{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T23:04:03Z","timestamp":1771455843702,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,24]],"date-time":"2019-12-24T00:00:00Z","timestamp":1577145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, we will solve the Bagley\u2013Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.<\/jats:p>","DOI":"10.3390\/e22010028","type":"journal-article","created":{"date-parts":[[2019,12,24]],"date-time":"2019-12-24T10:28:43Z","timestamp":1577183323000},"page":"28","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["An Investigation of Fractional Bagley\u2013Torvik Equation"],"prefix":"10.3390","volume":"22","author":[{"given":"Azhar Ali","family":"Zafar","sequence":"first","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowskiego St., 90-924 Lodz, Poland"},{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]},{"given":"Grzegorz","family":"Kudra","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowskiego St., 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowskiego St., 90-924 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s11071-010-9724-4","article-title":"Entropy analysis of integer and fractional dynamical systems","volume":"62","author":"Machado","year":"2010","journal-title":"Nonlinear Dynam."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lopes, A.M., and Machado, J.A.T. 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