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.

The lateral vibration of cracked isotropic thick beams is investigated. Generally, the analysis of thick beam based on line elements can be undertaken using either Timoshenko beam theory or a third order beam theory (TSDT) such as Reddy beam theory. TSDT is superior to Timoshenko beam theory, as it eliminates the need for a shear correction coefficient. However, there is no available solution for a cracked beam based on Reddy beam theory which is the main focus of this paper. The investigated beam is divided into several elements where their stiffness and mass matrices are derived using Reddy beam theory. Each element has two nodes with 3 degrees of freedom (1 lateral and 2 rotational) at each node. The crack was modeled using two rotating springs with stiffnesses that varied with crack depth. These elements are used to connect the adjacent beam elements. The impact of boundary conditions, slenderness ratio, crack location, and crack depth are investigated. In addition, experimental and finite element analyses of cracked steel beams is undertaken to validate the results of the present model. The results show that the maximum deviation between the analytical and the experimental results is less than 3% up to the third mode shape.

Purpose Mechanical properties of 1D nanostructures are of great importance in nanoelectromechanical systems (NEMS) applications. The free vibration analysis is a non-destructive technique for evaluating Young's modulus of nanorods and for detecting defects in nanorods. Therefore, this paper aims to study the longitudinal free vibration of a stepped nanorod embedded in several elastic media. Methods The analysis is based on Eringen’s nonlocal theory of elasticity. The governing equation is obtained using Hamilton’s principle and then transformed into the nonlocal analysis. The dynamic stiffness matrix (DSM) method is used to assemble the rod segments equations. The case of a two-segment nanorod embedded in two elastic media is then deeply investigated. Results The effect of changing the elastic media stiffness, the segments stiffness ratio, boundary conditions and the nonlocal parameter are examined. The nano-rod spectrum and dispersion relations are also investigated. Conclusion The results show that increasing the elastic media stiffness and the segment stiffness ratio increases the natural frequencies. Furthermore, increasing the nonlocal parameter reduces natural frequencies slightly at lower modes and significantly at higher modes.