":"10.3390\/axioms14060444","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T03:52:28Z","timestamp":1749181948000},"page":"444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Improving Stability and Reducing Vibrations of the Smooth and Discontinuous Oscillator Using a Proportional\u2013Derivative Controller"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8333-4222","authenticated-orcid":false,"given":"M. N.","family":"Abd El-Salam","sequence":"first","affiliation":[{"name":"Basic Sciences Department, Higher Technological Institute, 10th of Ramadan City 44634, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4497-9778","authenticated-orcid":false,"given":"Rageh K.","family":"Hussein","sequence":"additional","affiliation":[{"name":"Physics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia"}]},{"given":"Sabry M.","family":"El-Shourbagy","sequence":"additional","affiliation":[{"name":"Basic Sciences Department, Higher Technological Institute, 10th of Ramadan City 44634, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.jsv.2007.07.041","article-title":"Effects of delayed feedback control on nonlinear vibration absorber system","volume":"308","author":"Zhao","year":"2007","journal-title":"J. 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