{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T06:26:54Z","timestamp":1775543214501,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:00:00Z","timestamp":1688601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study addresses the nonlinear forced vibration of a functionally graded (FG) nanobeam subjected to mechanical impact and electromagnetic actuation. Two symmetrical actuators were present in the mechanical model, and their mechanical behaviors were analyzed considering the symmetry in actuation. The model considered the longitudinal\u2013transverse vibration of a simple supported Euler\u2013Bernoulli beam, which accounted for von K\u00e1rm\u00e1n geometric nonlinearity, including the first-order strain\u2013displacement relationship. The FG nanobeam was made of a mixture of metals and ceramics, while the volume fraction varied in terms of thickness when a power law function was used. The nonlocal Eringen theory of elasticity was used to study the simple supported Euler\u2013Bernoulli nanobeam. The nonlinear governing equations of the FG nanobeam and the associated boundary conditions were gained using Hamilton\u2019s principle. To truncate the system with an infinite degree of freedom, the coupled longitudinal\u2013transverse governing equations were discretized using the Galerkin\u2013Bubnov approach. The resulting nonlinear, ordinary differential equations, which took into account the curvature of the nanobeam, were studied via the Optimal Auxiliary Functions Method (OAFM). For this complex nonlinear problem, an explicit, analytical, approximate solution was proposed near the primary resonance. The simultaneous effects of the following elements were considered in this paper: the presence of a curved nanobeam; the transversal inertia, which is not neglected in this paper; the mechanical impact; and electromagnetic actuation. The present study proposes a highly accurate analytical solution to the abovementioned conditions. Moreover, in these conditions, the study of local stability was developed using two variable expansion methods, the Jacobian matrix and Routh\u2013Hurwitz criteria, and global stability was studied using the Lyapunov function.<\/jats:p>","DOI":"10.3390\/sym15071376","type":"journal-article","created":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T01:41:36Z","timestamp":1688694096000},"page":"1376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Longitudinal\u2013Transverse Vibration of a Functionally Graded Nanobeam Subjected to Mechanical Impact and Electromagnetic Actuation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2968-5309","authenticated-orcid":false,"given":"Nicolae","family":"Herisanu","sequence":"first","affiliation":[{"name":"Department of Mechanics and Strength of Materials, University Politehnica Timisoara, 300006 Timisoara, Romania"},{"name":"Center for Advanced and Fundamental Technical Research, Department of Electromechanics and Vibration, Romanian Academy, 300223 Timisoara, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2340-1280","authenticated-orcid":false,"given":"Bogdan","family":"Marinca","sequence":"additional","affiliation":[{"name":"Department of Applied Electronics, University Politehnica Timisoara, 300006 Timisoara, Romania"}]},{"given":"Vasile","family":"Marinca","sequence":"additional","affiliation":[{"name":"Department of Mechanics and Strength of Materials, University Politehnica Timisoara, 300006 Timisoara, Romania"},{"name":"Center for Advanced and Fundamental Technical Research, Department of Electromechanics and Vibration, Romanian Academy, 300223 Timisoara, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A., and Ford, R.G. 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