Investigation of stepped nano beam vibrations according to nonlocal elasticity theory

In this study, vibrations of stepped nanobeams were investigated according to Eringen's nonlocal elasticity theory. Eringen's nonlocal elasticity theory was used to capture the nanoscale effect. The stepped nanostructure chosen within the scope of the study is subject to the limits of Euler Bernoulli beam theory. The integro differential equations of state of the stepped nanobeam before and after the step were found using the Hamilton principle, which is one of the energy methods. In addition, by adding damping and forcing terms to the state equations, it is aimed to examine the vibration movements of the system under stress. Then, the obtained equations were subjected to nondimensionalization process to make them independent from the physical structure and dimensions of the material. Multi-time scale method, which is one of the perturbation methods, has been applied to solve dimensionless state equations. The solution is considered in two stages. In the first stage, the first order terms obtained from the perturbation expansion formed the linear problem. By solving the linear problem, the natural frequency equations were obtained and the first three mode values for different conditions were found and presented in tables. In the second stage, the solution of the second order of the perturbation expansion was made and nonlinear terms emerged as corrections of the linear problem from this solution. The effects of nonlinear terms on natural frequencies are calculated for different conditions. The main aim of the study is to examine the nano-sized and stepped beams. For this purpose, while the findings of the research were obtained, various non-local parameter values were obtained to capture the nano-scale effect, and frequency-response and nonlinear frequency-amplitude curves corresponding to the 1st Mode values of the beam for different step radius ratios and step locations were obtained to capture the step effect. In order to strengthen the accuracy of the data obtained as a result of vibration analysis, the results of the study were compared with the results of the nano beam study with Eringen's non-local elasticity theory in the literature and with the results of the study with other classical beam theories.