Dynamic analysis of rotationally restrained nanobeams made of viscoelastic functionally graded materials

Abstract

In this study, the free vibration behavior of functionally graded viscoelastic nonlocal Euler-Bernoulli beams under elastic boundary conditions is examined. First, the equations of motion are derived using the nonlocal theory and beam theory, incorporating the viscoelastic model. The material properties of the beam are then graded functionally. To solve the problem, the Fourier sinus series is chosen as the vertical displacement function, and the higher-order derivatives of this series are computed using Stokes' transformations. The boundary conditions are also integrated into the formulation, resulting in an eigenvalue problem that provides the most general solution for non-rigid boundary conditions. The results obtained are compared with existing studies in the literature, confirming their accuracy and convergence. Furthermore, the findings are presented through tables and figures, revealing key insights such as a curvilinear decrease in the real values of the fundamental frequencies as the viscous damping parameter increases, while the imaginary values increase linearly. It is also observed that the influence of the viscous damping parameter on the fundamental frequencies diminishes as the nonlocal length scale parameter grows.