Free vibration analysis of multistepped nonlocal Bernoulli-Euler beams using dynamic stiffness matrix method

The free vibration of multistepped nanobeams is studied using the dynamic stiffness matrix method. The beam analysis is based on the Bernoulli-Euler theory, and the nanoscale analysis is based on the Eringen's nonlocal elasticity theory. The nanobeam is attached to linear and rotational elastic supports at the start, end, and intermediate boundary conditions. The effect of the nonlocal parameter, boundary conditions, and step ratios on the nanobeam natural frequency is investigated. The results of the dynamic stiffness matrix methods are validated by comparing selected cases with the literature, which give excellent agreement with those literatures. The results show that the dimensionless natural frequency parameter is inversely proportional to the nonlocal parameters except in the first mode for clamped-free boundary conditions. Also, the gap between every two consecutive modes decreases with the increasing of the nonlocal parameter.