Multimodal approach for the analysis of geometrically nonlinear free vibrations of a carbon nanotube resting in a linear and nonlinear elastic medium subjected to thermal and magnetic fields

Abstract The aim of this paper is to investigate the problem of geometrically nonlinear free transverse vibrations of a carbon nanotube embedded in an elastic medium and subjected to thermal and magnetic effects, using Eringen's nonlocal elasticity theory applied to an Euler–Bernoulli beam. After solving the linear problem with Newton–Raphson's iterative method, the nonlinear equilibrium equation is addressed using a technique based on Hamilton's principle and spectral analysis for nonlinear free vibrations with large displacement amplitudes. This method illustrates Benamar's adaptation of the Rayleigh–Ritz technique. While this approach has been previously utilized for nonlinear transverse vibrations in continuous structures like beams, plates, and shells, it is being applied here for the first time to address vibrations in nanostructures. To validate this approach, comparisons are made between the current results and those obtained using alternative methods. The findings demonstrate good agreement between the current solutions and the previously obtained results.