Size-dependent vibration and static analyses of a nanobeam made of time-dependent material attached with viscoelastic boundaries using three different beam theories

In this study, free vibration analysis of viscoelastic nanobeams under viscoelastic boundary conditions has been carried out separately for Euler–Bernoulli, Timoshenko, and Levinson beam theories. First, the non-local theory and the viscoelastic model have been established, and then, the equations of motion have been obtained using Hamilton's principles. Higher-order Fourier series obtained by Stokes’ transforms have been used to solve the problem. With the inclusion of boundary conditions in the problem, an eigenvalue problem has been constructed from which the frequencies for each beam theory can be obtained. The results have been presented in graphs and tables, and some important results have been obtained; for example, the effect of damping decreases as the non-local length scale parameter increases, damping has more effect in large modes, and the influence of viscous damping parameter of Euler–Bernoulli beam theory is more than other beam theories.