Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

In this study, an eigen‐value problem for deformable boundary conditions in nonlocal elasticity is described using Levinson beam theory. Firstly, the nanobeam has been modeled by placing two springs that can be deformed in the downward direction. These springs control the amount of downward displacement at the ends. In the analytical solution, the displacement points are defined by two coefficients and the interior part of the nanobeam deflection is expressed by Fourier sine series. Stokes’ transformation is preferred to enforce the boundary conditions to the desired point. After the mathematical operations, a matrix of coefficients including the general elastic spring constants has been found. The eigenvalues of this coefficient matrix give the frequencies of the Levinson nanobeam. The effect of some parameters on the free vibration frequencies is shown in a series of graphs and tables.