Enhanced low-mass detection in carbon nanotube sensors through geometric non-linearity and magnetic field effects: A multimodal analysis of transverse vibrations

This paper investigates the free transverse vibrations, both linear and geometrically non-linear, of a carbon nanotube (CNT) carrying a point mass and subjected to thermal and magnetic fields, with the aim of analyzing the frequency shift induced by various parameters. The influence of the magnetic field on the curvatures is also studied. The theoretical framework combines Eringen’s nonlocal elasticity theory with von Kármán’s geometric non-linearity assumptions within the Euler-Bernoulli beam model. The governing non-linear equations of motion are derived via Hamilton’s principle. The linear frequency problem is first solved through a frequency sweep strategy combined with Newton-Raphson iterations. For the non-linear case, the equilibrium equation is handled using Hamilton’s principle in conjunction with spectral analysis. An approximate multimodal procedure, focused on the dominant mode and employing a so-called second formulation, is applied to solve the non-linear equations. The validity of the proposed approach is assessed by comparing the present results with those obtained from alternative methods, showing a strong agreement and confirming the reliability of the formulation.